deerlab.bg_hom3dex_phase#

bg_hom3dex_phase = <deerlab.model.Model object>#

Phase shift from a homogeneous distribution of spins with excluded volume

Parameters:
tarray_like

Time vector, in microseconds.

concscalar

Spin concentration

rexscalar

Exclusion radius

lamscalar

Pathway amplitude

Returns:
Bndarray

Dipolar background vector.

Notes

Parameter Table

Name

Lower

Upper

Type

Frozen

Unit

Description

conc

0.01

5e+03

nonlin

No

μM

Spin concentration

rex

0.01

20

nonlin

No

nm

Exclusion radius

lam

0

1

nonlin

No

Pathway amplitude

Model

../_images/model_scheme_bg_hom3dex.png

This implements the phase-shift arising from a hard-shell excluded-volume model, with spin concentration c_s (in μM) and the radius of the spherical excluded volume R_\mathrm{ex} (in nm).

The expression for this model is

B(t) = \exp \Bigg(- i c_\mathrm{s}\lambda_k \bigg( V_\mathrm{ex} \mathrm{Im}\{\mathcal{K}_0(t, R_\mathrm{ex})\} + \mathcal{I}_\mathrm{C}(t) \bigg)

where \mathcal{I}_\mathrm{C}(t) is an integral without analytical form given by

\mathcal{I}_\mathrm{C}(t) = \frac{4\pi}{3} D\,t \int_0^1 \mathrm{d}z~(1 - 3z^2) ~ \mathrm{C_i}\left( \frac{D\,t (1 - 3z^2)}{R_\mathrm{ex}^3 } \right)

where \mathrm{C_i} is the cosine integral function and D is the dipolar constant

D = \frac{\mu_0}{4\pi}\frac{(g_\mathrm{e}\mu_\mathrm{B})^2}{\hbar}