A theory of the specific heat capacity of solids put forward by Peter Debye in 1912, in which it was assumed that the specific heat is a consequence of the vibrations of the atoms of the lattice of the solid.

2. Writer: Syed Ahsan Raza (syedahsanraza63@gmail.com)

7. D
ω
↔
0

11. kx
ky
kz
:-nx,ny,nz =1,2,3,4.....
k2 = k2
x + k2
y + k2
z
k2 = (n2
x + n2
y + n2
z )(
𝜋2
𝐿2)
k2 = (
𝑛2
𝜋2
𝐿2 )
k = (
𝑛𝜋
𝐿
) ► n = (
𝑘𝐿
𝜋
)
n = (
𝑘𝐿
𝜋
)....eq 5 k = (
𝜔
𝑣
)
n = (
𝜔𝐿
𝜋𝑣
)....eq 6
This shows that we will get 1 frequency
(w) for every n value
every combination will give us a
frequency nx,ny,nz

12. we should only consider
positive area of the sperical
area as frequency can only
be positive
n is positive if =
1
8
(
4
3
n3𝜋)
0→ω→ =
1
8
(
4
3
n3𝜋)
ω→ω+dω→ =
𝜋
6
(3 n2dn)=
𝜋
2
( n2dn)
n = (
𝜔𝐿
𝜋𝑣
)
z(ω)dω=
𝜋
2
[(
𝜔𝐿
𝜋𝑣
)2d(
𝜔𝐿
𝜋𝑣
)] =(
ω2
𝐿3
2𝜋2
𝑣3)

13. z(𝜔)d𝜔 =(
𝜔2
𝑣
2𝜋2
𝑣3)
z(𝜔)d𝜔 =(
𝜔2
𝑣
2𝜋2)[
2
𝑣3
𝑡
+
1
𝑣3
𝑙
]

14. 0
𝜔𝑑
𝑧(𝜔)𝑑𝜔 = 3𝑁 . . . . . . . . (9)
Putting equation 8 into 9
𝑉
2𝜋2 [
2
𝑣3
𝑡
+
1
𝑣3
𝑙
] 0
𝜔𝑑
(𝜔)2𝑑𝜔 = 3N
𝑉
2𝜋2 [
2
𝑣3
𝑡
+
1
𝑣3
𝑙
][
𝜔𝑑
3
3
] = 3N
𝜔𝑑
3
=
18𝑁𝜋2
𝑉
[
2
𝑣3
𝑡
+
1
𝑣3
𝑙
]−1
𝜔𝑑 = [
18𝑁𝜋2
𝑉
]
1
3[
2
𝑣3
𝑡
+
1
𝑣3
𝑙
]
−1
3
[
2
𝑣
−
3
𝑡
+
1
𝑣
−
3
𝑙
] =
18𝑁𝜋2
𝑉𝜔𝑑
2 ....10
z(𝜔)d𝜔 =(
𝜔2
𝑣
2𝜋2)[
2
𝑣3
𝑡
+
1
𝑣3
𝑙
]
z(𝜔)d𝜔 =(
𝜔2
𝑣
2𝜋2)
18𝑁𝜋2
𝑉𝜔𝑑
2 ] . . . 11

15. 𝐸
= ℏ𝜔 [
1
2
+
1
𝑒
ℏ𝜔
𝑘𝑇 −1
]
Same as Einstein Model
𝐸
𝑧(𝜔)𝑑𝜔 = V
V = 0
𝜔𝑑
𝐸
𝑧(𝜔)𝑑𝜔
𝐶𝑣 =
𝑑𝑉
𝑑𝑇
=
𝑑
𝑑𝑇 𝐸
𝑧(𝜔)𝑑𝜔
𝐶𝑣 =
𝑑
𝑑𝑇
9𝑁
𝜔𝑑
3 ℏ𝜔 [
1
2
+
1
𝑒
ℏ𝜔
𝑘𝑇 −1
] 𝜔2𝑑𝜔
𝐶𝑣 =
9𝑁
𝑘𝑇2𝜔𝑑
3 ℏ2
0
𝜔𝑑 𝜔4𝑒
ℏ𝜔
𝑘𝑇
[𝑒
ℏ𝜔
𝑘𝑇 −1 ]
2 𝑑𝜔

16. dx =
ℏ𝐝𝝎
𝒌𝑻
Let
ℏ𝝎
𝒌𝑻
= x 𝜽𝒅 =
ℏ𝝎𝒅
𝒌
𝐶𝑣 = 9NK [
𝑇
𝜃𝑑
]3
0
𝜃𝑑
𝑇
𝑥4 𝑒𝑥 𝑑𝑥
(𝑒𝑥 − 1)
𝑁𝑘 = 𝑅
𝐶𝑣 = 9NK [
𝑇
𝜃𝑑
]3
0
𝜃𝑑
𝑇
𝑥4 𝑒𝑥 𝑑𝑥
(𝑒𝑥 − 1)2
0 → 𝜔𝑑
x =
ℏ𝜔
𝑘𝑇
→
ℏ𝜔𝑑
𝑘𝑇
when x = 0 and 𝜔 = 0
when 𝜔 = 𝜔𝑑
𝐶𝑣 = 9R [
𝑇
𝜃𝑑
]3
0
𝜃𝑑
𝑇
𝑥4 𝑒𝑥 𝑑𝑥
(𝑒𝑥 − 1)2

17. High and Low Temperature LimitsThe
integral in Equation (12) cannot be
evaluated in closed form. But the high
and low temperature limits can be
assessed.

18. High Temperature LimitFor the high
temperature case where T≫TD , the value of x
is very small throughout the range of the
integral. This justifies using the approximation
to the exponential ex≈1+x and reduces
equation

19. 1. Solid is not continuous it is discrete
2. Crystal structure avoidence
3. Sound have same velocity at all wavelength
4. Free electron ignorance.
5. 𝜃𝑑 = Constant