MAT-29

Bragg diffraction

n λ = 2 d sinθ. Constructive interference from crystal planes.

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CrystallographyBragg 1913

Governing equation

n\lambda=2d\sin\theta

$d \in \left[0.8, 6\right]$

where

n: Order (—)
\lambda: Wavelength (Å)
d: Plane spacing (Å)
\theta: Bragg angle (°)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

n\lambda=2d\sin\theta

$d \in \left[0.8, 6\right]$

$d = 0.8$ $\theta = 74.26$

Along the curve

②

Domain of the variable — blue

d_{\min} = 0.8,\quad d_{\max} = 6

$d \in \left[0.8, 6\right]$

$d = 0.8$ $d = 0.8$

Along the curve

③

Response curve — green

n\lambda=2d\sin\theta

$d \in \left[0.8, 6\right]$

$d = 0.8$ $\theta = 74.26$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\theta}{\mathrm{d}d}

$d \in \left[0.8, 6\right]$

$d = 0.833$ $\theta^{\prime} = -174.6$

Along the curve

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-29 — Bragg diffraction) is the form associated with Bragg 1913. Working symbols: $n$ , $\lambda$ , $d$ $\rightarrow$ $\theta$ . Bragg treated lattice planes as mirrors. Path difference 2 d sinθ is an integer number of wavelengths.

Purpose

Purpose: compute

\theta

from

n

\lambda

d

in Materials via

n\lambda=2d\sin\theta

n λ = 2 d sinθ. Constructive interference from crystal planes. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n = 1.000\,\mathrm{—}

\lambda = 1.540\,\mathrm{Å}

d = 2.050\,\mathrm{Å}

, the governing relation

n\lambda=2d\sin\theta

yields

\theta = 22.06\,\mathrm{^{\circ}}

. Two planes, an incident beam, a diffracted kick. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Order n1 —Wavelength \lambda1.54 ÅPlane spacing d2.05 Å

Outputs

Bragg angle \theta22.06 °

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Narration of this film

Two planes, an incident beam, a diffracted kick.

Bragg treated lattice planes as mirrors. Path difference 2 d sinθ is an integer number of wavelengths.

Reading speed

Watch on YouTube