# onion

Onion shell model with constant, linear or exponential density

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
sld_core Core scattering length density 10-6-2 1
sld_solvent Solvent scattering length density 10-6-2 6.4
n_shells number of shells None 1
sld_in[n_shells] scattering length density at the inner radius of shell k 10-6-2 1.7
sld_out[n_shells] scattering length density at the outer radius of shell k 10-6-2 2
thickness[n_shells] Thickness of shell k 40
A[n_shells] Decay rate of shell k None 1

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

This model provides the form factor, $$P(q)$$, for a multi-shell sphere where the scattering length density (SLD) of the each shell is described by an exponential, linear, or constant function. The form factor is normalized by the volume of the sphere where the SLD is not identical to the SLD of the solvent. We currently provide up to 9 shells with this model.

NB: radius represents the core radius $$r_0$$ and thickness[k] represents the thickness of the shell, $$r_{k+1} - r_k$$.

Definition

The 1D scattering intensity is calculated in the following way

$P(q) = [f]^2 / V_\text{particle}$

where

\begin{align*} f &= f_\text{core} + \left(\sum_{\text{shell}=1}^N f_\text{shell}\right) + f_\text{solvent} \end{align*}

The shells are spherically symmetric with particle density $$\rho(r)$$ and constant SLD within the core and solvent, so

\begin{align*} f_\text{core} &= 4\pi\int_0^{r_\text{core}} \rho_\text{core} \frac{\sin(qr)}{qr}\, r^2\,\mathrm{d}r &= 3\rho_\text{core} V(r_\text{core}) \frac{j_1(qr_\text{core})}{qr_\text{core}} \\ f_\text{shell} &= 4\pi\int_{r_{\text{shell}-1}}^{r_\text{shell}} \rho_\text{shell}(r)\frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r \\ f_\text{solvent} &= 4\pi\int_{r_N}^\infty \rho_\text{solvent}\frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r &= -3\rho_\text{solvent}V(r_N)\frac{j_1(q r_N)}{q r_N} \end{align*}

where the spherical bessel function $$j_1$$ is

$j_1(x) = \frac{\sin(x)}{x^2} - \frac{\cos(x)}{x}$

and the volume is $$V(r) = \frac{4\pi}{3}r^3$$. The volume of the particle is determined by the radius of the outer shell, so $$V_\text{particle} = V(r_N)$$.

Now lets consider the SLD of a shell defined by

$\begin{split}\rho_\text{shell}(r) = \begin{cases} B\exp\left(A(r-r_{\text{shell}-1})/\Delta t_\text{shell}\right) + C & \mbox{for } A \neq 0 \\ \rho_\text{in} = \text{constant} & \mbox{for } A = 0 \end{cases}\end{split}$

An example of a possible SLD profile is shown below where $$\rho_\text{in}$$ and $$\Delta t_\text{shell}$$ stand for the SLD of the inner side of the $$k^\text{th}$$ shell and the thickness of the $$k^\text{th}$$ shell in the equation above, respectively.

For $$A > 0$$,

\begin{align}\begin{aligned}f_\text{shell} &= 4 \pi \int_{r_{\text{shell}-1}}^{r_\text{shell}} \left[ B\exp \left(A (r - r_{\text{shell}-1}) / \Delta t_\text{shell} \right) + C \right] \frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r\\&= 3BV(r_\text{shell}) e^A h(\alpha_\text{out},\beta_\text{out}) - 3BV(r_{\text{shell}-1}) h(\alpha_\text{in},\beta_\text{in}) + 3CV(r_{\text{shell}}) \frac{j_1(\beta_\text{out})}{\beta_\text{out}} - 3CV(r_{\text{shell}-1}) \frac{j_1(\beta_\text{in})}{\beta_\text{in}}\end{aligned}\end{align}

for

\begin{align*} B&=\frac{\rho_\text{out} - \rho_\text{in}}{e^A-1} &C &= \frac{\rho_\text{in}e^A - \rho_\text{out}}{e^A-1} \\ \alpha_\text{in} &= A\frac{r_{\text{shell}-1}}{\Delta t_\text{shell}} &\alpha_\text{out} &= A\frac{r_\text{shell}}{\Delta t_\text{shell}} \\ \beta_\text{in} &= qr_{\text{shell}-1} &\beta_\text{out} &= qr_\text{shell} \\ \end{align*}

where $$h$$ is

$h(x,y) = \frac{x \sin(y) - y\cos(y)}{(x^2+y^2)y} - \frac{(x^2-y^2)\sin(y) - 2xy\cos(y)}{(x^2+y^2)^2y}$

For $$A \sim 0$$, e.g., $$A = -0.0001$$, this function converges to that of the linear SLD profile with $$\rho_\text{shell}(r) \approx A(r-r_{\text{shell}-1})/\Delta t_\text{shell})+B$$, so this case is equivalent to

\begin{align*} f_\text{shell} &= 3 V(r_\text{shell}) \frac{\Delta\rho_\text{shell}}{\Delta t_\text{shell}} \left[\frac{ 2 \cos(qr_\text{out}) + qr_\text{out} \sin(qr_\text{out}) }{ (qr_\text{out})^4 }\right] \\ &{} -3 V(r_\text{shell}) \frac{\Delta\rho_\text{shell}}{\Delta t_\text{shell}} \left[\frac{ 2\cos(qr_\text{in}) +qr_\text{in}\sin(qr_\text{in}) }{ (qr_\text{in})^4 }\right] \\ &{} +3\rho_\text{out}V(r_\text{shell}) \frac{j_1(qr_\text{out})}{qr_\text{out}} -3\rho_\text{in}V(r_{\text{shell}-1}) \frac{j_1(qr_\text{in})}{qr_\text{in}} \end{align*}

For $$A = 0$$, the exponential function has no dependence on the radius (so that $$\rho_\text{out}$$ is ignored this case) and becomes flat. We set the constant to $$\rho_\text{in}$$ for convenience, and thus the form factor contributed by the shells is

$f_\text{shell} = 3\rho_\text{in}V(r_\text{shell}) \frac{j_1(qr_\text{out})}{qr_\text{out}} - 3\rho_\text{in}V(r_{\text{shell}-1}) \frac{j_1(qr_\text{in})}{qr_\text{in}}$

The 2D scattering intensity is the same as $$P(q)$$ above, regardless of the orientation of the $$q$$ vector which is defined as

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

NB: 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.