Seeking SciNote, Physics: Why don't electrons fall into the nucleus?
If electrons are negative and protons are positive, why don’t the electrons get sucked into the center of the atom?
Great question! It’s such a great question that we’re going to have to take a couple of cracks at the answer to it.
Let’s start by doing something that physicists do all the time: sidestep a difficult question by first answering a more straightforward one. In this case, let’s ask the question, “If the Earth and the Sun are both massive bodies, why doesn’t gravity suck the earth into the Sun?"
First, let me quickly justify myself. Physicists love to wax on and on about the symmetry of physical law (Richard Feynman had an incredible lecture series called ”The Character of Physical Law“ and symmetry plays a big part), and this is a situation where that waxing is appropriate. When we look at the laws for electricity and gravity side-by-side, it jumps out that they’re almost identical:
Look at that! There’s a constant multiplied by a couple of factors all divided by the distance squared. This means that, mathematically, the two laws can be treated in the same way – and an answer using one law can usually be analogized to the other law. So by answering why the Earth doesn’t fall into the Sun, we can make progress answering why electrons don’t fall into the nucleus.
Now, most people with a bit of physics in their background will tell you that the Earth does, in fact, get sucked into the Sun. Then they’ll proudly put their nose in the air and be entirely unhelpful when you tell them that that doesn’t answer your question. If they’re pedantic (and, admittedly, I am a little bit), they’ll tell you that the Sun and Earth are pulled towards one another with the same force – then they’ll do the nose thing. So let’s be more detailed, and hopefully more helpful. Imagine (here I go answering another question again. I promise I’ll get back to yours, but often the simplest questions are the hardest to answer) that you’re standing atop a really high mountain with a cannon. This thought experiment is stolen right from Newton, by the way, so you know it’s a good one. If you shoot the cannon, the ball will fly some distance along the curved Earth before it falls down to the ground. If you shoot it with a little more speed (and evacuate the air from the Earth, because a physicist will sigh in despair if air resistance comes into this), the ball will travel farther along the curved Earth, maybe going a quarter or a half of the way around the planet if you shot it hard enough, before it falls to the ground because of gravity. But now imagine that you shoot the cannon even harder, and the ball goes all the way around the Earth and hits you in the back of the head. When you’re done tumbling down the side of the mountain, you’ll realize that the ball spent the whole time in the air being pulled towards the ground by gravity, but it was moving so quickly sideways that it kept missing the ground as it fell. Like in this image:
The last trajectory is what we call an orbit. Now imagine that you shoot a planet fast enough around a star, and that’s how the Earth stays going around the Sun. We’re constantly being pulled towards the center of the Solar System but we’re going sideways so quickly that we miss the Sun and stay in a stable loop around it. If you’re curious how this works, this game might help you understand. You’ll notice in that game that it’s very hard to make a circular orbit but relatively easy to make one in a squashed circle – what we call an ellipse. The Earth and the Sun actually both orbit in ellipses around their common center of mass, as it’s called. But since the Sun is so incredibly much more massive than the Earth is, we can get away with saying that the Sun doesn’t move and the Earth goes around it in an ellipse. And if you’re wondering how quickly the Earth is moving, Monty Python made a song about it.
Now, finally, on to your question. Remember how gravity and electromagnetism (labeled "Coulomb’s law” up in that picture) had the same sort of behavior? A question that can be answered by the gravity law can help us answer a question that deals with the electromagnetism law. So our next attempt at answering your question is going to be that the electron goes around the proton(/nucleus, but I’m going to keep saying “proton” to keep things simple) like the Earth goes around the Sun. That is, it’s falling towards the nucleus and swinging sideways in such a way that it follows an elliptical orbit around the center of charge, so to speak. If you work through the math, you’ll find out that this works really well for the electron-proton system, just like it did for the Earth-Sun system, with two slight problems that make everything fall apart.
First, the electron and the proton have equal charge, so their center of charge should be halfway between the two particles and they should both be in circular orbits around that center.[ 1] When we do the experiment, it seems that the nucleus is always in the center, unmoving, with the electrons going around it. But okay, there’s a resolution to this one, so it on its own doesn’t make everything fall apart. Protons are a thousand or so times more massive than electrons, and forces, even ones that happen because of electrical charges, move masses.[2] So even with equal forces, the acceleration of the proton is around a thousand times less than that of the electron, and maybe that’s the reason that the proton seems not to move. It’s like the Sun being more massive than the Earth again, making it look like the Sun doesn’t move at all even though it does. If you do this for a hydrogen atom – that is, if you fix the proton’s position and have the electron go around it – you get what’s called the Bohr radius, which is exactly the measured distance between the electron and the proton. It’s incidentally the distance I used to figure out the number in that first footnote. So everything is perfect and we don’t even need to mention that second problem that makes everything fall apart, right? Right.
Electrons radiate energy when they move. This is how radio towers move; they jiggle the electrons in the tower and generate radio waves, which are a form of energy. I promise there’s some math to back this up, but common sense should tell you that if an orbit depends on how quickly you’re able to move sideways, and if you’re losing energy by moving sideways, your orbit won’t last very long. An orbit is only stable if the object doing the orbiting has enough energy to stay in orbit, and electrons should quickly lose all of that energy. We’re back where we started, in other words. Why doesn’t the electron spiral into the nucleus?
Quantum mechanics comes to the rescue, like a superhero powered by differential equations and complex numbers [3]. You see, I lied to you a little bit up above. Well, okay, not as much lied as avoided the whole truth. There’s a problem with the analogy I gave with the Sun and the Earth. Well, okay, there are a couple of problems with it. I promise I’m a real physicist and that I’m not making things up as I go along; this is just how we have to approach questions in physics sometimes. The first problem is that the inverse-square law of gravity isn’t the full gravitational law; general relativity is. But that’s hardly an issue because Earth can’t tell the difference between the two from 150 million kilometers away from the Sun. Still, it should be mentioned. Second, electrons are not little balls of charge orbiting around the nucleus in the same way that Earth is a ball of mass orbiting around the Sun. Electrons are – well, that’s a tricky sentence to finish. Language developed to describe the world with which we interact on a day-to-day basis, and electrons are very, very different from that world. I can only tell you in a confusing analogy how they act. Electrons sometimes act like waves, meaning that they have a certain wavelength as they move. Now, while it’s not quite right to think of them as little balls wiggling up and down in space as they move, that’s the image that’s going to help most if you keep it in mind.
Now think of if you and a friend have ever shaken both ends of a rope. Most ways you shake it, the rope doesn’t do anything very interesting. But if you shake it a certain way, with a certain frequency (or, really, a certain set of frequencies called the resonance frequencies), you get what’s called a standing wave on the rope, where certain parts seem to stay still and other parts go up and down in neat ways. It’s like this.
For an electron to orbit the nucleus, its wave has to be closed, meaning that it has to close in on itself, like is shown here:
I know this is a lot of levels of abstraction, with the electron going from being like a planet to like a wave to a wave going around in a circle. But I promise this is the last one. The Cosmos reboot had an excellent animation showing a good way of thinking about this. Electrons might radiate some energy as they move, but their orbits can’t get any smaller because the wave has to remain closed. And this is what makes the electron orbits stable. If it got a little smaller, then the wave wouldn’t close on itself and the electron couldn’t occupy that orbit. So the orbits have to be in specific places at specific distances away from the nucleus, and the smallest an orbit can be is one wavelength.
It’s important to keep in mind that that’s only a way of picturing the electron but it’s not what the electron is actually doing – we know that from careful experiments. The electron isn’t a particle or a wave or a cloud around the nucleus like you’ll sometimes hear. It’s all of those and none of them because its behavior escapes our language and our mental imagery. Nonetheless, this imagery and language is spot-on for describing a way of thinking about why an electron stays in orbit around the proton.
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Image Sources:
Inverse square laws image: http://www.myidea.net/ideas/node/578
Newton’s cannonball: http://www.pas.rochester.edu/~blackman/ast104/newtongrav.html
More information on Newton’s cannon: http://en.wikipedia.org/wiki/Newton%27s_cannonball it has some excellent .gifs that help a lot.
Game: http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/kepler6.htm
Solar system center of mass: http://en.wikipedia.org/wiki/Barycentric_coordinates_%28astronomy%29
Electron orbit: http://ircamera.as.arizona.edu/astr_250/Lectures/Lecture_08.htm
Notes:
[1] You might wonder why I’ve decided to ignore gravity after going on and on about it. The electromagnetic force between the electron and the proton is 2.29 x 10^39 times more powerful than the gravitational force between them. That’s 2,290,000,000,000,000,000,000,000,000,000,000,000,000 times stronger. So while there is gravity around the nucleus, we can safely ignore it because electromagnetism much, much stronger between elementary particles. It’s also the force that keeps you from falling to the center of the Earth, but you’ll have to ask another question to learn about that.
[2] This is because Newton’s Second Law is F = m*a, meaning that acceleration is the force divided by the mass. Charge literally doesn’t enter into the equation.
[2] There is no better kind of superhero.
Answered by Brendan C., Expert Leader
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