Wavenumber definition

The rotational microwave spectrum of a diatomic molecule has absorption lines (expressed as reciprocal wavenumbers cm ) at 20, 40, 60, 80 and 100 cm . Calculate the rotational partition function at 100 K from its fundamental definition, using kT/h= 69.5 cm" at 100 K. 

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

Periodicity in space means that it repeats at regular intervals, known as the wavelength, A. Periodicity in time means that it moves past a fixed point at a steady rate characterised by the period r, which counts the crests passing per unit time. By definition, the velocity v = A/r. It is custom to use the reciprocals of wavelength 1/X — (k/2-ir) or 9, known as the wavenumber (k = wave vector) and 1/t — v, the frequency, or angular frequency u = 2itv. Since a sine or cosine (harmonic) wave repeats at intervals of 2n, it can be described in terms of the function... 

The Fourier transform introduces the wavenumber vector , which has units of 1 /length. Note that, from its definition, the velocity spatial correlation function is related to the Reynolds stresses by... 

By definition, the turbulent kinetic energy k can be found directly from the turbulent energy spectrum by integrating over wavenumber space ... 

From this definition, it can be observed that T,(k. t) is the net rate at which turbulent kinetic energy is transferred from wavenumbers less than k to wavenumbers greater than k. In fully developed turbulent flow, the net flux of turbulent kinetic energy is from large to small scales. Thus, the stationary spectral energy transfer rate Tu(k) will be positive at spectral equilibrium. Moreover, by definition of the inertial range, the net rate of transfer through wavenumbers /cei and kdi will be identical in a fully developed turbulent flow, and thus... 

The banded coherency spectrum has an analogous definition (Yeung and Pope 1993) where the energy spectra are replaced with the spectra integrated over a finite wavenumber band. From the spectral time-evolution equations, it is easily shown that, in the absence of mean scalar gradients, the time evolution of the coherency spectrum is governed by 1 dPap Tap 1 Taa 1 Tpp ... 

This equation is, of course, simply a definition of Ro(kx). Its usefulness arises from the explicit separation of the geometrical and Rayleigh components of the reflectance. Moreover, the fraction (k — kfy/ik — k ) in (7.26) contributes significantly only when kx is close to kp. The Rayleigh wavenumber is always greater than the bulk shear wavevector ks by 5-10 per cent or so (Table 6.2), and since R(kx) = 1 for kx > ks, again cf. Fig. 6.3(b)i, the approximation may be made that... 

To meet these requirements we first observe that the wavenumber and the phase are coupled to the angles of the velocity C given by expression (64). In this way the angle of any transmitted EMS wave in medium II can be expressed in terms of the angles ex and (3. In analogy with the classical analysis on total reflection, which includes phase differences [35], we introduce a complex form of the angle ex of an EMS wave. The definitions... 

Guidelli model of, 899 Habib and Bockris, 899 at the interface, importance of, 918 -ion interaction energy, 924 -metal interactions, 896 chemical forces, 897, 972 lateral forces, 897 monomers of, definition, 899 orientation of, 898 Parsons model of, 899 and potential of the electrode. 900. 924 preferential orientation of, 912 and solvent excess entropy, 912 the "three-state water model 898, 899 Wave nature of electrons, 788 Wavenumber, 799 Waves... 

Units of spectral moments specified by other workers may differ from those implied here because frequency in wavenumbers, v, is often replaced by angular frequency, . Also, normalization by the product of densities, Q2, is sometimes suppressed, especially if many-body interactions are considered. Furthermore, factors of h/2kT or hc/2kT have been used to the right of Eq. 3.6 in the definition of yo-... 

We note that in the definition of moments, we have used frequency v in units of cm-1. Other units of frequency are sometimes chosen instead of the experimentalist s standard use of frequency in wavenumber units (cm-1), angular frequencies co = 2ncv are the most likely choice of theorists, which leads to different dimensions of the moments, and to the appearence of factors like powers of 2nc. We note, moreover, that in the early days of collisional induction studies the zeroth moment yo was defined without the hyperbolic cotangens function, Eqs. 5.6. For the vibrational bands (hco 3> kT), the old and new definitions are practically identical. However, for the ro to translational band substantial differences exist. The old definition of yo was never intended to be used for the far infrared [314] only Eq. 5.6 gives total intensity in that case. 

In spectroscopy the unit cm-1 is almost always used for wavenumber, and term values and wavenumbers always refer to the reciprocal wavelength of the equivalent radiation in vacuum. The symbol c in the definition E/hc refers to the speed of light in vacuum. 

The precise definition of resolution depends on the lineshape, but usually resolution is taken as the full line width at half maximum intensity (FWHM) on a wavenumber, 8v, or frequency, 8v, scale. 

The definition of the ET reaction coordinates according to Eq. [3] allows a direct connection between the activated ET kinetics and steady-state optical spectroscopy. In a spectroscopic experiment, the energy of the incident light with the frequency v (v is used for the wavenumber) is equal to the donor-acceptor energy gap... 

Equation [139] is not very practical because an accurate definition of the average wavenumber, Vav = Vav/c, demands knowledge of the emission spectrum for which Eq. [136] provides a direct route to the transition dipole. But Eq. [139] can be used in approximate calculations by assuming Vav = Vem-... 

In each case the attenuation of sound can be formally represented by defining a complex wavenumber, where the sound attenuation coefficient is the imaginary part of the wavenumber. The complex wavenumber also leads to the definition of a complex sound speed and a complex dynamic elastic modulus. 

Birefringent retarders are made of anisotropic uniaxial crystals such as calcite, a-quartz, sapphire, zinc sulphide, cadmium selenide and others. As Eq. (3.2-14) shows, the produced retardation is exact for a definite wavenumber or for its multiples. Between them all remaining possible polarization states are passed through. 

Any object at a temperature above absolute zero emits thennal radiation, it is a thermal radiator. Ideally, its atoms or molecules are in a thennal equilibrium, the entire ensemble has a definite temperature. In contrast to lasers, thermal radiation sources produce non-coherent radiation. Its quanta have a random phase distribution, both spatially and temporarily. Planck s law defines the. spectral radiance of a black body the radiant power per solid angle, per area, and per wavelength L j (Eq. 3.3-2) or per wavenumber L j (Eq. 3.3-3) ... 

Spectroscopy as a two-dimensional analytical procedure provides ideally a number of independent channels of information, each at a definite wavenumber u with a width of Ai>. Let and define the upper and the lower limit of the usable intensity range, respectively, and a the standard deviation of the measured intensity values. On the assumption that these values obey a Gaussian distribution the potential amount of information Ipot which a spectrometer may transfer during a time r is given by (Eckschlager, 1969 Doerffel and Hildebrandt, 1969 Kaiser, 1970 Doerffel and Eckschlager, 1981) ... 

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