Blackbody Radiation

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Blackbody Radiation

• Václav PoteilBrno University of Technology

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Blackbody Definition

• Perfect absorber of radiation. Theoretical object
  that absorbs 100 of the radiation that hits it
  regardless the wavelength and angle. Therefore it
  reflects no radiation and appears perfectly
  black.
• Perfect emitter of radiation. At a particular
  temperature the black body emits the maximum
  amount of energy possible for that temperature
• Emits a definite amount of energy at each
  wavelength for a particular temperature. Standard
  black body radiation curves can be drawn for each
  temperature, showing the energy radiated at each
  wavelength. Maximum wavelength emitted by a black
  body radiator is infinite.
• The energy a blackbody radiates is characteristic
  of the radiation system only and not dependent
  upon the type of radiation which is incident upon
  it.
• All objects above absolute zero emit radiation.

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Blackbodies Around Us

• Every objects emits radiation although we do not
  normally notice it - our eyes are only sensitive
  to a very small portion of the electromagnetic
  spectrum.  An object must be quite hot for it to
  emit visible light. Objects at around room
  temperature emit mainly invisible infrared
  radiation.
• No material has been found to absorb all incoming
  radiation, but carbon in its graphite form
  absorbs about 97.
• The planets and stars can be regarded as
  blackbodies. Most of the light directed at a star
  is absorbed. Most approximate blackbodies are
  solids but stars are an exception because the gas
  particles in them are so dense they are capable
  of absorbing the majority of the radiant energy.
• A simple example of a black body radiator is the
  furnace. If there is a small hole in the door of
  the furnace heat energy can enter from the
  outside. Inside the furnace this is absorbed by
  the inside walls. The walls are very hot and are
  also emitting thermal radiation. This may be
  absorbed by another part of the furnace wall or
  it may escape through the hole in the door. This
  radiation that escapes may contain any
  wavelength. The furnace is in equilibrium as when
  it absorbs some radiation it emits some to make
  up for this.

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Blackbody Radiation

• A bit of history

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19th Century - Classical Physics

• 1860 Kirchoffs Law
• Kirchhoff states that good absorbers are also
  good radiators. A term blackbody is coined, to
  describe body that absorbs all of the incident
  radiant energy and that as a consequence of his
  law, must be the most efficient radiator.
• 1879 1884 Stefan-Boltzmann Law
• In 1879 Stefan concludes by experimental
  measurements that the total amount of energy
  radiated by a blackbody is proportional to the
  fourth power of its absolute temperature. In 1884
  Boltzmann reaches the same conclusion by applying
  thermodynamic relationships.
• 1894 Wiens Displacement Law
• Wien lays down the general form of equation for
  the spectral distribution of blackbody radiation.
  Unfortunately it agrees with the experimental
  data only at short wavelengths and at low
  temperatures. However useful implications are
  found that relate temperature to the wavelength
  at which the maximum amount of energy is
  radiated.

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20th century Quantum Physics

• 1900 Ultraviolet Catastrophe (Rayleigh Jeans
  Law)
• From the laws of classical physics Rayleigh
  derives an expression that fits the experimental
  data at long wavelengths and high temperatures.
  It predicts that the energy increases without
  limit as wavelength decreases,thus world should
  be filled with X-rays and Gamma Rays. Jeans only
  points out a numerical error in Rayleighs
  expression.
• 1900 Plancks Radiation Law
• Planck observes that Rayleigh-Jeans law seems
  valid at long wavelength while the Wien law seems
  equally valid at short wavelengths. In October
  he proposes his radiation formula that would be
  correct at all wavelength. At the urging of
  fellow scientists he undertakes theoretical
  derivation of his formula to obtain an expression
  for spectral energy distribution of BBR, which
  however results in Rayleigh-Jeans Law, already
  known to be an error. Planck concludes that the
  laws of classical physics are inadequate to
  describe processes at atomic level. He introduces
  the idea that the amplitudes of atomic
  oscillators can increse only in discrete steps,
  differing by the quantity hn , called a quantum
  of energy. On December 14th he announces
  derivation of his radiation law. Thus in less
  than two months the inadequate theories of
  classical physics have been bolstered by the
  concepts of quantum physics.

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Max Karl Ernst Ludwig Planck 23.4.1858
4.10.1947

• Planck gave no explanation of why the energy
  was quantized. Even though others overtook this
  revolutionary concept Planck himself did not
  believe it was physically real and thought he
  only developed a mathematical trick that worked.
  For his discovery he was awarded the Nobel Prize
  for Physics in 1918.

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Blackbody Radiation

• Underlying Physics

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Kirchhoffs Law

• Explains basic properties of objects in
  thermodynamic equilibrium with the surrounding
  environment.
• The incident radiation flux F can be divided to
  three componentsF FAbsorbed FReflected
  FTransmittedThree coefficients describe
  properties of objects exposed to radiation flux
  coefficient of absorption (a), coefficient of
  reflectance (b) and coefficient of transmissivity
  (t). a r t 1
• We can also conclude good absorber (receiver)
  is a good emmiter (source)

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Kirchhoffs Law

• Examples coefficients
• Blackbody a 1, r t 0,
• Greybody a konst lt 1, r 1 a, t 0
• Antireflection coating a t 1, r 0,
• Mirror r 1, a t 0,
• Fully transparent
• t 1, a r 0,
• Generally
• 0 ! (a, r, t ) ! 1

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Steffan-Boltzman Law

• Stefan-Boltzmann's law states that the rate that
  a body emits radiation per unit area is directly
  proportional to the body's absolute temperature
  to the fourth power. This law can be derived from
  the Planck radiation law in its energetical form
  (similarly for the photonic form).
• where se 5,67 . 10-8 W.m-2.K-4 is
  Stefan-Boltzmanns constant

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Steffan-Boltzman Law
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Stefan-Boltzmann in practice...

• The Stefan-Boltzmann constant gives the power P
  radiated from one square meter of black surface
  at temperature T.
• The Stefan-Boltzmanns law enables us  to
  determine the temperature of stars by measuring
  their color spectrum
• Sun has an effective blackbody temperature of
  5555K, thus it radiates 160,000 times more energy
  than the earth (T4).
• The visible part of spectrum contains about 40
  of its radiation. It's spectrum isn't as smooth
  as the "ideal. It is pitted and bumpy due to
  real world conditions including absorption of the
  radiation en route to the earth.
• We receive only 1/2 billion of the sun's total
  energy. This is enough to light New York City
  street lights for 2 days by collecting light for
  12 hours on one football field.

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Wien Displacement Law

• The maximum of spectral intensity changes
  according to the blackbody temperature. The
  corresponding wavelength can be determined from
  Planck radiation law by finding the local
  extremes of the respective functions.
• For its energetical form
• For its photonic form

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Wien Law in Practice

• Wien's displacement law says that the wavelength
  of the maximum emitted radiation is inversely
  proportional to the absolute (K) temperature.
  Allows us to determine temperatures of stars
  depending on its color. Something that glows blue
  hot is much warmer that one that glows red hot (x
  our psychological perception)
• If we put temperatures of the earth (280K) and
  the sun (5700K), we will see that lambda max is
  about 10µm and .50µm, respectively. Whereas the
  earth emits mostly infrared radiation, the sun
  emits mostly visible light. This is no accident -
  evolution has adapted us to see most efficiently
  in the light most readily available.
• This shift in the frequency at which radiant
  power is at its maximum is important for
  harnessing solar energy, such as in a greenhouse.
  We need glass which will allow the solar
  radiation in, but not let the heat radiation out.
  This is feasible because the two radiations are
  in very different frequency ranges - 5700K and,
  say, 300K - and there are materials transparent
  to light but opaque to infrared radiation.

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Planck Radiation Law

• Spectral specific radiant intensity of a
  blackbody at temperature T K can be expressed
  in the energetic form of Plack Radiation Law
• Or
• Models the blackbody radiation curve earlier
  observed in practice. Describes the energy
  density per unit time per unit wavelength.
• C1 W.m2, C2 K.m . 1st and 2nd radiation
  constant

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Planck Radiation Law

• By diving the previous equation by the energy of
  foton (Eh.c/l) we obtain the Plack Radiation Law
  in its photon form
• photon.s-1.sr-1 .cm-2.mm-1
• Where h 6,6256 . 10-34 J.s stands for
  Planck constant, Kb 1,3805 . 10-23J.K-1
  is Boltzmann constant
• c is speed of light in the vacuum

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Planck Radiation Law
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Planck Radiation Law in Practice

• Integrated over wavelength Plancks Radiation Law
  gives total radiated power
• P esAT4
• Radiated Power Stefan-Boltzmanns Law
• Take its derivative to find the peak of the
  distribution
• Wien Displacement Law
• Relate radiated power to energy density in
  photons
• Relate time to temperature to get radiative
  cooling time.

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Blackbody Radiation Curves

• These graphs shows how the black body radiation
  curves change at various temperatures. These all
  have their peak wavelengths in the infra-red part
  of the spectrum (fig.1) The graph shows that as
  the temperature increases, the peak wavelength
  emitted by the black body decreases and moves
  toward the visible part of the spectrum. Also, it
  shows that as temperature increases, the total
  energy emitted increases, because the total area
  under the curve increases. Note that none of the
  graphs touch the x-axis so they emit at every
  wavelength. This means that some visible
  radiation is emitted even at these lower
  temperatures and at any temperature above
  absolute zero, a black body will always emit some
  visible light. (fig.2).

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Planck vs. Rayleigh-Jeans

• For low frequencies..
• For wavelengths

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Blackbody Radiation

• Now back to practice..

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Real Bodies

• Stefan-Boltzmann, Wiens as well as Plancks Law
  were deduced for blackbodies. However, real
  objects have worse radiation properties than the
  blackbodies. Thus their non-dimensional
  coefficient of emissivity e is
• The emissivity of blackbody does not depend on
  direction of the radiation. Generally, selective
  sources are directionally dependent.

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Real Bodies

• The coefficient of emissivity generally depends
  on
• Type of the material of the source
• Properties of its surface (smooth, machined)
• Wavelength (spectral dependence)
• Temperature of the source
• Direction of radiation
• There are three basic types of thermal radiation
  sources
• Blackbody e (l) 1 on all wavelengths
• Graybody e (l) lt 1 constant on all wavelengths
• Selective radiator e (l) f (l). Changes by the
  wavelength. This relation can generally b very
  complicated.

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Real Bodies
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Blackbody in Practice

• Blackbody in practice is realized by heating a
  cavity with a very small orifice to a desired
  temperature. The inner surface is coated with
  black color. Thus radiation entering the cavity
  through the orifice is incident on its blackened
  inner surface and is partly absorbed and partly
  reflected. The reflected component is again
  incident at another point on the inner surface
  and gets partly absorbed and partly reflected.
  This process of absorption and reflection
  continues until the incident beam is totally
  absorbed by the body. The inner walls of the
  heated cavity also emit radiation, a part of
  which can come out through the orifice. This
  radiation has the characteristics of blackbody
  radiation - the spectrum of which can be analyzed
  by an infra-red spectrometer.

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Blackbody in Practice

• Blackbody sources are used for calibration of
  infrared imaging systems and thermometers
• Different designs for different blackbody
  temperatures, according to the use of the
  calibrated device.

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Summary (yes you have done it-)

• It is time to recap that in this lecture you have
  learned and now should understand ...
• what a Blackbody is and why this concept is
  important for infrared system engineering.
• how the underlying physical principles were
  researched.
• the basic physical laws and equations and their
  implications for practice.
• how real bodies differ from the ideal black
  bodies.
• how blackbodies can be modeled.

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References

• Ale Drastich Netelevizní Zobrazovací Systémy,
  Vutium 2001
• Richard D. Hudson Infrared System Engineering,
  Wiley-Interscience 1968
• E.L. Dereniak, G.D. Boreman Infrared Detectors
  and Systems, Wiley-Interscience 1996
• Internet

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Useful tools

• Excel Macro
• Contains calculation of Plancks curve,
  Ultraviolet Catastrophe and Radiation Power
  Output by temperature
• Java Applet
• Plots Plancks curve for a given temperature.
  Provides illustration of Steffan-Boltzmann Law
  and Wiens Law.
• Java Applet
• Plots Plancks curves for different
  temperatures.

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Thanks for your patience