From 0a5b19ebc7de725578678b4e31c8b500ad375065 Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Wed, 14 Jan 2026 15:32:11 +0100 Subject: [PATCH 1/9] adding fibonacci.py --- sasmodels/quadratures/fibonacci.py | 68 ++++++++++++++++++++++++++++++ 1 file changed, 68 insertions(+) create mode 100644 sasmodels/quadratures/fibonacci.py diff --git a/sasmodels/quadratures/fibonacci.py b/sasmodels/quadratures/fibonacci.py new file mode 100644 index 00000000..54f604f8 --- /dev/null +++ b/sasmodels/quadratures/fibonacci.py @@ -0,0 +1,68 @@ +# octahedron model +# Note: model title and parameter table are inserted automatically +r""" +This module provides a function to generate quasi-uniformly distributed points on the unit sphere +using the Fibonacci quadrature. These points can be used for numerical integration over the sphere, +which is particularly useful for orientational averaging in scattering calculations. +It also includes a plotting function to visualize the distribution of points on the sphere. +The only parameter is the number of points to generate. Around 500 points provide a good balance +between accuracy and computational efficiency for most applications. +""" + + +import matplotlib.pyplot as plt +import numpy as np + + +def fibonacci_sphere(npoints_fibonacci: int): + """ + Generates npoints quasi-uniformly distributed on the unit sphere + in Cartesian coordinates (x,y,z) and their associated weights. + Parameters + ---------- + npoints : int + Number of points to generate. + Returns + ------- + points : ndarray, shape (npoints, 3) + Cartesian coordinates of the points on the unit sphere. + weights : ndarray, shape (npoints,) + Weights associated with each point for integration on the sphere. + """ + + indices = np.arange(0, npoints_fibonacci, dtype=float) + 0.5 + phi = np.arccos(1.0 - 2.0 * indices / npoints_fibonacci) + theta = 2.0 * np.pi * indices / ((1.0 + 5.0 ** 0.5) / 2.0) + + x = np.cos(theta) * np.sin(phi) + y = np.sin(theta) * np.sin(phi) + z = np.cos(phi) + + weights = np.full(len(x), 1.0 / npoints_fibonacci) + + return np.column_stack((x, y, z)), weights + +def plot_fibonacci_sphere(npoints_fibonacci=500, figsize=(7, 7)): + """ + 3D representation of Fibonacci points on the unit sphere. + Parameters + ---------- + npoints : int + Number of points to generate and display. + figsize : tuple + Size of the figure. + + """ + pts, w = fibonacci_sphere(npoints_fibonacci) + + fig = plt.figure(figsize=figsize) + ax = fig.add_subplot(111, projection="3d") + ax.scatter(pts[:, 0], pts[:, 1], pts[:, 2], s=10, alpha=0.6) + + # Axes + ax.set_xlabel("x") + ax.set_ylabel("y") + ax.set_zlabel("z") + ax.set_title(f"Fibonacci points on the unit sphere, ({npoints_fibonacci} total points)") + ax.set_box_aspect([1, 1, 1]) + plt.show() From 310b92db5497b1a5f3c6a3927a7635f4800b959a Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Wed, 14 Jan 2026 15:45:23 +0100 Subject: [PATCH 2/9] adding the nanoprims file --- sasmodels/models/nanoprisms_fibonacci.py | 148 +++++++++++++++++++++++ 1 file changed, 148 insertions(+) create mode 100644 sasmodels/models/nanoprisms_fibonacci.py diff --git a/sasmodels/models/nanoprisms_fibonacci.py b/sasmodels/models/nanoprisms_fibonacci.py new file mode 100644 index 00000000..4fe16ed2 --- /dev/null +++ b/sasmodels/models/nanoprisms_fibonacci.py @@ -0,0 +1,148 @@ +r""" +This model provides the form factor P(q) for nanoprims with different cross-sections. +It contains both the form factor for a reference orientation and the 1D form factor after orientation average +using the Fibonacci quadrature. This quadrature provides a quasi-uniform distribution of points on the unit sphere +using the golden ratio. The only new parameter is the number of points to generate on the unit sphere. +The default value (500 points) provides a good balance between accuracy and computational efficiency. + +Definition +---------- + +expliquer les paramètres, donner les formules etc (voir article) + + +.. math:: + + formule + +.. figure:: img/nanoprisms_intensity_plot.png + +Validation +---------- + +Validation of the code is made using numerical checks. +Comparisons with Debye formula calculations were made using DebyeCalculator library (https://github.com/FrederikLizakJohansen/DebyeCalculator). +Good agreement was found at q < 0.1 1/Angstrom. + +References +---------- + +1. Jules Marcone, Jaime Gabriel Trazo, Rahul Nag, Claire Goldmann, Nicolas Ratel-Ramond, Cyrille Hamon and Marianne Imperor-Clerc (2025) J. Marcone et al. , + Form factor of prismatic particles for small-angle scattering analysis, JAC 2025 + +1. Wei-Ren Chen et al. "Scattering functions of Platonic solids". + In: Journal of Applied Crystallography - J APPL CRYST 44 (June 2011). + DOI: 10.1107/S0021889811011691 + +2. Croset, Bernard, "Form factor of any polyhedron: a general compact + formula and its singularities" In: J. Appl. Cryst. (2017). 50, 1245–1255 + https://doi.org/10.1107/S1600576717010147 + +3. Wuttke, J. Numerically stable form factor of any polygon and polyhedron + J Appl Cryst 54, 580-587 (2021) + https://doi.org/10.1107/S160057672100171 + +Authorship and Verification +---------------------------- + +* **Authors:** Marianne Imperor-Clerc (marianne.imperor@cnrs.fr) + Helen Ibrahim (helenibrahim1@outlook.com) ? + Jules Marcone () + Sara Mokhtari (smokhtari@insa-toulouse.fr) + +* **Last Modified by:** MIC **Date:** 11 December 2025 + +* **Last Reviewed by:** SM **Date:** 10 December 2025 + +""" + + +name = "nanoprisms" +title = "nanoprisms model with Fibonacci quadrature" +description = """ + nanoprisms with Fibonacci quadrature""" + +category = "plugin" + +import numpy as np +from numpy import inf +from prismformfactors import * + +from sasmodels.quadratures.fibonacci import fibonacci_sphere + +# ["name", "units", default, [lower, upper], "type", "description"], +parameters = [["sld", "1e-6/Ang^2", 126., [-inf, inf], "sld", + "nanoprism scattering length density"], + ["sld_solvent", "1e-6/Ang^2", 9.4, [-inf, inf], "sld", + "Solvent scattering length density"], + ["n_sides", "", 5, [3, 50], "", + "n_sides"], + ["R_ave", "Ang", 500, [0., inf], "volume", + "Average radius"], + ["L", "Ang", 5000, [0., inf], "volume", + "length"], + ] + + +def form_volume(nsides,Rave,L): + nsides=int(nsides) + edge = functions.edge_from_gyration_radius(nsides,Rave) + return functions.volume_nanoprism(nsides,edge,L) + +def Iqabc(qa,qb,qc,nsides,Rave,L): # proportionnal to the volume**2 + nsides=int(nsides) + edge = functions.edge_from_gyration_radius(nsides,Rave) + intensity=formfactor.I_nanoprism([qa,qb,qc],nsides,edge,L) + return intensity + + +def Iq(q, sld, sld_solvent, nsides:int, Rave, L, npoints_fibonacci:int= 500): + """ + Integration over the unit sphere with Fibonacci points. + Each point has an equal weight = 1/npoints_fibonacci + + Parameters + ---------- + q : float ou array + Norm of the scattering vector + sld, sld_solvent : + Contrast of scattering length density + nsides, Rave, L : + Geometrical parameters of the prism + npoints_fibonacci : int + Number of Fibonacci points on the sphere + Returns + ------- + Iq : ndarray + Scattering intensity averaged over all orientations + time_fib : float + Execution time of the function in seconds + n_points_total : int + Total number of points used in the quadrature + """ + + nsides = int(nsides) + q = np.atleast_1d(q) + q_unit,w = fibonacci_sphere(npoints_fibonacci) # shape (npoints, 3) + + # Projections + qa = q[:, np.newaxis] * q_unit[:, 0][np.newaxis, :] + qb = q[:, np.newaxis] * q_unit[:, 1][np.newaxis, :] + qc = q[:, np.newaxis] * q_unit[:, 2][np.newaxis, :] + + # Compute intensity + intensity = Iqabc(qa, qb, qc, nsides, Rave, L) # shape (nq, npoints) + + # Uniform average over the sphere + integral = np.sum(w[np.newaxis, :] * intensity, axis=1) + integral = np.mean(intensity, axis=1) + + + return integral * (sld - sld_solvent)**2 + + +Iq.vectorized = True + + + + From 29befd9dd0d2494b674cf30d92e9a4e684fc4e9c Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Wed, 14 Jan 2026 15:48:17 +0100 Subject: [PATCH 3/9] rename the file --- sasmodels/models/{nanoprisms_fibonacci.py => nanoprisms.py} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename sasmodels/models/{nanoprisms_fibonacci.py => nanoprisms.py} (100%) diff --git a/sasmodels/models/nanoprisms_fibonacci.py b/sasmodels/models/nanoprisms.py similarity index 100% rename from sasmodels/models/nanoprisms_fibonacci.py rename to sasmodels/models/nanoprisms.py From 357d9f90140738211ddeb5fe4849b2ec982cc5ba Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Fri, 16 Jan 2026 14:35:18 +0100 Subject: [PATCH 4/9] Add nanoprisms model and related files --- sasmodels/models/.gitignore | 1 + sasmodels/models/img/fibonacci_sphere.png | Bin 0 -> 422270 bytes sasmodels/models/img/nanoprisms_geometry.jpg | Bin 0 -> 228110 bytes sasmodels/models/nanoprisms.py | 478 ++++++++++++++++--- 4 files changed, 417 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model provides the form factor P(q) for nanoprims with different cross-sections. -It contains both the form factor for a reference orientation and the 1D form factor after orientation average -using the Fibonacci quadrature. This quadrature provides a quasi-uniform distribution of points on the unit sphere -using the golden ratio. The only new parameter is the number of points to generate on the unit sphere. -The default value (500 points) provides a good balance between accuracy and computational efficiency. +This model provides the form factor for nanoprisms with different cross-sections. +Orientation averaging is done by using the Fibonacci quadrature. +This quadrature provides a quasi-uniform distribution of points on the unit sphere +using the golden ratio. The number of points to generate on the unit sphere is set to 500 points, it usually provides +a good balance between accuracy and computational efficiency. Definition ---------- -expliquer les paramètres, donner les formules etc (voir article) +We consider particles having the shape of a right prism of length L and a cross section made of a regular polygon with n sides as illustrated in the figure below. +.. figure:: img/nanoprisms_geometry.png + +The size of a regular polygon can be characterized by its edge length E or by R, the circumradius of the polygon shown in black in the figure, with : + +.. math:: + + R = \frac{E}{2\sin(\pi/n)} + + +The apothem is the radius of the dashed circle, it is defined as: + +.. math:: + + R\cos(\pi/n) + +For comparison purposes, it is convenient to introduce an average radius R_ave (shown in red in the figure). +The area of the n-sided regular polygon is given by : .. math:: - formule + A = \pi R^{2} \, \mathrm{sinc}(\frac{2\pi}{n}\right) + = \pi R_{\mathrm{ave}}^{2} + +and the volume of the nanoprism is therefore given by: + +.. math:: + + V = L \times {A} + +where R_ave is the radius of the equivalent disc having the same area as the n-sided polygon. +It is also the squared average of the distance from the center of the polygon to any point of its perimeter. It is related to the circumradius R by : + +.. math:: + + R_{\mathrm{ave}}^{2} = R^{2} \, \mathrm{sinc}(\frac{2\pi}{n}\right) + +Form factor for a prism: Following Wuttke's expression, the form factor :math:`F(\mathbf{q})` for any right prism can be decomposed into the product of two factors. +One factor corresponds to the component :math:`\mathbf{q}_{\perp}` of the scattering vector that is perpendicular to the cross section and depends +only on the length :math:`L`. The other factor is coplanar with the cross section and involves the component :math:`\mathbf{q}_{\|}`; +it depends on the number of sides :math:`n` and the edge length :math:`E` of the polygon. +The perpendicular factor is: + +.. math:: + + f_{\perp}(\mathbf{q}_{\perp}, L) + = \mathrm{sinc}( + \frac{(\mathbf{q}_{\perp} \cdot \hat{\mathbf{n}})\, L}{2} + ) + +where :math:`\hat{\mathbf{n}}` is the direction normal to the cross section. +The length :math:`L` gives rise to a standard sinc function for the form factor. + +On the other hand, the parallel factor for a regular :math:`n`-sided polygon +of circumradius :math:`R` can be expressed as + +.. math:: + + f_{\|}(\mathbf{q}_{\|}, n, R) + = + \frac{2}{i q_{\|}^{2}} + \sum_{j=1}^{n} + \hat{\mathbf{n}} \cdot + \left( \mathbf{q}_{\|} \times \mathbf{E}_{j} \right) + \, + \mathrm{sinc}( \mathbf{q}_{\|} \cdot \mathbf{E}_{j} \right) + \exp( + i \mathbf{q}_{\|} \cdot \mathbf{M}_{j}) + +In the sum over all edges, :math:`\mathbf{M}_{j}` is the vector joining the center of the polygon to the middle of the jth edge andand +:math:`\mathbf{E}_{j}` is the half-edge vector. + +The scattered intensity for one prism is given by: + +.. math:: + + I(\mathbf{q}, n, R, L) + = + \left| F(\mathbf{q}, n, R, L) \right|^{2} + = + \left| + f_{\perp}(\mathbf{q}_{\perp}, L) + f_{\|}(\mathbf{q}_{\|}, n, R) + \right|^{2} + +Orientation averaging: The 1D form factor corresponds to the orientation average with all the possible orientations having the same probability. +Instead of rotating the shape through all the possible orientations in the integral, +it is equivalent to integrate the 3D scattering vector over a sphere of radius q with the shape in its reference orientation. + +.. math:: + + P(q) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \, + \int_0^{\frac{\pi}{2}} A_P^2(q) \, \sin\theta \, d\theta \, d\phi + +The sphere is sampled using Fibonacci quadrature to provide a quasi-uniform distribution of points on the unit sphere. +The repartition of the points is computed using the golden ratio (see fibonacci.py). -.. figure:: img/nanoprisms_intensity_plot.png + +.. figure:: img/fibonacci_sphere.png + + Fibonacci sphere using 5810 points. Validation ---------- -Validation of the code is made using numerical checks. -Comparisons with Debye formula calculations were made using DebyeCalculator library (https://github.com/FrederikLizakJohansen/DebyeCalculator). +The model has been tested against experimental data obtained on gold nanoprisms with pentagonal cross-section (see J. Marcone et al. JAC 2025). +Moreover, comparisons with Debye formula calculations were made using DebyeCalculator library (https://github.com/FrederikLizakJohansen/DebyeCalculator). Good agreement was found at q < 0.1 1/Angstrom. References ---------- -1. Jules Marcone, Jaime Gabriel Trazo, Rahul Nag, Claire Goldmann, Nicolas Ratel-Ramond, Cyrille Hamon and Marianne Imperor-Clerc (2025) J. Marcone et al. , - Form factor of prismatic particles for small-angle scattering analysis, JAC 2025 +1. Jules Marcone et al. "Form factor of prismatic particles for small-angle scattering analysis". + In: J. Appl. Cryst. (2025) 58, 543-552 + DOI: https://doi.org/10.1107/S1600576725000676 -1. Wei-Ren Chen et al. "Scattering functions of Platonic solids". - In: Journal of Applied Crystallography - J APPL CRYST 44 (June 2011). - DOI: 10.1107/S0021889811011691 +2. Wei-Ren Chen et al. "Scattering functions of Platonic solids". + In: Journal of Applied Crystallography - J. Appl. Cryst. (June 2011). + DOI: https://doi.org/10.1107/S0021889811011691 -2. Croset, Bernard, "Form factor of any polyhedron: a general compact +3. Croset, Bernard, "Form factor of any polyhedron: a general compact formula and its singularities" In: J. Appl. Cryst. (2017). 50, 1245–1255 https://doi.org/10.1107/S1600576717010147 -3. Wuttke, J. Numerically stable form factor of any polygon and polyhedron +4. Wuttke, J. Numerically stable form factor of any polygon and polyhedron J Appl Cryst 54, 580-587 (2021) https://doi.org/10.1107/S160057672100171 @@ -46,103 +141,362 @@ ---------------------------- * **Authors:** Marianne Imperor-Clerc (marianne.imperor@cnrs.fr) - Helen Ibrahim (helenibrahim1@outlook.com) ? - Jules Marcone () + Jules Marcone (jules.marcone@universite-paris-saclay.fr) Sara Mokhtari (smokhtari@insa-toulouse.fr) * **Last Modified by:** MIC **Date:** 11 December 2025 -* **Last Reviewed by:** SM **Date:** 10 December 2025 +* **Last Reviewed by:** SM **Date:** 16 January 2026 """ - - -name = "nanoprisms" -title = "nanoprisms model with Fibonacci quadrature" -description = """ - nanoprisms with Fibonacci quadrature""" - -category = "plugin" - import numpy as np from numpy import inf -from prismformfactors import * from sasmodels.quadratures.fibonacci import fibonacci_sphere +name = "nanoprisms" +title = "nanoprisms of different cross-sections" +description = """ + Model for nanoprisms of different cross-sections with orientation averaging using the Fibonacci quadrature""" +category = "plugin" # ["name", "units", default, [lower, upper], "type", "description"], parameters = [["sld", "1e-6/Ang^2", 126., [-inf, inf], "sld", "nanoprism scattering length density"], ["sld_solvent", "1e-6/Ang^2", 9.4, [-inf, inf], "sld", "Solvent scattering length density"], - ["n_sides", "", 5, [3, 50], "", - "n_sides"], - ["R_ave", "Ang", 500, [0., inf], "volume", + ["nsides", "", 5, [3, 50], "volume", + "nsides"], + ["Rave", "Ang", 500, [0., inf], "volume", "Average radius"], ["L", "Ang", 5000, [0., inf], "volume", - "length"], + "length"] ] +### Functions for geometrical calculations: volume, surface of the cross section, average radius, edge length etc. +def form_volume(nsides, Rave, L): + """ + Computes the volume of a nanoprism given its number of sides, average radius and length. + Parameters + ---------- + nsides : int + Number of sides of the regular polygon cross-section. + Rave : float + Average radius of the regular polygon cross-section. + L : float + Length of the nanoprism. + Returns + ------- + volume : float + Volume of the nanoprism. + """ + nsides = int(nsides) + edge = edge_from_gyration_radius(nsides,Rave) + return volume_nanoprism(nsides,edge,L) -def form_volume(nsides,Rave,L): - nsides=int(nsides) - edge = functions.edge_from_gyration_radius(nsides,Rave) - return functions.volume_nanoprism(nsides,edge,L) +def edge_from_gyration_radius(nsides:int, gyr): + """ + Computes the edge length of an n-sided regular polygon of gyration ratio gyr. + Parameters + ---------- + nsides : int + Number of sides of the regular polygon. + gyr : float + Gyration radius of the regular polygon. + Returns + ------- + edge : float + Edge length of the regular polygon. + """ + nsides = int(nsides) + return (gyr * 2 * np.sin(np.pi/nsides)) / np.sqrt(np.sinc(2/nsides)) + +def volume_nanoprism(nsides:int, edge, L): + """ Computes the volume of a nanoprism with a cross-section + in the shape of an n-sided regular polygon of edge length "edge", and length of the nanoprism "L". + Parameters + ---------- + nsides : int + Number of sides of the regular polygon cross-section. + edge : float + Edge length of the regular polygon cross-section. + L : float + Length of the nanoprism. + Returns + ------- + volume : float + Volume of the nanoprism. + """ + radius = radius_from_edge(nsides, edge) + surface = surface_from_radius(nsides, radius) + return surface * L + +def surface_from_radius(nsides:int, radius): + """ + Computes the area of an n-sided regular polygon of circumradius radius. + Parameters + ---------- + nsides : int + Number of sides of the regular polygon. + radius : float + Circumradius of the regular polygon. + Returns + ------- + area : float + Area of the regular polygon. + """ + return nsides * (radius**2) * np.sin(np.pi/nsides) * np.cos(np.pi/nsides) + +def radius_from_edge(nsides:int, edge): + """ + Computes the circumradius of an n-sided regular polygon of edge length edge. + Parameters + ---------- + nsides : int + Number of sides of the regular polygon. + edge : float + Edge length of the regular polygon. + Returns + ------- + radius : float + Circumradius of the regular polygon. + """ + return edge/(2*np.sin(np.pi/nsides)) + +def shape_generator(number_of_sides, radius): + """ + Computes the list of vertices of a n-sided regular polygon, of circumradius "radius". + Parameters + ---------- + number_of_sides : int + Number of sides of the regular polygon. + radius : float + Circumradius of the regular polygon. + Returns + ------- + vertices : list + List of the vertices of the regular polygon. + """ + vertices = [[radius*np.cos(2*step*np.pi/number_of_sides), radius*np.sin(2*step*np.pi/number_of_sides)] for step in range(0,number_of_sides)] + return vertices + +def edgecenters_generator(vertices:list): + """ + Computes the list of edge centers of a 2D shape defined by its vertices. + Parameters + ---------- + vertices : list + list of the vertices of the 2D-shape each listed as a list of the 2D coordinates of the shape in the plane + Returns + ------- + edgecenter : list + List of the edge centers of the 2D shape. + Preconditions + ------------- + a loop, all in the plane + """ + extended_vertices = [] + vertices + extended_vertices.append(vertices[0]) + edgecenter = [] + for i in range(len(vertices)): + coordinates=[] + for j in range(2): + coordinates.append((extended_vertices[i+1][j]+extended_vertices[i][j])/2) + edgecenter.append(coordinates) + return edgecenter + +def halfedges_generator(vertices:list): + """ + Computes the list of half-edges of a 2D shape defined by its vertices. + Parameters + ---------- + vertices : list + list of the vertices of the 2D-shape each listed as a list of the 2D coordinates of the shape in the plane + Returns + ------- + halfedge : list + List of the half-edges of the 2D shape. + Preconditions + ------------- + a loop, all in the plane + """ + extended_vertices = [] + vertices + extended_vertices.append(vertices[0]) + halfedge = [] + for i in range(len(vertices)): + coordinates=[] + for j in range(2): + coordinates.append((extended_vertices[i+1][j]-extended_vertices[i][j])/2) + halfedge.append(coordinates) + return halfedge + +# useful function sinc(x) = sin(x)/x +def sinc(x): + """ + Computes the cardinal sinus of x since in numpy sinc is defined with pi*x for the argument. + Parameters + ---------- + x : float or array + Input value(s). + Returns + ------- + sinc_x : float or array + Cardinal sinus of x. + """ + return np.sinc(x/np.pi) + +# useful functions for scalar product +def scalar_product(u,v): + """ + Computes the scalar product of two vectors u and v since numpy.dot does not work with different shapes of arrays. + Parameters + ---------- + u and v : list + lists of the coordinates of the vectors. + Returns + ------- + scalar_product : float + Scalar product of the two vectors. + """ + somme = 0 + for i in range(len(u)): + somme += u[i]*v[i] + return somme + +# Form factor and intensity calculation +# Reminder: the form factor is defined as: parallel factor (= complex function) * perpendicular factor (= sinc function) +# see documentation and Reference [1] (Jules Marcone et al. "Form factor of prismatic particles for small-angle scattering analysis") +def parallel_factor(vertices:list, q, c): # gives the area of the polygon at q==[0,0] + """ + Computes the parallel form factor of a 2D shape defined by its vertices at a specific in-plane scattering vector q. + Parameters + ---------- + vertices : list + list of the vertices of the 2D-shape each listed as a list of the + 2D coordinates of the shape in the plane + q : list + listed as a list of coordinates in the plane (q is q//) + c : float + an arbitrary constant + Returns + ------- + parallel_factor : complex + Parallel form factor of the 2D shape at the specific in-plane scattering vector q. + Preconditions + ------------- + a loop, all in the plane + """ + qmodulus2 = q[0]**2+q[1]**2 + qmodulus2 = np.asanyarray(qmodulus2) #conversion to array + cutoff = 10**-30 + qmodulus2_cutoff = np.where(qmodulus2==0, cutoff, qmodulus2) # replace by cutoff if equal to 0 + #This case starts occuring for computations of the prism formfactor, where during the + #calculation of the orientational average, q// may be equal to 0. + #if qmodulus2==0: + # return 0 + edgecenters = edgecenters_generator(vertices) + halfedges = halfedges_generator(vertices) + sum = 0 + for i in range(len(vertices)): + qEj = scalar_product(q, halfedges[i]) # scalar_product(q, halfedges[i]) + triple_product = q[0]*halfedges[i][1]-q[1]*halfedges[i][0] + #The exp(iqRj) is rewritten as a sum of cos+isin because of the way math.exp() works (not allowing complex as input) + qRj = scalar_product(q, edgecenters[i]) + sum += triple_product * (sinc(qEj)*(np.cos(qRj)+np.sin(qRj)*1J)-c) + return (2/(1J*qmodulus2_cutoff)*sum) + +def Amp_nanoprism(q,nsides,edge,L): # From factor in 3D of the nanoprism q with three components + """ + Computes the form factor amplitude of a prism with a n-sided regular polygon cross-section and length L for a specific three dimensional q. + Takes in account the parallel factor (complex function) and the perpendicular factor (sinc function). + Parameters + ---------- + q : list + listed as a list of three components [qa,qb,qc] + nsides : int + Number of sides of the regular polygon cross-section. + edge : float + Edge length of the regular polygon cross-section. + L : float + Length of the nanoprism. + Returns + ------- + Amp_nanoprism : complex + Form factor amplitude of the nanoprism at the specific three dimensional q. + """ + qa, qb, qc = q[0], q[1], q[2] + qab = [qa,qb] + radius = radius_from_edge(nsides,edge) + vertices = shape_generator(nsides,radius) +# A=parallel_factor(vertices,qab,0.)*sinc(qc*L/2)/surface_from_radius(nsides,radius) + perpendicular_factor = sinc(qc*L/2) + A = parallel_factor(vertices,qab,0.)*perpendicular_factor # parallel form factor * perpendicular form factor + return A def Iqabc(qa,qb,qc,nsides,Rave,L): # proportionnal to the volume**2 + """ + Calls the function that computes the edge length and the scattered intensity. + Parameters + ---------- + qa, qb, qc : float or array + components of the scattering vector q + nsides : int + Number of sides of the regular polygon cross-section. + Rave : float + Average radius of the regular polygon cross-section. + L : float + Length of the nanoprism. + Returns + ------- + Iqabc : float or array + Scattered intensity of the nanoprism at the specific three dimensional q components. + """ nsides=int(nsides) - edge = functions.edge_from_gyration_radius(nsides,Rave) - intensity=formfactor.I_nanoprism([qa,qb,qc],nsides,edge,L) + edge = edge_from_gyration_radius(nsides, Rave) + # intensity = I_nanoprism([qa,qb,qc], nsides, edge, L) + A = Amp_nanoprism([qa,qb,qc], nsides, edge, L) + intensity = (np.abs(A))**2 + intensity = intensity * (L)**2 # multiplication by the volume at the power 2 (? volume or L only ?) return intensity - def Iq(q, sld, sld_solvent, nsides:int, Rave, L, npoints_fibonacci:int= 500): """ - Integration over the unit sphere with Fibonacci points. - Each point has an equal weight = 1/npoints_fibonacci - + Computes the scattering intensity I(q) of nanoprisms averaged over all orientations using the Fibonacci quadrature. + The number of points on the sphere is set by npoints_fibonacci. Each point has an equal weight = 1/npoints_fibonacci. Parameters ---------- q : float ou array Norm of the scattering vector - sld, sld_solvent : + sld, sld_solvent : Contrast of scattering length density - nsides, Rave, L : + nsides, Rave, L : Geometrical parameters of the prism npoints_fibonacci : int - Number of Fibonacci points on the sphere + Number of Fibonacci points on the sphere, set to 500 by default + (higher number increases accuracy but also computation time, 500 is usually a good compromise) Returns ------- Iq : ndarray Scattering intensity averaged over all orientations - time_fib : float - Execution time of the function in seconds - n_points_total : int - Total number of points used in the quadrature """ - nsides = int(nsides) q = np.atleast_1d(q) q_unit,w = fibonacci_sphere(npoints_fibonacci) # shape (npoints, 3) - # Projections qa = q[:, np.newaxis] * q_unit[:, 0][np.newaxis, :] qb = q[:, np.newaxis] * q_unit[:, 1][np.newaxis, :] qc = q[:, np.newaxis] * q_unit[:, 2][np.newaxis, :] - # Compute intensity intensity = Iqabc(qa, qb, qc, nsides, Rave, L) # shape (nq, npoints) - # Uniform average over the sphere integral = np.sum(w[np.newaxis, :] * intensity, axis=1) integral = np.mean(intensity, axis=1) - - - return integral * (sld - sld_solvent)**2 - + return integral * (sld - sld_solvent)**2 * 10**-4 # Convert from [1e-12 A-1] to [cm-1] Iq.vectorized = True - - - +tests = [ + [{"background": 0, "scale": 1, "nsides": 4, "Rave": 10, "L":200,"sld": 1., "sld_solvent": 0.}, + 0.01, 5.62789], + [{"background": 0, "scale": 1, "nsides": 4, "Rave": 10, "L":200,"sld": 1., "sld_solvent": 0.}, + [0.01, 0.1], [5.62789, 0.73696]], +] From 11595a90f4479c48b71c0530f45d1b74be86bf76 Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Fri, 16 Jan 2026 14:46:48 +0100 Subject: [PATCH 5/9] wrong name when calling the image of the nanoprims, not png but jpg --- sasmodels/models/nanoprisms.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/sasmodels/models/nanoprisms.py b/sasmodels/models/nanoprisms.py index 2aa19fef..df16e6fa 100644 --- a/sasmodels/models/nanoprisms.py +++ b/sasmodels/models/nanoprisms.py @@ -10,7 +10,7 @@ We consider particles having the shape of a right prism of length L and a cross section made of a regular polygon with n sides as illustrated in the figure below. -.. figure:: img/nanoprisms_geometry.png +.. figure:: img/nanoprisms_geometry.jpg The size of a regular polygon can be characterized by its edge length E or by R, the circumradius of the polygon shown in black in the figure, with : From 7c7121049a562dc13dc819a725dae06b040a2f42 Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Fri, 16 Jan 2026 15:29:31 +0100 Subject: [PATCH 6/9] fixing the documentation (formulas) --- sasmodels/models/nanoprisms.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/sasmodels/models/nanoprisms.py b/sasmodels/models/nanoprisms.py index df16e6fa..6f576e6f 100644 --- a/sasmodels/models/nanoprisms.py +++ b/sasmodels/models/nanoprisms.py @@ -30,7 +30,7 @@ .. math:: - A = \pi R^{2} \, \mathrm{sinc}(\frac{2\pi}{n}\right) + A = \pi R^{2} \, \mathrm{sinc}\left(\frac{2\pi}{n}\right) = \pi R_{\mathrm{ave}}^{2} and the volume of the nanoprism is therefore given by: @@ -44,7 +44,7 @@ .. math:: - R_{\mathrm{ave}}^{2} = R^{2} \, \mathrm{sinc}(\frac{2\pi}{n}\right) + R_{\mathrm{ave}}^{2} = R^{2} \, \mathrm{sinc}\left(\frac{2\pi}{n}\right) Form factor for a prism: Following Wuttke's expression, the form factor :math:`F(\mathbf{q})` for any right prism can be decomposed into the product of two factors. One factor corresponds to the component :math:`\mathbf{q}_{\perp}` of the scattering vector that is perpendicular to the cross section and depends @@ -74,7 +74,7 @@ \hat{\mathbf{n}} \cdot \left( \mathbf{q}_{\|} \times \mathbf{E}_{j} \right) \, - \mathrm{sinc}( \mathbf{q}_{\|} \cdot \mathbf{E}_{j} \right) + \mathrm{sinc}\left( \mathbf{q}_{\|} \cdot \mathbf{E}_{j} \right) \exp( i \mathbf{q}_{\|} \cdot \mathbf{M}_{j}) From c96a4a3a8af57632f815d0ee5e6f4d5870a66762 Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Wed, 4 Feb 2026 14:00:11 +0100 Subject: [PATCH 7/9] Updating the nanoprism.py file with new normalization, renaming functions, deleting useless functions --- sasmodels/models/nanoprisms.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/sasmodels/models/nanoprisms.py b/sasmodels/models/nanoprisms.py index 6f576e6f..f441793a 100644 --- a/sasmodels/models/nanoprisms.py +++ b/sasmodels/models/nanoprisms.py @@ -490,7 +490,7 @@ def Iq(q, sld, sld_solvent, nsides:int, Rave, L, npoints_fibonacci:int= 500): # Uniform average over the sphere integral = np.sum(w[np.newaxis, :] * intensity, axis=1) integral = np.mean(intensity, axis=1) - return integral * (sld - sld_solvent)**2 * 10**-4 # Convert from [1e-12 A-1] to [cm-1] + return integral * (sld - sld_solvent)**2 * 1e4 # Convert to [cm-1] Iq.vectorized = True From faf12b90ba03e217725eecc2329c6b5ba017b4f4 Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Wed, 4 Feb 2026 14:35:14 +0100 Subject: [PATCH 8/9] wrong file updated, changes are still: normalization, removing of useless functions, renaming functions --- sasmodels/models/nanoprisms.py | 87 ++++++++++++---------------------- 1 file changed, 29 insertions(+), 58 deletions(-) diff --git a/sasmodels/models/nanoprisms.py b/sasmodels/models/nanoprisms.py index f441793a..52500176 100644 --- a/sasmodels/models/nanoprisms.py +++ b/sasmodels/models/nanoprisms.py @@ -153,12 +153,13 @@ from numpy import inf from sasmodels.quadratures.fibonacci import fibonacci_sphere +from sasmodels.special import sas_sinx_x name = "nanoprisms" title = "nanoprisms of different cross-sections" description = """ Model for nanoprisms of different cross-sections with orientation averaging using the Fibonacci quadrature""" -category = "plugin" +category = "shape:polyhedron" # ["name", "units", default, [lower, upper], "type", "description"], parameters = [["sld", "1e-6/Ang^2", 126., [-inf, inf], "sld", "nanoprism scattering length density"], @@ -190,8 +191,12 @@ def form_volume(nsides, Rave, L): Volume of the nanoprism. """ nsides = int(nsides) + edge = edge_from_gyration_radius(nsides,Rave) - return volume_nanoprism(nsides,edge,L) + radius = radius_from_edge(nsides, edge) + surface = surface_from_radius(nsides, radius) + return surface * L + def edge_from_gyration_radius(nsides:int, gyr): """ @@ -210,25 +215,6 @@ def edge_from_gyration_radius(nsides:int, gyr): nsides = int(nsides) return (gyr * 2 * np.sin(np.pi/nsides)) / np.sqrt(np.sinc(2/nsides)) -def volume_nanoprism(nsides:int, edge, L): - """ Computes the volume of a nanoprism with a cross-section - in the shape of an n-sided regular polygon of edge length "edge", and length of the nanoprism "L". - Parameters - ---------- - nsides : int - Number of sides of the regular polygon cross-section. - edge : float - Edge length of the regular polygon cross-section. - L : float - Length of the nanoprism. - Returns - ------- - volume : float - Volume of the nanoprism. - """ - radius = radius_from_edge(nsides, edge) - surface = surface_from_radius(nsides, radius) - return surface * L def surface_from_radius(nsides:int, radius): """ @@ -329,22 +315,8 @@ def halfedges_generator(vertices:list): halfedge.append(coordinates) return halfedge -# useful function sinc(x) = sin(x)/x -def sinc(x): - """ - Computes the cardinal sinus of x since in numpy sinc is defined with pi*x for the argument. - Parameters - ---------- - x : float or array - Input value(s). - Returns - ------- - sinc_x : float or array - Cardinal sinus of x. - """ - return np.sinc(x/np.pi) -# useful functions for scalar product +# Useful functions for scalar product def scalar_product(u,v): """ Computes the scalar product of two vectors u and v since numpy.dot does not work with different shapes of arrays. @@ -362,9 +334,10 @@ def scalar_product(u,v): somme += u[i]*v[i] return somme -# Form factor and intensity calculation +# Form factor and intensity calculations # Reminder: the form factor is defined as: parallel factor (= complex function) * perpendicular factor (= sinc function) # see documentation and Reference [1] (Jules Marcone et al. "Form factor of prismatic particles for small-angle scattering analysis") + def parallel_factor(vertices:list, q, c): # gives the area of the polygon at q==[0,0] """ Computes the parallel form factor of a 2D shape defined by its vertices at a specific in-plane scattering vector q. @@ -386,12 +359,12 @@ def parallel_factor(vertices:list, q, c): # gives the area of the polygon at q== a loop, all in the plane """ qmodulus2 = q[0]**2+q[1]**2 - qmodulus2 = np.asanyarray(qmodulus2) #conversion to array + qmodulus2 = np.asanyarray(qmodulus2) # conversion to array cutoff = 10**-30 qmodulus2_cutoff = np.where(qmodulus2==0, cutoff, qmodulus2) # replace by cutoff if equal to 0 - #This case starts occuring for computations of the prism formfactor, where during the - #calculation of the orientational average, q// may be equal to 0. - #if qmodulus2==0: + # This case starts occuring for computations of the prism formfactor, where during the + # calculation of the orientational average, q// may be equal to 0. + # if qmodulus2==0: # return 0 edgecenters = edgecenters_generator(vertices) halfedges = halfedges_generator(vertices) @@ -401,35 +374,36 @@ def parallel_factor(vertices:list, q, c): # gives the area of the polygon at q== triple_product = q[0]*halfedges[i][1]-q[1]*halfedges[i][0] #The exp(iqRj) is rewritten as a sum of cos+isin because of the way math.exp() works (not allowing complex as input) qRj = scalar_product(q, edgecenters[i]) - sum += triple_product * (sinc(qEj)*(np.cos(qRj)+np.sin(qRj)*1J)-c) + sum += triple_product * (sas_sinx_x(qEj)*(np.cos(qRj)+np.sin(qRj)*1J)-c) return (2/(1J*qmodulus2_cutoff)*sum) -def Amp_nanoprism(q,nsides,edge,L): # From factor in 3D of the nanoprism q with three components +def Fqabc(qa, qb, qc,nsides, Rave,L): # From factor in 3D of the nanoprism q with three components """ Computes the form factor amplitude of a prism with a n-sided regular polygon cross-section and length L for a specific three dimensional q. Takes in account the parallel factor (complex function) and the perpendicular factor (sinc function). Parameters ---------- - q : list - listed as a list of three components [qa,qb,qc] + qa, qb, qc : float + compenents of the scattering vector q nsides : int Number of sides of the regular polygon cross-section. - edge : float - Edge length of the regular polygon cross-section. + Rave : float + Average radius. L : float Length of the nanoprism. Returns ------- - Amp_nanoprism : complex + Fqabc : complex Form factor amplitude of the nanoprism at the specific three dimensional q. """ - qa, qb, qc = q[0], q[1], q[2] + qab = [qa,qb] + edge = edge_from_gyration_radius(nsides, Rave) radius = radius_from_edge(nsides,edge) vertices = shape_generator(nsides,radius) -# A=parallel_factor(vertices,qab,0.)*sinc(qc*L/2)/surface_from_radius(nsides,radius) - perpendicular_factor = sinc(qc*L/2) - A = parallel_factor(vertices,qab,0.)*perpendicular_factor # parallel form factor * perpendicular form factor + perpendicular_factor_value = sas_sinx_x(qc*L/2) * L + parallel_factor_value = parallel_factor(vertices,qab,0.) + A = parallel_factor_value * perpendicular_factor_value return A def Iqabc(qa,qb,qc,nsides,Rave,L): # proportionnal to the volume**2 @@ -451,11 +425,8 @@ def Iqabc(qa,qb,qc,nsides,Rave,L): # proportionnal to the volume**2 Scattered intensity of the nanoprism at the specific three dimensional q components. """ nsides=int(nsides) - edge = edge_from_gyration_radius(nsides, Rave) - # intensity = I_nanoprism([qa,qb,qc], nsides, edge, L) - A = Amp_nanoprism([qa,qb,qc], nsides, edge, L) - intensity = (np.abs(A))**2 - intensity = intensity * (L)**2 # multiplication by the volume at the power 2 (? volume or L only ?) + A = Fqabc(qa, qb, qc, nsides, Rave, L) + intensity = (np.abs(A))**2 # intensity is proportional to volume return intensity def Iq(q, sld, sld_solvent, nsides:int, Rave, L, npoints_fibonacci:int= 500): @@ -490,7 +461,7 @@ def Iq(q, sld, sld_solvent, nsides:int, Rave, L, npoints_fibonacci:int= 500): # Uniform average over the sphere integral = np.sum(w[np.newaxis, :] * intensity, axis=1) integral = np.mean(intensity, axis=1) - return integral * (sld - sld_solvent)**2 * 1e4 # Convert to [cm-1] + return (integral) * (sld - sld_solvent)**2 * 10**-4 Iq.vectorized = True From 76ec20d341efd62c144fddaaddf1d1377e6641df Mon Sep 17 00:00:00 2001 From: sara-mokhtari Date: Wed, 11 Feb 2026 16:46:16 +0100 Subject: [PATCH 9/9] debbuging ruff --- sasmodels/models/nanoprisms.py | 36 +++++++++++++++++----------------- 1 file changed, 18 insertions(+), 18 deletions(-) diff --git a/sasmodels/models/nanoprisms.py b/sasmodels/models/nanoprisms.py index 52500176..02360d43 100644 --- a/sasmodels/models/nanoprisms.py +++ b/sasmodels/models/nanoprisms.py @@ -55,7 +55,7 @@ .. math:: f_{\perp}(\mathbf{q}_{\perp}, L) - = \mathrm{sinc}( + = L \mathrm{sinc}( \frac{(\mathbf{q}_{\perp} \cdot \hat{\mathbf{n}})\, L}{2} ) @@ -78,7 +78,7 @@ \exp( i \mathbf{q}_{\|} \cdot \mathbf{M}_{j}) -In the sum over all edges, :math:`\mathbf{M}_{j}` is the vector joining the center of the polygon to the middle of the jth edge andand +In the sum over all edges, :math:`\mathbf{M}_{j}` is the vector joining the center of the polygon to the middle of the jth edge and :math:`\mathbf{E}_{j}` is the half-edge vector. The scattered intensity for one prism is given by: @@ -94,22 +94,22 @@ f_{\|}(\mathbf{q}_{\|}, n, R) \right|^{2} -Orientation averaging: The 1D form factor corresponds to the orientation average with all the possible orientations having the same probability. -Instead of rotating the shape through all the possible orientations in the integral, +Orientation average: The 1D form factor corresponds to the orientation average with all the possible orientations having the same probability. +Instead of rotating the shape through all the possible orientations, it is equivalent to integrate the 3D scattering vector over a sphere of radius q with the shape in its reference orientation. -.. math:: - - P(q) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \, - \int_0^{\frac{\pi}{2}} A_P^2(q) \, \sin\theta \, d\theta \, d\phi - The sphere is sampled using Fibonacci quadrature to provide a quasi-uniform distribution of points on the unit sphere. -The repartition of the points is computed using the golden ratio (see fibonacci.py). +The repartition of the N points is computed using the golden ratio (see fibonacci.py). +Each point of the quadrature on the unit sphere correspond to a vector :math:`\mathbf{u}_{j}`. +In the sum, all weights :math:`w_j` are taken identical and equal to :math:`\frac{1}{N}`. +.. math:: + + P(q) = \sum_{j=1}^{N} w_j I(q\mathbf{u}_{j}, n, R, L) .. figure:: img/fibonacci_sphere.png - Fibonacci sphere using 5810 points. + Fibonacci sphere using N=5810 points. Validation ---------- @@ -141,7 +141,7 @@ ---------------------------- * **Authors:** Marianne Imperor-Clerc (marianne.imperor@cnrs.fr) - Jules Marcone (jules.marcone@universite-paris-saclay.fr) + Jules Marcone (julesmarcone@gmail.com) Sara Mokhtari (smokhtari@insa-toulouse.fr) * **Last Modified by:** MIC **Date:** 11 December 2025 @@ -158,7 +158,7 @@ name = "nanoprisms" title = "nanoprisms of different cross-sections" description = """ - Model for nanoprisms of different cross-sections with orientation averaging using the Fibonacci quadrature""" + Model for nanoprisms of different cross-sections with orientation average using the Fibonacci quadrature""" category = "shape:polyhedron" # ["name", "units", default, [lower, upper], "type", "description"], parameters = [["sld", "1e-6/Ang^2", 126., [-inf, inf], "sld", @@ -200,13 +200,13 @@ def form_volume(nsides, Rave, L): def edge_from_gyration_radius(nsides:int, gyr): """ - Computes the edge length of an n-sided regular polygon of gyration ratio gyr. + Computes the edge length of an n-sided regular polygon of average radius Rave=gyr. Parameters ---------- nsides : int Number of sides of the regular polygon. gyr : float - Gyration radius of the regular polygon. + average radius of the regular polygon. Returns ------- edge : float @@ -377,14 +377,14 @@ def parallel_factor(vertices:list, q, c): # gives the area of the polygon at q== sum += triple_product * (sas_sinx_x(qEj)*(np.cos(qRj)+np.sin(qRj)*1J)-c) return (2/(1J*qmodulus2_cutoff)*sum) -def Fqabc(qa, qb, qc,nsides, Rave,L): # From factor in 3D of the nanoprism q with three components +def Fqabc(qa, qb, qc,nsides, Rave,L): # Form factor in 3D of the nanoprism for (qa, qb, qc) scattering vector """ - Computes the form factor amplitude of a prism with a n-sided regular polygon cross-section and length L for a specific three dimensional q. + Computes the form factor amplitude of a prism with a n-sided regular polygon cross-section and length L for (qa, qb, qc) scattering vector. Takes in account the parallel factor (complex function) and the perpendicular factor (sinc function). Parameters ---------- qa, qb, qc : float - compenents of the scattering vector q + components of the scattering vector q nsides : int Number of sides of the regular polygon cross-section. Rave : float