Qz-axis

Qx = and Qz = 7r/d are the corresponding wave vectors of the instability along the x- and 2 -axis and is the electric field potential defined in (5.12). In view of this, the following dispersion relation  

For a sheet with isotropy perpendicular to the z axis, X44 = X55, so that the torque Qz about an axis perpendicular to the symmetry axis is given by 

Typical stability diagram for a 3D ion trap. The value at = 1 along the qz axis is qz = 0.908. At the upper apex, = 0.149 998 and = 0.780 909. Drawn with data taken from March R.E and R.J. Hughes, Quadrupole Storage Mass Spectrometry, Chemical Analysis Vol. 102, Wiley Interscience, 1989. 

The CO scan of the (200) lattice node, obtained for the section along the qx axis at Qz = 0, is shown in Figure 7.32. As we illustrated in the previous example dealing with the study of lithium niobate, the increase in peak width is a result, in particular, of the relative disorientations of the crystals. Therefore, the width of this distribution yields an estimate of this mosaicity. For this film, we obtained a relative disorientation of 0.2°. 

The elements V2,-m describe the spatial interaction in a reference axis system of choice and are connected to Qz.m by a rotation as shown in Eq. (4). The spin tensor elements 72, are given in terms of the Cartesian angular momentum operators by t 

The reciprocal space intensity is displayed in Fig. 7. It has the shape of a rod along the Qz axis. Any modulation of the platelet density profile along the z axis yields a form factor  

It is important to note that, while P is a scalar density, Qz is one component of a vector density, the components transverse to the quantization axis being defined in a similar way. Analogous transition densities, in which KK is replaced by KL, are required in discussing transitions between states K and L of the system, but are not needed in the present work. 

As no DC voltage is applied, the 3D trap will be operated along the qu axis, because in the absence of DC voltage, au = 0. As already explained, qz is given by the following equation  

For transversely isotropic sheets, a much simpler formula applies for torsion around the symmetry axis 3, where the torque Qz is given by 

For a transversely isotropic sheet (with 3 direction as axis of symmetry), a similar expression describes torsion about an axis perpendicular to the symmetry axis. In this case, 44 = S55 and the torque Qz is given by 

Problem 3-27. Oscillating Cylinder as a Viscometer. Consider a cylinder immersed in a large bath of fluid with kinematic viscosity v that rotates sinusoidally about its axis with angular velocity Qz = sin( >t). The cylinder has a radius R, length L, and L/R p> 1. It is 

This equation is from Eq. (2.60) with e = —2 as [ly, Jy] = —i2Ky. Thus, a 45° pulse can convert a state with single quantum coherence Jy) into a Ky state that contains quadrupole order Qz and double quantum coherence Kz- Note that ly and Qy are invariants of motion since they commute with the above r.f. Hamiltonian. Thus, in this case the coefficients a2 and ay are now time independent [Q, Ky are used instead of Qz Kz in Eq. (2.68)]. Precession diagrams (Fig. 2.3) also may be drawn for a quadrupole interaction with spin Hamiltonian Hq = ujqQz [Eq. (2.34) with r = 0]. In this case, there are four invariants of motion, Iz Qz Kz, and J. An initial Ix state will evolve under Hq = ujqQz by precessing about the Qz axis at a frequency ujq in the /a — Jx plane, while an initial Jy state will precess in the — ly plane at a frequency ujq. However, an initial ly state will precess in the — Jy plane at a frequency —ojq according to 

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