Physical-quantities.md

@@ -44,19 +44,19 @@ The total energy `E_tot` is computed as the sum of all these contributions.
 # Densities 
 Densities are computed according to formulas:  (**TODO: there are missing ^2 in some formulas**)
 * `nu`:  
-![\nu(r) = \frac{1}{2}\sum_{|E_n|<E_c} u_{n,\uparrow}(r)v_{n,\downarrow}^{*}(r)( f_{\beta}(-E_n)-f_{\beta}(E_n) )](https://render.githubusercontent.com/render/math?math=%5Cnu(r)%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Csum_%7B%7CE_n%7C%3CE_c%7D%20u_%7Bn%2C%5Cuparrow%7D(r)v_%7Bn%2C%5Cdownarrow%7D%5E%7B*%7D(r)(%20f_%7B%5Cbeta%7D(-E_n)-f_%7B%5Cbeta%7D(E_n)%20))
+$`\nu(r) = \frac{1}{2}\sum_{|E_n|<E_c} u_{n,\uparrow}(r)v_{n,\downarrow}^{*}(r)( f_{\beta}(-E_n)-f_{\beta}(E_n) )`$
 * `rho_a`:  
-![n_{\uparrow}(r)= \sum_{|E_n|<E_c}|u_{n,\uparrow}(r)|^2 f_{\beta}(E_n)](https://render.githubusercontent.com/render/math?math=n_%7B%5Cuparrow%7D(r)%3D%20%5Csum_%7B%7CE_n%7C%3CE_c%7D%7Cu_%7Bn%2C%5Cuparrow%7D(r)%7C%5E2%20f_%7B%5Cbeta%7D(E_n))
+$`n_{\uparrow}(r)= \sum_{|E_n|<E_c}|u_{n,\uparrow}(r)|^2 f_{\beta}(E_n)`$
 * `rho_b`:  
-![n_{\downarrow}(r) = \sum_{|E_n|<E_c}|v_{n,\downarrow}(r)| f_{\beta}(-E_n)](https://render.githubusercontent.com/render/math?math=n_%7B%5Cdownarrow%7D(r)%20%3D%20%5Csum_%7B%7CE_n%7C%3CE_c%7D%7Cv_%7Bn%2C%5Cdownarrow%7D(r)%7C%20f_%7B%5Cbeta%7D(-E_n))  
+$`n_{\downarrow}(r) = \sum_{|E_n|<E_c}|v_{n,\downarrow}(r)|^2 f_{\beta}(-E_n)`$
 * `tau_a`:  
-![\tau_{\uparrow}(r) = \sum_{|E_n|<E_c}|\nabla u_{n,\uparrow}(r)| f_{\beta}(E_n)](https://render.githubusercontent.com/render/math?math=%5Ctau_%7B%5Cuparrow%7D(r)%20%3D%20%5Csum_%7B%7CE_n%7C%3CE_c%7D%7C%5Cnabla%20u_%7Bn%2C%5Cuparrow%7D(r)%7C%20f_%7B%5Cbeta%7D(E_n))  
+$`\tau_{\uparrow}(r) = \sum_{|E_n|<E_c}|\nabla u_{n,\uparrow}(r)|^2 f_{\beta}(E_n)`$  
 * `tau_b`:  
-![\tau_{\downarrow}(r) = \sum_{|E_n|<E_c}|\nabla v_{n,\downarrow}(r)| f_{\beta}(-E_n)](https://render.githubusercontent.com/render/math?math=%5Ctau_%7B%5Cdownarrow%7D(r)%20%3D%20%5Csum_%7B%7CE_n%7C%3CE_c%7D%7C%5Cnabla%20v_%7Bn%2C%5Cdownarrow%7D(r)%7C%20f_%7B%5Cbeta%7D(-E_n))  
+$`\tau_{\downarrow}(r) = \sum_{|E_n|<E_c}|\nabla v_{n,\downarrow}(r)|^2 f_{\beta}(-E_n)`$  
 * `j_a_x`, `j_a_y`, `j_a_z`:  
-![\vec{j}_{\uparrow}(r) = -\sum_{|E_n|<E_c} \textrm{Im}[u_{n,\uparrow}(r)\nabla u_{n,\uparrow}^*(r)] f_{\beta}(E_n)](https://render.githubusercontent.com/render/math?math=%5Cvec%7Bj%7D_%7B%5Cuparrow%7D(r)%20%3D%20-%5Csum_%7B%7CE_n%7C%3CE_c%7D%20%5Ctextrm%7BIm%7D%5Bu_%7Bn%2C%5Cuparrow%7D(r)%5Cnabla%20u_%7Bn%2C%5Cuparrow%7D%5E*(r)%5D%20f_%7B%5Cbeta%7D(E_n))  
+$`\vec{j}_{\uparrow}(r) = -\sum_{|E_n|<E_c} \textrm{Im}[u_{n,\uparrow}(r)\nabla u_{n,\uparrow}^*(r)`$  
 * `j_b_x`, `j_b_y`, `j_b_z`:  
-![\vec{j}_{\downarrow}(r) = \sum_{|E_n|<E_c} \textrm{Im}[v_{n,\downarrow}(r)\nabla v_{n,\downarrow}^*(r)] f_{\beta}(-E_n)](https://render.githubusercontent.com/render/math?math=%5Cvec%7Bj%7D_%7B%5Cdownarrow%7D(r)%20%3D%20%5Csum_%7B%7CE_n%7C%3CE_c%7D%20%5Ctextrm%7BIm%7D%5Bv_%7Bn%2C%5Cdownarrow%7D(r)%5Cnabla%20v_%7Bn%2C%5Cdownarrow%7D%5E*(r)%5D%20f_%7B%5Cbeta%7D(-E_n))  
+$`\vec{j}_{\downarrow}(r) = \sum_{|E_n|<E_c} \textrm{Im}[v_{n,\downarrow}(r)\nabla v_{n,\downarrow}^*(r)`$ 
 In these formulas $`E_{n}`$ denotes quasi-particle energy and $`E_c`$ is energy cut-off scale. Fermi distribution function $`f_{\beta}(E)=1/(\exp(\beta E)+1)`$ is introduced to model temperature $T=1/\beta$ effects.  
 
 Densities are accessible for user through structure [wslda_density](https://gitlab.fizyka.pw.edu.pl/gabrielw/wslda/-/tree/public/hpc-engine/wslda_potdens.h):