Units.md

+[[_TOC_]]
+
+# Units
 W-SLDA Toolkit uses natural units:
-* $`m=1`$,
-* $`\hbar=1`$,
-* $`k_B=1`$.
+* $`m=1`$ - mass of particle,
+* $`\hbar=1`$ - Plank's constant divide by $`2\pi`$,
+* $`k_B=1`$ - Boltzmann constant, 
\ No newline at end of file
+* $`dx=a`$ - lattice spacing.
+
+The lattice spacing is defined through [predefines.h](https://gitlab.fizyka.pw.edu.pl/wtools/wslda/-/blob/public/st-project-template/predefines.h)
+```c
+#define DX 1.0
+```
+It is *recommended* to work with lattice spacing `DX=1`, however, the code allows to use of other values, in order to support the functionality of [automatic interpolations](Automatic interpolations).
+
+# Dimensionless units
+When reporting values, it is convenient to use dimensionless units. See here for the [definition of reference scales](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 time interval $`t\varepsilon_F=1`$ particles at Fermi level pass distance of the order of average interparticle distance. In literature this time scale is also called _interaction time scale_, meaning typical time scale for scattering events. |
+
+# Converting to SI units 
+To compare with experimental data typically one needs to convert results to SI units. In order to do this, one needs to provide SI standards for mass, length, and time. 
+|standard | standard type| SI | code | 
+| -------| ---- | ----- | -----| 
+|mass | explicit: mass of particle | $`m\,\left[\textrm{kg}\right]`$ | 1 |
+| length | explicit: distance between two lattice points | $`l\,\left[\textrm{m}\right]`$ | 1 (*)|
+| time | implicit: time units is defined through requirement that $`\hbar=1`$ | $`\left[\textrm{s}\right]`$ | 1 |
+
+(*) It is assumed that in [predefines.h](https://gitlab.fizyka.pw.edu.pl/wtools/wslda/-/blob/public/st-project-template/predefines.h) `DX=1`. If not, and `DX=a`, one needs to rescale $`l\rightarrow l/a`$. 
+
+Below we provide conversion factors: 
+
+|qunatity| 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** |
+|$`t=1`$ | $`0.947\,\mu\textrm{s}`$|
+
+