Answer:

Introduction:

When X-rays are passed through a crystal, they diffract. The diffraction pattern provides information about the atomic arrangement in the crystal. The angle of diffraction and wavelength of X-ray can be used to calculate the length of the unit cell.

Given:

Wavelength of X-ray, λ = 154 ×10^-12m
Angle of diffraction, θ = 11.2°
Plane of diffraction, (111) and (123)

Calculation of length of unit cell:

The Bragg's law can be used to calculate the length of the unit cell.

Bragg's law is given by:

nλ = 2dsinθ

Where,
n = order of reflection (n = 1 for first order reflection)
d = interplanar spacing
θ = angle of diffraction
λ = wavelength of X-ray

For (111) plane,

d111 = a/√(h^2+k^2+l^2)

Where,
a = length of side of the unit cell
h, k, l = Miller indices of the plane

Putting the values in Bragg's law,

1(154 ×10^-12m) = 2(a/√3)sin11.2°

a = (nλ/2sinθ)√(h^2+k^2+l^2) .....(1)

a = (1 × 154 ×10^-12m)/(2sin11.2°/√3)

a = 0.404 nm

Therefore, the length of the side of the unit cell is 0.404 nm.

Calculation of angle of reflection:

For (123) plane,

d123 = a/√(h^2+k^2+l^2)

Putting the values of h, k, l in the above equation, we get:

d123 = a/√(1^2+2^2+3^2) = a/√14

Using Bragg's law,

nλ = 2dsinθ

For first order reflection, n = 1

1λ = 2dsinθ

sinθ = λ/2d

sinθ = λ/(2a/√14)

θ = sin^-1(λ/(2a/√14))

Putting the values in the above equation, we get:

θ = sin^-1[(154 ×10^-12m)/(2(0.404 ×10^-9m)/√14)]

θ = 23.4°

Therefore, the angle of reflection from (123) plane is 23.4°.

Explanation:

When X-rays are passed through a crystal, they diffract. The Bragg's law can be used to calculate the length of the unit cell. For (111) plane, the interplanar spacing (d111) is given by a/√(h^2+k^2+l^2), where a is the length of side of the unit cell and h, k, l are Miller indices of the plane. Putting the values in Bragg's law, the length of the side of the unit cell is calculated to be 0.404 nm.

For (123) plane, the interplanar spacing