Bragg’s law states that the incidence and scattered waves in x-ray diffraction make the same angle with a lattice plane.

X-ray is a form of electromagnetic radiation, which has an alternating electric field component. When a beam of X-ray encounters a charged particle like an electron in an atom, its rapidly alternating electric field causes the electron to oscillate and become a dipole. Since an accelerated charge emits electromagnetic radiation, the accelerated electron (sinusoidal change of velocity with time) reemits X-ray at the same frequency as the incident radiation. This phenomenon is known as ** scattering**.

X-ray diffraction is a technique that relies on the interference of X-rays, which are scattered by atoms in a crystal, for analysing properties of the crystal. As the overall electron density of an atom is approximately spherical, X-ray scattered by the atom is consequently spherical (see diagram below). This is analogous to the atoms being the source of spherical wavelets, as stated in Huygens’ principle. The scattered X-rays therefore interfere with one another to form a distinct pattern.

In 1913, William Bragg and his son Lawrence Bragg considered the specular waves scattered by atoms in a crystal, i.e. ** incident and scattered waves that make the same angle with a lattice plane** (see diagram below). They proposed that scattered specular X-ray waves, from atoms separated by an interplanar distance

*d*, interfere constructively when the difference between their respective total path lengths, which is

_{hkl}*AB + BC*, is an integer multiple of the wavelength of the X-ray.

So, according to Bragg’s law:

where *n* ∈ and *n* is called the order of diffraction.

Since both the order of diffraction and the common factor between planes are integers, we can substitute eq5 where *d _{hkl }*=

*nd*in eq10:

_{nh,nk,nl}Combining eq7 where for a primitive cubic unit cell and eq11, we have:

where *n* is now the common factor between planes. This eliminates the need to describe a signal in terms of the order of diffraction. In other words, each signal now refers to the first order diffraction from the (*nh nk nl*) plane. Eq12 is often written simply as:

because it is understood that *h = nh*, *k = nk* and *l = nl*. The significance of eq12 and eq13 is that a set of lattice planes, with a specific Miller index (*hkl*) in a solid that is composed of a particular unit cell dimension, scatters X-rays that interfere constructively at a specific specular angle. Therefore, eq12 and similar equations for other unit cell types allow us to determine unit cell dimensions, if the angle of diffraction and the Miller index (*hkl*) are known. This is accomplished experimentally via X-ray diffraction techniques, one of which is powder X-ray diffraction.

###### Question

What if we consider a scattered X-ray vector in a different direction with respect to the one used in Bragg’s law, e.g. one that points below the lattice plane?

###### Answer

You get a different constructive interference (diffraction) formula, e.g. the Laue equations, i.e. you have a different mathematical approach using different criteria to describe the same diffraction pattern.