Powder Diffraction Pattern Simulator

It predicts where diffraction peaks will appear for a given crystal lattice. You supply the lattice parameter, the radiation wavelength and the lattice type, and the simulator lists every allowed hkl reflection with its d-spacing, 2-theta angle, multiplicity and relative intensity, then plots the resulting pattern.

First compute d = a / sqrt(h squared + k squared + l squared), then apply Bragg's law: sin theta = lambda / (2d). The measured angle is 2-theta = 2 times theta. For example a = 4.0495 angstroms, hkl = 111 and lambda = 1.5406 angstroms give d = 2.3380 angstroms and 2-theta = 38.474 degrees.

Systematic absences are reflections whose structure factor cancels to exactly zero because of lattice centring or symmetry elements. Body-centred cubic allows only h+k+l even, face-centred cubic allows only all-even or all-odd hkl, and diamond cubic further removes all-even reflections whose h+k+l is not a multiple of four.

Look at the ratio of the first few h2+k2+l2 values. Face-centred cubic gives 3, 4, 8, 11, 12, so the first two peaks are a close pair. Body-centred cubic gives 2, 4, 6, 8, 10, an evenly spaced series starting with a single isolated peak. Simple cubic gives 1, 2, 3, 4, 5, 6, 8.

Both have h2+k2+l2 = 27, and in a cubic lattice d-spacing depends only on that sum, so both reflections diffract at exactly the same angle. A powder diffractometer cannot separate them, so this simulator merges them into a single line and adds their multiplicities together.

Multiplicity is the number of symmetry-equivalent planes that diffract at the same angle. In a cubic crystal the 100 family has multiplicity 6, the 111 family has 8, the 110 family has 12, and a general hkl family has 48. Higher multiplicity means more crystallites contribute, so the peak is stronger.

They are geometric estimates, not full-pattern intensities. The model includes multiplicity, the Lorentz-polarization factor and the lattice structure factor, but excludes the atomic scattering factor, thermal Debye-Waller attenuation, preferred orientation and absorption. Expect the correct peak positions and roughly the right pattern of strong versus weak lines.

The Lorentz-polarization factor, LP = (1 + cos squared 2-theta) / (sin squared theta times cos theta), corrects for the geometry of a powder diffractometer and for the partial polarization of the scattered X-rays. It rises sharply at very low and very high angles, which is why it strongly shapes relative peak heights.

Cu K-alpha at 1.5406 angstroms is the laboratory default and is assumed by most published patterns. Mo K-alpha at 0.7107 angstroms reaches many more reflections in the same angular range and is preferred for heavily absorbing samples. Synchrotron work typically uses a tunable wavelength between 0.5 and 1.5 angstroms.

Bragg's law needs sin theta = lambda sqrt(h2+k2+l2) / 2a to be at most 1. If a is smaller than about half the wavelength, even the first reflection is out of reach and no peaks exist. Either use a shorter wavelength or check that the lattice parameter was entered in angstroms.

No, it assumes a cubic cell where d = a / sqrt(h2+k2+l2). Tetragonal, hexagonal and lower-symmetry cells need a separate d-spacing expression with more than one cell parameter, which splits families such as 100 and 001 into separate peaks.

Peak positions move inversely with a. Doubling the lattice parameter halves every d-spacing value in the denominator of Bragg's law, so all peaks shift to lower 2-theta. Small changes in a produce the biggest angular shifts at high 2-theta, which is why lattice-parameter refinement relies on high-angle peaks.