Let's consider the following energy dispersion relationship for a charge carrier in a periodic crystal lattice: E(K)=cos2(K)
The expression for effective mass m* is 

4cos(2k)
 

h2/4sec(2k)
 

2sin(2k)/h2
 

None of the above

To find the effective mass \(m^*\) from the given energy dispersion relationship \(E(K) = \cos(2K)\), we need to use the definition of effective mass in terms of the second derivative of the energy with respect to the wavevector \(K\):

\[ m^* = \frac{\hbar^2}{\left. \frac{d^2E}{dK^2} \right|_{K=K_0}} \]

where \(\hbar\) is the reduced Planck's constant.

First, let's compute the first derivative of \(E(K)\) with respect to \(K\):

\[ \frac{dE}{dK} = \frac{d}{dK} \cos(2K) = -2\sin(2K) \]

Next, we compute the second derivative:

\[ \frac{d^2E}{dK^2} = \frac{d}{dK}(-2\sin(2K)) = -2 \cdot 2 \cos(2K) = -4\cos(2K) \]

Now, substituting this into the effective mass formula:

\[ m^* = \frac{\hbar^2}{-4\cos(2K)} \]

This simplifies to:

\[ m^* = -\frac{\hbar^2}{4\cos(2K)} \]

Since effective mass is usually taken as a positive quantity, we take the magnitude:

\[ m^* = \frac{\hbar^2}{4|\cos(2K)|} \]

Comparing with the given options:

1. \(4\cos(2K)\)
2. \(\frac{\hbar^2}{4\sec(2K)}\)
3. \(\frac{2\sin(2K)}{\hbar^2}\)

None of these options match our derived formula exactly. Therefore, the correct answer is:

**None of the above**

Do you want details or have any questions? Here are some relative questions that may expand your understanding:

1. How is the concept of effective mass used in semiconductor physics?
2. What are the physical implications of having a negative effective mass?
3. How does the effective mass affect the mobility of charge carriers in a crystal?
4. What is the significance of the wavevector \(K\) in the context of crystal lattices?
5. How does band structure influence the effective mass of electrons and holes?
6. Can you provide examples of materials where effective mass plays a crucial role in their properties?
7. How does the effective mass relate to the density of states in a material?
8. What are the limitations of using the effective mass approximation in complex materials?

**Tip:** When dealing with energy dispersion relations, always remember to check the units and physical significance of each term to ensure that derived expressions make sense in the context of the problem.

Learn how to calculate effective mass from the energy dispersion relationship E(K) = cos(2K). Understand the mathematical derivation using second derivatives and the physical implications of effective mass in crystal lattices. Suitable for advanced undergraduate students and researchers.

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