Bragg-Lane

Figure 4.21 Definition of direction cosines g and. (a) Lane case, (b) Bragg case...

The diffraction spots are expected where transmitted and diffracted electron beams intersect with the detector. The basis for diffraction pattern geometry analysis, thus, is the crystal reciprocal lattice and the Lane diffraction condition, or the equivalent Bragg s law, for diffraction ... 

Lane equations Equations that, like the Bragg equation, express the conditions for diffraction in terms of the path difference of scattered waves. Laue considered the path length differences of waves that are diffracted by two atoms one lattice translation apart. These path differences must be an integral number of wavelengths for diffraction (that is, reinforcement) to occur. This condition must be true simultaneously in all three dimensions. 

While Max von Lane used crystals to perform an experiment with x-rays, William H. Bragg (1862-1942) and his son William Lawrence Bragg (1890-1971) used x-rays to determine the structures of crystals. In 1912 and 1913 the Braggs developed and applied the diffraction equation that bears their name ... 

Yet, the limitation of Lane s method consists in estabhshing the multiple reflections, the reflections generated by the families of nh, nk, nl) planes, so generating the so-called Cruikshank s problem this arises by rewriting of the Bragg s law imder a mirltiple form ... 

FIGURE 5.33 Reflection power for Bragg s case for absorbent crystal after Lane (1960) James (1965). 

Max von Lane, a German physicist, was the first to suggest the use of x rays for the determination of crystal structure. Soon afterward, in 1913, the British physicists WiUiam Bragg and his son Lawrence developed the method on which modem crystal-structure determination is based. They realized that the atoms in a crystal form reflecting planes for x rays, and from this idea they derived the fundamental equation of crystal-structure determination. 

FIGURE 21.15 William Henry Bragg (1862-1942) and William Lawrence Bragg (1890-1971) were the father-and-son team that took von Lane s idea and expressed it in a simple mathematical form so it could be applied to any crystalline solid. Their Nobel Prize in physics followed the year after von Lane s. At an age of 25 years, William Lawrence Bragg is the youngest person to be named a Nobel laureate. 

It can be rightly said that The reciprocal lattice is as important in crystal structure analysis as the walking stick of a blind man moving in a narrow lane having frequent turns. It is extremely difficult if not impossible to picture the different intersecting crystal planes satisfying the Bragg s reflection in three-dimensional lattice from the two-dimensional array of spots or lines. 

The individual variants of the lattice model differ fi om each other in the way the spatial distribution of the molecules of the individual components is taken into account. The simplest solution is the Bragg-Williams (B-W) approach which assumes a random distribution of molecules within the bulk phase. The thermodynamical meaning of this assumption is that the mixture is regular. In the adsorption layer, however, it is only in two dimensions (i.e., within the individual sublayers that a statistical distribution of molecules is assumed). Pioneering work in this field was published by Ono [92-94] and Ono and Kondo [95,96]. The method was later applied to the description of L/G interfaces by Lane and Johnson [97] and later taken up by Altenberger and Stecki [98]. Analytic isotherm equations have also been derived from the above... 

Figure 5.46. Monochromatic Lane photographs (Mo Ka) of a-[(CH3)2(C2H5)2N][Ni(dmit)2]2 obtained by using a low-temperature IP system at 17 K. The arrows indicate the weak superlattice spots. The strong Bragg spots were broadened by the drastic structural phase transition at around 240 K. Debye rings from the sample folder (Cu wire) are also observable. (Reproduced with permission of The American Physical Society, from Ref 43.)...

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