I Bdrn Solid Stte Physics Crystl vibrtions nd the clssicl theory: The ssmption will be mde to consider tht the men eqilibrim position of ech ion is t Brvis lttice site The ions oscillte bot this men position Ths the instntneos position of the ion r will devite from its men position by the displcement, s shown in figre 59 This mens tht r r O Figre 59: The instntneos position of n ion is defined in terms of its men position the Brvis lttice vector nd the ionic displcement The sttic lttice model s compred to the dynmic model: As mentioned before tht the pir of toms in noble gses, for exmple, seprted by r contribtes potentil energy r to the totl potentil energy of the whole crystl where the ssmption is mde tht these toms re fixed t their Brvis lttice sites with zero inetic energies The totl potentil energy is the sm of ll potentil energies of the pirs of toms nd hs the expression: N 0 03
I Bdrn Solid Stte Physics Now if or ssmption is chnged sch tht llowing the toms to oscillte bot their men positions, the bove reltion mst be modified nd written s: r r However, the new expression of the totl potentil energy depends on the dynmicl vribles Therefore the Hmiltonin of the system is expressed s: P H, where P is the momentm of the tom whose eqilibrim position is nd its tomic mss is The hrmonic Approximtion: This pproximtion is bsed on the concept tht the toms will hve very smll devitions from their men positions In sch circmstnces the totl potentil energy my be expnded sing the Tylor's theorem, tht is, 3 r r r r r r 3! Pt r, nd insert the bovementioned dynmicl definition of the totl potentil energy we my get: 3 04
I Bdrn Solid Stte Physics 05 N ] [ 4 3 O The first term is the potentil energy t eqilibrim nd cn be given the expression: 0 eq N The second term represents the liner term which hs the coefficient This coefficient gives s the net force on ny tom in eqilibrim Ths this term mst vnish becse the net force on ny tom in sch conditions is zero All higher order terms cn be ten s corrections to eq The most importnt term mong sch high order terms is the qdrtic term qd or clled hrm, nmely, qd ] [ 4 or ] [ ] [ 4,,, z y x qd, where r r r r
I Bdrn Solid Stte Physics Note: In severl dynmicl pplictions the constnt term eq cn be neglected becse it is independent of the 's nd P 's nd the totl potentil energy trns ot to be the qdrtic term only Frther corrections to the totl potentil energy my be considered Sch corrections re lie the third nd forth order terms nd they re clled the nhrmonic terms Those re treted s smll pertrbtions on the dominnt hrmonic term Vibrtions of crystls with montomic bsis in onedimension: In order to stdy the elstic vibrtions of crystl when one tom in the primitive cell is considered, we need to now the direction by which the elstic wve my propgte long The propgtion of sch wve cses the entire plnes of toms to move in phse with two displcements: Either one is prllel to the direction of wve vector clled longitdinl vector b Or the other is perpendiclr to the direction of wve vector clled trnsverse vector However ech wve vector my hve three modes, one in longitdinl polriztion nd two trnsverse polriztions 06
I Bdrn Solid Stte Physics Let s te liner rry of toms long the x-xis s n exmple Ech tom with mss nd seprted by distnce x= n, where n is n integer The displcement of the n th ion long the x-xis from its eqilibrim position is defined by n, s shown in figre 60 If the ssmption is mde tht only neighboring ions will interct, then hrmonic pproximtion will be dopted where the qdrtic term in the expnded elstic potentil energy my be considered, nmely: qd C n [ n { n } ] Here it is ssmed tht the force on n tom in the plne of toms n csed by the displcement of the plne of toms n+ is proportionl to the chnge in displcements [n+-n] The x elstic constnt C, where is the elstic potentil energy of the two intercting ions seprted by distnce x Note: It mst be emphsized tht if only nerest-neighbor forces re ept, the hrmonic pproximtion for the -D Brvis lttice describes model in which ech ion is tied to its neighbor by perfect springs of spring constnt C 07
I Bdrn Solid Stte Physics n- n- n n+ n+ n b Figre 60: The displcement n of n ion from its eqilibrim position t n, t ny instnt b The hrmonic pproximtion for -D Brvis lttice describes model in which ech ion is tied to its nerest-neighbor by perfect springs with spring constnt C The eqtions of motion cn be simply obtined from the bove form of elstic potentil energy s: qd n C[ n { n } { n } ] n Considering the norml mode of oscilltions the sme s wve trveling on continos string, the displcement of the n th tom my be represented by: n, t i nt e If we re seeing soltions of sch form to the bove eqtions of motion, then these soltions mst stisfy the eqtions of motion, ie: e C[e e e i nt i nt i { n} t i { n} t i i C[ e e ] C cos, ], 08
I Bdrn Solid Stte Physics where Or C cos, C cos C sin, cos sin 4C sin This is the dispersion reltion for norml modes of liner chin of toms nd it is depicted in figre 6 4C Figre 6: The dispersion plot for montomic liner chin with only nerest-neighbor interctions The bondry of the first Brillion Zone: The bondry of the first Brilloin zone cn be checed ot s follows: The first derivtive with respect to of the lst reltion is d d C zero, ie 0 Since sin sin = 0 t d d 09
I Bdrn Solid Stte Physics Two extreme limits of or wvelength: i For very smll vles of or very long wvelengths, ie <<, cos cn be expnded sch tht cos Ths min nd the dispersion reltion C give min Conclsion: The elstic or sond wves hve freqencies tht re directly proportionl to in this region of the dispersion crve The slope of this liner reltion gives s the velocity of sond, nmely: C v sond Some physics might be extrcted from this, if we define the liner mss density or mss per nit length of the chin of toms s, then C my represent the tension of the chin T This will obviosly give s physicl pictre similr to the problem of wve trveling in string where the velocity of wve is written s T v Ths the freqency in or rel problem cn be expressed s v sond, s shown in figre6 ii When lrge vles of re considered mx t 4C, nd mx 0
I Bdrn Solid Stte Physics Figre 6: The liner prt of the dispersion plot t very smll vles of for montomic liner chin with only nerest-neighbor interctions emr: There re no norml modes tht cn hve nglr freqency greter thn mx Limittions on the wve vector within the first Brilloin zone: Since the dispersion crve is periodic in in intervls of /, tomic displcements re the sme for norml modes This mens tht the most convenient wy to specify the bondry conditions is to join the two remote ends of the chin bc together by one more of the sme springs tht connect internl toms s shown in figre 63 If we te the toms to occpy sites,, N, where N represents the totl nmber of toms, then we cn solve the eqtion of motion N times where n=,, N provided tht the bondry conditions re: = [N+], nd N = 0
I Bdrn Solid Stte Physics in This reqires tht e which my lso reqire tht n This is the well-nown Born-von Krmn periodic N bondry condition L=N Figre 63: The Born-von bondry condition A mssless rigid rod of length L=N connects the ion on the extreme right with the spring on the extreme left Notes: Any two modes with wve vector tht differ by / re the sme nd the smllest wve vector for ny given mode is restricted to the rnge The mximm wve vector At the bondries mx mx of the first Brilloin zone, the soltion e in - t does not represent trveling wve, bt stnding wve This mens tht in sch stnding wve, lternte toms oscillte in opposite phses, becse n = e in = - n, where the phse fctor e -it is ignored here Ths n is either eql + or - which depends on whether n is n even integer where the wve moves to the right or odd integer where the wve moves to the left
I Bdrn Solid Stte Physics Stnding wve nd Brgg reflection of x-rys: When the Brgg condition is stisfied stnding wve is set p nd no more trveling wve propgtion does exist in lttice Sbstitting the vles mx into the Brgg lw d sin = m will give d sin = m, where = Here the bove condition my be stisfied when d=, =/ nd m= Conclsion: Only wve lengths longer thn re reqired to represent the lttice motion This occrs by hving vles of within the first Brilloin zone ie within limits mx 3