Lecture 21 - Lattice Dynamics: frequency distribution in 1D...

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Unformatted text preview: Lattice Dynamics: frequency distribution in 1D lattice with structure A or AB? - + 1 1 j j j a ( 29 - = = +- 2 2 1 2 2 2 N N j j j j j m H t f from where we derive the equation of motion (see HW#9) ( 29 +- +- gg 1 1 2 coupled harmonic oscillators j j j j m = f = j for a harmonic time dependence, t j e y i ( 29 ( 29 +- +- +- - = +- gg 1 1 2 1 1 2 2 j j j j t t t t j j j j m m e y e y e y e y i i i i = f f = = - g gg 2 t j j t j j e y e y i i i ( 29 +-- +- = 2 1 1 2 j j j j m y y y y f = j (let's see if it works) we propose a sol y ution j e i ( 29 ( 29 ( 29 +-- = +- 1 1 2 2 j j j j m e e e e i i i i f ( 29 - = +- 2 2 j j m e e e i i i-i fe ( 29 - = +- 2 2 m e e i-i f ( 29 =- 2cos 2 f - = 2 2 4sin 2 m = 2 2 4 sin 2 m f = = max maximum is for sin 1 2 2 m f = max sin 2 =- =-- =-- 2 2 2 where we used cos 2 2 cos 1 1 2 sin 2 2 cos 2 2 4 sin 2 2 2 a ( 29 = i where we used e cos sin i ( 29 t+ j back to the displacement (t)=e function , j i ( 29 ( 29 ( 29 + + = = = = = = 2 t+ t+ t j 2 j' 2 + t j + + 2 e e ( ) ( ) e ) e ( j j j j t t t e i i i i i ( 29 = = 2 for - ' , we obtain equal values of j j j t ( 29 2 is the associatedto to the repetition of t a lattice...
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Lecture 21 - Lattice Dynamics: frequency distribution in 1D...

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