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General relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford

Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford

2011

Section VI

Unlettered 3

For the <Lennard-Jones potential>,
$$
V'(x)=\frac{24\epsilon}{x}\left[\left(\frac\sigma x\right)^6-2\left(\frac\sigma x\right)^{12}\right].
$$
Its minimum and harmonic <spring constant> are
$$
a=2^{1/6}\sigma,\qquad
K=4\epsilon\left(\frac{156\sigma^{12}}{a^{14}}-\frac{42\sigma^6}{a^8}\right)=\frac{72\epsilon}{a^2}.
$$
Using the <monatomic-chain dispersion relation>,
$$
\boxed{v=\sqrt{\frac{72\epsilon}{m}},\qquad
\omega_{\max}=2\sqrt{\frac Km}=\frac2{2^{1/6}\sigma}\sqrt{\frac{72\epsilon}{m}},\qquad f_{\max}=\frac{\omega_{\max}}{2\pi}.}
$$
The maximum is at $|k|=\pi/a$. Taking the <argon> atomic mass as $m\simeq39.95\,\mathrm u=6.63\times10^{-26}\,\mathrm{kg}$ and $\epsilon=1.602\times10^{-21}\,\mathrm J$ gives
$$
a\simeq0.382\,\mathrm{nm},\qquad K\simeq0.792\,\mathrm{N\,m^{-1}},
$$
$$
\boxed{v\simeq1.32\times10^3\,\mathrm{m\,s^{-1}},\qquad\omega_{\max}\simeq6.91\times10^{12}\,\mathrm{s^{-1}},\qquad f_{\max}\simeq1.10\,\mathrm{THz}.}
$$
These are the stipulated one-dimensional nearest-neighbour model estimates; a three-dimensional <argon> crystal has additional neighbours and <phonon polarizations>.

Solution

Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section VI / 6 / Unlettered 3 / Solution

Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section VI / 6 / Unlettered 3 / Solution

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Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section VI / 6 / Unlettered 3 / Solution

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