Qz-axis

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. 

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... 

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 ... 

For simplicity, we assume a 2D-periodic structure ( 2D-crystallinity ) in the monolayer film floating on the subphase The molecules are arranged in identical unit cells which form a regular lattice. Then, in GID, Bragg diffraction occurs when the lateral scattering vector Qhor, c.f. Fig. 1.1b, coincides with a reciprocal lattice vector Ghk The scattering is concentrated in so-called Bragg Rods (parallel to the Qz axis), defined by two Laue conditions or, in vector notation, by the equation Qhor=Ghk- By constrast, XR... 

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 ... 

Figure 2 Examples of global and local axis systems, (a) Molecular axis system for a homonuclear diatomic. Wth this system, all central multipoles with k 0 otl odd will be zero, and no S functions with k 0 oxl odd can appear in a molecule-molecule expansion of U(R, Q). The atomic multipoles Q q (all / 0 allowed) on the two atoms will be related by Qio = (-l) Qio- ( ) Local atomic axis system for a homonuclear diatomic molecule. With this definition QJq = Qio- (c) Molecular axis system for water. The nonzero atomic multipole moments for the O atom would be QoO> QlO) QzOj Qz2c = (Q22 + Qz-2)1 > QsOJ Q32c = (Qs2 + on the hydrogen atoms Qjo = Qoo. Qio = Qio> Qiic = (-Q11 + Qi-i) = -Qiic...

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 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°. 

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... 

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. 

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... 

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

Thus, XR and GID measure different parts of the monolayer structure factor, corresponding to different projections of the monolayer structure. XR corresponds to the projection of the monolayer density onto the z-axis and includes also scattering from the sub-phase. GID - with Qz —0, i.e., grazing exit as well as grazing incidence - measures the structure of the 2D-crystalline part of the monolayer, as projected onto the x-y plane. Finally, measurement of the GID signal Ihk(Qz) versus Qz (so-called Bragg Rod scans) gives three-dimensional information about the 2D-crystalline part of the monolayer. 

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