WAV-06

Bragg diffraction

2 d sin θ = n λ.

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DiffractionBragg 1913

Governing equation

2d\sin\theta = n\lambda

$\lambda \in \left[0.1, 5\right]$

where

d: Plane spacing (Å)
\lambda: Wavelength (Å)
n: Order (—)
\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

2d\sin\theta = n\lambda

$\lambda \in \left[0.1, 5\right]$

$\lambda = 0.1$ $\theta = 1.146$

Along the curve

②

Domain of the variable — blue

\lambda_{\min} = 0.1,\quad \lambda_{\max} = 5

$\lambda \in \left[0.1, 5\right]$

$\lambda = 0.1$ $\lambda = 0.1$

Along the curve

③

Response curve — green

2d\sin\theta = n\lambda

$\lambda \in \left[0.1, 5\right]$

$\lambda = 0.1$ $\theta = 1.146$

Along the curve

④

Rate of change — amber

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

$\lambda \in \left[0.1, 5\right]$

$\lambda = 0.131$ $\theta^{\prime} = 11.46$

Along the curve

Lecture brief

Historical brief

d’Alembert’s wave equation, Snell, the thin-lens maker, Doppler and Bragg interference are the classical optics-and-sound toolkit. The lab is propagation, image and shift. This sheet (WAV-06 — Bragg diffraction) is the form associated with Bragg 1913. Working symbols: $d$ , $\lambda$ , $n$ $\rightarrow$ $\theta$ . Constructive interference from successive crystal planes requires the path difference 2 d sin θ to be an integer number of wavelengths.

Purpose

Purpose: compute

\theta

from

d

\lambda

n

in Waves & optics via

2d\sin\theta = n\lambda

2 d sin θ = n λ. Use it when a real waves & optics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

d = 2.500\,\mathrm{Å}

\lambda = 1.540\,\mathrm{Å}

n = 1.000\,\mathrm{—}

, the governing relation

2d\sin\theta = n\lambda

yields

\theta = 17.939\,\mathrm{^{\circ}}

. Specular reflection from planes of spacing d. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Plane spacing d2.5 ÅWavelength \lambda1.54 ÅOrder n1 —

Outputs

Bragg angle \theta17.939 °

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Narration of this film

Specular reflection from planes of spacing d.

Constructive interference from successive crystal planes requires the path difference 2 d sin θ to be an integer number of wavelengths.

Reading speed

Watch on YouTube