Rotating coordinate system

Figure 1 shows the detailed steps of the measurement, from the perspective of a coordinate system rotating with the applied radiofrequency giq = yA)- The sample is in the magnetic field, and is placed inside an inductor of a radiofirequency circuit... 

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 special class of transformation, known as symmetry (or unitary) transformation, preserves the shape of geometrical objects, and in particular the norm (length) of individual vectors. For this class of transformation the symmetry operation becomes equivalent to a transformation of the coordinate system. Rotation, translation, reflection and inversion are obvious examples of such transformations. If the discussion is restricted to real vector space the transformations are called orthogonal. 

The most expensive parts of a conventional NSE instrument are the main solenoids providing the precession field. A closer look at Bloch s equation of motion for the spins (Eq. 2.11) shows that in a coordinate system that rotates with the precession frequency around Bg the spin is stationary, the coordinate system rotation is equivalent to the addition of - to all magnetic fields. By this means the large precession field inside the main coils may be transformed to zero - zero field spin-echo). The flippers are viewed as elements rotating... 

A useful equivalent way of looking at the resonance phenomenon is in a coordinate system rotating with the angular frequency of Hi about the z axis which coincides with the direction of Ho, the applied static field 44)-In this rotating coordinate system the effective fields are Hi, which is now time-independent, and a field Ho — u/y, where w is the angular frequency... 

If individual atomic coordinate systems are used, as is common when chemical constraints are applied in the least-squares refinement, they must first be rotated to have a common orientation. The transformation of the population parameters under coordinate-system rotation is described in section D.5 of appendix D (Cromer et al. 1976, Su 1993, Su and Coppens 1994). 

This representation is in block form, and is obviously reducible. Consider another coordinate system, rotated in the a — y plane by 45°. Verify that in this new coordinate system the formulas giving the effect of cr are a —y and y —s- —x. Find the matrix relating the two coordinate systems and verify that a similarity transformation applied to the matrices of this new representation produces the old representation. How does this demonstrate the reducibility of the new representation ... 

The direction cosines of each of the nine angles that describe the coordinate-system rotation are defined, for example, as... 

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. 

The factors C and C > are both about V2, and are needed to preserve the normalization. We must end up with the same number of combinations as the number of atomic orbitals we used. This can be understood by analogy with describing the distance between two particles in a plane by two different coordinate systems, rotated from one another by 45° (Figure 6.7). 

Figure 1.3 Sketch of an infinitesimal tetrahedron whose three faces coincide with the x-y, x-z, and y-z planes of the original (unprimed) Cartesian coordinate system. The third slant face appears to be oriented such that the axis x is normal to the ar ea of the slant face. The two remaining axes, y and z lie in the plane of the slant face but are orthogonal to one another as well as to the x -axis. Hence, x, j/, and z define a Cartesian coordinate system rotated with respect to the original one.

This flow can be obtained in the Taylor device, consisting of four rotating cylinders [474, 475]. Note that flow 2° is the same as flow 3° but in a different coordinate system (rotated about the Z-axis by 45° counterclockwise). 

Transport of light component is to be described in terms of its mass velocity, the vector J, with component in the radial direction and in the axial. In the coordinate system rotating at angular velocity w, the angular component Jg is zero. 

The solvent is assumed to be in solid body rotation at an angular speed (o, and the solute is assumed to move circumferentially with the solvent. A single solute is considered, that is, a binary mixture, and a cylindrical coordinate system rotating with the angular speed (o is adopted. The solute concentration is then a function only of the time t and radial distance r from the rotation axis. The continuity (diffusion) equation (Eq. 3.3.15) can therefore be written... 

The point does not move, while the coordinate system rotates by angle —a. 

In particular, the component r, which is equal to the population difference of the two sublevels, plays the same role as the longitudinal magnetization in the hf experiment. The component T2 is, in the present instance, related to the expectation value of S, (the x component of the magnetic moment). This analogy suggests that the motion of the vector r can best be represented by transforming to a primed coordinate system rotating at the frequency u) about, as shown below ... 

An object at rest in a coordinate system rotating with the earth is subject to a centrifugal force directed oirt-ward from the earth s axis and a gravitational force directed toward the earth s center. An observer or an instrument on the earth carmot distinguish between these two forces. Therefore they are combined into a resultant force... 

We obtain the same new coordinates if the point remains still while the coordinate system rotates in the opposite direction (i.e. by angle —a). 

Let us choose the direction for which we wish to determine Young s modulus E as the x l axis of a coordinate system rotated with respect to the crystal coordinate system. Let us assume in addition that with respect to this primed coordinate system there is only one component of stress, namely T[, all others being zero. This normal stress T[ causes a strain S[ given by... 

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