Download Princeton University, Physics 311/312 Blackbody Radiation, Page 1

Princeton University, Physics 311/312
Blackbody Radiation, Page 1
BLACKBODY RADIATION
Introduction
A “black body” is an object that absorbs all radiation falling on it. According to
Kirchoff’s Laws, such a perfect absorber must also be a perfect radiator; no other body at
the same temperature can radiate more energy per unit area in any given range of wavelengths. A small hole leading into a heated cavity makes a good experimental realization
of a black body. The flux of energy radiated from the hole is proportional to the radiant
energy density inside the cavity, which will be very nearly in thermal equilibrium.
The spectral distribution of black body radiation was a subject of great interest in the
19th century. The spectrum could be partially understood by means of thermodynamic
arguments, and many physicists felt that a complete understanding should be possible
using the techniques of statistical mechanics. The Stefan-Boltzmann Law, derived from
thermodynamics, states that the power per area radiated by a black body is proportional
to the fourth power of the temperature,
P = σT 4 ,
where σ = 5.7 × 10−8 Watts/K4 m2 . The proportionality can be verified by this experiment, although the magnitude of the constant cannot be determined.
A more detailed relation, also derivable from thermodynamics, is Wien’s displacement
law. It states that the surface brightness of black body radiation must be of the form
Bλ (T ) = T 5 f (λT )
where f is some unknown function of the product of the wavelength and temperature.†
Note that this law contains the Stefan-Boltzmann law, so that verifying it verifies Stefan’s
Law as well.
Attempts to derive the form of the function f (λT ) from classical statistical mechanics
met with total failure. Wien derived a fairly good approximation by making some plausible
(but unsupportable) assumptions; his approximation predicted too small an energy density
at long wavelengths, however. Rayleigh and Jeans used classical statistical mechanics to
derive a formula that fit the data in the long wavelength region but diverged in the shortwavelength limit. Its integral also diverged, meaning that it predicted an infinite radiation
density. This difficulty was referred to as “the ultraviolet catastrophe.”
Planck managed to patch up Wien’s formula by including a “−1” in the denominator
(see equation below), making it agree with experiments over the entire wavelength range.
† Bλ has units of Power/(area wavelength steradian) and is called a surface brightness,
a brightness, an intensity, or a radiance. Mathematically, it is a distribution. One often
sees the wavelength replaced by frequency.
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When he tried to justify this new formula, he found that he was forced to assume a proportionality between the energy radiated by an “atomic oscillator” and the frequency of the
radiation. This was the origin of the quantum hypothesis. Planck’s formula, expressed as
power per unit area per unit solid angle in wavelength range dλ, or simply the “brightness,”
is
2hc2
dλ .
Bλ (T )dλ = 5 hc/λkT
λ (e
− 1)
It contains one new constant, h, now known as Planck’s constant. It was of course a
triumph when the same value of this constant appeared in Einstein’s photoelectric equation
a few years later.
The Black Body
The blackbody used for this experiment is an oven with an effective cavity emissivity
near 1.0. An on/off control loop (servo system) supplies current to heaters to keep the
temperature constant. It takes about six hours to reach a temperature that is stable to
±5 K. While doing the experiment, check the temperature every twenty minutes or so.
The temperature is sensed with a platinum/platinum-rhodium thermocouple. The
voltage across a junction of two dissimilar metals is a reproducible function of temperature.
Typically, a reference junction is submerged in an ice bath though in this case we measure
its temperature an compensate for it. The calibration for the thermocouple is mounted on
top of the oven.
Infra-red Spectrometer
The spectrometer is a prism monochromator using a Wadsworth mirror that ensures
the prism is always used at minimum deviation. It is similar to the one described by
Harnwell and Livingood (1933). The prism is made of fluorite (calcium fluoride) with an
apex angle of 59.642◦ ± 0.002◦ . The focusing elements are concave mirrors. The slit widths
are set at 0.25 millimeters and are not adjustable. The spectrometer is mounted inside a
sealed tank with chemical absorbers to minimize the absorption of infra-red light by CO2
and H2 O molecular bands. The window through which the radiation enters the tank is
made of fluorite. The prism is rotated by a micrometer tangent screw that turns it by
0.002◦ per division.
Detector
The detector is a bolometer made of a strip of palladium leaf 10−5 cm thick and
blackened with acetylene soot. Radiation entering the spectrometer is chopped at about
25 Hz by a sectored rotating wheel; when radiation is absorbed by the blackened palladium,
its temperature increases and thus its electrical resistance rises. A battery drives a steady
l0 mA current through the palladium, so the increased resistance causes a larger voltage
drop across the strip. The resulting AC signal is capacitively coupled to a transformer and
then to a Keithley high gain preamplifier and a Princeton Applied Research JB-4 lock-in
amplifier.
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Schematic of experiment. Certain wavelengths of black-body radiation emitted from
the oven are selected by the spectrometer and detected by the bolometer. To make
sure that only radiation from the oven is counted, a chopper modulates it at 25 Hz.
A lockin amplifiers is then used to “lock-in” on the 25 Hz component of the total
detected signal.
Spectrometer Acceptance
Black body radiation is focused onto the entrance slit of the spectrometer so that
the area of the slit is the effective radiating area of the black body. The solid angle is
determined by the aperture of the prism, as viewed from the entrance slit. The signal is
proportional to these two quantities and to Bλ (T )∆λ, where ∆λ is the wavelength range
passed by the spectrometer. This range is proportional to the slit width, but also depends
on the dispersion, dn/dλ, of the prism. The bigger dn/dλ is, the more the spectrum is
spread out along the plane of the exit slit, and the smaller will be the range of wavelengths
that pass through the exit slit to the bolometer. A formula for the wavelength range, ∆λ,
is given by Taylor (1959) and is computed by the program INDEX ETC.C. The second
term in Taylor formula refers to diffraction effects that become negligible with slits as wide
as the ones in this spectrometer.
Experimental Procedure
1 Direct the Helium Neon laser beam (λ = 632.8 nm) into the spectrometer. It is
bright enough that you can see the light scattered from the mirror surfaces. Set the
micrometer to give the prism angle which causes the beam to go through the exit slit.
Also note the relative humidity in the spectrometer.
2 Now you can set up the PAR JB-4 lock-in amplifier. Turn on the chopper motor to
provide a reference signal. Set the lock-in to the SEL EXT (selective external) mode,
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because this guarantees that the reference signal to the phase-sensitive detector will
be symmetrical.
3 Tune the main frequency dial to maximize the meter reading with its switch set to
REF. Keep the LEVEL set low enough that the meter reading is 75 or lower. After
the peak is found, set the LEVEL to give a meter reading of 50–80.
4 Feed the reference signal through a 10 dB attenuator into the signal channel of the
lock-in. Set the meter switch to SIG and tune the FREQ TRIM knob for maximum
reading. Adjust the SIGNAL LEVEL to a convenient point.
5 Be sure that the preamplifier and the bolometer bias current are turned on, and
set the meter switch to “OUT ×1” with the signal and reference cables connected
for normal operation. The laser provides a strong signal, and the PHASE may be
adjusted to maximize it. The SIGNAL LEVEL must, of course, be set appropriately.
The FILTER time constant must be set to keep noise manageable; if you use a long
time constant, adjustments must be made very slowly.
6 Adjust the micrometer (always moving toward increasing readings) to maximize the
signal from the laser. This gives a fiducial setting that, when combined with the
characteristics of the prism, will enable you to convert micrometer readings to angles
of incidence on the prism—and then to wavelengths.
7 Start heating the oven. You should take data at three or four temperatures. In the
region of the peak, points should be fairly close together; toward the long wavelength
tail they can be farther apart. Judge the appropriate spacings by how much the signal
is changing. Use a long enough filter time constant to keep noise reasonable, and allow
enough time for the readings to settle down after changing the prism angle. If you
make a rough plot of your raw data, you will be able to see how things are going while
there is still an opportunity to take another point or two, as might be needed.
Data Analysis
You will probably want to use a computer to analyze your data for this experiment.
A program called REDUCE BBODY DATA.C has been written for this purpose, and is
available from L. Page. Some documentation for the existing program is in the code. If
you use this program, you should make sure that you understand how it works. You should
feel free to copy the program and modify it to suit your needs more precisely. If you need
help using the computer, ask!
Your write-up for this experiment should include three plots. The first is just plot
for one temperature of the raw data – micrometer reading vs. lockin output. The second
is a plot of intensity (corrected for spectrometer acceptance) versus wavelength for each
of the three temperatures at which you took data. You’ll find that there’s an advantage
to plotting all curves on one set of coordinate axes, as this will best show the effects of
the molecular absorption bands of CO2 and H2 O. In a third plot, present all of your data
points on a “universal plot,” to verify Wien’s displacement law. For the abscissa of this
plot, use λ/λm , where λm is the wavelength at the peak of the spectral distribution. You
can calculate λm from the temperature by using λm = 2897.79/T , where the wavelength is
in microns and the temperature is in Kelvin. You should also use your data to derive this
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result (with error bars). The proper choice of ordinate should cause all of your points to
fall on a common curve. A tabulation from Planck’s formula is available in the laboratory.
After an appropriate normalization, compare the data in the universal plot with Planck’s
law.
Questions to ponder
• Can a perfect mirror at 1000K radiate? What would you measure if the inside of the
oven were lined with mirrors?
• How much power is being radiated into the spectrometer by the oven?
• How do the optics work? Why is the Wadsworth prism/mirror needed?
• What is the effect of diffraction on the measured spectrum? What is the effect of
absorption by the window and prism?
• How was the frequency of the chopper chosen?
• How does the ‘steradian’ in the brightness formula relate to the instrument? Does it
change as the prism rotates?
• Why use a lock-in amplifier?
• Is the bolometer linear? How would you check?
• How does a thermocouple work? Could you draw a schematic of the circuit?
• At what wavelength is the peak intensity of a 300K black body?
• How would you “verify” Wien’s law?
References
Consult the texts by Harnwell and Livingood or Richtmeyer, Kennard, and Lauritsen
(or other texts) for the theory of black body radiation. Harnwell and Livingood also
describe an experiment that is pretty close to what we have. The “Taylor Manual,” which
relates prism angles to wavelengths, is available in the library or you can get a program
from L. Page. Coblentz (1914) presents a description of early research on black body
radiation at the National Bureau of Standards.
Harnwell and Livingood, 1933. Experimental Atomic Physics, (New York: McGraw
Hill), pp. 41-82.
Richtmeyer, Kennard, and Lauritsen, 1955. Introduction to Modern Physics, 5th
edition (New York: McGraw Hill), pp. 106–132.
Tables for the Black Body Radiation Experiment, Advanced Lab.
The Taylor Manual of Advanced Experiments in Physics, pp. 117-119, Wesley (1959).
Coblentz, 1914. Bulletin of the National Bureau of Standards, 10, 1.
Revised January, 1990, J. H. Taylor
Revised Sept., 1995, L. Page