Long wavelength limit

In summary, in chapter 4 -Phonons I of "Introduction to Solid State Physics" by C. Kittel, there is a long wavelength limit that can be applied when Ka << 1. This is due to the Taylor approximation of the cosine function, where the higher order terms can be disregarded depending on the desired level of accuracy. The frequencies that can allow this long wavelength limit to hold may vary depending on the specific application.
  • #1
Yu-Ting
In the "Introduction to Solid State Physics" by C. Kittel, there is a long wavelength limit in chapter 4 -Phonons I.

When Ka << 1 we can expand cos Ka ≡ 1 - ½ (Ka)2

the dispersion relation will become ω2 = (C/M) K2 a2

Does anyone know what frequencies can allow this long wavelength limit to hold?
 
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  • #2
I am not sure that there is a standard, since it likely depends on your application.
This is simply the Taylor approximation of the cosine function:
## \cos x = 1 - \frac12 x^2 + \frac{1}{24}x^4 - \frac{1}{720}x^6 ...##
Therefore, you can cut off the higher order terms whenever you feel they are small enough to be insignificant.
For example, when x = .1, your third term is less than .00001. Maybe this is small enough to disregard for your application. Maybe you need your error to be less than 10^-12. Then you should only apply the approximation when x is on the order of 10^-3.
 

1. What is the "long wavelength limit"?

The long wavelength limit refers to the maximum wavelength that can be observed or measured in a given system or experiment. This limit is determined by the capabilities of the equipment being used and can vary depending on the specific system or experiment.

2. How is the long wavelength limit calculated?

The long wavelength limit is typically calculated by considering the sensitivity and resolution of the equipment being used. This limit can also be affected by external factors, such as atmospheric conditions or background noise.

3. Why is the long wavelength limit important in scientific research?

The long wavelength limit is important because it sets the boundary for what can be observed and measured in a given system or experiment. It allows scientists to determine the range of wavelengths that can be studied and helps to ensure accurate and reliable data.

4. Can the long wavelength limit be extended?

Yes, the long wavelength limit can be extended by using more sensitive equipment or by implementing advanced techniques to reduce background noise. However, there are physical limitations that cannot be overcome, such as the wavelength range of the electromagnetic spectrum.

5. How does the long wavelength limit relate to the concept of resolution?

The long wavelength limit is directly related to the resolution of a system or experiment. A system with a smaller long wavelength limit will have a higher resolution, meaning it can distinguish between smaller differences in wavelengths. On the other hand, a system with a larger long wavelength limit will have lower resolution and may not be able to distinguish between closely spaced wavelengths.

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