Problem 51 - Chemistry and Chemical Reactivity

Question

The solid-state structure of silicon is shown below. (a) Describe this crystal as \(\mathrm{pc}, \mathrm{bcc},\) or fcc. (b) What type of holes are occupied in the lattice? (c) How many Si atoms are there per unit cell? (d) Calculate the density of silicon in \(\mathrm{g} / \mathrm{cm}^{3}\) (given that the cube edge has a length of \(543.1 \mathrm{pm}\) ). (e) Estimate the radius of the silicon atom. (Note: The Si atoms on the edges do not touch one another.

Step-by-Step Solution

Answer

(a) fcc; (b) None; (c) 8 Si atoms; (d) Density ≈ 2.33 g/cm³; (e) Radius ≈ 117.6 pm.

Step 1: Determine the Crystal Structure

The solid-state silicon crystal lattice is of the diamond cubic structure, which is a variant of the face-centered cubic (fcc) crystal. Therefore, the crystal can be described as fcc.

Step 2: Identify the Type of Holes Occupied

In the diamond cubic structure of silicon, each silicon atom is covalently bonded to four other silicon atoms. This means no interstitial holes like octahedral or tetrahedral are occupied, unlike in ionic crystals.

Step 3: Count Silicon Atoms per Unit Cell

The diamond cubic structure includes atoms at all the corners of the cube and additional atoms on each face center and in the cube's interior. Each corner is shared among eight unit cells, and each face center is shared with two, while the atoms in the interior belong wholly to one unit cell. Specifically, this results in 8 corner atoms contributing 1/8 each, 6 face-centered atoms contributing 1/2 each, and 4 atoms wholly contained within the cell, totaling 8 Si atoms in each unit cell.

Step 4: Calculate the Density of Silicon

To find the density, use the formula \( \text{Density} = \frac{\text{Mass of Unit Cell}}{\text{Volume of Unit Cell}} \). The mass of the unit cell can be calculated by multiplying the number of atoms per unit cell with the atomic mass of silicon (28.0855 u) and converting it into grams. Volume is derived from the cube edge, which is \( (543.1 \times 10^{-10} \, \text{cm})^3 \). Calculating this gives the density.

Step 5: Estimate the Radius of Silicon Atoms

In a covalently bonded diamond cubic structure, the atoms do not touch across the face diagonal but along the diagonal between opposite corners. The diagonal length equals \( 4r \), leading to \( r = \frac{d}{4} \), where \( d \) is the diagonal of the cube. Calculate \( d \) using the Pythagorean theorem in three dimensions \( d = \sqrt{3} \cdot a \), and solve for \( r \).

Key Concepts

crystal structure types

In solid-state physics, understanding crystal structure types is crucial as it informs us about the arrangement of atoms in a material. A few common types are simple cubic (pc), body-centered cubic (bcc), and face-centered cubic (fcc). Each type has a distinct pattern in which atoms are spaced. Silicon, the element in focus here, crystallizes in a diamond cubic structure, which is a variant of the fcc lattice. This means that in silicon's crystal structure, atoms are arranged in a way that each atom is covalently bonded to four others, creating a three-dimensional grid with additional atoms at certain key positions, adding complexity. The diamond cubic structure is unique because, unlike ionic compounds, it doesn't involve hole occupancy like tetrahedral or octahedral voids. This makes silicon's crystal lattice significant, especially in semiconductor technology.

unit cell calculations

The unit cell is the smallest repeating structure in a crystal lattice and is vital for understanding the material's overall structure and properties. In the case of silicon with a diamond cubic structure, the unit cell involves atoms at its corners, along its face centers, and within its interior. Calculating the number of atoms in such a unit cell is crucial:

• Each corner atom is shared by eight neighboring unit cells, contributing \(1/8\) of an atom per unit cell.
• Face-centered atoms are shared with two unit cells, each contributing \(1/2\) of an atom per unit cell.
• Additional atoms within the unit cell are fully counted as 1 whole atom each.

For silicon's diamond cubic structure, this results in 8 corner atoms \( imes 1/8 + \) 6 face-centered atoms \( imes 1/2 + \) 4 wholly internal atoms, equating to a total of 8 silicon atoms per unit cell. Understanding this assists in deeper calculations of material properties like density.

density of materials

Density is a fundamental property that measures how much mass is packed into a given volume, providing insights into material characteristics and applications. For silicon, determining density involves combining the number of silicon atoms per unit cell with their atomic mass. First, you calculate the total mass of the unit cell by multiplying the 8 atoms by the atomic mass of silicon (28.0855 u), converting these atomic mass units into grams. Next, find the volume of the unit cell using the given cube edge length, which for silicon is \( 543.1 \, \text{pm} \) or \( 543.1 imes 10^{-10} \, \text{cm} \). The cube's volume is then \( (543.1 imes 10^{-10})^3 \. \) Finally, applying the formula \( \text{Density} = \frac{\text{Mass of Unit Cell}}{\text{Volume of Unit Cell}} \, \) you get the density of silicon. This property is essential for understanding how silicon will perform in practical applications, such as in microelectronics.

atomic radius estimation

Estimating the atomic radius in a crystal structure like silicon's involves understanding how atoms are geometrically arranged. In the diamond cubic crystal, atoms align not directly across the cube face but along the diagonals defined by opposite corners. This means that the length of the cube's diagonal equates to four times the atomic radius, \(4r\). To calculate this diagonal, use the Pythagorean theorem adjusted for three dimensions, \(d = \sqrt{3} \cdot a\), where \(a\) is the length of one cube edge. Solving for \(r\), the radius is \(\frac{d}{4}\). This step helps in accurately modeling the nanometer-scale features, crucial for how silicon transistors work, enabling further technological advancements.