PHY3707 Assignment 1 solutions 2026
, 1.6 Angle Between Tetrahedral Bonds of Diamond
Given:
The angle between tetrahedral bonds in diamond is the same as the angle between the
body diagonals of a cube.
We must determine this angle using elementary vector analysis.
Choose the Body Diagonal Vectors
Consider a cube of side length 𝑎.
The body diagonals run from one corner to the opposite corner.
Let us take two body diagonal vectors from the origin:
⃗⃗⃗⃗
𝑑1 = 𝑎(1,1,1)
⃗⃗⃗⃗
𝑑2 = 𝑎(1, −1, −1)
These represent two tetrahedral bond directions.
Use the Dot Product Formula
The angle between two vectors is given by:
⃗⃗⃗⃗
𝑑1 ⋅ ⃗⃗⃗⃗
𝑑2
cos 𝜃 =
⃗⃗⃗⃗1 ∣∣ 𝑑
∣𝑑 ⃗⃗⃗⃗2 ∣
Compute the Dot Product
⃗⃗⃗⃗
𝑑1 ⋅ ⃗⃗⃗⃗
𝑑2 = 𝑎2 [(1)(1) + (1)(−1) + (1)(−1)]
= 𝑎2 (1 − 1 − 1)
= −𝑎2
, 1.6 Angle Between Tetrahedral Bonds of Diamond
Given:
The angle between tetrahedral bonds in diamond is the same as the angle between the
body diagonals of a cube.
We must determine this angle using elementary vector analysis.
Choose the Body Diagonal Vectors
Consider a cube of side length 𝑎.
The body diagonals run from one corner to the opposite corner.
Let us take two body diagonal vectors from the origin:
⃗⃗⃗⃗
𝑑1 = 𝑎(1,1,1)
⃗⃗⃗⃗
𝑑2 = 𝑎(1, −1, −1)
These represent two tetrahedral bond directions.
Use the Dot Product Formula
The angle between two vectors is given by:
⃗⃗⃗⃗
𝑑1 ⋅ ⃗⃗⃗⃗
𝑑2
cos 𝜃 =
⃗⃗⃗⃗1 ∣∣ 𝑑
∣𝑑 ⃗⃗⃗⃗2 ∣
Compute the Dot Product
⃗⃗⃗⃗
𝑑1 ⋅ ⃗⃗⃗⃗
𝑑2 = 𝑎2 [(1)(1) + (1)(−1) + (1)(−1)]
= 𝑎2 (1 − 1 − 1)
= −𝑎2
