PHY3707 Assignment 3 Solutions 2026

PHY3707 Assignment 3 Solutions 2026
UNISA
Due: Friday, 19 June 2026, 3:00 PM
Dear PH3707 student

Kindly find the attached Assignment 3 with unique
code 197029.

, PHY3707 – Assignment 3 (Unique Number: 197029)

Question 1

In a linear chain all the atoms have equal mass but are connected alternatively by
springs of force constants 𝒇𝟏 and 𝒇𝟐 . Derive the frequency-wavevector relation for
this chain. Are there still two branches? Explain.

Define the lattice

Consider a one-dimensional chain of identical atoms of mass 𝑀.

The springs alternate as:

𝑓1 , 𝑓2 , 𝑓1 , 𝑓2 , …


Since the spring constants alternate, the primitive cell contains two atoms.

Let

𝑢𝑛 = displacement of atom A
𝑣𝑛 = displacement of atom B


in the 𝑛𝑡ℎ unit cell.

The distance between adjacent atoms is 𝑎.



Equations of motion

For atom 𝑢𝑛 :

𝑑2 𝑢𝑛
𝑀 = −𝑓1 (𝑢𝑛 − 𝑣𝑛 ) − 𝑓2 (𝑢𝑛 − 𝑣𝑛−1 )
𝑑𝑡 2
Therefore,

𝑑 2 𝑢𝑛
𝑀 = −(𝑓1 + 𝑓2 )𝑢𝑛 + 𝑓1 𝑣𝑛 + 𝑓2 𝑣𝑛−1
𝑑𝑡 2


For atom 𝑣𝑛 :

𝑑2 𝑣𝑛
𝑀 = −𝑓1 (𝑣𝑛 − 𝑢𝑛 ) − 𝑓2 (𝑣𝑛 − 𝑢𝑛+1 )
𝑑𝑡 2
Thus,

𝑑2 𝑣𝑛
𝑀 = −(𝑓1 + 𝑓2 )𝑣𝑛 + 𝑓1 𝑢𝑛 + 𝑓2 𝑢𝑛+1
𝑑𝑡 2