Angular intensity functions

The intensities and angular positions of layer lines 1 and 2 are also shown in Figure 2. Again, the noise in the positions calculated is lowest at points corresponding to the peaks of intensity. For these layer lines there are also systematic variations in the position with subsequent peaks falling at different positions relative to that which would be expected if there were exactly 49 subunits in the 69 X axial repeat. This layer line splitting was used to calculate the relative contributions of the Bessel function terms on layer lines 1 and 2. 

Mie scattering functions are generally presented in terms of the intensity parameters for Mie scattering, also known as the angular intensity functions (0) and i2(0). The subscripts of these functions indicate perpendicular and plane polarization, respectively. Besides being functions of the scattering angle 0, ij(0) and i2(0) are functions of the particle properties m and a [e.g., Lowan (1948) or Denman et al. (1966)]. 

For a given a and m, the angular intensity functions are related to the scattered intensity coefficients q1(0) and q2(0) by the expressions... 

At this point, it is known what to do to obtain a molecular intensity function with coefficients that are independent of s. It is also known how to remove the problem of having to use data the angular range of which is limited. One main problem remains. The manner in which it was solved will be seen to have had a significant effect on the course of crystal structure analysis. 

A common phenomenon observed upon adsorption of molecules is the decrease in the elastic peak intensity as a function of coverage. Figures 15A, B show the intensity and angular width of the elastic peak as a function of HCN coverage on Pd. The intensity correlates well with layer formation on the surface, much like RHEED oscillations. Even though the scattering mechanism is decidedly dipolar , the... 

However, any practical, quantitative use of Eq. (9) presupposes that fully normalized distribution functions /icp,rcp(0) are available. If normalization by the mean of the two angular intensities is included, as in the Kuhn g-factor Eq. (1), we obtain ... 

The correlator (6) is of the utmost importance because its generating function enters into an expression which describes the angular dependence of intensity of scattering of light or neutrons [3]. It is natural to extend expression (6) for the two-point chemical correlation function by introducing the w-point correlator ya1... (kl...,kn l) which equals the joint probability of finding in a macromolecule n monomeric units Maj.Ma> divided by (n-1) arbitrary sequences... 

Unlike the wave function, the electron density can be experimentally determined via X-ray diffraction because X-rays are scattered by electrons. A diffraction experiment yields an angular pattern of scattered X-ray beam intensities from which structure factors can be obtained after careful data processing. The structure factors F(H), where H are indices denoting a particular scattering direction, are the Fourier transform of the unit cell electron density. Therefore we can obtain p(r) experimentally via ... 

In Section 1.5 we shall use the wave function (1.17) and the machinery for handling angular momentum (Section 1.4) for the computation of the intensities of spectroscopic transitions. 

There should exist a correlation between the two time-resolved functions the decay of the fluorescence intensity and the decay of the emission anisotropy. If the fluorophore undergoes intramolecular rotation with some potential energy and the quenching of its emission has an angular dependence, then the intensity decay function is predicted to be strongly dependent on the rotational diffusion coefficient of the fluorophore.(112) It is expected to be single-exponential only in the case when the internal rotation is fast as compared with an averaged decay rate. As the internal rotation becomes slower, the intensity decay function should exhibit nonexponential behavior. 

Although the probability of absorption of TIR evanescent energy by a fluorophore of given orientation decreases exponentially with distance z from a dielectric surface, the intensity of the fluorescence actually viewed by a detector varies with z in a much more complicated fashion. Both the angular pattern of the emitted radiation and the fluorescent lifetime are altered as a function of z by the proximity of the surface. 

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