Frequency centroid

Currently, as rapid progress of computer technology, AE waveforms can be recorded readily as well as the parametric features. Thus, such waveform-based features as peak frequency and frequency centroid are additionally determined in real time from the fast Fourier transform (FFT) of recorded waveforms. AE parametric features are thus extracted and provide good information to correlate the failure behavior of materials. 

The key feature of A3.8.18 is that the centroids of the reaction coordinate Feymnan paths are constrained to be at the position q. The centroid g particular reaction coordinate path q(x) is given by the zero-frequency Fourier mode, i.e. 

Specifically, following the rate expression of QTST in Eq. (4-1) and assuming the quantum transmission coefficients the dynamic frequency factors are the same, the kinetic isotope effect between two isopotic reactions L and H is rewritten in terms of the ratio of the partial partition functions at the centroid reactant and transition state... 

An example of the complexity of the frequency variations in the cubic cell is given in the extreme by the splitting of a peak into a doublet for ions excited to very large orbits. A similar phenomenon was noted by Marshall (47). In the example reported here, the ions were excited by a 0.385 volt peak-to-base RF burst in a 0.0254 m cubic cell with a 1 volt trap and a magnetic field of 1.2 T. Two maxima are discernible for ion-cyclotron-orbit sizes larger than the noptimaln orbit size at about 760 psec excitation time. Local centroids are measurable one increases by ca. 50 Hz and the other... 

We wish to point out, that by use of a suitable fiber which further broadens the spectrum, this fs laser frequency measurement technique has now been simplified to a setup with a single laser, as described elsewhere in this volume [6]. With the technique of Fig. 6, the 15 — 25 transition frequency was measured twice, first with a GPS referenced commercial Cs clock [29], and second with a transportable Cs atomic fountain clock constructed by A. Clairon and coworkers in Paris [30]. A total of 614 spectral lines was recorded in the latter measurement during ten days, and fitted with the described line shape model [13]. After adding a correction of 310 712 233(13) Hz to account for the hyperfine splitting of the 15 and 25 levels, we obtain for the hyperfine centroid [28] ... 

Although various interchanges of laser wavelengths, power meters, etc. will be made to control systematics, the basic technique for the measurement is indicated in fig. 5. The lasers are set on laser lines approximately equidistant from, but on either side of the resonance centroid, and balanced in power. The lasers are chopped in anti-phase and the difference signal, S(u> 1) — S(u>2) is recorded. The beam velocity is varied till the zero-crossing (where the signals are equal) is found. The resonance centroid (in the ion s rest frame) is then obtained from the relativistic Doppler formula and the mean of the two laser frequencies. 

Both definitions are natural since wq turns out to be the ratio of the microwave frequency w and the Kepler firequency H of the Rydberg electron, and Sq is the ratio of the microwave field strength and the field strength experienced by an electron in the noth Bohr orbit of the hydrogen atom. Motivated by the above discussion we have redrawn the results obtained by Bayfield and Koch (1974) and present them in Fig. 7.2 as an ionization signal (in arbitrary units) versus the scaled field strength defined in (7.1.3). For no in (7.1.3) we chose no = 66, the centroid of the band of Rydberg states present in the atomic beam. 

For another perspective we mention a second approach of which the reader should be aware. In this approach the dividing surface of transition state theory is defined not in terms of a classical mechanical reaction coordinate but rather in terms of the centroid coordinate of a path integral (path integral quantum TST, or PI-QTST) [96-99] or the average coordinate of a quanta wave packet. In model studies of a symmetric reaction, it was shown that the PI-QTST approach agrees well with the multidimensional transmission coefScient approach used here when the frequency of the bath is high, but both approaches are less accurate when the frequency is low, probably due to anharmonicity [98] and the path centroid constraint [97[. However, further analysis is needed to develop practical PI-QTST-type methods for asymmetric reactions [99]. 

The variation is quite complicated since the various quantities appearing in Eq. (128) such as c, and w, are themselves functions of the coefficients a/. The averages are constrained (cf. Eq. (121)) to the centroid defined by the aj s. To circumvent this complication one may use a self-consistent-field approach. Given an initial choice of the coefficients, one varies the Lagrangian in Eq. (128), ignoring the dependence of the c/s, and the frequencies u) and ojq on the transformation coefficients, to find a new set of transformation coefficients. The new set is then used to generate a new set of c/s, etc., and the procedure is repeated until convergence is obtained. 

The focus of this article, however, is on a more specialized topic in path integration— the path centroid perspective. One of the many interesting ideas suggested by Feynman in his formulation and application of path integrals was the notion of the path centroid variable [1], denoted here by the symbol q. The centroid is the imaginary time average of a particular closed Feynman path q(j), which, in turn, is simply the zero-frequency Fourier mode of that path, that is. 

The equations given above have a straightforward interpretation in the context of quadratic reference systems. By substituting the centroid-constrained propagator for the LHO into Eq. (2.26), and using a general LHO frequency co such that W q +q) = V q -i-q)- m(a q with = mj8(fl -b w ), the renormalized LHO frequency ca is defined by... 

It should be noted that these equations are to be solved for each position of the centroid q. The frequency in Eq. (2.27) is the same as the effective frequency obtained for the optimized LHO reference system using the path-integral centroid density version of the Gibbs-Bogoliubov variational method [1, pp. 303-307 2, pp. 86-96], Correspondingly, Eqs. (2.27) and (2.28) are exactly the same as those in the quadratic effective potential theory [1,21-23], The derivation above does not make use of the variational principle but, instead, is the result of the vertex renormalization procedure. The diagrammatic analysis thus provides a method of systematic identification and evaluation of the corrections to the variational theory [3],... 

Since is a convolution expression, the self-consistent equation for d is not local in Fourier space, and therefore one can no longer seek a single effective frequency solution as in Eq. (2.27). Therefore, this diagrammatic analysis demonstrates that the optimized LHO reference system is the best possible quadratic potential with which to approximate an anharmonic potential, a fact reached independently from the GB variational perspective. Further corrections to the centroid density are thus beyond an effective potential description [3]. 

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