Thermal equilibrium spectrum

System (A8.2)-(A8.4) defines completely the time variation of orientation and angular velocity for every path X(t). One can easily see that (A8.2)-(A8.4) describe the system with parametrical modulation, as the X(t) variation is an input noise and does not depend on behaviour of the solution of (Q(t), co(r). In other words, the back reaction of the rotator to the collective motion of the closest neighbourhood is neglected. Since the spectrum of fluctuations X(t) does not possess a carrying frequency, in principle, for the rotator the conditions of parametrical resonance and excitation (unrestricted heating of rotational degrees of freedom) are always fulfilled. In reality the thermal equilibrium is provided by dissipation of rotational energy from the rotator to the environment and... 

Clearly, if a situation were achieved such that exceeded Np, the excess energy could be absorbed by the rf field and this would appear as an emission signal in the n.m.r. spectrum. On the other hand, if Np could be made to exceed by more than the Boltzmann factor, then enhanced absorption would be observed. N.m.r. spectra showing such effects are referred to as polarized spectra because they arise from polarization of nuclear spins. The effects are transient because, once the perturbing influence which gives rise to the non-Boltzmann distribution (and which can be either physical or chemical) ceases, the thermal equilibrium distribution of nuclear spin states is re-established within a few seconds. 

Local thermal equilibrium (LTE) is an assumption that allows for the molecules to be in equilibrium with at least a limited region of space and remains an assumption when using the Boltzmann law for the relative populations of energy levels. The LTE assumption notwithstanding, observation of a series of transitions in the spectrum and measurement of their relative intensities allows the local temperature to be determined. We shall see an example of this in Section 4.4 where the Balmer temperature of a star is derived from the populations of different levels in the Balmer series. 

Figure 12.3 outlines the essential features of the PASADENA/PHIP concept for a two-spin system. If the symmetry of the p-H2 protons is broken, the reaction product exhibits a PHIP spectrum (Fig. 12.3, lower). If the reaction is carried out within the high magnetic field of the NMR spectrometer, the PHIP spectrum of the product consists of an alternating sequence of enhanced absorption and emission lines of equal intensity. This is also true for an AB spin system due to a compensating balance between the individual transition probabilities and the population rates of the corresponding energy levels under PHIP conditions. The NMR spectrum after the product has achieved thermal equilibrium exhibits intensities much lower than that of the intermediate PHIP spectrum. 

The relative abundances of these product ions change dramatically as the ion source pressure increases from El conditions to 25 Pa. Above 100 Pa, the relative concentrations stabilize at the levels represented by the Cl spectrum of methane reagent gas (Pig. 7.3). [4,19] Portunately, the ion source pressure of some 10 Pa in Cl practice is in the plateau region of Pig. 7.3, thereby ensuring reproducible Cl conditions. The influence of the ion source temperature is more pronounced than in PI because the high collision rate rapidly effects a thermal equilibrium. 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

For each series of measurements about 50 g of solvent was transferred quantitatively in the dry box to the cell by pouring it into the dilution bulb this was the minimum amount required to fill the cell bulb. The cell was removed from the dry box, placed in the oil bath, and connected to the bridge. Time was allowed for the attainment of thermal equilibrium then at least three resistance measurements were made at five-min intervals, and the average value was calculated. The cell was removed from the bath and returned to the dry box. Dilute stock solution was quantitatively added to the cell by means of a weighing buret. The contents of the cell were carefully mixed, and the resistance of the solution was measured as before. The procedure just described was repeated several times with the dilute stock solution and then with the concentrated stock solution. About ten concentrations with a hundredfold range were obtained. A portion of the final solution in the cell (the most concentrated solution) was removed, and the infrared spectrum taken no absorption band indicative of traces of water was observed at 3600 cm-1. It was necessary to obtain the densities of... 

Fig. 3.10. The rototranslational absorption spectrum of H2-H2 pairs, recorded in equilibrium hydrogen (i.e., para-Fh and ortho-FL concentrations are in thermal equilibrium proportions) at three temperatures 77.4 K ( ), 195 K (x), and 293 K ( ). Various pressures from 10 to 100 atmospheres and a 3 m absorption path length were used after [37],...

In emission spectrometry, the sample is the infrared source. Materials emit infrared radiation by virtue of their temperature. KirchhofF s law states that the amounts of infrared radiation emitted and absorbed by a body in thermal equilibrium must be equal at each wavelength. A blackbody, which is a body having infinite absorptivity, must therefore produce a smooth emission spectrum that has the maximum possible emission intensity of any body at the same temperature. The emissivity, 8, of a sample is the ratio of its emission to that of a blackbody at the same temperature. Infrared-opaque bodies have the same emissivity at all wavelengths so they emit smooth, blackbody-like spectra. On the other hand, any sample dilute or thin enough for transmission spectrometry produces a structured emission spectrum that is analogous to its transmission spectrum because the emissivity is proportional to the absorptivity at each wavelength. The emissivity is calculated from the sample emission spectrum, E, by the relation... 

Figure 22-(a) shows the DD/MAS spectrum in the resonance range of a-methyl-ene carbon at 0 °C. This spectrum represents the thermal equilibrium state of this sample, because it was obtained by a single pulse sequence with the repetition time of 600 s longer than 5 times the longest Tic in the system. The spectrum (b) is that of the crystalline component, which was obtained with use of Torchia s pulse sequence [53]. In the equilibrium spectrum, the noncrystalline contribution (amorphous plus interfacial) can be seen upfield to the crystalline component. Figure 23 shows the elementary line shapes of the amorphous and crystalline-amorphous interphases that comprise the noncrystalline resonance. 

Molecular Conformation of sPP gel. Figure 27 shows the DD/MAS 13C NMR spectrum of sPP gel. This spectrum was obtained by a single-pulse sequence (tt /2—FIDdd-tt)ii with the repetition time ty more than 5 times the longitudinal relaxation time Tic. Hence, this spectrum reflects the thermal equilibrium state of the gel. For comparison, the spectrum of the bulk ttgg crystal of this sample... 

I is the effective moment of inertia of a dipole (we consider here a linear molecule), determined by the relation (149). The spectral function L(z), calculated for thermal equilibrium, is linearly related to the spectrum C° of the dipolar autocorrelation function (ACF) C°(f) (VIG, p. 137 GT, p. 152) as... 

Figure 4 REMPI spectra of desorbed CO for the B1 + (V =0) X1 + (v" = 0) transition (a) (2 + 1)REMPI spectrum from Pt(00 1) [30] and (b) (1 + E)REMPI spectrum from 0 03 (000 1) [29], Solid lines in (a) indicate simulated curves at thermal equilibrium with various temperatures.

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