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
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
