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Choose an original simulated model: a primitive cubic lattice with one identical point scatterer per primitive unit cell, edge $a=0.400\mathrm{nm}$, and incident X-ray wavelength $\lambda =0.200\mathrm{nm}$. These are stipulated lengths, not measurements of sodium chloride. Suppose orientation information has already assigned the two observed reflections the Miller indices $(100)$ and $(110)$. The detector deflection is $2\theta$, where $\theta$ is the Bragg angle.

Use the existing interplanar spacing formula and Bragg's law, with higher diffraction orders absorbed into the reflection indices:The simulated $(100)$ reflection has $\sin \theta =1/4$, hence $2\theta =28.9550^{\circ }$. The $(110)$ reflection has $\sin \theta =\sqrt{2}/4$, hence $2\theta =41.4096^{\circ }$. Their squared sines are $0.0625$ and $0.125$, in ratio $1:2$. Substituting the unrounded angles independently givesAgreement tests a proposed indexing and scale together; disagreement would prompt checks of the indexing, wavelength, angular zero, or the assumed unit cell geometry.

Two peak positions alone do not establish the indexing. Assigning the same two angles to $(110)$ and $(200)$ would give $N=2,4$ and $a=0.400\sqrt{2}\simeq 0.565685\mathrm{nm}$ instead. More reflections, orientation information and systematic absences help discriminate such assignments. An unobserved reflection can be inaccessible, weak or extinguished by the atomic basis; absence is not automatically evidence for a larger interplanar spacing.

Absolute scale also requires measurement calibration of the wavelength. To first order, differentiation of the inferred length giveswith angular changes in radians. A stipulated detector-zero error $\delta (2\theta )=+0.100^{\circ }$, with correct wavelength, biases the first inferred edge downward by approximately $0.338\%$ and the second by $0.231\%$. A common angular offset therefore need not cancel when two inferred lengths are compared. These are model sensitivity estimates, not historical instrument uncertainties.

A lattice plane is a plane containing a two-dimensional periodic array of lattice points. Parallel lattice planes form equally spaced geometrical sheets; an atomic basis can place chemical atoms between them.

With primitive lattice translation vectors $a_i$, write $r=x{a}_1+y{a}_2+z{a}_3$. For integers $h,k,l$ with greatest common divisor one, the plane equationsgive a complete family labeled by Miller indices $(hkl)$. In a primitive cubic lattice, $(110)$ means $x+y=m$, parallel to $a_3$.^[ref]

Three Miller indices $(hkl)$ specify a lattice plane orientation in chosen unit cell axes. Choose a parallel plane away from the origin, express its intercepts in axis units, take reciprocals, and clear fractions. An infinite intercept gives index zero; a negative intercept gives a negative index, written with an overbar.^[ref]

For example, intercepts $(2a,3b,\infty )$ have reciprocal ratios $(1/2,1/3,0)$, so the orientation is $(320)$ after multiplying by six. Relative to the same axes, intercepts $(a,-b,\infty )$ give $(1\bar{1}0)$. Choosing a different parallel member changes the common intercept scale, not the orientation.

Primitive axes require relatively prime Miller indices for complete families. Centered conventional unit cells need not, as the FCC lattice $(200)$ example shows. Reflection labels can also retain higher diffraction orders. Use the reciprocal lattice for the normal vector with oblique axes; the direct vector $h{a}_1+k{a}_2+l{a}_3$ need not be perpendicular.

Consider a chosen primitive cubic lattice $a{\mathbb{Z}}^3$ with $a=0.400\mathrm{nm}$ and one identical point scatterer per primitive unit cell. Retain the incident wavelength $\lambda =0.154\mathrm{nm}$. This is an ideal point model. Its reciprocal lattice vectors are $q=(2\pi /a)(h,k,l)$ 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 satisfyThe reflection labels $(hkl)$ 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 $0.200000\mathrm{nm}$. Their calculated deflections $2\theta$ are $22.197435^{\circ }$, $31.594744^{\circ }$, $38.952948^{\circ }$ and $45.287481^{\circ }$. Equal steps in $S$ give equal steps in ${\sin }^2\theta$, not equal steps in $2\theta$. 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]