Bragg

Bragg scattering Coherent elastic scattering of monochromatic neutrons by a set of crystal planes. 

The spectrum of the secondary emission, that is, the intensity of X-ray radiation as a function of wavelength is established using a crystal analyzer based on Bragg s law. 

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

The treatment of such order-disorder phenomena was initiated by Gorsky (1928) and generalized by Bragg and Williams (1934) [5], For simplicity we restrict the discussion to the synnnetrical situation where there are equal amounts of each component (x = 1/2). The lattice is divided into two superlattices a and p, like those in the figure, and a degree of order s is defined such that the mole fraction of component B on superlattice p is (1 +. s)/4 while that on superlattice a is (1 -. s)/4. Conservation conditions then yield the mole fraction of A on the two superlattices... 

Nix and Shockley [6] gave a detailed review of the status of order-disorder theory and experiment up to 1938, with emphasis on analytic improvements to the original Bragg-Williams theory, some of which will be... 

Figure A2.5.21. The heat eapaeity of an order-disorder alloy like p-brass ealeulated from various analytie treatments. Bragg-Williams (mean-field or zeroth approximation) Bethe-1 (first approximation also Guggenheim) Bethe-2 (seeond approximation) Kirkwood. Eaeh approximation makes the heat eapaeity sharper and higher, but still finite. Reprodueed from [6] Nix F C and Shoekley W 1938 Rev. Mod. Phy.s. 10 14, figure 13. Copyright (1938) by the Ameriean Physieal Soeiety.

W L Bragg [7] observed that if a crystal was composed of copies of identical unit cells, it could then be divided in many ways into slabs with parallel, plane faces whose distributions of scattering matter were identical and that if the pathlengths travelled by waves reflected from successive, parallel planes differed by integral multiples of the... 

Figure Bl.8.2. Bragg s law. Wlien X = 2d sin 0, there is strong, constructive interference. (B) THE RECIPROCAL LATTICE...

Figure Bl.8.3. Ewald s reciprocal lattice construction for the solution of the Bragg equation. If Sj-s. is a vector of the reciprocal lattice, Bragg s law is satisfied for the corresponding planes. This occurs if a reciprocal lattice point lies on the surface of a sphere with radius 1/X whose centre is at -s..

The amplitude and therefore the intensity, of the scattered radiation is detennined by extending the Fourier transfomi of equation (B 1.8.11 over the entire crystal and Bragg s law expresses die fact that this transfomi has values significantly different from zero only at the nodes of the reciprocal lattice. The amplitude varies, however, from node to node, depending on the transfomi of the contents of the unit cell. This leads to an expression for the structure amplitude, denoted by F(hld), of the fomi... 

Figure Bl.8.4. Two of the crystal structures first solved by W L Bragg. On the left is the stnicture of zincblende, ZnS. Each sulphur atom (large grey spheres) is surrounded by four zinc atoms (small black spheres) at the vertices of a regular tetrahedron, and each zinc atom is surrounded by four sulphur atoms. On the right is tire stnicture of sodium chloride. Each chlorine atom (grey spheres) is sunounded by six sodium atoms (black spheres) at the vertices of a regular octahedron, and each sodium atom is sunounded by six chlorine atoms.

Unfortimately for modem crystallographers, all of tlie crystal stmctiires that could be solved by the choose-the-best-of-a-small-niunber-of-possibilities procedure had been solved by 1920. Bragg has been quoted as saying that the pyrite stmcture was very complicated , but he wrote, in about 1930, It must be realized, however, that (cases having one or two parameters) are still extremely simple. The more typical crystal may have ten, twenty, or forty parameters, to all of which values must be assigned before the analysis of the stmcture is complete. This statement is read with amusement by a modem crystallographer, who routinely works with hundreds and frequently with thousands of parameters. 

Bragg W L 1913 The diffraction of short electromagnetic waves by a crystal Proo. Camb. Phil. Soo. 17 43-58... 

Bragg W L 1913 The structure of some crystals as indicated by their diffraction of X-rays Proo. R Soo. A 89 248-60... 

Figure B2.1.1 Femtosecond light source based on an amplified titanium-sapphire laser and an optical parametric amplifier. Symbols used P, Brewster dispersing prism X, titanium-sapphire crystal OC, output coupler B, acousto-optic pulse selector (Bragg cell) FR, Faraday rotator and polarizer assembly DG, diffraction grating BBO, p-barium borate nonlinear crystal.

Fig. 1.5 A bubble raft illustrating the nature of a dislocation. The region of misfit near Y can be seen. (After Bragg and Nye )...

Bradyrhizobium Bragg effects Bragg equation Braggite Bragg mirrors Bragg reflector Bragg s law Braids Brain... 

X-rays are collected and analy2ed in ema in one of two ways. In wds, x-rays are dispersed by Bragg diffraction at a crystal and refocused onto a detector sitting on a Rowland circle. This arrangement is similar to the production of monochromati2ed x-rays for xps described above. In the other approach, edx, x-rays are all collected at the same time in a detector whose output scales with the energy of the x-ray (and hence, Z of the material which produces the x-ray.) Detectors used for ema today are almost exclusively Li-drifted Si soHd-state detectors. 

G. Melt2 and W. W. Morey, "Bragg Grating Formation and GermanosiHcate Fiber Photosensitivity," Proceedings of the SPIE Workshop of on Photoinduced Self-Organisation in OpticalPiber May 10—11, 1991, SPIE, Quebec City, Canada, pp. 185—189. 

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