Summary Chapter Van der Waals-London Interaction: [1]

Transcription

1 Smmar hapr 3 In hapr 3 sdid h inracion of aoms in a solid. W larnd ha h inracion consiss of a rplling and an aracing rm. W can ndrsand h cohsi nrg, h mling poin, blk modls from h inracion nrg. W discssd h: 1. Van dr Waals-London Inracion: [1] This prssion is ofn rfrrd o as h Lnnard-Jons ponial. is h disanc bn o nighboring aoms. Th firs rm is h rplsi rm and originas from h Pali clsion principl. To lcrons ha occp h sam spac canno ha h sam afncion, or ha h sam qanm nmbrs. So if h lcron disribion of aoms ih closd shlls orlap, on of h o lcrons nd o b parial promod o noccpid high nrg sas, so lcron orlap incrass h oal nrg of h ssm and gis hs a rplsi conribion o h inracion nrg. On somims modls h rplsi nrg b an ponnial form i.. / [] Th nd rm of h lnnard-jons ponial is h araci rm. This rm originas from h lcrosaic inracion bn h aoms. To nighboring aoms can b considrd o b o lcrosaicall copld harmonic oscillaors. This combind ssm has a lor grond sa nrg han h oal nrg of o non-copld oscillaors. Th diffrnc is largr h closr h aoms ar oghr, so his rsls in an araci rm. Van dr Waals inracions plas a crcial rol in h formaion of solids of nobl gasss.. Ionic inracion: / q [3] Th firs rm is h rplsi rm originaing from h Pali clsion principl and h nd rm is h araci rm originaing from h lcrosaic inracion. Th nd rm is ofn rfrrd o a h Madlng nrg. No ha h lcrosaic inracion in an ionic crsal is mch mor long rang han h Van dr Waals inracion mniond ndr (1). This p of inracion is imporan o dscrib h forcs ha kp h aoms of sals oghr. Abo gin qaions can b sd o calcla h oal cohsi nrg of a marial. To do so ha o kno h crsal srcr as on nds o sm p h () conribions of all aom/ion combinaions. So for a Xnon solid h oal cohsion nrg is gin b h folloing prssion:

2 o N 4 [13] pi pi Whr p i is h disanc bn aom I and. o is an sima of h oal cohsi nrg as i mliplis h nrg of h ih aom ih.5n, i.. h oal nmbr of aoms. Th facor.5 maks sr ha do no con h inracion nrg dobl. As boh h rplsi and araci rms in h Lnnard-Jons ponial ar r local on normall onl sms or n nighbors. For an fcc srcr h smmaion or h n nighbors ar gin b: i For h hcp srcr h rms ar: i 1-6 p p [14] i 1-6 p p [15] And for h bcc srcr h rms ar: i i 1-6 p p [16] i No ha sricl spaking qaion [13] gis h dpndnc of h cohsi nrg on h laic disanc. Th primnal cohsi nrg of h ssm is h minimm of qaion [13]. So on firs drmins h minimm of [13] o find h qilibrim consan and hn plgs o ino qaion [13] o drmin h cohsi nrg of h crsal. So for an fcc laic find: N [17] o o o 4 For an ionic crsal h cohsi nrg prssion is don or mor han s h nars nighbors. No ha h Madlng inracion rm gos 1/r and is hs mch lss local han h araci rm of h Lnnard-Jons ponial: ' / 1 q [18] p i Whr h firs smmaion is onl or h nars nighbors and h nd smmaion is or all aom pairs in h crsal. W can simplif his prssion b inrodcing for h nmbr of nars nighbors and inrodcing h Madlng consan for h smmaion or p i, i.. [19] p i

3 sling in: o 1 N / q [] Noic ha h Madlng consan onl dpnds on h gomr of h laic and is indpndn of h ionic radis or ionic charg. Th Madlng consans for diffrn crsal srcrs ar gin in h on pag 65: Nal = , sl = , ZnS = (inc blnd), ZnS (Wri, 4 of h 8 rahdral oids occpid b S)=1.6413, af (flori, all rahdral oids occpid b F), [1] On can find h cohsi nrg similarl as in h cas of h Van dr Waals solid, i.. s h driai of qaion [] qal o ro o find h laic paramr and hn rplac h rplsi rm in qaion [1] ih q / o. This rsls in a cohsi nrg qal o: o Nq o 1 o [] W also discssd ohr bonds, i.. coaln, mal and hdrogn bonds, qaliail. In h nd par of h chapr discssd lasic srains, basd on Hook s las b no in 3D inclding sharing srains and srsss bond h connional normal srains and srsss. Alhogh h planaion h inrodcion o h srain and srss lingo in Kil is r Enginring lik, I hink i is imporan all ha a good ndrsanding of rminolog. Disorion of h laic is dfind in rms of h disorion of h original orhogonal coordina ssm, ˆ, ˆ, ˆ in a n coordina ssm ', ', ' : ' 1 ˆ ˆ ˆ ' ˆ 1 ˆ ˆ ' ˆ ˆ 1 ˆ Whr dfin h dformaion. No ha,, and ar h fracional changs of lngh of ˆ, ˆ, ˆ. So imagin ha o dra h ni cors of a righ handd coordina ssm on o ndformd obc, hn o ill dform h obc and drmin h ac coordinas of h ni cors afr dformaion. W larnd ha h ffc of h dformaion dscribd b [3] on h posiion of an aom originall a r ˆ ˆ ˆ is gin b h displacmn cor: r' r ' ˆ ' ˆ ˆ Or in rms of h dformaion paramrs: [3] [4]

4 r ˆ ˆ r r ˆ r ˆ r ˆ ˆ [5] So h rlaion bn displacmn,, and dformaion paramr is: [6] W can no s h dfiniion of normal srain and shar srain o find rlaions bn h srain and h displacmn fncions,,, and. Normal srain in,, and, i..,, and, ar dfind as h chang in lngh in,, and dircion. Look o h figr blo and noic ha afr dformaion h -coordina of B changs ih / and s h rlaions of qaions [6]: [7] Sharing srains is dfind as h chang in angl, i.. +for h figr blo. From h figr blo s ha: an an [8] 1 1 Similarl for h ohr dircions find: [9] So s concld from his ha for h shar srains: [3]

5 W dfind h dilaion as h fracional incras of olm and fond for i: So smmariing hr ar onl 6 indpndn srain componns: i..,,,,, and. W inrodcd h srss componns, X, X, X, Y, Y, Y, Z, Z, Z hr h capial indicas h dircion of h forc and h sbscrip indicas h normal o h plan o hich h forc is applid. So X, Y, and Z ar normal srsss, and h ohrs ar shar srsss. W sa ha bcas of qilibrim h sm of h orqs shold b ro hich ld o h folloing qaliis: Y =Z Z =X X =Y So hr is onl 6 indpndn srss componns in a marial. Th rlaion bn h 6 srain componns and h 6 srss componns can b dscribd b a 66 mari. S qaions (37) and (38)

6 in h book. Th lmns of h S-mari ar rfrrd o as lasic complianc consans, and h lmns of h -mari ar rfrrd o as h lasic siffnss consans or modli of lasici. No ha h -mari is h 3D qialn of h spring consan. Alhogh h -mari has 36 componns, is off-diagonal componns ar smmric and =. W ralid his from h fac ha h lasic ponial nrg of h oal crsal consiss of all possibl rms of.5 rms. So h ngai driai of h ponial nrg ill ha similar rms in and. So hr ar onl (36-6)/+6=1 indpndn lasic siffnss consans for an arbirar marial. W pc ha for marials ha ha smmr bcas of h crsal srcr, h oal nmbr of indpndn lasic siffnss consans is considrabl lor. Kil dris h cas for cbic crsals. H shos on pag 78 ha bcas h for hr-fold roaion ais in h [1] dircion h nmbr of indpndn lasic siffnss consans is lss s hr, i.., 1, and. Th oal lasic siffnss mari for a cbic crsal is gin b: X Y Z Y Z X `1 [31] No ha Kil dos no prsn h siaions for h isoropic cas of for marials ha ha a crsal srcr ih a lor smmr. A smmar for h diffrn crsal srcrs discssd in chapr 1 is displad on h n pag. In gnral onl h riclinic crsal srcr has 1 indpndn siffnss consans. An isoropic sampl has onl o indpndn lasic siffnss consans. Th nrg dnsi of a cbic crsal ha is comprssd qall in all hr dimnsions i.. ===.33 hr d is h dialaion is gin b: B [3] 6 Whr B is ofn rfrrd o as h blk modls.

7 KEY TO NOTATION TILINI (1) MONOLINI (13) OTHOHOMBI (9) BI (3) (7) TETAGONAL (6) (7) TIGONAL (6) HEXAGONAL (5) ISOTOPI ()

8 In h hird par of chapr 3 discssd h lasic a propagaion in cbic crsals. W sd a cbic crsal ssm as ha simplifid h a-qaions. W sard off b assming a crain polariaion and hn appling Non s nd la o a cbic sgmn of h marial. For a polariaion in h -dircion ha o considr all srsss in h -dircion, i.. X, X, and X : X X X Y [33] sing h siffnss mari, i.. qaion [31], gis s h folloing a-qaion: 1 [34] No sing h srain displacmn rlaion of qaions 7 and 9 gis: 1 [35] W drid similar qaions for as polarid in h and dircions, i.. 1 [36] And 1 [37] W sd hos a-qaions and a plan a rial solions on pag 8 and 83 o dri h disprsion rlaion of as ih diffrn polariaions raling in diffrn dircions in h crsal. W fond ha for longidinal as propagaing in h [1] dircion h a loci is gin b: s [38] For h ransrs a h loci is gin b: s [39]

9 In boh cass h spd is proporional o h sqar roo of a componn of h lasic siffnss mari and h dnsi of h marial. W concldd in class ha i is no possibl o drmin h compl lasic siffnss mari from sond a masrmns in h [1] dircion of a cbic crsal. W also calclad h disprsion for as propagaing in h [] dircion of h crsal and larnd ha from spd masrmns of h longidinal a, h ransrs a polarid prpndiclar o h (1) plan, and h ransrs a polarid paralll o h (1) plan, on cold drmin all hr lasic siffnss consans proidd h dnsi of h marial is knon.