\n", " The 1st Brillouin zone of the selected cell is shown on the left. \n", diff --git a/notebook/band-theory/theory/theory_free_electron.ipynb b/notebook/band-theory/theory/theory_free_electron.ipynb index 0da9063..6a0a7b6 100644 --- a/notebook/band-theory/theory/theory_free_electron.ipynb +++ b/notebook/band-theory/theory/theory_free_electron.ipynb @@ -21,11 +21,7 @@ "source": [ "## **Introduction**\n", " \n", - "Here, we employ the empty lattice approximation for the electrons in a periodic \n", - "solid system. Using it, we compute and plot the electronic band structure for three \n", - "type of cells (simple cubic, FCC and BCC). We get the path for the band structure \n", - "from the seekpath\n", - "package. " + "This notebook gives a brief overview of the empty lattice approximation (free-electron model) of electrons in a metallic solid." ] }, { @@ -35,23 +31,22 @@ "source": [ "## Empty lattice approximation\n", "\n", - "In the empty lattice approximation, the electrons move \"freely\" in the \n", - "periodic potential. There is no electron-electron interaction. \n", + "In the empty lattice approximation, the electrons move \"freely\" in a weak, periodic potential. Electron-electron interactions are neglected.\n", "The eigenfunctions of the Schrödinger equation for free electrons are:\n", "\n", - "$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\vec{r}}$$ \n", + "$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\cdot \\vec{r}}$$ \n", "\n", - "When $\\vec{k'}$ lies outside the 1st Brillouin zone, the plane wave \n", + "When $\\vec{k}$ lies outside the 1st Brillouin zone, the plane wave \n", "can be written as:\n", " \n", - "$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\vec{r}}e^{i\\vec{G} \\vec{r}} = e^{i(\\vec{k}+\\vec{G})\\vec{r}}$$ \n", + "$$\\large \\psi(\\vec{r}) = e^{i\\vec{k}\\cdot \\vec{r}}e^{i\\vec{G} \\cdot\\vec{r}} = e^{i(\\vec{k}+\\vec{G})\\cdot \\vec{r}}$$ \n", " \n", - "where, $\\vec{k}$ vector is inside the first Brillouin zone and $\\vec{G}$ \n", + "where $\\vec{k}$ vector is inside the first Brillouin zone and $\\vec{G}$ \n", "is a reciprocal lattice vector. The dispersion is:\n", " \n", "$$\\large E = \\frac{\\hbar^2(\\vec{k}+\\vec{G})^2}{2m}$$\n", "\n", - "Please read more at the [Wikipedia](https://en.wikipedia.org/wiki/Empty_lattice_approximation)." + "You can read more about this model on [Wikipedia](https://en.wikipedia.org/wiki/Empty_lattice_approximation)." ] }, { @@ -74,12 +69,11 @@ " band structure. Adpoted from Wikipedia\n", "\n", " \n", - "Molecular diagrams can present the discrete energy levels for the \n", - "molecular systems. In contrast, solid system always have a very large \n", - "number of the orbitals. It leads to the energy levels to close together. Hence, the energy levels in solid are considered as continuous energy bands.\n", + "Molecular diagrams can represent the discrete energy levels in\n", + "molecular systems. In contrast, solid-state systems always have a macroscopic number of orbitals present. This leads to the energy levels merging together and forming a continuum. Hence, the energy levels in solid are considered to be continuous energy bands.\n", "\n", - "Since the wavevector is in three dimensions ($k_x$, $k_y$ and $k_z$), \n", - "it is difficult to plot the bands as a function of the wavevector (4 dimensional plotting). Usually, the bands are plotted along the straight lines, which connects high symmetry points (see, e.g. Figure 2).\n", + "Since the wavevector is three dimensional, having components $k_x$, $k_y$ and $k_z$, \n", + "it is difficult to plot the bands as a function of the wavevector. Therefore, the energy bands are typically plotted along straight lines, connecting high symmetry points of the Brillouin zone (see, e.g. Figure 2).\n", "\n", "