**Density of states in 1D, 2D, and 3D**

__In
1-dimension__

The density of state for 1-D is defined as the number of electronic or quantum
states per unit energy range per unit length and is usually denoted by

Where dN is the number of quantum states present in the energy range between E and E+dE

Here factor 2 comes
because each quantum state contains two electronic states, one for spin up and
other for spin down.

Equation (1) can be written as

From eq. (2), we have As we know

So

By using Eqs.
(4)and (5), eq. (3) becomes

This result is shown plotted in the figure.

It shows that all the states up to Fermi-level

**In
2-Dimensional**

The density of state for 2D is defined as the number of electronic or quantum
states per unit energy range per unit area and is usually defined as

Area (A)

Area
of the 4^{th} part of the circle in K-space

Here factor 2 comes
because each quantum state contains two electronic states, one for spin up and
other for spin down.

So And

By using eqns.
(10)and (11), eq. (9) becomes

This result is shown in
figure.

__In 3-Dimension __

Volume Volume of the 8

^{th}part of the sphere in K-space

Here factor 2 comes
because each quantum state contains two electronic states, one for spin up and
other for spin down.

Eq. (12) can be written as

From eq. (13) we have As we knowSo And

By using Eqs.
(15)and (16), eq. (14) becomes

__For Band theory of Solids__- Click Here
## 2 Comments

Hi, I am a year 3 Physics engineering student from Hong Kong. I cannot understand, in the 3D part, why is that only 1/8 of the sphere has to be calculated, instead of the whole sphere.

ReplyDeleteHope someone can explain this to me. Many thanks.

I think this is because in reciprocal space the dimension of reciprocal length is ratio of 1/2Pi and for a volume it should be (1/2Pi)^3. It has written 1/8 th here since it already has somewhere included the contribution of Pi. i hope this helps.

DeleteIf you have any doubt, please let me know