Homework #6. 1. Continuum wave equation. Show that for long wavelengths the equation of motion,, reduces to the continuum elastic wave equation dt
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1 Hoework #6 Continuu wave equation Show that for long wavelength the equation of otion, d u M C( u u u, reduce to the continuu elatic wave equation u u v t x where v i the velocity of ound For a, u u i all elacing u with u (x, we have u u(a, u u(( a, and d u u( t Thu t u u u u(( a u( a a a x x u or u(( a u(( a u( a a x d u Then M C( u u u becoe u( t u u( t u M C a or v t x t x with Ca v M Note the a denity i v Ca M Y M /a and the Yong odulu i Y C / a Then Diatoic chain Conider the noral ode of a linear chain in which the force contant between nearet-neighbor ato are alternately C and C Let the ae be equal, and let the nearet-neighbor earation be a/ Find (K at K and K /a Sketch in the dierion relation by eye Thi roble iulate a crytal of diatoic olecule uch a H C C C C C The ato can be earated into two grou: The firt grou ha ring C at the left and ring C at right, and it otion can be decribed by u Aex( ikx it
2 The econd grou ha ring C at the left and ring C at right, and it otion can be decribed by v Bex( ikx it Then the equation of otion for thee two grou are: d u C( v ( u C v u d v C( u ( v C u v Subtitute u Aex( ikx it and v Bex( ikx it into thee two equation, ika / ika / A C( Be A C( Be A ; ika / ika / B C( Ae B C( Ae B ika / ika / (C A C(e e B or ika / ika / C(e e A (C B To have non-zero olution, we have ika / ika / det (C A C(e e B / / ( ( ika ika C e e A C B (C C ( co ka, (C C co ka C ( co ka At k=, C, A B ; and, A B At k=/a, C, A ib ; and C, A ib a a k
3 Atoic vibration in a etal Conider oint ion of a M and charge e iered in a unifor ea of conduction electron The ion are iagined to be in table equilibriu when at regular lattice oint If one ion i dilaced a all ditance r fro it equilibriu oition, the retoring force i largely due to the electric charge within the here of radiu r centered at the equilibriu oition Take the nuber denity of ion (or of conduction electron a /(4, which define (a Show / that the frequency of a ingle ion et into ocillation i ( e / M (b Etiate the value of thi frequency for odiu, roughly (c Fro (a, (b, and oe coon ene, etiate the order of agnitude of the velocity of ound in etal (a For a uniforly charged here of charge denity, the electrical field at r 4 4 e e within the here i E( r r r Now, o r 4 / 4 er E( r e r e The force acting on the ion i F ee Kr with K Then K e M M (b For Sodiu, 4 M 67 g 8 g Taking =Å, we have e ( 48 4 ~ (/ 4 M (c v ~ a ~ c / 4 Soft honon ode Conider a line of ion of equal a but alternating in charge, with e e( a the charge on the th ion The interatoic otential i the u of two contribution: ( a hort-range interaction of force contant C that act between nearet-neighbor only, and ( a coulob interaction between all ion (a Show that the contribution of the coulob interaction to the atoic force contant i C ( e / a, where a i the equilibriu nearet-neighbor ditance (b C Fro (/ M C ( co Ka, here C include both nearet neighbor and other neighbor, how that the dierion relation ay be written a / in Ka ( ( co Ka, where 4 / M and e / a (c Show that i negative (untable ode at the zone boundary Ka if 475 or 4/7 (, where i a ieann zeta function Show further that the eed of ound at all Ka i iaginary if
4 (ln 7 Thu goe to zero and the lattice i untable for oe value of Ka in the interval (, if Notice that the honon ectru i not that of a diatoic lattice becaue the interaction of any ion with it neighbor i the ae a that of any other ion (a The force between two ion i e / r u u + a a+u + -u For two ion at th and (+th ite, their earation ditance i a without ion dilaceent, and i a u u after the th and (+th ion are dilaced by u and, reectively Then the force between the two ion will change by an u ( e ( e ( e aount of ( u u Here ( a u u ( a ( a account for the ign change of the two charge (b The equation of otion for the th ion i d u ( e ( u ( ( u u u u u u u ( a Let u Aex( ikx it, and ubtitute it into the above equation ( e ika (co ka ( e 4 ka in a ika ( co ka a ( ( a ( ( e ( a (co ka ( 4 e Define and, then a ka ( in ( co ka ( co ka ( e ( co ka ika e ika (
5 (c At ka, 5 co( ka ( 6 If / 6 475, For ka, ka ka ( and in ( ( ka 5 ( ( ka ka ( ln Then If / ln 7, Strong Coulob interaction ake the lattice untable 5 For a D lattice, if k k n / a, (a Show that k and k decribe the ae elatic wave (b For a ecial cae of k / a and k 7 / a, ake a lot of co( k x and co( k x veru x / a Confir the concluion of (a fro the lot (a For u Aex( ikx it, u Aex( ikx it, u / u ex[( ik k x] For k k n / a and x a, u / u ex( i n] (b Note the two curve cro at x/a=integer x/a
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