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Lattice Vibration I - Coggle Diagram
Lattice Vibration I
Heat Capacity of Solids
The Energy given to lattice vibrations is the dominant contribution to the heat capacity in most solids
Classical statistical mechanics explained the heat capacity of insulators at room temperature fairly well but failed for lower temperatures and it totally failed for metals.
Metals were expected to have much higher heat capacity than insulators because of many free electrons but it turn out that a metal's heat capacity at room temperature is similar to that of insulator
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Dulong and petit found experimentally that for many solid at room temperature,
Cv = 3R = 25 JK-1mol-1
Debye Model
Just like Einstein model, it also recovers the Dulong-Petit law at thih
The maximum vibration frequency is determined by Debye frequency, WD
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Einstein Model
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In Three dimensions, we replace N by 3N, since each atom has three degrees of freedom.
High Temperature Limit
This obviously goes to 1 as x = 0. This yields the classical result of Dulong and Petit, Cv = 3Nk
Low Temperature Limit
In this expression, Cv = 0 as T= 0 as required. )The Exponential always wins.)
In Einstein model, the high temperature behaviour is good, the behaviour at T=0 is good, the low temperature behaviour is not very good.
While we measure Cv directly propotional to T^3, but this expression at low temperature Cv direct propotional e^-hw/kt
Born von Karman Model
Born-von Karman boundary conditions are periodic boundary conditions which impose the restriction that a wave function must be periodic on a certain Bravais lattice.
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