# Bragg's Equation

Thus, Path difference = WY + YZ

= XY sinθ + XY sinθ

= 2XY sinθ

= 2d sinθ

So, nλ = 2d sinθ

This equation is Known as Bragg's equation.

where, n = 1,2,3...(diffrection order)

λ = wavelength of X-rays

d = distance between planes

θ = angle at which interference occurs.

## How to Utilize Bragg's Equation to Determine Crystal Structure

The X-ray diffraction processes used for crystals are of two types-the rotating crystal process and powder process. In the latter, a powdered sample is used in place of crystal and hence simple exposure of X-rays is sufficient and so no rotation is necessary because powdered sample has crystals arranged in all possible orientations.**In rotating crystal process**, a narrow beam of X-ray strikes a crystal mounted on the turn table. The crystal is rotated so as to increase glancing angle at which the X-rays are incident at the exposed face of the crystal. The intensities of the reflected rays are measured on the recording device. The angle of the maximum reflections is θ of Bragg's equation. The process is repeated for each face of the crystal. The lowest angle at which the maximum deflection occurs corresponds to n = 1, called Ist order reflection and so on. Generally, the angle of 1st order reflection is taken as θ in order to set the n value equal to 1.

If the θ value of the Ist order reflection from three faces viz., 100, 111 & 112 of NaCl crystal be 5.9°, 8.4° and 5.2° respectively; then from Bragg's equation, we get-

d = (nλ/2)sin

^{−1}θ

As n and λ are the same in each case,

Hence, d ∝ sin

^{−1}θ

or, d ∝ 1/sinθ

Therefore, d value in three faces are in the ratio-

d

_{100}: d

_{111}: d

_{112}= (1/sin 5.9) : (1/sin 8.4) : (1/sin 5.2)

or, d

_{100}: d

_{111}: d

_{112}= (1/0.103) : (1/0.146) : (1/0.091)

or, d

_{100}: d

_{111}: d

_{112}= 1.000 : 0.704 : 1.154

The ratio is closer to that exists in FCC. Hence, NaCl lattice has FCC structure of Na

^{+}ions interlocked with similar pattern of Cl

^{−}ions.

# Bragg's Law Calculator

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