diff --git a/notebook/band-theory/theory/theory_brillouin_zone.ipynb b/notebook/band-theory/theory/theory_brillouin_zone.ipynb index 31fe602..8eb7cd8 100644 --- a/notebook/band-theory/theory/theory_brillouin_zone.ipynb +++ b/notebook/band-theory/theory/theory_brillouin_zone.ipynb @@ -21,9 +21,9 @@ "source": [ "## **Introduction**\n", "\n", - "The concept of the reciprocal space and reciprocal lattice was first introduced by [Paul Peter Ewald](https://en.wikipedia.org/wiki/Paul_Peter_Ewald) (German crystallographer and physicist) in 1921 [[Annalen der Physik, 369, 3, 253 (1921)]](https://onlinelibrary.wiley.com/doi/10.1002/andp.19213690304). The concept of reciprocal space is very helpful to understand the X-ray diffraction. It is also one of the core concept in the solid state physics.\n", + "The concept of the reciprocal space and the reciprocal lattice was first introduced by [Paul Peter Ewald](https://en.wikipedia.org/wiki/Paul_Peter_Ewald) (German crystallographer and physicist) in 1921 [[Annalen der Physik, 369, 3, 253 (1921)]](https://onlinelibrary.wiley.com/doi/10.1002/andp.19213690304). The concept is invaluable, for example, when one is trying to interpret X-ray diffraction patterns. It is also one of the core conceptual tools employed in solid state physics.\n", "\n", - "The Wigner–Seitz construction is a method to build the primitive cell. In the reciprocal space, the Wigner-Seitze primitive cell is called the 1st Brillouin zone." + "The Wigner–Seitz construction is a method to build the primitive cell in real space. Its counterpart in reciprocal space (i.e. transformation of the Wigner-Seitz cell to reciprocal space) is called the 1st Brillouin zone." ] }, { @@ -33,20 +33,20 @@ "source": [ "## **Reciprocal space and reciprocal lattice**\n", "\n", - "Let's define $\\vec{a}_1$, $\\vec{a}_2$ and $\\vec{a}_3$ as a set of primitive vectors for the direction of the lattice. The lattice vector $\\vec{R}_l$ can be written as:\n", + "Let's define $\\vec{a}_1$, $\\vec{a}_2$ and $\\vec{a}_3$ as a set of primitive vectors describing the real-space lattice. The lattice vector $\\vec{R}_l$ can be written as:\n", "\n", "$$\\vec{R}_l = (l_1\\vec{a}_1 + l_2\\vec{a}_2 + l_3\\vec{a}_3)$$\n", "\n", "The volume of the lattice cell is computed as:\n", "\n", - "$$\\Omega = \\vec{a}_1 (\\vec{a}_2 \\times \\vec{a}_3)$$\n", + "$$\\Omega = \\vec{a}_1 \\cdot (\\vec{a}_2 \\times \\vec{a}_3)$$\n", "\n", - "Now, let's define another set of primitive vectors: $\\vec{b}_1$, $\\vec{b}_2$, $\\vec{b}_3$. If they satisfy the relation with $\\vec{a}_1$, $\\vec{a}_2$, $\\vec{a}_3$ as:\n", + "Now, let's define another set of primitive vectors, $\\vec{b}_1$, $\\vec{b}_2$ and $\\vec{b}_3$ which satisfy the relationship\n", "\n", - "$$\\vec{a}_i \\vec{b}_j = 2\\pi \\delta_{ij} $$\n", + "$$\\vec{a}_i \\cdot \\vec{b}_j = 2\\pi \\delta_{ij} $$\n", "\n", - "$\\delta_{ij}$ is the Dirac delta function, which equals to 1 when $i=j$ and equals\n", - "to 0 when $i \\neq j$, both sets of primitive vectors are reciprocal lattice vectors to each others. The primitive vectors $\\vec{b}_1$, $\\vec{b}_2$, $\\vec{b}_3$ can be constructed as:\n", + "Here, $\\delta_{ij}$ is the Dirac delta function, which equals to 1 when $i=j$ and equals\n", + "to 0 when $i \\neq j$. Such as set of vectors $\\{\\vec{b}_1$, $\\vec{b}_2$, $\\vec{b}_3\\}$ are referred to as the reciprocal lattice vectors corresponding to $\\{\\vec{a}_1, \\vec{a}_2,\\vec{a}_3\\}$. The reciprocal lattice vectors can be explicitly constructed as:\n", "\n", "$$\\vec{b}_1 = 2\\pi \\frac{\\vec{a}_2 \\times \\vec{a}_3}{\\vec{a}_1 (\\vec{a}_2 \\times \\vec{a}_3)}$$\n", "$$\\vec{b}_2 = 2\\pi \\frac{\\vec{a}_3 \\times \\vec{a}_1}{\\vec{a}_1 (\\vec{a}_2 \\times \\vec{a}_3)}$$\n", @@ -60,21 +60,20 @@ "source": [ "## **Wigner–Seitz cell**\n", "\n", - "For the soild state systems, there are many ways to construct the primitive cell\n", + "For soild-state systems, there are many ways to construct the primitive cell\n", "according to the symmetry of the lattice.\n", - "The most common method is called Wigner-Seitz cell, which is named after two physicists [Eugene Wigner](https://en.wikipedia.org/wiki/Eugene_Wigner) and \n", + "The most common method is called the Wigner-Seitz construction, which is named after two physicists [Eugene Wigner](https://en.wikipedia.org/wiki/Eugene_Wigner) and \n", "[Frederick Seitz](https://en.wikipedia.org/wiki/Frederick_Seitz).\n", - "The Wigner-Seitz cell around a lattice point is closer than any other lattice points, which is the defination of the Wigner-Seitz cell.\n", + "The Wigner-Seitz cell is defined as the locus of point in real space which are closer to a given point on the crystalline lattice than any of the other lattice points.\n", "\n", - "For a 2D lattice grid (as shown in Figure 1), connect one random lattice point to all its nearest \n", - "neighbor lattice points (all the blue line). Then bisection each blue line with a line (the red line) and \n", - "the smallest polyhedron is the boundary of the Wigner-Seitz cell. The animination of the construction method\n", - "is in the right of Figure 1. We can use the same method to construct the Wigner-Seitz cell in three dimension lattice space.\n", + "For a 2D lattice (as shown in Figure 1), the Wigner-Seitz cell can be produced via the following steps. First, connect one random lattice point to all its nearest \n", + "neighbor lattice points (the blue lines). Then bisect each blue line with another line (the red line). The smallest polyhedron formed in this way is the boundary of the Wigner-Seitz cell. An animination perorming this construction\n", + "is illustrated in the right of Figure 1. We can use the same method to construct the Wigner-Seitz cell for a three dimensional lattice.\n", "\n", "
\n", " \"drawing\"\n", " \"drawing\"\n", - "
Figure 1. Demonstration of the construction of the Wigner-Seitz cell in two \n", + "
Figure 1. Demonstration of the construction of the Wigner-Seitz cell in a two \n", " dimensional system. Figure and animination are adpoted from \n", " Wikipedia.
\n", "
" @@ -91,14 +90,14 @@ "[Léon Brillouin](https://en.wikipedia.org/wiki/L%C3%A9on_Brillouin).\n", "The 1st Brillouin zone is the Wigner-Seitz primitive cell in the **reciprocal lattice**.\n", "As shown in Figure 2, the pink area shows the 1st Brillouin zone. The second Brillouin zone\n", - "is constructed by drawing perpendicular bisectors of the reciprocal lattice group to the second \n", - "nearest lattice points. The region of the second Brillouin zone is the polyhedron without the \n", - "area of the 1st Brillouin zone, which is the yellow area in the Figure 2. The green and red areas together are the 3rd Brillouin zone.\n", + "is constructed by drawing perpendicular bisectors of the group of reciprocal lattice vectors connecting to the second \n", + "nearest neighbour lattice points. The region corresponding to the second Brillouin zone is the polyhedron with the \n", + "area of the 1st Brillouin zone removed; this is the yellow area in the Figure 2. The green and red areas together constitute the 3rd Brillouin zone.\n", "\n", "
\n", " \"drawing\"\n", "
Figure 2. Demonstration of the construction of the 1st, 2nd and 3rd \n", - " Brillouin zone for two-dimensional lattice.
\n", + " Brillouin zones for a two-dimensional lattice.\n", "
" ] }, @@ -127,7 +126,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.12" + "version": "3.10.12" } }, "nbformat": 4,