sarkas.potentials.qsp

Module for handling Quantum Statistical Potentials.

Potential

Quantum Statistical Potentials are defined by three terms

\[U(r) = U_{\rm pauli}(r) + U_{\rm coul} + U_{\rm diff} (r)\]

where

\[U_{\rm pauli}(r) = k_BT \ln (2) e^{ - 4\pi r^2/ \Lambda_{ab}^2 }\]

is due to the Pauli exclusion principle,

\[U_{\rm coul}(r) = \frac{q_iq_j}{4\pi \epsilon_0} \frac{1}{r}\]

is the usual Coulomb interaction, and \(U_{\rm diff}(r)\) is a diffraction term.

There are two possibilities for the diffraction term. The most common is the Deutsch Potential

\[U_{\rm deutsch}(r) = \frac{q_aq_b}{4\pi \epsilon_0} \frac{e^{- 2 \pi r/\Lambda_{ab}} }{r}.\]

The second most common form is the Kelbg potential

\[U_{\rm kelbg}(r) = - \frac{q_aq_b}{4\pi \epsilon_0} \frac{1}{r} \left [ e^{- 2 \pi r^2/\Lambda_{ab}^2 } - \sqrt{2} \pi \dfrac{r}{\Lambda_{ab}} \textrm{erfc} \left ( \sqrt{ 2\pi} r/ \Lambda_{ab} \right ) \right ].\]

In the above equations the screening length \(\Lambda_{ab}\) is the thermal de Broglie wavelength between the two charges defined as

\[\Lambda_{ab} = \sqrt{\frac{2\pi \hbar^2}{\mu_{ab} k_BT}}, \quad \mu_{ab} = \frac{m_a m_b}{m_a + m_b}\]

Note that in Ref. [Hansen and McDonald, 1981] the DeBroglie wavelength is defined as

\[\Lambda_{ee} = \sqrt{ \dfrac{\hbar^2}{2 \pi \mu_{ee} k_{B} T}},\]

while in statistical physics textbooks is defined as

\[\Lambda_{ee} = \sqrt{ \dfrac{2 \pi \hbar^2}{\mu_{ee} k_{B} T}} .\]

The latter will be used in Sarkas. The difference is in the factor of \(2\pi\), i.e. the difference between a wave number and wave length.

Potential Attributes

The elements of the sarkas.potentials.core.Potential.pot_matrix are:

pot_matrix[0] = qi*qj/4*pi*eps0
pot_matrix[1] = 2pi/deBroglie
pot_matrix[2] = e-e Pauli term factor
pot_matrix[3] = e-e Pauli term exponent term
pot_matrix[4] = Ewald parameter
pot_matrix[5] = Short-range cutoff

Functions

pretty_print_info(potential)	Print potential specific parameters in a user-friendly way.
update_params(potential, species)	Create potential dependent simulation's parameters.