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3. For a 1D monoatomic chain, we can write equation of motion as $m(\ddot{\delta x_n}) = F_n = \kappa(\delta x_{n+1} + \delta x_{n-1} - 2\delta x_n)$ Where, $\delta x_n = x_n - x_n^{eq}$ and $x_n^{eq} = na$. Show that by using solution of this equation which describes the normal modes as waves $\delta x_n = Ae^{i\omega t - ikx_n^{eq}} = Ae^{i\omega t - ikna}$ we can obtain the dispersion relation as shown below. $\omega = 2\sqrt{\frac{\kappa}{m}} \left| \sin \left(\frac{ka}{2}\right) \right|.$

          3. For a 1D monoatomic chain, we can write equation of motion as
$m(\ddot{\delta x_n}) = F_n = \kappa(\delta x_{n+1} + \delta x_{n-1} - 2\delta x_n)$
Where, $\delta x_n = x_n - x_n^{eq}$ and $x_n^{eq} = na$. Show that by using solution of this equation which
describes the normal modes as waves
$\delta x_n = Ae^{i\omega t - ikx_n^{eq}} = Ae^{i\omega t - ikna}$
we can obtain the dispersion relation as shown below.
$\omega = 2\sqrt{\frac{\kappa}{m}} \left| \sin \left(\frac{ka}{2}\right) \right|.$

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3. For a 1D monoatomic chain, we can write equation of motion as
m(δ̈ ̈ẍn̈) = Fn = κ(δ xn+1 + δ xn-1 - 2δ xn)
Where, δ xn = xn - xn^eq and xn^eq = na. Show that by using solution of this equation which
describes the normal modes as waves
δ xn = Ae^iω t - ikxn^eq = Ae^iω t - ikna
we can obtain the dispersion relation as shown below.
ω = 2√((κ)/(m))| sin((ka)/(2)) |.

Added by Brenda E.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

11/13/2022

Step 1

Step 1: Write the equation of motion and assume a plane-wave normal-mode solution: m \ddot{\delta x_n} = \kappa(\delta x_{n+1} + \delta x_{n-1} - 2\delta x_n), with trial solution \delta x_n = A e^{i\omega t - i k n a}. Show more…

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For a 1D monoatomic chain, we can write the equation of motion as mÏ‰Â²xâ‚™ = Fâ‚™ = Kâ‚™â‚Šâ‚xâ‚™â‚Šâ‚ + Kâ‚™â‚‹â‚xâ‚™â‚‹â‚ - 2Kâ‚™xâ‚™, where x = x_eq and x_eq = na. Show that by using the solution of this equation, which describes the normal modes as waves, we can obtain the dispersion relation as shown below.

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