Question

Germanium has a cubic unit cell with a side edge of \(565 \mathrm{pm} .\) The density of germanium is \(5.36 \mathrm{g} / \mathrm{cm}^{3}\) What is the crystal system adopted by germanium?

Short answer

The crystal system adopted by Germanium is Diamond Cubic.

Step-by-step solution

Calculate the number of atoms in a unit cell

Germanium has a diamond cubic crystal structure. Therefore, there are 8 atoms in a unit cell.

Calculate the volume of the unit cell

The edge length is given in picometres (pm), we need to convert this first to centimetres (cm). \(1 \mathrm{pm} = 1 \times 10^{-10} \mathrm{cm}\) So, the edge length is \(565 \times 10^{-10} \mathrm{cm}\). The volume (V) of the cube would be \(V = a^3 = (565 \times 10^{-10})^3 \mathrm{cm}^3\)

Calculate the Theoretical density (ρ)

The theoretical density can be calculated as \(\rho = \frac{n \times m}{V \times N_A}\) where \(n\) is the number of atoms in the unit cell, \(m\) is the molar mass of germanium, \(V\) is the volume of unit cell, \(N_A\) is Avogadro's number (\(6.022 \times 10^{23} \, \text{atoms/mol}\)). Substituting the respective values we get the theoretical density of germanium. If this matches the given density, then the crystal structure assumed (i.e., diamond cubic in this case) is correct.

Key concepts

Diamond Cubic

The diamond cubic crystal structure is a specific arrangement of atoms in a space lattice. It's known for its strength and stability, making it ideal for materials like diamond and germanium. In this structure, each atom is covalently bonded to four other atoms, forming a tetrahedral shape. This structure can also be viewed as two interpenetrating face-centered cubic (FCC) lattices that are offset by a quarter of a body diagonal.

Some key features of the diamond cubic structure include:

• It has a low density compared to other structures due to the amount of empty space.
• Atoms are arranged in layers where each layer is rotationally shifted relative to the ones above and below it, creating a systematic and repeating pattern.
• Complexity in terms of symmetry and packing leads to unique physical properties, such as high melting points and substantial electrical insulation.

Understanding this structure helps in materials science, especially when analyzing semiconductor devices and materials, like germanium.

Unit Cell

In crystallography, the unit cell is the smallest repeating unit that reflects the overall symmetry and structure of a crystal system. You can think of it as a 3D blueprint that represents the entire lattice. In the case of germanium which adopts the diamond cubic structure, the unit cell contains eight atoms.

This means that while germanium's unit cell may be viewed as a small component, it encapsulates the periodicity and geometric arrangement of atoms present in the larger crystal. Importantly, the calculation of parameters such as volume and the number of atoms in a unit cell is crucial. This helps to determine the crystal's density and helps confirm the macroscopic properties of the material.

Here's why the unit cell is significant:

• It provides a scaffold to predict how atoms will be arranged in the crystal.
• Calculating the unit cell dimensions allows material scientists to deduce important properties like density, thermal expansion coefficients, and more.
• Understanding changes at the unit cell level can link to macroscopic changes in material properties.

The simplicity or complexity of a unit cell provides insights into the characteristics and future applications of the material.

Germanium Properties

Germanium is a crucial semiconductor material with significant applications in electronics and photonics. Displaying a diamond cubic crystal structure, germanium shares several properties with silicon, yet it offers unique benefits.

Key properties of germanium include:

• Density: With a density of 5.36 g/cm³, germanium is relatively dense among semiconductors, which correlates to its atomic mass and the packing efficiency of its diamond cubic structure.
• Semi-conducting: Germanium has a bandgap of about 0.67 eV, making it a useful semiconductor, although it is less prevalent than silicon for many modern applications.
• Thermal Conductivity: While its thermal conductivity is lower than that of silicon, germanium efficiently dissipates heat, beneficial for certain electronics applications.

Moreover, germanium's ability to efficiently respond to infrared radiation allows it to be used in optical devices and infrared cameras. Understanding these properties not only assists in practical application in technology but also ensures its effective use and implementation in varied technological domains.