Bragg’s law was first proposed by Sir William Bragg and his son Sir Lawrence Bragg. They studied the diffraction of X-rays on various surfaces and defined the nature of x-rays diffraction on the crystal surface.

Basically, the law provides a relationship between the X-rays shooting on a crystal surface and its reflection from the surface.

**Statement of Brag’s Law:**

Suppose x-rays fall on a crystal plane with an incidence angle **θ**; then the rays are scattered from the crystal surfaces. During this process, some of the rays behave as if they were reflected from the surface, making the same angle **θ**. Hence, making the crystal plane behave like a mirror.

## Bragg’s Equation

After observing the nature of x-rays diffraction, Bragg’s equation is created.

```
nλ = 2dSinθ
```

Here,

**d**is the distance between two reflecting planes in a crystal**θ**is the incidence angle**λ**is the wavelength of x-rays**n**is an integer

Bragg’s equation gives us two important observation

- Angle of Incidence is equal to Angle of Reflection on a crystal plane.
- The path difference between two reflecting rays is equal to an integer number of the wavelength.

### Derivation of Bragg’s Equation

When monochromatic X-rays are incident upon a crystal, atoms in different layers act as a source of scattering radiation of the same wavelength, as shown in the above figure.

The intensity of the reflected beam will be maximum at a certain incident angle when the path difference between two reflected wave from two different planes is an integral multiple of the wavelength of X-rays.

That is, for maximum intensity,

```
path difference = nλ
where
```**n** is an integer: **1, 2, 3,...** and **λ** is the wavelength.

From triangles **BGE** and **BEH** in the above figure,

```
EH = GE = BEsinθ
or, EH = GE = dsinθ
```

**GE** is also the path difference between planes. Hence,

```
path difference = dsinθ
```

Combining two values of path difference, we get

```
nλ = dsinθ
```

This the Bragg’s equation of diffraction of x-rays.

## Application of Bragg’s Law

**1. Identify Crystal’s Atomic Structure**

When x-rays fall on the crystal, atoms inside the crystal diffract the radiation. Now, by observing the pattern of diffracted rays, we can identify the structure of crystals.

**2. Neutron and Electron Diffraction**

We can apply the diffraction nature of Bragg’s law to both the neutron and electron diffraction processes.

### Example: Find the spacing of atomic planes in the crystal using Bragg’s law

Suppose x-rays of wavelength **3.6 * 10 ^{-11} m** undergo first-order reflection at an angle of

**4.8°**from the crystal. Find the spacing between atomic planes.

Given,

- Wavelength (λ) = 3.6 * 10
^{-11}m - Glancing angle(θ) = 4.8°
- Order of diffraction (n) = 1

Suppose the space between atomic planes be, **d**. According to Bragg’s equation,

```
nλ = dsinθ
or, 1 * 3.6 * 10
```^{-11} = d * sin4.8
or, d * 0.08367784 = 3.6 * 10^{-11}
or, d = 3.6 * 10^{-11} / 0.08367784333
or, d = 4.3 * 10^{-10} m

Hence the spacing of the atomic planes is **4.3 * 10 ^{-10} m**.

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