Question

The relation between d-spacing formula and Bragg's equation for a cubic crystal for the first-order reflection is given by the relation: \[{\mathbf{sin}}\;{\mathbf{\theta }} = \dfrac{{\mathbf{\lambda }}}{{{\mathbf{2a}}}}{({{\mathbf{h}}^{\mathbf{2}}} + {{\mathbf{k}}^{\mathbf{2}}} + {{\mathbf{l}}^{\mathbf{2}}})^{\dfrac{{\mathbf{1}}}{{\mathbf{2}}}}}\]
State whether the given statement is true or false.
A.True
B.False

Answer 1

Hint: To answer this question you should recall the concept of spacing in a crystalline solid. Use the formula of Bragg’s equation to find the d-spacing. The meaning of a cubic crystal must also be known
The formula used:
 \[{\text{d}} = \dfrac{{\text{a}}}{{\sqrt {{{\mathbf{h}}^{\mathbf{2}}} + {{\mathbf{k}}^{\mathbf{2}}} + {{\mathbf{l}}^{\mathbf{2}}}} }}\;{\text{ }}\;{\text{ }}\;....\left( i \right)\]
where \[a = \] lattice parameter and \[{\text{h,\k,\l}} = \;\] Miller indices and
\[{\text{n}}\lambda = 2{{\text{d}}_{{\text{hkl}}}}\sin \theta \;\;{\text{ }}\;{\text{ }}\;(ii)\]
where \[{{\text{d}}_{{\text{hkl}}}}\] = distance between the atomic layers, \[{\text{n}}\] = an integer, \[\lambda \] = wavelength of the incident X-ray beam and \[\sin \theta \;\] = glancing angle

Complete step by step answer:
We know that for a cubic crystal lattice the value of \[n\] in Bragg’s equation is = 1. Bragg’s law is used to determine the angles of coherent and incoherent scattering from a crystal lattice. The movement of the charged particles resulting from X rays radiates waves which are slightly blurred due to different effects and this phenomenon is known as Rayleigh scattering. This forms the basis of diffraction analysis. The phenomenon of diffraction is the result of wave interference, and this analysis is known as Bragg diffraction.
Substituting the value of equation (i) in (ii), we have:
 \[{\mathbf{sin}}\;{\mathbf{\theta }} = \dfrac{{\mathbf{\lambda }}}{{{\mathbf{2a}}}}{({{\mathbf{h}}^{\mathbf{2}}} + {{\mathbf{k}}^{\mathbf{2}}} + {{\mathbf{l}}^{\mathbf{2}}})^{\dfrac{{\mathbf{1}}}{{\mathbf{2}}}}}\].
This matches with the statement given in the question, hence, the given statement is true.

Hence the correct option is option A.

Note:
The value of d-spacing is equal to the distance between planes of atoms that give rise to different diffraction peaks in an interference pattern. In the pattern each peak results from a corresponding d-spacing. The diffractogram can be used to develop a 3D coordinate system to describe the planes of atoms with their direction. The value of d-spacing has the dimensions of length, hence usually reported in Ångstroms.