Physics 272 Laboratory Experiments
Dr. Greg Severn
Spring 2016
How big is Planck’s constant? Planck found that the energy spectrum had to
be tiled with discrete energy chunks in order for the average energy per electromagnetic mode to solve the so-called ultra-violet catastrophe. But this involved
invoking, ultimately, a new physical quantity: the quanta of electromagnetic radiation, now called the photon, with an energy quantum of = hν, where h is a
universal constant, Planck’s constant. Planck determined its magnitude by fitting
ideal blackbody radiation curves to actual data. We will do this too. You will
need to assess the correspondence between your model predictions and your experimental measurements (discrepancy) and, separately, assess the extent to which the
measured quantities are reliable in themselves (uncertainty). You will also get the
spectrum of a hot light bulb filament, and try to answer the thorny question of how
the filament and the sun look so alike even though the filament is much colder.
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Spectroscopic measurements of heat radiation, and estimation of
Planck’s constant
3.1
Introduction
In our first spectroscopic experiment of the semester we examine heat radiation. Blackbody radiation was the subject
in crisis which led to the quantum revolution. Quantum theory arose not because of the quantization of matter, but
rather because of the quantization of light. It really is true, as G.P. Thomson once said1 that it is “...seldom that
a scientific conception is born in its final form, or owns a single parent. More often it is the product of a series of
minds, each in turn modifying the ideas of those that came before, and providing material for those that come after.”
The notion of the quantum of radiation had a number of prominent refiners, Einstein chief among them, but the idea
had just one parent, and it is just that the radiation law, and the universal constant that makes the radiation law
apply to the electromagnetic radiation in thermal equilibrium with a surface at temperature T , that is, blackbody
radiation, are named after Planck. In this laboratory, we will collect the heat radiation from the Sun and from an
electric light bulb filament. The Sun is the hottest blackbody radiator in the local neighborhood of the cosmos and
radiates at a rate of ∼ 1026 W, the light bulb, about 10 W. Samples of heat radiation spectra for these is are given
below in figure 1. We will attempt to fit a Planck’s Law radiation curve to the solar data, playing with Planck’s
constant more or less as an adjustable parameter in order to estimate its magnitude from actual data.
Studies performed by the pre-quantum scientists of the 19th Century, of heat radiation intensity data showed
that the wavelength corresponding to the peak of the radiation intensity shortened inversely with temperature,
λmax T = 2.898 × 10−3 m · K,
(1)
a relation known as Wein’s displacement law and that total energy radiated, integrated over all wavelengths, was
proportional to the fourth power of the temperature
Z ∞
ρ(λ)dλ = 5.67 × 10−8 T 4 ,
(2)
o
known as the Stefan-Boltmann law, where the constant where the constant (the Stefan-Boltzman constant) has units
of W/(m2 K 4 ). But the constants had no basis in theory for Planck’s predecessors. Worse, the best of classical
1 From
Nobel Lectures, Physics 1922-1941, Elsevier Publishing Company, Amsterdam, 1965
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spectral energy density (arbitrary units)
2.4
2.2
2.0
1.8
Tungsten Lamp 2856 K
Planck black body theory curves,
1.6
T just about right
T too hot
1.4
T too cold
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
0
500
1000
1500
2000
2500
3000
wavelength (nm)
Figure 1: On top, a measurement of the spectrum of skylight. The Fraunhofer lines (here, A-K)are obvious as narrow
spectral dips in the spectrum. Which arise from molecular absorption in the atmosphere of the sun, and which from
that from the atmosphere of the earth? On the bottom, a spectral energy density measurements from a hot (2856
K) tungsten filament compared with theoretical Planck radiation curves for a variety of temperatures.
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conceptions held that radiation law applying to cavity radiation (the best approximation of an ideal blackbody
radiator), was
8πν 2 kB T
dν,
(3)
ρν dν =
c3
known as the Rayleigh-Jeans law, which used the equipartition theorem to assign and average energy of kB T to
each cavity mode (standing wave) for which the number of modes per unit volume, per unit frequency was known
to be 8πν 2 dν/c3 . Brilliant, and nicely in agreement with the data in the long wavelength limit, but it was horribly
wrong in the short wavelength limit. (The radiant power increases without limit at short wavelength-even a room
temperature object should be able to melt things, all things, in its vicinity from the short wavelength heat radiation,
theoretically , that is. Well, the theory was scandalously wrong–the ultraviolet catastrophe, it was called.
Enter Planck. The theoretical solution to this problem was first derived by him in 1900 and finally understood by
him in 1909 as suggesting that idealized model oscillators within the material absorb and emit radiation in quantum
jumps of discrete bundles of energy in the amount = hν, It was a radical departure from classical concepts. Planck
held that the energy of the bundle was proportional to its frequency rather than the amplitude of the electromagnetic
wave. This left a constant of proportionality to be determined by fitting data for various black body curves. The
constant is a universal constant and is now called Planck’s constant, h = 6.626 × 10−34 J − sec. In Planck’s theory,
the distribution of heat radiation, that is, of electromagnetic energy, per unit volume, per unit frequency, that would
be in equilibrium with the oscillators in the surface of the material at temperature T was
ρν dν =
8πhν 3
1
dν,
c3 ehν/kB T − 1
(4)
ρλ dλ =
8πch
1
dλ.
λ5 ehc/λkB T − 1
(5)
or, expressed as a function of wavelength
Comparing our data with a version of this latter expression of Planck’s energy distribution, since our data is obtained
experimentally as a function of wavelength is the principle part of the experiment. For use in modeling our data,
we will restate ρλ using constants with a view to using data to help us get a feel for the new universal (quantum)
constant h,
C1
1
ρλ dλ = 5 C /λT
dλ,
(6)
λ e 2
−1
by adjusting the value of C1 to adjust the amplitude of the theory curve to fit the data, and C2 to fit the peak and
the shape of the theory curve to the best fit the data. The constant C2 = hc/kB , according Planck’s theory, which
has a value of 1.439 × 10−2 m · K. The scanning monochromator however furnishes wavelength data in units of nm,
and so the value of this constant in those units would be 109 times bigger (think about this!), 1.439 × 107 nm · K.
There are of course non-idealities associated with real blackbody radiators that are not a part of Planck’s model.
But the model did more than explain simple blackbody radiation curves- it contributed new ideas to the stock of
concepts used to understand the physical universe, concepts of over arching importance. So when the observed
notches in the solar blackbody spectrum were looked at with high resolving power, absorption dips were discovered
which could be understood in terms of single photon absorption events, Planck’s light quanta at work again. Josef
Fraunhofer (c. 1814) was the first to observe the notches. After Planck, and the advent of quantum mechanics, one
understood the Fraunhofer lines as the excitation of quantized molecular and atomic energy states. And these can
be explored experimentally while measuring the blackbody spectrum of the Sun.
3.2
Procedure
We will use the Ocean Optics USB4000 to measure the emission spectrum of two hot bodies, the sun (really the
sky), and a hot (light bulb) filament. Using a multi-mode fiber optic cable that couples light into a fixed grating
spectrometer, equipped with a CCD detector, collect as much direct sunlight as possible. Do this for both light
sources. The arrangement of the apparatus just mentioned is shown below in figure 2. Your instructor will indicate
how to collect the data using the Ocean View software.
The software will allow us to capture the spectrum rather directly, obtaining graphs and data files of the spectrum.
Be careful to record the filenames. Once you have obtained these data files (bring a flash drive, or email the files
to yourself and your co-conspirators), it’s all analytical work after that! Once we have collected data files for the
wavelength scans for 300-1100nm, you will want to compare the result to that of an ideal blackbody curve as
dicussed above. Of course there are experimental “issues” that one must take care of before one can really make a
good comparison. Do the following
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Figure 2: The Ocean Optics USB4000 set up. The light source is skylight in the vicinity of the sun. In the second
part of the experiment, the light source will be a hot light bulb filament.
1. This is no small part of the lab as you will soon see (in more ways than one). You can see the published spectral
response function of the CCD detector grating on our public course website[2]. The curve fitting parameters
for this function has been splined with a 5th order polynomial fit of the form
y = Intercept + B1 ∗ x1 + B2 ∗ x2 + B3 ∗ x3 + B4 ∗ x4 + B5 ∗ x5 ,
(7)
, where y is the normalized response function, strictly for the interval 400-1000nm, and x is the wavelength in
nm. Obtain the curve-fitting parameters from the instructors! If only it was a flat function wavelength! But
no, it has to be non-linear! I will try to post a file that describes the detector so you can see what I mean. The
detector is not uniformly sensitive to all wavelengths of light. So, what the CCD records is actually a distortion
of the signal it receives. Not only this, but the grating itself varies with wavelength in efficiency. This too has
to be factored in (or out, depending on the point of view). Our simplistic model for the data is this:
Ir (λ)
=
It (λ) ∗ ccdsr (λ) ∗ gtngsr (λ)
(8)
It (λ)
=
Ir (λ)/(ccdsr (λ) ∗ gtngsr (λ)),
(9)
where Ir (λ), It (λ) and the measured and true intensities, respectively, measured as functions of wavelength,
and ccdsr (λ) ∗ gtngsr (λ), are the response functions, respectively, of the ccd and the grating. The operations
may be completed in the excel files furnished for this experiment. This statement is also true of our eyes.
We will come to this presently. But, to unfold the true(er) signal from the CCD signal, simply divide the
received signal by the normalized spectral response function. One can do this by using the spectral response
curve fitting function, and evaluating it at every wavelength measured. Replot the the two signals on the same
graphs. The CCD signal is what the CCD “sees”, and the unfolded signal is ‘reduced’, by which we will mean
something closer to the real signal as it would be measured by an ideal detector. Record your guess: how did
you think the reduced signal would look? Hotter or colder?
2. Use Wien’s law to estimate T for the data. This will be the temperature used in the and compare this with
the T from the solar black body theory curve. Do all of these measures agree? Explain the differences in their
estimates, citing physical effects.
3. Now plot the theory curve (eq.4) on top of these two curves. You may rescale the peak of equation so that the
theory curve and the experimental data agree, roughly, around the peak.
4. Next, and this will be the primary measurement, vary the constant C2 for the best fit. Systematically vary C2
over a given range (given by the instructor), and from this calculate h ± ∆h. This is the main result of the
lab.
5. Obtain one spectrum for a hot filament, over the same range of wavelengths, and perform the same data
reduction as the the sky spectrum with this difference: take the ’known value’ for C2 , and vary the filament
temperature to obtain a best fit. What’s the best estimate of the filament temperature? With what precision
do you know it? This is the second most important result of the lab. Record your subjective estimate
of the color of the filament.
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3.3
Questions and Calculations
1. Show that eq. (5) comes directly from eq. (4) using the substitution, ν = c/λ. How must we replace dν?
2. Is the percentage discrepancy between h about as big as the percentage uncertainty? How did you estimate
the percentage uncertainty? Furnish quantitative values from your work. Describe also ∆T in the same way.
Indicate if it is assessed in the same way as ∆h.
3. Speculate about the reason for the notches (dips) in the sky spectrum. What physical process can cause them?
State your thinking in your own words. These were first observed by Fraunhofer in the late 18th C.
4. The actual sky spectrum data is systematically below the theory curve in both short wavelength and the long
wavelength ranges, on either side of the peak. Speculate about possible physical processes that could account
for this. Comment briefly about the implications of these effects on global warming.
5. How is it that the filament look like the color of the sun, even though it is less than 1/2 the temperature of the
Sun?
3.4
Ocean Optics USB4000 UV-VIS Spectrometer
The manual for the device and the software will soon be posted, stay tuned, and pay attention to the directions for
use given in lab.
References
[1] See our course web site, experiments folder, for links, and also the public course lab website as well.
[2] See also the document titled, “CCD spectral response function”.
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