# 'Month of Equations': What Does The Zeeman Effect Really Mean?

### October 14: The Zeeman Effect

Meaning of Equation

When an external magnetic field is applied to a quantum system, sharp spectral lines split into multiple closely spaced lines.

A lot to explain, I know. This single line meaning of equation isn't enough to explain this beautiful phenomenon. Just like previous equations in this series, we'll begin from scratch and develop the concept. So before I begin with Zeeman effect, let me first explain what are spectral lines.

We know what Niels Bohr said about the atom. He said that the electrons are orbiting the nucleus in fixed orbits and when an electron jumps from higher energy level to lower energy level, it emits photons carrying some energy. Now those photons can be in visible, infrared, X-rays etc. depending on the difference of the two energy levels. These are known as the emission spectral lines and in a spectrometer, they look like this:

Did you notice something? The spectral lines are very sharp which means that definite amount of energy is being emitted. Each spectral line corresponds to a jump of an electron from one energy level to another energy level. The image below shows the origin of spectral lines in hydrogen atom.

So far, so good! Let us move on to next mini-concept. We know that electrons are revolving around the nucleus of an atom. Can you relate this to something? What happens when electrons move? Answer is current. Current is nothing but moving charge. So you can compare the motion of an electron around the nucleus as a current loop and every moving charge produces a magnetic field and thus a magnetic dipole moment. Don't be afraid, magnetic dipole moment is nothing but a vector that is the product (cross) of area of loop and the current in the loop. If you feel that things got complicated here, just keep this in mind: An electron revolving around the nucleus is a current loop - This current produces a magnetic field - There is a quantity called dipole moment that is the product of current and area of loop.

Now we have the tools to build our concept of Zeeman effect. When an external magnetic field is applied to this system, something strange happens. Now, each spectral line is split into multiple lines. Something like this:

What happens is that when the magnetic field is turned on, it starts interacting with the magnetic dipole moment of the system. It tries to align it in its own direction and hence there comes an associated potential energy into the picture. As shown below, magnetic moment is pointing towards red arrow. If I apply magnetic field from left to right, it will try to align this whole loop in its own direction. In this scenario, some work is done to complete this task. Thus the energy levels split and now we have multiple energy levels and thus multiple spectral lines. I deliberately didn't go into mathematical details of this phenomenon as it would make the things quite complex. If you want to go into such details, see this.

Why did I say its a beautiful concept? Has it got any applications? Yes! of course. The Zeeman effect is used to determine the magnetic field of Sun and other stars. The formula contains magnetic field which can be known by plugging the values of different parameters.

The last image that shows the beauty of this effect is shown below:

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