the interplanar spacing
Pick a family of parallel lattice planes, say the (111) sheets, and imagine them stacked like the pages of a closed book. The interplanar spacing, written d or d_hkl, is simply the perpendicular gap between one page and the next. It is one of the most useful numbers in crystallography because it is exactly what a diffraction experiment measures, and it links the abstract indices (hkl) to a real distance in angstroms.
For a cubic crystal there is a beautifully simple formula: d = a / sqrt(h^2 + k^2 + l^2), where a is the edge length of the cube. So if a is 4 angstrom, the (111) spacing is 4 / sqrt(3) = 2.31 angstrom, while the (200) spacing is 4 / 2 = 2.00 angstrom. Notice the pattern: the bigger the indices, the more finely the crystal is sliced, and the smaller the spacing. The widest-spaced planes have the smallest indices.
This d-spacing is the d that appears in Bragg's law, n lambda = 2 d sin theta, which is why measuring the angles of diffraction peaks lets you work backwards to the spacings, and from the spacings to the size and shape of the unit cell. In reciprocal-space language the spacing is the inverse of the length of the reciprocal-lattice vector for that plane, 1/d — a hint of why widely spaced planes in real space correspond to points near the origin in the diffraction pattern.
Face-centred cubic nickel has a = 3.52 angstrom. The (111) spacing is 3.52 / sqrt(3) = 2.03 angstrom, and (200) is 3.52 / 2 = 1.76 angstrom. Feeding d = 2.03 angstrom and copper radiation (lambda = 1.54 angstrom) into Bragg's law gives the first-order (111) peak at theta = 22.3 degrees, or 2-theta = 44.5 degrees — right where nickel's strongest peak is measured.
The d-spacing turns plane indices into a measurable distance, and Bragg's law reads it off the peak angle.
Larger indices mean smaller spacing, not larger. The (100) family is more widely spaced than (200), even though both share an orientation.