def main(Qstr):
Q = float(Qstr)
if shape == 'sphere':
print("exact", NORM * sp.sas_3j1x_x(Q * RADIUS)**2)
print("gauss-20", *gauss_quad_2d(Q, n=20))
print("gauss-76", *gauss_quad_2d(Q, n=76))
print("gauss-150", *gauss_quad_2d(Q, n=150))
print("gauss-500", *gauss_quad_2d(Q, n=500))
print("gauss-1025", *gauss_quad_2d(Q, n=1025))
print("gauss-2049", *gauss_quad_2d(Q, n=2049))
print("gauss-20 usub", *gauss_quad_usub(Q, n=20))
print("gauss-76 usub", *gauss_quad_usub(Q, n=76))
print("gauss-150 usub", *gauss_quad_usub(Q, n=150))
#gridded_2d(Q, n=2**8+1)
gridded_2d(Q, n=2**10 + 1)
#gridded_2d(Q, n=2**12+1)
#gridded_2d(Q, n=2**15+1)
if shape not in ('paracrystal', 'core_shell_parallelepiped'):
# adaptive forms on models for which the calculations are fast enough
print("dblquad", *scipy_dblquad_2d(Q))
print("semi-romberg-100", *semi_romberg_2d(Q, n=100))
print("romberg", *scipy_romberg_2d(Q))
with mp.workprec(100):
print("mpmath", *mp_quad_2d(mp.mpf(Qstr), shape))
plot_2d(Q, n=200)
def sphere(qab, qc):
q = sqrt(qab**2 + qc**2)
return sas_3j1x_x(q * radius)
def Iq(q, sld, sld_shell, sld_poly, sld_solvent, radius, t_shell, poly_sig,
C_infty, M0, Mn, nu, v):
# Bond angles
theta0 = 68.0 * pi / 180.0
# Kuhn length
b = C_infty * 1.54 / cos(theta0 / 2.0)
# Deg. of polymerization
N = (Mn / M0) * cos(theta0 / 2.0) / C_infty
# Volume of core regions:
Rcoreshell = radius + t_shell
Vcore = 4.0 / 3.0 * pi * radius**3
Vcoreshell = 4.0 / 3.0 * pi * (radius + t_shell)**3
# Number of grafted chains per core:
Ng = poly_sig * 4.00 * pi * (0.1 * Rcoreshell) * (0.1 * Rcoreshell)
Vtotal = Vcoreshell + Ng * N * v
# One over excl. vol. parm.:
onu = 1.0 / nu
o2nu = 1.0 / 2.0 / nu
# Propagator function:
Ea = sas_sinx_x(q * (Rcoreshell))
# Polymer size variable
Usub = (q * b)**2 * N**(2 * nu) / 6.0
# Form factor amplitude of core-shell sphere:
with errstate(divide='ignore'):
Fs = (sld - sld_shell) * Vcore * sas_3j1x_x(q * radius) + (
sld_shell - sld_solvent) * Vcoreshell * sas_3j1x_x(q * Rcoreshell)
# Form factor amplitude of the polymer:
with errstate(divide='ignore'):
Fp = N * v * o2nu * power(
Usub, -o2nu) * sas_gamma(o2nu) * sas_gammainc(o2nu, Usub)
# Form factor of the polymer (Pp(q) is not simply Fp(q)^2!!):
with errstate(divide='ignore'):
Pp = (N * v)**2 * (onu * power(Usub, -o2nu) * sas_gamma(o2nu) *
sas_gammainc(o2nu, Usub) - onu * power(Usub, -onu) *
sas_gamma(onu) * sas_gammainc(onu, Usub))
# Combine all terms to form intensity:
#
# Term 1: Core-shell particle:
inten = Fs * Fs
# Term 2: Polymer
inten = inten + Ng * (sld_poly - sld_solvent) * (sld_poly -
sld_solvent) * Pp
# Term 3: Particle/polymer crossterm:
inten = inten + 2.0 * Ng * (sld_poly - sld_solvent) * Fs * Ea * Fp
# Term 4: Polymer/polymer crossterm:
inten = inten + Ng * (Ng - 1) * (sld_poly - sld_solvent) * (
sld_poly - sld_solvent) * Fp * Ea * Ea * Fp
with errstate(divide='ignore'):
inten = inten * 1.0e-4 / Vtotal
return inten
def Iq(q,
sld,
sld_poly,
sld_solvent,
radius=60,
poly_sig=0.50,
rg=40,
nu=0.5,
v_poly=30):
"""
:param q: Input q-value
:param sld: Core scattering length density
:param sld_poly: Polymer scattering length density
:param sld_solvent: Solvent scattering length density
:param radius: Core radius
:param poly_sig: Polymer grafting density
:param rg: Grafted polymer radius of gyration
:param nu: Grafted polymer excluded volume parameter
:param v_poly: Volume of one polymer
:return: Calculated intensity
"""
# Number of grafted chains per core:
Ng = poly_sig * 4.00 * pi * (0.1 * radius) * (0.1 * radius)
# Volume of core regions:
Vcore = 4.0 / 3.0 * pi * radius**3
Vtotal = Vcore + Ng * v_poly
# One over excl. vol. parm.:
onu = 1.0 / nu
o2nu = 1.0 / 2.0 / nu
# Propagator function:
Ea = sas_sinx_x(q * radius)
# Polymer size variable
Usub = (q * rg)**2 * (2.0 * nu + 1.0) * (2.0 * nu + 2.0) / 6.0
# Form factor amplitude of core-shell sphere:
with errstate(divide='ignore'):
Fs = 3.0 * (sld - sld_solvent) * Vcore * sas_3j1x_x(q * radius)
# Form factor amplitude of the polymer:
with errstate(divide='ignore'):
Fp = o2nu * power(Usub, -o2nu) * sas_gamma(o2nu) * sas_gammainc(
o2nu, Usub)
# Form factor of the polymer (Pp(q) is not simply Fp(q)^2!!):
with errstate(divide='ignore'):
Pp = onu * power(Usub, -o2nu) * sas_gamma(o2nu) * sas_gammainc(
o2nu, Usub) - onu * power(
Usub, -onu) * sas_gamma(onu) * sas_gammainc(onu, Usub)
# Combine all terms to form intensity:
#
# Term 1: Core-shell particle:
inten = Fs * Fs
# Term 2: Polymer
inten = inten + Ng * v_poly * v_poly * (sld_poly - sld_solvent) * (
sld_poly - sld_solvent) * Pp
# Term 3: Particle/polymer crossterm:
inten = inten + 2.0 * Ng * v_poly * (sld_poly - sld_solvent) * Fs * Ea * Fp
# Term 4: Polymer/polymer crossterm:
inten = inten + Ng * (Ng - 1) * v_poly * v_poly * (
sld_poly - sld_solvent) * (sld_poly - sld_solvent) * Fp * Ea * Ea * Fp
with errstate(divide='ignore'):
inten = inten * 1.0e-6 * 1.0e-6 * 1.0e8 / Vtotal
return inten
def main():
import argparse
parser = argparse.ArgumentParser(
description="asymmetric integration explorer",
formatter_class=argparse.ArgumentDefaultsHelpFormatter,
)
parser.add_argument('-s',
'--shape',
choices=SHAPES,
default='parallelepiped',
help='oriented shape')
parser.add_argument('-q',
'--q_value',
type=str,
default='0.005',
help='Q value to evaluate')
parser.add_argument('pars',
type=str,
nargs='*',
default=[],
help='p=val for p in shape parameters')
opts = parser.parse_args()
pars = {k: v for par in opts.pars for k, v in [par.split('=')]}
build_shape(opts.shape, **pars)
Q = float(opts.q_value)
if opts.shape == 'sphere':
print("exact", NORM * sp.sas_3j1x_x(Q * RADIUS)**2)
# Methods from quadpy, if quadpy is available
# AlbrechtCollatz: [1-5]
# BazantOh: 9, 11, 13
# HeoXu: 13, 15, 17, 19-[1-2], 21-[1-6], 23-[1-3], 25-[1-2], 27-[1-3],
# 29, 31, 33, 35, 37, 39-[1-2]
# FliegeMaier: 4, 9, 16, 25
# Lebedev: 3[a-c], 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35,
# 41, 47, 53, 59, 65, 71, 77 83, 89, 95, 101, 107, 113, 119, 125, 131
# McLaren: [1-10]
# Stroud: U3 3-1, U3 5-[1-5], U3 7-[1-2], U3 8-1, U3 9-[1-3],
# U3 11-[1-3], U3 14-1
quadpy_method(Q, "AlbrechtCollatz:5")
quadpy_method(Q, "HeoXu:39-2")
quadpy_method(Q, "FliegeMaier:25")
quadpy_method(Q, "Lebedev:19")
quadpy_method(Q, "Lebedev:131")
quadpy_method(Q, "McLaren:10")
quadpy_method(Q, "Stroud:U3 14-1")
print("gauss-20 points=%d => %.15g" % gauss_quad_2d(Q, n=20))
print("gauss-76 points=%d => %.15g" % gauss_quad_2d(Q, n=76))
print("gauss-150 points=%d => %.15g" % gauss_quad_2d(Q, n=150))
print("gauss-500 points=%d => %.15g" % gauss_quad_2d(Q, n=500))
print("gauss-1025 points=%d => %.15g" % gauss_quad_2d(Q, n=1025))
print("gauss-2049 points=%d => %.15g" % gauss_quad_2d(Q, n=2049))
print("gauss-20 usub points=%d => %.15g" % gauss_quad_usub(Q, n=20))
print("gauss-76 usub points=%d => %.15g" % gauss_quad_usub(Q, n=76))
print("gauss-150 usub points=%d => %.15g" % gauss_quad_usub(Q, n=150))
#gridded_2d(Q, n=2**8+1)
gridded_2d(Q, n=2**10 + 1)
#gridded_2d(Q, n=2**12+1)
#gridded_2d(Q, n=2**15+1)
# adaptive forms on models for which the calculations are fast enough
SLOW_SHAPES = {
'fcc_paracrystal',
'bcc_paracrystal',
'sc_paracrystal',
'core_shell_parallelepiped',
}
if opts.shape not in SLOW_SHAPES:
print("dblquad", *scipy_dblquad_2d(Q))
print("semi-romberg-100", *semi_romberg_2d(Q, n=100))
print("romberg", *scipy_romberg_2d(Q))
with mp.workprec(100):
print("mpmath", *mp_quad_2d(mp.mpf(opts.q_value)))
plot_2d(Q, n=200)