PHN-006 Write up on
BLACKBODY RADIATION
Only the quantum theory of light can explain its origin
Wite up by
Tetarwal Rajneesh
enrol no. 22116094
ECE(O8)
THE REASON I CHOSE THIS TOPIC:
Blackbody radiation is a fundamental concept in physics and serves as a
foundation for various fields, including thermodynamics, quantum
mechanics, and astrophysics. By understanding the principles of blackbody
radiation, I can gain a deeper understanding of these interconnected areas
of study. The study of blackbody radiation played a pivotal role in the
quantum revolution, marking a profound shift in our understanding of the
physical world. By examining the behavior of blackbody radiation, physicists
encountered discrepancies with classical theories, sparking a scientific
revolution. This led to the development of quantum mechanics, a
groundbreaking framework that fundamentally transformed our
understanding of the microscopic world. After learning about this
background of blackbody radiation I became very curious to dive into this
topic hoping it would lit a spark in me .
HOW IS THIS TOPIC RELEVANT TO THE COURSE:
Blackbody radiation is highly relevant to quantum physics as it played a
crucial role in the development and formulation of quantum mechanics. The
study of blackbody radiation led to the realization that classical theories,
such as classical electromagnetism and classical statistical mechanics,
failed to accurately describe the observed behavior of radiation emitted by a
black body.
INTRODUCTION:
In 1894, the German physicist Planck embarked on the study of blackbody
radiation. In 1899, he derived Wien's law from thermodynamics. However, in the
same year, the German physicist Ruth discovered biases in Wien's law and
experimental results in the high-frequency region. It wasn't until 1900 that Planck,
through relentless efforts, formulated the well-known Planck blackbody radiation
formula. However, this formula was solely based on experimental data and lacked a
reasonable theoretical explanation.
After several months of work, Planck finally made a breakthrough. He realized that
the only feasible assumption for a rational interpretation of the formula was that
energy emission and absorption occur in discrete increments, rather than as a
continuous change. In other words, during radiation emission and absorption,
energy is not infinitely divisible, but rather exists in minimum units. Planck
referred to these units as energy quanta or quanta, which was a bold idea. This
hypothesis opened the door to a new era and had significant implications for the
scientific community in the 20th century. The study of blackbody radiation marked
an epoch-making milestone and ushered in a new era of scientific research. Black
body is considered as an ideal body with which all the bodies’ radiation is
compared. The concept of blackbody is an idealization as perfect black bodies do
not exist in nature.
A PRACTICAL BLACKBODY :
A black body is a theoretical object that absorbs all incident electromagnetic
radiation, regardless of its frequency or energy. The
radiation emitted by such a body is referred to as black
body radiation. To conceptualize a black body, we often use
an idealized model—a cavity with opaque walls and a tiny
opening. When light enters this cavity through the small
hole, it undergoes multiple reflections, making it highly
improbable for the light to escape. This behaviour effectively
traps the light inside the cavity, providing a practical
representation of a black body.
PERFECT BLACKBODY:
In the 19th century, Kirchhoff introduced the theoretical concept of a perfect black
body, which consisted of a surface layer that absorbs all incident radiation with
infinitesimally small thickness. However, Planck identified several significant
limitations to this idea. He outlined three requirements that a black body must
meet:
1. The body must allow radiation to enter but not reflect it.
2. The body must have a minimum thickness that is sufficient to absorb the
incoming radiation and prevent its re-emission.
3. The body must satisfy stringent limitations on scattering to prevent radiation
from entering and bouncing back out.
What Was Observed: Two Laws
The first quantitative conjecture based on experimental observation of hole
radiation was:
Stefan’s Law (1879):
the total power P radiated from one square meter of black surface at
temperature T goes as the fourth power of the absolute temperature:
P=σT4,
σ=5.67×10−8 watts/sq.m./K4
Five years later, in 1884, Boltzmann derived this T4 behaviour from theory: he
applied classical thermodynamic reasoning to a box filled with electromagnetic
radiation, using Maxwell’s equations to relate pressure to energy density.
Wien’s Displacement Law (1893):
As the oven temperature varies, so does the frequency at which the emitted
radiation is most intense. In fact, that frequency is directly proportional to the
absolute temperature:
fmax∝T
BLACKBODY RADIATION EXPERIMENTAL OBSERVATIONS:
In 1895, at the University of Berlin, Wien and Lummer punched a small hole in the
side of an otherwise completely closed oven, and began to measure the radiation
coming out.
The beam coming out of the hole was passed through a diffraction grating, which
sent the different wavelengths/frequencies in different directions, all towards a
screen. A detector was moved up and along the screen to down find how much
radiant energy was being emitted in each frequency range. They found a radiation
intensity/frequency curve close to this
They were also able to confirm both Stefan’s Law P = σT 4 and Wien’s
Displacement Law by measuring the black body curves at different temperatures.
THE BLACKBODY CURVE:
Rayleigh’s Sound Idea: Counting Standing Waves:
In 1900, a few months before Planck's ground breaking work, Lord Rayleigh
adopted a more straightforward approach to understanding the radiation inside an
oven. He disregarded the notion of oscillators within the walls and instead
considered the radiation to be a collection of standing waves within a cubic
enclosure, specifically electromagnetic oscillators. In contrast to the unclear nature
of wall oscillators, these standing electromagnetic waves provided a clearer and
more comprehensible framework for analysis.
The possible values of wavelength are:
λ = 2a, a,
2a
,…
3
So the allowed frequencies are
f=
c
c
c
c
= , 2( ), 3( ), …
λ 2a
2a
2a
c
apart. We define the spectral
2a
density by stating that number of modes between
These allowed frequencies are equally spaced
f and f + Δf = N(f)Δf
Δf is large compared with the spacing between successive
frequencies. Evidently for this one-dimensional exercise N(f) is a constant equal
where we assume that
to
.
2a
, each mode corresponds to an integer point on the real axis in units
c
c/2a
The amplitude of oscillation as a function of time is:
y = Asin
2πx
λ
sin2πft
more conveniently written
y = Asinkxsinωt, where k =
2π
, ω = 2πf, so ω = ck.
λ
The allowed values of k(called the wave number) are:
K =
2π
λ
=
π 2π 3π
,
,
,
a
a
a
… f=
ck
.
2π
The generalization to three dimensions is simple: in a cubical box of side
a,
an allowed standing wave must satisfy the boundary conditions in all three
directions. This means the choices of wave numbers are:
2π
π 2π 3π
= ,
,
,…
λx
a a
a
2π
π 2π 3π
ky =
= ,
,
,
λy
a a
a
kx =
kz =
2π
π 2π 3π
= ,
,
,…
λz
a a
a
That is to say, each modes is labeled with three positive integers:
π
(kx, ky, kz) = (l, m, n)
a
and the frequency of the mode is:
f=
Ck
= (C⁄2π)√k 2x + k 2y + k 2z
2π
For infrared and visible radiation in a reasonable sized oven, frequency intervals
c
measured experimentally are far greater than the spacing
2a
of these integer points. Just as in the one-dimensional example, these modes fill
a
the three-dimensional k -space uniformly, with density ( )3
π
but now this means the mode density is not uniform as a function of frequency.
The number of them between f and f + Δf
in units
= N(f)Δf
(π/a)3, of the spherical shell of radius k =
and restricted to all components of
1/8.
k
is the volume in
2πf
c
,
thickness
k-space,
Δk =
2πΔf
c
being positive (like the integers), a factor of
Including a factor of 2 for the two polarization states of the standing
electromagnetic waves, the density of states as a function of frequency in an oven
of volume
,
V = a3 is:
(4πk 2 )Δk
1
a 3
2π 3 2
N(f)Δf = × 2 ×
= 14 × ( ) × 4π ( ) f Δf
π 3
8
π
c
(a)
giving the density of radiation states in the oven
𝑁(𝑓)𝛥𝑓 =
What about Equipartition of Energy?
8𝑣𝜋𝑓2 𝛥𝑓
𝑐3
.
A central result of classical statistical mechanics is the equipartition of energy: for
a system in thermal equilibrium, each degree of freedom has average energy
½ kBT. ( kB being Boltzmann's constant.) Thus molecules in a gas have average
3
kinetic energy 2 kBT, ½ kBT for each direction, and a simple one-dimensional
harmonic oscillator has total energy kBT: ½ kBT kinetic energy and ½ kBT potential
energy.
Comparing now the formula for the number of modes N(f)Δf in a small interval Δf
8𝑣𝜋𝑓 2 𝛥𝑓
𝑁(𝑓)𝛥𝑓 =
𝑐3
with Planck’s formula for radiation energy intensity in the same interval:
𝜌(𝑓, 𝑇) 𝛥𝑓 =
8𝑣𝜋𝑓2 𝛥𝑓
𝑐3
ℎ𝑓
ℎ𝑓
ⅇ 𝑘𝐵𝑇
−1
for the low frequency modes ℎ𝑓 ≪kBT we can make the approximation
ⅇ
ℎ𝑓
𝑘𝐵 𝑇
− 1 ≅ ℎ𝑓 /kBT
and it follows immediately that each mode has energy kBT, in line with classical
predictions.But things go badly wrong at high frequencies! The number of modes
increases without limit, the energy in these high frequency modes, though, is
decaying exponentially as the frequency increases. Ehrenfest later dubbed this the
ultraviolet catastrophe. Rayleigh’s sound approach apparently wasn’t so sound
after all something crucial was missing.
A nice example of black body radiation is that left over from the Big Bang. It has
been found that the intensity pattern of this background radiation in the Universe
follows the black body curve very precisely, for a temperature of about three
degrees above absolute zero.
MY LEARNING FROM THIS :
Through my research, I explored how quantum physics emerged as a response to
the limitations of classical theories and the intriguing ideas that scientists
developed to address these challenges. I had an enjoyable experience delving into
various sources, including websites, research papers, and videos, to gain a
comprehensive understanding of the topic. Discovering the connection between
Wien's law and Planck's ground breaking concept of energy quantization has been
fascinating, as it marked a significant shift in my scientific understanding.
REFERENCES :
•
•
Black Body Radiation Michael Fowler, University of Virginia
-Black Body Radiation (virginia.edu)
Original Research Article, School of Physics and Information Technology,
Shandong, China
Researchandverificationofblackbodyradiationlaw (ipindexing.com)
•
•
CONCEPTS OF MODERN PHYSICS, SIXTH EDITION -AURTHUR BEISER
Wikipedia -Black-body radiation - Wikipedia