Bragg’s Law

In 1913, Sir William Bragg and his son Sir Lawrence Bragg worked on a mathematical relation to determine inter-atomic distances from x-rays diffraction pattern. Bragg’s law states that

“The intensity of reflected beam from the crystal lattice at certain angle will be maximum if the path difference between the two reflected waves from two different planes is an integral multiple of the wavelength of incident X-ray.”

If path difference is $d$ and $\theta$ is the angle of diffraction, then $$2d\sin \theta =n\lambda$$

This relation is known as Bragg’s law.

Derivation of Bragg’s Law

Consider parallel monochromatic X-rays $AB$ and $PQ$ having wavelength $\lambda$ are incident on the atomic planes of a crystal making a glancing angle $\theta$. Let the distance between two successive planes be $d$. The ray $AB$ is reflected by atom $B$ of the first atomic plane along $BC$ while the ray $PQ$ is reflected by atom $Q$ of the second atomic plane along $QR$.

$BM$ and $BN$ are the perpendiculars drawn on the ray $PQ$ and $QR$ respectively. Then, the path difference between the rays $ABC$ and $PQR$ is $MQ+QN$.

According to the theory of interference, the reflected x-rays $BC$ and $QR$ will be intense if the path difference $(MQ+QN)$ is an integral multiple of wavelength $(\lambda )$ i.e.

$$MQ+QN=n\lambda \text{ \_\_\_\_(1)}$$ $$\text{Where, }n=1,2,3,\dots$$

In right angled triangle $BMQ$,

$$\sin \theta =\frac{MQ}{BQ}$$ $$MQ=BQ\sin \theta =d\sin \theta \text{ \_\_\_\_(2)}$$

In right angled triangle $BNQ$,

$$\sin \theta =\frac{NQ}{BQ}$$$$NQ=BQ\sin \theta =d\sin \theta \text{ \_\_\_\_(3)}$$

[Trigonometric Functions and their Relations]

From equations $(1)$, $(2)$ and $(3)$,

$$d\sin \theta +d\sin \theta =n\lambda$$ $$2d\sin \theta =n\lambda \text{ \_\_\_\_(4)}$$

This is known as Bragg’s law.

The value of $n$ is the order of reflection. For first order of reflection, $n=1$, for second order of reflection, n=2 and so on.

Lattice Planes or Bragg’s Planes

Bragg thought that atoms inside the crystal are arranged over different sets of parallel planes separated by a fixed distance. These atomic planes are called lattice planes or Bragg’s planes.

By measuring the value of glancing angle (angle of diffraction) for a known value of wavelength of a monochromatic x-ray and for a particular order of reflection, the spacing between the lattice planes can be calculated.

$$\text{Since, }\sin \theta \le 1$$ $$\text{So, }n\lambda \le 2d$$

Due to this, Bragg’s law is applicable because the inter-molecular spacing is of the order of few angstroms and wavelength of x-rays is also of same order. It is not valid for visible light because visible light ranges from $3800$ to $7800$ $\AA$.

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