Bragg's Law

Q: What does the letter $n$ mean?

It is the order of the peak. The first peak has $n = 1$, the second has $n = 2$, and so on. It counts whole wavelengths in the extra path. It is always a whole number, so it cannot be 1.5. It is not the refractive index used in optics.

In one sentence: X-rays bounce off the evenly spaced layers of a crystal, and the bounces add into one strong peak only at certain angles.

$n\lambda = 2d\sin\theta$

Read it out loud: “n times lambda equals 2 times d times sine theta. n is a whole number: 1, 2, 3 and so on. Lambda is the wavelength, the distance from one wave crest to the next. d is the distance between neighboring flat layers of atoms in the crystal. Theta is the glancing angle, the small angle between the beam and those layers. Sine theta is a number between 0 and 1 that depends on the angle.”

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.

Powers of ten, like $10^{-10}$ m: X-ray wavelengths are tiny, so scientists write them with a power of ten. The small raised number, the exponent, says how many times to multiply by ten. A negative exponent means divide by ten that many times, so the number is small. $10^{-10}$ is one ten-billionth. An X-ray wavelength of $1.50 \times 10^{-10}$ m means 1.50 with the decimal point moved 10 places to the left: 0.00000000015 m.
Greek letters $\lambda$ (lambda) and $\theta$ (theta): Greek letters are extra names for quantities, because the Latin alphabet runs out. The letter $\lambda$ is said LAM-duh, and it is the usual name for a wavelength. The letter $\theta$ is said THAY-tuh, and it usually names an angle. Each letter here stands for one number, and the letter names do not change what the formula means.
Sine and inverse sine, $\sin\theta$ and $\arcsin$: The sine of an angle is a number from 0 to 1 that grows as the angle grows from 0 to 90 degrees. On a right triangle, it is the length of the side opposite the angle divided by the longest side. A calculator gives $\sin 30^\circ = 0.5$ . The inverse sine, written arcsin, works backward: if the sine is 0.3, arcsin(0.3) gives the angle, about 17.5 degrees.
Degrees and radians: An angle can be measured in degrees, where a full turn is 360 degrees, or in radians, where a full turn is about 6.283 radians. Both are correct, but a calculator must be set to the unit you are using. In degree mode, $\sin 30^\circ = 0.5$ . In radian mode, typing 30 gives a very different number, so a wrong setting is one of the most common mistakes.
Crystal planes: A crystal is a solid whose atoms sit in a regular, repeating pattern, a bit like tiles on a floor. Some slices through that pattern contain many atoms in one flat sheet. These sheets are called planes. Parallel planes repeat again and again, and $d$ is the distance between neighboring planes. Table salt and many minerals form crystals.
Waves that add or cancel: A wave has a crest, a high point, and a valley, a low point. When two copies of a wave meet with their crests lined up, they add into a bigger wave. When a crest meets a valley, they cancel. Two waves that travel different distances end up in step when the difference is a whole number of wavelengths. That idea is the heart of this formula.
Nanometer (nm) and the meter: The meter, m, is the base unit of length. A nanometer, nm, is one billionth of a meter, which is $10^{-9}$ m. So 0.150 nm is $0.150 \times 10^{-9}$ m. Atoms are a few tenths of a nanometer across, which is why nanometers are a natural unit for crystal spacings.

What each symbol means

Symbol	Meaning
$n$	The order of the peak. It is a whole number, 1, 2, 3 and so on, that counts how many full wavelengths fit into the extra path. The first peak has $n = 1$ . It has no unit, because it is only a count. It is not the refractive index, which is a different quantity written with the same letter in optics.
$\lambda$	The wavelength of the incoming wave, in meters (m). It is the distance from one crest to the next. For X-rays it is usually about $10^{-10}$ m, which is one ten-billionth of a meter, about the same as the spacing between atoms. The example below uses $\lambda = 0.150$ nm.
$d$	The spacing between neighboring parallel planes of atoms in the crystal, in meters (m). Typical values are a fraction of a nanometer, which is $10^{-10}$ m to $10^{-9}$ m. The crystal sets this distance. Measuring the peaks lets scientists work out $d$ when the wavelength is known.
$\theta$	The glancing angle: the angle between the incoming beam and the flat planes, measured along the planes rather than from a line sticking straight out of the surface. It is an angle, so it is measured in degrees or radians. The example below finds $\theta = 17.5$ degrees for the first peak.

The formula piece by piece

$\theta$: The glancing angle. It is the angle the beam makes with the flat planes of the crystal. A small angle means the beam runs almost along the planes, and a larger angle means it hits them more directly. The formula finds the angles at which a strong peak appears.
$\sin\theta$: The sine of the glancing angle. It is a number from 0 to 1 that depends on the angle. It appears because the extra distance traveled is a side of a right triangle, and the sine turns the angle into the length of that side compared with the longest side.
$2d$: Twice the spacing between neighboring planes. The wave goes down to the lower plane and comes back up, so the distance between the planes is counted twice. The factor 2 is the two trips, down and back.
$2d\sin\theta$: The extra distance a wave travels after reflecting from a lower plane compared with an upper one. Two trips of length $d\sin\theta$ each give $2d\sin\theta$ . This is the path difference that decides whether the reflections add or cancel.
$n\lambda$: A whole number of wavelengths. With $n = 1$ it is one wavelength, with $n = 2$ it is two wavelengths, and so on. When the extra distance equals this amount, the reflections arrive in step and add into a strong peak.
$n\lambda = 2d\sin\theta$: The whole formula in one sentence: a whole number of wavelengths equals twice the plane spacing times the sine of the glancing angle. A strong peak appears only at the angles that make this statement true.

What the formula is doing, and why

What goes in. A beam of waves, such as X-rays, hits a crystal. The crystal has flat layers of atoms, called planes, spaced a fixed distance $d$ apart. Each plane reflects a small part of the wave, the way a window reflects a little light. The beam arrives at a glancing angle $\theta$ , a shallow angle between the beam and the planes. The formula asks at which angles these reflections add up into a strong peak.

What happens: the extra path. A wave reflected from a lower plane travels farther than one reflected from the plane above it. Picture the trip down to the lower plane. The spacing $d$ and the angle $\theta$ form a right triangle, and the extra length of that trip is $d\sin\theta$ . The trip back up adds the same amount, so the extra distance is $2d\sin\theta$ in total.

Why whole numbers of wavelengths? When the extra distance equals exactly one wavelength, two wavelengths, or any whole number $n$ of wavelengths, the reflections arrive in step. Their crests line up and add into a strong peak. At other angles the reflections arrive partly out of step and cancel, so no strong peak appears. This is why a push on a swing builds up only when each push arrives at the right moment.

Why the sine has a limit. The sine of any angle is never bigger than 1. So $n\lambda$ can be no bigger than $2d$ . If $n\lambda$ is bigger than $2d$ , no angle can satisfy the rule, and that order of peak cannot exist for that crystal. This is why a crystal shows only a few peaks for a given wavelength.

Where it applies. The rule needs a regular crystal, with planes that repeat at a fixed spacing. Glass and liquids lack such planes, so they do not give sharp peaks like this. Measuring the angles of the peaks lets scientists find $d$ when the wavelength is known, or the wavelength when $d$ is known. X-ray crystallography uses this to work out how atoms are arranged in minerals and in protein crystals.

An everyday picture

Think of a staircase where each step is one layer of atoms. Imagine a sound pulse bouncing off each step. A pulse that bounces off a lower step travels a little farther than one from a higher step. When the extra distance is a whole number of wavelengths, all the bounces return in step and add into one loud echo in that direction. At other directions they return out of step and cancel, so the echo is weak.

Worked example: First and second peaks from an X-ray beam

An X-ray beam has wavelength $\lambda = 0.150$ nm, which is $1.50 \times 10^{-10}$ m. It hits a crystal whose atomic planes are $d = 0.250$ nm apart, which is $2.50 \times 10^{-10}$ m. Find the glancing angle of the first peak, $n = 1$ , and the second peak, $n = 2$ . Both lengths are in nanometers, so we can work with those numbers directly.

Rearrange the formula to find the sine. We want the angle, so we first need its sine. Divide both sides of $n\lambda = 2d\sin\theta$ by $2d$ to get $\sin\theta = \frac{n\lambda}{2d}$ . Now the unknown angle stands alone on one side, and every other piece is a number we know.
Put in the numbers for the first peak. For the first peak $n = 1$ . Both $\lambda$ and $d$ are in nanometers, so the same $10^{-9}$ factor appears on both and cancels when we divide: $\sin\theta = \frac{1 \times 0.150}{2 \times 0.250} = \frac{0.150}{0.500} = 0.300$ . If we use meters instead, $\frac{1.50 \times 10^{-10}}{2 \times 2.50 \times 10^{-10}}$ , the $10^{-10}$ parts divide to $10^{0} = 1$ , and we get the same 0.300.
Check that the sine is allowed. A sine can never be larger than 1, and 0.300 fits inside that range. This quick test matters, because a sine above 1 would mean no peak can form at any angle, as the last step will show.
Find the angle with the inverse sine. We ask which angle has a sine of 0.300. The inverse sine gives $\theta = \arcsin(0.300) \approx 17.46^\circ$ , which rounds to about 17.5 degrees. The calculator must be in degree mode, or the number it gives will not be an angle in degrees.
Second peak, n = 2. Now the extra path is two whole wavelengths, so $\sin\theta = \frac{2 \times 0.150}{2 \times 0.250} = \frac{0.300}{0.500} = 0.600$ . Doubling $n$ doubles the sine, so the second peak sits at a larger angle than the first. The inverse sine gives $\theta = \arcsin(0.600) \approx 36.87^\circ$ , which is about 36.9 degrees.
Third peak, and why the peaks stop. For $n = 3$ we get $\sin\theta = \frac{3 \times 0.150}{0.500} = 0.900$ , so $\theta \approx 64.2^\circ$ . For $n = 4$ we get $\sin\theta = \frac{4 \times 0.150}{0.500} = 1.200$ . A sine of 1.2 is impossible, so no fourth peak exists for this crystal and this wavelength. The rule is that $n\lambda$ must not be larger than $2d$ .
Read the result with units. The sine has no unit, so each answer is an angle. The nanometers cancel in the first step, because the wavelength and the spacing were in the same unit to start with. Only the angle is left as the answer, and it is measured in degrees.

Answer: The first peak appears at a glancing angle of about 17.5 degrees, and the second at about 36.9 degrees. A third peak appears near 64.2 degrees. A fourth would need a sine above 1, so it cannot form with this wavelength and this spacing.

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

Common mistakes

Measuring the angle from the surface normal, the line that sticks straight out of the surface. The glancing angle is measured between the beam and the flat planes, so a beam that runs nearly along the planes has a small angle.
Forgetting the 2 in $2d\sin\theta$ . The extra path has two trips, down and back up. Without the 2, the sine comes out twice as large, so the angle comes out too large, or the calculator reports that no angle exists.
Entering degrees into a calculator set to radians, or the reverse. The number 17.5 in radian mode is an angle of about 1003 degrees, so its sine is a completely different number. Check the mode before you calculate.
Assuming a peak appears at every angle. Only the angles that satisfy the rule give a strong peak. At the other angles the reflections cancel.

Where you will meet it

X-ray crystallography, which finds how atoms are arranged in crystals, minerals and protein crystals.
Electron diffraction, which confirmed that electrons behave as waves. In 1927 Clinton Davisson and Lester Germer saw electrons scatter into peaks at the angles this rule predicts.
Neutron diffraction, which can locate light atoms such as hydrogen, atoms that X-rays find hard to see.
Materials science, where the positions of diffraction peaks identify which crystal a sample contains and show how well it is made.

Questions people ask

Why do X-rays reflect from inside the crystal?

The beam enters the crystal, and every layer of atoms reflects a little of it, not only the top layer. The reflections from many layers add into a strong peak only at the angles that match the rule. At the other angles they cancel.

What does the letter $n$ mean?

It is the order of the peak. The first peak has $n = 1$ , the second has $n = 2$ , and so on. It counts whole wavelengths in the extra path. It is always a whole number, so it cannot be 1.5. It is not the refractive index used in optics.

Why does the rule also work for electrons?

Electrons behave as waves too, and their wavelength depends on how fast they move. When electrons move at the right speed, their wavelength is close to the spacing between atoms in a crystal. Then the electrons diffract like X-rays, and the same formula works.

Double-Slit Interference: Waves from two narrow openings overlap, and at certain places they add into bright bands while at others they cancel into dark ones.
De Broglie Wavelength: Every moving object has a wave that goes with it, and the heavier or faster the object moves, the shorter that wave is.
Planck-Einstein Relation (E = hf): E = hf says each packet of light carries energy in proportion to its frequency: higher frequency means more energy per packet, and brightness does not change it.