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The diffuse intensity of an alloy with α₁ = -0.46

The diffuse intensity of an alloy with α₁ = -0.46
The diffuse intensity of an alloy with α₁ = -0.46. The diffuse part of the scattering across the wavevectors an 12 × 12 block can be asked about, one square per wavevector with darkness the intensity. The marked squares are where the average structure scatters — the sharp part, which is what a Bragg reflection is. The diffuse maximum here is at (0.50, 0.50), which is the zone boundary: the alloy is trying to alternate, and a crystal that succeeded would put a sharp reflection exactly there. Summed over every wavevector, the intensity is exactly one per site whatever the correlations are — order moves scattering about, it does not create it.

The diffuse part of the scattering across the wavevectors an 12 × 12 block can be asked about, one square per wavevector with darkness the intensity. The marked squares are where the *average* structure scatters — the sharp part, which is what a Bragg reflection is. The diffuse maximum here is at (0.50, 0.50), which is the zone boundary: the alloy is trying to alternate, and a crystal that succeeded would put a sharp reflection exactly there. Summed over every wavevector, the intensity is exactly one per site whatever the correlations are — order moves scattering about, it does not create it.

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The diffuse intensity of an alloy with α₁ = -0.46. The diffuse part of the scattering across the wavevectors an 12 × 12 block can be asked about, one square per wavevector with darkness the intensity. The marked squares are where the average structure scatters — the sharp part, which is what a Bragg reflection is. The diffuse maximum here is at (0.50, 0.50), which is the zone boundary: the alloy is trying to alternate, and a crystal that succeeded would put a sharp reflection exactly there. Summed over every wavevector, the intensity is exactly one per site whatever the correlations are — order moves scattering about, it does not create it. How it is known

The average scatters sharply and the rest does not

A crystal whose lattice is perfect and whose occupation is not scatters in two parts: the average structure gives Bragg reflections, and the variance is spread over everything between them. The split is exact, the total is one unit per site whatever the disorder does, and an ensemble of n arrangements mislays exactly a fraction 1/n of it.

 How much one site knows about another, by separation. The Warren–Cowley parameters: the average of the product of the occupations of two sites a given vector apart, over every pair in every configuration. The value at the origin is exactly one — a site always agrees with itself — and it falls away with distance, alternating in sign where the alloy prefers unlike neighbours. These numbers are the whole of what the diffuse scattering measures: its intensity at a wavevector is their Fourier transform, computed here separately and agreeing to the last bits of the arithmetic. Nothing about them requires the crystal to be ordered, and their falling away is what short-range order means. How it is known

The order a diffuse pattern measures

Where a diffuse maximum sits says what the crystal is trying to become, and its shape is the Fourier transform of how much each site knows about its neighbours. The correlations are a small array of numbers, the intensity is their transform, and neither route to the other loses anything.

 A mode with no dipole, landing in a phase that may have one. Two marks per row: the first is filled when the mode itself carries a dipole — the displacements, weighted by charge, summing to something other than zero — and the second when the class of the phase it produces permits a polarisation at all. A row with the first empty and the second filled is an improper case: nothing about the transition was about becoming polar, and the phase that results may be polar anyway, so a polarisation appears as a side effect at second order in an order parameter that is about something else. The zone-boundary rows are where these occur; at the zone centre in the plane there are none, because the only two-dimensional order parameters available there are the polarisation itself. Symmetry at work

The polarisation nobody asked for

A mode whose displacements cancel exactly can still leave a phase whose class permits a polarisation. The crystal then becomes polar as a side effect of a transition that was about something else — and in the plane, at the zone centre, the arithmetic says this cannot happen at all.

 Diffuse scattering gathered under every Bragg peak. The one-phonon thermal diffuse scattering of a square lattice with springs to nearest and next-nearest neighbours, over the reciprocal plane from minus 2.5 to 2.5 in each index, with the Bragg reflections marked. The intensity rises as one over the squared distance to every reflection, because the acoustic phonons' frequencies vanish there, and it grows with the square of the scattering vector, so the lobes are strongest far from the origin. The lobes are not round: a small step from (1,0) along k gives 2.7 times the intensity of the same step along h, because it probes transverse phonons, which are softer, and that ratio is what an experiment reads the elastic constants from. How it is known

A warm plane crystal has no Bragg peaks

Thermal motion is disorder that moves, and in space it costs a crystal a little of each Bragg peak and gathers it into diffuse scattering underneath. In a plane it costs more than a little. Computed exactly from the phonons of a harmonic square lattice, an atom's mean-square wandering grows by the same amount every time the crystal doubles in size, so no Debye–Waller factor survives, and every Bragg peak fades as a power of the crystal's size — a power proportional to the temperature times the square of the reflection's distance from the origin.

 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. How it is known

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.

 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. How it is known

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.

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