DENSITY OF ENERGY STATES

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 =
??????����� �� �����?????? ������ ���??????��� � ��� �+��
??????����� �� �ℎ� �����

N(E) dE =
�(�) ��
??????
…………(1)
Let us consider a sphere of radius “n” in space with quantum numbers nn, ny and nz.
n
2
= nx
2
+ ny
2
+ nz
2

Number of energy states within a sphere of radius n =
4
3
π�
3
Consider sphere of radius ‘n’ due to one octant n =
1
8
.(
4
3
π�
3
)
IIIly sphere of radius ‘n+dn’ n + dn =
1
8
.[
4
3
π (�+??????�)
3
]
Therefore no of energy states available in n and n+dn is
D(E)dE =
1
8
.[
4
3
π (�+??????�)
3
−
4
3
π�
3
]
D(E)dE =
1
8
.[
4
3
π (�
3
+??????�
3
+3�
2
??????�+3�??????�
2
−�
2
)

]
dn
2
and dn
3
is very very small it can be neglected
D(E)dE =
1
8
.[
4
3
π ( 3�
2
??????�)

]
D(E)dE = [
??????�
2
??????�
2
] ---- (1)

From Schodinger wave equation, allowed energy level
E =
�
2
ℎ
2

8�??????
2
Differentiating the above equation,
dE =
ℎ
2

8�??????
2 2 �d�
�d� =
8�??????
2

2ℎ
2 dE
�
2
=
(8�??????
2
E)


(ℎ
2)

� =
(8�??????
2
E)
1/2

(ℎ
2)1/2

2. Applying the value of n and ndn in equ (1)
�(�)??????�=
??????
2
×
(8�??????
2
�)
1/2
(ℎ
2
)
1/2
8�??????
2
2ℎ
2 ??????�

�(�)??????�=
??????
4
×
(8�??????
2
)
3/2
(ℎ
2
)
3/2 E
1/2
??????�
�(�)??????�=
??????
4h
3
× (8�)
3/2
L
3
E
1/2
??????�
??????= ??????
3
= 1
�(�)??????�=
??????
4h
3
× (8�)
3/2
E
1/2
??????�
This energy states accommodate two electron based on Pauli’s exclusion principle
(one is spin up and one is spin down)
�(�)??????�=2 x
??????
4h
3
× (8�)
3
2 L
3
E
1
2 ??????�

�(�)??????�=
??????
2h
3
× (8�)
3
2 L
3
E
1
2 ??????�