# Interference 2D Centered Square Lattice

Scattering from cylinders distributed along a squared centered lattice.

• The particles are cylinders with radii and heights of 3 nm.
• Their spatial distribution is composed of two square lattices, shifted by half a lattice length in both directions:
• The first square lattice is centered at the origin, with a lattice length of 25 nm.
• The second one, with the same lattice spacing and the same type of particles at its nodes is initialized at x = y = lattice length/2 = 12.5 nm.
• The lattices' base vectors are parallel to the axes of the reference cartesian frame.
• The wavelength is equal to 1 Å.
• The incident angles are αi = 0.2° and Φi = 0°.
Real-space model:
Intensity Image:
Python Script:
"""
2D lattice with disorder, centered square lattice
"""
import numpy
import bornagain as ba
from bornagain import deg, angstrom, nm

def get_sample():
"""
Returns a sample with cylinders on a substrate,
forming a 2D centered square lattice
"""
# defining materials
m_ambience = 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)

# collection of particles
interference = ba.InterferenceFunction2DLattice.createSquare(25.0*nm)
pdf = ba.FTDecayFunction2DCauchy(300.0*nm/2.0/numpy.pi,
100.0*nm/2.0/numpy.pi)
interference.setDecayFunction(pdf)

particle_layout = ba.ParticleLayout()
position1 = ba.kvector_t(0.0, 0.0, 0.0)
position2 = ba.kvector_t(12.5*nm, 12.5*nm, 0.0)
cylinder_ff = ba.FormFactorCylinder(3.*nm, 3.*nm)
cylinder = ba.Particle(m_particle, cylinder_ff)
basis = ba.ParticleComposition()
particle_layout.setInterferenceFunction(interference)

# assembling the sample
air_layer = ba.Layer(m_ambience)
substrate_layer = ba.Layer(m_substrate)

multi_layer = ba.MultiLayer()

print(multi_layer.treeToString())
return multi_layer

def get_simulation():
"""
Create and return GISAS simulation with beam and detector defined
"""
simulation = ba.GISASSimulation()
simulation.setDetectorParameters(200, -2.0*deg, 2.0*deg,
200, 0.0*deg, 2.0*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.getIntensityData()

if __name__ == '__main__':
result = run_simulation()
ba.plot_intensity_data(result)