deerlab.dd_shellvoidshell#

dd_shellvoidshell = <deerlab.model.Model object>#

Particles uniformly distributed on a spherical shell and on another concentric spherical shell separated by a void.

Parameters:

rarray_like: Distance axis, in nanometers.
radiusscalar: Inner shell radius.
thickness1scalar: Inner shell thickness.
thickness2scalar: Outer shell thickness.
separationscalar: Shell-shell separation.

Returns:

Pndarray: Distance distribution.

Notes

Parameter List

Name	Lower	Upper	Type	Frozen	Unit	Description
`radius`	0.1	20	nonlin	No	nm	Inner shell radius
`thickness1`	0.1	20	nonlin	No	nm	Inner shell thickness
`thickness2`	0.1	20	nonlin	No	nm	Outer shell thickness
`separation`	0.1	20	nonlin	No	nm	Shell-shell separation

Model

$P(r) = \left(R_1^3((R_3^3 - R_1^3)P_\mathrm{BS}(r|R_1,R_3) - (R_4^3 - R_1^3)P_\mathrm{BS}(r|R_1,R_4)) + R_2^3((R_4^3 - R_2^3)P_\mathrm{BS}(r|R_2,R_4) - (R_3^3 - R_2^3)P_\mathrm{BS}(r|R_2,R_3)) \right)/((R_4^3 - R_3^3)(R_2^3 - R_1^3))$

with

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

and

$R_1 = R$

$R_2 = R + w_1$

$R_3 = R + w_1 + d$

$R_4 = R + w_1 + d + w_2$

where $R$ is the inner shell’s radius, $w_1$ and $w_2$ denote the thickness of the inner and outer shells, respectively, and $d$ is the shell-shell separation.

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)

../_images/deerlab-dd_shellvoidshell-1.png