Curvas de dispersión#
Cadena monoatómica#
(378)#\[ U=\sum_n \frac{1}{2} K \left[u(n)-u(n+1)\right]^2\]
(379)#\[ m\vec{a}=\vec{F}\rightarrow M\ddot{u}(n)=-\frac{\partial U}{\partial u(n)}=-K\left[2u(n)-u(n-1)-u(n+1)\right]\]
(380)#\[ \ddot{\vec{u}}=\mathcal{H}\vec{u}\]
(381)#\[\ddot{u}(n)=-\frac{K}{M}\left[2u(n)-u(n-1)-u(n+1)\right]\]
(382)#\[\begin{split}\mathcal{D}=
\begin{pmatrix}
\cdots & K/M & -2K/M & K/M & 0 & 0 &\cdots \\
\cdots & 0 & K/M & -2K/M & K/M & 0 & \cdots \\
\cdots & 0 & 0 & K/M & -2K/M & K/M & \cdots \\
\end{pmatrix}\end{split}\]
(383)#\[\mathcal{D}_{ij}=\mathcal{D}_{ji}\]
(384)#\[u(n)\rightarrow u(n)+d\]
(385)#\[\begin{split} \frac{d^2}{dt^2} \left(u(n)+d\right)=\ddot{u(n)}&=-\frac{K}{M}\left[2u(n)+2d-u(n-1)-d-u(n+1)-d\right]\\
&=-\frac{K}{M}\left[2u(n)-u(n-1)-u(n+1)\right]\end{split}\]
(386)#\[\ddot{u}(n)=\sum_{n^\prime} \mathcal{D}(n-n^\prime) u(n^\prime)\]
(387)#\[u(n)=\epsilon e^{i(kna-\omega_k t)}\]
(388)#\[\ddot{\vec{u}}_n=\frac{d^2}{dt^2} u(n)= -\omega_k^2 \epsilon e^{i(kna-\omega_k t)} = -\omega_k^2 u(n)\]
(389)#\[\ddot{\vec{u}}=-\omega_k^2 \epsilon e^{i(kna-\omega_k t)} =\sum_{n^\prime} \mathcal{D}(n-n^\prime) \epsilon e^{i(kn^\prime a-\omega_k t)}\rightarrow \omega_k^2=-\sum_{n^\prime} \mathcal{D}(n-n^\prime)e^{-ik(n-n^\prime)a} \]
(390)#\[\omega_k^2=-\sum_{n^\prime} \mathcal{D}(n^\prime)e^{ikn^\prime a}\]
(391)#\[k\rightarrow \frac{2\pi}{a} \frac{m}{N}, m\in (0,N-1)\]
(392)#\[k\in(-\frac{\pi}{a},\frac{\pi}{a})\]
(393)#\[\begin{split}\mathcal{D}=
\begin{pmatrix}
\cdots & K/M & -2K/M & K/M & 0 & 0 &\cdots \\
\cdots & 0 & K/M & -2K/M & K/M & 0 & \cdots \\
\cdots & 0 & 0 & K/M & -2K/M & K/M & \cdots \\
\end{pmatrix}\end{split}\]
(394)#\[\omega_k^2=-\underbrace{\frac{K}{M}e^{-ika}}_{n=-1}+\underbrace{\frac{2K}{M}e^{ik0a}}_{n=0}-\underbrace{\frac{K}{M}e^{ika}}_{n=1}=\frac{K}{M}\left(2-2\cos ka\right)=2\frac{K}{M}\left(1-\cos ka\right)\]
(395)#\[ T_a u(n)=u(n+1)=\epsilon e^{i(k(n+1)a+\omega_kt)} = e^{ika}\epsilon e^{i(kna+\omega_kt)}=e^{ika} u(n)\]
(396)#\[T_{\vec{R}} u(\vec{r}) = e^{i\vec{k}\vec{R}} u(\vec{R})\]
(397)#\[\begin{split}\omega^2(k+g)&=-\sum_{n} \mathcal{D}(n)e^{i(k+g)na}=\underbrace{-\sum_{n} \mathcal{D}(n)e^{ikna}e^{i2\pi mn}}_{g=\frac{2\pi}{a}m, m\in\mathcal{Z}}\\
&=-\sum_{n} \mathcal{D}(n)e^{ikna}=\omega^2(k)\end{split}\]
(398)#\[s=v_k=\frac{d\omega_k}{dk}=a\sqrt{\frac{K}{M}}\cos\frac{ka}{2}\rightarrow \omega_k \approx s k \rightarrow s(k=0)=a\sqrt{\frac{K}{M}}\]
(399)#\[s=v_k=\frac{d\omega_k}{dk}=a\sqrt{\frac{K}{M}}\cos\frac{ka}{2}\rightarrow s(k=\pi/a)=0\]
Cadena con motivo#
(400)#\[ U=\sum_n \frac{1}{2}K_1 \left(v_n-u_n\right)^2+ \frac{1}{2}K_2 \left(u_{n+1}-v_n\right)^2\]
(401)#\[M_1 \frac{d^2 u_n}{dt^2} = -\frac{\partial U}{\partial u_n} = -K_1 v_n -K_2 v_{n-1} + (K_1+K_2) u_n\]
(402)#\[M_2 \frac{d^2 v_n}{dt^2} = -\frac{\partial U}{\partial v_n} = -K_1 u_n -K_2 u_{n+1} + (K_1+K_2) v_n\]
(404)#\[\begin{split}\begin{pmatrix} u_n\\v_n \end{pmatrix}=\begin{pmatrix}\epsilon_u\\ \epsilon_v\end{pmatrix} e^{i(kna-\omega_kt)}=\vec{\epsilon}e^{i(kna-\omega_kt)}\end{split}\]
(404)#\[\begin{split}\begin{pmatrix} u_n\\v_n \end{pmatrix}=\begin{pmatrix} \epsilon_u\\\epsilon_v \end{pmatrix}e^{i(kna-\omega_kt)}\rightarrow
\begin{cases}
M_1 \frac{d^2 u_n}{dt^2} &= -K_1 v_n -K_2 v_{n-1} + (K_1+K_2) u_n\\
M_2 \frac{d^2 v_n}{dt^2} &= -K_1 u_n -K_2 u_{n+1} + (K_1+K_2) v_n
\end{cases}\end{split}\]
(405)#\[\begin{split}-M_1 \omega_k^2 \epsilon_u &= -K_1 \epsilon_v -K_2e^{-ika} \epsilon_v + (K_1+K_2) \epsilon_u\\
-M_2 \omega_k^2 \epsilon_v & = -K_1 \epsilon_u -K_2e^{ika} \epsilon_u + (K_1+K_2) \epsilon_v\end{split}\]
(406)#\[\begin{split}\begin{pmatrix}
M_1 & 0\\
0 & M_2
\end{pmatrix} \omega_k^2 \begin{pmatrix} \epsilon_u\\\epsilon_v \end{pmatrix} =
\begin{pmatrix}
-(K_1+K_2) & K_1+K_2e^{-ika} \\
K_1+K_2e^{ika} & -(K_1+K_2)
\end{pmatrix}
\begin{pmatrix} \epsilon_u\\\epsilon_v \end{pmatrix}
\rightarrow M\omega_k^2 \vec{\epsilon}=K_k\vec{\epsilon}\end{split}\]
(407)#\[K_{\vec{k}}=\sum_{\vec{\Delta \vec{R}} K_{\Delta\vec{R}} e^{i\vec{k}\vec{R}}} \rightarrow M^{-1} K_k \vec{\epsilon} = \omega_k^2 \vec{\epsilon} \longrightarrow \mathcal{D}_k \vec{\epsilon} = \omega_k^2 \vec{\epsilon}\]
(408)#\[\mathcal{D}_k \vec{\epsilon} = \omega_k^2 \vec{\epsilon} \Leftrightarrow K_k\vec{\epsilon}=M\omega_k^2 \vec{\epsilon}\]
(409)#\[\begin{split} \left[\begin{pmatrix}
-(K_1+K_2) & K_1+K_2e^{-ika} \\
K_1+K_2e^{ika} & -(K_1+K_2)
\end{pmatrix}+
\begin{pmatrix}
M_1 & 0\\
0 & M_2
\end{pmatrix}\omega_k^2
\right]\vec{\epsilon}=\begin{pmatrix}
0\\ 0
\end{pmatrix}\end{split}\]
(410)#\[\begin{split}\left\vert K_k+M\omega_k^2\right\vert = 0 \rightarrow
\begin{vmatrix}
-(K_1+K_2)+M_1\omega_k^2 & K_1+K_2e^{-ika} \\
K_1+K_2e^{ika} & -(K_1+K_2)+M_2\omega_k^2
\end{vmatrix}=0\end{split}\]
(411)#\[M_1 M_2 \omega_k^4-(K_1+K_2)(M_1+M_1)\omega_k^2+2K_1 K_2 (1-\cos ka) =0\]
(412)#\[\begin{split} \begin{pmatrix} u_n\\v_n \end{pmatrix}=\begin{pmatrix} \epsilon_u\\\epsilon_v \end{pmatrix}e^{i(kna-\omega_kt)}\rightarrow
\omega_k^2 = \frac{(K_1+K_2)(M_1+M_1)\pm\sqrt{(K_1+K_2)^2(M_1+M_1)^2-8M_1 M_2 K_1 K_2 (1-\cos ka)}}{2M_1 M_2}\end{split}\]
(413)#\[\vec{k} \parallel \vec{\epsilon}\]
(414)#\[\vec{k} \perp \vec{\epsilon}\]
Vibrations in 2D and 3D#
(415)#\[\begin{split}\begin{pmatrix} u_n\\v_n \end{pmatrix}=\begin{pmatrix} \epsilon_u\\\epsilon_v \end{pmatrix}e^{i(kna-\omega_kt)}\rightarrow
\begin{pmatrix} \vec{u}_n\\\vec{v}_n \end{pmatrix}=\begin{pmatrix} \epsilon_u^x\\\epsilon_u^y\\\epsilon_v^x\\\epsilon_v^y \end{pmatrix}e^{i(kna-\omega_kt)}\end{split}\]
Definición#
The divergence is an operator that is applied on a vector field.
(416)#\[\vec{\nabla}\cdot \vec{f}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}\]
As can be seen, the result of the applying the divergence is a scalar field.
Interpretation#
The divergence measures the difference between the number of field lines that enter and leave from/to a particular point.
Ejemplo
Fig. 208 The divergence measures the number of lines entering/exiting a point.#
A positive divergence is associated with a majority of lines coming out (leaving) from a point. This point is a source of field lines. On the other hand, a point with a negative divergence is seen to come from a majority of lines entering into a point. This point is called a sink of field line.