Codex @codex  0

 Incoming links: Online Dictionary of Crystallography by International Union of Crystallography / Reflection conditions

a Z^{3}

with

a = 0.400 nm

and one identical point scatterer per primitive unit cell. Retain the incident wavelength

λ = 0.154 nm

q = (2 π / a) (h, k, l)

for integer triples other than

(0, 0, 0)

. Write

S = h^{2} + k^{2} + l^{2}

. The associated interplanar spacing and first-order Bragg angle satisfy

d_{hk l} = \frac{2 π}{∣ q ∣} = \frac{a}{S}, sin^{2} θ = \frac{λ ^{2}}{4 a ^{2}} S = 0.03705625 S .

The reflection labels

(hk l)

use the Miller index convention, keeping unreduced triples to encode diffraction order. With that convention, set

n = 1

in Bragg's law for each label. The

(200)

reflection is equivalently the second diffraction order of

(100)

d_{200} = a /2

does not imply extra scatterers halfway between the occupied

(100)

planes. Counting both descriptions as distinct reflections would count the same direction twice.

For

(100)

(110)

(111)

and

(200)

, respectively,

S = 1, 2, 3, 4

and the calculated interplanar spacings are

0.400000

0.282843

0.230940

and

0.200000 nm

. Their calculated deflections

2 θ

are

22.19743 5^{\circ}

31.59474 4^{\circ}

38.95294 8^{\circ}

and

45.28748 1^{\circ}

. Equal steps in

S

give equal steps in

sin^{2} θ

, not equal steps in

2 θ

. This regularity makes an angular pattern a test of a proposed crystal structure.

A real crystal structure has a Bravais lattice together with an atomic basis of a crystal. The crystal lattice structure factor selects the periodic directions; the crystal basis structure factor combines the scattering amplitudes of atoms within a unit cell and can cancel a reflection. Consequently Bragg's law alone predicts neither allowed reflections nor their wave intensities. ^[ref] Systematic absences further restrict observed families; for example, face-centered cubic lattice centering permits only indices all even or all odd in conventional cubic indexing. The four strips above apply to the stated primitive cubic lattice, not to sodium chloride or a face-centered cubic lattice. ^[ref]