Show that when the temperature is such that T Debye temperature, the specific

And then the N and Boltzmann's constant come from considering the number of atoms and the energy distribution at low temperatures, respectively. In summary, for low temperatures, the specific heat can be written as C_{V}\proptoNk_{B}(T/\Theta_{D}) due to the relationship between temperature and energy distribution, and the number of atoms in the system.
  • #1
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Homework Statement



Consider phonons propagating on a one-dimensional chain of N identical atoms of mass M interacting by nearest-neighoour spring constants of magnitude C.

a) Show that the Debye frequency can be written as [itex]\omega[/itex][itex]_{D}[/itex]=[itex]\pi[/itex](C/M)[itex]^{1/2}[/itex].
b) Show that when the temperature is such that T<<[itex]\Theta[/itex][itex]_{D}[/itex], where [itex]\Theta[/itex][itex]_{D}[/itex]=[itex]\hbar[/itex][itex]\omega[/itex][itex]_{D}[/itex]/k[itex]_{B}[/itex] is the Debye temperature, the specific heat can be written as C[itex]_{V}[/itex][itex]\propto[/itex]Nk[itex]_{B}[/itex](T/[itex]\Theta_{D}[/itex])

The Attempt at a Solution



I have done part a), but for part b):

my notes say that for T<<[itex]\Theta[/itex][itex]_{D}[/itex], C[itex]_{V}[/itex][itex]\approx[/itex]([itex]\frac{T}{\Theta_{D}}[/itex])[itex]^{3}[/itex]

so how can C[itex]_{V}[/itex][itex]\propto[/itex]Nk[itex]_{B}[/itex](T/[itex]\Theta_{D}[/itex])

when there is a power of 3? And where do the N and Boltzmann's constant come from?

I have also looked in other places, but nowhere has told me why the specific heat can be written in the form they ask you to show it can be written in.

Thanks if you help.
 
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  • #2


Is it possible to say that for low T, x=x^3, so that gets rid of the power of 3?
 

1. What is the Debye temperature?

The Debye temperature is a characteristic temperature for a solid material that represents the maximum temperature at which all lattice vibrations (phonons) are active. It is named after Peter Debye, who developed the concept in 1912.

2. How is the Debye temperature related to specific heat?

The Debye temperature is directly related to the specific heat of a material. At temperatures below the Debye temperature, the specific heat of a material increases as the temperature decreases. At temperatures above the Debye temperature, the specific heat remains constant.

3. How is the Debye temperature determined experimentally?

The Debye temperature can be determined experimentally by measuring the heat capacity of a material at low temperatures and extrapolating the data to determine the temperature at which the specific heat becomes zero. This temperature is known as the Debye temperature.

4. What factors affect the Debye temperature?

The Debye temperature is affected by the density and composition of a material. Materials with higher densities and heavier atoms tend to have higher Debye temperatures. Additionally, the Debye temperature can be affected by external factors such as pressure and strain.

5. Why is the Debye temperature important in materials science?

The Debye temperature is an important parameter in materials science as it provides information about the lattice vibrations and thermal properties of a material. It is also used in the study of phase transitions and the behavior of materials at low temperatures.

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