Lecture: Experimental Solid State Physics Today s Outline

Lctur: Exprimntl Solid Stt Physics Tody s Outlin Structur of Singl Crystls : Crystl systms nd Brvis lttics Th primitiv unit cll Crystl structurs with multi-tomic bsis Ttrhdrl nd octhdrl voids in lttics Th clos pcking of sphrs Lttic plins Th Rciprocl Lttic Exprimntl Scttring Mthods

Crystl systms & Brvis Lttics

Crystl systms & Brvis lttics Dscription of crystls by bsis vctors: Th vctors, b, nd c r th Bsis Vctors of th crystl, if vry tom cn b rchd by linr combintion (Trnsltion Vctor) : T m1 + mb + m3c Th lttic is clld Trnsltion Lttic. Th vctors, b, nd c r forming th Unit Cll. Th volum of th unit cll is V E ( b) c {prlll-pipdl product or Grmn: Spt-Produkt } A lttic sit cn b occupid by just on tom (th xmpl bov), but cn lso b occupid by group of toms. This is th cs in.g. Clcit (CCO 3, Grmn: Klk-Spt ) or Dimond...

Crystl systms & Brvis lttics Dscription of crystls by bsis vctors: Atomic Bsis In Clcit (CCO 3, Grmn: Klk-Spt ), th C + ions form fc-cntrd trigonl lttic... Ds mrkirt dn höchstn Punkt! C + Ion (fc-cntrd trigonl lttic)

Crystl systms & Brvis lttics Dscription of crystls by bsis vctors: Atomic Bsis In Clcit (CCO 3, Grmn: Klk-Spt ), th C + ions form fc-cntrd trigonl lttic...... with CO 3 groups in btwn. Th Bsis Vctors r pointing only onto th C + ions... Ds mrkirt dn höchstn Punkt! C + Ion (fc-cntrd trigonl lttic) CO 3 - Ions Bsis vctor: trnsltion lttic

Crystl systms & Brvis lttics Dscription of crystls by bsis vctors: Atomic Bsis In Clcit (CCO 3, Grmn: Klk-Spt ), th C + ions form fc-cntrd trigonl lttic...... with CO 3 groups in btwn. Th Bsis Vctors r pointing only onto th C + ions...... nd dditionlly, n Atom Bsis dfins th positions of th othr toms, blonging to on lttic sit! Ds mrkirt dn höchstn Punkt! C + Ion (fc-cntrd trigonl lttic) CO 3 - Ions Bsis vctor: trnsltion lttic Atomic Bsis

Crystl systms & Brvis lttics Dscription of crystls by bsis vctors: Atomic Bsis In Clcit (CCO 3, Grmn: Klk-Spt ), th C + ions form fc-cntrd trigonl lttic...... with CO 3 groups in btwn. Th Bsis Vctors r pointing only onto th C + ions...... nd dditionlly, n Atom Bsis dfins th positions of th othr toms, blonging to on lttic sit! In Dimond, th Atom Bsis consist of Crbon toms, ch forming fcc lttic.

Crystl systms & Brvis lttics Symmtry Symmtry is th min ftur of crystls. Thr r diffrnt typs of symmtry : Trnsltion symmtry (discussd bfor) Invrsion symmtry Mirror Symmtry Rottion Symmtry In trnsltion lttics it is only possibl to hv rottion xis with -, 3-, 4-, nd 6- fold symmtry! This is bcus closd plin cn only b filld by rctngls, tringls, squrs, nd hxgons! C C3 C4 C6 This symmtry rducs ll possibl crystl lttics to 7 Crystl systms nd 14 Brvis Lttics...

Th 7 Crystl systms nd 14 Brvis lttics triclinic b c α β γ monoclinic b c α γ 90 β orthorhombic b c α β γ 90 hxgonl b c α β 90 γ 10 trigonl b c α β γ 90 ttrgonl b c α β γ 90 cubic b c α β γ 90 bsis cntrd bsis cntrd fc cntrd body cntrd body cntrd Primitiv lttics: 1 tom pr unit cll fc cntrd fc cntrd

Th primitiv unit cll

Crystl systms & Brvis lttics Th primitiv unit cll Th primitiv unit cll lwys contins on tom! Obviously, th non-primitiv Brvis lttics lik hxgonl, body-cntrd cubic (bcc), nd fc-cntrd cubic (fcc) contin mor thn on tom. But it is possibl to construct primitiv unit cll (with rducd symmtry) for bcc,... bcc: tom pr unit cll Construction of th primitiv unit cll Bsis vctors of th primitiv unit cll ' b ' c' ( + + ) ( + ) x ( + ) x x y y y z z z

Crystl systms & Brvis lttics Th primitiv unit cll Th primitiv unit cll lwys contins on tom! Obviously, th non-primitiv Brvis lttics lik hxgonl, body-cntrd cubic (bcc), nd fc-cntrd cubic (fcc) contin mor thn on tom. But it is possibl to construct primitiv unit cll (with rducd symmtry) for bcc, fcc,... fcc: 4 tom pr unit cll Construction of th primitiv unit cll Bsis vctors of th primitiv unit cll ' b ' c' ( + ) y ( + ) x ( + ) x z z y

Crystl systms & Brvis lttics Th primitiv unit cll Th primitiv unit cll lwys contins on tom! Obviously, th non-primitiv Brvis lttics lik hxgonl, body-cntrd cubic (bcc), nd fc-cntrd cubic (fcc) contin mor thn on tom. But it is possibl to construct primitiv unit cll (with rducd symmtry) for bcc, fcc, nd hxgonl. hxgonl: 3 tom pr unit cll Construction of th primitiv unit cll Bsis vctors of th primitiv unit cll ' ' c' b' b ' b c' c

Crystl systms & Brvis lttics Th primitiv unit cll: Construction of th Wignr-Sitz-Cll Thr r mny possibilitis to construct primitiv cll. Th simplst on is th on with stright bordr lins. This primitiv unit cll is clld Wignr-Sitz-Cll. For construction, th connction lins to th nxt nighbours will b cut t th hlf of it s lngth by prpndiculr lin. Th closd connctions of ths prpndiculr lins will giv th Wignr-Sitz-Cll. Construction of th Wignr- Sitz-Cll for d lttic Wignr-Sitz-Cll for bcc lttic Wignr-Sitz-Cll for fcc lttic

Crystl structurs with multi-tomic bsis

Crystl structurs with multi-tomic bsis Dimond: Dimond consists of fcc lttic with bsis of C-toms. Hnc, dimond consists of fcc lttics, shiftd by T ( 1 / 4, 1 / 4, 1 / 4 ). Evry toms is in ttrhdrl nvironmnt of 4 othr toms sp3 hybridistion! Bttr to s in this pictur (ctully ZnS) on fcc lttic (.g. Zn) is shiftd into th othr fcc lttic (.g. S) Img: Univrsity of Florid, http://itl.chm.ufl.du/041_f97/lc_indx.html

Crystl structurs with multi-tomic bsis Sodium Chlorid (NCl) : NCl consists of fcc lttic with bsis of 1 N-toms nd 1 Cl-tom. Both, th N + ions nd th Cl - ions form fcc lttic. Both fcc lttics r shiftd by T ( 1 /, 1 /, 1 / ). If N nd Cl would b idnticlly, th lttic would b simpl-cubic with lttic constnt of /!

Crystl structurs with multi-tomic bsis Csium Chlorid (CsCl) : Lik in NCl, CsCl hs bsis of 1 Cs-toms nd 1 Cl-tom. But, th Cs + ion is lrgr thn th N + ion. Thrfor, CsCl prfrs to crystllis in simpl-cubic lttics. Tht mns both, th Cs + ions nd th Cl - ions from primitiv-cubic lttic (pcc). Both pcc lttics r shiftd by T ( 1 /, 1 /, 1 / ). If Cs nd Cl would b idnticlly, th lttic would b body-cntrd cubic with th sm lttic constnt of!

Ttrhdrl nd octhdrl voids in lttics

Ttrhdrl nd octhdrl voids in lttics Ttrhdrl nd octhdrl voids in fcc lttic: In fcc lttic, two typs of voids cn b filld with multi-tom bsis: Ttrhdrl voids nd octhdrl voids. Img: Univrsity of Florid, http://itl.chm.ufl.du/041_f97/lc_indx.html

Ttrhdrl nd octhdrl voids in lttics Ttrhdrl nd octhdrl voids in fcc lttic: In fcc lttic, two typs of voids cn b filld with multi-tom bsis: Ttrhdrl voids nd octhdrl voids. Filling ll of th octhdrl voids lds to th NCl structur with lmnt rtio of 1 : 1 N : Cl Img: Univrsity of Florid, http://itl.chm.ufl.du/041_f97/lc_indx.html

Ttrhdrl nd octhdrl voids in lttics Ttrhdrl nd octhdrl voids in fcc lttic: In fcc lttic, two typs of voids cn b filld with multi-tom bsis: Ttrhdrl voids nd octhdrl voids. Filling hlf of th ttrhdrl voids lds to th ZnS Zink-Blnd structur with lmnt rtio of 1 : 1 N : Cl Img: Univrsity of Florid, http://itl.chm.ufl.du/041_f97/lc_indx.html

Ttrhdrl nd octhdrl voids in lttics Ttrhdrl nd octhdrl voids in fcc lttic: In fcc lttic, two typs of voids cn b filld with multi-tom bsis: Ttrhdrl voids nd octhdrl voids. Filling ll of th ttrhdrl voids lds to th CF structur with lmnt rtio of 1 : C : F Img: Univrsity of Florid, http://itl.chm.ufl.du/041_f97/lc_indx.html

Th clos pcking of sphrs

Th clos pcking of sphrs How to chiv th closst pcking of sphrs : Th scond tom lyr B ( ) will b fittd into th voids of th first tom lyr. Th third tom lyr C ( ) will hv two possibilitis : (1) Using th sm positions lik th first lyr A A hxgonl hcp structur will b build, with lyr squnc of AB-AB-... () Using th 3 rd possil position A cubic fcc structur will b build, with lyr squnc of ABC-ABC-... hcp fcc

Th clos pcking of sphrs Finlly: Som xmpls of crystl structurs of lmntl crystls Th crystllistion structur of som lmnts of th PSE

Lttic plins

Lttic Plins Th Millr Indics Millr Indics (hkl) r idntifying lttic plins in crystl. Thy cn b clcultd from th 3 intrsctions of th plin with th coordint xis: S 1 : m 1 S : m b / S 3 : m 3 c. Thn, th rciprocl of m 1, m, nd m 3 hv to b multiplid with th smllst numbr p in mnnr, tht coprim * numbrs h, k, nd l will b gind: p p p h, k, l m m m 1 3 Millr Indics in squrd brckts [hkl] r idntifying th dirction in crystl, which is prpndiculr to th lttic plin (hkl). [100] [010] [111] [110] * coprim tilrfrmd

Lttic Plins Th Millr Indics Millr Indics (hkl) r idntifying lttic plins in crystl. Thy cn b clcultd from th 3 intrsctions of th plin with th coordint xis: S 1 : m 1 S : m b / S 3 : m 3 c. Thn, th rciprocl of m 1, m, nd m 3 hv to b multiplid with th smllst numbr p in mnnr, tht coprim * numbrs h, k, nd l will b gind: p p p h, k, l m m m 1 3 Incrsing th intrsction with th y-xis, (10) (310) (410) th Millr Indics of th othr xis (in (110) z this xmpl of th x-xis) will incrs! Th Millr Indx gts mor nd mor lik th (100) plin. y x (100) * coprim tilrfrmd

Lttic Plins Th Millr Indics: Invrtd dirctions Ngtiv Millr Indics rflct nd invrsion of th dirction, nd thy r writtn [111] instd of [-1-1-1]. This is of importnc, bcus in htrognous mtrils, on surfc is trmintd by on lmnt nd th rvrsd surfc is trmintd by th othr [111] lmnt. For xmpl th GAs[111] surfc is As-trmintd, whrs th GAs[111] surfc is G-trmintd! [111] As 3- G 3+ This cn lso b sn in ZnO for th Zn-trmintd [0001]-surfc nd th O-trmintd [0001]-surfc. Bcus ZnO is hxgonl crystl systm, thr r 4 Millr Indics usd...

Lttic Plins Th Millr Indics: Hxgonl Millr Indics In hxgonl crystl systms r 4 Millr indics usd to dscrib th lttic plins nd dirctions. This is bcus in hxgonl systm, th bsis lttic vctors in x- nd y-dirction r not orthogonl (s right). Thrfor, quivlnt lttic plins will hv nonquivlnt Millr Indics! But this is not vry convnint. Lt s show this for th sid-plins in hxgonl lttic : Th sid-plns would hv th Millr Indics [110], [100], y y c' ' b' [100] [110] x 1 x Pltzhltr!!!

Lttic Plins Th Millr Indics: Hxgonl Millr Indics In hxgonl crystl systms r 4 Millr indics usd to dscrib th lttic plins nd dirctions. This is bcus in hxgonl systm, th bsis lttic vctors in x- nd y-dirction r not orthogonl (s right). c' Thrfor, quivlnt lttic plins will hv nonquivlnt Millr Indics! But this is not vry convnint. Lt s show this for th sid-plins in hxgonl lttic : b' ' Th sid-plns would hv th Millr Indics [110], [100], [010], [010] [100] x [110] y 1 y x Pltzhltr!!!

Lttic Plins Th Millr Indics: Hxgonl Millr Indics In hxgonl crystl systms r 4 Millr indics usd to dscrib th lttic plins nd dirctions. This is bcus in hxgonl systm, th bsis lttic vctors in x- nd y-dirction r not orthogonl (s right). c' Thrfor, quivlnt lttic plins will hv nonquivlnt Millr Indics! But this is not vry convnint. Lt s show this for th sid-plins in hxgonl lttic : b' ' Th sid-plns would hv th Millr Indics [110], [100], [010], nd th rvrsd [110], [100], [010]! [010] [100] [110] But it is not obvious, tht.g. [110] nd [100] r quivlnt dirctions in th crystl! [110] y [100] x [010] Pltzhltr!!!

Th Rciprocl Lttic

Th Rciprocl Lttic Dfinition nd proprtis of th rciprocl lttic At first, th dfinition of th rciprocl bsis vctors, which dfin th rciprocl lttic : {V E is th volum of th unit cll} Th rciprocl bsis vctor * is orthogonl to th plin, dfind by b nd c : Th rciprocl lttic of primitiv-cubic lttic is gin primitiv-cubic lttic! For bsis vctors in bcc nd fcc lttics th following pplis : Th rciprocl lttic of bcc lttic is fcc lttic, nd vic-vrs! ( ) ( ) ( ) ( ) b V c c V b c b V c b c b E E E * * * π π π π ij i j πδ * ( ) ( ) ( ) y x z x z y c b ' ' ' + + + fcc primitiv ( ) ( ) ( ) z y x z y x z y x c b * * * + + + + π π π fcc rciprocl ( ) ( ) ( ) z y x z y x z y x c b ' ' ' + + + + bcc primitiv bcc rciprocl ( ) ( ) ( ) y x z x z y c b ' ' ' + + + π π π

Th Rciprocl Lttic Th dvntgs of th rciprocl lttic Th following proprtis of th rciprocl lttic r of lrg bnfit for scttring in lttics : ❶ Th following rciprocl lttic vctor G is orthogonl to th lttic plin (hkl) : G h + * * * + k b l c ❷ Th bsolut vlu G is invrsly proportionl to th lttic plin distnc d hkl : π G d hkl This mks it possibl to clcult vry sily th impuls diffrnc k btwn th incoming (mttr)wv k 0 nd scttrd (mttr)wv k s : 0 k k k s G For scttring only th lmntry cll in rciprocl spc is importnt. This cll is clld Brillouin Zon nd it cn b constructd lik th Wignr-Sitz-Cll...

Th Rciprocl Lttic Construction of th first Brillouin zon lik Wignr-Sitz-Cll Construction in rciprocl spc Middl-Vrticl lins btwn th nxt nighbouring rciprocl lttic points Shown r th first Brillouin zons of Fc-Cntrd-Cubic (fcc), Body-Cntrd-Cubic (bcc), nd hxgonl lttic. fcc Points with high symmtry r mrkd : - Γ cntr of th Brillouin zon - X crossing of singl-indxd dirction (.g. [100], [010], [001], [100],...) with th Brillouin zon bordr. - K crossing of doubl-indxd dirction (.g. [110], [101], [110],...) with th Brillouin zon bordr. - L crossing of room digonls (.g. [111], [111], [111],...) with th Brillouin zon bordr. bcc hxgonl

Th Rciprocl Lttic Enrgy nd Frmi surfcs in th rciprocl lttic A vctor k in th rciprocl lttic rprsnts th k-vctor of (lctron) wv. Th impuls of th wv rprsntd by th k-vctor is Enrgy nd impuls r lso connctd,.g. fr lctrons follows : Tht mns: Th lngth k of th k-vctor rprsnts n nrgy, nd sphr round th origin in th rciprocl spc dfins sphr of constnt nrgy! Th most importnt nrgy is th Frmi nrgy, up to wr stts r filld with lctrons. Tht lds to th so-clld Frmi Surfcs wht r simpl sphrs for fr lctrons. All stts (rciprocl lttic point) insid th Frmi Surfc r occupid! E p m ikr ψ p h k h k m

Exprimntl Scttring Mthods

Exprimntl Scttring Mthods Gnrl rmrks to scttring mthods Scttring is possibl with photons, lctrons, nutrons, nd vn toms s prob. Bcus of diffrnt proprtis, th diffrnt probs r suitd for spcil purpos : Photons Elctrons Nutron H-Atoms 1.4 kv Enrgy for (d Brogli-) wv lngth of 1Å 150V 8 mv 1 mv bulk snsitiv surfc snsitiv bulk snsitiv surfc snsitiv λ d Brogli h m E kin

Exprimntl Scttring Mthods Rmmbr: Brgg Rflction Brgg rflction is dfind t crtin lttic plins nd long crtin dirctions! Msurmnt: Th rflctnc for prticls (hr: nutrons) with fixd wv lngth (hr λ dbrogli 1.16Å) prpndiculr to crtin crystl dirction (hr: long [110] dirction of CF crystl) will show pks whn th Brgg-condition is fulfilld. d sinϑ m λ C F

Exprimntl Scttring Mthods Gnrlisd Brgg diffrction: Lu diffrction Lu diffrction is th gnrlisd form of th Brgg diffrction Lu diffrction is not rstrictd to lttic plns, but it considrs th scttring on singl toms in lttic. Th pth diffrnc btwn th scttrd wvs is d cosϑ + d cosϑ' d ( n n') For constructiv intrfrnc, this pth diffrnc must b m λ (with m intgr) : By rplcing th distnc of th scttring toms d by th Brvis vctor R, th intrfrnc condition will b xpndd to th whol crystl : R ( k k ') π This now mns, tht th diffrnc btwn th two wv vctor du to scttring (k - k ) multiplid by Brvis lttic vctor must b n intgr of π! This is idnticl to th dmnd, tht (k - k ) must b vctor G of th rciprocl lttic, wht lds to th Lu condition : d ( n n') m λ d ( k k ') πm k π λ k ' k G m

Exprimntl Scttring Mthods Lu Condition: Wht dos it mns? Agin th Lu condition : k ' k It cn b sn from th Ewld Construction (Ewld Sphr), whthr th rsulting vctor k k is vctor of th rciprocl lttic : k In th rciprocl spc, circl round th k wv vctor k is drwn. Th rdius is givn by th wv lngth r π/λ. If th circl primtr xctly crosss rciprocl lttic point, th Lu condition is fulfilld, nd rflx in dirction of k is obsrvd. This is normlly not th cs, for fixd k (mns: fixd dirction nd fixd wv lngth of th incidnt wv). Thrfor, on hs to continuously chng ithr th incidnc ngl (dirction of k) or th incidnc wv lngth (lngth of k)! k Vry complx structurs lik protins cn b invstigtd (s xmpl on top). G

Exprimntl Scttring Mthods Dby-Schrrr Mthod Vrition of th incidnc ngl cn vry sily b rlisd by using smpl of crystllin powdr. Th powdr grins of µm siz r sufficint to produc scttring rflxs. Th rndom orinttion of th smll crystls simults th incidnc ngl vrition. Bcus th th rndom distribution of th micro-crystls, th smpl is lso symmtric vrsus rottion round th incidnt bm. Thrfor, th Dby-Schrrr Mthod producs cons round th incidnt bm.

Exprimntl Scttring Mthods Dby-Schrrr Mthod: Clcultion of th lttic constnt From th gomtry of th pprtus, th diffrction ngl (ttntion: th clcultd dflction ngl is ϑ!) cn b clcultd. From th first-ordr Brgg qution th lttic plin distnc d hkl cn b clcultd : d hkl sinϑ λ In orthogonl lttics, from d hkl th lttic constnts cn b obtind vi : 1 d hkl h k + + b Cn b drivd from formuls givn on slid Th dvntg of th rciprocl lttic! l c Hint: Th following rflxs r occurring t th cubic lttics: primitiv: 100, 110, 111, 00, 10, 11, 0, 300, 1, 310, 311,, 30,... bcc: 110, 00, 11, 0, 310,, 31, 400, 411, 330, 40,... fcc: 111, 00, 0, 311,, 400, 331, 40, 4, 333...