 Home

Codex Wiki

List of universities

University of Oxford

Course of the University of Oxford

Undergraduate course of the University of Oxford

Physics course of the University of Oxford

Year 3 of the physics course of the University of Oxford

Condensed matter physics course of the University of Oxford

Past exam of the condensed matter physics course of the University of Oxford

2013

Substitute the traveling <normal mode> $u_j=u\exp[i(kja-\omega t)]$. Combining the opposite-neighbour phases gives
$$
m\omega^2=2K(1-\cos ka)+2G(1-\cos2ka),
$$
so the positive-frequency branch is
$$
\boxed{\omega_0(k)=2\sqrt{\frac{K\sin^2(ka/2)+G\sin^2(ka)}{m}},
\qquad |k|\leq\pi/a.}
$$
For positive <spring constants> this is a stable acoustic branch. At long wavelengths,
$$
\omega_0^2\simeq\frac{a^2(K+4G)}m k^2,\qquad
\boxed{c_s=a\sqrt{\frac{K+4G}{m}}.}
$$
The next-nearest-neighbour bond spans twice the distance, hence contributes the factor four. Relative to $G=0$, positive $G$ increases the sound speed by $\sqrt{1+4G/K}$. The chain still has one acoustic branch and no optical branch.

The group speed near $k=0$ and the maximum frequency need not change in the same way. The zone-edge value is always $\omega_0(\pi/a)=2\sqrt{K/m}$, because the next-nearest-neighbour sites then move together. For $G>K/4$ the frequency maximum moves inside the zone, with $\cos(ka)=-K/(4G)$; the small-$k$ sound speed nevertheless keeps increasing.

\Image[/past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2013-further-neighbour-phonons-and-viscous-decay.png]
{title=One acoustic branch with further-neighbour springs, and three viscous-decay regimes}
{description=Original calculated dispersions for stated values of $G/K$, all normalized using the same $K,m,a$. The right panel shows a single normal coordinate with initial displacement one and initial <velocity> zero, for damping ratios $\gamma/\omega_0=0.2,1,2$. It illustrates that a nonoscillatory overdamped response does not have the same envelope as an underdamped <phonon>.}

Solution

Past exam of the condensed matter physics course of the University of Oxford / 2013 / 3 / b / Solution

Past exam of the condensed matter physics course of the University of Oxford / 2013 / 3 / b / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the condensed matter physics course of the University of Oxford / 2013 / 3 / b / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)