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 $\phi_i = 0^{\circ}$.

Real-space model

Intensity image

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"""
Spheres on two hexagonal close packed layers
"""
import numpy
import bornagain as ba
from bornagain import deg, angstrom, nm


def get_sample():
    """
    Returns a sample with spheres on a substrate,
    forming two hexagonal close packed layers.
    """
    m_air = ba.HomogeneousMaterial("Air", 0.0, 0.0)
    m_substrate = ba.HomogeneousMaterial("Substrate", 6e-6, 2e-8)
    m_particle = ba.HomogeneousMaterial("Particle", 6e-4, 2e-8)

    radius = 10.0*nm
    sphere_ff = ba.FormFactorFullSphere(radius)
    sphere = ba.Particle(m_particle, sphere_ff)
    particle_layout = ba.ParticleLayout()

    pos0 = ba.kvector_t(0.0, 0.0, 0.0)
    pos1 = ba.kvector_t(radius, radius, numpy.sqrt(3.0)*radius)
    basis = ba.ParticleComposition()
    basis.addParticles(sphere, [pos0, pos1])
    particle_layout.addParticle(basis)

    interference = ba.InterferenceFunction2DLattice.createHexagonal(radius*2.0)
    pdf = ba.FTDecayFunction2DCauchy(10*nm, 10*nm)
    interference.setDecayFunction(pdf)

    particle_layout.setInterferenceFunction(interference)

    air_layer = ba.Layer(m_air)
    air_layer.addLayout(particle_layout)
    substrate_layer = ba.Layer(m_substrate, 0)
    multi_layer = ba.MultiLayer()
    multi_layer.addLayer(air_layer)
    multi_layer.addLayer(substrate_layer)
    return multi_layer


def get_simulation():
    """
    Returns a GISAXS simulation with beam and detector defined.
    """
    simulation = ba.GISASSimulation()
    simulation.setDetectorParameters(200, -1.0*deg, 1.0*deg,
                                     200, 0.0*deg, 1.0*deg)
    simulation.setBeamParameters(1.0*angstrom, 0.2*deg, 0.0*deg)
    return simulation


def run_simulation():
    """
    Runs simulation and returns intensity map.
    """
    simulation = get_simulation()
    simulation.setSample(get_sample())
    simulation.runSimulation()
    return simulation.result()


if __name__ == '__main__':
    result = run_simulation()
    ba.plot_simulation_result(result)
HexagonalLatticesWithBasis.py