Large Particle Form Factor

This example demonstrates, that for large particles (~1000nm) the contribution to the scattered intensity from the form factor oscillates rapidly within one detector bin and analytical calculations (performed for the bin center) give completely a wrong intensity pattern. In this case Monte-Carlo integrations over detector bin should be used.

The simulation generates four plots using different sizes of the particles, (radius=10 nm, height=20nm) or (radius=1 μm, height=2 μm), and different calculation methods: analytical calculations or Monte-Carlo integration. The other parameters are identical:

  • The sample is made of a monodisperse distribution of cylinders, deposited randomly on a substrate.
  • There is no interference between the scattered waves.
  • The wavelength is equal to 1 Å.
  • The incident angles are αi = 0.2° and Φi = 0°.
Real-space model: 
Intensity Image: 
Python Script: 
Large cylinders in DWBA.

This example demonstrates that for large particles (~1000nm) the formfactor
oscillates rapidly within one detector bin and analytical calculations
(performed for the bin center) give completely wrong intensity pattern.
In this case Monte-Carlo integration over detector bin should be used.
import bornagain as ba
from bornagain import deg, angstrom, nm
from matplotlib import pyplot as plt

default_cylinder_radius = 10*nm
default_cylinder_height = 20*nm

def get_sample(cylinder_radius, cylinder_height):
    Returns a sample with cylindrical particles on a substrate.
    # 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
    cylinder_ff = ba.FormFactorCylinder(cylinder_radius, cylinder_height)
    cylinder = ba.Particle(m_particle, cylinder_ff)
    particle_layout = ba.ParticleLayout()
    particle_layout.addParticle(cylinder, 1.0)

    air_layer = ba.Layer(m_ambience)
    substrate_layer = ba.Layer(m_substrate)

    multi_layer = ba.MultiLayer()
    return multi_layer

def get_simulation(integration_flag):
    Returns a GISAXS simulation with defined beam and detector.
    If integration_flag=True, the simulation will integrate over detector bins.
    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)
    simulation.getOptions().setMonteCarloIntegration(integration_flag, 50)
    return simulation

def run_simulation():
    Run simulation and plot results 4 times: for small and large cylinders,
    with and without integration

    fig = plt.figure(figsize=(12.80, 10.24))

    # conditions to define cylinders scale factor and integration flag
    conditions = [
        {'title': "Small cylinders, analytical calculations",
         'scale': 1,   'integration': False},

        {'title': "Small cylinders, Monte-Carlo integration",
         'scale': 1,   'integration': True},

        {'title': "Large cylinders, analytical calculations",
         'scale': 100, 'integration': False},

        {'title': "Large cylinders, Monte-Carlo integration",
         'scale': 100, 'integration': True}

    # run simulation 4 times and plot results
    for i_plot, condition in enumerate(conditions):
        scale = condition['scale']
        integration_flag = condition['integration']

        sample = get_sample(default_cylinder_radius*scale,
        simulation = get_simulation(integration_flag)
        result = simulation.getIntensityData()

        # plotting results
        plt.subplot(2, 2, i_plot+1)
        plt.subplots_adjust(wspace=0.3, hspace=0.3)


        plt.text(0.0, 2.1, conditions[i_plot]['title'],
                 horizontalalignment='center', verticalalignment='center',

if __name__ == '__main__':