Correct the structure-factor and coupling formulas of the Bloch-wave tutorial

docs/user_guide/tutorials/blochwave.ipynb

@@ -108,10 +108,10 @@
     "The structure factor is calculated using the independent atom model, as a superposition of atomic scattering factors:\n",
     "\n",
     "$\n",
-    "F(\\mathbf{g}) = \\sum_{j} f_j(g) \\exp(-2\\pi i \\mathbf{g} \\cdot \\mathbf{r}_j)\n",
+    "F(\\mathbf{g}) = \\frac{1}{\\Omega} \\sum_{j} f_j(g) \\exp(2\\pi i \\mathbf{g} \\cdot \\mathbf{r}_j)\n",
     "$\n",
     "\n",
-    "where $f_j(g)$ is the atomic scattering factor of the $j$'th atom. The Bloch-wave code uses the same parametrizations of the scattering factors as multislice, see our description in our [walkthrough on potentials](walkthrough:potentials). \n",
+    "where $f_j(g)$ is the atomic scattering factor of the $j$'th atom, $\\mathbf{r}_j$ its position and $\\Omega$ the volume of the unit cell. The Bloch-wave code uses the same parametrizations of the scattering factors as multislice, see our description in our [walkthrough on potentials](walkthrough:potentials). \n",
     "\n",
     "Here, we create a `StructureFactor` object representing the structure factor. We provide the two required keyword arguments: `atoms` we created above, and the maximum spatial frequency `g_max` given in $\\mathrm{Å}^{-1}$ defines the maximum spatial frequency of the structure factor. Increasing `g_max` is equivalent to making the sampling (pixel size) smaller in a multislice simulation. As we describe later on, the $k$-vector of the simulated diffraction patterns, is half that of the structure factor.\n",
     "\n",
@@ -259,10 +259,10 @@
     "We can get the potential from the structure factor using a 3D Fourier transform\n",
     "\n",
     "$\n",
-    "V(\\mathbf{r}) = \\frac{4 \\pi \\epsilon_0}{2 \\pi a_0 e} \\int F(g) \\exp(2\\pi i \\mathbf{g}\\cdot \\mathbf{r}) \\mathrm{d}\\mathbf{r} \\quad ,\n",
+    "V(\\mathbf{r}) = \\frac{1}{\\kappa} \\sum_{\\mathbf{g}} F(\\mathbf{g}) \\exp(-2\\pi i \\mathbf{g}\\cdot \\mathbf{r}) \\ , \\qquad \\kappa = \\frac{4 \\pi \\epsilon_0}{2 \\pi a_0 e} \\ ,\n",
     "$\n",
     "\n",
-    "where $\\epsilon_0$ is the vaccum permitivity, $a_0$ is the Bohr radius and $e$ is the electron charge. Note that we use the unit convention from ASE, as described [here](appendix:conventions).\n",
+    "so that $V_{\\mathbf{g}} = F(\\mathbf{g}) / \\kappa$ is the Fourier coefficient of the potential. Here $\\epsilon_0$ is the vacuum permittivity, $a_0$ is the Bohr radius and $e$ is the electron charge. Note that we use the unit convention from ASE, as described [here](appendix:conventions).\n",
     "\n",
     "You can get the full 3D potential as a NumPy array using the `.get_potential_3d` method. The projected potential as `PotentialArray` object, which may be used in a normal *ab*TEM multislice simulation, can be obtained using `get_projected_potential`. This is currently only implemented for orthogonal unit cells."
    ]
@@ -308,7 +308,7 @@
     "We can approximate the inclusion of phonons by providing a `thermal_sigma`, which will apply a Debye-Waller factor to the structure factor, effectively, blurring the potential in real space\n",
     "\n",
     "$\n",
-    "F(\\mathbf{g}) = \\sum_{j} f_j(g) \\exp(-2\\pi i \\mathbf{g} \\cdot \\mathbf{r}_j) \\exp(-2 \\pi^2 \\sigma^2 g) .\n",
+    "F(\\mathbf{g}) = \\frac{1}{\\Omega} \\sum_{j} f_j(g) \\exp(2\\pi i \\mathbf{g} \\cdot \\mathbf{r}_j) \\exp(-2 \\pi^2 \\sigma_j^2 g^2) .\n",
     "$"
    ]
   },
@@ -529,9 +529,11 @@
     "where\n",
     "\n",
     "$\n",
-    "U_{g} = F_g.\n",
+    "U_{g} = \\frac{2 m e}{h^2} V_g = \\frac{\\sigma}{\\pi \\lambda} V_g = \\frac{\\sigma}{\\pi \\lambda \\kappa} F_g \\ ,\n",
     "$\n",
     "\n",
+    "with $m$ the relativistic electron mass, $\\sigma$ the interaction parameter, $\\lambda$ the wavelength and $V_g = F_g / \\kappa$ the Fourier coefficient of the potential.\n",
+    "\n",
     "The scattering matrix is given as\n",
     "\n",
     "$\n",