Lecture 21

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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lattice Dynamics: frequency distribution in 1D lattice with structure A or AB? ξ ξ ξ- + 1 1 j j j a µ ( 29 ξ ξ ξ- = ∂ = +- ∂ ∑ ∑ 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 ξ ξ ξ ξ +- +- gg 1 1 2 coupled harmonic oscillators j j j j m = f ϖ ξ = j for a harmonic time dependence, t j e y i ( 29 ( 29 ϖ ϖ ϖ ϖ ξ ξ ξ ξ ϖ +- +- +- → - = +- 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 ϖ ϖ ξ ϖ ξ ϖ = = - g gg 2 t j j t j j e y e y i i i ( 29 ϖ +-- +- → = 2 1 1 2 j j j j m y y y y f φ = j (let's see if it works) we propose a sol y ution j e i ( 29 ( 29 ( 29 φ φ φ φ ϖ +-- = +- 1 1 2 2 j j j j m e e e e i i i i f ( 29 φ φ φ φ ϖ- = +- 2 2 j j m e e e i i i-i fe ( 29 φ φ ϖ- = +- 2 2 m e e i-i f ( 29 φ =- 2cos 2 f φ ϖ- = 2 2 4sin 2 m φ ϖ = 2 2 4 sin 2 m f φ ϖ ϖ = ⇒ = max maximum is for sin 1 2 2 m f φ ϖ ϖ = max sin 2 α α φ φ =- =-- =-- 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 φ φ φ ± = ± i where we used e cos sin i ( 29 ϖ φ ξ t+ j back to the displacement (t)=e function , j i ( 29 ( 29 ( 29 π ϖ φ φ ϖ φ π φ ϖ φ ϖ φ π π ξ ξ ξ + + = = = = = = 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 π ξ φ ∆ = = 2 for - ' , we obtain equal values of j j j t ( 29 π λ ξ φ 2 is the associatedto to the repetition of t a lattice...
View Full Document

{[ snackBarMessage ]}

### Page1 / 14

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online