Question

X rays from a copper X-ray tube \((\lambda=154 \mathrm{pm})\) were diffracted at an angle of 14.22 degrees by a crystal of silicon. Assuming first-order diffraction \((n=1\) in the Bragg equation), what is the interplanar spacing in silicon?

Short answer

The interplanar spacing in the silicon crystal is approximately \(3.14 \times 10^{-10}\) meters, or \(314\ \text{pm}\).

Step-by-step solution

Analyzing the given data

The given data are: Wavelength of X-rays (\(\lambda\)) = \(154 \mathrm{pm}\) (1 picometer = \(10^{-12}\) meters) Angle of diffraction (\(\theta\)) = 14.22 degrees Order of diffraction (\(n\)) = 1

Convert angle to radians

To calculate the sine function, we need to have the angle (\(\theta\)) in radians. We can convert the degrees to radians using the following formula: Radians = Degrees × \(\frac{\pi}{180}\) So, \(\theta\) in radians = 14.22 × \(\frac{\pi}{180}\)

Apply Bragg's law

Now, we can apply Bragg's law to solve for the interplanar spacing \(d\): Bragg's law: \(n\lambda=2d\sin{(\theta)}\) Rearrange the formula to solve for \(d\): \(d=\frac{n\lambda}{2\sin{(\theta)}}\) Substitute the given values: \(d=\frac{1 \times 154 \times 10^{-12}}{2 \times \sin(14.22 \times \frac{\pi}{180})}\)

Calculate the interplanar spacing

Calculate the value of \(d\): \(d=\frac{154 \times 10^{-12}}{2 \times \sin(14.22 \times \frac{\pi}{180})} \approx 3.14 \times 10^{-10} \mathrm{m}\) The interplanar spacing in the silicon crystal is approximately \(3.14 \times 10^{-10}\) meters, or \(314\ \text{pm)\).

Key concepts

Bragg equation

The Bragg equation, which is fundamental to the field of X-ray crystallography, is elegantly simple yet powerful in its ability to reveal the underlying structure of crystalline materials. It is expressed as \(n\lambda = 2d\sin(\theta)\), where \(n\) represents the order of diffraction, \(\lambda\) the wavelength of the incident X-ray beam, \(d\) the interplanar spacing between the crystal planes, and \(\theta\) the angle of incidence that produces the observed diffraction pattern.

This equation establishes a direct relation between the angle at which X-rays are diffracted by a particular set of lattice planes in a crystal and the distance between those planes. By measuring the angles and wavelengths of the diffracted beams, scientists can deduce the distances between the planes, thereby gaining insight into the crystal's structure. One must be aware, however, that the equation assumes an ideal crystal with no defects and does not account for more complex phenomena such as multiple scattering or absorption.

X-ray diffraction

X-ray diffraction (XRD) is a powerful analytical technique often used to identify the crystallographic structure of materials. When X-rays strike a crystalline substance, they are scattered in many directions. If these scattered waves interfere constructively, they create a diffraction pattern which is unique to the structure of the crystal.

In a standard XRD experiment, a monochromatic beam of X-rays irradiates a powdered or solid crystalline sample. The crystalline structure causes the beam to diffract in specific directions. By measuring the angles and intensities of these diffracted beams—and using the Bragg equation—scientists can produce a three-dimensional picture of the electron density within the crystal. This allows for the determination of not only the size and shape of the unit cell of the crystal lattice but also the positions of atoms within the cell.

XRD has a wealth of applications including the characterization of crystalline materials, determination of phase purity, and the study of crystal defects.

Crystallography

Crystallography is a scientific discipline that studies the arrangement of atoms in crystalline solids. The atomic structure of a crystal can be thought of as a three-dimensional pattern where the atoms are positioned in repetitive arrangements. This structure is reflected in the crystal's physical properties and symmetry.

Using the principles of crystallography, scientists are able to determine the positions of atoms in a crystal, the chemical bonds, the disorder and the various other macroscopic properties. Techniques such as X-ray diffraction are central to this field, allowing researchers to delve into the microscopic world to retrieve information about the macroscopic world.

Modern crystallography includes the study of more than just crystals. It encompasses the study of the solid state and the characterization of materials, which may not exhibit perfect crystal structures. Nonetheless, the fundamental principles derived from Bragg's Law and the detailed understanding of interatomic spacings remain at the heart of the discipline.