DEBYE MODEL FOR HEAT CAPACITY IN SOLIDS



PHYS430 STAT. THERMODYNAMICS

DEBYE MODEL FOR HEAT CAPACITY IN SOLIDS

Eren ÇANGA

1274927

Submitted to: Doç. Dr. Altuğ ÖZPİNECİ

DEBYE MODEL FOR HEAT CAPACITY IN SOLIDS

1. INTRODUCTION

The amount of energy required to raise the temperature of one kilogram of the substance by one kelvin. The SI unit for specific heat capacity is the joule per kilogram kelvin, J·kg-1·K-1.By heat capacity, it is often referred that heat capacity at constant volume, which is more fundamental than the heat capacity at constant pressure.The heat capacity at constant volume is defined as

Eqn. 1 Cv= T[pic]= ([pic]

Where S is the entropy, U is the energy, and T is temperature.

The experimental facts about the heat capacity of solids are these:

1. In room temperature range the value of the heat capacity of nearly all monoatomic solids is close to 3Nk, or 25 J mol-1 deg -1.

2. At lower temperatures the heat capacity drops rapidly and approaches zero as T3 in insulators and as T in metals.If metal becomes semiconductor, the drop is faster than T. [1]

[pic]

Figure 1.Heat capacity vs. temperature of Mg2SiO4, Mg (diamonds), Si (circles), O (triangles). [2]

The Debye model is developed by Peter Debye in 1912.He estimated the phonon contribution to the heat capacity in solids. The Debye model treats the vibration of the lattice as phonons in a box, in contrast to Einstein model, which treats the solid as non-interacting harmonic oscillators. The Debye model predicts the low temperature depence of heat capacity T3 that confirms the experimental results. Moreover, it covers the high temperature limits like the Einstein model. [2]

2. EINSTEIN MODEL

The average energy of an oscillator of frequency [pic],[pic]is[pic]. For N oscillators in one dimension, all having same frequency, the thermal energy is [1]

Eqn. 2 [pic]

Where [pic] , k boltzman constant and [pic] is the thermal avarage of the number of phonons in an elastic wave of given frequency.

Then the heat capacity of oscillator is

Eqn. 3 [pic]

(a) At high temperatures [pic] and [pic]

Eqn. 4 [pic]

independent of ω. Also C = dU/dT = 3R, in agreement with experiment.

(b) At low temperatures [pic] and [pic]

Eqn 5. [pic]

And C can be found as follows,

Eqn. 6 [pic]

Therefore, it can be seen that Einstein model succesfully predicts that C decreases with decreasing T. However, exponential decrease is not observed; if low frequencies are present, then [pic] will be small, much smaller than kT even at low temperatures; C will remain at 3kT to much lower frequencies and the fall off is not as dramatic as predicted by the Einstein model.

3.DEBYE MODEL

Debye model uses wide spectrum of frequencies to describe the complicated pattern of lattice vibrations. It is assumed that hypothetical oscillators generate simple sine waves throughout the crystal and these will displace the atoms away from their equilibrium positions by an amount equal to the amplitude of the sine wave at that point. If whole set of such oscillators generates sine waves of certain frequencies and amplitudes, then it can be predicted that the superposition of such waves will simulate the complicated pattern of the actual atomic vibrations. The distribution of oscillators is quasi-continuous in ω hence integration can be used instead of summation. [1]

Eqn. 7 [pic]

[pic]

In order to go further there are two problems to solve:

1. Density of states function is required

2. Need to set the range of frequencies over which the integration is to be performed, i.e. the cut-off or limiting frequency needs to be determined

Density of States

For standing waves, and considering a cube of material of side L, the appropriate boundary condition for vibrational waves reflected from mechanically free surfaces is that an antinode of the vibration amplitude should exist at each surface. This corresponds to there being an integral number of half-wavelengths of the standing wave along the length of the cube[3]. The allowed values of the standing wave vectors are given by

Eqn. 8

[pic]

[pic]

Figure 1 Schematic 1D illustration of a standing wave set up between the free surfaces of a cube of an elastic continuum with antinodes at the free surfaces.[3]

Each allowed standing-wave solution of the wave equation consistent with the boundary conditions is represented by a point in the reciprocal space containing the k-vectors. The spacing between allowed k-values is (ks=(/L, and so the volume of k-space corresponding to the one k-value (standing-wave state) is

Eqn. 9

[pic].

The number of k-values contained in a unit volume of k-space is

Eqn. 10

[pic]

where V=L3 is the sample volume. The density of k states is uniform in k space and depends only on the sample size. For large samples, k can be taken as a continuous variable rather than as a discrete quantity.

Eqn. 11

[pic].

The density of states for a mode i in terms of frequency, gi((), may be obtained using the linear dispersion relation valid for long-wavelength acoustic modes, giving[3]

Eqn. 12

[pic],

where vi is an appropriate sound velocity for mode i. However, three acoustic modes, one longitudinal (LA) mode and two degenerate transverse (TA) modes, can propagate in continuous elastic media, and so the total vibrational density of states is given by

Eqn. 13

[pic]

where vo is an appropriate average of the LA- and TA-mode velocities. This quadratic frequency dependence is the so called Debye density of vibrational states.

Eqn. 15

[pic],

the Debye frequency is given by

Eqn. 16

[pic].

Now N can be found that,

Eqn. 17

[pic]

Eqn. 18

[pic], (see Eqn. 12)

Eqn. 19

[pic]

Eqn. 20

[pic]

[pic]

Eqn. 21

[pic]

Eqn. 22

[pic]

Eqn. 23

[pic]

Eqn. 24

[pic]

This is called as Debye temperature

Eqn. 25

[pic]

x3 for three branches, i.e. 2 transverse acoustical and 1 longitudinal acoustic

Eqn. 26

[pic]

Debye Specific Heat Capacity

Eqn. 27

[pic]

Eqn. 28

[pic]

Eqn. 29

[pic]

Debye specific heat capacity for a single acoustic branch. However, there are three branches, i.e. two transverse acoustic and one longitudinal acoustic branch. The total specific heat is therefore [3]:

Eqn. 30

[pic]

At low temperature

Eqn. 31

[pic]

Eqn. 32

[pic]

Eqn. 33

[pic]

Debye T3 approximation at low temperature as in the experimental results

At high temperature

Eqn. 34

[pic]

Eqn. 35

[pic]

[pic]

Figure parision between Debye and Einstein model [4]

CONCLUSION

For sufficiently low temperatures, the Debye approximation should be quite good, as here only long wavelength acoustic modes are excited. These are just the modes that may be treated as in elastic continuum with macroscopic elastic constants. The energy of short wavelength modes is too high to be populated at low temperatures.

REFERENCES

[1] Charles Kittel, Solid State Physics, 1976, John Wiley & Sons Inc.

[2]

[3] Introduction to Lattice Dynamics, Martin T. Dove, Cambridge topics in mineral physics and chemistry lecture notes

[4]

-----------------------

( = 2L, k=(/L, n=1

(=2L, k=2((/L), n=2

(=2/3 L, k=3 (/L, n=3

0

L

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