| Understanding Kirchhoff's Laws.
The Parable Explained: Electrons in Orbits.
To understand the spectra of the elements, and then later to apply this new understanding to the spectra of the stars, we first have to note a piece of important physics which was a long time in the developing. This important fact is that an atom is mostly empty space, with negatively-charged electrons orbiting a massive, positively-charged nucleus. I will not describe how this was worked out (although that is a fascinating story in itself), but just accept the conclusion.
To start with, I remind you that an atom contains some number of protons (positively-charged particles) in a centrally-located lump called the nucleus, as well as some neutrons (which have no charge) which also reside in the nucleus. The number of protons determines what kind of element it is: hydrogen, the simplest atom, has one proton; helium has two; carbon has six; oxygen has eight; uranium has ninety-two; and so on. The number of protons is called the atomic number .
Around this nucleus, relatively far out, one finds the negatively-charged electrons. Their presence gives rise to a host of electrical effects and helps, in part, to explain the chemical bonds which hold together molecules, crystals, and other substances. In an atom of ordinary matter, there are as many electrons as there are protons in the nucleus, so the atom is electrically neutral. But electrons can be relatively easily pulled off, leaving the atom ionised. The resulting ions are positively charged.
A Simple Visualisation.For our purposes here, we can visualise an atom as being rather like the solar system. Here are the relevant features: In the solar system, relatively low-mass planets orbit around the massive sun. (The sun is about one thousand times as massive as Jupiter, the biggest planet.) In the atom, each proton and neutron has about two thousand times the mass of an electron. This means that we can treat the solar system/atom as though there is an unmoving heavy lump (the sun/the nucleus) at the center around which the lighter pieces orbit. In the solar system, the planets are attracted to the sun by the force of gravity. Their sideways motions keep them from falling into the sun. In the atom, the electrons are attracted to the central nucleus by the electric force, which is trillions of times stronger than gravity.
One Important Difference: No Well-Defined Orbits.In the solar system, the planets move around the sun in well-defined orbital paths. For instance, if you come back a century from now, you can predict exactly where the Earth will be. The electrons, however, do not behave in this way: the modern science of quantum mechanics tells us that we can only assign a certain probability to the chances of finding an electron in a particular location. (For instance, in a hydrogen atom there is a high probability that the electron will be quite close to the proton at the center, a much lower probability that it will be found far away.) Thus it is wrong to think of the electrons as being like little billiard balls orbiting a central massive object in pre-ordained paths. Nevertheless, this is what I want you to visualise, because what I will describe below is most easily understood in these terms. Please bear in mind, however, that we are oversimplifying considerably in this discussion!
And One Critical Difference: Quantized Orbits.Despite the warning I have just sounded, I ask you now to imagine the atom as looking like a small solar system, with some number of electrons (only one, for hydrogen, of course) orbiting the central nucleus, like planets orbiting the sun, in some simple path, which you can take to be a circle for our purposes. The critical difference I introduce now is to tell you that, even if we allow ourselves this hugely simplified picture, there is an additional remarkable restriction: Planets can orbit the sun anywhere they like. That is, with big enough rockets, we could move the Earth itself so that it orbited in some new path, somewhere between Mars and Jupiter, say - or anywhere else. But the electrons in the atom can only orbit in certain fixed orbits, either `here' or `there' but not in between. The electron orbits are said to be quantized. In general, how does one orbit differ from another? To understand this, think of the solar system again. If you wanted to move the Earth out to a larger orbit, you would have to do some real work to drag it farther away from the sun's gravity. On the other hand, a planet farther out, if moving too slowly to maintain its orbit, has a natural tendency to fall in towards the sun, moving faster as it does so - and thus converting some of its `gravitational potential energy' to the energy of motion. In short, you would have to provide energy to pull a planet farther out from the sun; but if you let something fall in towards the sun, that converts some of the gravitational potential energy to other forms which you could use (just as water falling at Niagara Falls can be used to turn turbines and generate electricity). The very process of falling in provides energy. Thus the planets differ in the total energy associated with their motions and positions, and to change a planet's orbit can either require energy (you have to do work to move the planet farther out, `battling' the sun's gravity) or liberate it (the planet falls closer in and releases energy). This analysis holds for the atom too: the electrons are in orbits which are associated with different total energies. To move an electron from one orbit to a bigger orbit requires us to provide some amount of energy (feed energy into the atom, as it were); but if the electron changes from one orbit to a smaller orbit, this leads to a release of energy (which can come right out of the atom, as we will see). For the planets around the solar system, there are various attributes in addition to total energy that we can associate with an orbit - any non-circularity (the ellipticity) of the orbit, for example. Orbits can differ in their total energy and their total angular momentum, and this is as true for electrons `orbiting' a nucleus as it is for planets or comets orbiting the sun. The details, however, are not critical to a qualitative understanding of how the spectra are produced by atoms, so let us simply continue to visualise an atom with a massive central lump and electrons orbiting happily in circles. The most imporant thing to remember is that there are only a limited number of allowed circular orbits which any given electron can occupy!
Why Quantized?There are two ways of asking this question: (i) what do I mean by the word `quantized' itself? and (ii) why does the quantization of some quantity happen within atoms? Let us address these issues in turn. Why do we use the word `quantized?' Well, first let me remind you what it means, by considering a few examples. Your weekly pay cheque is quantized: it may be $351.26 (or more, or less) but it is never expressed in units of fractions of a cent. All dollar amounts are quantized in units of at least a penny. Likewise, your final mark in this course may be 75, or 76; but it will not be 75.34119. The `quantum' in the course (the unit, from the Latin word for `how much' ) is a single percentage point. By contrast, the air temperature outside may be 10 degrees, or 10.3, or 10.671, or anything at all: it is what is called a continuous variable (although of course we usually only quote it to the nearest degree: ``the low tonight will be 7 below''). Your car can travel at 80 km/h, or 81, or something in between - 80.729931102 km/h, for instance. There is no physical law that permits it to be precisely 80, or precisely 81, but forbids any value in between. Contrast that to the gears in your car: you can be in first gear, or second gear, or third gear; but there is no 'gear 1.3743'. The electrons in the atom are quantized in certain ways, of which the most interesting for our purposes is the energy - just as a planet orbiting the sun has a total energy which depends on its position, the size of its orbit. The manifestation is that (for our purposes, although I emphasise that this is a much over-simplified way of thinking of this) the electrons are constrained to move in one of the allowed fixed orbits which we are visualising.
A Homespun Analogy.To understand what comes next, I ask you to consider another of my fanciful analogies, one which I presented in class. In the lecture, I showed you a cartoon of a person on a bicycle, riding around and around on a `wedding cake' arrangement of platforms. Think about the perilous situation of the cyclist for a moment: He is perfectly safe to ride around on his present level (and indeed he could even come to a stop without any problem). But if his attention wanders, he might ride over the edge and go crashing down to the level below, thanks to the Earth's gravity. If this happens, his gravitational potential energy is converted to the `kinetic' energy of motion as he falls; then, on impact, his kinetic energy is converted to other forms: noise, breaking of bones, the bending of the bicycle frame, and so on. Conversely, if he wishes to ride on a higher platform, some agency has to provide the needed energy to lift him up to that level. There is no purpose in lifting him merely halfway: he cannot ride around in empty space, but has to be put up safely onto a higher shelf. This is the analogy I want you to visualise for an individual electron orbiting the nucleus. It may seem strange to introduce yet another analogy, but I believe that (i) the `solar system' model allows you to understand that the orbits themselves are a consequence of the dominant central attractive body (the sun == the nucleus), whereas (ii) the bicycle analogy will help you understand the consequences of the quantized orbits and how they differ in energy.
The Inevitability and Promptness of the Fall.The cyclist is unlike the atom in one very important way. If he keeps his wits about him, he may ride around indefinitely on one of the higher shelves, never falling off. By contrast, any electron which finds itself in a large (energetic) orbit has a natural tendency to speedily drop back down into lower energy orbits, shedding its excess energy as it goes. (It is as though the cyclist inevitably rides off every shelf he lands on, and within a few seconds is right down on the ground, with his bicycle all mangled from the crashes.) When the electron is in the smallest available orbit, the atom is said to be in the `ground state' - easily remembered as being analogous to our poor cyclist down at ground level. As noted, an electron in a high-energy (large) orbit drops down very promptly to lower orbits, perhaps in a few trillionths of a second. And, as I said, it ``sheds its excess energy as it goes.'' You will not be surprised to learn, finally - and here is the punch line at last! - that the energy comes out in the form of photons , our familiar packets of energy. In the case of the cyclist, the energy leads to his injury; in the atom, the energy is merely released, leaving the electron quite intact.
A Diversity of Paths.If you like, you can visualise the poor cyclist toppling from the tenth shelf down to the ninth, then from there to the eighth, and so on, suffering a bunch of relatively small crashes until he winds up on the ground. On the other hand, he might have the bad fortune to ride off the tenth shelf in a way which makes him clear all the others and fall straight to the ground, at which time the impact will release a lot of energy (as much as all the other impacts combined). Intermediate steps are possible: say, a fall from the tenth to the sixth shelf, then to the third, second, and ground. The electron can do this, too, with various probabilities (not every jump is equally likely).
The Spacing of the Shelves.A final point is that the `shelves' in the atom are not equally spaced. As I showed you in class, again with respect to the `wedding cake' arrangments of shelves on which the cyclist is riding, the bottom step is the big one, with successive upward steps being smaller in size. (That is, the lowest shelf is a very long way above the ground, and the second shelf is fairly far above the first; but the spacing becomes gradually less and less as we climb.) You can see what this means. If the cyclist is on the tenth shelf and falls down to the ninth, not much energy will be released, and little damage will be done. But if he falls from the second level to the ground, a lot of energy will be released and he may come to real harm. Now we are ready, at last, to understand Kirchhoff's laws!
Kirchhoff's Second Law: Why Diffuse Gases Produce Emission Lines.Let us think about hydrogen gas for the moment. (Since it has only one electron per atom, it is easy to visualise.) If we start with a cloud of hydrogen atoms, the natural tendency is for any electrons in big orbits to drop down to smaller orbits within their atoms, and in short order we should have all the atoms in the ground state, with nothing of consequence happening. It is not quite that placid, however. In real gases, there is some measureable temperature: the atoms are moving around with some speed, and will run into one another. When this happens, some of the energy in the collision may `bump' the electron up into a higher orbit (just as a speeding car could bump into our beleaguered cyclist and toss him up onto a high shelf). Once up there, the electron will promptly fall back down to some lower orbital level (typically within trillionths of a second, as noted above). In doing so, it emits a photon which carries off the difference in energy between the orbits. There are a variety of transitions, or jumps, available (say, from the sixth orbital to the third, second, and ground; or straight to the ground in one jump). Moreover, each atom will `make up its own mind' about which jumps to make, subject to the probabilities alluded to earlier. Thus a host of hydrogen atoms will yield a host of photons - but only of certain energies! Only the wavelengths corresponding to photons of the allowed energies (the differences from orbit to orbit) will be produced. A jump from the third to the second level produces a red photon of a characteristic wavelength (or frequency, or energy). A jump from the fifth level to the second, however, produces a more energetic blue photon -- again of fixed, determinable energy. This explains the characteristic spectrum of hydrogen emission lines. You can see that a continuous energy supply is needed to keep the cloud of hydrogen gas glowing. Once an electron falls down to the smallest orbit (the atom returns to the ground state), that atom will emit no more light until it once again gets bumped up to a higher energy level. (When this happens, the gases are said to be `collisionally excited.') The extent to which this happens depends on the temperature, of course This can be accomplished by the heating of the gas. Alternatively, in a discharge tube (like a neon lamp), an electric current provides a stream of electrons which pass through the gas from one electrode to another. As these free-flying electrons bump into the atoms, the collisional energy can bump the other electrons - those within the neon atoms - to higher levels. Finally, we can also understand why each element has a characteristic emission spectrum. The electrons in the helium atom, for instance, move in orbits which are of different size and spacing than those in hydrogen because they are orbiting a nucleus of a different sort (in particular, it has two protons rather than hydrogen's one). In addition, each of the electrons of helium is influenced by the presence of the other electron in the atom. The interplay between the component electrons can be very complex in the heavier elements, which contain many orbiting electrons, and the spectra may be less simple in appearance, but the fundamental quantum physics is the same.
Ionization.You can imagine that very energetic collisions might lead to such vigorous collisions that electrons can be knocked right off the atoms. Indeed, this does happen, yielding a gas which contains positively charged ions and free electrons, making the gas what is called a plasma. This is what we find in the sun, where the temperatures are so high that essentially all the atoms are fully ionized (even things like uranium, which has ninety-two positively-charged protons, cannot hang onto its electrons deep in the sun's interior). High in the atmosphere of the Earth, there is a region called the ionosphere: fast-moving cosmic rays (particles moving through space, most of them electrons) and particles in the `solar wind' run into the outer parts of the Earth's atmosphere and ionise some of the gases, although not fully (that is, the oxygen atoms will have lost some but not all of their eight electrons).
Kirchhoff's Third Law: The Formation of Absorption Lines.Consider a great host of photons of all energies (i.e. light of all wavelengths) moving through space in the direction of a group of isolated atoms. To simplify things, visualise only hydrogen atoms; and let us further imagine that the hydrogen gas is quite cool, so that all the atoms are in the ground state, with each electron in the lowest energy orbit. What will happen to the photons as they pass through this gas cloud? The vast majority of the photons will pass by completely unscathed, so an observer on the far side would see the light come through pretty much unaffected. But think of a photon with just the right energy to make the electron in one of the atoms jump up from the ground state to some higher orbit. It may do so, being `absorbed' in the process. Does this mean that light of that particular wavelength is completely lost from the original beam? Not necessarily, because you must remember that the electron, now in a higher orbit, will very promptly drop back down, emitting its excess energy in the form of a photon. But does this mean that the beam is left unchanged? Again, not necessarily. There are two things to consider: Any re-emitted photon will set off in some completely random direction, so that, from the observer's point of view, there is indeed a net loss of light at that particular wavelength. (Some of the photons are re-emitted in the direction of original travel, but most go off in other directions.) In short, the observer will indeed note that there is light `missing' at that wavelength. This explains why we see an absorption line in the spectrum! The second point is that the photons emitted may not be the same as those absorbed. Suppose, for instance, that a fairly energetic photon bumps the electron from the ground state up to the third level. You might get the electron dropping right back down to the ground state, giving back exactly the same kind of photon (moving in some possibly new direction). Alternatively, however, you might find that the electron drops back down in stages, first from the third level to the second (giving off one photon of modest energy); then from the second to the first (giving a second photon of somewhat larger energy, since this step is larger). Of course, the two photons together have to add up to the energy of the one originally absorbed. The net effect of all this is that the original photons at that wavelength are reduced in number as seen by an observer for whom the light is passing through the cloud. Note in particular that it is now easy to understand why the pattern of lines in the absorption-line spectrum is the same as what we would see in emission if we had the diffuse gas just on its own. I hope you agree that this quantum mechanical understanding of the form and function of the atom, coupled with our understanding of photons as little packets of energy which can be absorbed or emitted, explains in a very natural way all of Kirchhoff's laws. It also allows you to understand, qualitatively at least, how we can work out the compositions of the stars from the invaluable information in their spectra. And indeed we can do even more, as we will see.
Why Are There Sources of Continuous Radiation At All?Having read the foregoing sections, you may wonder why there are any "thermal radiators" at all! (Remember that this term refers to objects which emit a continuous spectrum of light at a great variety of wavelengths, as described by the Planck law.) Since every object consists of atoms of ordinary matter, doesn't this mean that we should see nothing but emission lines? When we heat a wire in a light bulb to incandescence, why doesn't it give off the few well-defined wavelengths which are characteristic of the tungsten metal of which it is made? Use of a prism will reveal instead that it is indeed a continuous spectrum, with light of a broad range of colours. The answer has to do with the density of the material. (You may recall that Kirchhoff's Second Law says specifically that low-density gases give an emission-line spectrum when heated.) In oversimplified terms, the explanation is that the emission of a photon cannot take place in idealised isolation whenever atoms are closely packed together. The energies which are released as the electron leaps from orbit to orbit or from one energy state to another are strongly influenced by the presence of other atoms, electrons, protons, and so on, with the net result that the well-defined energy you naively expect can be "smeared out" to some other value. Energy is also released when the hot, dense material changes in other ways -- for instance, the way in which a hot lump of metal is 'jiggling' internally may change, with a net release of energy in the form of photons. These changes are more complex to describe and understand than the simple changes of orbits of electrons within atoms, but must be considered in working out the net emission of energy. The standard Planck form of the continuous spectrum can be understood from the summed effect of just such processes. Only in rarefied gases, where the atoms are well-separated, do we get simple emission-line spectra. That's why neon lights are in vacuum tubes! Previous chapter:Next chapter
0: Physics 015: The Course Notes, Fall 2004 1: Opening Remarks: Setting the Scene. 2: The Science of Astronomy: 3: The Importance of Scale: A First Conservation Law. 4: The Dominance of Gravity. 5: Looking Up: 6: The Seasons: 7: The Spin of the Earth: Another Conservation Law. 8: The Earth: Shape, Size, and State of Rotation. 9: The Moon: Shape, Size, Nature. 10: The Relative Distances and Sizes of the Sun and Moon: 11: Further Considerations: Planets and Stars. 12: The Moving Earth: 13: Stellar Parallax: The Astronomical Chicken 14: Greek Cosmology: 15: Stonehenge: 16: The Pyramids: 17: Copernicus Suggests a Heliocentric Cosmology: 18: Tycho Brahe, the Master Observer: 19: Kepler the Mystic. 20: Galileo Provides the Proof: 21: Light: Introductory Remarks. 22: Light as a Wave: 23: Light as Particles. 24: Full Spectrum of Light: 25: Interpreting the Emitted Light: 26: Kirchhoff's Laws and Stellar Spectra. 27: Understanding Kirchhoff's Laws. 28: The Doppler Effect: 29: Astronomical Telescopes: 30: The Great Observatories: 31: Making the Most of Optical Astronomy: 32: Adaptive Optics: Beating the Sky. 33: Radio Astronomy: 34: Observing at Other Wavelengths: 35: Isaac Newton's Physics: 36: Newtonian Gravity Explains It All: 37: Weight: 38: The Success of Newtonian Gravity: 39: The Ultimate Failure of Newtonian Gravity: 40: Tsunamis and Tides: 41: The Organization of the Solar System: 42: Solar System Formation: 43: The Age of the Solar System: 44: Planetary Structure: The Earth. 45: Solar System Leftovers: 46: The Vulnerability of the Earth: 47: Venus: 48: Mars: 49: The Search for Martian Life: 50: Physics 015 - Parallel Readings.
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(Monday, 23 April, 2018.)