Ewald Sphere Geometry Calculator
Compute the Ewald sphere radius, the reciprocal lattice vector magnitude, and check whether a reflection can diffract at a given wavelength.
🔵 What is Ewald Sphere Geometry?
The Ewald sphere is a geometric construction in reciprocal space, a sphere of radius k = 2π/λ centered on the crystal, that turns the diffraction condition into a simple visual rule: a reflection diffracts exactly when its reciprocal lattice point touches the sphere's surface. This calculator computes the Ewald sphere radius, the reciprocal lattice vector magnitude for a candidate reflection, and checks whether that reflection lies within the limiting sphere, the maximum reach possible at that wavelength.
Crystallographers use the Ewald construction to visualize which reflections are geometrically accessible before ever computing individual Bragg angles. A single-crystal diffractometer operator uses it to plan which crystal orientations will bring specific reflections into diffracting position. A structural biologist collecting X-ray data on a protein crystal relies on the limiting sphere (radius 2k) to know the absolute best resolution achievable at their beamline's wavelength, no amount of extra exposure time can reveal reflections beyond that geometric limit. A student learning diffraction theory uses the Ewald sphere as the geometric bridge connecting reciprocal lattice vectors directly to Bragg's law.
A common point of confusion is thinking the Ewald sphere and Bragg's law are separate ideas. They are the same physics expressed differently: the condition that a reciprocal lattice point of magnitude |G|=2π/d touches a sphere of radius k is algebraically identical to nλ = 2d sinθ. Another subtlety is the limiting sphere, radius 2k, not k, which sets the true resolution boundary, since a reflection needs |G| ≤ 2k (not just ≤ k) to ever touch the sphere at some crystal orientation.
This calculator computes k, |G|, and the observability check directly, along with the Bragg angle and scattering vector magnitude when the reflection is observable, useful for crystallography and solid-state physics coursework and for understanding diffraction geometry beyond the algebra of Bragg's law alone.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Cu-Kα X-Rays, Silicon's (111) Planes
λ = 1.5406 Å (Cu-Kα), d = 3.13 Å (close to silicon's real (111) spacing)
Example 2 — A Reflection Outside the Limiting Sphere
λ = 1.5406 Å (Cu-Kα), d = 0.6 Å (a very fine spacing)
Example 3 — Mo-Kα X-Rays (Shorter Wavelength)
λ = 0.71 Å (approximate Mo-Kα), d = 2.0 Å
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Ewald sphere?
The Ewald sphere is a geometric construction in reciprocal space with radius k = 2π/λ, where λ is the incident wavelength. A reflection at reciprocal lattice point G diffracts if and only if that point lies exactly on the sphere's surface, this single visual condition is equivalent to Bragg's law.
How do you calculate the Ewald sphere radius?
k = 2π/λ, where λ is the wavelength of the incident X-rays, neutrons, or electrons. For Cu-Kα X-rays (λ=1.5406 Å), the Ewald sphere radius is k = 2π/1.5406 ≈ 4.0784 Å⁻¹.
What is the limiting sphere in crystallography?
The limiting sphere has radius 2k (twice the Ewald sphere radius) and marks the absolute maximum reciprocal lattice vector magnitude that could ever diffract at a given wavelength. Any reflection with |G| > 2k (equivalently d < λ/2) can never touch the Ewald sphere at any crystal orientation, so it is permanently unobservable at that wavelength.
How is the Ewald sphere related to Bragg's law?
They describe the same physics in different languages. The Ewald sphere condition, that reciprocal point G touches the sphere, is mathematically identical to nλ=2d sinθ. Both reduce to the same relationship: |G| = 2π/d = 2k sinθ = (4π/λ)sinθ, which rearranges directly to Bragg's law.
What is the scattering vector Δk?
The scattering vector Δk (also written Q or G) is the difference between the diffracted and incident wave vectors, Δk = k' - k. Its magnitude is |Δk| = 2k sinθ, and when the Bragg condition is satisfied, |Δk| exactly equals the reciprocal lattice vector magnitude |G| = 2π/d for that reflection.
Why can't a reflection with d < λ/2 ever be observed?
Because its reciprocal lattice vector magnitude |G| = 2π/d exceeds 2k, the diameter of the Ewald sphere. No matter how the crystal is rotated, a point farther from the origin than the sphere's diameter can never intersect the sphere's surface, so no rotation brings that reflection into diffracting condition at that wavelength.
How do I get more reflections into range: shorter or longer wavelength?
Shorter wavelength. A smaller λ gives a larger k = 2π/λ, which enlarges the Ewald sphere and its limiting sphere (radius 2k), bringing more distant reciprocal lattice points within reach. This is why high-resolution structure determination uses short-wavelength, high-energy radiation.
What units does the Ewald sphere radius use?
Inverse angstroms (Å⁻¹) in the 2π convention used here, matching the reciprocal lattice vector convention used throughout this calculator suite. Some crystallography references use the 1/λ convention (without 2π) instead, giving a numerically different but equally valid radius.
Does the Ewald sphere apply to neutron and electron diffraction too?
Yes. The Ewald sphere construction applies to any wave-like probe, X-rays, neutrons (via their de Broglie wavelength), and electrons, the only difference between techniques is the numerical value of λ, which sets the Ewald sphere radius k = 2π/λ.
How is this different from the Bragg Law Calculator?
The Bragg Law Calculator solves nλ=2d sinθ directly for whichever variable is unknown. This calculator frames the same physics in reciprocal space, computing the Ewald sphere radius and checking the limiting-sphere resolution condition explicitly, useful for understanding which reflections are geometrically accessible before solving for any specific angle.