Black body radiation

1. ORIGIN OF QUANTUM THEORY
BLACK BODY RADIATION
1. INTRODUCTION
2. BLACK BODY RADIATION AND ITS SPECTRUM
3. STEFAN’S LAW AND WIEN’S LAW OF RADIATION
4. RAYLEIGH-JEANS LAW
5. FAILURE OF CLASSICAL THEORY TO EXPLAIN BLACK
BODY RADIATION
6. CONCLUSION
Ref: Perspective of Modern Physics By A Beiser
Modern Physics By B L Theraja
Physics of atom By J B Rajam

2. BLACK BODY RADIATION
Thermal Radiation: This refers to em radiation
mainly in the IR region, by means of which heat
energy is exchanged between bodies.
Radiant Energy: All bodies at all times are
continuously emitting energy by virtue of their
temperature. This energy is called the radiant energy
or thermal radiation. It travels like visible light in the
form of electro-magnetic waves with the velocity of
light. These waves can be transmitted through
vacuum or any medium like air.

3. Emissive power: The emissive power of a body at a
particular temperature and for a given wavelength,
is defined as the radiant energy emitted per unit
time, per unit surface area of the body within a unit
wavelength range. If E d is the power radiated per
unit area then E is the emissive power.
Absorptive power: The absorptive power of a body
at a given temperature and for a given wavelength,
is defined as the ratio of radiant energy absorbed
per s , by unit surface area of the body to the total
energy falling per unit time on the same area.

4. BLACK BODY
A body whose absorptivity is unity for all
wavelengths is a perfect black body or simply a black
body.
Or
A body which absorbs all the incident radiation
completely irrespective of wavelength falling on it,
reflecting none and transmitting none, is called a
black body.
A black body is only an ideal concept. Lamp
black or platinum black is the nearest to such a body.

5. Fery’s Black body :The Hollow Chamber
or Hohlraum
The hollow chamber is a good approximation
of a black body. It has a tiny aperture through
which radiation is emitted, and is immersed in a
heat bath to keep it at constant temperature.
The radiation emitted can
be detected and analyzed
with a spectrometer in
order to obtain the
spectral distribution of
the emitted energy.

6. 1. It is found that as the temperature of a body is
raised , the body emits radiation of all wavelengths.
The color emitted by it becomes richer in waves of
shorter length.
2. The wavelength at different portions of the
spectrum can be calculated by the formula for
dispersion of prism.
3. By measuring the intensity at various
wavelengths in the whole spectrum of black body
radiation , a graph can drawn between intensity and
wavelength.
4. The fig shows plots of black body spectrum at
different temperatures.

7. The spectrum of the
black body radiation is
plotted for three
different temperatures,
2000 K, 1750 K, and
1250 K.
It is observed that:
1. as the temperature of the body rises, the intensity
of radiation for each wavelength increases.
2. for any one temperature, energy is distributed
continuously among the various wavelengths and is
maximum for a particular wavelength.

8. 3. the point of maximum intensity shifts towards the
shorter wavelengths as the temperature increased.
4. the total energy of radiation for given temperature
is given by the area under the curve. The area
increases according to the forth power of absolute
temperature.

9. LAWS OF BLACK BODY RADIATION
1. Kirchhoff’s law: It states that, at a given
temperature, the ratio of the emissive power to the
absorptive power for a given wavelength is the same
for all bodies and is equal to the emissive power of a
perfectly black body.
E=e/a= constant

10. STEFAN-BOLTZMANN LAW
According to Stefan’s law, the total amount of
heat radiated by a perfectly black body per second per
unit area is directly proportional to the fourth power
of its absolute temperature
ie E  T4 or E =  T4
where  is a constant, called Stefan’s constant and its
value is 5.67 x 10 –8 W/m2 K4
.

11. If a black body A at
absolute temperature
T is surrounded by
another black body B
at absolute
temperature To , then
amount of heat lost by black body A =  T4
amount of heat absorbed by black body B from
black body A =  To
4
hence net amount of heat lost by A per second
per sq centimeter =  (T4 - To
4). This is also known as
Stefan’s Boltzmann law

12. WEIN’S DISPLACEMENT LAW
Wein found that as the temperature of a
black body is elevated, the spectrum retains its
general shape , but the maximum shifts towards
shorter wavelength side so that the wavelength of the
most intense radiation is inversely proportional to
the absolute temperature.
ie mT = constant = 0.2898 cm K.
This is known as Wein’s displacement law.

13. If the wavelength of maximum emission of the spectral
distribution of the black body is plotted over 1/T, one
obtains a straight line.
Wien's second law does a pretty good job of simulating the
behavior of the black body spectrum at short wavelengths.
It fails at longer wavelengths.

14. Wein obtained an expression for the
monochromatic energy density  within an
isothermal black body enclosure in the wavelength 
and +d as
 d = C1-5/ e C2/T d [Note:  (c/4)=E]
where  is the wavelength , T is absolute temperature.
This formula is essentially empirical and contains two
adjustable constants C1 and C2. By adjusting these two
constants Wein could explain the nature of black body
curves except at longer wavelengths.

15. RAYLEIGH-JEANS LAW
Rayleigh derived the radiation law based on
the following assumptions:
1. The radiant energy is due to atomic oscillators
capable of assuming all values of energy and having
an average thermal energy = kT
2. Standing e m waves exists between any two points
of the enclosure.
3. In thermal equilibrium , the average energy of the
wave equals that of the oscillators, i.e., kT.

16. Just like a string can vibrate to produce a
fundamental and a whole series of overtones, there
should be many modes of vibration present in the
standing waves of radiation in the cavity space. The
number of modes of vibration dn per unit volume of
space in the wavelength range  to +d is
dn = 8 d/4
if we multiply the above equation by the
average energy per mode (kT) then it gives the
Rayleigh-Jeans law
ie   d = = 8 kT d/4

17. This law fits well for longer wavelengths but at
shorter wavelengths it tends towards infinity. This is
referred to as “ uv catastrophe” which is predicted
from classical physics, but obviously not observed.
If the failure of Wein’s law
was too bad, that of R-J
law presented a crisis.
Thus classical theory was
unable to explain the black
body radiation
phenomenon.

18. The Planck Hypothesis
In order to explain the frequency
distribution of radiation from a hot cavity
(blackbody radiation) Planck proposed that the
atomic oscillators or resonators emit or absorb
energy in discrete units; each unit is referred to
as a quantum. The energy of a quantum is
proportional to the frequency.
To avoid the crisis presented by Rayleigh-
Jeans Law (the ultraviolet catastrophe), Planck
argued that the higher modes would be less
populated.

19. The quantum idea was soon seized to
explain the photoelectric effect, became part of
the Bohr theory of discrete atomic spectra, and
quickly became part of the foundation of modern
quantum theory.

20. PLANCK’S LAW OF RADIATION
Planck was led to consider the possibility of violation
of the law of equipartition of energy on which the
classical theory is based.
Classical laws give satisfactory results at low
frequencies
The average total energy approaches kT as 
tends to zero
The discrepancy at high frequencies could be
eliminated if there exists, for some reason ,a cutoff
such that E tends to zero as  tends to infinity.
kT



21. Planck realised that the average energy of standing
waves is a function of frequency E() having above
two properties. This is in contrast to law of
equipartition of energy
Equipartition law arises from the classical theory. The
average energy
where P (E) is the probility of finding a given particle
of a system in the range E and E+dE and is called the
Boltzmann distribution function
kT


kT
e
P
d
P
d
P kT
/
)
(
;
)
(
)
( 

















22. Planck’s Argument:
Planck’s great contribution came when he realised
that he could obtain the required cutoff (E=0) by
modifying the calculation of E from P(E) by treating
the energy E as if it were a discreate variable
instead of a the continuous variable.
This can be done by writing the energy equation in
terms of sum instead of an integral.

23. Further, Planck assumed that the average energy Ē at a
given frequency could take on only certain discreate
values that are integral multiples of the basic quantum.
i.e. as the set of
allowed values of energy.
Thus Planck discovered that he could obtain
when E is small----- law frequencies and average
energy E = 0 when E is large ----- high frequencies.
Therefore he needed to make E an increasing
function of .
..........
,.........
3
,
2
,
,
0 




kT



24. Numerical work showed him that he could take the
simplest possible relation between E and  having
above properties. He assumed, E or  E=h
where h is the Planck’s constant=6.63 x 10 –34 Js
According to Planck the average energy E is
which satisfies Planck’s argument.
The energy density is given then by
This equation does agree with the experimental
results.
1
/


 kT
h
e
h






 
 d
e
h
d kT
h
1
8
/
4




25. Thus Plank’s quantum concept is: Radiation is not
emitted or absorbed in continuous amounts but in
discrete bundles of energy equal to h
These bundles or packets of radiant energy are called
quanta or photons.
PROPERTIES OF PHOTON
1. Energy of a photon is represented by E=nh where
n= 1,2,….
According to quantum mechanics E=(n+1/2) h
The limiting value of photon energy is 1/2h but not

26. 2. Energy of the photon is independent of intensity
3. Mass and momentum: Photon has zero rest mass
E = h = mc2, therefore m=h/c2
And momentum p = mc= (h/c2).c=h/c= h/
4. Photons are electrically neutral and hence are
unaffected by electric and magnetic fields.

27. Derivation of Planck’s law of radiation
kT
e
P
d
P
d
P
kT
/
)
(
)
(
)
(

















In classical mechanics
the average energy of
oscillator is given by
where
is Boltzmann distribution function
Planck modified this approach by writing above in
terms of sum as

28. 




















kT
nh
kT
nh
e
e
nh
P
P
/
/
)
(
)
(
)
(



Put E = nh
and P(E) = e-nh/kT
Expanding this series we
obtain the following
expression for the average
energy
1
/


 kT
h
e
h



29. And therefore the energy density is the number of
modes of vibration X the average energy per mode
i.e.




 
 d
e
h
d kT
h
1
8
/
4



This is Planck’s
law of radiation
which explains
the correct
behaviour of
black body
radiation
phenomenon.