## Monte-Carlo integration for large qr

This example demonstrates, that for large particles (~$1000$ nm) 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 $=20$ nm) or (radius $=1$ $\mu$m, height $=2$ $\mu$m), 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 0.1 nm.
• The incident angles are $\alpha_i = 0.2 ^{\circ}$ and $\varphi_i = 0^{\circ}$.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105  #!/usr/bin/env python3 """ Large cylinders in DWBA. This example demonstrates that for large particles (~1000nm) the form factor 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 angstrom, ba_plot as bp, deg, nm from matplotlib import pyplot as plt default_cylinder_radius = 10*nm default_cylinder_height = 20*nm def get_sample(cylinder_radius, cylinder_height): # Define materials m_vacuum = ba.RefractiveMaterial("Vacuum", 0, 0) m_substrate = ba.RefractiveMaterial("Substrate", 6e-6, 2e-8) m_particle = ba.RefractiveMaterial("Particle", 6e-4, 2e-8) # Define particle layout cylinder_ff = ba.Cylinder(cylinder_radius, cylinder_height) cylinder = ba.Particle(m_particle, cylinder_ff) particle_layout = ba.ParticleLayout() particle_layout.addParticle(cylinder) # Define layers vacuum_layer = ba.Layer(m_vacuum) vacuum_layer.addLayout(particle_layout) substrate_layer = ba.Layer(m_substrate) # Define sample sample = ba.MultiLayer() sample.addLayer(vacuum_layer) sample.addLayer(substrate_layer) return sample def get_simulation(sample, integration_flag): """ Returns a GISAXS simulation with defined beam and detector. If integration_flag=True, the simulation will integrate over detector bins. """ beam = ba.Beam(1, 1*angstrom, ba.Direction(0.2*deg, 0)) n = bp.simargs['n'] det = ba.SphericalDetector(n, -1.2*deg, 1.2*deg, n, 0, 2*deg) simulation = ba.ScatteringSimulation(beam, sample, det) simulation.options().setMonteCarloIntegration(integration_flag, 50) if not "__no_terminal__" in globals(): simulation.setTerminalProgressMonitor() return simulation def simulate(): ret = [] # conditions to define cylinders scale factor and integration flag conditions = [{ 'title': "Small cylinders, analytical calculations", 'scale': 1, 'integration': False, 'zmin': 1e-5, 'zmax': 1e2 }, { 'title': "Small cylinders, Monte-Carlo integration", 'scale': 1, 'integration': True, 'zmin': 1e-5, 'zmax': 1e2 }, { 'title': "Large cylinders, analytical calculations", 'scale': 100, 'integration': False, 'zmin': 1e-5, 'zmax': 1e10 }, { 'title': "Large cylinders, Monte-Carlo integration", 'scale': 100, 'integration': True, 'zmin': 1e-5, 'zmax': 1e10 }] # run simulation 4 times for i_plot, condition in enumerate(conditions): scale = condition['scale'] integration_flag = condition['integration'] sample = get_sample(default_cylinder_radius*scale, default_cylinder_height*scale) simulation = get_simulation(sample, integration_flag) result = simulation.simulate() result.setTitle(condition['title']) ret.append(result) return ret if __name__ == '__main__': bp.parse_args(sim_n=201, fontsize=12) results = simulate() bp.make_plot(results, 2) 
Examples/scatter2d/LargeParticlesFormFactor.py