### The Bohr Model

As discussed on the previous page, the Rutherford model faced a serious problem: it implied that electrons don’t always obey the laws of classical physics. If electrons obeyed Newton’s laws of motion, they would be knocked out of orbit by forces from neighboring atoms. And if they obeyed Maxwell’s equations, they would slow down and crash into the nucleus.

To explain the strange behavior of electrons in an atom, Danish physicist Niels Bohr (1885 - 1962) suggested that electrons follow a different set of laws, similar to the laws governing the behavior of standing waves. A standing wave is a wave that stays in the same place as it oscillates. The vibrating strings of musical instruments like guitars and violins are examples of standing waves. For another example, imagine that you and a friend are each holding one end of a stretched slinky spring. If you move your arms up and down in just the right way, a standing wave will form in the spring. The peaks and troughs of the wave will oscillate directly up and down, rather than travelling along the length of the spring. In order for this to work, however, the distance between you and your friend must be an exact multiple of the distance between the wave’s nodes (i.e. a multiple of half the wavelength). In other words, you could make the spring oscillate by halves, or by thirds, or by fourths (etc.) of the distance between you and your friend; but you can’t create a standing wave with just any arbitrary wavelength. According to Bohr, electrons act a bit like standing waves: they can only have certain specific “wavelengths.”

So, are electrons particles, or are they waves? Bohr gave an astonishing answer to this question. Electrons are neither waves nor particles, but they behave like both waves and particles!

To see what he meant by this, first imagine that electrons are like miniature slinky springs stretched in a circle that surrounds the nucleus of an atom. (We’ll call this circle the electron’s “orbit,” even though the electron isn’t really orbiting the nucleus.) The electron oscillates like a standing wave, and only certain wavelengths are possible for any given orbit. These possible wavelengths depend on the circumference of the orbit: the circumference must be an exact multiple of the distance between nodes. Since the circumference of a circle depends on its radius, this means that the electron’s possible wavelengths depend on its distance from the nucleus.

On the other hand, suppose electrons are particles that orbit the nucleus as the Rutherford model suggested. In that case, the electron could only move at certain specific speeds. Too fast and the electron will spiral outward and get away; too slow and it will spiral inward and crash into the nucleus. The allowable speeds of the orbiting electron depend on its distance from the nucleus, just as its allowable wavelengths depend on its distance from the nucleus.

Both speed and wavelength correspond to energy. The faster a particle is moving, the greater its kinetic energy. And the shorter the wavelength of a wave, the greater the wave’s energy. Now, here’s the crucial point. For any given orbit around the nucleus, a standing wave is only allowed to have certain energies, and a particle must likewise have a certain energy in order to follow that orbit. There are only a few possible orbits that allow standing waves and particles to have exactly the same energy. These very special orbits, called Bohr orbits, are the paths that electrons follow around the nucleus.

An electron may “jump” from a low-energy Bohr orbit to a high-energy one, or “fall” from a high-energy orbit to a low-energy one. But an electron cannot fall below the lowest Bohr orbit. The electron likes to behave as both a particle and a wave, and there are no lower orbits that permit it to do both. That’s why electrons don’t crash into the nucleus, despite the strong pull of the electromagnetic force. Thus, Bohr’s model avoids the central problem that beset the Rutherford model.

To summarize:

• According to Bohr, electrons aren’t really particles. At least, they’re not completely like particles. They’re kind of like waves, and kind of like particles.
• Electrons orbit at distances where the allowed speed for particles corresponds to the same energy as an allowed wavelength for standing waves:
• At any given distance from the nucleus, an electron must have a certain wavelength (and therefore a certain energy) in order to produce a stable standing wave.
• At any given distance from the nucleus, an electron must have a certain speed (and therefore a certain energy) to maintain a stable orbit.
• The Bohr orbits correspond to distances and energies that work for both waves and particles.
• The dual nature of electrons (i.e., the fact that they behave as both particles and waves) explains why an electron doesn’t crash into the nucleus, despite the strong electromagnetic pull between the electron and the nucleus.