Rotating reference frame

In a reference frame rotating about z with frequency go, the rf field Hi has two components (1) a dc component Hiy and (2) a component oscillating with frequency 2go. The 2G0-component has very little effect on and can be discarded. If the rf field is "on resonance" (go = goq), then, in the RRF, Hq disappear completely, and we are left with only the dc component of Hi,... 

Upon transforming to an interaction representation (i.e., a reference frame rotating at frequency w) in which the density matrix is defined by Equation 7, and invoking the rotating wave approximation which consists of dropping all high-frequency motions with respect to u . Equation 25 becomes... 

On rewriting the equation in a reference frame rotating at the resonance frequency of the optical and microwave transition and neglecting all coherence effects between singlet and triplet states we find an equation of motion of the form... 

The preceding argument shows that observers in dUfer-ent coordinate systems perceive different accelerations. Mathematically, this is expressed in the following way. First of all, the relationship between the derivative of the position vector r in an inertial reference ftame denoted by subscript 1 and its derivative in a reference frame rotating with angular velocity n relative to the inertial frame is... 

Figure 1.5 Components of a vector A in two reference frames rotated by an angle 9 from each other.

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

The Coriolis veclor lies in the same plane as the velocity vector and is perpendicular to the rotation vector. If the rotation of the reference frame is anticlockwise, then the Coriolis acceleration is directed 90° clockwise from the velocity vector, and vice versa when the frame rotates clockwise. The Coriolis acceleration distorts the trajectory of the body as it moves rectilinearly in the rotating frame. 

When dealing with the motions of rigid bodies or systems of rigid bodies, it is sometimes quite difficult to directly write out the equations of motion of the point in question as was done in Examples 2-6 and 2-7. It is sometimes more practical to analyze such a problem by relative motion. That is, first find the motion with respect to a nonaccelerating reference frame of some point on the body, typically the center of mass or axis of rotation, and vectorally add to this the motion of the point in question with respect to the reference point. 

Assume that origins of two Cartesian systems of coordinates are located at the same point and the frame of reference P rotates about a point 0 of the frame P with constant angular velocity co. Let us imagine two planes, one above another, so that the upper plane P rotates and, correspondingly, unit vectors iiand ji change their direction, Fig. 2.2b. Consider an arbitrary point p, which has coordinates x, y on the plane P and xi, yi on P, and establish relationships between these pairs of coordinates. For the radius vector of the point p in both frames we have... 

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... 

Since the software is also home built, a number of unique functions can be added to it. An example of such is the way that the data is plotted. In addition to the conventional data plotting scheme, the parametric plot of the real and the imaginary data on the xy-plane is supported. This option enables one to "see" the precession of the transverse nuclear magnetization in the rotating reference frame with respect to the carrier frequency. 

Note that the subscript axes indices are 1, 2, 3, and not x, y, z, to indicate that the term is diagonal in an axis system that is generally different from that which diagonalizes the g-tensor of system A and/or B. This means that a full characterization of the asymmetric exchange does not only require values for the three elements of the //-vector, but also three angles over which to rotate to get to the g-diagonalizing reference frame. For example, if we would take the latter to refer to the g-tensor of center A, then we should in general write... 

Rotating single-crystal measurements also permitted the extraction of the orientation of the magnetic tensor in the molecular reference frame and the experimental easy axis was found to coincide with the idealized tetragonal axis of the coordination dodecahedron of Dy. Crystal field calculations assuming idealized tetragonal symmetry permitted the reproduction of magnetic susceptibility data for gz = 19.9 and gxy 0 [121]. More elaborated calculations such as ab initio post Hartree-Fock CASSCF confirmed this simple analysis [119]. 

We describe as rigid-body rotation any molecular motion that leaves the centre of mass at rest, leaves the internal coordinates unaltered, but otherwise changes the positions of the atomic nuclei with respect to a reference frame. Whereas in a simple molecule, such as carbon monoxide, it is easy to visualize the two atoms vibrating about a mean position, i.e. with the bond length changing periodically, we may sometimes find it easier to see the vibration in our mind s eye if we think of one atom being stationary while the other atom moves relative to it. 

Figure 1. Precessing magnetic moment in (a) the laboratory reference frame and in (b-d) the rotating reference frame.

RF Field H, in the Rotating Reference Frame Ji precesses about H, with frequency yH,... 

Figure 2. Magnetic moment precessing about the rf field in the rotating reference frame.

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