Secular motion

Fig. 17.8 Illustration of the trapping principle in an ion trap. The effect of rotating the saddle potential in part (a) is a pseudopotential well illustrated in part (b). For particles with suitable mass (and charge) the particle motion in the pseudopotential is indicated by the black line. The motion is a combination of a secular motion in the pseudopotential well and a small amplitude micromotion at the frequency of rotation mf. If the particle motion is damped, the particle can come to rest at the bottom of the well...

Locatelli, U. and Giorgilli, A. (2000). Invariant tori in the secular motions of the three-body planetary systems. Cel. Mech., 78 47-74. 

The AMD also is no longer invariant, but its variation is small and thus limitations of the eccentricities similar to that observed in the secular motion (but different) exist. 

The amount of secular motion of the ion resulting from its finite kinetic energy cannot be tested by this method since the secular motion is not phase coupled to the trap voltage. However, the intensity modulation owing to this motion can be seen in a periodic modulation of the photon correlation signal. For all measurements presented here, this amplitude was below X/5. This correspnds to the ion having a temperature of approximately 1 mK and a mean... 

The signal to noise ratio observed in the experiment is shot noise limited. The signal in Fig. 3 corresponds to a rate of the scattered photons of about 10 s which is an upper limit since photons were lost from detection due to scattering into sidebands caused by the secular motion of the ion. In order to reduce... 

The Secular Motions of the Hydrogen Atom in an Electric Field... 

We now examine the secular motions caused by the electric field. The perihelion of the orbital ellipse alters its position relatively to the line of nodes, and the latter itself moves uniformly about the axis of the field. It follows from (5) that two periods of the perihelion motion occur during one revolution of the line of nodes. 

The secular motions of the orbit under the influence of the electric field are thus as follows while the line of nodes revolves once, the perihelion of the orbital ellipse performs two oscillations about the meridian plane perpendicular to the line of nodes. For a transit through this meridian plane in one direction, the total momentum J2°/27t is a maximum and consequently the eccentricity is a minimum for a transit in the other direction the eccentricity is a maximum. Since the component Js°/2it of the angular momentum in the direction of the field remains constant, the inclination of the orbital plane oscillates with the same frequency as the eccentricity. It has its maximum or minimum value when the perihelion passes through the equilibrium position, and it assumes both its maximum and minimum value twice during one revolution of the line of nodes. The major axis remains constant during this oscillation of orbital plane and perihelion (since Jj0 remains constant) the eccentricity varies in such a way that the electrical centre of gravity always remains in the plane... 

Bohr has given another and more illuminating method of calculating the secular motions of the hydrogen atom in an electric field.1 Using a similar method, Lenz and Klein succeeded in deducing the effect of the simultaneous influence of a magnetic field and of an electric field arbitrarily orientated with respect to it. 

These considerations provide a justification for our previous method of determining the secular perturbations ( 18) by regarding them as first approximations in a method of successive approximations. The higher approximations lead to periodic variations of the wfc° s and J7l° s, whose amplitudes are at most of the order of magnitude of A. Secular motions of wa°, J ° do not occur also in addition to the secular motions of wp°, 3p° which we recalculated in the first stage of the process, only periodic variations occur having frequencies of the same order of magnitude and amplitudes proportional to A. 

Further special cases can occur, e.g. when the secular motion de-... 

After calculating the unperturbed motion of the inner electron, we can find the secular motions of the remaining variables by introducing a new Hamiltonian function, the mean value of Hx taken over the unperturbed motion of the inner electron. The integration of the corresponding Hamilton-Jacobi equation is again performed by the methods of the theory of perturbations. 

The term secular growth in this context is a reference to the long-term growth of perturbations in celestial mechanics. For example, the precession of the Earth s polar axis occurs on a long period relative to its orbital motion and much longer period than its rotation, and so may be classed as a secular motion. In the context of molecular simulations, we use this to refer to accumulation of drift that takes the system steadily away from the energy surface. 

The stable solutions of (9.49) can be described as a superposition of two components A periodic micromovement of the ions with the RF tURp around a guiding center that itself performs slower harmonic oscillations with the frequency X2 in the x-y-plane and with the frequency co = 2f2 in the z-direction (secular motion) [1217]. The X- and z-components of the ion motion are... 

In addition, use of axial modulation scanning was also found to increase resolving power this effect is believed to arise as follows (March 1998). Immediately prior to ion ejection, as Vg is scanned upwards the axial secular motions of ions of a particular m/z value become resonant with the supplementary potential applied between the end caps, so that the axial excursions of the resonantly excited ions (and only these ions) increase in magnitude. As a result these ions escape from the space charge effects of the cloud of ions of higher m/z while the latter are stiU coUisionally cooled at the center of the trap. In this manner the ions that are resonantly excited become tightly... 

For a single trapped ion, its trajectory is a superposition of a fast oscillating motion (micromotion) with frequency Q. and a slow (secular) motion that can enclose a large area in the trap. The amplitude of the micromotion is proportional ioq fq< 1. From simulations it is known that when many ions are to be stored in the trap, a small q... 

FIGURE 18.10 Spatial separation of species, (a) Simulation of an ion crystal containing 50 laser-cooled (LC) Ba" " ions and 50 sympathetically cooled (SC) AF" " ions at 15 and 18 mK, respectively. Here, radio-frequency micromotion is included, (b) Cross-section of the crystal in (a). The micromotion is directed toward the electrodes (located on the x and y axes) and leads to a (slight) blurring of the AF" " ions in the radial direction, in addition to the blurring caused by secular motion and found also when the simulations are performed in a time-averaged pseudopotential. (From Zhang, C.B. et al., Phys. Rev. A, 76, 012719, 2007. With permission.)... 

A three-dimensional representation of an ion trajectory, shown in Figure 8, has the general appearance of a Lissajous curve composed of two fundamental frequency components, rUf O and >z,o of the secular motion. Higher-order (n) frequencies exist and the family of frequencies is described by given by... 

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