r""" Definition ---------- Similarly to the onion, 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 the error function, power-law, or exponential functions. The scattering intensity is computed by building a continuous custom SLD profile along the radius of the particle. The SLD profile is composed of a number of uniform shells with interfacial shells between them. .. figure:: img/spherical_sld_profile.png Example SLD profile Unlike the :ref:`onion` model (using an analytical integration), the interfacial shells here are sub-divided and numerically integrated assuming each sub-shell is described by a line function, with *n_steps* sub-shells per interface. The form factor is normalized by the total volume of the sphere. .. note:: *n_shells* must be an integer. *n_steps* must be an ODD integer. Interface shapes are as follows: 0: erf($\nu z$) 1: Rpow($z^\nu$) 2: Lpow($z^\nu$) 3: Rexp($-\nu z$) 4: Lexp($-\nu z$) 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 calculated 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 in $r$. 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 & \mbox{for } A \neq 0 \\ B \Big( \frac {(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{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 & \mbox{for } A \neq 0 \\ \rho_{\text{flat}_{i+1}} & \mbox{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 & \mbox{for } A \neq 0 \\ B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{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-shell. Thus B and C are determined. Once $\rho_{\text{inter}_i}$ is found at the boundary of the sub-shell 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=1}^{n_\text{steps}} \int_{r_j}^{r_{j+1}} \rho_{ \text{inter}_i } (r_j) \frac{\sin(qr)} {qr} r^2 dr \\ \approx 4 \pi \sum_{j=1}^{n_\text{steps}} \Big[ 3 ( \rho_{ \text{inter}_i } ( r_{j+1} ) - \rho_{ \text{inter}_i } ( r_{j} ) V (r_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_{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:: :nowrap: \begin{align*} V(a) &= \frac {4\pi}{3}a^3 && \\ a_\text{in} \sim \frac{r_j}{r_{j+1} -r_j} \text{, } & a_\text{out} \sim \frac{r_{j+1}}{r_{j+1} -r_j} \\ \beta_\text{in} &= qr_j \text{, } & \beta_\text{out} &= qr_{j+1} \end{align*} We assume $\rho_{\text{inter}_j} (r)$ is approximately linear within the sub-shell $j$. Finally the form factor can be calculated by .. math:: P(q) = \frac{[f]^2} {V_\text{particle}} \mbox{ 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} .. 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) Authorship and Verification --------------------------- * **Author:** Jae-Hie Cho **Date:** Nov 1, 2010 * **Last Modified by:** Paul Kienzle **Date:** Dec 20, 2016 * **Last Reviewed by:** Steve King **Date:** March 29, 2019 """ import numpy as np from numpy import inf, expm1, sqrt from scipy.special import erf name = "spherical_sld" title = "Spherical SLD intensity calculation" description = """ I(q) = background = Incoherent background [1/cm] """ category = "shape:sphere" SHAPES = ["erf(|nu|*z)", "Rpow(z^|nu|)", "Lpow(z^|nu|)", "Rexp(-|nu|z)", "Lexp(-|nu|z)"] # pylint: disable=bad-whitespace, line-too-long # ["name", "units", default, [lower, upper], "type", "description"], parameters = [["n_shells", "", 1, [1, 10], "volume", "number of shells (must be integer)"], ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "solvent sld"], ["sld[n_shells]", "1e-6/Ang^2", 4.06, [-inf, inf], "sld", "sld of the shell"], ["thickness[n_shells]", "Ang", 100.0, [0, inf], "volume", "thickness shell"], ["interface[n_shells]", "Ang", 50.0, [0, inf], "volume", "thickness of the interface"], ["shape[n_shells]", "", 0, [SHAPES], "", "interface shape"], ["nu[n_shells]", "", 2.5, [1, inf], "", "interface shape exponent"], ["n_steps", "", 35, [0, inf], "", "number of steps in each interface (must be an odd integer)"], ] # pylint: enable=bad-whitespace, line-too-long source = ["lib/polevl.c", "lib/sas_erf.c", "lib/sas_3j1x_x.c", "spherical_sld.c"] single = False # TODO: fix low q behaviour have_Fq = True radius_effective_modes = ["outer radius"] profile_axes = ['Radius (A)', 'SLD (1e-6/A^2)'] SHAPE_FUNCTIONS = [ lambda z, nu: erf(nu/sqrt(2)*(2*z-1))/(2*erf(nu/sqrt(2))) + 0.5, # erf lambda z, nu: z**nu, # Rpow lambda z, nu: 1 - (1-z)**nu, # Lpow lambda z, nu: expm1(-nu*z)/expm1(-nu), # Rexp lambda z, nu: expm1(nu*z)/expm1(nu), # Lexp ] def profile(n_shells, sld_solvent, sld, thickness, interface, shape, nu, n_steps): """ Returns shape profile with x=radius, y=SLD. """ n_shells = int(n_shells + 0.5) n_steps = int(n_steps + 0.5) z = [] rho = [] z_next = 0 # two sld points for core z.append(z_next) rho.append(sld[0]) for i in range(0, n_shells): z_next += thickness[i] z.append(z_next) rho.append(sld[i]) dz = interface[i]/n_steps sld_l = sld[i] sld_r = sld[i+1] if i < n_shells-1 else sld_solvent fun = SHAPE_FUNCTIONS[int(np.clip(shape[i], 0, len(SHAPE_FUNCTIONS)-1))] for step in range(1, n_steps+1): portion = fun(float(step)/n_steps, max(abs(nu[i]), 1e-14)) z_next += dz z.append(z_next) rho.append((sld_r - sld_l)*portion + sld_l) z.append(z_next*1.2) rho.append(sld_solvent) # return sld profile (r, beta) return np.asarray(z), np.asarray(rho) # TODO: no random parameter generator for spherical SLD. demo = { "n_shells": 5, "n_steps": 35.0, "sld_solvent": 1.0, "sld": [2.07, 4.0, 3.5, 4.0, 3.5], "thickness": [50.0, 100.0, 100.0, 100.0, 100.0], "interface": [50.0]*5, "shape": [0]*5, "nu": [2.5]*5, } tests = [ # Results checked against sasview 3.1 [{"n_shells": 5, "n_steps": 35, "sld_solvent": 1.0, "sld": [2.07, 4.0, 3.5, 4.0, 3.5], "thickness": [50.0, 100.0, 100.0, 100.0, 100.0], "interface": [50]*5, "shape": [0]*5, "nu": [2.5]*5, }, 0.001, 750697.238], ]