Concept

Debye waller factor — where it appears

The factor by which thermal motion reduces a Bragg reflection, the exponential of minus half the momentum transfer squared times the mean-square displacement. When that displacement grows without bound with the crystal's size, the factor no longer settles and the peak is lost.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as phonon, thermal diffuse scattering — the same set of essays touches all of them, so they are one junction rather than several.

Along the rod above a reflection, a free sheet loses its peak. The Bragg intensity per site squared of a warm square-lattice sheet at the in-plane reflection (1, 0) and at heights 0.1, 0.2 and 0.3 along the rod above it, against the sheet's size, on logarithmic axes. In the plane the peak falls as a power of the size whatever the sheet does out of it. Above the plane a free sheet's peak collapses faster and faster, to 2.3e-2 at size 128 and height 0.3; under tension every height falls as a power, to 0.38.

A flapping sheet keeps its peaks in the plane

A warm sheet one atom thick wanders out of its plane far more than within it: a free sheet with only bending stiffness has a mean-square height that grows with its area. That wandering cannot touch a reflection whose scattering vector lies in the plane, which fades exactly as it would if the sheet were held flat. Every reflection with a component along the normal loses its peak faster than any power of the size. Tension puts it back on a power law.

diffraction · Disorder

Every stacking reflection of a warm stack of sheets falls as a power of its size. The Bragg peak per site squared per layer squared of the stacking reflections (0, 0, l), l = 1 to 5, of a stack of flapping square-lattice sheets with bending stiffness 1 and coupling 0.1 between neighbouring layers, at kT/k₁ = 0.02, against the stack's width, with the number of layers growing as the width squared. Each order falls as a fixed power of the size, and the powers grow as the square of the order: 0.20, 0.80, 1.79 and 3.18 for the first four. The fifth falls as the fourth power, as fast as the crystal's volume grows, which is what a reflection with no peak at all does.

A stack lends each sheet a tension

A single warm sheet wanders so far out of its plane that every reflection with a component along its normal is lost. Stack the sheets and couple each to its neighbours, and inside the stack each sheet wanders only as a sheet under tension does, logarithmically. The stacking reflections survive as peaks that fall as powers of the crystal's size, and the power grows as the square of the order, so the fifth order of the stack computed here has no peak at all. A stack of finitely many sheets keeps even the first only out to a width growing as the square root of its thickness.

diffraction · Disorder

Named alongside it

The objects these essays reach for when they reach for this one.

Bragg peak Disorder Long-range order Phonon Thermal diffuse scattering

All concepts