The Lorentz Factor

Most X-ray diffraction experiments are done with an incident beam that is partially polarized either by use of monochromators or from the nature of the source itself e.g. synchrotron radiation). The degree of polarization, P, defined as  

In the case of neutron diffraction there is no polarization effect, but the Lorentz factor applies so that the neutron reflecting power is  

In some crystalline materials a phase transition on lowering the temperature may produce a modulated structure. This is characterized by the appearance of satellite or superstructure reflections that are adjacent reflections (called fundamental reflections) already observed for the high temperature phase. The superstructure reflections, usually much weaker than fundamental reflections, can in some cases be indexed by a unit cell that is a multiple of the high temperature cell. In such a case the term commensurate modulated structure is commonly used. However, the most general case arises when the additional reflections appear in incommensurate positions in reciprocal space. This diffraction effect is due to a distortion of the high temperature phase normally due to cooperative displacements of atoms, ordering of mixed occupied sites, or both. Let us consider the case of a displacive distortion. 

The positions of the atoms of the high temperature phase in the whole crystal can be written as R/y= R/ -l- Xy, where R/= /la -l- + fc is the position vector 

More general anharmonic modulations can be considered by extending the sum in Equation (29) to a higher number of waves, say D in such a case the D q vectors are linear combinations of the basic (rationally independent) d q vectors d D). In the following we considered only the harmonic model of dimension d. 

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The atomic amplitude functions take account of the atomic F- factor, the temperature factor, the Lorentz factor, and the polarization factor. 

Before going further, it may be noted that the flipping ratio does not depend either on the Lorentz factor or on absorption in the sample. Certain instrumental parameters such as the polarisation of the neutron beam for the two spin states, the half wavelength contamination of the neutron beam and the dead-time detector can readily be taken into account when analysing the data. On the other hand, the extinction which may occur in the scattering process is not so easy to assess, but must also be included [14]. Sometimes, it is even possible to determine the magnetisation density of twinned crystals [15]. 

Recently this approach was extended by inclusion of the isovector-scalar partner, the 5-meson, of the isoscalar scalar a-meson [22], Unfortunately the value of the coupling for the 5-meson cannot be determined well by fitting properties of stable nuclei. Also in its simplest, density independent form, the inclusion of the 5-meson leads to an even larger net value for po. This happens because of the presence of the Lorentz factor m /E in the scalar potential... 

The central assumption underlying the standard approach to tachyon theory is that the usual Lorentz transformation also applies to the superluminal case. One therefore simply takes the Lorentz factor [1 — (v/c)2)]1/2 and substitutes v > c into it [27,74]. This leads directly to an imaginary rest mass and propagation time for tachyons, with many difficulties of interpretation [74]. 

Owing to the Lorentz factor in formula (4.54), when v approaches c, we have b etf—meaning that a molecule can be excited by a very distant passing particle, which is in contradiction with reality. This is a consequence of the fact that in our derivation (as in Ref. 150) we made no allowance for the weakening of the interaction between a charged particle and molecule when b> a, which is due to the polarization of the medium. The account of dielectric properties of the medium should lead to finite values of b eff even at v = c. 

In the program Corspot, the Lorentz factor used was of the form for precession camera data,... 

It is very difficult to mount a specimen in an X-ray fibre camera such that it is precisely normal to the beam. Indeed, frequently the fibre must be tilted deliberately by a nominal amount to observe specific meridional reflections. Because the tilt angle p features in the Lorentz factor and other expressions given in section 7, it is necessary to obtain a value for it using the microdensitometer data. AXIS provides two methods for calculating /3 both are similar to procedures outlined by Fraser et al. (4) and, superficially, resemble those described in section 6 for calculating 5. The first method again requires the user to specify the positions of pairs of equivalent spots from a data file. This time, however, both members of a pair must be... 

There are other factors affecting the intensity of the peaks on a x-ray diffraction profile of a powdered sample. We have analyzed the structure factor, the polarization factor, and the broadening of the lines because of the dimensions of the crystallites. Now, we will analyze the multiplicity factor, the Lorentz factor, the absorption factor, the temperature factor, and the texture factor [21,22,24,26],... 

Abstract. The Coulomb interaction which occurs in the final state between two particles with opposite charges allows for creation of the bound state of these particles. In the case when particles are generated with large momentum in lab frame, the Lorentz factors of the bound state will also be much larger than one. The relativistic velocity of the atoms provides the oppotrunity to observe bound states of (-n+fx ), (7r+7r ) and (7x+K ) with a lifetime as short as 10-16 s, and to measure their parameters. The ultrarelativistic positronium atoms (.4oe) allow us to observe the effect of superpenetration in matter, to study the effects caused by the formation time of A e. from virtual e+e pairs and to investigate the process of transformation of two virtual particles into the bound state. 

The mechanism of Aab creation is the Coulomb interaction in the final state (between a+ and b ), formatting from two virtual particles a+ and b, the bound state Aab (Fig. 1). This mechanism, in principle, allows for creation of all types of bound states and if a+ and b are relativistic particles, then Aab will also be relativistic. For ultra-relativistic atoms, there are effects caused by final time of atom formation and new phenomena during atom interaction with matter. High value of the Lorentz factors of atoms also allows for the detection new short lived bound states An, Ao and A k, consisting accordingly from (7r+p ), (7r+7r ) and (tt+ K ) mesons and to measure their parameters. 

The Lorentz factor, a geometrical factor that describes how the crystal is moved through the diffraction condition. 

The polarizability a(-w w) is involved in several linear optical experiments including refractive index measurements. Equation (93) shows that the solute molecule experiences a local field which is larger than the macroscopic field by the cavity field factor/ " and by the reaction field factor f For typical media the magnitude of the productis of the order of 1.3-1.4. In the case of a pure liquid this product simplifies to the Lorentz factor L", (86), and (94) simphfies to (95)... 

The Lorentz and polarization corrections,often called Lp, are geometrical corrections made necessary by the nature of the X-ray experiment. The Lorentz factor takes into account the different lengths of time that the various Bragg reflections are in the diffracting position. This correction factor differs for each type of detector geometry. For example, the Lorentz correction for a standard four-circle diffractometer... 

The Lorentz factor takes into account two different geometrical effects and it has two components. The first is owing to finite size of reciprocal lattice points and finite thickness of the Ewald s sphere, and the second is due to variable radii of the Debye rings. Both components are functions of 0. 

Figure 1. The distributions of the Lorentz factor, pressure and particle density of the shocked gas as functions of self-similar variable All quantities are normalized to unity at the shock front where x = L Long-dashed lines correspond to the losses uniformly distributed in the downstream medium and decreasing with time as f 1. Short-dashed lines are for the case of localized losses with 5= rj= 0. The solid line — solution of Blandford and McKee (1976).

In the case of localized losses we treat them as discontinuities of the energy, momentum and particle number fluxes at the shock front, which are characterized by three parameters e, 5, r/ equal to the fractions of corresponding fluxes lost at the shock front in the front comoving frame. We obtain the following expressions for the pressure p2, number density n2 and the Lorentz factor 72 immediately behind the shock front ... 

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