Wavenumber determination

The use of the symbols F J) and B for quantities which may have dimensions of frequency or wavenumber is unfortunate, but the symbolism is used so commonly that there seems little prospect of change. In Equations (5.11) and (5.12) the quantity B is known as the rotational constant. Its determination by spectroscopic means results in determination of intemuclear distances and represents a very powerful structural technique. 

Make measurements of the transition wavenumbers in the rotational spectrum of silane from Figure 5.10 and hence determine the Si—H bond length. 

A close look at Figure 6.8 reveals that the band is not quite symmetrical but shows a convergence in the R branch and a divergence in the P branch. This behaviour is due principally to the inequality of Bq and Bi and there is sufficient information in the band to be able to determine these two quantities separately. The method used is called the method of combination differences which employs a principle quite common in spectroscopy. The principle is that, if we wish to derive information about a series of lower states and a series of upper states, between which transitions are occurring, then differences in wavenumber between transitions with a common upper state are dependent on properties of the lower states only. Similarly, differences in wavenumber between transitions with a common lower state are dependent on properties of the upper states only. 

From the following wavenumbers of the P and R branches of the 1-0 infrared vibrational band of H Cl obtain values for the rotational constants Bq, Bi and B, the band centre coq, the vibration-rotation interaction constant a and the intemuclear distance r. Given that the band centre of the 2-0 band is at 4128.6 cm determine cOg and, using this value, the force constant k. 

A publication by Elguero et al. 66BSF3744) discusses UV spectra of 170 pyrazoles determined in 95% ethanol. Rigorously, the UV substituent effects must be discussed using wavenumbers, since only wavenumbers (in cm ) are proportional to transition energies, AE. However, in order to have more familiar values, data on l-(2,4-dinitrophenyl)pyrazoles <66BSF3744) have been transformed from wavenumbers to wavelengths (in nm) (Table 20). 

Determination of the optical constants and the thickness is affected by the problem of calculating three results from two ellipsometric values. This problem can be solved by use of the oscillator fit in a suitable wavenumber range or by using the fact that ranges free from absorption always occur in the infrared. In these circumstances the thickness and the refractive index outside the resonances can be determined - by the algorithm of Reinberg [4.317], for example. With this result only two data have to be calculated. 

Cyc/o-Undecasulfur Su was first prepared in 1982 and vibrational spectra served to identify this orthorhombic allotrope as a new phase of elemental sulfur [160]. Later, the molecular and crystal structures were determined by X-ray diffraction [161, 162]. The Sn molecules are of C2 symmetry but occupy sites of Cl symmetry. The vibrational spectra show signals for the SS stretching modes between 410 and 480 cm and the bending, torsion and lattice vibrations below 290 cm [160, 162]. For a detailed list of wavenumbers, see [160]. The vibrational spectra of solid Sn are shown in Fig. 23. 

The time resolution of the instrument determines the wavenumber-dependent sensitivity of the Fourier-transformed, frequency-domain spectrum. A typical response of our spectrometer is 23 fs, and a Gaussian function having a half width... 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

In the time-domain detection of the vibrational coherence, the high-wavenumber limit of the spectral range is determined by the time width of the pump and probe pulses. Actually, the highest-wavenumber band identified in the time-domain fourth-order coherent Raman spectrum is the phonon band of Ti02 at 826 cm. Direct observation of a frequency-domain spectrum is free from the high-wavenum-ber limit. On the other hand, the narrow-bandwidth, picosecond light pulse will be less intense than the femtosecond pulse that is used in the time-domain method and may cause a problem in detecting weak fourth-order responses. 

Figure 3.17 presents ps-TR spectra of the olehnic C=C Raman band region (a) and the low wavenumber anti-Stokes and Stokes region (b) of Si-rra i-stilbene in chloroform solution obtained at selected time delays upto 100 ps. Inspection of Figure 3.17 (a) shows that the Raman bandwidths narrow and the band positions up-shift for the olehnic C=C stretch Raman band over the hrst 20-30 ps. Similarly, the ratios of the Raman intensity in the anti-Stokes and Stokes Raman bands in the low frequency region also vary noticeably in the hrst 20-30 ps. In order to better understand the time-dependent changes in the Raman band positions and anti-Stokes/Stokes intensity ratios, a least squares htting of Lorentzian band shapes to the spectral bands of interest was performed to determine the Raman band positions for the olehnic... 

Fig. 2 The experimentally determined potential energy V(), expressed as a wavenumber for convenience, as a function of the angle in the hydrogen-bonded complex H20- HF. The definition of

Fig. 5 The potential energy V tj>), expressed as a wavenumber, as a function of the angle for a H2S- HC1 and b H2S- C1F. These have been obtained using ab initio calculations, by the method discussed in the text. Several vibrational energy levels associated with the motion in

In either case, the information on the vibrational transition is contained in the energy difference between the excitation radiation and the inelastically scattered Raman photons. Consequently, the parameters of interest are the intensities of the lines and their position relative to the Rayleigh line, usually expressed in wavenumbers (cm 1). As the actually recorded emissions all are in the spectral range determined by the excitation radiation, Raman spectroscopy facilitates the acquisition of vibrational spectra through standard VIS and/or NIR spectroscopy. 

Rather than looking at the spectrum obtained from the secular determinant (5), we will here consider the spectrum SG for fixed wavenumber k and than average over k. One can write the spectrum in terms of a periodic orbit trace formula reminiscent to the celebrate Gutzwiller trace formula being a semiclassical approximation of the trace of the Green function (Gutzwiller 1990). We write the density of states in terms of the traces of SG, that is,... 

If we know the masses of the atoms involved in a vibration, and the wavenumber can be determined from a spectrum, then we can readily calculate a value for the force constant k. 

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