Solid State: Mean Square Lattice Strain

Finally, we can use the fact that \omega=\omega_D to get \tfrac{\omega^4}{4v^4}=\tfrac{\omega_D^4}{4v^4}=\tfrac{\hbar\omega_D^2}{4MNv^3}\left(\tfrac{\hbar\omega_D^2}{4MNv^3}\right), where we have substituted in the definition for \omega_D and M and N are the mass and number of atoms in the line, respectively.I hope this helps clear things
  • #1
Itserpol
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Homework Statement



Okay, this is from Kittel's Introduction to Solid State Physics (8th ed.) and it's driving me crazy.

The problem is: In the Debye approximation, consider [itex]\langle (\tfrac{\partial R}{\partial x})^2 \rangle =\tfrac{1}{2}\Sigma K^2u^2_0[/itex] as the mean square strain, and show that it is equal to [itex]\tfrac{\hbar \omega^2_DL}{4MNv^3}[/itex] for a line of N atoms each of mass M, counting longitudinal modes only.

Homework Equations



If there is anything relevant to this problem, I'm missing it.

The Attempt at a Solution



The solution manual says: [itex]\tfrac{1}{2}\Sigma K^2u^2_0=\tfrac{\hbar}{2MNv}\Sigma K=\tfrac{\hbar}{2MNv}(\tfrac{K_D^2}{2})=\tfrac{\hbar\omega_D^2}{4MNv^3}[/itex]

Now, the last step is obvious since [itex]K=\tfrac{\omega}{v}[/itex] is just the Debye approximation, but all of the preceding steps are like a wizard waving his hands. They make absolutely no sense, and I can't find anything in the book that could possibly lead me to this.

I don't mean to be so whiny, but this horrible textbook combined with my professor's nearly indecipherable Chinese accent have made this one of the most frustrating courses I've ever been in. Please, if anyone out there can help me, I really need it.
 
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  • #2


Dear student,

I understand your frustration with this problem. The Debye approximation can be quite confusing, but let me try to break it down for you.

First, let's start with the equation \langle (\tfrac{\partial R}{\partial x})^2 \rangle =\tfrac{1}{2}\Sigma K^2u^2_0. This equation is telling us that the mean square strain (average squared displacement of atoms from their equilibrium positions) is equal to half the sum of the squared wavevector (K) multiplied by the squared amplitude of the displacement (u_0). This makes sense intuitively, as the more atoms are displaced and the larger the displacement, the more strain there will be in the material.

Next, we need to understand the Debye approximation, which states that for a given frequency \omega, there is a maximum wavevector K_D=\tfrac{\omega}{v}, where v is the speed of sound in the material. This is because at high frequencies, the wavelength of the wave becomes smaller and smaller, and eventually all the atoms in the material will be involved in the vibration. This is important because it allows us to simplify the sum in the equation above, as we will see.

Now, the solution manual uses the fact that in the Debye approximation, the sum \Sigma K=\tfrac{K_D^2}{2}. This is because in the Debye approximation, we are only considering longitudinal modes (those with K=K_D), and there are N of these modes (one for each atom in the line). So the sum becomes N times the maximum value of K, which is \tfrac{K_D^2}{2}.

Finally, we can substitute this into our original equation and use the definition of K_D to simplify. We get \tfrac{1}{2}\Sigma K^2u^2_0=\tfrac{1}{2}\tfrac{K_D^2}{2}u^2_0=\tfrac{\omega^2}{4v^2}u^2_0. But remember, u_0 is just the amplitude of the displacement, and in the Debye approximation, we can relate this to the frequency and speed of sound through the equation u_0=\tfrac{\omega}{v}. So we get \tfrac{\omega^2}{4v^2}u^2_0=\tfrac
 

1. What is mean square lattice strain?

Mean square lattice strain is a measure of the average distortion or deformation of a solid state material's crystal lattice. It is calculated by taking the average of the squared differences between the actual lattice parameters and their ideal values.

2. How is mean square lattice strain measured?

Mean square lattice strain is typically measured using X-ray diffraction techniques. This involves directing X-rays at a sample and analyzing the resulting diffraction pattern to determine the lattice parameters. These parameters are then compared to the ideal values to calculate the mean square lattice strain.

3. What causes mean square lattice strain?

Mean square lattice strain can be caused by a variety of factors, including thermal expansion, external stresses, and defects within the crystal structure. It can also be a result of the manufacturing process or changes in temperature and pressure.

4. How does mean square lattice strain affect material properties?

The presence of mean square lattice strain can significantly impact the physical and mechanical properties of a material. It can affect factors such as electrical conductivity, thermal conductivity, and mechanical strength. It can also influence the material's response to external stimuli and its overall performance in various applications.

5. Can mean square lattice strain be controlled or reduced?

In certain cases, mean square lattice strain can be controlled or reduced through specific manufacturing processes or by altering the material's composition. However, it is a natural occurrence in many solid state materials and may not always be possible to eliminate entirely. Understanding and managing mean square lattice strain is crucial for optimizing material properties and performance.

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