# The Deriving Braggs Law Engineering Essay

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derived by the English physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1913 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence (Î˜, Î»). The variable d is the distance between atomic layers in a crystal, and the variable lambda is the wavelength of the incident X-ray beam (see applet); n is an integer.

This observation is an example of X-ray wave interference (Roentgenstrahlinterferenzen), commonly known as X-ray diffraction (XRD), and was direct evidence for the periodic atomic structure of crystals postulated for several centuries. The Braggs were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS and diamond. Although Bragg's law was used to explain the interference pattern of X-rays scattered by crystals, diffraction has been developed to study the structure of all states of matter with any beam, e.g., ions, electrons, neutrons, and protons, with a wavelength similar to the distance between the atomic or molecular structures of interest.

## How to Use this Applet

The applet shows two rays incident on two atomic layers of a crystal, e.g., atoms, ions, and molecules, separated by the distance d. The layers look like rows because the layers are projected onto two dimensions and your view is parallel to the layers. The applet begins with the scattered rays in phase and interferring constructively. Bragg's Law is satisfied and diffraction is occurring. The meter indicates how well the phases of the two rays match. The small light on the meter is green when Bragg's equation is satisfied and red when it is not satisfied.

The meter can be observed while the three variables in Bragg's are changed by clicking on the scroll-bar arrows and by typing the values in the boxes. The d and Î˜ variables can be changed by dragging on the arrows provided on the crystal layers and scattered beam, respectively.

## Deriving Bragg's Law

Bragg's Law can easily be derived by considering the conditions necessary to make the phases of the beams coincide when the incident angle equals and reflecting angle. The rays of the incident beam are always in phase and parallel up to the point at which the top beam strikes the top layer at atom z (Fig. 1). The second beam continues to the next layer where it is scattered by atom B. The second beam must travel the extra distance AB + BC if the two beams are to continue traveling adjacent and parallel. This extra distance must be an integral (n) multiple of the wavelength (Î») for the phases of the two beams to be the same:

nÎ» = AB +BC (2).

Fig. 1 Deriving Bragg's Law using the reflection geometry and applying trigonometry. The lower beam must travel the extra distance (AB + BC) to continue traveling parallel and adjacent to the top beam.

Recognizing d as the hypotenuse of the right triangle Abz, we can use trigonometry to relate d and ï± to the distance (AB + BC). The distance AB is opposite Î˜ so,

AB = d sinÎ˜(3).

Because AB = BC eq. (2) becomes,

nÎ» = 2AB (4)

Substituting eq. (3) in eq. (4) we have,

nÎ» = 2 d sinÎ˜ï€¬ï€ (1)

and Bragg's Law has been derived. The location of the surface does not change the derivation of Bragg's Law

## DIFFRACTION OF X-RAYS THROUGH CRYSTALS

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The nature of x-rays is electromagnetic i.e. they are electromagnetic waves. X-rays have very short wavelength of the order of 10 x 10 -10 m. Therefore it is not possible to produce interference fringes of x-rays by Young's double slit experiment or by thin film method. The reason is that the fringe spacing is

ï„ x = ï¬L/d and unless the slits are separated by a distance of 10 x 10 -10 m, the fringes so obtained will be closed together that they can not be observed.

How ever it is possible to obtain x-rays diffraction by making use of crystals such as rock salt in which the atoms are uniformly spaced in planes and separated by a distance of order of 2 A to 5A. Therefore, the diffraction of x-rays takes place when they incident on the surface of crystals.

## BRAGG'S EQUATION

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Consider a set of parallel lattice planes having spacing 'd' between each other as shown.

Consider two rays 'a' and 'b' incident on the surface of crystal of NaCl. After reflection, these rays reflected and are in phase. After reflection they interfere each other.

## Â

The path difference between the two reflected rays is given by:

## Â

Now the X-rays will interfere constructively if the path difference is an integral multiple of wavelength ï¬.

Thus,

## Â

This relation is known as Bragg's Law. The spacing of the atomic layers of crystals can be found from the density and atomic weight. Both 'm' and 'ï±' can be measured and hence the wave length of x-rays can be measured by using Bragg's equation.

## Â

Diffraction in crystals

1.8.1 Interference

Electromagnetic radiation displays interference and diffraction effects due to the nature of its waves. "Interference" is the property of waves to overlap each other and, under certain circumstances, to cancel out or amplify each other.

Amplification takes place when waves of identical wavelength have zero phase difference (coherence), i.e. when "wave maxima" and "wave minima" overlap in such a way that minima meet minima and maxima meet maxima. This is precisely the case when the phase difference Î”Î» is zero or a multiple of the wavelength Î», i.e.:

Î”Î» = nÎ»

n = 0, 1, 2, ...

"n" is referred to as the "order" (Fig. 12):

Fig. 12: Amplification resulting from the effects of interference

Where the phase difference is one half of the wavelength, that is where n = 1/2, 3/2, 5/2, ... , wave maxima coincide with wave minima resulting in total cancellation (Fig. 13). When a number of waves of the same wavelength propagating in the same direction interfere with each other under continuous phase shift, only the coherent among them will be amplified. In total, the rest will almost completely cancel each other out.

Fig. 13: Cancellation resulting from the effects of interference

1.8.2 Diffraction

From what we experience every day we know that light generally travels in straight lines. This corresponds with the notion of light as a beam of particles (photons, quanta). We know from ocean waves that when a wave series travels through a hole smaller than the wavelength, the waves exiting the hole spread out to the sides. Light displays the same wave characteristic. The deviation of light from its travel in a straight line is called diffraction.

There are numerous applications for the effects of diffraction. In wavelength dispersive XRF we are mainly interested in diffraction in reflection grids. Often used in the optical range (Î» = 380 - 750 nm) are mirror lattices produced by spacing grooves at equal distances in reflecting metal surfaces. This is not possible in the X-ray field because the wavelengths involved are around 2 to 5 orders of magnitude smaller (Î» = 0.02 - 11 nm). Very much smaller lattice distances, such as those found in natural crystals, are required for X-ray diffraction in the reflection grid.

Diffraction is a prerequisite for wavelength dispersive XRF. After excitation of the elements in the sample (by X-rays), a blend of element-characteristic wavelengths (fluorescence radiation) leaves the sample. There are now two methods in XRF of identifying these various wavelengths. Energy dispersive XRF calls on the assistance of an energy dispersive detector that is able to resolve the different energies. Wavelength dispersive XRF utilizes diffraction effects of crystal to separate the various wavelengths. The detector subsequently determines the intensity of a particular wavelength. The procedure will be covered in detail in the following sections.

1.8.3 X-ray Diffraction From a Crystal Lattice, Bragg's Equation

Crystals consist of a periodic arrangement of atoms or molecules that form a crystal lattice. In such an arrangement of atoms you generally find numerous planes running in different directions through the lattice points (atoms, molecules), and not only horizontally and vertically but also diagonally. These are called lattice planes. All of the planes parallel to a lattice plane are also lattice planes and are a set distance apart from each other. This distance is called the lattice plane distance "d."

When parallel X-rays strike a pair of parallel lattice planes, every atom within the planes acts as a scattering centre and emits a secondary wave. All of the secondary waves combine to form a reflected wave. The same occurs on the parallel lattice planes for only very little of the X-ray wave is absorbed within the lattice plane distance, d. All these reflected waves interfere with each other. If the amplification condition "phase difference = a whole multiple of wavelengths" (Î”Î» = nÎ») is not precisely met, the reflected wave will interfere such that cancellation occurs. All that remains is the wavelength for which the amplification condition is precisely met. For a defined wavelength and a defined lattice plane distance, this is only given with a specific angle, the Bragg angle (Fig. 14).

Fig. 14 Bragg's Law

Under amplification conditions, parallel, coherent X-ray light (rays 1, 2) falls on a crystal with a lattice plane distanced d and is scattered below the angle Î¸ (rays 1', 2'). The proportion of the beam that is scattered on the second plane has a phase difference of 'ACB' to the proportion of the beam that was scattered at the first plane. Following the definition of sine:

'AC'

Â =Â sinÎ¸

or

'AC'Â =Â dÂ sinÎ¸

d

The phase difference 'ACB' is twice that, so:

'ACB'Â =Â 2dÂ sinÎ¸

The amplification condition is fulfilled when the phase difference is a whole multiple of the wavelength Î», so:

'ACB'Â =Â nÎ»

This results in Bragg's Law:

nÎ»Â =Â 2dÂ sinÎ¸

## Bragg's equation

n = 1, 2, 3, ...

reflection order

Fig. 15a: 1st order reflection: Î» = 2d sin Î¸1

Fig. 15b: 2nd order reflection: 2Î» = 2d sin Î¸2

Fig. 15c: 3rd order reflection: 3Î» = 2d sin Î¸3

Figures 15a, b and c illustrate Bragg's Law for the reflection orders n = 1, 2, 3.

On the basis of Bragg's Law, by measuring the angle Î¸, you can determine either the wavelength Î», and thus chemical elements, if the lattice plane distance d is known or, if the wavelength Î» is known, the lattice plane distance d and thus the crystalline structure.

This provides the basis for two measuring techniques for the quantitative and qualitative determination of chemical elements and crystalline structures, depending on whether the wavelength Î» or the 2d-value is identified by measuring the angle Î¸ (Table 3):

Table 3: Wavelength dispersive X-ray techniques

Table 3: Wavelength dispersive X-ray techniques

## Instrument type

Î»

d

Î¸

X-ray flourescence

Spectrometer

d

Î»

Î¸

X-ray diffraction

Diffractometer

In X-ray diffraction (XRD) the sample is excited with monochromatic radiation of a known wavelength (Î») in order to evaluate the lattice plane distances as per Bragg's equation.