Installation

Install by using

pip install aptrajectories

Example

For usage, see the example Notebook notebook.

Theory

$Robin-Rollan{d}^{[1]}$ type simulations for APT trajectories.

Consider a perfect conductor isolated in space brought to electric potential $V_a$, as shown above.

$V_p$ be the electrostatic potential at point P on the surface. Coloumb's law makes it possible to write this potential from the distribution of charges in space (on the surface in this case):

$$V_p=\frac{1}{4\pi {\epsilon }_o}{\iint }_{S^{\prime }}\frac{\sigma (P^{\prime })}{|\overrightarrow{u}|},d{S}^{\prime }$$ $$\overrightarrow{u}=\overrightarrow{P^{\prime }P}$$

When the point P is displaced infinitesimally displace along the unit surface normal $\overrightarrow{n_P}$, the variation in the potential $V_P$ is given by $\frac{\partial V_P}{\partial n_P}=\frac{1}{4\pi {\epsilon }_o}{\iint }_{S^{\prime }}\frac{\partial (1/|\overrightarrow{u}|)}{\partial n_P}\sigma (P^{\prime }),d{S}^{\prime }$

$$\frac{\partial 1/|\overrightarrow{u}|}{\partial n_P}=-\frac{1}{|\overrightarrow{u}{|}^2}\frac{\partial |\overrightarrow{u}|}{\partial n_P}=-\frac{\cos (\theta )}{|\overrightarrow{u}{|}^2}=-\frac{\overrightarrow{n_P}\cdot \overrightarrow{u}}{|\overrightarrow{u}{|}^3}$$

where, $\theta$ is the angle between vectors $\overrightarrow{n_P}$ and $\overrightarrow{u}$. Then:

$$\frac{\partial V_P}{\partial n_P}=-\frac{1}{4\pi {\epsilon }_o}{\iint }_{S^{\prime }}\frac{\overrightarrow{n_P}\cdot \overrightarrow{u}}{|\overrightarrow{u}{|}^3}\sigma (P^{\prime }),d{S}^{\prime }$$

The surface electric field at point P is given by definition:

$$\overrightarrow{F_P}=-\overrightarrow{\mathrm{\nabla }}(V_p)=-\frac{\partial V_P}{\partial n_P}\overrightarrow{n_P}$$

The theory of electrostatics directly relates this field for the conductors to their charge surface density.

$$|\overrightarrow{F_P}|=\frac{\sigma (P^{\prime })}{2{\epsilon }_o}$$

Combining the above two equations gives: $\sigma (P)=\frac{1}{2\pi }{\iint }_{S^{\prime }}\frac{\overrightarrow{n_P}\cdot \overrightarrow{u}}{|\overrightarrow{u}{|}^3}\sigma (P^{\prime }),d{S}^{\prime }-(1)$

The above integral equation called Robin's equation, connects the surface density of charges at a point on the surface to the distribution of the charge density on the rest of the surface of the conductor. It is possible to solve iteratively by introducing a series of functions $f_n$ defined by:

$$f_{n+1}(P)=\frac{1}{2\pi }{\iint }_{S^{\prime }}\frac{\overrightarrow{n_P}\cdot \overrightarrow{u}}{|\overrightarrow{u}{|}^3}{f}_n(P^{\prime }),d{S}^{\prime }$$

with $f_o$ as any function defined on a convex surface, Robin showed that this would converge to equation (1).

We consider surface as a series of fractional point charges ${q_{i}}{i=1,2,...,N}$ placed on each surface atom over a small surface $s{at}$ .

The electrostatic equilibrium of the conductor can then be described by a local Robin equation as given in equation (1):

$$\frac{q_i}{s_{at}}=\frac{1}{2\pi }\sum _{k=1,k\ne i}^N{q}_k\frac{\overrightarrow{n_i}\cdot \overrightarrow{r_{i,k}}}{|\overrightarrow{r_{i,k}}{|}^3}$$

For a point M infinitesimally close to point i, but inside the conductor. The field generated is given by the right-hand side of the above equation pointing outward, whereas the field induced by the atom i points inward, thereby canceling each other out.

Now a sequence of charges is introdued $q_{i,n}$ on atom i, as follows: $\frac{q_{i,n+1}}{s_{at}}=\frac{1}{2\pi }\sum _{k=1,k\ne i}^N{q}_{k,n}\frac{\overrightarrow{n_i}\cdot \overrightarrow{r_{i,k}}}{|\overrightarrow{r_{i,k}}{|}^3}$

where $q_{i,0}$ being an arbitary real positive number. the unit vector for $i^{th}$ atom depends on the theoretical P nearest neighbors:

$$\vec{n_{i}} = -\frac{\sum_{j=1}^{P}\delta {j} \vec{r{ji}}}{|\sum_{j=1}^{P}\delta {j} \vec{r{ji}}|}$$

Reference

[1] Rolland, N., Vurpillot, F., Duguay, S., & Blavette, D. (2015). A Meshless Algorithm to Model Field Evaporation in Atom Probe Tomography. Microscopy and Microanalysis, 21(6). https://doi.org/10.1017/S1431927615015184