Solving the Density of States: Understanding dn/dE

In summary: There's a subtle difference in that "deriving" is finding the formula from first principles, while "differentiating" is finding the rate of change of a formula that is already given.
  • #1
Addez123
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Homework Statement
$$E = \frac{(n_x^2 + n_y^2 +n_z^2) \pi^2 \hbar^2}{2mL^2}$$
Find density of state
Relevant Equations
Quantum mechanics
$$n = \sqrt{n_x^2 + n_y^2 +n_z^2}$$
$$E = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$
$$n = \sqrt{ \frac{2mL^2E}{\pi^2 \hbar^2} }$$

This is all given by the textbook.
It's even as friendly as to say
$$\text{differential number of states in dE} = \frac{1}{8}4 \pi n^2 dn$$
$$D(E) = \frac{ \text{differential number of states in dE} }{dE} = \frac{1}{8}4 \pi n^2 \frac{dn}{dE}$$

Everything written above is what my textbook says when it tries to explain density of state.
Then it says: "Its left to the reader to show that this equation becomes:"
$$D(E) = \frac {m^{3/2}L^3}{\pi^2 \hbar^3 \sqrt{2}} E^{1/2}$$

What is dn/dE?
am I suppose to take the derivative of dn first??

If I do im left with
$$\frac{\pi mL^2E}{ \pi^2\hbar^2} \frac{1}{dE}$$
Now what?
Divide by a derivative?! What does that even mean!?
What am I suppose to do with the ##\frac{1}{dE}## term?

I've been stuck at this point for days now. No single youtube formula can explain the steps because everyone does it differently and involves other constants such as k etc. It's all very confusing.
 
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  • #2
Addez123 said:
Homework Statement:: $$E = \frac{(n_x^2 + n_y^2 +n_z^2) \pi^2 \hbar^2}{2mL^2}$$
Find density of state
Relevant Equations:: Quantum mechanics

$$n = \sqrt{n_x^2 + n_y^2 +n_z^2}$$
$$E = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$
$$n = \sqrt{ \frac{2mL^2E}{\pi^2 \hbar^2} }$$

This is all given by the textbook.
It's even as friendly as to say
$$\text{differential number of states in dE} = \frac{1}{8}4 \pi n^2 dn$$
$$D(E) = \frac{ \text{differential number of states in dE} }{dE} = \frac{1}{8}4 \pi n^2 \frac{dn}{dE}$$

Everything written above is what my textbook says when it tries to explain density of state.
Then it says: "Its left to the reader to show that this equation becomes:"
$$D(E) = \frac {m^{3/2}L^3}{\pi^2 \hbar^3 \sqrt{2}} E^{1/2}$$

What is dn/dE?
It's the derivative of n with respect to E. You are given the formula for n as a function of E. This is a fairly simple differentiation problem.
Addez123 said:
am I suppose to take the derivative of dn first??
No. See above.
Addez123 said:
If I do im left with
$$\frac{\pi mL^2E}{ \pi^2\hbar^2} \frac{1}{dE}$$
No, that's incorrect.
Addez123 said:
Now what?
Divide by a derivative?! What does that even mean!?
What am I suppose to do with the ##\frac{1}{dE}## term?

I've been stuck at this point for days now. No single youtube formula can explain the steps because everyone does it differently and involves other constants such as k etc. It's all very confusing.
 
Last edited:
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  • #3
Do you understand their "friendly " hint? Do you realize where the 1/8 comes from? You are trying to count the number of degenerateb states as n gets large by taking a continuum approximation.
Youtube formula? How about a book?
 
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  • #4
Mark44 said:
It's the derivative of n with respect to E. You are given the formula for n as a function of E. This is a fairly simple differentiation problem.

No. See above.

No, that's incorrect.
I literally can not explain how thankful I am for this response.
I dont know how I couldnt read dn/dE as ##\frac{d}{dE}(n)## but it just never clicked.

One final problem though. Their result has ##\sqrt{2}## in the denominator. I've done it twice but I get it in the numurator, isnt that correct?
 
  • #5
hutchphd said:
Do you understand their "friendly " hint? Do you realize where the 1/8 comes from? You are trying to count the number of degenerateb states as n gets large by taking a continuum approximation.
Youtube formula? How about a book?
The 1/8th is because we calculate all states as if it were in a cartesian coordinate system and since n cant be negative we only cover the first octant.

Its the surface of a sphere in the 1st octant * dn as they explain it.
 
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  • #6
Addez123 said:
One final problem though. Their result has ##\sqrt{2}## in the denominator. I've done it twice but I get it in the numurator, isnt that correct?
I get exactly their result. In my work I ended up with 2 in the denominator, and ##\sqrt 2## in the numerator. Simplifying gives ##\sqrt 2## in the denominator.
 
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  • #7
Mark44 said:
I get exactly their result. In my work I ended up with 2 in the denominator, and ##\sqrt 2## in the numerator. Simplifying gives ##\sqrt 2## in the denominator.
Ahh yes yes! You get 1/2 from derivating the sqrt(E)!
Now it all makes sense.

Unbelivably grateful, thanks alot :) :)
 
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  • #8
Addez123 said:
You get 1/2 from derivating the sqrt(E)!
Minor nit -- "derivating" is not a word in English, but "differentiating" is.
 
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Related to Solving the Density of States: Understanding dn/dE

What is the density of states (DOS) in the context of solid-state physics?

The density of states (DOS) refers to the number of electronic states that are available at a specific energy level in a material. It is a crucial concept in solid-state physics as it helps in understanding the distribution of electrons in a material, and it influences various physical properties such as electrical conductivity and thermal properties.

How is the density of states (DOS) calculated?

The density of states (DOS) is calculated using quantum mechanical principles. For a given energy, the DOS is determined by counting the number of available quantum states per unit volume and per unit energy interval. This often involves solving the Schrödinger equation for the system and considering the boundary conditions and symmetry of the material. Numerical methods and approximations, such as the tight-binding model or density functional theory (DFT), are frequently used for practical calculations.

What is the significance of dn/dE in the context of DOS?

The term dn/dE represents the derivative of the number of states (n) with respect to energy (E), which essentially gives the density of states (DOS). It indicates how the number of available electronic states changes with energy. A high value of dn/dE at a particular energy level means that there are many states available for electrons to occupy at that energy, which can significantly affect the material's electronic properties.

Why is understanding the density of states important for material science?

Understanding the density of states is vital for material science because it provides insights into the electronic structure of materials. The DOS affects how electrons are distributed in a material, which in turn influences its electrical, thermal, and optical properties. For example, in semiconductors, the DOS near the band edges determines the carrier concentration and mobility, which are critical for designing electronic devices.

How does the density of states vary between different types of materials (e.g., metals, semiconductors, insulators)?

The density of states varies significantly between different types of materials. In metals, the DOS at the Fermi level is typically high, allowing for a large number of free electrons that contribute to electrical conductivity. In semiconductors, the DOS is low at the Fermi level but increases near the conduction and valence band edges. Insulators have a large band gap with very few states available near the Fermi level, resulting in poor electrical conductivity. Each material's specific DOS profile is crucial for its unique electronic properties.

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