Themes > Science > Physics > Solid State Physics > Atomic Bonding and Crystal Structure > X-Ray Diffraction > X-ray Diffraction and Braggs Law

Since a beam of X-rays consists of a bundle of separate waves, the waves can interact with one another.  Such interaction is termed interference.  If all the waves in the bundle are in phase, that is their crests and troughs occur at exactly the same position (the same as being an integer number of wavelengths out of phase, nl, n = 1, 2, 3, 4, etc.), the waves will interfere with one another and their amplitudes will add together to produce a resultant wave that is has a higher amplitude (the sum of all the waves that are in phase. 


If the waves are out of phase, being off by a non-integer number of wavelengths, then destructive interference will occur and the amplitude of the waves will be reduced.  In an extreme case, if the waves are out of phase by a multiple of 1/2l (n/2l ), the resultant wave will have no amplitude and thus be completely destroyed.

The atoms in crystals interact with X-ray waves in such a way as to produce interference.  The interaction can be thought of as if the atoms in a crystal structure reflect the waves.  But, because a crystal structure consists of an orderly arrangement of atoms, the reflections occur from what appears to be planes of atoms.  Let's imagine a beam of X-rays entering a crystal with one of these planes of atoms oriented at an angle of q to the incoming beam of monochromatic X-rays (monochromatic means one color, or in this case 1 discreet wavelength as produced by the characteristic sprectra of the X-ray tube).

Two such X-rays are shown here, where the spacing between the atomic planes occurs over the distance, d.  Ray 1 reflects off of the upper atomic plane at an angle q equal to its angle of incidence.  Similarly, Ray 2 reflects off the lower atomic plane at the same angle q.  While Ray 2 is in the crystal, however, it travels a distance of 2a farther than Ray 1.  If this distance  2a is equal to an integral number of wavelengths (nl), then Rays  1 and 2 will be in phase on their exit from the crystal and constructive interference will occur.
If the distance  2a is not an integral number of wavelengths, then destructive interference will occur and the waves will not be as strong as when they entered the crystal.  Thus, the condition for constructive interference to occur is

nl = 2a

but, from trigonometry, we can figure out what the distance 2a is in terms of the spacing, d, between the atomic planes.

a = d sin q

or 2a = 2 d sin q

thus, nl = 2d sin q

This is known as Bragg's Law for X-ray diffraction.

What it says is that if we know the wavelength ,l , of the X-rays going in to the crystal, and we can measure the angle q of the diffracted X-rays coming out of the crystal, then we know the spacing (referred to as d-spacing) between the atomic planes.  

d = nl /2 sin q

Again it is important to point out that this diffraction will only occur if the rays are in phase when they emerge, and this will only occur at the appropriate value of n (1, 2, 3, etc.) and q.

In theory, then we could re-orient the crystal so that another atomic plane is exposed and measure the d-spacing between all atomic planes in the crystal, eventually leading us to determine the crystal structure and the size of the unit cell.


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