Instrumentation for Characterization of Nanomaterials (v11) 11. Crystal Potential

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1 Istumtatio o Chaactizatio o Naomatials (v). Cystal Pottial Dlta uctio W d som mathmatical tools to dvlop a physical thoy o lcto diactio om cystal. Idal cystals a iiit this, so th will b som iiitis lii about. Usually, th iiit quatity oly xists 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 x (somtims calld a Diac dlta uctio), is positiv iiity at just o plac o th umb li, wh x 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 x, is multiplid by x ad itatd ov all x, th sult is 0. 0 x x dx x So, say x. W ca immdiatly otic a al popty o x: x xdx Th ital om to ca b thouht o as a limit: lim L L x xl Dlta uctios a somtims calld uit impuls uctios. Foui tasom Th Foui tasom o a uctio x ca b thouht o as its qucy pstatio. Th Foui tasom is ivtibl, so w ca t x bac om. ix x dx x //Foui tasom x ix x d //ivs Foui tasom W a usi th ad symbols to pst th Foui tasom ad ivs Foui tasom opatios, which a both lia, mai that o ay uctios ad ad coicits ad.

2 Istumtatio o Chaactizatio o Naomatials (v) H is a xampl: xxa ia Notic that th dlta uctio ca b witt as: x ix d H is aoth xampl with two dlta uctios: x xaxa cos a Covolutio thom Covolutios o two uctios show up wh vy poit i o uctio is modiid by aoth uctio: x* x x xxdx x Say w ow th Foui tasoms o th two uctios x, x Th covolutio thom stats that th Foui tasom o thi covolutio quals th poduct o thi Foui tasoms: x* x Th-dimsioal vsios Th Foui tasom o a -D uctio ad th ivs FT a lim i lim d, i d I -D, th dlta uctio ca still b witt as a ital, ad has th sampli popty lim i lim d, 0 0d Th covolutio o two -D uctios is lim * d Th covolutio thom i -D bcoms * Piodic uctios A Foui sis is a th pstatio o a piodic uctio by a iiit sum o hamoics:

3 Istumtatio o Chaactizatio o Naomatials (v) i Th piodicity is uaatd: uvw uvw i uvw i i i uvw Wh discussi cystals, w ow w will b summi ov th RLs by pmuti th idics h amo all its, so w ca adopt a shothad otatio: i i 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 ix dx L 0, othwis Th -D vsio loos li x, lim i 0 d 0, othwis It is ot mo usul to di a omalizd, disct vsio lim 0 d 0, othwis i, This om avoids th puzzli iiitis. Poo o covolutio thom Do you d poo o th covolutio thom? H it is:

4 Istumtatio o Chaactizatio o Naomatials (v) 4 x xhx dx xhxx x x dx x x ix ix dx dx xhx x x x dx dx d d h x x i x i x d d h dx dx x x i x i xx ix d d h d h h Foui compots Say w di a omalizd Foui tasom usi th mthod o dii : lim i d Assum is piodic. Th i Evaluati th ivs: lim d i i lim i 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: i W ca ma som alizatios about th. Lt s loo at th complx cojuat o :

5 Istumtatio o Chaactizatio o Naomatials (v) * * i 5 It is usually a ood stati poit to assum * i * i * i i So i th cystal pottial is al, w ca always say: * is al. So Now lt s loo at th what happs i w ivt th cystal about th oii i I th oii is a ct o ivsio symmty, th i i i i So, i th cystal pottial is ctosymmtic, w ca say: I th cystal pottial is both al ad ctosymmtic, th all o its Foui coicits a al: * al Evaluati th cystal pottial by covolutio Th tools itoducd h so a a itdd to ma li asi. Fo xampl cosid a piodic aay o dlta uctios, locatd at th lattic poits o a cystal: X - Now ta th lctostatic pottial pottial: * X o just o uit cll. Th covolutio o ths ivs th cystal I dict spac, w t bac th xpctd sum o pottials o all uit clls - Th advata coms i cipocal spac, wh w ca us th covolutio thom X

6 Istumtatio o Chaactizatio o Naomatials (v) 6 Cystal uctio (lattic sum) Ca w say aythi i paticula about lim N X N - Its Foui compots a: X? Fo a iiit cystal X N N lim lim i lim lim i 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 N -i, a RL N v Nv 0, othwis So th Foui sis pstatio o th cystal uctio is vy simpl: i X X v i Uit-cll pottials Th total cystal pottial is a sum ov uit-cll pottials: lim N 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 -d m atoms m W aud that th idividual pottials o isolatd atoms had sphical symmty. So m m si 4 d 0

7 Istumtatio o Chaactizatio o Naomatials (v) 7 Thus, th Foui tasom o th uit-cll pottial is: lim d -d m atoms m atoms m m m i m id Th si o th xpot o th phas acto is somtims opposit by covtio. Plas ba with m. Evaluati th Foui compots o th cystal pottial W saw that is th covolutio * X so is th poduct: X