Frequency spectrum calculations

Therefore, the problem of frequency spectrum calculation is how to solve the coefficient by Equation 7. Separate the odd item and even item of can get... 

Actual frequency spectrum calculations were carried out by Smith (1948) and Houston (1948). Fig. III.2 reproduces Smith s results. This curve was calculated making use of the lattice constant, two elastic constants, and one Raman frequency. The model for calculation included first and second neighbor forces. Recalculation of this frequency distribution with improved elastic constants by Victor (1962) shows that heat capacities calculated in this way are at elevated temperatures about... 

A detailed discussion of the frequency spectrum has been made by Yoshimori and Kitano (1956). Komatsu (1958, 1964), and Young and Koppel (1965), where reviews of other and earlier papers also can be found. Fig. III.4 reproduces the frequency spectrum calculated by Young and Koppel (1965) using the root sampling technique (Section II.4.2). Its agreement with the data of Table III.3 between 100 and 1000° K is within experimental error. Some main features of the spectrum are also supported qualitatively by neutron scattering. The force constants... 

Fig. 12b). Since practically the same spectral shape is obtained at Q-band (35 GHz) (Fig. 12c), the commonly used criterion stating that the shape of an interaction spectrum is frequency-dependent fails to apply in this case. Actually, outer lines arising from the exchange interaction are visible on the spectrum calculated at Q-band (Fig. 12c), but these lines would be hardly detectable in an experimental spectrum, because of their weak intensity and to the small signal-to-noise ratio inherent in Q-band experiments. In these circumstances, spectra recorded at higher frequency would be needed to allow detection and study of the spin-spin interactions. 

H atomic spectrum Calculation of the allowed transition frequencies in the... 

Additionally, with the inclusion of computers as part of an instrument, mathematical manipulation of data was possible. Not only could retention times be recorded automatically in chromatograms but areas under curves could also be calculated and data deconvoluted. In addition, computers made the development of Fourier transform instrumentation, of all kinds, practical. This type of instrument acquires data in one pass of the sample beam. The data are in what is termed the time domain, and application of the Fourier transform mathematical operation converts this data into the frequency domain, producing a frequency spectrum. The value of this methodology is that because it is rapid, multiple scans can be added together to reduce noise and interference, and the data are in a form that can easily be added to reports. 

Fig. 13.21 shows another example of oscillatory burning of an RDX-AP composite propellant containing 0.40% A1 particles. The combustion pressure chosen for the burning was 4.5 MPa. The DC component trace indicates that the onset of the instability is 0.31 s after ignition, and that the instability lasts for 0.67 s. The pressure instability then suddenly ceases and the pressure returns to the designed pressure of 4.5 MPa. Close examination of the anomalous bandpass-filtered pressure traces reveals that the excited frequencies in the circular port are between 10 kHz and 30 kHz. The AC components below 10 kHz and above 30 kHz are not excited, as shown in Fig. 13.21. The frequency spectrum of the observed combustion instability is shown in Fig. 13.22. Here, the calculated frequency of the standing waves in the rocket motor is shown as a function of the inner diameter of the port and frequency. The sonic speed is assumed to be 1000 m s and I = 0.25 m. The most excited frequency is 25 kHz, followed by 18 kHz and 32 kHz. When the observed frequencies are compared with the calculated acoustic frequencies shown in Fig. 13.23, the dominant frequency is seen to be that of the first radial mode, with possible inclusion of the second and third tangential modes. The increased DC pressure between 0.31 s and 0.67 s is considered to be caused by a velocity-coupled oscillatory combustion. Such a velocity-coupled oscillation tends to induce erosive burning along the port surface. The maximum amplitude of the AC component pressure is 3.67 MPa between 20 kHz and 30 kHz. - ...

Application of Eq. (2.58) to calculate the temperature factors requires knowledge of the full frequency spectrum of the crystal throughout the Brillouin zone. Such information is only available for relatively simple crystal structures such as Al, Ni, KC1, and NaCl (Willis and Pryor 1975, p. 13ff.). Agreement between theory and experiment for such solids is often quite reasonable. 

Fig. 21.7 Photoexcitation spectrum of the 6sl5d D2 —> (6p3/2 d) 7 = 3 transition in Ba as a function of the frequency of the third laser. All three lasers are circularly polarized in the same sense. The broken line is the spectrum calculated from Eqs. (21.15) and (21.16). The energy level inset is not to scale (from ref. 9).

A method which circumvents many of the disadvantages of the transmission line and cavity perturbation technique was pioneered by Stuchley and Stuchley (1980). This technique calculates the dielectric parameters from the microwave characteristics of the reflected signal at the end of an open-ended coaxial line inserted into a sample to be measured. This technique has been commercialized by Hewlett Packard with their development of a user-friendly software package (Hewlett Packard 1991) to be used with their network analyzer (Hewlett Packard 1985). This technique is outstanding because of its simplicity of automated execution as well as the fact that it allows measurements to be made over the entire frequency spectrum from 0.3 MHz to 20 GHz. 

Figure 14. The low-frequency loss spectrum calculated from rigorous formulas (135) for the spectral function. G = 2.4,g — 2.5,y — 0.2, P — it/8 u — 3.5 (solid line), 5.5 (dashed line), and 10 (dash-and-dotted line). Vertical lines mark estimated V, values, given by Eq. (34c).

Figure 7.X2. (a) Lattice low-temperature vibration spectrum of solid rt-hex-ane. Vertical lines indicate Raman active frequencies, (b) Calculated dispersion curves of low-frequency vibrations. (From Takeuchi et al. [1980].)...

As described in earlier sections, any two material bodies will interact across an intermediate substance or space. This interaction is rooted in the electromagnetic fluctuations— spontaneous, transient electric and magnetic fields—that occur in material bodies as well as in vacuum cavities. The frequency spectrum of these fluctuations is uniquely related to the electromagnetic absorption spectrum, the natural resonance frequencies of the particular material. In principle, electrodynamic forces can be calculated from absorption spectra. 

Since almost all equations used in impedance methods are derived assuming linearity, it is important to have some means of verifying this supposition. The Kramers-Kronig relations2 link Z with Z" and allow the calculation of values for Z" at any frequency from a knowledge of the full frequency spectrum of Z, and vice versa. 

The convolution theorem states diat /, g and h are Fourier transforms of F, G and H. Hence linear filters as applied directly to spectroscopic data have their equivalence as Fourier filters in die time domain in other words, convolution in one domain is equivalent to multiplication in die other domain. Which approach is best depends largely on computational complexity and convenience. For example, bodi moving averages and exponential Fourier filters are easy to apply, and so are simple approaches, one applied direct to die frequency spectrum and die other to die raw time series. Convoluting a spectrum widi die Fourier transform of an exponential decay is a difficult procedure and so die choice of domain is made according to how easy the calculations are. 

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