r""" This model calculates an empirical functional form for SAS data using SpericalSLD profile Similarly to the OnionExpShellModel, this model provides the form factor, P(q), for a multi-shell sphere, where the interface between the each neighboring shells can be described by one of a number of functions including error, power-law, and exponential functions. This model is to calculate the scattering intensity by building a continuous custom SLD profile against the radius of the particle. The SLD profile is composed of a flat core, a flat solvent, a number (up to 9 ) flat shells, and the interfacial layers between the adjacent flat shells (or core, and solvent) (see below). .. figure:: img/spherical_sld_profile.gif Exemplary SLD profile Unlike the model (using an analytical integration), the interfacial layers here are sub-divided and numerically integrated assuming each of the sub-layers are described by a line function. The number of the sub-layer can be given by users by setting the integer values of npts_inter. The form factor is normalized by the total volume of the sphere. Definition ---------- The form factor $P(q)$ in 1D is calculated by: .. math:: P(q) = \frac{f^2}{V_\text{particle}} \text{ where } f = f_\text{core} + \sum_{\text{inter}_i=0}^N f_{\text{inter}_i} + \sum_{\text{flat}_i=0}^N f_{\text{flat}_i} +f_\text{solvent} For a spherically symmetric particle with a particle density $\rho_x(r)$ the sld function can be defined as: .. math:: f_x = 4 \pi \int_{0}^{\infty} \rho_x(r) \frac{\sin(qr)} {qr^2} r^2 dr so that individual terms can be calcualted as follows: .. math:: f_\text{core} = 4 \pi \int_{0}^{r_\text{core}} \rho_\text{core} \frac{\sin(qr)} {qr} r^2 dr = 3 \rho_\text{core} V(r_\text{core}) \Big[ \frac{\sin(qr_\text{core}) - qr_\text{core} \cos(qr_\text{core})} {qr_\text{core}^3} \Big] f_{\text{inter}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr f_{\text{shell}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{flat}_i } \frac{\sin(qr)} {qr} r^2 dr = 3 \rho_{ \text{flat}_i } V ( r_{ \text{inter}_i } + \Delta t_{ \text{inter}_i } ) \Big[ \frac{\sin(qr_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) - q (r_{\text{inter}_i} + \Delta t_{ \text{inter}_i }) \cos(q( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } ) ) } {q ( r_{\text{inter}_i} + \Delta t_{ \text{inter}_i } )^3 } \Big] -3 \rho_{ \text{flat}_i } V(r_{ \text{inter}_i }) \Big[ \frac{\sin(qr_{\text{inter}_i}) - qr_{\text{flat}_i} \cos(qr_{\text{inter}_i}) } {qr_{\text{inter}_i}^3} \Big] f_\text{solvent} = 4 \pi \int_{r_N}^{\infty} \rho_\text{solvent} \frac{\sin(qr)} {qr} r^2 dr = 3 \rho_\text{solvent} V(r_N) \Big[ \frac{\sin(qr_N) - qr_N \cos(qr_N)} {qr_N^3} \Big] Here we assumed that the SLDs of the core and solvent are constant against $r$. The SLD at the interface between shells, $\rho_{\text {inter}_i}$ is calculated with a function chosen by an user, where the functions are Exp: .. math:: \rho_{{inter}_i} (r) = \begin{cases} B \exp\Big( \frac {\pm A(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A \neq 0 \\ B \Big( \frac {(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A = 0 \\ \end{cases} Power-Law .. math:: \rho_{{inter}_i} (r) = \begin{cases} \pm B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) ^A +C & \text{for} A \neq 0 \\ \rho_{\text{flat}_{i+1}} & \text{for} A = 0 \\ \end{cases} Erf: .. math:: \rho_{{inter}_i} (r) = \begin{cases} B \text{erf} \Big( \frac { A(r - r_{\text{flat}_i})} {\sqrt{2} \Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A \neq 0 \\ B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) +C & \text{for} A = 0 \\ \end{cases} The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD is continuous at the boundaries of the interface as well as each sub-layers. Thus B and C are determined. Once $\rho_{\text{inter}_i}$ is found at the boundary of the sub-layer of the interface, we can find its contribution to the form factor $P(q)$ .. math:: f_{\text{inter}_i} = 4 \pi \int_{\Delta t_{ \text{inter}_i } } \rho_{ \text{inter}_i } \frac{\sin(qr)} {qr} r^2 dr = 4 \pi \sum_{j=0}^{npts_{\text{inter}_i} -1 } \int_{r_j}^{r_{j+1}} \rho_{ \text{inter}_i } (r_j) \frac{\sin(qr)} {qr} r^2 dr \approx 4 \pi \sum_{j=0}^{npts_{\text{inter}_i} -1 } \Big[ 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } ( r_{j} ) V ( r_{ \text{sublayer}_j } ) \Big[ \frac {r_j^2 \beta_\text{out}^2 \sin(\beta_\text{out}) - (\beta_\text{out}^2-2) \cos(\beta_\text{out}) } {\beta_\text{out}^4 } \Big] - 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } ( r_{j} ) V ( r_{ \text{sublayer}_j-1 } ) \Big[ \frac {r_{j-1}^2 \sin(\beta_\text{in}) - (\beta_\text{in}^2-2) \cos(\beta_\text{in}) } {\beta_\text{in}^4 } \Big] + 3 \rho_{ \text{inter}_i } ( r_{j+1} ) V ( r_j ) \Big[ \frac {\sin(\beta_\text{out}) - \cos(\beta_\text{out}) } {\beta_\text{out}^4 } \Big] - 3 \rho_{ \text{inter}_i } ( r_{j} ) V ( r_j ) \Big[ \frac {\sin(\beta_\text{in}) - \cos(\beta_\text{in}) } {\beta_\text{in}^4 } \Big] \Big] where .. math:: V(a) = \frac {4\pi}{3}a^3 a_\text{in} ~ \frac{r_j}{r_{j+1} -r_j} \text{, } a_\text{out} ~ \frac{r_{j+1}}{r_{j+1} -r_j} \beta_\text{in} = qr_j \text{, } \beta_\text{out} = qr_{j+1} We assume the $\rho_{\text{inter}_i} (r)$ can be approximately linear within a sub-layer $j$ Finally form factor can be calculated by .. math:: P(q) = \frac{[f]^2} {V_\text{particle}} \text{where} V_\text{particle} = V(r_{\text{shell}_N}) For 2D data the scattering intensity is calculated in the same way as 1D, where the $q$ vector is defined as .. math:: q = \sqrt{q_x^2 + q_y^2} .. figure:: img/spherical_sld_1d.jpg 1D plot using the default values (w/400 data point). .. figure:: img/spherical_sld_default_profile.jpg SLD profile from the default values. .. note:: The outer most radius is used as the effective radius for S(Q) when $P(Q) * S(Q)$ is applied. References ---------- L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum Press, New York, (1987) """ import numpy as np from numpy import inf name = "spherical_sld" title = "Sperical SLD intensity calculation" description = """ I(q) = background = Incoherent background [1/cm] """ category = "sphere-based" # pylint: disable=bad-whitespace, line-too-long # ["name", "units", default, [lower, upper], "type", "description"], parameters = [["n_shells", "", 1, [0, 9], "volume", "number of shells"], ["npts_inter", "", 35, [0, inf], "", "number of points in each sublayer Must be odd number"], ["radius_core", "Ang", 50.0, [0, inf], "volume", "intern layer thickness"], ["sld_core", "1e-6/Ang^2", 2.07, [-inf, inf], "", "sld function flat"], ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "", "sld function solvent"], ["func_inter0", "", 0, [0, 4], "", "Erf:0, RPower:1, LPower:2, RExp:3, LExp:4"], ["thick_inter0", "Ang", 50.0, [0, inf], "volume", "intern layer thickness for core layer"], ["nu_inter0", "", 2.5, [-inf, inf], "", "steepness parameter for core layer"], ["sld_flat[n_shells]", "1e-6/Ang^2", 4.06, [-inf, inf], "", "sld function flat"], ["thick_flat[n_shells]", "Ang", 100.0, [0, inf], "volume", "flat layer_thickness"], ["func_inter[n_shells]", "", 0, [0, 4], "", "Erf:0, RPower:1, LPower:2, RExp:3, LExp:4"], ["thick_inter[n_shells]", "Ang", 50.0, [0, inf], "volume", "intern layer thickness"], ["nu_inter[n_shells]", "", 2.5, [-inf, inf], "", "steepness parameter"], ] # pylint: enable=bad-whitespace, line-too-long source = ["lib/librefl.c", "lib/sph_j1c.c", "spherical_sld.c"] profile_axes = ['Radius (A)', 'SLD (1e-6/A^2)'] def profile(n_shells, radius_core, sld_core, sld_solvent, sld_flat, thick_flat, func_inter, thick_inter, nu_inter, npts_inter): """ Returns shape profile with x=radius, y=SLD. """ z = [] beta = [] z0 = 0 # two sld points for core z.append(0) beta.append(sld_core) z.append(radius_core) beta.append(sld_core) z0 += radius_core for i in range(1, n_shells+2): dz = thick_inter[i-1]/npts_inter # j=0 for interface, j=1 for flat layer for j in range(0, 2): # interation for sub-layers for n_s in range(0, npts_inter+1): if j == 1: if i == n_shells+1: break # shift half sub thickness for the first point z0 -= dz#/2.0 z.append(z0) #z0 -= dz/2.0 z0 += thick_flat[i] sld_i = sld_flat[i] beta.append(sld_flat[i]) dz = 0 else: nu = nu_inter[i-1] # decide which sld is which, sld_r or sld_l if i == 1: sld_l = sld_core else: sld_l = sld_flat[i-1] if i == n_shells+1: sld_r = sld_solvent else: sld_r = sld_flat[i] # get function type func_idx = func_inter[i-1] # calculate the sld sld_i = intersldfunc(func_idx, npts_inter, n_s, nu, sld_l, sld_r) # append to the list z.append(z0) beta.append(sld_i) z0 += dz if j == 1: break z.append(z0) beta.append(sld_solvent) z_ext = z0/5.0 z.append(z0+z_ext) beta.append(sld_solvent) # return sld profile (r, beta) return np.asarray(z), np.asarray(beta)*1e-6 def ER(n_shells, radius_core, thick_inter0, thick_inter, thick_flat): total_thickness = thick_inter0 total_thickness += np.sum(thick_inter[:n_shells], axis=0) total_thickness += np.sum(thick_flat[:n_shells], axis=0) return total_thickness + radius_core demo = { "n_shells": 4, "npts_inter": 35.0, "radius_core": 50.0, "sld_core": 2.07, "sld_solvent": 1.0, "thick_inter0": 50.0, "func_inter0": 0, "nu_inter0": 2.5, "sld_flat":[4.0,3.5,4.0,3.5], "thick_flat":[100.0,100.0,100.0,100.0], "func_inter":[0,0,0,0], "thick_inter":[50.0,50.0,50.0,50.0], "nu_inter":[2.5,2.5,2.5,2.5], } #TODO: Not working yet tests = [ # Accuracy tests based on content in test/utest_extra_models.py [{"n_shells":4, 'npts_inter':35, "radius_core":50.0, "sld_core":2.07, "sld_solvent": 1.0, "sld_flat":[4.0,3.5,4.0,3.5], "thick_flat":[100.0,100.0,100.0,100.0], "func_inter":[0,0,0,0], "thick_inter":[50.0,50.0,50.0,50.0], "nu_inter":[2.5,2.5,2.5,2.5] }, 0.001, 0.001], ]