In their book \(X\) -rays and crystal structures (which begins 'It is now two years since Dr. Laue conceived the idea \(\ldots\) ) the Braggs give a number of simple examples of X-ray analysis. For instance, they report that the reflection from (100) planes in KCl occurs at \(5^{\circ} 23\) ', but for \(\mathrm{NaCl}\) it occurs at \(6^{\circ} 0^{\prime}\) for X-rays of the same wavelength. If the side of the \(\mathrm{NaCl}\) unit cell is \(564 \mathrm{pm}\), what is the side of the \(\mathrm{KCl}\) unit cell? The densities of \(\mathrm{KCl}\) and \(\mathrm{NaCl}\) are \(1.99 \mathrm{~g} \mathrm{~cm}^{-3}\) and \(2.17 \mathrm{~g} \mathrm{~cm}^{-3}\) respectively. Do these values support the \(\mathrm{X}\) ray analysis?

Bragg's Law is a fundamental principle in X-ray crystallography that explains how X-rays scatter off of crystal planes. This law states that for X-rays to constructively interfere, the path difference between rays scattered from adjacent planes must be an integer multiple of the wavelength. The equation is given by:
\[n\lambda = 2d\sin(\theta)\]
where \(n\) is the order of reflection which is typically 1 in many practical cases, \(\lambda\) is the wavelength of the X-ray, \(d\) is the distance between the crystal planes in the crystal lattice, and \(\theta\) is the angle of incidence – the angle at which the X-ray hits the crystal plane. When the conditions outlined by Bragg's Law are met, the reflected X-rays from the crystal planes can be detected, and information about the crystal structure can be inferred.

Crystal planes are the imaginary planes that pass through the lattice points of a crystal structure. These planes have a particular orientation and spacing in a crystal, represented by Miller indices in the form (hkl). In X-ray crystallography, X-rays interact with these planes to produce diffraction patterns. By studying these patterns, scientists can determine the positions of atoms within the material and the distance between the planes, referred to as interplanar spacing \(d\). This spacing is crucial for applying Bragg's Law as it directly correlates with the diffraction angle \(\theta\) and the order of reflection \(n\). By analyzing the diffraction pattern, researchers can elucidate the three-dimensional arrangement of atoms within the crystal.

The unit cell is the smallest repeating unit in the crystal lattice that can demonstrate the entire structure of the crystal when repeated in all three spatial dimensions. The unit cell is characterized by its lattice parameters: lengths of the cell edges (\(a\text{, }b\text{, }c\)) and the angles between them (\(\alpha\text{, }\beta\text{, }\gamma\)). Crystals can be classified into different systems, such as cubic, tetragonal, orthorhombic, etc., based on the symmetry properties of their unit cells. For cubic crystals like NaCl and KCl, the unit cell is a cube so the edge lengths are equal (\(a = b = c\)) and the angles are all 90 degrees (\(\alpha = \beta = \gamma = 90^\circ\)). In X-ray crystallography, the determination of the unit cell dimensions is a critical step towards understanding the overall crystal structure.

The angle of incidence in X-ray crystallography refers to the angle at which the incoming X-ray beam meets the crystal plane. It is denoted as \(\theta\) and is a vital part of Bragg's Law. In the context of the exercise provided, the angle of incidence is the specific angle given for the X-rays reflecting from the crystal planes of KCl and NaCl. Adjusting the angle of incidence and monitoring the angular position where maximum reflection (a peak in intensity) occurs allows researchers to measure the interplanar spacing \(d\) and indirectly infer the size of the unit cell. For accurate results, it is critical to measure this angle precisely in experiments.

Density calculation in the context of crystallography involves determining the theoretical density of a crystal from its unit cell dimensions and molecular formula. The formula for density \(\rho\) of a crystalline substance is given by:
\[\rho = \frac{Z\cdot M}{N_A\cdot a^3}\]
where \(Z\) is the number of formula units in the unit cell, \(M\) is the molar mass of the substance, \(N_A\) is Avogadro's number, and \(a\) is the length of the unit cell's edge. After calculating the theoretical density, it can be compared to empirical density values to validate crystallographic data. In the case of NaCl and KCl, comparisons between calculated and observed densities will support or challenge the validity of the X-ray crystallographic analysis performed to determine the unit cell dimensions.