In the X-ray diffraction pattern recorded for a simple cubic solid (lattice parameter a = 1 Å) using X rays of wavelength 1 Å, the first order diffraction peak(s) would appear for thea)(100) planesb)(112) planesc)(210) planesd)(220) planesCorrect answer is option 'A'. Can you explain this answer?

Question

In the X-ray diffraction pattern recorded for a simple cubic solid (lattice parameter a = 1 &Aring;) using X rays of wavelength 1 &Aring;, the first order diffraction peak(s) would appear for thea)(100) planesb)(112) planesc)(210) planesd)(220) planesCorrect answer is option 'A'. Can you explain this answer?

Answer

To calculate the positions of the diffraction peaks for a simple cubic lattice, we use the Bragg equation:

nλ = 2dsinθ

where n is the order of the diffraction peak, λ is the wavelength of the incident X-rays, d is the interplanar spacing, and θ is the angle between the incident X-rays and the planes of the crystal lattice. For a simple cubic lattice, the interplanar spacing is given by:

d = a/√(h^2 + k^2 + l^2)

where a is the lattice parameter, and h, k, and l are the Miller indices that describe the orientation of the planes.

For the (100) plane, h = 1, k = 0, and l = 0, so

d = a/√(1^2 + 0^2 + 0^2) = a

For the (110) plane, h = 1, k = 1, and l = 0, so

d = a/√(1^2 + 1^2 + 0^2) = a/√2

For the (111) plane, h = 1, k = 1, and l = 1, so

d = a/√(1^2 + 1^2 + 1^2) = a/√3

We can use these values of d to calculate the positions of the diffraction peaks for different orders of diffraction (n). For example, for n = 1, we have:

λ = 2d sinθ

For the (100) plane, θ = 0, so sinθ = 0, and the first diffraction peak occurs at λ = 2a.

For the (110) plane, sinθ = 1/√2, so the first diffraction peak occurs at λ = 2a/√2.

For the (111) plane, sinθ = 1/√3, so the first diffraction peak occurs at λ = 2a/√3.

We can continue this pattern for higher orders of diffraction (n = 2, 3, etc.), and we will find that the positions of the diffraction peaks follow a series of equidistant lines in the X-ray diffraction pattern. The spacing between these lines is proportional to the lattice parameter (a), so we can use the positions of the diffraction peaks to determine the value of a for the simple cubic lattice.

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