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33 {\LARGE {\bf Materials Physics I}\\}
38 {\Large\bf Tutorial 2 - proposed solutions}
44 \item $r_i=r_{i0}+u_i$\\
45 $\rho=r_2-r_1=r_{20}+u_2-r_{10}-u_1=(r_{20}-r_{10})+(u_2-u_1)
47 \item $\Phi-\Phi_0=\frac{D}{2}(\rho-\rho_0)^2
48 =\frac{D}{2}(\rho^2+\rho_0^2-2\rho_0\rho)$\\
49 $\rho^2=\rho_0^2+\sigma^2+2\rho_0\sigma$
50 $\Rightarrow$ $\rho=\sqrt{\rho_0^2+\sigma^2+2\rho_0\sigma}$\\
51 $\Rightarrow$ $\Phi-\Phi_0=\frac{D}{2}
52 [2\rho_0^2+\sigma^2+2\rho_0\sigma-
53 2\rho_0\sqrt{\rho_0^2+\sigma^2+2\rho_0\sigma}]$
55 \item $\sigma \parallel \rho_0$:
57 \item \begin{flushleft}
58 \includegraphics[height=6cm]{elongation_p01.eps}
59 \includegraphics[height=6cm]{elongation_p02.eps}
60 \includegraphics[height=6cm]{elongation_p03.eps}
62 \item $\sigma = \sigma_{\parallel}$:\\
63 $\rho_0 \sigma_{\parallel} = |\rho_0| |\sigma_{\parallel}|$\\
64 $\Phi-\Phi_0=\frac{D}{2}\left(2\rho_0^2+\sigma_{\parallel}^2+
65 2\rho_0\sigma_{\parallel}-
66 2\rho_0\sqrt{(\rho_0+\sigma_{\parallel})^2}\right)
67 =\frac{D}{2}\sigma_{\parallel}^2$
69 \item $\sigma \perp \sigma_0$:
71 \item \begin{flushleft}
72 \includegraphics[height=5.3cm]{elongation_n01.eps}
73 \includegraphics[height=5.3cm]{elongation_n02.eps}
74 \includegraphics[height=5.3cm]{elongation_n03.eps}
76 \item $\sigma=\sigma_{\perp}$:\\
77 $\sigma_{\perp} \rho_0 = 0$\\
78 $\Phi-\Phi_0=\frac{D}{2}\left[2\rho_0^2+\sigma_{\perp}^2-
79 2\rho_0\sqrt{\rho_0^2+\sigma_{\perp}^2}\right]$
81 \item $\sigma_{\perp} = \alpha \rho_0$, $\alpha \ll 1$\\
82 $\sqrt{\rho_0^2+\sigma_{\perp}^2}=
83 \sqrt{\rho_0^2+\alpha^2\rho_0^2}=
84 \rho_0\sqrt{1+\alpha^2}\stackrel{Taylor}{=}
85 \rho_0(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)$\\
86 $\Rightarrow \Phi-\Phi_0=
87 \frac{D}{2}\left[\rho_0^2\left(2+\alpha^2-
88 2(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)\right)\right]=
89 \frac{D}{2}\left[\rho_0^2(\frac{\alpha^4}{4}+\ldots)\right]$\\
90 $\Rightarrow \Phi-\Phi_0\stackrel{\alpha\ll 1}{=}
91 \frac{D}{2}\rho_0^2\frac{\alpha^4}{4}=
92 \frac{D}{2}\sigma_{\perp}^2\frac{\alpha^2}{4}$
93 \item $\sigma_{\parallel}$, $\sigma_{\perp} \ll \rho_0$\\
94 $\Rightarrow$ potential contribution of $\sigma_{\perp}$
95 compared to contribution of $\sigma_{\parallel}$
99 \item As long as the displacements and thus the elongation is small
100 compared to the equilibrium state the change in the potential
101 due to the perpendicular elongation is negligible small.
102 \item Regarding a possible existence of perpendicular elongation
103 the model of the linear chain is unproblematic.
104 \item In a real crystal couplings in other directions exist.
105 These can only be neglected if they are small compared to the
106 coupling of the considered direction.
112 \item \begin{itemize}
114 Atom type 1: $M_1$, $u_s$ (elongation of atom $s$ of type 1)\\
115 Atom type 2: $M_2$, $v_s$ (elongation of atom $s$ of type 2)\\
116 Lattice constant: $a$, Spring constant: $C$
117 \item Equations of motion:\\
118 $M_1\ddot{u}_s=C(v_s+v_{s-1}-2u_s)$\\
119 $M_2\ddot{v}_s=C(u_{s+1}+u_s-2v_s)$
121 $u_s=u\exp(i(ska-\omega t))$\\
122 $v_s=v\exp(i(ska-\omega t))$
123 \item Solution of the equation system:\\
124 $-\omega^2M_1u\exp(i(ska-\omega t))=
125 C\exp(-i\omega t)[v\exp(iska)+v\exp(i(s-1)ka)-2u\exp(iska)]$\\
126 $\Rightarrow -\omega^2M_1u=Cv(1+\exp(-ika))-2Cu$\\
127 $-\omega^2M_2v\exp(i(ska-\omega t))=
128 C\exp(-i\omega t)[u\exp(i(s+1)ka)+u\exp(iska)-2v\exp(iska)]$\\
129 $\Rightarrow -\omega^2M_2v=Cu[\exp(ika)+1]-2Cv$\\
130 Non trivial solution only if determinant of coefficients
131 $u$ and $v$ is zero.\\
135 2C-M_1\omega^2 & -C[1+\exp(-ika)]\\
136 -C[1+\exp(ika)] & 2C-M_2\omega^2
140 4C^2+M_1M_2\omega^4-2C\omega^2(M_2+M_1)-
141 \underbrace{C^2(1+\exp(ika))(1+\exp(-ika))}_{
142 C^2(\underbrace{1+1+\exp(ika)+\exp(-ika)}_{
143 2+2\cos(ka)=2(1+\cos(ka))})}$\\
145 M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$
148 \item \begin{eqnarray}
149 \omega^2&=&C\left(\frac{2C(M_1+M_2)}{2M_1M_2}\right)\pm
150 \sqrt{\frac{4C^2(M_1+M_2)^2}{4M_1^2M_2^2}-
151 \frac{2C^2(1-cos(ka))}{M_1M_2}} \nonumber \\
152 &=&C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)\pm
153 \sqrt{C^2\frac{(M_1+M_2)^2}{M_1^2M_2^2}-
154 \frac{1}{M_1M_2}2C^2(1-cos(ka))} \nonumber \\
155 &=&C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)\pm
156 C\sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2-
157 \frac{2(1-\cos(ka))}{M_1M_2}} \nonumber
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214 $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$ (Taylor)\\
215 Optical branch: $\omega^2\approx
216 2C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)$\\
217 Acoustic branch: $\omega^2\approx
218 \frac{C/2}{M_1+M_2}k^2a^2$\\
220 Optical branch: $u/v = - M_2/M_1$ (out of phase)\\
221 \item $k=\pm \pi/a$:\\
222 $\rightarrow \omega^2=2C/M_2,2C/M_1$