Basic lengths in SNSs

length scales

• inter-atomic distance is $a_{inter-atomic} \simeq 2.4 A$

The nearest neighbor distance is 0.235 nm which is about 5x of Bohr radius. And the diameter of a silicon atom is $2.35 A$ (lattice size of Si is $5.4 A$)

• medium-range order is $\xi = \frac{\Lambda}{(\hbar c_s^3 \rho)^{1/2}} \simeq \frac{1.6*10*10^{-12} erg}{(13*10^{-10})^{1/2}} \simeq 44~\dot{A}$,

where $\Lambda = 10eV= 1.6*10^{-11} erg$

by matching energy scales of phonon and dipole interaction energy $\frac{\hbar c_s}{\xi} \simeq \frac{\Lambda^2}{c_s^2 \rho \xi^3} $

• fundamental glasson length $a_{gl} = 2 \frac{\Lambda^2 m_{gl}}{\hbar^2 \rho c_t^2}$

by matching energy scales of localised kinetic energy and dipole interaction energy $\frac{\hbar^2}{2 m_{gl} \xi^2} \simeq \frac{\Lambda^2}{c_s^2 \rho \xi^3} $

taking $m_{gl}=5*10^{-26} g$, we get

$a_{gl} \simeq 2 \frac{(1.6*10^{-11}erg)^2 5*10^{-26}g }{(10^{-26} erg*s)^2 (2g/cm^3) (4*10^5)^2} \simeq 80 \dot{A} $

we get the mass scale $m_{gl}=5*10^{-26} g$ without any particular model