deerlab.dd_shell#

dd_shell = <deerlab.model.Model object>#

Uniform distribution of particles on a spherical shell

Parameters:

rarray_like: Distance axis, in nanometers.
radiusscalar: Inner shell radius.
thicknessscalar: Shell thickness.

Returns:

Pndarray: Distance distribution.

Notes

Parameter List

Name	Lower	Upper	Type	Frozen	Unit	Description
`radius`	0.1	20	nonlin	No	nm	Inner shell radius
`thickness`	0.1	20	nonlin	No	nm	Shell thickness

Model

$P(r) = \left(R_2^6 P_\mathrm{B}(r|R_2) - R_1^6 P_\mathrm{B}(r|R_1) - 2(r_2^3 - r_1^3)P_\mathrm{BS}(r|R_1,R_2)\right)/(R_2^3 - R_1^3)^2$

with

$P_\mathrm{BS}(r|R_i,R_j) = \frac{3}{16R_i^3(R_j^3 - R_i^3)}\begin{cases} 12r^3R_i^2 - r^5 \quad \text{for} \quad 0\leq r < \min(2R_i,R_j - R_i) \\ 8r^2(R_j^3 - R_i^3) - 3r(R_j^2 - R_i^2)^2 - 6r^3(R_j - R_i)(R_j + R_i) \quad \text{for} \quad R_j-R_i \leq r < 2R_i \\ 16r^2R_i^3 \quad \text{for} \quad 2R_i\leq r < R_j - R_i \\ r^5 - 6r^3(R_j^2 + R_i^2) + 8r^2(R_j^3 + R_i^3) - 3r(R_j^2 - R1_2)^2 \quad \text{for} \quad \max(R_j-R_i,2R_i) \leq r < R_i+R_j \\ 0 \quad \text{for} \quad \text{otherwise} \end{cases}$

$P_\mathrm{B}(r|R_i) = \begin{cases} \frac{3r^5}{16R_i^6} - \frac{9r^3}{4R_i^4} + \frac{3r^2}{R_i^3} \quad \text{for} \quad 0 \leq r < 2R_i \\ 0 \quad \text{for} \quad \text{otherwise} \end{cases}$

and

$R_1 = R$

$R_2 = R + w$

where $R$ is the inner shell radius, and $w$ is the shell thickness.

References

[1]

D.R. Kattnig, D. Hinderberger, Analytical distance distributions in systems of spherical symmetry with applications to double electron-electron resonance, JMR, 230, 50-63, 2013

Examples

Example of the model evaluated at the start values of the parameters:

(Source code, png, hires.png, pdf)