For which set of crystallographic planes will a first-order diffraction peak occur at a diffraction angle of \(46.21^{\circ}\) for BCC iron when monochromatic radiation having a wavelength of \(0.0711 \mathrm{~nm}\) is used?

Bragg's Law relates the angle of diffraction (\(\theta\)), the wavelength of the radiation (\(\lambda\)), and the interplanar spacing (\(d_{hkl}\)) for a crystal. It is given by the equation: $$2d_{hkl} \sin{\theta} = n\lambda$$ Where \(n\) is the order of diffraction.

For a BCC lattice, the interplanar spacing formula is given by: $$\frac{a}{d_{hkl}} = \sqrt{h^2 + k^2 + l^2}$$ Where \(a\) is the lattice parameter, and (h, k, l) are the Miller indices of a set of crystallographic planes. As given, for BCC iron, the lattice parameter is \(a = 0.2866 \mathrm{~nm}\).

We can rearrange Bragg's Law to obtain a formula for \(d_{hkl}\) in terms of the given angle and wavelength. Since the problem asks for the first-order diffraction (\(n = 1\)), we can simplify Bragg's Law as follows: $$d_{hkl} = \frac{\lambda}{2\sin{\theta}}$$ Substituting the given values, we get: $$d_{hkl} = \frac{0.0711 \mathrm{~nm}}{2\sin{46.21^{\circ}}} = 0.1048 \mathrm{~nm}$$

Now, we can substitute this value for \(d_{hkl}\) into the interplanar spacing equation obtained in step 2: $$\sqrt{h^2 + k^2 + l^2} = \frac{a}{d_{hkl}} = \frac{0.2866 \mathrm{~nm}}{0.1048 \mathrm{~nm}} = 2.7316$$ Squaring both sides, we get: $$h^2 + k^2 + l^2 = 7.4611$$

We need to find the set of integer values (h, k, l) that satisfy the equation \(h^2 + k^2 + l^2 = 7.4611\). Since the expression in step 4 does not exactly equal an integer value, we can round up to the nearest integer value as an approximation. Therefore, we can try to find a set of indices (h, k, l) that satisfy: $$h^2 + k^2 + l^2 = 8$$ There are many possible sets of Miller indices that could satisfy this equation, but since we are dealing with a BCC lattice, we need to ensure that the indices are either all odd or all even. One such set that satisfies this condition is (1, 1, 1). To confirm this, let's plug the values for h, k, and l into our equation: $$1^2 + 1^2 + 1^2 = 3$$ In this case, the equation does not hold. The other possible set of indices that satisfies the condition of having all even numbers is (2, 0, 0): $$2^2 + 0^2 + 0^2 = 4$$ As the sum is closer to 8, we can conclude that the set of crystallographic planes for which a first-order diffraction peak will occur for BCC iron at a diffraction angle of \(46.21^{\circ}\) using monochromatic radiation with a wavelength of \(0.0711 \mathrm{~nm}\) is (2, 0, 0).