Rotating coordinate frame

Equations (5.12a-c) are the Bloch equations in the rotating coordinate frame. 

Transformation into the rotating coordinate frame If the received signal... 

FIg. 3.3.11 The WAHUHA pulse cycle, (a) Timing diagram, (b) Time-evolution operators in the rotating coordinate frame, (c) Rotations in the rotating coordinate frame (RCF). 

To gain some insight into the theory of multi-pulse NMR, the WAHUHA sequence (t, -Fx, t, —y, T, T, -Fy, t, —x, t) is applied in the rotating coordinate frame to a pair of spins coupled by the dipole-dipole interaction [Sch9]. The pulses are considered to be infinitely short. Their flip angle is 90°, and they are applied in -Fx, —y, -Fy, and —x directions of the rotating coordinate frame. The density matrix p to- -6x) after completion of a WAHUHA cycle is obtained from the density matrix p(ta) before application of the cycle by transformation with the evolution operator U(i) of the cycle. 

The average propagator is introduced to describe the echo attenuation in the displacement experiments of Fig. 5.4.4 in terms of the particle displacement R along the gradient direction during the gradient-pulse delay A. To this end (5.4.26) is transformed to the rotated coordinate frame of Fig. 5.4.5 with axes defined in (5.4.21). By the same derivation used for (5.4.22) one arrives at... 

From these relations it follows that is related to the angular momentum modulus, and that the pairs of angle a, P and y, 8 are the azimuthal, and the polar angle of the (J ) and the (L ) vector, respectively. The angle is associated with the relative orientation of the body-fixed and space-fixed coordinate frames. The probability to find the particular rotational state IMK) in the coherent state is... 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

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... 

In Section 5.2 the set of internal state variables k was introduced. In the referential theory, a similar set of referential internal state variables K will be introduced in the same way without further physical identification at this stage. It will merely be assumed that each member of the set K is invariant under the coordinate transformation (A.50) representing a rigid rotation and translation of the coordinate frame. 

Since S, S, E, E, and by assumption K and K are all invariant under a rotation of coordinate frame, it is easily verified that C, , , , and si are similarly invariant, and that all the equations of this section are objective. 

The objectivity of the spatial stress rate relation (5.154) may be investigated by applying the coordinate transformation (A.50) representing a rotation and translation of the coordinate frame. The spatial strain and its convected rate are indifferent by (A.58) and (A.62). So are the stress and its Truesdell rate. It is readily verified from (5.151), (5.152), and the fact that K has been assumed to be invariant, that k and its Truesdell rate are also indifferent. Using these facts together with (A.53) in (5.154) with c and b given by (5.155)... 

The factors in Q on each side cancel, so that the stress rate relation is invariant to a rotation of coordinate frame, and is objective. Note that it is the special dependence of c and b on which makes the stress rate relation objective. If... 

It is expected that constitutive equations should be invariant to relative rigid rotation and translation between the material and the coordinate frame with respect to which the motion is measured, a property termed objectivity. In order to investigate this invariance, the coordinate transformation... 

Consequently, the stretching tensor and the convected rate of spatial strain are indifferent, but the spin tensor is not, involving the rate of rotation of the coordinate frame. From (A.24) and (A.26)... 

This projection is constant on a free path and changes only at collision moments (tk) when rotation of the GF z axis takes place Jo(tk + 0) = 12q Tq0Jq (tk — 0). The collision operator t is expressed in terms of the operator D, which rotates the coordinate frame [23] ... 

Fig. 1.2 Behavior of the magnetization in a simple echo experiment. Top a free induction decay (FID) follows the first 90° pulse x denotes the phase of the pulse, i.e., the axis about which the magnetization is effectively rotated. The 180° pulse is applied with the same phase the echo appears at twice the separation between the two pulses and its phase is inverted to that of the initial FID. Bottom the magnetization vector at five stages of the sequence drawn in a coordinate frame rotating at Wo about the z axis. Before the 90° pulse, the magnetization is in equilibrium, i.e., parallel to the magnetic field (z) immediately aftertbe 90° pulse, it has been rotated (by90° ) into the transverse (x,y) plane as it is com-...

Recently, Lipton et al. [25] have used zinc-67 NMR to investigate [Zn(HB(3,5-(CH3)2pz)3)2] complexes which have been doped with traces of paramagnetic [Fe(HB(3,4,5-(CH3)3pz)3)2]. The low-temperature Boltzmann enhanced cross polarization between XH and 67Zn has shown that the paramagnetic iron(II) dopant reduces the proton spin-lattice relaxation time, Tj, of the zinc complexes without changing the proton spin-lattice relaxation time in the Tip rotating time frame. This approach and the resulting structural information has proven very useful in the study of various four-coordinate and six-coordinate zinc(II) poly(pyrazolyl)borate complexes that are useful as enzymatic models. 

We present here some very general exact results, which hold for arbitrary reorientation mechanisms of any molecule in an equilibrium isotropic fluid (but not a liquid crystal). A coordinate frame (R) is rigidly attached to the molecule of interest. Its orientation in the laboratory frame (L) is defined by the Euler rotation = (affy) that carries a coordinate frame from coincidence with the laboratory frame L to coincidence with the molecular frame R/ The conditional probability per unit Euler volume [( (0r at time t must depend only on the Euler rotation A = 1 (i.e., rotate first by < 0 then... 

The orientation of the coordinate frame R fixed in a particular rod with respect to the laboratory frame L is specified by the Euler rotation, = (afiy), as before. Under the preceding assumptions, a diffusion equation for the probability density [/(if, /)] is derived,<29) namely,... 

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