COSTO BENEFICIO DE LOS RECURSOS PROPUESTOS.

Bragg’s Law has formed an integral part of the analysis of collagenous matrix materials presented in this thesis. The origins of the law are presented below to gain more of an understanding of the concepts involved and how it came about. X-ray diffraction of a crystalline solid was found to produce surprising patterns of reflected X-rays by William Lawrence Bragg and his father William Henry Bragg in 1913 (Thomas, 1990). Bragg explained this pattern by modelling the crystal as a set of parallel planes separated by a constant parameter d. Intense peaks, otherwise known as Bragg peaks, occur when scattered waves interfere constructively. Bragg’s law describes the condition for constructive interference from successive planes of the crystalline lattice and is illustrated in Figure 2.22 where n is an integer, λ is the wavelength of the incident wave, d is the spacing between the planes in the atomic lattice, and θ is the angle between the incident ray and the scattering planes.

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This illustration (Figure 2.22) showcases a crystalline solid approached by two X- ray beams with identical wavelengths. The beams are scattered off two different atoms with the lower beam traveling an extra distance of 2dsinθ. Constructive interference occurs when this extra length is equal to an integer (n) multiple of the wavelength of the incident X-ray (λ).

The roots of Bragg’s law really go back to 1912 when Laue and a small group of scientists made the ground-breaking discovery of X-ray diffraction by crystals. Walter Friedich, Paul Knipping, and Max von Laue submitted a one page report to the Bavarian Academy of Science stating they were “engaged since April 21, 1912 with experiments about the interference of X-rays passing through crystals” (Eckert, 2012). The report laid their claims on the discovery prior to it being formally communicated in a published paper and was backed up by a sketch of the experiment apparatus (Figure 2.23) and some exposures that were sent in with the report.

Figure 2.23. Friedich and Knipping’s improved experimental set-up (Ewald, 1962). Reprinted from Fifty Years of X-ray Diffraction, P. P. Ewald, Laue’s Discovery of X-ray Diffraction by Crystals, pg 41, copyright 1962, with permission from Springer.

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Figure 2.24. Zincblende Laue photographs along (a) four-fold and (b) three-fold axes (Ewald, 1962). Reprinted from Fifty Years of X-ray Diffraction, P. P. Ewald, Laue’s Discovery of X-ray Diffraction by Crystals, pg 42, copyright 1962, with permission from Springer.

It was Laue who started to develop a theory based on the assumption that X-rays were electromagnetic radiation. Laue surmised it might be possible for a crystal irradiated with X-rays to give off diffraction effects. On 8th _{June 1912 at a meeting}

of the Bavarian Academy, Laue presented an introduction on the theory of diffraction by a three-dimensional lattice (Thomas, 1990). The exciting and truly

(b)

(a)

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ground-breaking discovery quickly gained Max von Laue a Nobel Prize in 1914 for “his discovery of the diffraction of X-rays by crystals” (Nobelprize.org, 2014). Unfortunately, Laue went wrong when he considered the experimental results from ZnS (Figure 2.24). He fixated on an erroneous notion that the observed effects were associated with X-rays arising in the crystal. Laue also assumed the crystal had a basic cubic structure with one molecule per unit cell. However, despite the flawed assumptions behind his theory, Laue established the wave like nature of X-rays through his explanation of diffraction as the phenomenon causing the observed spots for ZnS.

Sir Lawrence Bragg and his father William Henry Bragg were very much interested in the nature of X-rays and Laue’s paper intrigued them. Bragg was able to piece together a collection of several different pieces of knowledge from lectures he had attended that culminated in Bragg’s law and provided the answer to Laue’s ZnS spots.

Figure 2.25. Change of shape of X-ray reflections as the photographic plate was moved further away from the crystal (Thomas, 1990).

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X-ray reflections were round when the plate was close to the crystal. These reflections that were round became elliptical in the horizontal direction when the photographic plate was moved further away from the crystal (Figure 2.25). Bragg pointed out that an incident cone of X-rays of continuously varying wavelengths that are reflected by lattice planes would come to focus in the vertical direction but would spread out in the horizontal direction (Thomas, 1990).

The idea that sheets of atoms in the crystals reflected formless X-ray pulses came to Bragg. If regarded in such a way then, similar to Wilson’s treatment of a diffraction grating (using a diffraction grating white light could be a succession of formless pulses which the lines of a diffraction grating convert into a train of waves (Thomas, 1990)), a wave train would be formed from the pulses reflected from successive equidistant sheets. Since the path difference between the waves of the reflected train is 2d sin θ where θ is the glancing angle at which the radiation falls on the planes and d is their spacing, it followed immediately that the wavelengths (λ) of the different orders of reflection would be given by nλ=2dsinθ

where n is an integer. Bragg then drew upon the idea that lattices could be both simple cubic or face-centred lattices from a lecture by Gosling on crystal structures. Laue had previously tried to describe the observations of ZnS based on a simple cubic structure. Bragg endeavoured to explain the ZnS diffraction photographs assuming the structure is a face-centred cubic lattice. Everything fell into place. Bragg showed that the Laue pictures were made by a continuous range of X-ray wavelengths, a kind of white radiation, and that X-ray diffraction could be used to get information about the crystal structure. This was the start of X-ray analysis of crystals. Sir Lawrence Bragg and his father were awarded a Nobel Prize in Physics in 1915 for “their services in the analysis of crystal structure by means of X-rays” (Nobelprize.org, 2014).

Bragg’s Law made it possible for crystal structures to be explored and defined with collagen being a prime example of this. Collagen fibrils showcase axial periodicity. This d banding pattern includes a gap region and an overlap region in every d spacing period. SAXS patterns of collagen produce Bragg peaks due to this highly repetitive structure. As such it is possible to determine the d spacing of the collagen fibrils from the location of the Bragg peaks.

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