# spherical_sld

Sperical SLD intensity calculation

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
n_shells number of shells None 1
sld_solvent solvent sld 10-6-2 1
sld[n_shells] sld of the shell 10-6-2 4.06
thickness[n_shells] thickness shell 100
interface[n_shells] thickness of the interface 50
shape[n_shells] interface shape None 0
nu[n_shells] interface shape exponent None 2.5
n_steps number of steps in each interface (must be an odd integer) None 35

The returned value is scaled to units of cm-1 sr-1, absolute scale.

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.

Unlike the <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.

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)


Definition

The form factor $$P(q)$$ in 1D is calculated by:

$P(q) = \frac{f^2}{V_\text{particle}} \text{ where } f = f_\text{core} + \sum_{\text{inter}_i=0}^N f_{\text{inter}_i} + \sum_{\text{flat}_i=0}^N f_{\text{flat}_i} +f_\text{solvent}$

For a spherically symmetric particle with a particle density $$\rho_x(r)$$ the sld function can be defined as:

$f_x = 4 \pi \int_{0}^{\infty} \rho_x(r) \frac{\sin(qr)} {qr^2} r^2 dr$

so that individual terms can be calculated as follows:

\begin{align}\begin{aligned}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]\end{aligned}\end{align}

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:

$\begin{split}\rho_{{inter}_i} (r) &= \begin{cases} B \exp\Big( \frac {\pm A(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A \neq 0 \\ B \Big( \frac {(r - r_{\text{flat}_i})} {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A = 0 \\ \end{cases}\end{split}$

Power-Law

$\begin{split}\rho_{{inter}_i} (r) &= \begin{cases} \pm B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) ^A +C & \mbox{for } A \neq 0 \\ \rho_{\text{flat}_{i+1}} & \mbox{for } A = 0 \\ \end{cases}\end{split}$

Erf:

$\begin{split}\rho_{{inter}_i} (r) = \begin{cases} B \text{erf} \Big( \frac { A(r - r_{\text{flat}_i})} {\sqrt{2} \Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A \neq 0 \\ B \Big( \frac {(r - r_{\text{flat}_i} )} {\Delta t_{ \text{inter}_i }} \Big) +C & \mbox{for } A = 0 \\ \end{cases}\end{split}$

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 $$P(q)$$

\begin{align}\begin{aligned}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]\end{aligned}\end{align}

where

\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

$P(q) = \frac{[f]^2} {V_\text{particle}} \mbox{ where } V_\text{particle} = V(r_{\text{shell}_N})$

For 2D data the scattering intensity is calculated in the same way as 1D, where the $$q$$ vector is defined as

$q = \sqrt{q_x^2 + q_y^2}$

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)