Orbital Frequencies

A more model-dependent way to constrain neutron star structure has to do with measurements of orbital frequencies in the accretion disk near the neutron star. Suppose that the frequency of some observed phenomenon could be identified with an orbital frequency vor >, and that this phenomenon lasted many cycles. The orbital radius f 0rb is clearly greater than the stellar radius R. In... 

Here j = cJ/GM2 is a dimensionless spin parameter, where J is the stellar angular momentum. If in a particular case one believes that the observed frequency is in fact the orbital frequency at the ISCO, then the mass is equal to the upper limit given above. 

Figure 3. Constraints from orbital frequencies. The 1330 Hz curve is for the highest kilohertz quasi-periodic oscillation frequency yet measured (for 4U 0614+091, by van Straaten et al. 2000). The 1500 Hz curve shows a hypothetical constraint for a higher-frequency source. Other lines are as in Figure 1. All curves are drawn for nonrotating stars the constraint wedges would be enlarged slightly for a rotating star (see Miller, Lamb, Psaltis 1998).

This picture is that described by the BO approximation. Of course, one should expect large corrections to such a model for electronic states in which loosely held electrons exist. For example, in molecular Rydberg states and in anions, where the outer valence electrons are bound by a fraction of an electron volt, the natural orbit frequencies of these electrons are not much faster (if at all) than vibrational frequencies. In such cases, significant breakdown of the BO picture is to be expected. 

We have thus far considered the probability of superelastic scattering on a single orbit. To obtain the scattering rate, or autoionization rate, we simply multiply this probability by the orbital frequency, 1/n3.4 Once again we find that T oc 1/n3 and that T decreases with increasing Z. The scattering description we have just given is a two channel description. This picture, when many channels are present, forms the basis of multichannel quantum defect theory.5... 

Ion-cyclotron resonance Trapping of ions in cubic cell under influence of trapping voltage and magnetic field. Orbital frequency related inversely to m/z value. 

The quadrupole ion trap (QJT) is about the size of a small fist and consists of a ring electrode and two hyperbolic end electrodes (see March and Todd68 for a detailed theory of operation and history of development). Like the linear ion trap (LIT, see below), the QJT operates at relatively high pressure (10 3 torr) with a helium buffer gas that assists the ions to maintain a stable orbital frequency. The buffer gas also serves as the collision gas for collision-induced dissociation (CID) during MS/MS experiments. 

Hysteresis between a fixed point and a strange attractor) Consider the Lorenz equations with <7 = 10 and b = 8/3. Suppose that we slowly turn the r knob up and down. Specifically, let r = 24.4 -h sin or, where ft) is small compared to typical orbital frequencies on the attractor. Numerically integrate the equations, and plot the solutions in whatever way seems most revealing. You should see a striking hysteresis effect between an equilibrium and a chaotic state. 

Thus, an electron at infinity has W 0 and therefore / = = 0. At finite distances its frequency of revolution is finite. If we assume that the electron, in coming from infinity to a non-radiating orbit where its frequency is / emits light whose frequency is the mean of the orbital frequencies / > and / (this is the correspondence with classical. physics) we have... 

The polarizability of an atom or molecule describes the response of the electron cloud to an external field. The atomic or molecular energy shift KW due to an external electric field E is proportional to i for external fields that are weak compared to the internal electric fields between the nucleus and electron cloud. The electric dipole polarizability a is the constant of proportionality defined by KW = -0(i /2. The induced electric dipole moment is aE. Hyperpolarizabilities, coefficients of higher powers of , are less often required. Technically, the polarizability is a tensor quantity but for spherically symmetric charge distributions reduces to a single number. In any case, an average polarizability is usually adequate in calculations. Frequency-dependent or dynamic polarizabilities are needed for electric fields that vary in time, except for frequencies that are much lower than electron orbital frequencies, where static polarizabilities suffice. 

Table 2 General relationships between the clay mineral distribution and the three main Earth s orbital frequency bands according to latitude, from cross-correlation spectral analysis of X-ray diffraction data on North Atlantic cores...

The length of the mean free path and, consequently, the required vacuum is related to the overall size of most instruments, e.g., 10 Torr for a quadrupole. The vacuum requirement of 10 Torr for the ICR and orbitrap may sean surprising at first glance. Consider If the orbit that an ion follows in an ICR ceU has a diameter of 3 cm, the circumference of the orbit is 2jtr or 10 cm given that the orbital frequency of an ion of ndz 400 is -350,000 Hz (in a 9.4 Tesla magnet), the distance that an ion travels per second is 350,000 Hz x 10 cm = 3,500,000 cm = 3.5 km (over 2 miles). Obviously, a vacuum of 10 Torr is required to avoid unwanted collisions. 

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