Broad line approximation

In practice this condition may be fulfilled not only in excitation, e.g. by means of a pulsed laser or a continuous dye laser with insufficient frequency selectivity, but also by means of fines from a continuous gas laser working in simultaneous axial mode u>i (multimode) generation regime see Fig. 3.10(a). Let Au>i = u>i+1 — uii = itc/L denote the mode separation in a laser, L being the resonator length. Then, as pointed out in [110, 127, 231], broad line approximation works if Awj is smaller than the width of the Bennet holes r en [268, 320] in the absorption contour see Fig. 3.10(6). The positions of the Bennet holes are determined by the condition ujq — w/ + kv = 0, where luq is the central transition frequency, k is the wave vector and v is the velocity of the absorbing particle. The broad fine approximation is valid if the following conditions are fulfilled (see Fig. 3.10) ... 

Table 3.7), i.e., at first glance the broad line approximation works unsatisfactorily. One must, however, keep in mind the instability of the modes due to the phase change under conditions of freely running modes (without synchronization). Broadening of the Bennet dips due to saturation of optical transition also takes place [268]. This makes the broad line approximation more realistic considering that the width of the amplification contour of the laser is 6 109 s-1. 

Thus, broad line approximation, being sufficiently realistic in a number of cases, permits us to introduce one dynamic constant Tp to characterize the absorption process. It is this fact which makes separation of dynamic and angular variables following (2.1) possible. This approximation will always be assumed in the course of our further discussion. 

The system of equations obtained, (5.22) and (5.23), in broad line approximation in many cases allows us to carry out the analysis of non-linear optical pumping of both atoms and molecules in an external magnetic field. Some examples will be considered in Section 5.5, among them the comparatively unexplored problem of transition from alignment to orientation under the influence of the dynamic Stark effect. But before that we will return to the weak excitation and present, as examples, some cases of the simultaneous application of density matrix equations (5.7) and expansion over state multipoles (5.20). 

This is a good place to mention that in our efforts to preserve a unified approach we have restricted ourselves to the approximation that all molecules are affected by light independently of their velocities ( broad line approximation ). This makes it possible to factorize, in a natural way, the calculations into two parts the dynamic and the angular (geometric) parts, the latter forming the subject of the present book. 

In the linear approximation there is a direct Fourier relationship between the FID and the spectrum and, in the great majority of experunents, the spectrum is produced by Fourier transfonnation of the FID. It is a tacit assumption that everything behaves in a linear fashion with, for example, imifonn excitation (or effective RF field) across the spectrum. For many cases this situation is closely approximated but distortions may occur for some of the broad lines that may be encountered in solids. The power spectrum P(v) of a pulse applied at Vq is given by a smc fiinction 18]... 

In their subsequent analysis Baker and Bleaney (ibidem) decided to ignore the last term on the assumption that gdl 3b hv. Although this is a reasonable approximation for lanthanide and actinide integer-spin ions doped in single crystals, it is not usually an acceptable assumption for the broad-line spectra from metalloproteins. Furthermore, the assumption of a A-distribution around zero (i.e., D 0 but all other zero-field interaction parameters are zero) is equally untenable for biomolecules. Therefore, we go for a later extension of the theory, based on a full Equation 12.9 and on (A) 0, for application to metalloproteins (Hagen 1982b). 

Variations in broad line NMR are primarily sensitive to motion. Consequently, changes in Av and AM2 are indications of motional processes in the system. If, as a first approximation, the relaxation process can be expressed by a single correlation time, xc, the following relation between NMR parameters and tc can be established ... 

High resolution l3C NMR is also used in the determination of the composition of the dispersed phase in cured rubber modified epoxies in order to analyze the chemical structure of the mobile segments 152). In this case quantitative analysis is possible because the areas under each peak are approximately equal to the number of carbons contributing to the peak, and the intensities of the broad lines from the rigid phase are very low, almost indistinguishable from the baseline noise. The structure of crosslinked networks based on poly(3,4-pyrrolidinediethylene), synthesized by different methods, was determined from gels swollen in water and chloroform 153). 

Absorption bands in crystal field spectra are not sharp lines. Instead, as the spectra illustrated in figs 3.1, 3.2 and 3.3 show, they contain rather broad envelopes approximating gaussian profiles which at half peak-height may have full widths ranging from <100 cm-1 to 1,000-2,000 cm-1. Several factors lead to broadened absorption bands and they are discussed below. 

This approximation is sufficient for broad lines and particle diameters <5 nm. The particle size is determined in the direction normal to the lattice plane indicated by the Miller index of the line used for profile determination. The formula featuring a constant accounting for the fact that the breadth is measured in radians (as is 20) is only valid for spherical particles. 

XH nmr spectrum has not been measured because of the low solubility of the complex. The 1H nmr spectrum of the triisopropylphosphine analog, tratts-[PdH(BH4) P(i-Pr)3 2], shows five broad lines centered at t23.2 (TMS) and these five lines may be considered to be the most intense of seven lines, if JPH is approximately the same as JHbh4 (ca. 9 Hz). 

It is clear, however, from the discussion involving Eqs. (7)—(9) and from the sudden approximation sum rule that the spectrum associated with the photoionization of a core electron should not, in fact, necessarily consist only of a single line some data observed for RbCl and RbF (40) are shown in Fig. 16. The narrow peaks are the Rb 4s24 6(1S) - -4s14 >6(2S) excitation and the broad peaks, approximately equal in intensity, arise from multiple electron excitation , that is, the production of final states such as 4s24 4 s(2S), where n > 5. Even though the photoemission event is just a one-electron dipole process, multiple excitation can occur because the wavefunction of the instantaneous intermediate state of the (TV—1)-electron ion [Eq. (7)] has overlap with wavefunctions of such multiply excited states that is, i has components which are eigenfunctions, n(N—1), of multiply... 

At the same time very often the real optical field interacting with atoms ha.s rather broad spectral profile, width of which is broader or comparable with the inhomogeneous width of the atomic transition. In this case, a broad spectral line approximation for quantum density matrix approach has proved to be verj- rewai d-ing. This approximation was introduced in the 1960s by C. Cohcn-Taimoudji for excitation of atoms with ordinai-y light sources [10]. This was an era before lasers. Later on it was adjusted for application for exedtation of atoms wdth multimode lasers [11] and for excitation of molecules in the case of large angular momentum states [3, 12]. 

Another potential application for LEDs is in illumination. The requirements for devices that serve as illumination sources are somewhat different than the monochromatic OLEDs described above. OLEDs targeted for RGB displays have to give electroluminescent spectra with a relatively narrow line shape centered on the peak wavelength. On the other hand, an illumination source is meant to approximate the blackbody solar spectrum and needs to have a broad line shape with roughly equal intensity across the entire visible spectrum. Therefore, in order to attain complete coverage across the visible spectrum, an OLED used for illumination purposes typically employs multiple emitters are that are either co-deposited into a single emissive layer or distributed into different layers or regions of the device. A number of the different device architectures have been reported to achieve efficient white EL and are discussed below. 

We have observed additionally the wider resonances -125 ppm (AV / 4 kHz) assigned to two (of four) protons of the ethylene bridge, and a very broad line at -405 ppm (AV /2> 10 kHz) of the iminc protons. The latter signal was detected for all complexes at approximately the same field position. The resonances of the 3 and 6 aromatic protons of complex 2 arc masked by those of the residual protons of watcr-d and DMSO-d, The "H spectrum of complex 5 displays the peaks of the 3 and 6 dcuierons at 2,0 and -1.9 ppm. respectively [70]. Some spectral data for other mangancse(salcn) complexes arc collected in Table I. 

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