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Radial paracrystal

The radial paracrystal structure is used to model cumulative disorder of interparticle distances. It is called radial to stress the fact that it only takes into account the radial component of the scattering vector.

Each circle on the plot above represents the area where the probability to find a particle, given a particle at the origin, is above some arbitrary threshold. The growing size of the areas emphasizes the fact the our knowledge about next neighbor’s location decreases with the distance to the origin.

The BornAgain user manual (Chapter 3.5, Paracrystal) details the theoretical model and gives some links to the literature.

RadialParacrystal class

The radial paracrystal is parameterized by the position distribution of the nearest neighbor centered at the peak distance. On the plot below the half-width of this distribution is marked as $\omega$ and the center is marked with “Peak Distance”. The probability distributions of finding other particles to the right are deduced from accumulating position uncertanties of previous particles in the chain.

To create a radial paracrystal, use the following constructor:

structure = ba.RadialParacrystal(particle, peak_distance, damping_length)

Arguments:

particle        # particle or Mixture placed at lattice sites
peak_distance   # average distance to the next neighbor, in nanometers
damping_length  # damping/coherence length, in nanometers

The parameter damping_length is used to introduce finite size effects by applying a multiplicative coefficient equal to $exp \left(-\frac{peak\_distance}{damping\_length}\right)$ to the Fourier transform of the probability density of a nearest neighbor. A value of 0 means no damping-length correction.

Probability distribution

To account for next neighbor position uncertainty, assign a probability distribution to the structure with setProbabilityDistribution(pdf):

structure = ba.RadialParacrystal(particle, 10*nm, 1000*nm)
structure.setProbabilityDistribution(ba.Profile1DCauchy(30*nm))

The following distributions are available:

ba.Profile1DCauchy(omega)
ba.Profile1DGauss(omega)
ba.Profile1DGate(omega)
ba.Profile1DTriangle(omega)
ba.Profile1DCosine(omega)
ba.Profile1DVoigt(omega, eta)

The parameter omega sets the half-width of the distribution in nanometers. In the case of the pseudo-Voigt distribution, an additional dimensionless parameter eta balances between the Gaussian and Cauchy profiles.

Domain size

The radial paracrystal provides a way to calculate the scattering from a finite portion of the paracrystal using the setDomainSize(nm) method. The resulting behaviour is similar to the case when damping_length is used.

structure = ba.RadialParacrystal(particle, 10*nm, 0)
structure.setProbabilityDistribution(ba.Profile1DCauchy(30*nm))
structure.setDomainSize(10000*nm)

Particle mixtures

For particle mixtures (DA/SSCA approximations), use the Mixture class:

mix = ba.Mixture()
mix.addParticle(particle_1, 0.8)
mix.addParticle(particle_2, 0.2)

structure = ba.RadialParacrystal(mix, 18*nm, 1000*nm)
structure.setProbabilityDistribution(profile)

The mixture can be used anywhere a particle is accepted. Its vertical position is controlled by the layer method (deposit2D or suspend2D) and by any translations already applied to the particles.

Example

The radial paracrystal model is used to demonstrate the differences in size-distribution models.