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1D paracrystal

In this example we simulate the scattering from a grating with cumulative spacing disorder. This is done by using Paracrystal1D together with very long boxes.

Paracrystal1D describes a chain of particles along one in-plane direction, with distances between neighbors drawn from a probability distribution. Position uncertainties accumulate with the distance along the chain, as in the ideal one-dimensional paracrystal model. The structure factor depends on the projection of the scattering vector onto the chain direction.

  • For the perfectly ordered counterpart of this structure, see the 1D lattice.
  • For the isotropic, angular-averaged paracrystal, see the radial paracrystal.

Scattering intensity

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#!/usr/bin/env python3
"""
Simulation of grating using very long boxes and cumulative 1D paracrystal order.
Monte-carlo integration is used to get rid of
large-particle form factor oscillations.
"""
import bornagain as ba
from bornagain import angstrom, ba_plot as bp, deg, nm


def get_sample():
    """
    A sample with a grating on a substrate,
    modelled by very long boxes forming a 1D paracrystal.
    """

    # Materials
    particle_color = (0.86, 0.24, 0.18)
    particle_mat = ba.RefractiveMaterial("Particle", particle_color, 6e-5, 2e-8)
    substrate_color = (0.28, 0.57, 0.82)
    substrate_mat = ba.RefractiveMaterial("Substrate", substrate_color, 6e-6, 2e-8)
    vacuum = ba.Vacuum()

    # Particles
    box_length = 10000*nm
    ff = ba.Box(10*nm, box_length, 10*nm)
    particle = ba.Particle(particle_mat, ff)
    particle_rotation = ba.RotationZ(45*deg)
    particle.rotate(particle_rotation)

    # Interference functions
    layout = ba.Paracrystal1D(particle, 30*nm, 45*deg, 1/box_length, 1000*nm)
    profile = ba.Profile1DGauss(4*nm)
    layout.setProbabilityDistribution(profile)

    # Layers
    layer_1 = ba.Layer(vacuum)
    layer_1.deposit2D(layout)
    layer_2 = ba.Layer(substrate_mat)

    # Sample
    sample = ba.Sample()
    sample.addLayer(layer_1)
    sample.addLayer(layer_2)

    return sample


def get_simulation(sample):
    beam = ba.Beam(1e10, 1*angstrom, 0.2*deg)
    n = 101
    det = ba.SphericalDetector(n, -1*deg, 1*deg, n, 0, 2*deg)
    simulation = ba.ScatteringSimulation(beam, sample, det)
    simulation.options().setMonteCarloIntegration(True, 100, seed=0)
    if not "__no_terminal__" in globals():
        simulation.setTerminalProgressMonitor()
    return simulation


if __name__ == '__main__':
    sample = get_sample()
    simulation = get_simulation(sample)
    result = simulation.simulate()
    ba.showSample3D(sample, sample_size=300*nm, seed=0)
    bp.plot_datafield(result, intensity_min=1e-3, unit_aspect=1)
    bp.plt.show()
auto/Examples/scatter2d/Paracrystal1DGrating.py

Paracrystal1D constructor

The structure is created with

structure = ba.Paracrystal1D(particle, length, xi, linear_density,
                             damping_length)

Arguments:

particle        # particle or Mixture placed at lattice sites
length          # mean nearest-neighbor distance, in nanometers
xi              # rotation of the lattice with respect to x-axis, in radians
linear_density  # number of lattice lines per nanometer in the transverse axis
damping_length  # damping/coherence length, in nanometers

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

Probability distribution

The distribution of nearest-neighbor distances must be assigned with setProbabilityDistribution(pdf) before the structure can be used in a simulation:

structure = ba.Paracrystal1D(particle, 30*nm, 0, 1/box_length, 1000*nm)
structure.setProbabilityDistribution(ba.Profile1DGauss(4*nm))

The available distributions are listed in the radial paracrystal reference. A zero-width profile represents the ordered limit: together with a finite damping_length, the structure factor then equals that of a Crystal1D with decay function ba.Profile1DCauchy(damping_length).

Domain size

The scattering from a finite portion of the chain can be calculated using the setDomainSize(nm) method. The resulting behaviour is similar to the case when damping_length is used.

Position variance

Independent (non-cumulative) positional disorder can be added with setLateralPositionVariance(value), where value is given in nm$^2$. It is applied through a Debye-Waller factor, as in Crystal1D.