4 problems on infinite 1D or 2D lattices

Jun 2nd, 2016
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1. For a linear diatomic lattice dispersion curve the maximum frequency of the acoustic phonon branch is 1012 Hz. The forbidden gap is 4 meV wide. Calculate the lowest allowed frequency of the optical branch. Calculate the force constant  if the lighter of the two types of atom in the lattice has a mass number of 23

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1. For a linear diatomic lattice dispersion curve the maximum frequency of theacoustic phonon branch is 1012 Hz. The forbidden gap is 4 meV wide. Calculatethe lowest allowed frequency of the optical branch.Calculate the force constant if the lighter of the two types of atom in the latticehas a mass number of 23.The acoustic branch is1 11 14 sin 2 ka ) ( )2 }m Mm MMmThe maximum frequency of the acoustic branch is achieved when ka = /2 with1 11 14 2 {( ) ( ) 2 }m Mm MMm1 11 1 {( ) ( )}m Mm M2M2f M 101222M 6.28 1012On the other hand, the forbidden gap equal to difference between the minimumallowed energy of optical branch and the maximum allowed energy of theacoustic branch. The optical branch is 2 {(1 11 14 sin 2 ka ) ( )2 }m Mm MMmand the lowest allowed frequency occurs at ka = /2 with1 11 1 2 {( ) )}m Mm M 2 / mtherefore, we have 2 {( 22 4meVmM224 10 3)mM6.582 10 -16 6.08 1012 Hz(the lowest allowed optical frequnecy is26.081m 1012 Hz 2222 0.968 1012 Hz 1.00 1012 Hz2M 1.97 1012 Hzm = 23 x1.66x10-27 kg = 3.82x10-26 kg2 2 1.97 1012 Hzm 1.24 10131 m (1.24

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