Spectral functions translation

The translational spectral function, g(v), may be considered a (very diffuse) spectral line centered at zero frequency which arises from transitions between the states of relative motion of the interacting pair. It is the free-state analog of the familiar vibrational and rotational transitions of bound systems, with the difference that the motion is here aperiodic the period goes to zero due to the lack of a restoring force. The negative fre-... 

At constant temperature, the observed widths of the spectral functions decrease with increasing mass of the collisional pair. This fact is a simple consequence of the mean translational energy of a pair, jm v = kT, which is the same for all pairs. The interaction time is roughly proportional to the reciprocal root mean square speed, and thus to the square root of the reduced mass. 

According to Eq. 3.5, yi may be considered the total absorption in the translational band. We, however, prefer to consider Mq the total intensity, Eq. 3.4 with n = 0, because the spectral function g(v) is more closely related to the emission (absorption) process than a(v). For rare gas mixtures, we have the relationships of Eqs. 3.7. In other words, yo may be considered a total intensity of the spectral function, g(v), and the ratio yi /yo is a mean width of the spectral function (in units of cm-1). Both moments increase with temperature as Table 3.1 shows. With increasing temperature closer encounters occur, which leads to increased induced dipole moments and thus greater intensities. 

Liquids. The translational absorption profiles of a 2% solution of neon in liquid argon have been measured at various temperatures along the coexistence curve of the gas and liquid phases [107]. Figure 3.8 shows the symmetrized spectral function at four densities. At the lowest density (479 amagat for T = 145 K curve at top) the profile looks much like the binary spectral function seen in Fig. 3.2, especially the nearexponential wing for frequencies v > 25 cm-1. With increasing density the intercollisional dip develops at low frequencies, much like the dips seen at much lower densities in Fig. 3.5 - only much broader. 

The simplified translational and orientational spectral functions Lq and L, pertinent to elastic vibration of H-bonded molecules, are represented each in the form of a Lorentz line ... 

Accounting for (120), we finally derive the expression for the translational spectral function ... 

Turning to a particular case of frozen translations, in which the spectral function is given by (153) and the effective reorientation frequency by (154d), one can see that, if the rotary constant c exceeds a certain critical value... 

This result, when substituted into the expressions for C(t), yields expressions identical to those given for the three cases treated above but with one modification. The translational motion average need no longer be considered in each C(t) instead, the earlier expressions for C(t) must each be multiplied by a factor exp(- co2t2kT/(2mc2)) that embodies the translationally averaged Doppler shift. The spectral line shape function 1(G)) can then be obtained for each C(t) by simply Fourier transforming ... 

The MD simulations show that second shell water molecules exist and are distinct from freely diffusing bulk water. Freed s analytical force-free model can only be applied to water molecules without interacting force relative to the Gd-complex, it should therefore be restricted to water molecules without hydrogen bonds formed. Freed s general model [91,92] allows the calculation of NMRD profiles if the radial distribution function g(r) is known and if the fluctuation of the water-proton - Gd vector can be described by a translational motion. The potential of mean force in Eq. 24 is obtained from U(r) = -kBT In [g(r)] and the spectral density functions have to be calculated numerically [91,97]. 

After Fourier transformation, the time domain of the Green s function is translated to frequency dependency. We start with the resulting spectral representation of the one-electron propagator... 

Some systems cannot be well described by translational motion, so instead they require a model based on rotational diffusion. The commonly used model is one where the nuclei and radical form a bound complex, then this complex rotates to modulate dipolar coupling.74 Here, the overall correlation time consists of the rotational correlation time of the solvent complex, xT, and the exchange rate of molecules in and out of the complex, tm, where 1 /tc 1 /tr I 1 /tm. The form of this spectral density function is simpler4,25 ... 

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