The Debye model and Einstein models of heat capacity, Lecture notes of Solid State Physics

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Solid State Physics

Lecture 8 – The Debye model

Advanced Technology Institute and Department of Physics

Recap from Lecture 7

• Concepts of “temperature” and thermal equilibrium are based on the idea that individual particles in a system have some form of motion
• Heat capacity can be determined by

considering vibrational motion of

atoms

• We considered two models:
  • Dulong-Petit (classical)
  • Einstein (quantum mechanical)
• Both models assume atoms act

independently – this is made up for in

the Debye model ( today )

Dulong-Petit

A more realistic model…

• Both the Einstein and Dulong-Petit models treat each

atom independently. This is not generally true.

• When an atom vibrates, the force on adjacent atoms

changes causing them to vibrate (and vice-versa)

• Oscillations can be broken down into modes

Nice animations here: http://www.phonon.fc.pl/index.php

1D case

3D case

Java applet: http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html

Debye model

• Basic idea similar to Einstein model, with one key difference:

Einstein:

Energy of system = Phonon Energy x Average number of phonons

Debye:

Energy of system = Phonon Energy x Average number of phonons x number of modes

The number and type of Einstein: number of modes = number of atoms modes are the key difference

Debye: each mode has its own k value (and hence frequency)

Modes: Quantum mechanics

Modes are quantised in units of where the fundamental frequency of each mode is

The Einstein model assumed that each oscillator has the same frequency

Debye theory accounts for different possible modes (and therefore different )

Modes with low will be excited at low temperatures and will contribute to the heat capacity. Therefore heat capacity varies less abruptly at low T compared with Einstein model

 





Low frequency modes correspond to multiple atoms vibrating together (sound or acoustic waves)

Standing waves: revision

L

Consider a vibrating string

Lowest (fundamental) frequency

More generally n = 1

n = 2

n = 3

n = 4

 L 2



n

L n L λ

2

2

  



Other results follow:

L

vn ω πf L

v vn f





    2  2

2

L

π n k





 

Standing waves in 2D crystals

L

L

Fundamental mode (2D)

Each component of the wave is quantised separately and added in quadrature

L

π x kx ^ ky 

y (^) Magnitude of k-vector for mode

L

π k kx ky 2

2 2   

Corresponding angular frequency

vk

v ω  πf   

2 2  L

ω v

  2

In both cases so these two modes are degenerate

As frequency increases, more and more states share the same frequency & energy (called DEGENERACY )

Standing waves in 2D crystals:

Degeneracy

x

y

L

L L

L

L

kx

L

ky 

L

kx 

L

ky

L

π L L k^2

2 2 ^  

  

  

  

    L

π L L k^2

2 2 ^  

  

  

  

  

L

ω v

  5

Number of States in 3D

In 3D we consider the number of states within a sphere of radius k

Sphere “volume” =

“volume” of k-state =

  dk

Vk g k dk 2

2 



kx

ky

kz

3^ k 3

4  k

3

3

l



l



l

 l



k-state

Number of States in 3D

  dk

Vk g k dk 2

2

2 



kx

ky

kz

k

l



l

 l



k-state

We know that

Hence

i.e. the number of standing waves (modes) increases as ^2

Sound can propagate with 2 transverse and 1 longitudinal wave in a solid  total no. of states = 3g(  )d 

ω  vk  dω  vdk

  (^)  

   d v

V g d 2 3

2

2



Phonon animations here: http://www.phonon.fc.pl/index.php

Some crystal modes of vibration

Debye model: Total average energy of System

From earlier: Energy of system

Phonon energy

Average no. of phonons

No. of = x x modes

 

   

    



  d

k T

g g d

k T

E

B B

 

  



 

 

  



 

max max

(^0 0) exp 1

3 3

exp 1

1









(NB: Ignoring zero-point energy)

Integrate over all modes

This is perhaps the most useful parameter in the Debye theory

• It allows us to predict the heat capacity at any temperature
• It provides an indication of the temperature at which we approach the classical limit of the Dulong-Petit theory

From earlier, we know that and

Therefore,

The Debye Temperature D

B

D k

ω θ max

 

31 2 max 6  

  

  V

N ω v π

(^13)

(^13)

2

2

max (^6)

6

  

  

 

 

  



 V

k N

V

N

v D B 





 

So if we know N/V then we can predict the speed of sound in a solid

The Debye Temperature D: examples

High D corresponds to a large max

Large max implies large forces, low max implies weak bonds

Diamond D = 2230K Dulong-Petit poor fit at room temperature. Strongly bonded

Iron D = 457K Dulong-Petit reasonable fit at room temperature.

Lead D = 100K Dulong-Petit good fit at room temperature. Weakly bonded