Rotating coordinate system frame

Instead, suppose the x and y axes were themselves precess-ing clockwise (when viewed from above) around the z axis at the same frequency the nuclear spins are precessing. Further suppose we, the observers, were precessing around the z axis at the same frequency. To differentiate this rotating coordinate system from the fixed (i.e., laboratory frame) system, we will use labels and z to represent the three rotating axes (the z axis is coincident on and equivalent to the z axis). To us rotating observers, the rotating axes and B, appear stationary, and M will rotate in the plane perpendicular to B. These relationships are shown in Figure 2.9,... 

Example 1 Most discussions involving the Bloch model introduce the concept of the rotating frame. The concept of a rotating coordinate system is a familiar one because in real life positions and motion are referred to the earth, a coordinate system that is rotating. Similarly rather than refer the motion of the magnetization vectors to the fixed laboratory coordinate system, it is simpler to refer their motion to a rotating frame of reference which rotates at the NMR transmitter frequency of the nucleus under study. 

Thus the acceleration in the rotating frame equals the siun of the net force per unit mass that would be present in an inertial system and the two apparent forces due to the rotation of the coordinate system. When Newton s law is expressed in a rotating coordinate system, the Coriolis and centripetal accelerations are seen as additional forces per unit mass. 

Figure 1. The space-fixed (ATZ) and body-fixed (xyz) frames. Any rotation of the coordinate system XYZ) to (xyz) may be performed by three successive rotations, denoted by the Euler angles (a, 3, y), about the coordinate axes as follows a) rotation about the Z axis through an angle a(0 < a < 2n), (b) rotation about the new yi axis through an angle P(0 < P < 7i), (c) rotation about the new zi axis through an angle y(0 Y < 2n). The relative orientations of the initial and final coordinate axes are shown in panel (d).

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

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. 

Consider a rotation of the earth around the z-axis in which every particle, elementary volume, of the earth moves along the horizontal circle with the radius r. Our first goal is to find the distribution of forces inside the earth and with this purpose in mind we will derive an equation of motion for an elementary volume of the fluid. Let us introduce a Cartesian system of coordinates with its origin 0, located on the z-axis of rotation. Since this frame of reference is an inertial one, it does not move with the earth, we can write Newton s second law as... 

The values of the rhombicity parameters are conventionally limited to the range 0 < EjD < 1/3 without loss of generality. This corresponds to the choice of a proper coordinate system, for which /)zz (in absolute values) is the largest component of the D tensor, and /) is smaller than Dyy. Any value of rhombicity outside the proper interval, obtained from a simulation for instance, can be projected back to 0 < EID < 1/3 by appropriate 90°-rotations of the reference frame, that is, by permutations of the diagonal elements of D. To this end, the set of nonconventional parameters D and EID has to be converted to the components of a traceless 3x3 tensor D using the relationships... 

Laboratory frame model A means of visualising the processes taking place in an NMR experiment by observing these processes at a distance, i.e., with a static coordinate system. See Rotating frame model. 

In (3.1) not all tensors are necessarily coaxial or diagonal. If the principal axes system of the g tensor is chosen as the molecular coordinate system eM, g has diagonal form. The laboratory frame eL is then related to eM by the rotation matrix R according to... 

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. 

Eq. (1.19) described the precession of M about the total magnetic field B using a coordinate system with fixed axes x, y, and z. Correspondingly, eq. (1.29) describes the magnetization vector M as it precesses about the effective field Beff [7] in a coordinate system rotating with frequency m = 2 7t v about the z axis and symbolized as the x , y, z frame of reference with the rotating unit vectors and k. 

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. 

Fig. 2.3. Components of transverse magnetization with different chemical shifts in a fixed coordinate system x, y, z (a) and in the rotating frame x, v, z (b).

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by... 

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