Gabriel WlazłowskiUnits
W-SLDA Toolkit uses natural units:
-
m=1- mass of particle, -
\hbar=1- Plank's constant divide by2\pi, -
k_B=1- Boltzmann constant, -
dx=a- lattice spacing.
The lattice spacing is defined through predefines.h
#define DX 1.0It is recommended to work with lattice spacing DX=1; however, the code allows the use of other values, in order to support the functionality of automatic interpolations. Through the lattice spacing, the standard of the code length unit is defined.
Dimensionless units
When reporting values, it is convenient to use dimensionless units. See here for the definition of reference scales.
| quantity | dimensionless units | interpretation |
|---|---|---|
| energy | E/E_{\textrm{ffg}} |
In literature it is called (generalized) Bertsch parameter \xi. |
| distance | xk_F |
Fermi momentum k_F defines the average distance between particles. Thus xk_F=1 means a distance of the order of interparticle distance. |
| time | t\varepsilon_F |
In the time interval t\varepsilon_F=1, particles at the Fermi level pass a distance of the order of the average interparticle distance. In the literature, this time scale is also called the interaction time scale, referring to the typical time scale for scattering events. |
Converting to SI units
To compare with experimental data, typically one needs to convert results to SI units. To do this, one needs to provide SI units for mass, length, and time.
| standard | standard type | SI | code |
|---|---|---|---|
| mass | explicit: mass of particle | m\,\left[\textrm{kg}\right] |
1 |
| length | explicit: via lattice spacing | l\,\left[\textrm{m}\right] |
1 (*) |
| time | implicit: time units is defined through requirement that \hbar=1
|
\left[\textrm{s}\right] |
1 |
(*) The lattice spacing defines how the distance between two lattice points relates to the unit of length in which the code operates. If DX=1 then the lattice spacing itself serves as the standard for the code unit length.
Below, we provide conversion factors:
| quantity | symbol | code value | SI value |
|---|---|---|---|
| wave-vector | k |
1 | \frac{1}{l}\,\left[\frac{\textrm{1}}{\textrm{m}}\right] |
| momentum | p=\hbar k |
1 | \frac{\hbar}{l}\,\left[\frac{\textrm{kg m}}{\textrm{s}}\right] |
| energy | E=\frac{p^2}{2m} |
1 | \frac{\hbar^2}{ml^2}\,\left[J\right] |
| velocity | v=\frac{p}{m} |
1 | \frac{\hbar}{ml}\,\left[\frac{\textrm{m}}{\textrm{s}}\right] |
| time | t=\frac{l}{v} |
1 | \frac{ml^2}{\hbar}\,\left[\textrm{s}\right] |
Example
Consider gas of {}^{6}\textrm{Li}. Moreover, let us assume that lattice spacing is 0.1\,\mu\textrm{m}. Then we have
| standard | value |
|---|---|
m |
9.988\times 10^{-27}\,\textrm{kg} |
l |
0.1\,\mu\textrm{m} |
\hbar |
1.055\times 10^{-34}\,\textrm{J s} |
| code value | SI value |
v=1 |
0.106\,\frac{\textrm{m}}{\textrm{s}} |
t=1 |
0.947\,\mu\textrm{s} |