Bragg Equation Calculator

Bragg’s Law relates diffraction maxima to lattice plane spacing:

• n·λ = 2·d·sin(θ)
• θ = asin(n·λ / (2·d)) when solving for angle
• λ = 2·d·sin(θ) / n when solving for wavelength
• d = n·λ / (2·sin(θ)) when solving for spacing
• n = 2·d·sin(θ) / λ when solving for order

Angles must be physically valid: n·λ / (2·d) ≤ 1. If you enter 2θ, the calculator uses θ = (2θ)/2.

1. Select what you want to calculate: θ, λ, d, or n.
2. Choose whether your instrument reports θ or 2θ, and set degrees or radians.
3. Enter the known values with correct units (especially for λ and d).
4. Pick output units and decimal places, then press Calculate.
5. Use Download CSV or Download PDF to save your result.

What Bragg’s Law Measures

Bragg’s law links a diffraction peak to the geometry of crystal planes. When X‑rays reflect from parallel planes separated by distance d, the path difference becomes an integer multiple of the wavelength. Peaks appear when n·λ = 2·d·sin(θ) is satisfied.

Typical X‑ray Wavelengths Used

Many powder diffractometers use copper Kα radiation at about 1.5406 Å, while molybdenum Kα is about 0.7107 Å. Shorter wavelengths often shift peaks to smaller angles for the same spacing. Enter the correct λ to match your tube or monochromator.

Understanding d‑Spacing and Planes

The spacing d corresponds to the distance between lattice planes labeled by Miller indices (hkl). Common crystalline materials produce d‑spacings roughly from 1 Å to 5 Å, though larger spacings occur in layered compounds and polymers.

Why Instruments Report 2θ

In most XRD geometries, the detector scans through an angle of 2θ while the incident beam and sample define the scattering angle θ. A peak listed at 2θ = 40° corresponds to θ = 20°. This calculator accepts either form.

Choosing the Diffraction Order n

The order n is typically an integer, most often 1. Higher orders can appear, but many patterns are indexed assuming first order reflections. If you solve for n and get a value near 2, 3, or 4, it can hint at higher‑order scattering.

Angle Limits and Valid Solutions

A real solution requires n·λ/(2·d) ≤ 1 because sin(θ) cannot exceed 1. For example, with λ = 1.5406 Å and d = 1.0 Å, first order would demand sin(θ) = 0.7703, which is valid, but smaller d can break this condition.

Improving Accuracy in Practice

Use consistent units for λ and d, and ensure angles are in degrees or radians as selected. For higher precision, keep more decimal places for 2θ peak positions and wavelength. In laboratory scans, peaks commonly fall within 2θ ≈ 20°–80°, where angular errors strongly affect computed d‑spacing. Averaging repeated scans can reduce peak uncertainty below 0.02° in many setups, improving d estimates. Record temperature and sample prep too.

Worked Interpretation Example

Suppose a peak occurs at 2θ = 40° using Cu Kα (λ = 1.5406 Å) with n = 1. Then θ = 20° and d = n·λ/(2·sin(θ)) ≈ 1.5406/(2·0.342) ≈ 2.25 Å. That spacing can be compared to reference patterns to identify phases or verify lattice changes.

1) What is the difference between θ and 2θ?

θ is the Bragg angle inside the equation. Many diffractometers report 2θ because the detector moves through twice the Bragg angle. This tool converts 2θ to θ automatically.

2) Can the diffraction order n be non‑integer?

In classical Bragg diffraction, n is an integer. If you calculate n and obtain a non‑integer, it usually indicates inconsistent inputs, wrong wavelength, or a peak assignment issue.

3) What does it mean if n·λ/(2·d) is greater than 1?

It means no real θ exists because sin(θ) cannot exceed 1. Recheck units, reduce n, confirm λ, or consider that the chosen d does not correspond to that peak.

4) Which units should I use for λ and d?

Use any length unit you like, but keep λ and d consistent. Å and nm are common in crystallography. The calculator converts units internally, then returns results in your chosen output units.

5) Should I enter angles in degrees or radians?

Most XRD peak lists are in degrees, so degrees are typical. Use radians if your workflow already uses radians. The calculator converts your input to radians for the trigonometric functions.

6) Why are Ångströms used so often?

Atomic plane spacings are around 1–5 Å, making Å a convenient scale. Using Å avoids many leading zeros that appear when using meters for crystal lattice distances.

7) How can I estimate d from a measured peak quickly?

Select “Spacing d,” enter λ, n, and the measured θ or 2θ, then calculate. For rough checks, first order n=1 is often assumed unless you have evidence of higher‑order reflections.

What Bragg’s Law Measures

Bragg’s law links a diffraction peak to the geometry of crystal planes. When X‑rays reflect from parallel planes separated by distance d, the path difference becomes an integer multiple of the wavelength. Peaks appear when n·λ = 2·d·sin(θ) is satisfied.

Typical X‑ray Wavelengths Used

Many powder diffractometers use copper Kα radiation at about 1.5406 Å, while molybdenum Kα is about 0.7107 Å. Shorter wavelengths often shift peaks to smaller angles for the same spacing. Enter the correct λ to match your tube or monochromator.

Understanding d‑Spacing and Planes

The spacing d corresponds to the distance between lattice planes labeled by Miller indices (hkl). Common crystalline materials produce d‑spacings roughly from 1 Å to 5 Å, though larger spacings occur in layered compounds and polymers.

Why Instruments Report 2θ

In most XRD geometries, the detector scans through an angle of 2θ while the incident beam and sample define the scattering angle θ. A peak listed at 2θ = 40° corresponds to θ = 20°. This calculator accepts either form.

Choosing the Diffraction Order n

The order n is typically an integer, most often 1. Higher orders can appear, but many patterns are indexed assuming first order reflections. If you solve for n and get a value near 2, 3, or 4, it can hint at higher‑order scattering.

Angle Limits and Valid Solutions

A real solution requires n·λ/(2·d) ≤ 1 because sin(θ) cannot exceed 1. For example, with λ = 1.5406 Å and d = 1.0 Å, first order would demand sin(θ) = 0.7703, which is valid, but smaller d can break this condition.

Improving Accuracy in Practice

Use consistent units for λ and d, and ensure angles are in degrees or radians as selected. For higher precision, keep more decimal places for 2θ peak positions and wavelength. In laboratory scans, peaks commonly fall within 2θ ≈ 20°–80°, where angular errors strongly affect computed d‑spacing. Averaging repeated scans can reduce peak uncertainty below 0.02° in many setups, improving d estimates.

Worked Interpretation Example

Suppose a peak occurs at 2θ = 40° using Cu Kα (λ = 1.5406 Å) with n = 1. Then θ = 20° and d = n·λ/(2·sin(θ)) ≈ 1.5406/(2·0.342) ≈ 2.25 Å. That spacing can be compared to reference patterns to identify phases or verify lattice changes.

1) What is the difference between θ and 2θ?

θ is the Bragg angle inside the equation. Many diffractometers report 2θ because the detector moves through twice the Bragg angle. This tool converts 2θ to θ automatically.

2) Can the diffraction order n be non‑integer?

In classical Bragg diffraction, n is an integer. If you calculate n and obtain a non‑integer, it usually indicates inconsistent inputs, wrong wavelength, or a peak assignment issue.

3) What does it mean if n·λ/(2·d) is greater than 1?

It means no real θ exists because sin(θ) cannot exceed 1. Recheck units, reduce n, confirm λ, or consider that the chosen d does not correspond to that peak.

4) Which units should I use for λ and d?

Use any length unit you like, but keep λ and d consistent. Å and nm are common in crystallography. The calculator converts units internally, then returns results in your chosen output units.

5) Should I enter angles in degrees or radians?

Most XRD peak lists are in degrees, so degrees are typical. Use radians if your workflow already uses radians. The calculator converts your input to radians for the trigonometric functions.

6) Why are Ångströms used so often?

Atomic plane spacings are around 1–5 Å, making Å a convenient scale. Using Å avoids many leading zeros that appear when using meters for crystal lattice distances.

7) How can I estimate d from a measured peak quickly?

Select “Spacing d,” enter λ, n, and the measured θ or 2θ, then calculate. For rough checks, first order n=1 is often assumed unless you have evidence of higher‑order reflections.