Hexagonal lattices with basis

Scattering from two layers of spheres distributed along a hexagonal close packed structure.

  • The sample is made of spherical particles deposited on a substrate.
  • These $10$-nanometer-radius particles are distributed along a hexagonal close packed structure:
    • each layer is generated using a two-dimensional hexagonal lattice with a lattice length of $20$ nm and its $a$-axis parallel to the $x$-axis of the reference Cartesian frame.
    • the vertical stacking is done by specifying the position of a “seeding” particle for each layer: $(0,0,0)$ for the first layer and $(R,R,\sqrt{3}R)$ for the second layer, $R$ being the radius of the spherical particle.
  • The wavelength is equal to $1$ $\unicode{x212B}$.
  • The incident angles are $\alpha_i = 0.2 ^{\circ}$ and $\varphi_i = 0^{\circ}$.

Real-space model

Intensity image

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#!/usr/bin/env python3
"""
Spheres on two hexagonal close packed layers
"""
import bornagain as ba
from bornagain import deg, nm, kvector_t


def get_sample():
    """
    Returns a sample with spheres on a substrate,
    forming two hexagonal close packed layers.
    """

    # Define materials
    material_Particle = ba.HomogeneousMaterial("Particle", 0.0006, 2e-08)
    material_Substrate = ba.HomogeneousMaterial("Substrate", 6e-06, 2e-08)
    material_Vacuum = ba.HomogeneousMaterial("Vacuum", 0.0, 0.0)

    # Define form factors
    ff_1 = ba.FormFactorFullSphere(10.0*nm)
    ff_2 = ba.FormFactorFullSphere(10.0*nm)

    # Define particles
    particle_1 = ba.Particle(material_Particle, ff_1)
    particle_2 = ba.Particle(material_Particle, ff_2)
    particle_2_position = kvector_t(10.0*nm, 10.0*nm, 17.3205080757*nm)
    particle_2.setPosition(particle_2_position)

    # Define composition of particles at specific positions
    particle_3 = ba.ParticleComposition()
    particle_3.addParticle(particle_1)
    particle_3.addParticle(particle_2)

    # Define 2D lattices
    lattice = ba.BasicLattice2D(20.0*nm, 20.0*nm, 120.0*deg, 0.0*deg)

    # Define interference functions
    iff = ba.InterferenceFunction2DLattice(lattice)
    iff_pdf = ba.FTDecayFunction2DCauchy(10.0*nm, 10.0*nm, 0.0*deg)
    iff.setDecayFunction(iff_pdf)

    # Define particle layouts
    layout = ba.ParticleLayout()
    layout.addParticle(particle_3, 1.0)
    layout.setInterferenceFunction(iff)
    layout.setWeight(1)
    layout.setTotalParticleSurfaceDensity(0.00288675134595)

    # Define layers
    layer_1 = ba.Layer(material_Vacuum)
    layer_1.addLayout(layout)
    layer_2 = ba.Layer(material_Substrate)

    # Define sample
    sample = ba.MultiLayer()
    sample.addLayer(layer_1)
    sample.addLayer(layer_2)

    return sample


def get_simulation(sample):
    beam = ba.Beam(1.0, 0.1*nm, ba.Direction(0.2*deg, 0*deg))
    detector = ba.SphericalDetector(200, -1*deg, 1*deg, 200, 0*deg, 1*deg)
    simulation = ba.GISASSimulation(beam, sample, detector)
    return simulation


if __name__ == '__main__':
    import ba_plot
    sample = get_sample()
    simulation = get_simulation(sample)
    ba_plot.run_and_plot(simulation)
HexagonalLatticesWithBasis.py