The density of energy states (D(E)) is defined as the number of energy states per unit volume in an energy interval. It is used to calculate the number of charge carriers per unit volume of any solid. D(E) can be derived from quantum mechanics by considering the number of allowed energy states within spheres of increasing radius in momentum space. The final expression for D(E) is proportional to the square root of the energy and depends on material properties such as the effective mass of charge carriers and volume of the material.

1. DENSITY OF ENERGY STATES
It is defined as the number of energy states per unit volume in an energy interval of
metal, It is used to calculate the number of charge carriers per unit volume of any solid.
N(E) dE =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑛𝑒𝑟𝑔𝑦 𝑠𝑡𝑎𝑡𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐸 𝑎𝑛𝑑 𝐸+𝑑𝐸
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑒𝑡𝑎𝑙
N(E) dE =
𝐷( 𝐸) 𝑑𝐸
𝑉
…………(1)
Let us consider a sphere of radius “n” in space with quantum numbers nn, ny and nz.
n2 = nx2 + ny2 + nz2
Number of energy states within a sphere of radius n =
4
3
π𝑛3
Consider sphere of radius ‘n’ due to one octant n =
1
8
. (
4
3
π𝑛3
)
IIIly sphere of radius ‘n+dn’ n + dn =
1
8
. [
4
3
π (𝑛 + 𝑑𝑛)3
]
Therefore no of energy states available in n and n+dn is
D(E)dE =
1
8
. [
4
3
π (𝑛 + 𝑑𝑛)3
−
4
3
π𝑛3
]
D(E)dE =
1
8
. [
4
3
π (𝑛3
+ 𝑑𝑛 3
+ 3𝑛 2
𝑑𝑛 + 3𝑛𝑑𝑛 2
− 𝑛 2
) ]
dn2 and dn3 is very very small it can be neglected
D(E)dE =
1
8
. [
4
3
π ( 3𝑛2
𝑑𝑛) ]
D(E)dE = [
𝜋𝑛 2 𝑑𝑛
2
] ---- (1)
From Schodinger wave equation, allowed energy level
E =
𝑛 2ℎ 2
8𝑚𝐿 2
Differentiating the above equation,
dE =
ℎ 2
8𝑚𝐿 2 2 𝑛d𝑛
𝑛d𝑛 =
8𝑚𝐿 2
2ℎ 2
dE
𝑛2
=
(8𝑚𝐿 2E)
(ℎ 2)
𝑛 =
(8𝑚𝐿 2E)1/2
(ℎ 2)1/2

2. Applying the value of n and ndn in equ (1)
𝐷( 𝐸) 𝑑𝐸 =
𝜋
2
×
(8𝑚𝐿2
𝐸)
1/2
(ℎ2)1/2
8𝑚𝐿2
2ℎ2 𝑑𝐸
𝐷( 𝐸) 𝑑𝐸 =
𝜋
4
×
(8𝑚𝐿2)
3/2
(ℎ2)3/2 E1/2
𝑑𝐸
𝐷( 𝐸) 𝑑𝐸 =
𝜋
4h3
× (8𝑚)3/2 L3E1/2 𝑑𝐸
𝑉 = 𝐿3
= 1
𝐷( 𝐸) 𝑑𝐸 =
𝜋
4h3 × (8𝑚)3/2
E1/2
𝑑𝐸
This energy states accommodate two electron based on Pauli’s exclusion principle
(one is spin up and one is spin down)
𝐷( 𝐸) 𝑑𝐸 = 2 x
𝜋
4h3 × (8𝑚)
3
2 L3
E
1
2 𝑑𝐸
𝐷( 𝐸) 𝑑𝐸 =
𝜋
2h3
× (8𝑚)
3
2 L3
E
1
2 𝑑𝐸