Advanced Solid State Physics

0 1 2 3 4 5 6 7 8

x 10−6

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

wavelength / m

internal energy density u / J/m

3

T=2000 K

T=2200 K

Figure 5: Planck radiation - internal energy as a function of wavelength atT= 2000K

Out of these resultsPlanck’s radiation lawcan be derived. From the internal energyu=

∫∞
0 u(ω)dω
the energy density can be found with substitutingx=k~BωT:

u(ω) =~ω

ω^2

c^3 π^2

1

e

k~ω

BT− 1

dω

This is Planck’s radiation law in terms ofω. The first term is the energy of one mode, the second
term is the density of modes/states, the third term is theBose-Einstein factor, which comes from
the geometric series. Planck’s law can also be written in terms of the wavelengthλby substituting
x=λkhcBT, which is more familiar (see also fig. 5):

u(λ) =

8 πhc

λ^5

1

e

hc

λkBT − 1

dλ

The position of the peak of the curve just depends on the temperature - that isWien’s displacement
law. It is used to determine the temperature of stars.

λmax=

2897 , 8 · 10 −^6 mK

T

4.1 Thermodynamic Quantities

After an integration over all frequencies (all wavelengths) we get the thermodynamic quantities. The
free energyF has the form

F=−kBTln(Z) =

− 4 σV T^4

3 c

J.