Intensity Boltzmann factor

If the spectmm is observed in absorption, as it usually is, and at normal temperatures the intensities of the transitions decrease rapidly as v" increases, since the population of the uth vibrational level is related to Nq by the Boltzmann factor... 

Here the relative intensities of the components of each branch are determined by the Boltzmann factor Correlation function K (t, J), corresponding to Gq(a>, J), is obviously the correlation function of a transition matrix element in Heisenberg representation... 

The interpretation of the stress dependent intensities is that the stress raises the energy of those B—H configurations with their axis along the direction of stress. The H has sufficient thermal energy at 100 K to reorient (Fig. 20b) the different orientations are populated according to their (stress-dependent) Boltzmann factors. Because the H can move at the measurement temperature (100 K) on the time scale of a Raman measurement (a few minutes) Herrero and Stutzmann (1988b) were able to estimate an upper limit for the barrier for H-motion. These authors assumed that the rate limiting step for H motion obeys first order kinetics and obtained Eb < 0.3 eV. 

Radiation thermometry (visual, photoelectric, or photodiode) 500-50,000 Spectral intensity I at wavelength A Planck s radiation law, related to Boltzmann factor for radiation quanta Needs blackbody conditions or well-defined emittance... 

Infrared spectroscopy 100-1500 Intensity 1 of rotational lines of light molecules Boltzmann factor for rotational levels related to I Also Doppler line broadening useful, principal applications to plasmas and astrophysical observations, proper sampling, lack of equilibrium, atmospheric absorption often problems... 

As the intensity of summation bands do not depend on Boltzmann factors, most of the extra frequencies mentioned in the last two paragraphs will persist at least to some degree at low temperatures, i.e. they will contribute to a temperature independent residual band width. 

On pp. 31 Iff., a preliminary discussion of the symmetry of induced line profiles was given. The spectral lines encountered in collision-induced absorption show a striking asymmetry which is described roughly by a Boltzmann factor, Eq. 6.59. However, it is clear that at any fixed frequency shift, the intensity ratio of red and blue wings is not always given exactly by a Boltzmann factor, for example if dimer structures of like pairs shape the profile, or more generally in the vibrational bands. We will next consider the latter case in some detail. 

For spectra corresponding to transitions from excited levels, line intensities depend on the mode of production of the spectra, therefore, in such cases the general expressions for moments cannot be found. These moments become purely atomic quantities if the excited states of the electronic configuration considered are equally populated (level populations are proportional to their statistical weights). This is close to physical conditions in high temperature plasmas, in arcs and sparks, also when levels are populated by the cascade of elementary processes or even by one process obeying non-strict selection rules. The distribution of oscillator strengths is also excitation-independent. In all these cases spectral moments become purely atomic quantities. If, for local thermodynamic equilibrium, the Boltzmann factor can be expanded in a series of powers (AE/kT)n (this means the condition AE < kT), then the spectral moments are also expanded in a series of purely atomic moments. 

For transitions in absorption the Boltzmann distribution of molecules over the vibrational levels of the ground state implies that a majority of bands observed will emanate from the lowest, zero-point level transitions from higher levels will be weakened in proportion to the Boltzmann factor exp [ — Jiv/lcT], Hot bands arising from excited vibrational levels can be identified by studying the effect of temperature on the relative intensities. At ordinary temperatures the Boltzmann factor decreases approximately tenfold for each 500 cm-1 of vibrational... 

So the separation from the vf1 fundamental is just X13, the coupling constant These bands are, of course, hot bands, their intensity depending on the Boltzmann factor. Since X13 can be either positive or negative such hot bands may appear on either the low or the high frequency side of v 1. Furthermore, since the coupling constant X13 is usually relatively small such bands can make a considerable contribution to the apparent width and intensity of the main v 1 band. (For additional examples see12) and13).)... 

These bands are partly hot and, of course many other such combinations could be imagined if for the lower frequency we used vp instead of v . The calculations of Robertson17) which take account of both the Franck-Condon and Boltzmann factors substantiate these conditions. For the HC1 complex at 300 °K the (0,1)- (1,0) difference band, the (0,0)->(l, 0) fundamental and the (Q, 0)-+(1,1) summation band which have relative intensities 0.39, 0.80 and 1,00 in this order receive contibutions from (other) hot bands of 0.16, 0.135 and 0.54 respectively. This means that the summation band can be about as sensitive to temperature as the difference band. 

The relative weakness of the subbands is very likely connected with the frequencies of v and vp which are, of course, higher than for the HC1 complex. This diminishes the intensity of the hot bands due to the higher Boltzmann factor. Arnold and Millen varied temperature from room temperature to 70 °C and found that the intensity of the 3300 cm-1 band increased with increasing temperature more than that of the other bands confirming its assignment as a difference tone. It would be desirable to measure this spectrum at lower temperatures as well but, to the writer s knowledge, this has not been done. 

The other lines in Fig. 11 are vibrational satellites. Their intensities depend on the Boltzmann factor and therefore on the frequencies of the vibrations that are involved. Knowing the temperature these frequencies can be computed from the observed intensities. The vibrations which stand a good chance of causing satellites are, of course, the ones of low frequency. Table 9 is a part of Legon, Millen and... 

Rogers Table 1 88) who assigned all the satellites to v , vp and their various overtones and combinations. From the intensities they obtained v = 197 15 cm-1 and vp = 91 + 20 cm-1 in fair agreement with their values obtained by Thomas 78) from the vibrational spectrum. They did not observe satellites with the other bridge bending mode, 555 cm-1, presumably because of its low Boltzmann factor. 

This difference ist due mainly to experimental difficulties nitrogen resonances fall within the low-frequency range, 1 to 5 MHz, while most of the chlorine resonances, especially in covalent compounds, are located in the 30-40 MHz band.Thus the Boltzmann factor hv/kT for nitrogen is seven times smaller and the line intensity is accordingly reduced, Moreover, the splitting of the nitrogen resonance line by the asymmetry of the electric field gradient tensor reduces the line intensity by a factor of more than 2. 

Theoretically 3), the intensities of the three lines are equal however, the Boltzmann factor, possible anisotropy of the crystalline sample and, most of all, the relaxation times can modify them to the extent that some lines remain unobserved (cf. Ref. 77 in Five-membered heterocycles ). 

As mentioned, the line intensities are equal exept for the contribution of the Boltzmann factor, the relaxation times and the orientation of the sample crystallites relative to the radio-frequency magnetic field.They thus deserve no further mention. 

Some compounds, however, display no resonance at this temperature, e. g. ICN 10) and piperazine 12) in the former compound, strains may explain the absence of the lines observed at higher temperatures. Operation at liquid nitrogen temperature is convenient because it enhances the Boltzmann factor and the tank coil quality factor. On the other hand, long relaxation times may reduce line intensities below observability, so that it may be better to operate at a slightly higher temperature where relaxation times are shorter while the Boltzmann factor and the coil quality are not much decreased13). 

In contrast to the CH3CN situation, the spectra of interstellar ammonia give considerable insight into excitation and de-excitation mechanisms. From the observed intensities of the interstellar ammonia lines it has been derived that the excitation temperature 7 12, determined from the relative intensities of the (1,1) and the (2,2) lines, is notably lower than the excitation temperature r13 determined from the intensities of the (1,1) and (3,3) lines. Thus the (3,3) level shows an excess population over the (1,1), (2,2) levels. In other words, ortho-ammonia is not in equilibrium with para-ammonia. However, a more detailed study of the two para-ammonia levels (1,1) and (2,2) also reveals that their relative populations are not given by a simple Boltzmann factor for each of them. The (1,1) level has population in excess over the Boltzmann distribu-... 

The expressions can also be used to calculate relative intensities for emission from, for example, the v = 43 level (excited by an argon ion laser in Exp. 39) to various v" levels. In this case, the Boltzmann factor in section 5 can be set equal to 1, since only one upper state is involved. The emission experiment shows dramatic intensity variations for different v" values, and quantitative relative peak intensity values can be measured from the few narrow rotational lines seen for each vibrational transitioa... 

Figure 6 Energy transfer diagram illustrating Rayleigh and Raman scattering (top), and Raman spectra for CCfr excited at room (298 K) and hqnid-N2 (77 K) temperatures by Ar+ ion laser radiation of Xq = 488.0 nm or vo = 20 492 cm (bottom). The number above the peaks is the Raman shift, Av = vq — Wc cm. Since the fraction of molecules occupying excited states depends on the Boltzmann factor (kT = 207 cm at 298 K), the intensities of anti-Stokes bands fall off rapidly with decreasing temperature (kT = 54 cm at 77 K) and increasing vibrational frequency Vk...

These expressions for the intensities contain three factors that depend on the temperature, namely the degree of ionization, the Boltzmann factors and the partition functions. In particular, ay can be written as a function of the electron number density and the Saha function as ... 

At zero stress, a moderate temperature rise of samples containing the >(H,0) centres results in the observation of two thermalized spectra >2(14,0) and >3(H,0) with 2p i lines at 8.224 and 7.76meV, respectively [185], If it is assumed that the final states of the lines of these spectra are the same as those of >i(H,0), one deduces (H, O) - >2 (H, O) and >1 (H, O) - >3 (H, O) energy separations of 2.512 and 2.98 meV, respectively. Thus, the thermal population of the Is ( >2) and Is ( >3) states corresponding to these energy differences should result in intensity ratios between the >2 and >3 spectra on the one hand and the >1 spectrum on the other hand, smaller than those actually measured. A fit of the measured intensity ratios to the splitting deduced from the realistic Boltzmann factors gives only 1.57 and 1.94 meV for... 

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