Probability distribution Lorentzian

There are a number of other common probability distributions in addition to the Gaussian distribution, including the binomial distribution, the Poisson distribution, and the Lorentzian distribution. However, the Gaussian distribution is generally used in discussing experimental errors, and we return to this topic in Chapter 11. 

Gaussian and Poisson distributions are related in that they are extreme forms of the Binomial distribution. The binomial distribution describes the probability distribution for any number of discrete trials. A Gaussian distribution is therefore used when the probability of an event is large (this results in more symmetric bell-shaped curves), whereas a Poisson distribution is used when the probability is small (this results in asymmetric curves). The Lorentzian distribution represents... 

Assume that a random variable, x, is governed by the probability distribution (a version of the Lorentzian... 

Still assuming that a Lorentzian distribution of vibrational energies and the dipole approximation are employed. In this expression is the IR transition momenL Mu is the Raman transition probability, is the resonant mode frequency and is the natural line width of the transition. Since sum-frequency active modes must be both IR- and Raman-active, any vibrational mode that has an inversion centre cannot be sum-frequency-active. This result coupled with the coherent nature of sum-frequency generation precludes any sum-frequency response from bulk isotropic media. 

Under these conditions the singlet amplitude is distributed according to a Lorentzian distribution over the molecular eigenstates. Exciting with a broad (white) laser (or at least with a laser that completely spans the interaction width), one then sees in the fluorescence first the Fourier transform of the Lorentzian distribution, that is, an exponential decay. The density of /c> was, however, not taken to be so high as to dilute the singlet amplitude effectively to zero. It was taken to be intermediate, which meant that each ME still had enough radiative probability so as to radiate independently,... 

The Lorentzian distribution, which describes the resonances of nuclear reactions —in particular how the probability of interaction (cross section, see Chap. 4)... 

To provide a more quantitative description of the surface defect structure and distribution, we use a model for randomly distributed surface defects (Lu and Lagally 1982). With this model, the rocking curve width and its variation with Q can be calculated from a few parameters that describe the probability of encountering a defect (i.e., a step) and the phase change in encountering that defect. (For specular reflectivity, this phase change reflects the height difference across the step.) Within this model, an approximately Lorentzian line shape is reproduced. 

Fig. 20. Distribution of (a) field components and (b) field magnitude (c) shows die resulting static Kubo-Toyabe functions. Solid lines refer to Gaussian, dashed lines to Lorentzian field distributions. B is the most probable field and HWHM the half width at half maximum of the distributions.

Condition (ii) can be fulfilled in other crystal structures on occasion as well. An AFM state is usually a condition, since, as mentioned, the contact field will not vanish in a FM material (but is present only in conducting compounds). The important point is that in an AFM spontaneous spin precession can be absent although LRO of the spin system exists. The strict consequence of (ii) would be a non-depolarizing pSR signal in ZF. But in nearly all cases the field distribution (iii) exists and Lorentzian Kubo-Toyabe patterns are seen instead. It is important to realize that the width of the Lorentzian Kubo-Toyabe patterns is not simply connected to the size of the magnetic moments the concentration and nature of faults enters dominantly. A randomness of these faults (though probable) is not required since the muon positions woidd in any case be randomly distributed relative to them. We finally point out that the width of the field distribution is rather small (about 8 G for UAs) and in many cases not significantly different from that produced by nuclear dipoles. A distinction between the two can be cumbersome in some cases. 

Lorentzian distribution respectively) one can see that the estimates for the critical randomness in the Lorentzian case are about 25% less than the estimates for the rectangular one. This is probably due to the long tails of the Lorentzian distribution. 

Fermi level. In a coarse approximation the p-level distribution can be assumed to correspond to that of a Fermi gas of free electrons, or even simpler, to that of a rectangular shaped distribution of equally spaced states (Richtmyer et al. 1934). Assuming equal transition probabilities and a Lorentzian lifetime broadening, the K absorption coefficient follows an arctan curve as a function of energy (cf. fig. 7, Ho) ... 

Hamiltonian = matrix element of the Hamiltonian H I = nuclear spin I = nuclear spin operator /r( ), /m( ) = energy distributions of Mossbauer y-rays = Boltzmann constant k = wave vector L(E) = Lorentzian line M = mass of nucleus Ml = magnetic dipole transition m = spin projection onto the quantization axes = 1 — a — i/3 = the complex index of refraction p = vector of electric dipole moment P = probability of a nuclear transition = tensor of the electric quadrupole q = eZ = nuclear charge R = reflectivity = radius-vector of the pth proton = mean-square radi-S = electronic spin T = temperature v =... 

---