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Diffstat (limited to 'source/know/concept/density-of-states/index.md')
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diff --git a/source/know/concept/density-of-states/index.md b/source/know/concept/density-of-states/index.md index 60b81d5..1c3b511 100644 --- a/source/know/concept/density-of-states/index.md +++ b/source/know/concept/density-of-states/index.md @@ -8,15 +8,15 @@ categories: layout: "concept" --- -The **density of states** $g(E)$ of a physical system is defined such that -$g(E) \dd{E}$ is the number of states which could be occupied -with an energy in the interval $[E, E + \dd{E}]$. -In fact, $E$ need not be an energy; +The **density of states** $$g(E)$$ of a physical system is defined such that +$$g(E) \dd{E}$$ is the number of states which could be occupied +with an energy in the interval $$[E, E + \dd{E}]$$. +In fact, $$E$$ need not be an energy; it should just be something that effectively identifies the state. In its simplest form, the density of states is as follows, -where $\Gamma(E)$ is the number of states with energy -less than or equal to the argument $E$: +where $$\Gamma(E)$$ is the number of states with energy +less than or equal to the argument $$E$$: $$\begin{aligned} g(E) @@ -25,9 +25,9 @@ $$\begin{aligned} If the states can be treated as waves, which is often the case, -then we can calculate the density of states $g(k)$ in -$k$-space, i.e. as a function of the wavenumber $k = |\vb{k}|$. -Once we have $g(k)$, we use the dispersion relation $E(k)$ to find $g(E)$, +then we can calculate the density of states $$g(k)$$ in +$$k$$-space, i.e. as a function of the wavenumber $$k = |\vb{k}|$$. +Once we have $$g(k)$$, we use the dispersion relation $$E(k)$$ to find $$g(E)$$, by demanding that: $$\begin{aligned} @@ -37,14 +37,14 @@ $$\begin{aligned} = g(k) \dv{k}{E} \end{aligned}$$ -Inverting the dispersion relation $E(k)$ to get $k(E)$ might be difficult, +Inverting the dispersion relation $$E(k)$$ to get $$k(E)$$ might be difficult, in which case the left-hand equation can be satisfied numerically. -Define $\Omega_n(k)$ as the number of states with -a $k$-value less than or equal to the argument, -or in other words, the volume of a hypersphere with radius $k$. -Then the $n$-dimensional density of states $g_n(k)$ +Define $$\Omega_n(k)$$ as the number of states with +a $$k$$-value less than or equal to the argument, +or in other words, the volume of a hypersphere with radius $$k$$. +Then the $$n$$-dimensional density of states $$g_n(k)$$ has the following general form: $$\begin{aligned} @@ -54,14 +54,14 @@ $$\begin{aligned} } \end{aligned}$$ -Where $D$ is each state's degeneracy (e.g. due to spin), -and $k_{\mathrm{min}}$ is the smallest allowed $k$-value, -according to the characteristic length $L$ of the system. -We divide by $2^n$ to limit ourselves to the sector where all axes are positive, -because we are only considering the magnitude of $k$. +Where $$D$$ is each state's degeneracy (e.g. due to spin), +and $$k_{\mathrm{min}}$$ is the smallest allowed $$k$$-value, +according to the characteristic length $$L$$ of the system. +We divide by $$2^n$$ to limit ourselves to the sector where all axes are positive, +because we are only considering the magnitude of $$k$$. -In one dimension $n = 1$, the number of states within a distance $k$ from the -origin is the distance from $k$ to $-k$ +In one dimension $$n = 1$$, the number of states within a distance $$k$$ from the +origin is the distance from $$k$$ to $$-k$$ (we let it run negative, since its meaning does not matter here), given by: $$\begin{aligned} @@ -69,9 +69,9 @@ $$\begin{aligned} = 2 k \end{aligned}$$ -To get $k_{\mathrm{min}}$, we choose to look at a rod of length $L$, +To get $$k_{\mathrm{min}}$$, we choose to look at a rod of length $$L$$, across which the function is a standing wave, meaning that -the allowed values of $k$ must be as follows, where $m \in \mathbb{N}$: +the allowed values of $$k$$ must be as follows, where $$m \in \mathbb{N}$$: $$\begin{aligned} \lambda = \frac{2 L}{m} @@ -79,9 +79,9 @@ $$\begin{aligned} k = \frac{2 \pi}{\lambda} = \frac{m \pi}{L} \end{aligned}$$ -Take the smallest option $m = 1$, -such that $k_{\mathrm{min}} = \pi / L$, -the 1D density of states $g_1(k)$ is: +Take the smallest option $$m = 1$$, +such that $$k_{\mathrm{min}} = \pi / L$$, +the 1D density of states $$g_1(k)$$ is: $$\begin{aligned} \boxed{ @@ -91,8 +91,8 @@ $$\begin{aligned} } \end{aligned}$$ -In 2D, the number of states within a range $k$ of the -origin is the area of a circle with radius $k$: +In 2D, the number of states within a range $$k$$ of the +origin is the area of a circle with radius $$k$$: $$\begin{aligned} \Omega_2(k) @@ -100,8 +100,8 @@ $$\begin{aligned} \end{aligned}$$ Analogously to the 1D case, -we take the system to be a square of side $L$, -so $k_{\mathrm{min}} = \pi / L$ again. +we take the system to be a square of side $$L$$, +so $$k_{\mathrm{min}} = \pi / L$$ again. The density of states then becomes: $$\begin{aligned} @@ -112,14 +112,14 @@ $$\begin{aligned} } \end{aligned}$$ -In 3D, the number of states is the volume of a sphere with radius $k$: +In 3D, the number of states is the volume of a sphere with radius $$k$$: $$\begin{aligned} \Omega_3(k) = \frac{4 \pi}{3} k^3 \end{aligned}$$ -For a cube with side $L$, we once again find $k_{\mathrm{min}} = \pi / L$. +For a cube with side $$L$$, we once again find $$k_{\mathrm{min}} = \pi / L$$. We thus get: $$\begin{aligned} @@ -130,17 +130,17 @@ $$\begin{aligned} } \end{aligned}$$ -All these expressions contain the characteristic length/area/volume $L^n$, +All these expressions contain the characteristic length/area/volume $$L^n$$, and therefore give the number of states in that region only. -Keep in mind that $L$ is free to choose; +Keep in mind that $$L$$ is free to choose; it need not be the physical size of the system. In fact, we typically want the density of states per unit length/area/volume, -so we can just set $L = 1$ in our preferred unit of distance. +so we can just set $$L = 1$$ in our preferred unit of distance. If the system is infinitely large, or if it has periodic boundaries, -then $k$ becomes a continuous variable and $k_\mathrm{min} \to 0$. -But again, $L$ is arbitrary, +then $$k$$ becomes a continuous variable and $$k_\mathrm{min} \to 0$$. +But again, $$L$$ is arbitrary, so a finite value can be chosen. |