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The Usefulness of Bragg Equation in XRay Crystallography

X-ray is an electromagnetic radiation lying between ultraviolet (UV) and gamma rays in the electromagnetic spectrum. It has high penetrating power and its wavelength is between 0.01 and 100A°. Diffraction can be observed in crystals when the wavelength is within the same range as the shortest inter-atomic distances (between ~0.5 and ~2.5 A°. This is the reason why X-rays can be used to study atomic and molecular structure of crystals.

There are two broad applications of X-rays in the characterization of materials:

(i) X-ray spectrometry - used for chemical analysis, only limited use in the characterization of pharmaceuticals

(ii) X-ray diffractometry – used to study of the structure of crystalline materials, extensively used to characterize pharmaceutical solids.

There are two principal applications of X-ray diffractometry :

i) X-ray crystallography is a tool used to elucidate structure of molecules or compound, use single crystals (defined shape, close to spherical, shining, clean, no cracks) to determine structure of crystalline phases.

Crystal is the product of a unit cell and a lattice, its molecules and atoms are arranged in ordered and uniform manner.

ii) X-ray powder diffractometry, the sample is usually in the form of a powder.

X-Ray crystallography is a technique that used the fact that X-Rays are diffracted by crystals. Based on the diffraction pattern obtained from the ordered assembly of molecules or atoms in the crystal, the electron density can be reconstructed and thus the arrangement of the atoms can be determined. It plays a vital role on drug development and design. The results from X-ray crystallographic studies provide accurate and reliable 3-dimensional structural parameters and the only method for determining the “absolute” configuration of a molecule.

Diffraction of X-ray by crystal

Diffraction is a scattering phenomenon. Basically, this is the process which is happened when X-rays are diffracted by crystal :

Schematic of basic X-ray diffraction of crystal experiment

1. X rays is produced by the impact of high-velocity electrons on a target of some pure metal (eg copper and molybdenum)

2. X-rays are collimated to get a parallel beam.

3. The X-ray beam is directed on a crystal of interest.

4. The beam strike on the crystal surface at angle θ and pass through it

5. The electrons surrounding the nuclei of the atoms in the crystal will scatter the X-rays in all directions. Some waves will reinforce/combine (interfere) with one another and results in phase difference, to form diffracted beams. The phase difference (in phase or out phase) is depends on the molecular structure of the crystals.

6. Following Bragg's law, constructive interference of two in-phase waves resulting in intense peaks (Bragg’s peaks) which produce the diffraction pattern as spots/marks/dots (or reflection) on the area detector (eg film, electronic detector face) like below :

To summarize it, strong intensities peaks (Bragg peaks) are obtained in the diffraction pattern when scattered beam complies to the the Bragg's equation.

7. Each atom in the crystal serves as a center for scattering of the waves, which then form the diffraction pattern (the magnitudes and phases of the waves contributed by each atom to the interference pattern). This data can be used to elucidate the crystal structure.

8. A diffraction pattern is obtained in a diffratogram plot of the intensity of diffracted waves versus scattering angle, θ like below :

Brag, Father and Son History

X Ray diffraction in crystals like above was explained and formulated by by English physicist William Lawrence Bragg and William Henry Bragg in 1913. The duo discovered that crystalline solids produced interesting patterns of reflected X-rays, which for certain specific wavelengths and incident angles, intense peaks of reflected radiation (Bragg peaks) were produced. Thus, Bragg establishes certain connection between the diffraction angle, θ (Bragg angle), wavelength of X-rays,  and interplanar spacing (d-spacing).

W. H. Bragg & W. L. Bragg

Although simple, Bragg's law provide the basis and fundamental theory of X-Ray crystallography. Due to their highly significant impact discovery, the father and son were awarded the Nobel Prize in physics in 1915 for determining crystal structures starting with sodium chloride, zinc sulphide and diamond.

Bragg formulation of X-ray diffraction

Bragg’s Law explains the scattering (diffraction) of X-rays from crystals and show the relationship between θ, λ, and d and describes the conditions under which diffraction would occur. It is assumed that a perfectly parallel and monochromatic X-ray beam, of wavelength  is incident on a crystalline sample at an angle θ. Diffraction will occur under a certain specific  and θ only where intense peaks of reflected radiation is produced.

Diffraction will occur if :

n = 2d sin 

where

θ = diffraction angle (angle of incidence between the beam and the layer)

d = d-spacing, distance/separation of the scattering layers (between the successive planes in the crystal lattice), expressed in A °

n = number of X-ray wavelength, order of reflection (a positive integer).

When an X-ray beam (A) strikes a crystal surface at angle θ, part of the beam is scattered by the surface layer atoms (P). The unscattered part of the beam (B) penetrates to the second layer of atoms (Q) where again a fraction is scattered, and the reaminder passes on to the third layer, R and so on. The cumulative effect of this scattering from the regularly spaced centres of the crystal will produce a diffraction pattern.

From the equation, if the wavelength,  of the X-rays going in to the crystal is known, and the angle θ of the diffracted X-rays coming out of the crystal can be measured, then d-spacing between the atomic planes can be calculated (d = n/2 sin q).

In Bragg's formula, the crystal surface are bombarded with an X-ray beam at an angle θ and the beam are scattered/diffracted with the same angle. Thus for a given d-spacing and , the first order peak (n = 1) will occur at a particular θ value. Similarly, the θ values for the second (n = 2) and higher order (n > 2) peaks can be predicted.

As we know, a beam of X-rays consists of a bundle of separate waves and the waves can interact/combine with each another.  This interaction is called interference. There area 2 types of wave interference in X-rays, constructive interference and destructive interference. When Bragg’s Law is satisfied and number of X-ray wavelength is a positive integer (eg n = 1, 2, 3, 4..etc), constructive interference is happened. It is the result of synchronized light waves that interfere/add together to increase the wave amplitude.

In contrast, if the two waves are out of phase, being off by a non-integer number of wavelengths, then destructive interference will occur (it cancel each other out) and the amplitude of the waves will be reduced and resulting in darkness.

Top - the constructive interference of two in-phase waves resulting in a new wave with double the amplitude

Bottom - the destructive interference of two completely out-of-phase waves in which the resultant wave has zero amplitude, i.e., the two waves extinguish one another

Conclusion

Bragg Equation is the fundamental theory behind X-Ray crystallography. It relates the angle of incidence between the beam and the layer (θ) to the wavelength of the X-rays (λ) and the separation of the scattering layers (d)

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