MTE5881 Chap.2 Bragg Diffraction and Reciprocal Space
Bragg Diffraction and Reciprocal Space
Bragg's law and the Ewald construction describe the same diffraction event from two viewpoints. Real-space planes reinforce waves when the path difference is an integer wavelength. Reciprocal space replaces that ray construction with a scattering vector normal to the planes, of magnitude equal to the inverse spacing.
Miller indices, reciprocal bases and crystal rotation then provide a precise language for predicting which reflections appear. A disciplined solution keeps theta separate from the detector's two-theta reading, checks the arcsine domain and retains units. The same checks connect a numerical spacing result to the geometric claim that rotation changes selected reciprocal points rather than the crystal structure itself.
What this chapter covers
- 01
Bragg condition
- 02
Theta and two-theta
- 03
Scattering vector
- 04
Miller indices
- 05
Reciprocal lattice
- 06
Ewald sphere
Worked example · free
Recover spacing from a measured angle
- +1Convert the detector angle to theta: 44.0 divided by 2 equals 22.0 degrees.
- +3Use d = wavelength divided by twice sine theta to obtain 0.2056 nm, rounded to 0.206 nm.
- +1Invert d to obtain 4.86 inverse nanometres and check the units.
Key terms
- Bragg angle
- The angle between the incident beam and diffracting planes used in Bragg's law.
- Miller indices
- An integer triplet derived from reciprocal plane intercepts.
- Friedel pair
- A reciprocal vector and its opposite partner.
- Reciprocal vector
- A vector normal to a plane family whose magnitude encodes the inverse interplanar spacing.
- Ewald construction
- A reciprocal-space selection picture identifying scattering vectors that satisfy elastic diffraction geometry.
- Scattering vector
- The vector difference between outgoing and incoming wavevectors for a diffraction event.
- Interplanar spacing
- The perpendicular repeat distance between adjacent planes belonging to the same indexed family.
Bragg Diffraction and Reciprocal Space FAQ
Why do theta and two-theta both appear?
Bragg's law uses the angle between the beam and planes, while many instruments report the full separation between incident and diffracted directions. Converting the reported detector angle before substitution prevents a systematic spacing error.
What does crystal rotation change?
Rotation changes the reciprocal lattice orientation relative to the Ewald sphere. Different reciprocal points can meet or leave the diffraction condition even though wavelength, plane spacings and atomic arrangement remain unchanged.
Why is reciprocal spacing inverse?
A plane family with smaller real-space separation produces a reciprocal vector of larger magnitude. This inverse relationship lets diffraction geometry organise many plane families compactly and predicts that smaller d generally appears farther from the reciprocal origin.
How do Miller indices relate to plane intercepts?
Take the intercepts in lattice units, invert the non-infinite values and reduce them to the smallest integers. A plane parallel to an axis has an infinite intercept and therefore a zero index for that direction.
What numerical checks belong after Bragg's law?
Confirm that the sine argument is no greater than one, distinguish theta from two-theta, state the diffraction order and retain compatible wavelength and spacing units. Then verify that smaller spacing maps to a larger scattering angle.
Are the Bragg and Ewald descriptions competing models?
No. They express the same elastic diffraction condition in different representations. The plane construction emphasises path difference, while reciprocal space emphasises whether the required scattering vector meets the geometric selection surface.
Exam move
Alternate numerical Bragg questions with verbal Ewald predictions. For every calculation, write whether the supplied angle is theta or two-theta, state the diffraction order and keep units visible. For every plane-index problem, reconstruct the intercepts after obtaining the Miller indices. Sketch one real-space plane stack beside one reciprocal vector and label d and one divided by d.
Finally, explain rotation without saying that the structure changed: the reciprocal lattice moves relative to the fixed selection construction. This combination of arithmetic and geometry is more robust than memorising isolated formulas. Build a four-line solution template: known quantities with units, governing relation, admissibility check and physical interpretation.
After solving, alter only one input and predict the direction of change before recalculating. Practise translating each plane statement into a reciprocal-vector statement and back again. Include one impossible numerical case so you recognise an arcsine argument above unity as a failed diffraction condition, not a calculator error.
Close by comparing a rotated crystal with a changed lattice parameter: both can move observed features, but they make different claims about the specimen. Add an indexing drill with finite, fractional and infinite plane intercepts. Invert them, clear common factors and reconstruct the original intercept pattern to verify the answer.
Keep parentheses for planes and square brackets for directions, and state why equal index numbers do not make those objects generally interchangeable outside the cubic shortcut. On an Ewald sketch, label the incident vector, scattered vector and their difference before placing the reciprocal point. Explain separately what wavelength, crystal rotation and lattice parameter would change.
End with one sentence distinguishing geometric permission to diffract from the structure factor that controls strength.
