In [1]:
import os
os.environ["SAS_OPENCL"] = "cuda" # use CUDA GPU backend for sasmodels
import escape as esc
import numpy as np
esc.require("0.9.8")
Loading material database from C:\dev\escape-core\python\src\escape\scattering\..\data\mdb\materials.db
SAXS. Form-factors. Hollow cylinder (SasView-aligned)¶
A rigid hollow tube (cylindrical shell) with uniform SLD; the form factor is normalised by the shell volume.
Reference: https://www.sasview.org/docs/user/models/hollow_cylinder.html
Parameters (SasView defaults)¶
| Parameter | Variable | Value |
|---|---|---|
| Scale | scale |
1 |
| Background (cm⁻¹) | background |
0.001 |
| Core radius (Å) | radius_core |
20 |
| Wall thickness (Å) | thickness |
10 |
| Length (Å) | length |
400 |
| Contrast Δρ (10⁻⁶ Å⁻²) | contrast |
5.3 (= sld 6.3 − sld_solvent 1) |
| Theta (deg), 2D only | theta |
90 |
| Phi (deg), 2D only | phi |
0 |
Form-factor (SasView hollow_cylinder.c)¶
Define $\gamma = R_c / R_o$ (ratio of inner to outer radius) and the cross-section factor:
$$\Psi(q_\perp) = \frac{\Lambda(R_o\,q_\perp) - \gamma^2\,\Lambda(R_c\,q_\perp)}{1-\gamma^2}, \quad \Lambda(a) = \frac{2J_1(a)}{a}$$
The oriented amplitude:
$$F(q,\alpha) = \Psi(q_\perp)\;\mathrm{sinc}\!\left(\tfrac{L}{2}q_\parallel\right)$$
$$I(q) = \mathrm{scale}\cdot\Delta\rho^2\,V_{\mathrm{shell}}\int_0^{\pi/2} F^2\,\sin\alpha\,d\alpha + \mathrm{background}$$
In [2]:
# ── Variables ──────────────────────────────────────────────────────────────
q = esc.var("Q")
alpha = esc.var("alpha") # angle between cylinder axis and q
# ── Parameters ─────────────────────────────────────────────────────────────
scale = esc.par("Scale", 1.0, scale=1e8, fixed=True)
radius_core = esc.par("Core radius", 20.0, units=esc.angstr)
thickness = esc.par("Thickness", 10.0, units=esc.angstr)
length = esc.par("Length", 400.0, units=esc.angstr)
contrast = esc.par("Contrast", 5.3, scale=1e-6, units=f"{esc.angstr}^-2")
background = esc.par("Background", 0.001, userlim=[0.0, 0.03])
# ── Geometry ───────────────────────────────────────────────────────────────
radius_outer = radius_core + thickness
half_length = 0.5 * length
v_shell = np.pi * (esc.pow(radius_outer, 2) - esc.pow(radius_core, 2)) * length
gamma_sq = esc.pow(radius_core / radius_outer, 2) # (R_c/R_o)^2
# ── Oriented amplitude ─────────────────────────────────────────────────────
# q_axial = q * cos(alpha): component along cylinder axis (sinc factor)
# q_radial = q * sin(alpha): component perpendicular to axis (Bessel factor)
q_axial = q * esc.cos(alpha)
q_radial = q * esc.sin(alpha)
# Psi = [Lambda(R_o*q_radial) - gamma^2 * Lambda(R_c*q_radial)] / (1 - gamma^2)
# Lambda(a) = 2*J1(a)/a = 2 * j1_over_x(a)
lam_outer = 2.0 * esc.j1_over_x(radius_outer * q_radial)
lam_inner = 2.0 * esc.j1_over_x(radius_core * q_radial)
psi = (lam_outer - gamma_sq * lam_inner) / (1.0 - gamma_sq)
F = psi * esc.sinc(half_length * q_axial)
# ── Powder average ─────────────────────────────────────────────────────────
i1d = (scale * esc.pow(contrast, 2) * v_shell
* esc.integral(esc.pow(F, 2) * esc.sin(alpha),
alpha, 0.0, np.pi / 2.0,
numpoints=61, maxiter=5, epsabs=1e-5)
+ background)
In [3]:
i1d.device = "gpu"
qs = np.linspace(0.001, 1.0, 300)
i1d.show(coordinates=qs).config(
title="Hollow cylinder — powder average (1D)",
xlog=True, ylog=True,
xlabel=f"Q [{esc.angstr}^-1]", ylabel="I(q) [cm^-1]")
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2D oriented scattering (qx, qy)¶
For a fixed orientation $(\theta, \phi)$ the amplitude is evaluated directly at detector coordinates. The cylinder axis unit vector is $\hat{\mathbf{u}} = (\sin\theta\cos\phi,\;\sin\theta\sin\phi,\;\cos\theta)$.
$$I_{\mathrm{2D}}(q_x,q_y) = \mathrm{scale}\cdot\Delta\rho^2\,V_{\mathrm{shell}}\,F^2(q_\parallel, q_\perp) + \mathrm{background}$$
In [4]:
qx = esc.var("qx")
qy = esc.var("qy")
theta = esc.par("Theta", 90.0, userlim=[0.0, 180.0], units="deg")
phi = esc.par("Phi", 0.0, userlim=[0.0, 360.0], units="deg")
deg = np.pi / 180.0
sin_t = esc.sin(theta * deg)
ux = sin_t * esc.cos(phi * deg)
uy = sin_t * esc.sin(phi * deg)
q_sq = esc.pow(qx, 2) + esc.pow(qy, 2)
q_par_2d = qx * ux + qy * uy
q_perp_2d = esc.sqrt(q_sq - esc.pow(q_par_2d, 2))
lam_outer_2d = 2.0 * esc.j1_over_x(radius_outer * q_perp_2d)
lam_inner_2d = 2.0 * esc.j1_over_x(radius_core * q_perp_2d)
psi_2d = (lam_outer_2d - gamma_sq * lam_inner_2d) / (1.0 - gamma_sq)
F_2d = psi_2d * esc.sinc(half_length * q_par_2d)
i2d = scale * esc.pow(contrast, 2) * v_shell * esc.pow(F_2d, 2) + background
i2d.device = "gpu"
xs = np.linspace(-1.0, 1.0, 300); ys = np.linspace(-1.0, 1.0, 300)
xv, yv = np.meshgrid(xs, ys)
coords_2d = np.column_stack([xv.flatten(), yv.flatten()]).flatten()
i2d.show(coordinates=coords_2d).config(
title="Hollow cylinder — oriented 2D SAXS (qx, qy)",
xlabel=f"qx [{esc.angstr}^-1]", ylabel=f"qy [{esc.angstr}^-1]",
cblog=True, colorscale="jet")
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SasView reference model & comparison¶
| ESCAPE parameter | SasView parameter | Notes |
|---|---|---|
contrast * 1e-6 |
sld - sld_solvent |
contrast in Å⁻² |
radius_core |
radius |
inner (core) radius (Å) |
thickness |
thickness |
wall thickness (Å) |
length |
length |
cylinder length (Å) |
In [5]:
import time
import matplotlib.pyplot as plt
from sasmodels.core import load_model
from sasmodels.data import empty_data1D
from sasmodels.direct_model import DirectModel
qs = np.linspace(0.001, 1.0, 300).copy()
kernel = load_model("hollow_cylinder")
f_sas = DirectModel(empty_data1D(qs), kernel)
sas_pars = dict(scale=1.0, background=0.001,
sld=6.3, sld_solvent=1.0,
radius=20.0, thickness=10.0, length=400.0)
f_sas(**sas_pars)
i1d.device = "gpu"; i1d(qs[:5])
def timeit(fn, n=5):
t0 = time.perf_counter()
for _ in range(n): result = fn()
return (time.perf_counter() - t0) / n * 1e3, result
t_sas, Iq_sas = timeit(lambda: f_sas(**sas_pars))
i1d.device = "gpu"
t_gpu, Iq_gpu = timeit(lambda: i1d(qs), n=3)
i1d.device = "cpu"
t_cpu, Iq_cpu = timeit(lambda: i1d(qs))
i1d.device = "gpu"
print(f"SASView GPU : {t_sas:.0f} ms")
print(f"ESCAPE GPU : {t_gpu:.0f} ms")
print(f"ESCAPE CPU : {t_cpu:.0f} ms ({len(qs)} q-pts)")
rel = np.max(np.abs((Iq_gpu - Iq_sas) / Iq_sas)) * 100
print(f"Max relative diff vs SasView: {rel:.2f}%")
esc.overlay(Iq_sas, Iq_gpu, Iq_cpu, coordinates=qs).config(
xlabel="Q (1/A)", ylabel="I(q) (1/cm)",
xlog=True, ylog=True,
fig_title=f"Hollow cylinder I(q) — {len(qs)} pts",
labels=["SASView", "ESCAPE GPU", "ESCAPE CPU"],
line_styles=["solid", "dash", "dot"],
line_widths=[2, 3, 3]
)
SASView GPU : 10 ms ESCAPE GPU : 1 ms ESCAPE CPU : 5 ms (300 q-pts) Max relative diff vs SasView: 20.90%
C:\Users\User\AppData\Local\Temp\ipykernel_34832\3329007158.py:16: UserWarning: Input array does not own its data (e.g. it is a view or slice); data will be copied
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