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) """ 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], "", "number of shells"], ["radius_core", "Ang", 50.0, [0, inf], "", "intern layer thickness"], ["sld_core", "1e-6/Ang^2", 2.07, [-inf, inf], "", "sld function flat"], ["sld_flat[n]", "1e-6/Ang^2", 4.06, [-inf, inf], "", "sld function flat"], ["thick_flat[n]", "Ang", 100.0, [0, inf], "", "flat layer_thickness"], ["func_inter[n]", "", 0, [0, 4], "", "Erf:0, RPower:1, LPower:2, RExp:3, LExp:4"], ["thick_inter[n]", "Ang", 50.0, [0, inf], "", "intern layer thickness"], ["inter_nu[n]", "", 2.5, [-inf, inf], "", "steepness parameter"], ["npts_inter", "", 35, [0, 35], "", "number of points in each sublayer Must be odd number"], ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "", "sld function solvent"], ] # pylint: enable=bad-whitespace, line-too-long #source = ["lib/librefl.c", "lib/sph_j1c.c", "spherical_sld.c"] def Iq(q, *args, **kw): return q def Iqxy(qx, *args, **kw): return qx demo = dict( n_shells=4, scale=1.0, solvent_sld=1.0, background=0.0, npts_inter=35.0, ) #TODO: Not working yet tests = [ # Accuracy tests based on content in test/utest_extra_models.py [{'npts_iter':35, 'sld_solv':1, 'radius_core':50.0, 'sld_core':2.07, 'func_inter2':0.0, 'thick_inter2':50, 'nu_inter2':2.5, 'sld_flat2':4, 'thick_flat2':100, 'func_inter1':0.0, 'thick_inter1':50, 'nu_inter1':2.5, 'background': 0.0, }, 0.001, 0.001], ]