Frame of reference, rotating

As described earlier, if a sample made up of nuclei with nuclear spin j is placed in a magnetic field, Bq, the nuclei start to precess around the direction of the applied field with a frequency Oq, known as the Larmor frequency, and their 

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Examples of even processes include heat conduction, electrical conduction, diflfiision and chemical reactions [4], Examples of odd processes include the Hall effect [12] and rotating frames of reference [4], Examples of the general setting that lacks even or odd synnnetry include hydrodynamics [14] and the Boltzmaim equation [15]. 

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. 

Suppose we adopt a rotating frame of reference with coordinates x, y, z such that the fixed field Hq lies along the z axis and the x, y coordinate system rotates about the z axis with the frequency of the field ff,. Let Hi be stationary along the x axis. 

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 i, j, k and ii, ji, ki are unit vectors in the inertial and rotating frames of reference, respectively. Performing a differentiation with respect to time we obtain for the velocity of the point p... 

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 ... 

Equations (3.75 and 3.76) describe the motion of a free particle in a rotating frame of reference, when the angular velocity is relatively small and the z-axis is directed along the plumb line. Certainly, a presence of terms with a> is related to the rotation of the earth. Besides, the gravitational field also contains a term with this frequency. It may be proper to emphasize again that in deriving these formulas we assumed... 

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... 

Here, ak is the isotropic chemical shift referenced in ppm from the carrier frequency co0, SkSA is the anisotropy and tfk SA the asymmetry of the chemical-shielding tensor, here also expressed in ppm. Note that for heteronuclear cases different reference frequencies co0 are chosen for different nuclei (doubly rotating frame of reference). The two Euler angles ak and pk describe the orientation of the chemical-shielding tensor with respect to the laboratory-fixed frame of reference. The anisotropy dkSA defines the width and the asymmetry t]kSA the shape of the powder line shape (see Fig. 11.1a). 

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)). 

Instead of the laboratory frame (x,y,z-axis), let us consider a rotating frame of reference which rotates with an angular frequency c% relative to the laboratory. In this frame of reference, the nuclear magnets appear to precess with the angular frequency [Pg.119]

Figure 2. The revolution of macroscopic magnetization in a rotating frame of reference (a), application of additional magnetic field (BL) along x -axis as a 90° pulse which tips the macroscopic magnetization into the x y -plane (b), as they precess in the x y -plane, the macroscopic magnetization diminishes because nuclear magnets actually precess at slightly different frequencies which causes dephasing.

In a rotating frame of reference (i.e., that of an observer anchored to the disc), the trajectory is nearly radial. Since this reference frame is most relevant when we consider the disc/fluid interaction, it is helpful to evaluate the flow on this basis. 

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