The Bohr atom

In this post, I hope to describe the Bohr model of the atom. In the next post, I will use the Bohr model (together with the nature of light discussed in the last few posts) to predict the existence of "spectral lines," which will finally bring me back to dark matter by explaining exactly how we measure the speed of those rotating galaxies (see the Dark Matter Intro link at the right if this is not familiar). Historically speaking, I'm presenting this material backwards, as the observation of spectral lines came first and the explanation came later, but I will proceed anyway.

Niels Bohr is in many ways the father of quantum physics, if not its prime mover. He came of age before the revolutionary wave of the 1920s, but almost all of the physicists involved in developing quantum mechanics (Heisenberg, Dirac, Pauli to name a few) spent some time at the Institute of Theoretical Physics he founded in 1921. His model of the atom was a precursor to all of the discoveries of quantum physics to follow.

So what is that model? Although it turns out not to be accurate, it's a pretty good start, and I would guess that the Bohr model is generally the picture most of us have in our minds for the atom today. Analogous to our picture of the solar system, the Bohr model imagines a dense, very small nucleus, surrounded by orbiting electrons (I took the picture from a website at Jefferson Lab).

By itself, that isn't all that interesting - the real theoretical interest of the Bohr atom was that the electrons are constrained to lie at specific orbits. In the solar system, the planets could theoretically lie at any radius - we could take the Earth, move it a little farther away from the sun, and it would still orbit contentedly (we might all be a bit colder, but the orbit would be fine). In Bohr's atom, an electron can't move to a slightly larger orbit; instead, it would have to jump to the next available orbit. The analogy is that the Earth can only be at our orbit or Mars' orbit, but nowhere in between.

To take this a step further, we can add energy considerations. The farther away from the nucleus, the more energy an electron has. Therefore, whenever an electron switches orbits, it gives up or takes in energy (depending on whether it's heading out or heading in). If it is only allowed to be in certain orbits, then the possible energy steps are discrete - it can only take in or give up a very specific amount of energy. For one more analogy, suppose my mom is on an elevator on the ground floor. If she goes up in the elevator, she can only get off on floors, she can't get off in between floors. And, as she goes up, she picks up a specific amount of energy (which she could give back if she were to jump out a window - she would be rather more regretful if she jumped out a 4th floor window as opposed to a 1st floor window because of all the energy she picked up by going up the elevator). The electrons in the orbit of the Bohr atom are like my mom on the elevator - they can't get off between floors and the amount of energy they can pick up or give out is discrete and fixed.

The very astute reader might see a similarity between this model and the photoelectric effect of Einstein from a few posts ago, when light could carry a specific and discrete amount of energy depending on its frequency. In fact, this is exactly the same thing - how does the electron gain or give up energy? By absorbing or emitting photons (in the photoelectric effect, an electron absorbs enough energy to jump off the surface of the metal, or be "freed" from the orbit). This has significant results for observations of atoms as we'll see in the next post. For now, though, I'll put up one more diagram, similar to the one above, containing the positively charged nucleus at the center, and an electron that can be in one of three "energy levels" (or floors) by emitting or absorbing a photon (the wavy line). Also, for a little interactive demonstration of the same thing, try this.

(Courtesy of the Department of Energy)

The Double-Slit Experiment (quantitative) - Part 2

(click on the picture for a larger view)

Now that we know what a sine wave is, we can understand the double-slit experiment. I need to start with a few definitions that I probably should have put in the last post: the wavelength is the length between two successive peaks in the wave (often represented by λ) and the amplitude is the height of the wave (we'll call it A). There is a symmetry property of the wave; if you shifted the wave to the right or left by its wavelength, it would look exactly the same. In fact, you could shift the wave by any integer times the wavelength, and you wouldn't be able to tell the difference. This will be important later on.

To understand the double slit experiment, we need to ask what happens when two waves overlap. The answer depends on their "phase," or where each wave is in its oscillation relative to the other. For example, suppose two waves are perfectly "in phase," so that when one wave is peaking, so is the other. When you add these two waves together, you'll get a wave that is twice as big in amplitude.

What about when the waves are "out of phase" so that one is all the way up when the other is all the way down? In that case, the waves destructively interfere so that the addition contains no wave at all.

This interference is the key to the double-slit experiment and allows us to predict the shape of the light pattern on the screen. When the light impinges on the slit, the waves that come come out the other side are initially in phase. If you look at the the screen directly across from the slit, you would see a dark spot. However, that bright spot will be banded by bright spots, which will in turn be banded by two more dark spots in a fringe pattern. To understand why, let's zoom in on the slit right where the light passes through (on this lovely diagram I stole from here).

Here, the distance between the slit and the wall is L and the slit separation is d. Where on the wall do you expect to see bright or dark spots? If we want there to be a bright spot on the wall, then we know the two waves must interfere constructively (the first case discussed above) or be in phase. A dark spot will appear when the waves are out of phase and interfere destructively. Let θ (again, angles are always θs) be the angle between the horizontal and the position of a given fringe on the wall). If we look high on the wall, the light that came out of the top slit doesn't have to go as far as the light that came out of the bottom slit (in other words, r₁ is bigger than r₂). This means that the bottom ray of light will have more time to trace out its wavelength and will drop out of phase with the top ray of light. The extra distance traveled by the bottom ray is equal to d*sin θ (remember that the sine function also related the sides of a triangle and notice that the light paths r₁ and r₂ form a triangle with the slit). Now, remember the symmetry of the wave - a wave that is shifted by its wavelength looks the same. So if the extra distance traveled by the bottom wave equaled exactly its wavelength, it would look identical to the top wave, and the waves would interefere constructively - a bright spot would appear on the wall. If, on the other hand, the extra distance traveled was exactly half a wavelength, so that the bottom wave had just enough time to get out of phase, the two waves would interfere destructively and a dark spot would appear on the wall.

This is exactly what happens - bright spots appear if d*sin θ = λ or some multiple of λ, while dark spots appear if d*sin θ = λ/2. Wave properties predict exactly the patterns that appear in the double-slit experiment, confirming that light is like a wave.

The Double-Slit Experiment (quantitative) - Part 1

I want to try and explain some of the math behind the double-slit experiment discussed in the last post. The goal here is not to explain the weird nature of light mathematically, which is beyond the scope of a blog. I do want to show how the double-slit experiment proves light behaves as a wave quantitatively and give an example of how math can be used to explain the results of an experiment.

After a brief discussion with my mom, I realize that I will have to start by explaining what the sine function is (hence the "Part 1" in the title. For the reader who knows what the sine function is, I apologize. Hopefully you will enjoy this post anyway [I always like reading about something I know, it's egotistically gratifying and maybe there will be some interest to be found here from a pedagogical standpoint]). From looking at some of the comments, I fear that just the mention of something called a sine function will cause eyes to glaze over, so let me try and explain why I think it's cool. Math is a language, and each additional element in the language expands the scope of what you can talk about. For example, English with just nouns and verbs would be a boring language ("I wrote"). This is like math with just multiplication. But when you include adverbs and adjectives, all of a sudden you can say something interesting ("I wrote a fascinating post on math, and everyone unanimously agreed that I was the best blogger around who specializes in explaining physics to his mother."). In math, it's the same way; the sine function is a tool that enables a discussion of a whole host of things that were previously unavailable, and in particular, waves.

Let's start with a circle like the one shown and draw a line from the center of the circle to the edge. I'm going to trace out the circumference of the circle with this line. At any one time, the angle between that line and the horizontal axis is θ (my mom will ask about the variable names again, for whatever reason angles are always given Greek letters, and θ is always the first one given), and the projection of that line on the horizontal and vertical axes are x and y respectively. I'm particularly interested in the vertical projection, y (hence the color). Initially, when θ=0, the line is entirely horizontal, and y=0. As θ increases, then so does y, until reaching its maximum value when the line is entirely in the vertical axis. Then y decreases before reaching 0 again, and then goes negative, before finally returning to where it started. We can imagine going around the circle again and getting the exact same thing.



Now plot y as a function of θ, and we get a wave.
This is the sine function - or more technically, the ratio of y to the radius of the circle (we could have performed a similar exercise for the x coordinate and obtained the cosine function, which is [clearly] a very close relative of the sine). It describes a wave as well as circular motion. It also represents a relationship between the sides of a triangle (the alert reader will have noticed that x, y and the radius created a triangle for each angle, suggesting that the sine of an angle relates the length of the sides of a triangle to the hypotenuse). Among many other things. All in all, it's really useful.

What is the nature of light?

The first non-introductory post was on the Doppler effect, which is relevant to dark matter because of its applications to galactic rotation curves and the evidence those curves provide for the existence of dark matter. I figure I can keep mining this particular bit of evidence for a series of posts explaining exactly how we measure galactic rotation curves and why that implies the existence of dark matter. To do that, I need to start with one of the mysteries of physics, the "wave-particle duality." I expect I'll need a few tries at explaining this (since I have some trouble understanding it myself), but we'll see how it goes.

A Google search for the wave-particle duality will give you many interesting discussions and it has been commented on throughout the years by people who are eminently smarter than I. Therefore, I will try not to be too clever, but instead just describe qualitatively what is going on. Essentially, fundamental particles like photons, electrons, etc. behave as both waves and particles. The debate between these two views of the world goes back as far as Newton (who suggested light was "corpuscular") and Huygens (who thought it was a wave), and the problem lies in the fact that there is evidence for both.

Light as a wave
In the 1800s, Young and Fresnel performed a series of double-slit experiments. In these experiments, a light source is directed towards a plate with either one or two very thin slits in it, and the experimentalist observes the resulting pattern that forms on a screen behind the slits. In the single-slit case, the picture that forms on the screen shows a diffraction pattern (i.e. alternating bright and dark spots) instead of a single bright spot as you might expect if the light was a point-like particle. In the double-slit case, the resulting pattern shows signs of interference (I got the nice illustration from the Wikipedia entry, which is very good as usual).

Both of these results can be explained if one thinks of the light as a wave - if a flat wave (like waves at the beach) comes across a barrier with a small hole, the part of the wave that passes through the hole begins propagating outward in all directions, with circular wave fronts (you can test this in a bathtub if you are interested). Therefore these results from experiments performed in the early 1800s seemed to prove definitively that light was a wave.

Light as a particle
Unfortunately, Einstein came along and in one year, 1905, published three of the most famous papers in physics history. And while his work on special and general relativity are monumental accomplishments, in the end, he received his Nobel prize for a theoretical description of the photoelectric effect that showed that light is a particle.

The photoelectric effect is the phenomenon that when light is shone on certain metals, the metal emits electrons. Naively, using the wave theory of light, one might think that the brighter the light (i.e. the bigger the wave), the more energetic the freed electrons would be. In fact, the energy of the electron is independent of the intensity of the light (although brighter light does correspond to more electrons). Instead, it is the frequency of the impinging light that determines the energy of the freed electron, and if the frequency is low enough, no electrons are freed at all, no matter how much light is blasted against the metal.

As we've already discussed, light has wave properties. It's wavelength, the distance between two wave peaks, determines the color of the light, which is how we see the many colors of the rainbow. It turns out that violet and blue light have shorter wavelengths while red light has a longer wavelength. Frequency is essentially how often a wave passes a given spot, and since the speed of light is constant, the frequency is inversely proportional to the wavelength, so violet and blue light have higher frequencies than red light. Therefore, by the photoelectric effect, one can imagine a metal for which the smallest amount of blue light would free electrons, but a Sun's worth of red light would have no effect.

Einstein figured out this dilemma by building on an idea of Max Planck's from a few years earlier. Basically, he suggested that light was actually a discrete particle (a quantum, or photon) with an energy proportional to its frequency (for the purposes of dark matter, this is really the only statement that you need to take away from this long discussion on light - it's energy is proportional to its frequency, which is in turn inversely proportional to its wavelength). If the photon has enough energy, an electron is released from the metal; if the photon does not have enough energy, then no electrons are released, no matter how bright the light. This proved that light was a particle.

Everything is both wave and particle
In the end, physicists have had to accept that light seems to have both properties. One might think that the slit experiments can be explained if the light was a wave made of multiple photons (like a water wave consists of many water molecules). However, experiments in the 20th century showed that even when the intensity of light hitting the slits was reduced to individual photons, the interference patterns still emerged. This is really, really weird. From our macroscopic experience, we would think that a single photon would pass through one slit or the other. Instead, it seems as if the individual photon actually passes through both. Basically, our physical intuition loses its way; individual light quanta are both waves and particles.

Another (brief) philosophical tangent

In a comment on the quantitative Doppler effect post, my mother had the following to say:

"Mom again. I have a feeling that however clearly you explain it, some people who have never taken advanced math in any form will never really understand it. I like much better the idea of the dark matter, the neutrinos racing through my finger-tips, etc. Perhaps you should select your subject-matter differently - when you say you are a physicist, what questions do people at cocktail parties ask you? I'm sure not about variables. The best stuff is the underground machine etc. You may think I am trivial or superficial-minded, but I speak honestly."

I understand her point, but I think I'd like to keep trying with the quantitative posts every now and then anyway. The problem I have with separating this blog from the quantitative aspects is that without the math behind it, physics is reduced to a matter of faith. It's nice to talk about neutrinos going through fingernails and underground mines, but at a cocktail party I inevitably have to say something like, "trust me" or "you'll just have to believe me." We can't actually feel neutrinos going through our fingernails. With this blog I'm trying to show that the rather general, romantic and literally intangible idea of dark matter is actually based on years and years of physics research and that most of what goes into it is fully understood and not a matter of faith at all (although, I will be the first to admit that much of it is speculative - after all, we don't truly know what we are looking for, just that it is there).

As physics arguments are generally expressed in mathematical terms, I think I'll keep the math arguments in there from time to time, even if my mom generally ignores them. I think she could understand it if she really wanted to and it was explained well (which this blog probably will not do as it's probably the wrong vehicle anyway), but in the end, it really doesn't matter to her if she can derive an expression for the Doppler effect or not, does it? The main point that I would be trying to illustrate, then, is that such a derivation exists and could be understood.

The Doppler effect (quantitative)

The first version of the qualitative post contained a paragraph at the end in which I did some real math (I have since removed that paragraph, as it appears in a different form in this post). My mother loved the bit about the tennis and thought she had really grasped the general idea; alas, when confronted with a paragraph containing algebraic variables, she felt somewhat bewildered and lost because I hadn't given it enough of an introduction. I was reminded that she hasn't really done any advanced math in several years. Mom, I apologize, and I'm going to take some time now to talk a bit about the philosophy of mathematics in physics because I do plan on using math in this blog whenever it's applicable (which will be, presumably, often, as this is a physics blog).

First, I've titled these two posts as "qualitative" and "quantitative." This is a standard and very useful division in physics. When discussing a subject "qualitatively," one is really trying to grasp the basic idea or gain a sense of intuition. Then, after gaining that first level of understanding, one begins to approach a problem "quantitatively" by actually working the math. I would not be surprised if many of the subjects I broach in this blog will be handled in a similar fashion.

Next, I want to say that one purpose of introducing abstract variables in a mathematical treatment of a problem is to generalize the solution. For example, in the last post, I tried to explain the Doppler effect using a very specific situation. If my mom is hitting tennis balls every two seconds, then decides to run towards the ball machine, she will have to hit tennis balls at a faster rate. We've solved the Doppler effect for that one case. But that situation doesn't apply if we wished to talk about listening to a police siren on a street corner or the rotational speeds of galaxies in an argument about dark matter. That's why the abstraction of math is so useful in physics - we can deal with the problem in such a way that we can use the same language and solution in each instance.

Finally, I'm going to be using variables to represent various parameters in the problem. Specifically, I'm going to use the following:
f₀: the frequency at which the balls are emitted by the ball machine
v: the speed or velocity of the tennis balls
v_r: the speed or velocity at which my mom runs
f: the frequency at which she hits each tennis ball.










The choice of these variables is completely arbitrary; I could have used anything to represent the quantities of interest. In general, however, we try to use variables that are easily associated with what they represent, like "v" for velocities and "f" for frequencies. The subscripts are then used to delineate different quantities of the same type. f₀ is given a "0" because in a sense, it is the original frequency or the initial frequency. My mom's velocity is given the subscript "r" because she is the receiver, in contrast to the source (if the ball machine were moving, I would have described its velocity as "v_s").

If there are any questions or comments about this introductory stuff, please do comment below.

Onto the problem. Given the variables defined above, I want to define a couple more terms. If the machine spits out balls at a frequency f₀, then the time between each ball, T₀, is 1/f₀ (in the example, I said that the time between each ball was 2 seconds, so the frequency is then 1/2 per second).

In the time elapsed before the next ball is fired, the previous ball has traveled a distance equal to its speed times the time elapsed, or v*1/f₀. This is the distance between each ball.

To determine the time between each ball that my mom observes, we will need the absolute distance between balls (v*1/f₀) and the speeds of both my mom and the balls. At the instant when my mom hits a ball, she is v*1/f₀ away from the next ball. However, she is still running towards that ball, and the ball is still moving towards her. Therefore, that distance will be covered by the combination of her running towards the ball and the ball moving towards her, i.e. with a velocity equal to the sum of her velocity and the ball's velocity, v+v_r. Therefore the time it takes for her to see the next ball is the total distance divided by the total velocity,


T = v * 1/f₀ * 1/(v+v_r).


To calculate the frequency at which she observes each ball, we invert that time, so


f = (v+v_r)/v*f₀.


In the example, f₀ = 1 per 2 seconds, v = 12 m/s, and v_r = 12 m/s, so f = (12+12)/12 * 1/2 = 1 per second, which is exactly what we saw. However, this equation is now general for any similar situation. The equation can be generalized still further to take into account a moving source as well, in which case


f = (v+v_r)/(v+v_s)*f₀.


Now we can talk about listening to sirens on a sidewalk or galactic rotation curves and use the same equation to represent all 3 situations.

The Doppler Effect (qualitative)

In my first post, I talked about how the Doppler effect is a shift in the observed frequency of a wave caused by the relative motion of a source and an observer. In this, my first detailed post, I will try to explain how that actually works. As my mom plays tennis, and this blog is ostensibly aimed at her, I'm going to use a rather tortured tennis analogy.

Suppose my mother is using a ball machine to practice her ground strokes. The ball machine spits out a tennis ball every 2 seconds and each ball moves at the same speed. To use some real numbers, a tennis court is about 24 meters long from baseline to baseline, and let's assume that the balls take 2 seconds to go from the machine to my mother standing at the other baseline (or the ball takes 1 second to get to the net, and then another second to get to my mom). Therefore, as my mom hits a ball, the ball machine is in the process of shooting the next one. As my mom continues to practice, she hits a stroke every 2 seconds.


In this video, my mom hits ground strokes every 2 seconds, and the next ball is released as she hits the previous stroke (this is some pretty great animation, huh? Unfortunately, the timing is not quite right, so it's not 2 seconds in real time).

Now, suppose she wanted to work on her volleying and therefore decides to run to the net. She hits a stroke (a nice approach shot, presumably), and then starts sprinting to the net as the ball machine spits out the next ball. assuming she runs really fast (like Usain Bolt), she can make it to the net in 1 second. When she gets there, the next ball from the ball machine will already be there to meet her (remember, the ball takes 1 second to get to the net, so while my mom was running in, the next ball was also heading towards the net). Instead of hitting a ball every 2 seconds, she'll hit this one after only 1 second, because she was moving relative to the ball machine. Then, once she's stationary at the net, she will once more see a ball every 2 seconds. In a sense, this is the Doppler effect. When my mom, the observer, was moving relative to the ball machine, the source, the frequency with which she hit balls changed (from once every 2 seconds to once every second). Then, when she was no longer moving, the frequency returned to its usual value.


In this one, my mom hits a ground stroke and runs to the net, at which point she is confronted with the next ball after only 1 second, instead of the usual 2. This is caused by the relative difference between her speed and the ball machine, or the Doppler effect (again, the timing isn't quite right, but you get the idea).

How to find it

In the last post, I said that dark matter could be a new type of particle that only interacts weakly, which is why we've never seen it before. The goal of my research is to build a very sensitive radiation detector and directly detect a WIMP (by observing the energy released on that rare occasion when a WIMP does interact with something in the detector). This is hard. Given our current limits on dark matter, we expect to see maybe a handful of events per year in our detector. That means we would run our detector continuously for an entire year, and we might see a single event that we could point to and say that it was a WIMP.

If that was the only requirement, building such a detector wouldn't be so hard. The difficulty arises in the fact that there is radiation flying all over the place all the time that is not associated with dark matter, and our detector is sensitive to that as well. This is known as background. It's as if you were trying to have a conversation with someone at a loud party who refused to raise his voice. Because of all the background conversations, it would be very hard to understand what that person was saying. A dark matter detector has a similar problem. For example, because of energetic particles passing through the atmosphere called cosmic rays and other ambient sources of radiation, a standard radiation detector (a Geiger counter is shown in the picture) goes off about 100 times per second. Or 10 million times a day. Or 3.7 billion times a year. And we want to be sensitive to 1 event. Imagine trying to hear what one particular person was saying when half of all the people on Earth were speaking at the same time and that's what dark matter experimentalists are trying to do.

How do we plan to do this? First, we will put our detector underground (in an active nickel mine in Sudbury, ON). This has been done with great success by neutrino experiments in the past – if the detector is underground, the earth helps shield the detector from cosmic rays, knocking the background down to say just the population of the USA. Second, we want to use very clean materials in our detector – if you can purify and clean everything very well, you can get rid of many sources of background that are always just lying around. Specifically, we plan to use liquid argon or liquid neon as our detector materials. These elements are very easily purified, so that we can remove anything that might produce radiation before filling our detector. Argon and neon have the great property that when exposed to radiation, they will “scintillate” or produce light. That will be our signal, in that we will look for flashes of light produced by a WIMP interacting in the liquid. In addition, the size of an argon or neon detector can be quite large, helping increase the size of our dark matter “target.”

Finally, we hope to reduce the majority of our backgrounds by using the timing of the light produced by an interaction. Most backgrounds in our detector are caused by radiation scattering off of electrons – these are called “electronic recoils.” A dark matter event would occur from a WIMP scattering off a nucleus, or a “nuclear recoil.” These two types of events have different time signatures in the scintillation light, and we can use the timing to tell them apart.

Our plan is to build a sensitive detector, eliminate all the backgrounds, and listen for that one interesting conversation.

What is dark matter

In the first post of this blog, I briefly discussed how galaxy rotation curves provide evidence for the existence of dark matter - I didn't really said anything about what dark matter actually is. We've only said that it exists, that it has mass (i.e. it interacts with gravity), and that it doesn't interact with light like every day matter. The truth is, even though dark matter is 85% of the total matter in the universe, we don't know what it is because we've never seen it directly. That's one of the reasons we're looking for it. There are a number of theories for what dark matter might be, but for now I'll just talk about one of the most popular, the Weakly Interacting Massive Particle or WIMP (again, a cute name that is quite a literal description of a particle that has mass and interacts weakly).

There are four forces in nature. The first is gravity, by which mass attracts other mass. The second is electromagnetism between charges, such that like charges repel and opposite charges attract (electromagnetism is also the interaction between matter and light). The third force is the “strong” force, which holds protons and neutrons together. The final force, and the one of interest here, is the “weak” force, which is involved in nuclear reactions. The weak force is weak mainly because its range is very small. You have to be really, really close to something to interact weakly. For example, there is a very light particle called the neutrino that only interacts weakly (neutrinos are too light to constitute dark matter). Neutrinos are produced in nuclear reactions, including nuclear reactors. As the Sun is basically a giant nuclear reactor, it emits neutrinos all the time - 60 billion solar neutrinos go through each one of our fingernails every second, but they just don’t hit anything; basically, because neutrinos only interact weakly, we are transparent to them.

A heavy particle that interacts weakly, or a WIMP, is exactly the kind of thing that could be the dark matter - we simply wouldn't have observed it before because the weak interaction is so rare.