reword smectic order parameter example

examples/Smectic.ipynb

@@ -5,7 +5,14 @@
    "metadata": {},
    "source": [
     "# Calculating Smectic Order Parameters\n",
-    "In systems with nematic (orientational) many other types of phases can exist. When position in exactly one dimension is ordered alongside orientations we get smectic phases. There are many different types of smectic phases. The simplest one of them is called smectic A. In smectic A phase the particles are organized in layers and layer surface normal is aligned with nematic director. The phase in which layer director does not align with particle's long axes is called smectic C phase. We shall focus on smecitc A phase in this tutorial. First lets construct a system with perfect smectic and nematic order. To do this particles must be placed in layers, meaning that their z-height is restricted but position in the xy plane is randomized. We set all the orientations to align with z axis. We start by importing relevant modules and writing helper functions for randomization of orientations and system construction."
+    "Systems with nematic order can exhibit more complex phases if positional order exists alongside orientational order. There are many different types of these phases, which are collectively known as smectic phases. The simplest smectic phase is called smectic A. In the smectic A phase, orientational order is nematic and positions are ordered into layers. This tutorial will focus on calculating the smectic order parameter for a smectic A phase."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The helper functions in the code block below will help us construct the systems we study in this notebook."
    ]
   },
   {
@@ -71,7 +78,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Next we construct the system using the above helper functions."
+    "Now, lets start by constructing a system with perfect smectic and nematic order. To create perfect nematic order, we align each orientation with the z-axis. To create perfect smectic order, particles are placed in layers by the z component of their position while the x and y components are randomized."
    ]
   },
   {
@@ -6079,11 +6086,11 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "To compute the smectic order parameter we use the following equation:\n",
+    "We can compute the smectic order parameter with the following equation:\n",
     "$$\n",
-    "P(d) = \\frac{1}{N} \\sum_j^N \\exp( \\frac{2\\pi i}{d} \\mathbf{n} \\cdot \\mathbf{r}_j) (6)\n",
+    "P(d) = \\frac{1}{N} \\sum_j^N \\exp( \\frac{2\\pi i}{d} \\mathbf{n} \\cdot \\mathbf{r}_j) (6),\n",
     "$$\n",
-    "where $d$ is the layer distance, $N$ is the number of particles and $\\mathbf{r}$ is the position of particle. It is clear that this value is a function of layer distance $d$. Fortunately it is well known that at the appropriate layer distance $d$, the value of $P$ must be at a global maximum. Thus to find the true value of the smectic order parameter $P$ we simply optimize it as a function of $d$ to find the layer distance as well. A good introduction in how to compute smectic order parameters can be found in appendix of [this work](https://pubs.aip.org/aip/jcp/article/138/20/204901/566257/An-atomistic-description-of-the-nematic-and).  "
+    "where $d$ is the layer distance, $N$ is the number of particles and $\\mathbf{r}$ is the position of particle. Because of the cyclial nature of complex exponentials, $P$ reaches a global maximum at specific values of the layer distance $d$. To find a value of $d$ which gives perfect smectic order, we maximize $P$ as a function of $d$. The code in the block below demonstrates this process. (A good introduction in how to compute smectic order parameters can be found in appendix of [this work](https://pubs.aip.org/aip/jcp/article/138/20/204901/566257/An-atomistic-description-of-the-nematic-and).)"
    ]
   },
   {
@@ -6131,7 +6138,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "As we can see the system has perfect nematic and perfect smectic order. The distance between the smectic layers determined from the smectic order parameter matches the layer thickness we used to construct the system in the first place. Now consider a system in which orientations are somewhat randomized and nematic order is not perfect, but smectic order still is. In this case smectic order parameter should still be close to 1, but nematic order should be much lower."
+    "As we can see, the system with the optimal spacing between the layers has both perfect nematic and perfect smectic order. We next consider a system with perfect smectic order, but imperfect nematic order. We achieve this by introducing orientational disorder, but keeping the layer spacing the same."
    ]
   },
   {
@@ -12661,7 +12668,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "now lets introduce noise in z position of our particles to get a more realistic system snapshot. Now both smecitc and nematic order should be between 0 and 1."
+    "Lastly, we demonstrate a system with imperfect layering and imperfect orientational order. In this system, both smectic and nematic order parameters are between 0 and 1."
    ]
   },
   {
@@ -19130,7 +19137,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.8"
+   "version": "3.7.12"
   }
  },
  "nbformat": 4,