I pointed a commercial spectrometer at the sky, see Fig. 1 Collecting this data was spectroscopy! A few important notes before we continue:
- This spectrometer has a dispersive element that was pre-calibrated, so the angle the light hits the detector is automatically converted to a wavelength.
- Spectrometers are not uniformly sensitive. This means that the detector you are using might be more sensitive to red light than blue light. Graphs like Fig. 1 are useful for identifying the wavelengths present in the light, but unless the detector has been calibrated to correct for wavelength sensitivity, the shape may be incorrect. In fact, the real spectrum of the sky goes to far longer wavelengths, but this spectrometer is not sensitive to light at those frequencies.
Fig. 1 The spectral components of light from the sun after it passes through the atmosphere
- When this type of spectrum was first taken in the 1800s, those dips were a mystery, and physicists love a good mystery. Whenever a physicist is presented with data, the first thing they ask themselves is, “Why does the data look the way it does?” If we understand the concepts, we should be able to produce a mathematical model to describe (and predict) the data.
- For this data, there are, at least, two things that need to be thought through. The first is the overall shape. The second is, what are those dips?
Historically, it was a fun journey to figure out the overall shape. Physics known in the 1800s predicted a far different shape. In fact, when challenged with describing the overall shape, physics failed hard. When physicists tried to predict the spectrum in Fig. 1, the model was close for large wavelengths but very, very wrong for small wavelengths. Starting at the larger wavelengths and moving to shorter wavelengths, the math says the spectrum should get larger and larger and larger instead of turning around and getting smaller, which would require infinite energy. The dramatic result of the math is this: the universe doesn’t exist. Whenever the model says the universe doesn’t exist, there is something we don’t understand.1 This result is now known as the “ultraviolet catastrophe.” It is a dramatic name, but the predictions from the mathematical model were also dramatic. We clearly needed a better model!
In 1901, after a lot of thought and model development, German physicist Max Planck eventually figured out that if light was composed of photons, then there would be way fewer higher-energy photons than lower-energy ones, which solved the problem. In 1905, German born physicist Albert Einstein built upon this idea with the photoelectric effect. The mathematical result of this new model predicted the shape in Fig. 1, which is modeled by the complicated formula:
(1)
where h = 6.626 × 10−34 Js is Planck’s constant, c is the speed of light, kB = 1.38 × 10−23 J/K is a constant known as Boltzmann’s constant,2 and T is the temperature of the object. For temperature, we use the unit kelvin, which is named after British mathematician, physicist, and engineer Lord William Thomson. The unit for Boltzmann’s constant is joule/kelvin. Using this model to fit the data from the spectrometer, we can measure the temperature of the surface of the sun to be about 5800 kelvin. We call M(λ, T ) “spectral radiant exitance”, and it tells us how much radiant energy per second is leaving the object per unit area per unit wavelength.3 The units for M(λ, T ) are W/m3. I should note that this equation is for a “perfect” blackbody, which doesn’t exist in real life. Not having a perfect blackbody is equivalent to saying there is no such thing as a room with perfectly parallel and reflective walls. However, it still does a pretty good job with modeling the spectrum emitted by objects.
It isn’t just the sun that emits a blackbody spectrum. A hot pan on the stove does, you do, and I do as well. The amount of light that is emitted as well as the peak wavelength of the blackbody spectrum only depends upon the temperature of the object. Remarkably, everything else in this model is either a constant or wavelength, which is our horizontal axis. Figure 2 shows a few different plots of the spectral radiant exitance for different temperatures. For objects near 5000 K, like the sun, they emit light that is visible to our human eye. For cooler objects, like us, the maximum spectral radiant exitance occurs more near 10,000 nm, which is 10μm. There are special cameras that can see this wavelength of light. You may have seen thermal imaging or watched the Predator movies. I did not include the vertical scale in this plot because the numbers are big and hard to interpret without really digging into the spectral radiant exitance formula. But hotter objects do emit more light. The maximum spectral radiant exitance for a 6000 K object is about 3 million times larger than for a 310 K object.
Fig. 2 Left: Blackbody spectrum for very hot objects. The surface of the sun is 5800 K. Right: Your blackbody spectrum, assuming you are human (310 K is about 98 F). Notice the horizontal axis is a very different scale
The Stefan–Boltzmann Law and Wien’s Displacement Law Imagine you have a blackbody with surface area A. Besides the spectrum of the blackbody, you can also measure the total power emitted by the object. The total power emitted is proportional to the area under a spectral radiant exitance graph. After some math (calculus), the total power emitted by a blackbody:
where σ is a constant known as the Stefan–Boltzmann constant,4 ϵ is the emissivity, and A is the surface area of the object. Emissivity is a number between 0 and 1 indicating how closely the object behaves like a blackbody. If it is a perfect blackbody, ϵ = 1. For a sphere, like the sun, the surface area is 4πR2, where R is the radius of the sphere. The emissivity of the sun is about 0.99, so it is nearly a blackbody.
Several years before Max Planck solved the ultraviolet catastrophe, German physicist Wilhelm Wien experimentally noticed that the wavelength of the maximum spectral radiant exitance graph, see Fig. 2.4, was inversely proportional to temperature. He discovered that a blackbody with a temperature of 6000 K had a maximum around λpeak = 484 nm, 5000 K had a maximum around 580 nm, 4000 K around 725 nm, etc. After studying the trend between temperature and peak wavelength, he determined:
(2)
where b = 2.898 × 106 nm K. This formula was later derived from the spectral radiant exitance formula. If you have the calculus background to find the maximum of a function and want to derive this formula, you should know that, unfortunately, the resulting equation is a transcendental equation and thus does not have an analytical solution. However, you can find a numerical solution.
What About Those Dips? Let’s return to the experimental data we took for the spectrum of the sky, see Fig. 1. Blackbody radiation explains the overall shape, but what about the dips? What do you think is making those dips? The answer is below the fun little puzzle in Fig. 3.
Fig. 32.5 A fun little puzzle to separate the question from the answer.
There are two sources of the dips: atoms and molecules in our atmosphere and the sun itself. For example, the dip near 750 nm is due to oxygen molecules in the atmosphere. There are photons coming from the sun that have the perfect energy to excite oxygen molecules. Because oxygen molecules absorb only photons with the perfect energy, some light is lost at very specific wavelengths before it reaches our spectrometer.
The second source for the dips is the sun itself. While the sun emits blackbody radiation, that light passes through gas at the surface of the sun. That gas absorbs light just like atoms and molecules in our atmosphere. If we wanted just the composition of the sun, we should collect this data from space!
This leads to one method of performing spectroscopy with a spectrometer. You take a light source, which can be a blackbody, a flashlight, or really any light source you want, and send it into your spectrometer. Record this spectrum. Next, place an atomic sample between the light source and the spectrometer and record this spectrum. Comparing the two spectra tells you which wavelengths got absorbed. Each wavelength that is lost from the light due to absorption is the same wavelength needed to excite the atom from one energy level to another. So measuring these absorption dips gives you information about the energy levels of the atoms.
1 I feel like this might be the understatement of the millennium.
2 Boltzmann’s constant was named by Max Planck in honor of the Austrian physicist and philosopher Ludwig Boltzmann.
3 The units are, admittedly, confusing! As a rough interpretation, you can think of these units as telling you how much light is being emitted from a blackbody at a given wavelength. If you look at Fig. 2.3, there is more light being emitted at 500 nm than 700 nm, so the spectral radiant exitance is larger at 500 nm. This is what we mean by “per unit wavelength.”
4 Named after the Carinthian Slovene physicist, mathematician, and poet Josef Stefan, who empirically found the relationship, and Austrian physicist and philosopher Ludwig Boltzmann, who derived the equation.
