Deriving Normalized Debye Model: Step-by-Step Guide

This is the normalized equation for the phonon spectrum in the Debye model. The process of normalizing a function involves dividing by its integral and this is how g(ω)=3ω2/ω3D was derived.
  • #1
Rajini
621
4
Hi Members,
We know that in Debye model, the phonon spectrum is given by g([tex]\omega[/tex])=Const.[tex]\omega^2[/tex]. And if we have N atoms and so 3N normal modes..so 3N=integral of g([tex]\omega[/tex])d[tex]\omega[/tex] and solving by simple integration rule we get Const. = 9N/([tex]\omega^3_D[/tex]).
Now my problem is how to get it Normalized equation..g([tex]\omega[/tex])=?(1).
In books i see g([tex]\omega[/tex])=3[tex]\omega^2[/tex]/[tex]\omega^3_D[/tex].
I just need to know how to derive g([tex]\omega[/tex])=3[tex]\omega^2[/tex]/[tex]\omega^3_D[/tex]?
Please explain me clearly every step for such derivation..And in general how to normalizes a function step by step ?
thanks for your time!
and eagerly waiting for your reply
 
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  • #2
..To normalize the function, you need to divide the function by its integral over all frequencies. The integral of g(ω) is 9N/ω3D. Dividing the function by this integral will give you the normalized form of g(ω). So, g(ω)=3ω2/ω3D.
 
  • #3


Hello,

Thank you for your question. Normalization is an important step in deriving the Debye model. Here is a step-by-step guide to deriving the normalized Debye model:

Step 1: Start with the Debye phonon spectrum equation: g(\omega) = Const. \omega^2

Step 2: Integrate the equation over all frequencies to get the total number of modes: 3N = \int_0^{\omega_D} g(\omega) d\omega

Step 3: Solve for the constant by dividing both sides by the integral: Const. = 3N / \int_0^{\omega_D} \omega^2 d\omega

Step 4: Use the integral rule \int x^n dx = x^(n+1) / (n+1) to solve the integral: Const. = 3N / (\omega_D^3 / 3)

Step 5: Simplify the expression to get the final equation for the constant: Const. = 9N / \omega_D^3

Step 6: Substitute the value of the constant back into the Debye phonon spectrum equation: g(\omega) = (9N / \omega_D^3) \omega^2

Step 7: Finally, we can normalize this equation by dividing both sides by the total number of modes (3N): g(\omega) = (9N / \omega_D^3) \omega^2 / (3N)

Step 8: Simplify the equation to get the normalized Debye model: g(\omega) = 3 \omega^2 / \omega_D^3

In general, to normalize a function, we divide it by its maximum value or by the total area under the curve. This ensures that the function is on a scale of 0 to 1, making it easier to compare with other functions. In the case of the Debye model, we divided by the total number of modes to get a normalized function that represents the probability of a phonon having a particular frequency.

I hope this explanation helps you understand how to derive the normalized Debye model step by step. Let me know if you have any further questions. Happy learning!
 

1. What is the Debye Model and why is it important in physics?

The Debye Model is a theoretical model used to describe the behavior of solids at low temperatures. It is important in physics because it helps us understand the properties of solids such as heat capacity, thermal conductivity, and electrical conductivity.

2. What is the purpose of deriving a normalized Debye Model?

The purpose of deriving a normalized Debye Model is to simplify the Debye Model by removing any physical constants and expressing it in terms of dimensionless variables. This makes it easier to compare and analyze the behavior of different types of solids.

3. What are the steps involved in deriving a normalized Debye Model?

The steps involved in deriving a normalized Debye Model include:1. Start with the Debye Model equation2. Substitute the Debye temperature with a dimensionless variable3. Use the definition of specific heat capacity to obtain an expression for the heat capacity4. Substitute the dimensionless variable into the heat capacity expression5. Simplify and rearrange the equation to obtain the normalized Debye Model

4. What are the assumptions made in the Debye Model?

The assumptions made in the Debye Model include:1. The atoms in the solid are arranged in a regular lattice structure2. The lattice vibrations can be treated as simple harmonic oscillators3. The lattice vibrations are quantized, meaning they can only have certain discrete energies4. The lattice vibrations do not interact with each other

5. How is the Debye temperature related to the properties of a solid?

The Debye temperature is related to the properties of a solid in the following ways:1. It determines the maximum frequency of lattice vibrations in a solid2. It is directly proportional to the melting temperature of a solid3. It is inversely proportional to the specific heat capacity of a solid4. It is related to the speed of sound in a solid

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