Zeeman Effect

In one sentence: A magnetic field shifts an atom’s energy levels by small equal steps, so one spectral line splits into several.

$\Delta E = m_\ell\,\mu_B\,B$

Read it out loud: “Delta E equals m sub l times mu sub B times B. The energy shift, delta E, equals the magnetic quantum number m sub l, which is a whole number, times the Bohr magneton, mu sub B, which is a fixed number, times the magnetic field strength B. Mu is the Greek letter said mew. Delta E is the energy shift, the change in energy caused by the field.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Magnetic field $B$ and the tesla: A magnetic field is the invisible push a magnet feels in the space around it. Its strength is written $B$ and measured in tesla, T. Earth’s magnetic field at the surface is about 25 to 65 microtesla, and many MRI scanners use 1.5 or 3 tesla. An older unit, the gauss, is 1/10,000 of a tesla, so 1 T equals 10,000 gauss.
Energy, joules and electron volts: Energy is the ability to make something happen. Its unit is the joule, written J. One electron volt, written eV, is the energy that one electron gains when it is pushed through one volt, which equals $1.602176634 \times 10^{-19}$ J. Energies inside atoms are a few electron volts, so eV is the natural unit for atoms, and joules are the unit used in the formulas.
Powers of ten, like $10^{-24}$: Very big and very small numbers are written as a short number times a power of ten. The exponent says how many times to multiply by 10, and a minus sign means divide. The power $10^{-24}$ is a decimal point followed by 23 zeros and then a 1. Multiplying powers adds the exponents, and dividing subtracts them, so $10^{-24} \div 10^{-19} = 10^{-5}$ .
The Greek letter $\mu$ and the magnetic moment: The Greek letter mu, $\mu$ , said mew, is often used for a magnetic moment. A magnetic moment is the strength of a tiny magnet, such as the magnet made by an electron’s orbiting charge. A bigger magnetic moment gives a bigger push or turn in a field. The subscript in $\mu_B$ names the Bohr magneton, a standard unit of this strength for an electron.
Spectral line: A spectral line is one sharp band of light at a single color. A spectrometer splits the light from a glowing gas into its colors, and each line marks light given off when an electron drops from one energy level to a lower one. When the energy levels split in a magnetic field, one line can split into several lines that are very close in color.
The magnetic quantum number $m_\ell$ and the subscript $\ell$: $m_\ell$ is the magnetic quantum number, a whole-number label from $-\ell$ to $+\ell$ . The letter $\ell$ is the orbital quantum number from the earlier page on angular momentum. The subscript $\ell$ is read ell, and it tells you that the label belongs to $\ell$ . For $\ell = 1$ the allowed labels are $-1$ , $0$ and $+1$ .
Frequency, hertz and Planck’s constant $h$: Frequency counts how many waves pass each second. Its unit is the hertz, written Hz, where 1 Hz is one wave per second. The prefix giga, G, means a billion, so 14 GHz is 14 billion hertz. Planck’s constant $h = 6.62607015 \times 10^{-34}$ J s links energy to frequency: a step in frequency equals a step in energy divided by $h$ .

What each symbol means

Symbol	Meaning
$\Delta E$	The energy shift of the level caused by the magnetic field, measured in joules (J) or electron volts (eV). The example gives $9.27 \times 10^{-24}$ J for $m_\ell = +1$ in a 1 tesla field, which is about $5.79 \times 10^{-5}$ eV.
$m_\ell$	The magnetic quantum number, a whole number from $-\ell$ to $+\ell$ . It has no unit, because it is a count of steps. Each allowed value gives one level, and that level moves up or down by that many steps of size $\mu_B B$ .
$\mu_B$	The Bohr magneton, a fixed constant equal to $9.2740100783 \times 10^{-24}$ J/T, which means joules per tesla. The unit J/T is energy per tesla. It is the energy step produced by one tesla of field for one step of $m_\ell$ .
$B$	The magnetic field strength, measured in tesla (T). Earth’s field is about 25 to 65 microtesla. Many laboratory magnets reach a few tesla, and the example uses 1 T. A bigger field makes each energy step bigger.

The formula piece by piece

$m_\ell$: The starting point is the magnetic number, a whole number that picks the level. Zero gives no shift. Positive numbers move the level up, and negative numbers move it down.
$\mu_B$: The Bohr magneton, the size of one energy step per tesla. It is a fixed constant, about $9.27 \times 10^{-24}$ J/T, so its value is the same for every atom.
$\mu_B B$: The energy step for one unit of $m_\ell$ in a field $B$ . The units work out: J/T times T gives J, so the product is an energy, and the field sets its size.
$\Delta E = m_\ell\,\mu_B\,B$: The energy shift of a level is its whole-number label times one energy step, and each step is the Bohr magneton times the magnetic field.
$\mu_B = \frac{e\hbar}{2m_e}$: The standard formula for the Bohr magneton, built from the electron charge $e$ , the reduced Planck constant $\hbar$ and the electron mass $m_e$ . It comes out to $9.274 \times 10^{-24}$ J/T.

What the formula is doing, and why

Why the levels move. When an atom sits in a magnetic field $B$ , its energy levels shift. The electron’s orbital motion makes a tiny current loop, which acts like a tiny magnet. The field pushes these tiny magnets into a few allowed directions, and each allowed value of $m_\ell$ corresponds to one level. The shift for that level is $\Delta E = m_\ell\,\mu_B\,B$ .

Why multiply? The shift is the number of steps, $m_\ell$ , times the size of one step, $\mu_B B$ . Double the field and every step doubles. Make $m_\ell$ twice as big and the shift doubles too. This is a straight-line rule, so the shift grows in step with both the field and the label. A negative $m_\ell$ gives a negative shift, which means the level moves down.

Where the step size comes from. The Bohr magneton $\mu_B = \frac{e\hbar}{2m_e}$ is built from the electron’s charge $e$ and its mass $m_e$ . Multiply the charge by $\hbar$ , and divide by twice the mass. A smaller electron mass would make the step bigger. The value is so small that the splitting needs a sensitive spectrometer to see.

Normal and anomalous. For $\ell = 1$ there are three levels, shifted by $-\mu_B B$ , $0$ and $+\mu_B B$ . A spectral line from a level with $\ell = 1$ down to one with $\ell = 0$ therefore splits into three lines in a field. This pattern is called the normal Zeeman effect, named after Pieter Zeeman, who observed the splitting of spectral lines in 1896. Real atoms also have electron spin, which adds its own magnetic effect and a more complicated pattern, called the anomalous Zeeman effect. This page covers the normal case, where spin is ignored.

An everyday picture

Think of a set of stairs inside the atom, where each step is the same height. The magnetic field sets the height of one step, and $m_\ell$ says how many steps up or down a level sits. A stronger field makes each step taller, so the levels spread further apart.

Worked example: The energy step in a 1 tesla field

A magnetic field of $B = 1$ T acts on an atom. Use $\mu_B = 9.2740100783 \times 10^{-24}$ J/T and the conversion $1\ \text{eV} = 1.602176634 \times 10^{-19}$ J. Find the energy step for $m_\ell = +1$ in joules and in electron volts, and then the matching frequency step.

Multiply the label by the Bohr magneton and the field. For $m_\ell = +1$ , $\Delta E = (+1) \times 9.2740100783 \times 10^{-24}\ \text{J/T} \times 1\ \text{T}$ . The unit T cancels, leaving joules. Multiplying by $+1$ and by 1 changes nothing, so $\Delta E = 9.27 \times 10^{-24}$ J.
Convert to electron volts. Divide by $1.602176634 \times 10^{-19}$ J per eV: $\frac{9.2740100783 \times 10^{-24}}{1.602176634 \times 10^{-19}} \approx 5.79 \times 10^{-5}$ eV. The front numbers give 5.788, and the exponents subtract, $-24 - (-19) = -5$ , so the result is $5.79 \times 10^{-5}$ eV.
Find the frequency step. A frequency step is the energy step divided by Planck’s constant: $\frac{\Delta E}{h} = \frac{9.27 \times 10^{-24}\ \text{J}}{6.62607015 \times 10^{-34}\ \text{J s}}$ . The joules cancel, leaving one over seconds, which is the hertz. The front numbers give 1.40, and the exponents subtract, $-24 - (-34) = +10$ . So the step is $1.40 \times 10^{10}$ Hz.
Say the frequency in plain terms. $1.40 \times 10^{10}$ Hz is 14 billion waves per second, which is 14 GHz. That is small compared with the frequencies of visible light, which are roughly $10^{14}$ to $10^{15}$ Hz, so the split lines are very close in color.
Compare with Earth’s field. Earth’s field at the surface is about 25 to 65 microtesla, and a microtesla is $10^{-6}$ T. A field of 1 T is therefore about 15,000 to 40,000 times stronger than Earth’s field, which is why the lab field makes the splitting easier to measure.

Answer: In a 1 T field each step is about $5.79 \times 10^{-5}$ eV. That is very small compared with atomic transition energies of a few electron volts, so the splitting needs a sensitive spectrometer to see. The frequency step is about $1.40 \times 10^{10}$ Hz, or 14 GHz.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using gauss or millitesla without converting. One tesla equals 10,000 gauss, so a field given in gauss must be divided by 10,000 first to get tesla.
Mixing joules and electron volts in the same sum. Convert both to one unit before you compare or add them.
Reading a negative $m_\ell$ as an error. A negative value means the level shifts down, which is a normal result.
Using the normal formula for a real atom with spin. Spin changes the pattern, and that case is the anomalous Zeeman effect.

Where you will meet it

Laboratory spectroscopy, where the splitting measured in a known field shows how far each level moves.
Astronomy, where the splitting of spectral lines from the Sun reveals the magnetic fields in sunspots.
Electron spin resonance, a close relative, where the Bohr magneton sets the scale of the energy steps of electron spins in a magnetic field.
Quantum mechanics textbooks, in problems about atoms in magnetic fields.

Questions people ask

What is a Bohr magneton?

It is a natural unit of magnetic strength for an electron, $\mu_B = \frac{e\hbar}{2m_e}$ , with the value $9.2740100783 \times 10^{-24}$ J/T. It sets the size of the energy steps in the Zeeman effect when the field is given in tesla.

Are the levels split when there is no magnetic field?

In this simple picture, no. Levels with different $m_\ell$ have the same energy until a field is applied. The field is what separates them.

Why is one tesla such a large field?

Earth’s magnetic field at the surface is about 25 to 65 microtesla. One tesla is about 15,000 to 40,000 times stronger, which is why labs use fields of about one tesla or more to make the splitting clear.

Orbital Angular Momentum Quantization: An electron’s turning motion around the center of an atom can only take certain fixed amounts, like steps on a staircase.
Spin One-Half: An electron has a built-in spin that, measured along any direction, gives only one of two values, called up or down.
Hydrogen Energy Levels: An electron in a hydrogen atom can sit only on certain energy steps, and when it jumps between steps it gives off or takes in light of one exact color.
Fine-Structure Constant: The fine-structure constant is a pure number that sets how strongly charged particles and light interact, and its value is close to 1 over 137.