Fine-Structure Constant

In one sentence: 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.

Read it out loud: “Alpha equals e squared divided by the quantity four pi times epsilon sub zero times h-bar times c. Alpha is the fine-structure constant, a pure number with no unit. Here e is the elementary charge, epsilon sub zero is the vacuum permittivity, h-bar is the reduced Planck constant, c is the speed of light, and pi is about 3.14159. The value of alpha is approximately one divided by 137.036.”

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.

Pure numbers and units that cancel

A pure number has no unit, such as meters, seconds or joules. The fine-structure constant is a pure number. It is built from quantities whose units cancel, so the result is a plain ratio, like 1 divided by 137. Because it has no unit, its value is the same whatever units you measure with, which is why physicists treat it as a basic number of nature.

Powers of ten, like $10^{-19}$ and $10^{-38}$, and squares like $e^2$

A short number times a power of ten is the usual way to write very small numbers. The power $10^{-38}$ is a decimal point followed by 37 zeros and then a 1. The raised 2 in $e^2$ means $e$ multiplied by itself, so $(1.602 \times 10^{-19})^2 = 2.567 \times 10^{-38}$. The power of ten is squared too: exponent $-19$ times 2 gives $-38$.

Electric charge, the coulomb and $e$

Electric charge is the property that makes charged particles push or pull on each other. Its unit is the coulomb, written C. The elementary charge $e$ is the charge of one proton, $1.602176634 \times 10^{-19}$ C, and it is exact by definition. One coulomb is about $6.24 \times 10^{18}$ elementary charges, so the charge of one electron is a very small share of a coulomb.

Permittivity $\varepsilon_0$ and the farad

The vacuum permittivity, written $\varepsilon_0$, said epsilon zero, tells how strong the electric force between charges is in empty space. Its unit is the farad per meter, F/m. A farad is the unit of capacitance, which is how much charge a device, such as the capacitor in a camera flash, can store for each volt across it. The value is $8.8541878128 \times 10^{-12}$ F/m.

Greek letters $\alpha$, $\varepsilon$, $\pi$ and $\hbar$

Alpha, $\alpha$, is said AL-fuh. Epsilon, $\varepsilon$, is said EP-sih-lon. Pi, $\pi$, is said pie, and it is about 3.14159, the ratio of a circle’s distance around to its width across. The symbol $\hbar$ is the reduced Planck constant, said h-bar. Greek letters are only names, so each page says what its letters stand for.

Joules and electron volts

Energy is measured in joules, J. One electron volt, eV, is the energy that an electron gains when it is pushed through one volt, which equals $1.602176634 \times 10^{-19}$ J. The hydrogen energy in the example is about 13.6 eV, which is $2.18 \times 10^{-18}$ J, a very small amount of energy written with a negative power of ten.

The speed of light $c$ and the unit m/s

The symbol $c$ is the speed of light in vacuum, exactly 299,792,458 meters per second, which is written m/s. Speed is distance divided by time, so m/s means meters per second. The product $\hbar c$ has units of joule meters, J m, which is energy times length, the same units as the top part of the formula.

The hydrogen atom

Hydrogen is the simplest atom, with one proton in its nucleus and one electron around it. The ground state is the lowest energy state of that electron. Its binding energy, the energy needed to pull the electron away, is about 13.6 eV. This simple atom is a common test case for physics formulas, because it can be solved exactly in the simplest model.

What each symbol means

The formula piece by piece

$e^2$

Start with the squared charge, the charge of one electron multiplied by itself. The electric force between two charges grows with the product of the charges, so the charge enters the formula squared.

$4\pi\varepsilon_0$

The standard factor for electric force in SI units. The force between two charges is their product divided by $4\pi\varepsilon_0$ and by the distance squared. Including this factor makes the ratio come out as a pure number.

$\hbar c$

The reduced Planck constant times the speed of light. Its unit is joule meters, which is energy times length. The top part, $\frac{e^2}{4\pi\varepsilon_0}$, has the same unit.

$\frac{e^2}{4\pi\varepsilon_0\,\hbar c}$

The top part divided by the bottom part. Both have the unit joule meters, so the units cancel, and what remains is a pure number. This ratio is the fine-structure constant.

$\alpha = \frac{e^2}{4\pi\varepsilon_0\,\hbar c} \approx \frac{1}{137.036}$

The fine-structure constant is the squared elementary charge divided by $4\pi$ times the vacuum permittivity times $\hbar$ times $c$, and that pure number is about one over 137.

What the formula is doing, and why

What alpha is. The fine-structure constant $\alpha$ is a dimensionless number, a pure number with no units. Its value is about $0.0072974$, or $1/137.036$. Because it has no units, it has the same value in any system of units. That is what makes it a fundamental constant: the number is part of how nature works, not an accident of the measuring tools.

Why this shape? The top part, $e^2$, is the strength of the electric force between two charges. The bottom part, $4\pi\varepsilon_0\hbar c$, combines the electric force law with the quantum scale and the speed of light. Why divide? Dividing one quantity with units of energy times length by another with the same units cancels every unit, so only a pure number is left. Division is the step that turns a list of units into a plain ratio.

Why it is small. A small $\alpha$ means light and charged particles interact weakly. Atomic physics works well with simple steps, because each correction is only a small nudge. Quantum electrodynamics, the theory of light and charge, uses $\alpha$ as the size of its correction steps. A much larger $\alpha$ would make those corrections big, and the simple picture would be far less useful.

What it sets. The constant sets the scale of many atomic effects. In hydrogen, the ground-state energy is $\frac{\alpha^2 m_e c^2}{2}$, which is about 13.6 eV. The square of $\alpha$ appears because the energy scales with the square of the electric interaction strength. The fine structure of spectral lines, the small splitting that gave the constant its name, is also set by $\alpha$.

Measured, not derived. Current theory does not predict the value of $\alpha$. It is measured in experiments, such as measurements of the electron’s magnetic moment and of how atoms recoil after absorbing light. The formula gives the same number when you put in the measured values of the constants.

An everyday picture

Think of $\alpha$ as a dial that sets how strongly charged particles and light exchange energy. At about 1/137 the dial sits low, so each interaction is a small nudge. That is why atoms can be described as a set of small steps that add up to the observed spectrum.

Worked example: Computing alpha from the constants, then the hydrogen energy

Use $e = 1.602176634 \times 10^{-19}$ C, $\varepsilon_0 = 8.8541878128 \times 10^{-12}$ F/m, $\hbar = 1.054571817 \times 10^{-34}$ J s, $c = 299792458$ m/s and $m_e = 9.1093837 \times 10^{-31}$ kg, which is the mass of an electron. First find $\alpha$, then use it to find the ground-state energy of hydrogen.

1. Square the charge for the top of the fraction. $e^2 = (1.602176634 \times 10^{-19})^2$. The number squares to about 2.56697, and the exponent doubles: $-19 \times 2 = -38$. So $e^2 \approx 2.56697 \times 10^{-38}$ C$^2$. Squaring means multiplying the number by itself.
2. Multiply the bottom of the fraction. The bottom is $4\pi\varepsilon_0\hbar c$. Multiply the front numbers: $4\pi \times 8.8541878128 \times 1.054571817 \times 2.99792458 \approx 351.767$. The powers of ten add: $-12 + (-34) + 8 = -38$. Tidy $351.767 \times 10^{-38}$ by moving the decimal point two places left and raising the exponent by two, which gives $3.51767 \times 10^{-36}$ J m.
3. Divide to get alpha. $\alpha = \frac{2.56697 \times 10^{-38}}{3.51767 \times 10^{-36}}$. Dividing the front numbers gives $2.56697 / 3.51767 \approx 0.729735$. The exponents subtract, $-38 - (-36) = -2$, so $\alpha \approx 0.729735 \times 10^{-2} = 0.00729735$. Its inverse is $\frac{1}{\alpha} \approx 137.036$.
4. Check that the units cancel. The top, $\frac{e^2}{4\pi\varepsilon_0}$, is an energy times a length, joule meters, and $\hbar c$ is also joule meters. Dividing joule meters by joule meters leaves no unit, so $\alpha$ is a pure number, the same in any system of units.
5. Find the electron’s rest energy. Use $m_e c^2 = 9.1093837 \times 10^{-31}\ \text{kg} \times (2.99792458 \times 10^{8}\ \text{m/s})^2$. The speed squared is about $8.98755 \times 10^{16}$ m$^2$/s$^2$, because the exponent doubles, $8 \times 2 = 16$. Multiplying gives $81.8711 \times 10^{-15}$ J, which tidies to $8.18711 \times 10^{-14}$ J.
6. Compute the hydrogen energy in joules. The ground-state energy is $\frac{\alpha^2 m_e c^2}{2}$. First $\alpha^2 = (0.00729735)^2 \approx 5.3251 \times 10^{-5}$. Then $\frac{5.3251 \times 10^{-5} \times 8.18711 \times 10^{-14}}{2}$. The exponents add, $-5 + (-14) = -19$, and $5.3251 \times 8.18711 / 2 \approx 21.799$, so the result is $2.1799 \times 10^{-18}$ J.
7. Convert to electron volts. Divide by $1.602176634 \times 10^{-19}$ J per eV: $\frac{2.1799 \times 10^{-18}}{1.602176634 \times 10^{-19}}$. The exponents subtract, $-18 - (-19) = +1$, and $2.1799 / 1.60218 \approx 1.3606$. So the energy is $13.606$ eV, about 13.6 eV.

Answer: The constant is $\alpha \approx 0.00729735$, or $1/137.036$, a pure number with no units. Using it, the hydrogen ground-state binding energy is about 13.6 eV, which matches the measured binding energy of the hydrogen ground state.

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

Common mistakes

• Leaving out the $4\pi$ in the SI formula. Without it the number is $4\pi$ times too large, about 12.6 times too large.
• Using $h$ where $\hbar$ belongs. Since $h = 2\pi\hbar$, that mistake makes $\alpha$ about $2\pi$ times too small, which is about 6.3 times too small.
• Thinking $\alpha$ has a unit or depends on the unit system. It is dimensionless, so its value is the same in every system of units.
• Confusing $\alpha$ with the fine-structure splitting itself. $\alpha$ is the constant, and the splitting is an effect that depends on it.

Where you will meet it

• The fine-structure splitting of hydrogen spectral lines, where the small splitting is set by $\alpha$.
• Quantum electrodynamics, the theory of light and charge, where $\alpha$ is the size of the steps in its corrections.
• Precision tests of the electron’s magnetic moment, which determine $\alpha$ with very high accuracy.
• Quick estimates of atomic sizes and energies, such as the Bohr radius, $a_0 = \hbar/(m_e c\,\alpha)$, which sets the typical distance of the electron from the nucleus in hydrogen.

Questions people ask

Why is it called the fine-structure constant?

It sets the size of the small splitting of spectral lines called fine structure. The splitting was observed in hydrogen, and $\alpha$ controls how large it is.

Is alpha really about 1 over 137?

Yes. The measured value is $1/137.036$, or about $0.0072974$. No accepted theory explains why it is close to 137, so the number is treated as a measured fact.

Does alpha change over time?

Experiments so far have not found a change. Researchers keep testing the value with precise measurements, so its value rests on measurements rather than on a definition.

Related formulas

• Bohr Radius: The Bohr radius is the typical size of a hydrogen atom, about 0.0529 nanometers, and it comes from combining a few fixed constants of nature.
• Hydrogen-Like Atom Energy: An atom or ion with only one electron has energy levels like hydrogen’s, but a larger nuclear charge pulls every level deeper, so the electron is held more tightly.
• Rydberg Formula: The Rydberg formula gives the exact color of the light that a hydrogen atom gives off when its electron drops from one energy level to a lower one.
• Zeeman Effect: A magnetic field shifts an atom’s energy levels by small equal steps, so one spectral line splits into several.

Sources

1. Fine-structure constant, Wikipedia
2. Fundamental physical constants, NIST