NANO 70-Nots Chapt -Diactd bams Dlta uctio W d som mathmatical tools to dvlop a physical thoy o lcto diactio. Idal cystals a iiit this, so th will b som iiitis lii about. Usually, th iiit quatity oly ists at a poit i spac - ith dict o cipocal - so w t iit umbs o this that w masu. Ad wh w adjust ou thoy lat o al, impct cystals, th iiit this bcom iit. A dlta uctio δ (somtims calld a Diac dlta uctio, is positiv iiity at just o plac o th umb li, wh = 0. It is zo vy wh ls. But this is a somwhat itactabl diitio. A btt diitio is basd o its sampli popty. That is, whv aoth uctio, such as (, is multiplid by δ ad itatd ov all, th sult is ( 0. ( 0 = δ d = So say ( =. W ca immdiatly otic a al popty o δ : = δ d = Th ital om to + ca b thouht o as a limit: lim L = = L L Dlta uctios a somtims calld uit impuls uctios. Foui tasom Th Foui tasom ( o a uctio ( ca b thouht o as its qucy pstatio. Th Foui tasom is ivtibl, s o w ca t ( bac om (. πi ( = d = I[ ] //Foui tasom = πi = ( d = I [ ( ] //ivs Foui tasom = W a usi th I ad I symbols to pst th Foui tasom ad ivs Foui tasom opatios, which a both lia, mai that I[ α +α ] =αi [ ] +αi [ ] o ay uctios ad ad coicits α ad α. H is a ampl:
NANO 70-Nots =δ( a ( = πia Notic that th dlta uctio ca b witt as: π δ =I ( = i d = H is aoth ampl with two dlta uctios: = [ δ( a +δ ( + a ] ( = cos( π a Covolutio thom Covolutios o two uctios show up wh vy poit i o uctio is modiid by aoth uctio: * = ( ( d = Say w ow th Foui tasoms o th two uctios ( = I { }, ( = I { } Th covolutio thom stats that th Foui tasom o thi covolutio quals th poduct o thi Foui tasoms: I { * } = ( ( Th-dimsioal vsios Th Foui tasom o a -D uctio ad th ivs FT a lim πi lim Ω ( = d, π i = d Ω I -D, th dlta uctio ca still b witt as a ital, ad has th sampli popty lim Ω π i lim δ = d Ω, ( 0 = δ( 0 d Th covolutio o two -D uctios is lim * = ( ( d Th covolutio thom i -D bcoms I { * } = ( ( Piodic uctios A Foui sis is a th pstatio o a piodic uctio by a iiit sum o hamoics: ( = πi
NANO 70-Nots Th piodicity is uaatd: ( + = uvw πi = ( ( + = uvw πi ( + uvw = πi πi uvw I cystalloaphy, w ow will b summi ov th RLs by pmuti th idics ( h amo all its, so w ca adopt a shothad otatio: i ( = = π πi I oth wods, th sum ov th RLs is a abbviatio o th sum ov Mill idics: Aoth dlta uctio It mas this cla i w di aoth dlta uctio to us i cipocal spac L, 0 lim πi = d L = 0, othwis Th -D vsio loos li lim πi = 0 d = 0, othwis, It is ot mo usul to di a omalizd, disct vsio lim 0 d = 0, othwis πi, = This om avoids th puzzli iiitis. Poo o covolutio thom Do you d poo o th covolutio thom? H it is:
NANO 70-Nots 4 = h = d ( h( ( =I { } = d πi = d d ( h( πi ( = d d d ( d h( = πi πi πi ( d d ( h( d d πi πi ( = d d ( h( ( ( = d ( h( ( ( = ( h( Foui compots Say w di a omalizd Foui tasom usi th mthod o dii : lim πi = d Assum ( is piodic. Th ( = π i Evaluati th ivs: i ( lim = π πi d lim πi( = d = I oth wods, o a piodic uctio, th Foui coicits o th RLs a th oly o-zo Foui compots. Foui compots o cystal pottial Th mai uctio I hav i mid i this discussio is th lctostatic pottial, o a cystal, o cystal pottial, o shot. Th cystal pottial ca b witt as: Φ ( = Φ πi W ca ma som alizatios about th Φ. Lt s loo at th compl cojuat o Φ( :
NANO 70-Nots [ Φ ( ] = ( Φ * * πi 5 It is usually a ood stati poit to assum Φ( is al. So Φ = [ Φ ] i Φ = ( Φ π * πi( i Φ = ( Φ π * πi * So i th cystal pottial is al, w ca always say: ( Φ * =Φ Now lt s loo at th what happs i w ivt th cystal about th oii Φ( = Φ πi I th oii is a ct o ivsio symmty, th Φ =Φ( Φ = Φ πi πi πi πi Φ = Φ So, i th cystal pottial is ctosymmtic, w ca say: Φ =Φ I th cystal pottial is both al ad ctosymmtic, th all o its Foui coicits a al: Φ =Φ = ( Φ * = al Evaluati th cystal pottial by covolutio Th tools itoducd h so a a itdd to ma li asi. Fo ampl cosid a piodic aay o dlta uctios, locatd at th lattic poits o a cystal: X = δ( - Now ta th lctostatic pottial φ o just o uit cll. Th covolutio o ths ivs th cystal pottial: Φ =φ * X ( I dict spac, w t bac th pctd sum o pottials o all uit clls Φ = φ( - Th advata coms i cipocal spac, wh w ca us th covolutio thom IΦ [ ] =Iφ [ ] I[ X ( ]
NANO 70-Nots 6 Cystal uctio (lattic sum Ca w say aythi ls about I[ X ( ]? Fo a iiit cystal lim N X = N δ - = Its Foui compots a: X N N lim lim πi lim lim πi = δ d = N N = = W ca just divid up spac ito uit-cll-sizd ios, with o lattic poit p io. I th uit cll volum is v, th = Nv X lim, = a RL N v Nv 0, = othwis N -πi = = = So th Foui sis pstatio o th cystal uctio is vy simpl: i X ( = X = v π πi Uit-cll pottials Th total cystal pottial is a sum ov uit-cll pottials: lim N Φ = φ - N = W ca usually assum that th uit-cll pottial is a sum ov atomic pottials, with atoms locatd at thi appopiat positios i th uit cll: m φ = φ -d m atoms ( m W aud that th idividual pottials o isolatd atoms had sphical symmty. So ( m ( m si π φ = 4π φ d π = 0
NANO 70-Nots 7 Thus, th Foui tasom o th uit-cll pottial is: lim φ = φ d = φ -d m atoms m atoms ( ( m m m πi m πid Th si o th pot o th phas acto is somtims opposit by covtio. Plas ba with m. Evaluati th Foui compots o th cystal pottial W saw that Φ is th covolutio Φ =φ * X ( So Φ is th poduct: Φ =φ( X Total wav uctio W ot assum th wav uctio abov th sampl is dscibd by a pla wav that psts th icidt bam: ψ ( = πi Blow th sampl, th total, tasmittd wav uctio is a collctio o diactd bams, icludi 0, which a ach pla wavs with dit amplituds ad tavli i dit dictios: ψ ( = Ψ πi usi th abbviatio = + + s I w ow th cystal stuctu ad oitatio, w also ow th. But w d a pocdu to id th bam amplituds Ψ. Oc w ow thos, th itsitis o th diactio spots will b I = Ψ