SOLID STATE PHYSICS. Crystal structure. (d) (e) (f)

Transcription

1 SOLID STAT PHYSICS y defto, sold state s that partcular aggregato form of matter characterzed by strog teracto forces betwee costtuet partcles (atoms, os, or molecules. As a result, a sold state materal has a depedet geometrc form ( cotrast to lquds, whch tae the form of the cotaer ad a varat volume ( cotrast to gases/vapors gve temperature ad pressure codtos. As temperature creases, a sold state materal ca evolve to aother aggregato form (lqud or gas. Sold state physcs studes the structural, mechacal, thermodyamc, electrcal, magetc, ad optcal propertes of (poly-crystalle ad o-crystalle solds (for example, amorphous materals, such as glass. Crystal structure The propertes of crystalle solds are determed by the symmetry of the crystalle lattce, because both electroc ad phooc systems, whch determe, respectvely, the electrc/ magetc ad thermal respose of solds, are very sestve to the regular atomc order of materals ad to ay (local or o-local perturbato of t. The crystalle structure ca be revealed by the macroscopc form of atural or artfcally-grow crystals (see the pctures below, or ca be ferred from the resultg debrs after cleavg a crystalle materal. (a (b (c (d (e (f Crystals of (a baryt, (b salt, (c hexagoal beryl, (d trgoal quartz, (e mooclc gypsum, ad apatte (f

2 Crystal Structure No-crystalle materals have o log-rage order, but at least ther optcal propertes are smlar to that of crystalle materals because the wavelegth of the cdet photos (of the order of μm s much larger tha the lattce costat of crystals ad so, photos see a effectve homogeeous medum. Other propertes of o-crystalle materals are derved based o cocepts proper to crystalle solds ad, therefore, the crystal structure s extremely mportat uderstadg the propertes of sold state materals. The macroscopc, perfect crystal s formed by addg detcal buldg blocs (ut cells cosstg of atoms or groups of atoms. A ut cell s the smallest compoet of the crystal that, whe staced together wth pure traslatoal repetto, reproduces the whole crystal. The perodcty of the crystalle structure that results ths way s cofrmed by X- ray dffracto expermets. The fgures below llustrate crystals whch the bass cossts of (a oe atom ad (b two atoms. (a (b The group of atoms or molecules that forms, by fte repetto, the macroscopc crystal s called bass. The bass s postoed a set of mathematcal/abstract pots that form the lattce (also called ravas lattce. So, a crystal s a combato of a bass ad a lattce. Although usually the bass cossts of oly few atoms, t ca also cota complex orgac or orgac molecules (for example, protes of hudreds ad eve thousads of atoms. I two dmesos, all ravas lattce pots R ma a ( m + ca be obtaed as superpostos of tegral multples of two o-collear vectors ad a a (m ad are arbtrary tegers. A bass cosstg of s atoms s the defed by the set of

3 Crystal Structure vectors r j m ja + ja, j,,,s, that descrbe the posto of the ceters of the bass atoms wth respect to oe pot of the ravas lattce. I geeral, m,. j j very pot of a ravas lattce s equvalet to every other pot,.e. the arragemet of atoms the crystal s the same whe vewed from dfferet lattce pots. The ravas lattce defed by ( s varat uder the operato of dscrete traslato Tpq pa + qa alog teger multples p ad q of vectors ad a, respectvely, because a T ( pq ( Rm Tpq + Rm R p+ m, q+ s aga a ravas lattce pot. I fact, sce the traslato operato s addtve,.e. T pq Tuv Tp+ u, q+ v TuvT pq, assocatve,.e., ad has a verse, such that T T ad T ( T T ( T T T, commutatve,.e. T pq uv m pq p, q pq uv m pqt uv T pq T p, q I wth I the detty trasformato, t follows that the traslatos form a abela (commutatve group. ecause codto ( s satsfed for all ravas lattce pots, ad a are called prmtve traslato vectors, ad the ut cell determed by them s called prmtve ut cell. The modulus of these vectors, a a ad a, are the lattce costats alog the a a respectve axes, ad the area of the ut cell two dmesos s S a. It s mportat a to otce that the set of vectors ad a s ot uque (see the fgures below, but all prmtve ut cells have the same area. a The prmtve ut cell covers the whole lattce oce, wthout overlap ad wthout leavg vods, f traslated by all lattce vectors. A equvalet defto of the prmtve ut cell s a cell wth oe lattce pot per cell (each lattce pot the fgures above belog to

4 Crystal Structure 4 four adjacet prmtve ut cells, so that each prmtve ut cell cotas 4 (/4 lattce pot. No-prmtve (or covetoal ut cells are larger tha the prmtve ut cells, but are sometmes useful sce they ca exhbt more clearly the symmetry of the ravas lattce. esdes dscrete traslatos, the ravas lattce s varat also to the pot group operatos, whch are appled aroud a pot of the lattce that remas uchaged. These operatos are: Rotatos by a agle π / about a specfc axs, deoted by C, ad ts multples, j C ( C j. Geometrc cosderatos mpose that,,, 4 ad 6, ad that repeatg the rotato tmes oe obtas C, where s the detty operato, whch acts as r r. Moreover, C π does ot represet a symmetry elemet. C D θ θ A The allowed values of ca be determed assumg that we apply a rotato wth a agle θ aroud a axs that passes frst through a pot A ad the through a adjacet lattce pot. The pots A ad are separated by the lattce costat a. If C ad D are the resultg pots, they should also be separated by a teger multple of a. rom the requremet that CD a + as( θ π / a a cosθ ma, or cosθ ( m /, wth m teger, t follows that m ca oly tae the values,,,, ad, the correspodg π /θ tag the values specfed above. As for traslatos, the rotatos also form a abela group. xamples of two-dmesoal fgures wth dfferet rotato symmetres: C C C 4 C 6

5 Crystal Structure 5 Iverso I, whch s defed by the operato r r f appled aroud the org. Reflecto σ j, whch ca be appled aroud the horzotal plae (j h, the vertcal plae (j v, or the dagoal plae (j d. Improper rotato S, whch cossts of the rotato the plae ormal to the rotato axs. Note that S I. C followed by reflecto Whe we combe the pot group symmetry wth the traslatoal symmetry, we obta the space-group symmetry. It s mportat to otce that the bass ca troduce addtoal symmetry elemets, such as helcodal symmetry axes ad gldg reflecto plaes. The fgure bellow represets several symmetry operatos: (a traslatos, (b rotato, (c verso, ad reflecto wth respect to a (d vertcal, ad (e horzotal plae. (a (b (c (d (e Crystal lattces are classfed accordg to ther symmetry propertes at pot group operatos. The fve ravas lattce types two dmesos are show the fgure below. These are: square lattce, for whch a a, ad γ 9, where γ s the agle betwee a ad a, rectagular lattce, for whch a, ad γ 9, a cetered rectagular lattce, whch s a rectagular lattce wth a addtoal lattce pot the ceter of the rectagle, hexagoal lattce, for whch a, ad γ 6 (or for a dfferet choce of the org, a oblque rectagular lattce (called also oblque lattce, for whch a a, ad γ 9, 6 (or.

6 Crystal Structure 6 Wth the excepto of the cetered rectagular lattce, all ut cells the fgure above are prmtve ut cells. The prmtve cell for the cetered rectagular lattce s a rhombus (see fgure at rght ad therefore ths ravas lattce s also called rhombc lattce, case whch ts prmtve ut cell has a, ad γ 9, 6 (or. a ach lattce type has a dfferet set of symmetry operatos. or all ravas lattce types two dmesos, the rotato axes ad/or reflecto plaes occur at lattce pots. There are also other locatos the ut cell wth comparable or lower degrees of symmetry wth respect to rotato ad reflecto. These locatos are dcated the fgure below.

7 Crystal Structure 7 I order to corporate the formato about the pot group symmetry the prmtve cell, the Wger-Setz cell s usually employed. Ths partcular prmtve ut cell s costructed by frst drawg les to coect a gve lattce pot to all earby lattce pots, ad the drawg ew les (or plaes, three-dmesoal lattces at the md pot ad ormal to the frst les. The Wger-Setz cell s the smallest area (volume eclosed by the latter les (plaes. A example of the costructo of a Wger-Setz cell for a twodmesoal oblque lattce s llustrated the fgure below. or a two-dmesoal square lattce the Wger-Setz cell s also a square. The Wger-Setz cell s always cetered o a lattce pot ad corporates the volume of space whch s closest to that lattce pot rather tha to ay other pot. θ r The faces of the Wger-Setz cell satsfy the relato r cosθ R /, where R s the dstace to the earest eghbor ad θ s the agle betwee r ad R. Ths relato ca be rewrtte as ( r R R or, sce the equato s equvalet to the replacemet of R wth R, r R + R, ad fally, ( r + R r. I other words, the faces of the Wger- Setz cell are determed by the tersecto betwee equal-radus spheres cetered at the earest-eghbor pots of the ravas lattce. I a smlar maer, three dmesos, all ravas lattce pots R mp ma + a + pa ( ca be obtaed as superpostos of tegral multples of three o-coplaar prmtve traslato vectors, ad a (m,, ad p are arbtrary tegers, ad the pot group a a

8 Crystal Structure 8 operatos are defed detcally. The volume of the prmtve ut cell, whch ths case s a parallelepped, s Ω ( a a. a There are 4 three-dmesoal ravas lattces, whch belog to 7 crystal systems, as ca be see from the fgure below, where the prmtve traslato vectors are deoted by a, b, c (wth respectve legths a, b, ad c, ad α, β, γ are the agles betwee b ad c, c ad a, ad a ad b, respectvely. These crystal systems, whch are dfferet pot groups edowed wth a sphercal symmetrc bass, are: cubc, for whch a b c, α β γ 9. It cossts of three o-equvalet spacegroup lattces: smple cubc, body-cetered cubc, ad face-cetered cubc. Ths s the crystal system wth the hghest symmetry ad s characterzed by the presece of four C axes (the dagoals of the cube tetragoal, for whch a b c, α β γ 9. It ecompasses the smple ad bodycetered ravas lattces ad cotas oe C 4 symmetry axs. orthorhombc, for whch a b c, α β γ 9. It corporates the smple, bodycetered, face-cetered, ad sde-cetered lattces ad has more tha oe C symmetry axs or more tha oe reflecto plae (actually, three such axes/plaes, perpedcular to each other. hexagoal, for whch a b c, α β 9, γ. It s characterzed by the exstece of a sgle C6 symmetry axs. The covetoal hexagoal ut cell (see the fgure at rght s composed of three prmtve cells. trgoal, for whch a b c, α β γ 9. It cotas a sgle C axs. mooclc, for whch a b c, α γ 9 β. It cludes the smple ad sdecetered lattces, ad has oe C symmetry axs ad/or oe reflecto plae perpedcular to ths axs. trclc, for whch a b c, α β γ 9. Ths s the crystal system wth the lowest symmetry. It s ot symmetrc wth respect to ay rotato axs or reflecto plae.

9 Crystal Structure 9 The relatos betwee these lattces ca be summarzed the fgure at the rght. The dfferet crystal systems have dfferet umbers of ut cell types because other possble ut cell types caot represet ew ravas lattces. or example, both the body-cetered ad the face-cetered mooclc lattces ca be reduced to the sde-cetered lattce by approprately choosg the prmtve traslato vectors. xamples of two sets of prmtve traslato vectors for a body-cetered cubc (bcc lattce are represeted the fgure below at left ad ceter, whle the fgure at rght dsplays a set of prmtve traslato vectors for a face-cetered cubc (fcc lattce.

10 Crystal Structure The prmtve traslato vectors for the left fgure above ca be expressed as a ( a / ( x + y, a ( a / ( x + y +, a ( a / ( x y +, (4 z z z whle those for the rght fgure are a ( a / ( x +, a ( a / ( y +, a ( a / ( z + (5 y z x ad the agles betwee these vectors are 6. A smple lattce has lattce pots oly at the corers, a body-cetered lattce has oe addtoal pot at the ceter of the cell, a face-cetered lattce has sx addtoal pots, oe o each sde, ad a sde-cetered lattce has two addtoal pots, o two opposte sdes. The smple lattces are also prmtve lattces ad have oe lattce pot per cell, sce the eght stes at the corers are shared by eght adjacet ut cells, so that 8 (/8. The o-smple lattces are o-prmtve. The volume of the prmtve ut cell these lattces s obtaed by dvdg the volume of the covetoal ut cell by the umber of lattce pots. I partcular, the body-cetered lattces have two pots per ut cell: the eght at the corers whch cotrbute wth 8 (/8, ad the oe the ceter, whch belogs etrely to the ut cell. The face-cetered lattces have 4 lattce pots per cell: those the corers cotrbute wth 8 (/8, ad those o the faces cotrbute wth 6 (/, sce they are shared by two adjacet cells. ally, the sde-cetered lattces have two lattce pots per cell: the pots at the corer cotrbute wth 8 (/8, ad those o the faces wth (/. The characterstcs of the cubc lattces wth sde a are summarzed the table below. If each lattce pot s expaded to a sphere wth a radus equal to half of the dstace betwee earest eghbors, such that adjacet spheres touch each other, the a pacg fracto ca be defed as the fracto betwee the volume of the spheres cotaed the covetoal ut

11 Crystal Structure cell ad the volume of the ut cell. Note that the volume betwee the spheres oe ca always sert smaller spheres, whch ca stad for other atom types. Smple ody-cetered ace-cetered Volume of a a a covetoal cell Lattce pots per 4 cell Volume of prmtve a a / a /4 cell Number of earest 6 8 eghbors Nearest-eghbor a a/ a/ dstace Number of secod 6 6 eghbors Secod-eghbor a a a dstace Pacg fracto π/6.54 π/8.68 π/6.74 The 4 ravas lattces corporate all possble crystalle structures; they result by tag to cosderato the space-group symmetry,.e. the symmetry at traslatos ad the pot group symmetry of the lattce (the symmetry wth respect to rotato, reflexo or verso. Whe the bass cossts of oly oe atom, the ravas lattce s detcal wth the crystalle structure. ut whe the bass s complex ad cossts of several atoms, say s, the crystalle structure ca be see as formed by the terpeetrato of s ravas lattces. The ravas lattces have always a verso ceter oe of the lattce pots, whereas such a verso ceter ca lac crystals wth complex bases. y coutg the pot groups of the possble dfferet crystals (whch have bases wth dfferet symmetres, oe eds wth crystalle classes that ca be accommodated by the 7 crystal systems. Also, there are space groups that result from the combato of the crystalle structures wth the traslatoal symmetry.

12 Crystal Structure Idex system for lattce pots, drectos ad plaes Whe the org of the prmtve traslato vectors s a lattce pot, aother lattce pot wth a posto R mp ma + a + pa s smply specfed by the set of umbers [[m,,p]]. A egatve teger m, or p s deoted by a sg placed o top of t. or example, [[ m p ]] stays for the lattce pot specfed by the tegers m, ad p, wth m, ad p postve umbers. I partcular, for the three-dmesoal prmtve ravas lattces the coordates of the lattce pot at the org are [[,,]], the other lattce pots dfferg oly through dscrete traslatos alog the three coordate axs. The umber of o-equvalet lattce pots a ravas lattce s gve by the umber of lattce pots per ut cell. I partcular, for the body-cetered lattce, the posto of the lattce pot at the ceter of the cube s deoted by [[/,/,/]], the three addtoal lattce pots face-cetered lattces havg coordates [[,/,/]], [[/,,/]], [[/,/, ]]. I a smlar maer, depedg o the set of opposte stes they ca occupy, the addtoal ste a face-cetered lattce has the coordates [[,/,/]], [[/,,/]] or [[/,/,]]. A drecto, by defto, passes through two lattce pots. To specfy a drecto a crystalle lattce, oe uses the symbol [mp], where m, ad p are three tegers determed by the followg rule: sce oe ca specfy a drecto by the coordates [[, p ]] ad [[ m, p ]] of two pots through whch t passes, the dces m, ad p m,, are defed as the smallest teger umbers that satsfy the proportoalty relatos m m m, p p, p p m p m, (6 p m or m : p : p ( m m : ( : ( p. (7 If oe of the tegers s egatve, the sg s placed o top of the teger. or example, [ m p ] stays for the drecto specfed by the tegers m, ad p. If the drecto s ot cosdered as a oreted axs but as a smple le, the drecto specfed by the tegers m,, ad p s the same as that specfed by m,, ad p (otherwse, the chage of all sgs meas a chage of drecto of the same le. If there are several equvalet drectos (equvalet, from the pot of vew of crystal symmetry, they are deoted as mp. A

13 Crystal Structure partcular stuato s ecoutered the hexagoal lattce, whch lattce drectos are labeled by four umbers (ths stuato s ot further dscussed ths course. xamples: The axs s the [] drecto. The a a examples are llustrated the fgure below. axs s the [ ] drecto. Other a a [] [] a [] [] I three-dmesoal lattces, the oretato of a crystal plae s determed by three o-collear pots the plae. If each pot s stuated o a dfferet crystal axs, the plae s specfed by the coordates of the pots terms of the lattce costats,, ad a. a a Aother way to specfy the oretato of a plae, whch s more useful for structure aalyss, volves the determato of three dces, called Mller dces, accordg to the rule: d frst the tercepts of the plae o the axes terms of lattce costats a, a, ad a, rrespectve of the ature (prmtve or o-prmtve of the ut cell. Tae the recprocal of these umbers. If fractoal, reduce these umbers to the smallest three tegers, say m,, p, wth the same rato. The result, symbolzed by (mp (or ( m p f the secod dex, for example, s egatve, s the Mller dex system of the plae. It s obvous that the Mller dex for a tercept at fty s zero. The faces of a cubc crystal, for example, are deoted by (, (, (, (, (, ad (. Moreover, the plae ( s parallel to (, but cuts the a axs at a /. If, from the pot of vew of crystal symmetry, there s a set of oparallel equvalet plaes, they are symbolzed as {mp}. or example, the set of faces of a cubc crystal s {}. Aga, for the hexagoal lattce there are four Mller dces stead of three. xamples of Mller dces are gve the fgures below.

14 Crystal Structure 4 ( ( ( Note that the Mller dces determe ot oly oe plae but a famly of parallel plaes, sce there s a fte umber of plaes wth the same dces, all of whch cut the coordate axes at s / m, s /, ad s / p, wth s teger. The plae that cuts the axes at / m, /, ad / p s the closest to the org from the famly of parallel plaes. Note also that the plaes wth Mller dces (sm,s,sp are parallel wth the plae (mp, but the dstace betwee them s s tmes smaller. or example, the set of plaes ( s parallel to but twce as close as the ( set of plaes. I cubc crystals, the plae (mp s perpedcular to the drecto [mp] wth the same dces, but ths result caot be exteded to other crystal systems. A example s gve the fgure below.

15 Crystal Structure 5 Smple crystal structures Oe of the most smple crystal structures ad, at the same tme, of geeral terest, s that of NaCl (sodum chlorde. It s llustrated below. The lattce s face-cetered cubc, wth a bass cosstg of oe Cl o (blue at [[]] ad a Na + o (gree at [[/,/,/]]. As ca be see from the fgure below, a ut cube cossts of four NaCl uts, wth Na + os at postos [[/,/,/]], [[,,/]], [[,/,]], ad [[/,,]] ad Cl os at [[]], [[/,/,]], [[/,,/]], ad [[,/,/]]. ach atom has as earest eghbors sx atoms of opposte d. xample of crystals wth ths structure ad ther lattce costats are gve below. Crystal a(å Crystal a (Å Crystal a (Å L 4. Kr 6.6 MgO 4. Lr 5.5 Agr 5.77 MO 4.4 NaCl 5.64 Ag 4.9 MgS 5. NaI 6.47 CaSe 5.9 PbS 5.9 KCl 6.9 ao 5.5 SrTe 6.47 Aother commo structure s that of CsCl (other crystals wth the same structure are gve the table below. The lattce s ths case smple cubc, wth a bass cosstg of oe Cs + o (red at [[]], ad oe Cl o (gree at [[/,/,/]]. The umber of earest eghbors (of opposte d s eght. Crystal a (Å Crystal a (Å Crystal a (Å AlN.88 CsCl 4. TlCl.8 CuZ (β-brass.94 Csr 4.9 Tlr.97 AgMg.8 CsI 4.57 TlI 4. The crystal structure of damod (ad also of S ad Ge semcoductors s represeted below.

16 Crystal Structure 6 Crystal a (Å C (damod.57 S 5.4 Ge 5.66 α-s (grey 6.49 It s a face-cetered cubc (fcc lattce wth a bass cosstg of two detcal atoms, wth coordates [[]] ad [[/4,/4,/4]]. Alteratvely, damod ca be vewed as beg formed from two terpeetratg fcc lattces, dsplaced by /4 of the volume dagoal. Sce the covetoal ut cell of the fcc lattce cotas 4 lattce pots, t follows that the covetoal ut cell of damod has 4 8 atoms. No prmtve cell exsts that cotas oly oe atom. I damod, each atom has 4 earest eghbors ad ext earest eghbors. It s usually ecoutered materals where the covalet bodg prevals. Note that, although a fcc lattce, the pacg fracto of the damod structure s oly.4. A closely related crystal structure to that of the damod s the cubc zc sulfde (zc blede structure. It dffers from damod that the two atoms of the bass are dfferet ( ths case, Z ad S. The covetoal ut cell cotas four molecules, the Z atoms (dar blue the fgure below beg placed at the postos [[]], [[,/,/]], [[/,,/]] ad [[/,/,]], whereas the S atoms (gree occupy the postos [[/4,/4,/4]], [[/4,/4,/4]], [[/4,/4,/4]], ad [[/4,/4,/4]]. ach atom s surrouded by four equally dstat atoms of the opposte d, placed the corers of a regular tetrahedro. Crystal a (Å Crystal a (Å Crystal a (Å SC 4.5 AlP 5.45 IAs 6.4 ZS 5.4 AlAs 5.66 ISb 6.48 ZSe 5.67 GaAs 5.65 SC 4.5 MS (red 5.6 GaSb 6. CuCl 5.4 CdS 5.8 GaP 5.45 Cur 5.69 CdTe 6.48 AgI 6.47 HgSe 6.8

17 Crystal Structure 7 Ule the damod structure, where there s a ceter of verso at the mdpot of every le betwee earest-eghbor atoms, such verso ceters are abset the zc blede structure. Ths s a example of addtoal symmetry operatos related to the bass of the crystal structure. The hexagoal close-paced (hcp crystal structure ca be obtaed from the hexagoal ravas lattce f the bass cossts of two atoms (blue ad red the fgure below, left ad f the atoms oe plae, whch touch each other, also touch the atoms adjacet plaes. The pacg fracto ths case s.74 (as fcc lattces, ad s maxmum. Ths crystal structure s foud the sold state of may elemets, as ca be see from the table below. The hcp structure ca be vewed as vertcal arragemet of two-dmesoal hexagoal structures, such as the sphercal atoms the secod layer are placed the depressos left the ceter of every other tragle formed by the ceters of the sphercal atoms the frst layer. The thrd layer of atoms s the placed exactly above the frst, the fourth above the secod, ad so o. Ths d of arragemet s called AA I a deal hcp structure, the heght betwee the frst ad the thrd layers (the heght alog the c axs the fgure below s c 8/ a.6a. ecause the symmetry of the hcp lattce s depedet of the rato c/a, real hcp structures ths rato ca tae values close to, but ot exactly detcal to the deal.6 value (see the table below. Crystal a (Å c/a Crystal a (Å c/a He.57.6 Mg..6 e.9.58 T Nd.66.6 Zr..59 Z Y Cd Gd α-co.6.6 Lu.5.58

18 Crystal Structure 8 If the c/a rato dffers cosderably from the deal.6 value, the hexagoal structure s o loger closepaced. Ths s the case of graphte, for example, whch s a o-closed-paced hexagoal structure of carbo atoms (see the fgure at rght, wth a.4å ad c.4 Å, whch mples that c/a.9. The fact that the hcp structure has the same pacg fracto as the fcc structure s easly explaed the fgure below. Suppose that we place the frst two plae of atoms as the hcp structure. If the atoms the thrd plae are postoed over the ceters of the tragles formed by the ceters of the atoms the frst plae that have o atoms from the secod plae above them, the resultg structure s fact a fcc. Ths vertcal arragemet s called ACAC The hcp ad fcc structures dffer oly by the vertcal arragemet (AA or ACAC of hexagoal plaes of atoms. A structure closely related to hcp s wurtzte, geerally ecoutered bary compoud semcoductors such as ZS (wurtzte, ZO, N, CdS, CdSe, GaN, AlN, but sometmes also terary compouds such as Al.5 Ga.5 N. I bary compouds (see the fgure at rght, each elemet has a hcp structure, ad the crystal s formed by terpeetratg two such structures, so that a atom oe hcp lattce s equallydstaced from the atoms the other hcp lattce. The crystal structure of the elemets the perodc table s dcated the fgure below. Note that several elemets ca suffer trastos from oe crystalle structure to aother depedg o the exteral codtos: temperature, pressure, etc. I the table below dhcp stads for double hexagoal closed-paced (the heght of the cell alog the drecto ormal to the hexagoal plaes s twce that the hcp structure

19 Crystal Structure 9 Lattce costats of some elemets that crystallze the fcc crystal structure: Crystal a (Å Crystal a (Å Crystal a (Å Crystal a (Å Crystal a (Å Ar 5.6 Au 4.8 Cu.6 N.5 Pt.9 Ag 4.9 Ca 5.58 Kr 5.7 Pb 4.95 Sr 6.8 Al 4.5 β-co.55 Ne 4.4 Pd.89 Xe 6. Lattce costats of some elemets that crystallze the bcc crystal structure: Crystal a (Å Crystal a (Å Crystal a (Å Crystal a (Å a 5.6 e 4.8 Mo.6 Rb.5 Cr 4.9 K 5.58 Na 5.7 Ta 4.95 Cs 4.5 L.55 Nb 4.4 V.9 W 6.8

20 Recprocal lattce The cocept of recprocal lattce s drectly coected wth the perodcty of crystalle materals ad of ther physcal propertes (such as charge desty, electrc feld dstrbuto, etc.. Sce the crystal s varat uder ay traslato wth a ravas lattce vector R mp ma + a + pa ( for ay tegers m, or p, ay fucto ϕ wth the same perodcty as the crystalle lattce must satsfy the relato ϕ r ϕ( r + R, ( ( mp where x, x, s a arbtrary posto vector wth coordates,, ad x measured r ( x x x wth respect to the (geerally o-orthogoal system of coordates determed by, a, a ad a. Ths meas that ϕ ( x, x, x ϕ( x + ma, x + a, x + pa ( or, for a fucto that ca be expaded a ourer seres ϕ ( x, x, x ϕ exp[ ( Gx + G x + Gx] (4 G, G, G t follows that, for ay m,, ad p, exp( mg a, exp( G a, exp( pg a. (5 Thus, G, wth,,, ca oly tae dscrete values G πs / a, (6 ad (4 ca be rewrtte as

21 Recprocal lattce ϕ ( r ϕ exp( G r (7 s, s, s where G s + (8 b + sb sb s a vector a coordate system defed by the vectors b,,,, such that b a πδ. (9 j j Smlar to the ravas lattces that are costructed startg wth the prmtve vectors a, oe ca defe a recprocal lattce terms of the prmtve vectors b, such that G (8 are pots the recprocal lattce. A recprocal lattce ca oly be defed wth respect to a gve drect lattce. As demostrated the followg, the G vectors have dmesos (ad meag of wavevectors related to plae waves wth the perodcty of the drect lattce. If the vectors a are chose ad the volume of the prmtve cell the drect space s Ω ( a a a, the vectors b ca be chose as b π / Ω( a, b π / Ω( a, b π / Ω( a. ( ( a ( a ( a It follows the that the volume of the prmtve cell of the recprocal lattce s gve by π Ω rec b ( b b ( / Ω. ( xamples of drect ad correspodg recprocal lattces two dmesos are gve the fgures below.

22 Recprocal lattce or a d ( x, a d ( x +, the vectors of the recprocal lattce are determed from y y codto (9, ad are foud to be b ( π / d( x, b ( π / d( x +. y y Whe x ad y are ot orthogoal, but x y ε (see the fgure above, for a dx cy ad a d x + cy, we obta (please chec! b c + dε d + cε π x π cd( ε cd( ε c dε d cε y, b π x + π y. cd( ε cd( ε I three dmesos, the recprocal lattces for the ravas lattces the cubc system are summarzed the table below Real space Recprocal space Lattce Lattce costat Lattce Lattce costat SC a SC π / a CC a CC 4π / a CC a CC 4π / a The recprocal lattce of a cubc lattce s also cubc sce, ths case, f x, y, z are orthogoal vectors of ut legth, ax, ay, a az ad Ω a, from ( t follows that b a a (π / a, b (π / a, b (π / z,.e. the recprocal lattce s smple cubc wth a x lattce costat π / a. a y a Aalogously, the recprocal lattce to the bcc lattce wth (see the frst course ( a / ( x + y, a ( a / ( x + y +, a ( a / ( x y +, ad Ω a / has z z z prmtve vectors b (π / a( x +, b (π / a( y +, b (π / a( z +,.e. s a fcc y z x lattce wth a volume (of the prmtve ut cell recprocal state of Ω (π / a, rec whereas the recprocal lattce of the fcc lattce, wth a ( a / ( x +, a ( a / ( y +, y z

23 Recprocal lattce 4 a ( a / ( z +, ad Ω a / 4 s a bcc lattce wth Ω 4(π / a ad prmtve vectors b x (π / a( x + y, b (π / a( x + y +, b (π / a( x y +. I both cases the z z rec z cubc structure of the recprocal lattce has a lattce costat of 4π / a. Observato: The recprocal lattce of a recprocal lattce s the drect lattce. ecause the product of a prmtve ravas lattce vector ad of a prmtve vector of the recprocal cell s a teger multple of π,.e. that G R π ( mh + pl, ( mp hl + for all tegers m,, p ad h,, l, t follows that exp( G R for ay vector R the ravas lattce ad ay vector G the recprocal lattce. Ths mples that the fucto exp( G r has the same perodcty as the crystal because exp[ G ( r + R] exp( G r exp( G R exp( G r. As a cosequece, cell exp( G r dv ( s depedet of the choce of the cell ad a traslato wth a arbtrary vector d should ot chage the value of the tegral. More precsely, f cell exp[ G ( r + d] dv exp( G r dv (4 cell the [exp( G d ] cell exp( G r dv, from whch t follows that cell exp( G r dv Ωδ (5 G, ad that the set of fuctos exp( G r form a complete, orthoormal bass for ay perodc fucto whch has the same perodcty as the crystal,.e. whch ca be wrtte as ϕ ( r ϕ exp( G r. (6 G G

24 Recprocal lattce 5 If the formula above s regarded as a ourer trasformato of the perodc fucto ϕ, the coeffcets precsely, sce ϕ G ca be retreved by performg a verse ourer trasformato. More cell ϕ( rexp( G' r dv ϕg G cell ϕg G cell exp[ ( G G' r] dv exp( G rexp( G' r dv ϕgωδ G GG' (7 t follows that Ω ϕ ϕ( rexp( G r dv. (8 cell G Relatos betwee the drect ad recprocal lattces Oe geometrcal property that ca be easly show s that the recprocal lattce vector G mp mb + b + pb (9 s perpedcular to the plae (actually, to the set of parallel plaes wth Mller dces (mp the ravas lattce. The closest plae to the org from the set of plaes (mp cuts the a coordate axes at a / m, a /, ad a /, respectvely. p To show that (mp s perpedcular to G t s suffcet to demostrate that G s perpedcular to two o-collear vectors the (mp plae, whch ca be chose as mp mp u a a / m, v a p a / m, ( / / ad satsfy, deed, the relatos u G v G ( mp mp because of (9. The, t follows that the ormal to the (mp plae that passes through the org ca be expressed as G G. ( mp mp / mp

25 Recprocal lattce 6 a /p a a / a a a /m A cosequece of ths result s that the dstace betwee two cosecutve plaes wth the same Mller dces (mp s versely proportoal to the modulus of. Sce we ca always draw a plae from the (mp famly through the org, the dstace betwee two successve plaes s equal to the dstace betwee the org ad the closest plae to org from the (mp famly. Ths dstace s obtaed by calculatg the projecto o the ormal to the (mp,.e. o ( t s foud that G mp G G, of ay of the vectors a / m, a /, or / p. Usg mp mp / mp a d mp a a a π. ( m p G mp So, d mp m b + b + p b + m( b π b + p( b b + pm( b. (4 b

26 Recprocal lattce 7 As already poted out the dscusso about Mller dces, the dstace betwee ay two plaes the famly (sm,s,sp, s s tmes smaller tha betwee ay two plaes the famly (mp. The two famles/sets of plaes are parallel. I partcular, for the smple, body-cetered ad face-cetered cubc ravas lattces wth the prmtve traslato vectors gve the Crystal Structure secto of the course, the dstace betwee two cosecutve plaes wth the same Mller dces s, respectvely, d sc mp d bcc mp d fcc mp a, (5a m + + p a, (5b ( + p + ( p + m + ( m + a (5c ( + p m + ( p + m + ( m + p Due to the form of (7, the vectors G of the recprocal lattce ca be uderstood as wavevectors of plae waves wth the perodcty of the lattce ad wavelegths π / G, smlar to wavevectors optcs that are perpedcular to wavefrots ad have dmesos related to the wavelegth λ as π / λ. The frst rllou zoe Aalogous to the Wger-Setz cell drect lattces, oe ca defe a prmtve ut cell the recprocal lattce that has the same symmetry as ths lattce. Ths prmtve ut cell s ow as the frst rllou zoe. The costructo of the frst rllou zoe s smlar to that of the Wger-Setz cell,.e. we draw les to coect a gve lattce pot the recprocal lattce to all earby lattce pots, ad the draw ew les (or plaes, three-dmesoal lattces at the md pot ad ormal to the frst set of les. These les (plaes are called ragg plaes sce (as we wll see later all vectors that fsh o these surfaces satsfy the ragg codto. The frst rllou zoe s the the area (volume recprocal space that ca be reached from the org, wthout crossg ay ragg plaes. Hgher-order rllou zoes, say the th rllou zoe, are the defed as the area (volume recprocal space that ca be reached from the org by crossg exactly ragg plaes. The costructo of the frst (lght blue, secod (lght brow ad thrd (dar blue rllou zoes for a two-dmesoal lattce s llustrated the fgure below. The ragg plaes eclosg the th rllou zoe correspod to the th order X-ray dffracto.

27 Recprocal lattce 8 Although hgher order rllou zoes are fragmeted, the fragmets, f traslated, loo le the frst rllou zoe. Ths process s called reduced zoe scheme. All rllou zoes, rrespectve of the order, have the same volume. The hgher-order rllou zoes for a two-dmesoal square lattce are llustrated the fgure below. As for Wger-Setz cells, the faces of the frst rllou zoe satsfy the relato G G /, where G s the dstace to the earest eghbor the recprocal space. Ths relato ca be rewrtte as G G or, sce the equato s equvalet to the replacemet of G wth G, we obta ( + G,.e. the frst rllou zoe s the tersecto of spheres wth the same radus cetered at earest eghbor pots the recprocal lattce.

28 Recprocal lattce 9 I partcular, sce the recprocal lattce of the bcc lattce s a fcc lattce, the frst rllou zoe of the bcc lattce (see the polyhedro the fgure a below s the Wger-Setz cell of the fcc. The reverse s also true: the frst rllou zoe of a fcc lattce (the trucated octahedro/rhombododecahedro fgure b below s the Wger-Setz cell of the bcc lattce. or certa ravas lattce, partcular bcc, fcc ad hexagoal, the pots of hghest symmetry the recprocal lattce are labeled wth certa letters. The ceter of the rllou zoe s all cases deoted by Γ. Other symmetry pots are deoted as follows (see also fgures: sc lattce: bcc lattce: fcc lattce: M ceter of a edge R corer pot X ceter of a face H corer pot jog four edges N ceter of a face P corer pot jog three edges K mddle of a edge jog two hexagoal faces L ceter of a hexagoal face U mddle of a edge jog a hexagoal ad a square face W corer pot X ceter of a square face

29 Recprocal lattce hexagoal lattce: A ceter of a hexagoal face H corer pot K mddle of a edge jog two rectagular faces L mddle of a edge jog a hexagoal ad a rectagular face M ceter of a rectagular face Dsperso relatos of electros ad phoos for dfferet crystal drectos use ths labelg (see the fgures below, the labels dcatg the drecto but also the symmetry of the crystal, sce dfferet labels are used for dfferet symmetres.

30 X-ray dffracto o crystalle structures The drect observato of the perodcty of atoms a crystalle materal reles o the X-ray or partcle (electro or eutro dffracto/scatterg o these spatally perodc structures, sce the wavelegth of the cdet beam s these cases comparable to the typcal teratomc dstace of a few Å. Optcal dffracto s ot sutable for ths purpose sce the wavelegth of photos s much too log (about μm comparso to the lattce costat (a few Agstroms. I a dffracto expermet, both the X-ray or partcle source ad the detector are placed vacuum ad suffcetly far away from the sample such that, for moochromatc radato, the cdet ad outgog X-ray or partcle beams ca be approxmated by plae waves. The X-rays ca be used ether trasmsso or reflecto cofguratos. The dffracto pcture offers formato regardg the symmetry of the crystal alog a certa axs. I partcular, the postos of the spots gve formato about the lattce ad the testy aalyss reveal the composto of the bass. The X-rays peetrate deeply the materal, so that may layers cotrbute to the reflected testy ad the dffracted pea testes are very sharp ( agular dstrbuto. To obta sharp testy peas of the scattered radato, the X-rays should be specularly reflected by the atoms oe plae.

31 X-ray dffracto λ hc / or X-rays, the wavelegth s determed from the relato hν hc / λ or, whch equals a few Å f s of the order of few ev. I fact, λ(å.4/(ev. X-rays are scattered mostly by the electroc shells of atoms a sold, sce the ucle are too heavy to respod. lectros ca also have de rogle wavelegths smlar to the lattce costats of crystals. I ths case ( h / λ / m, ad for a electro eergy of 6 ev, the correspodg wavelegth λ h / m s about 5 Å. Actually, f the etc eergy of the electros s acqured a accelerato voltage potetal U, such that eu, oe has λ(å.8/[u(v] /. or eutro dffracto we have to cosder a smlar relato, except that the electro mass m has to be replaced by the eutro mass M. The, λ(å.8/[(ev] /. Whe a wave teracts wth the crystal, the plae wave s scattered by the atoms the crystal, each atom actg le a pot source (Huyges prcple. ecause a crystal structure cossts of a lattce ad a bass, the X-ray dffracto s a covoluto of dffracto by the lattce pots ad dffracto by the bass. Geerally, the latter term modulates the dffracto by the lattce pots. I partcular, f each lattce pot acts as a coheret pot source, each lattce plae acts as a mrror. The X-rays scattered by all atoms the crystalle lattce terfere ad the problem s to determe the ravas lattce (cludg the lattce costats ad the bass from the terferece patters. The wave that s dffracted a certa drecto s a sum of the waves scattered by all atoms. Hgher dffracto testes wll be observed alog the drectos of costructve terferece, whch are determed by the crystal structure tself. G The dffracto of X-rays by crystals s elastc, the X-rays havg the same frequecy (ad wavelegth before ad after the reflecto. The path dfferece betwee two cosecutve plaes separated by d s A d sθ. rst-order costructve terferece occurs f

32 X-ray dffracto d sθ λ, ( codto ow as ragg s law. The ragg law s a cosequece of the perodcty of the crystal structure ad holds oly f λ d. Ths s the reaso why the optcal radato s ot sutable to detect the crystalle structure, but oly X-rays ad electro or eutro beams ca perform ths tas. Hgher order dffracto processes are also possble. The ragg relato determes, through the agle θ, the drectos of maxmum testy. These drectos are detfed as hgh-testy pots o the detecto scree, the posto of whch reveal the crystal structure. or example, f the sample has a cubc crystal structure oreted such that the drecto [] (the dagoal of the cube s parallel to the cdet beam, the symmetry of the pots o the detector scree wll reveal a C symmetry axs. O the cotrary, f the dffracto patter has a C6 symmetry axs, the crystal s hexagoal, f t has a C4 symmetry axs t s a tetragoal crystal, whereas t s cubc f t shows both a ad a C symmetry axs. C4 The ragg formula says othg about the testy ad wdth of the X-ray dffracto peas, assumes a sgle atom every lattce pot, ad eglects both dffereces scatterg from dfferet atoms ad the dstrbuto of charge aroud atoms. A closer loo at the teracto betwee the X-rays ad the crystal of volume V reveals that the ampltude of the scattered radato (whch s proportoal to the ampltude of the oscllato of the electrc ad magetc felds of the total dffracted ray s determed by the local electro cocetrato ( r exp( G r, whch s a measure of the stregth of the G G teracto, ad has the same perodcty as the crystalle lattce. The dffracto testy I. or elastc X-ray scatterg, the phase of the outgog beam, wth wavevector ', dffers from that of the comg beam that propagates wth a wavevector through exp[ ( ' r], so that ( rexp[ ( ' r] dv ( rexp( Δ r dv G exp[ ( G Δ r] dv ( G where Δ ' s the scatterg vector, whch expresses the chage wavevector. The result the above tegral depeds o the volume of the crystal. If the crystal has legth L ad N prmtve cells the drecto (,, of a orthogoal coordate system (f the

33 X-ray dffracto 4 crystal system s ot orthogoal, a trasformato of coordates to the x x, y x, z x axes should be performed, the tegral alog the drecto s gve by L / L / π exp ( s a Δξ x dx a s[ π ( s π ( s Δξ N Δξ ] L sc[ π ( s Δξ N ] ( where s, Δ ξ are the compoets of G ad Δ o the axs ad costat o the same drecto. The fucto ad teds to the Drac delta fucto for large x. Therefore, large-volume crystals scatterg occurs oly f a L / N s the lattce sc ( x s x / x has a maxmum value for x, Δ G, (4 case whch V G. (I fte-volume crystals there s a sort of ucertaty the agular rage of Δ aroud G for whch the scatterg ampltude taes sgfcat values: as the volume decreases, the agular rage creases. The above codto suggests that X-ray dffracto expermets reveal the recprocal lattce of a crystal, opposto to mcroscopy, whch exposes the drect lattce (f performed wth hgh-eough resoluto. ragg plae ' The dffracto codto + G + G. I partcular, the form Δ ' G ca be rewrtte as ' + G or h ' h + hg of the dffracto codto represets the mometum coservato law of the X-ray photo the scatterg process; the crystal receves the mometum hg. or elastc scatterg ' ad thus G + G,

34 X-ray dffracto 5 or G G /, equato that defes the faces of the frst rllou zoe (the ragg plaes. The geometrc terpretato of ths relato (see the fgure above s that costructve terferece/dffracto s the strogest o the faces of the frst rllou zoe. I other words, the frst rllou zoe exhbts all the wavevectors that ca be ragg-reflected by the crystal. The dffracto codto s equvalet to ragg s law, whch ca be wrtte for a certa set of plaes separated by the dstace G G d d mp as ( / λsθ π / d mp π, or, wth G m b + b + p (for the drecto of G wth respect to the set of plaes, b see the fgure llustratg the ragg law. The Laue codto The dffracto codto Δ G ca be expressed stll aother way: f we multply both terms of ths relato wth the prmtve traslato vectors of the drect lattce, we obta the Laue codtos a Δ πs, a Δ πs, a Δ πs, (5 where s are tegers. The Laue equatos have a smple geometrcal terpretato: Δ les smultaeously o a coe about,, ad a,.e. les at the commo le of tersecto of a a three coes. Ths codto s qute dffcult to satsfy practce. Moreover, aalogy to optcal dffracto expermets, the Laue codto ca be vewed as a codto of costructve terferece betwee waves dffracted by two atoms separated by a prmtve traslato vector or, by exteso, betwee waves dffracted by all atoms the crystal. At ragg reflecto, the radato scattered by all atoms arrves phase at the detector, ad testy peas are obtaed. The wald sphere The drecto of terferece peas ca be easly determed also va a smple geometrcal costructo suggested by wald. Namely, oe costructs a sphere (a crcle two dmesos see the red crcle the fgure above aroud a pot O the recprocal lattce chose such that the cdet wavevector wth O as org, eds o a arbtrary lattce pot A. The org of the wald sphere (or crcle s ot ecessary a lattce pot.

35 X-ray dffracto 6 O A The radus of the sphere (crcle s the waveumber of the cdet (ad outgog radato '. A maxmum testy s foud aroud a drecto ' f ad oly f the wald sphere (crcle passes through aother pot of the recprocal lattce. The drecto ' s determed by the org O of the wald sphere ad ths lattce pot o the surface (crcumferece, whch s separated from the tp of (from A by a recprocal lattce vector. It s possble that for certa cdece agles ad wavelegths of the X-rays o such preferetal drecto ' exsts. Therefore, to obta peas the scattered testy t s geeral ecessary to vary ether the wavelegth or the cdece agle of the comg X-rays such that a suffcet umber of recprocal lattce pots fd themselves o the wald sphere (crcle, order to determe uambguously the crystal structure. I the frst method, called Laue method, the radus of the wald sphere (crcle s vared cotuously (see, for example, the gree crcle

36 X-ray dffracto 7 the fgure above, whle the secod method, called the rotatg crystal method or Debye- Scherrer-Hull method, the wald sphere (crcle s rotated aroud the orgal lattce pot wth respect to whch the wald sphere (crcle was costructed. The result s represeted wth the dar blue crcle the fgure above. I aother dffracto method (the Debye-Scherrer method polycrystalle samples are used, whch are ether fxed or rotate aroud a axs. I ths case, the cdet beam s scattered by oly those crystalltes (radomly oreted wth plaes that satsfy the ragg codto. ecause the sample cotas crystalltes wth all oretatos, the dffracto patter o the scree s o loger formed from dscrete pots, but from cocetrc crcles. The fluece of the bass o the scattered ampltude If the Laue/dffracto codto Δ G s satsfed, a explct accout of the bass fluece mples that the assumpto of pot/sphercal sources at the lattce pots have to be modfed. I ths case, we have foud that V N ( rexp( G r dv NS, (6 cell G G where Ω ( rexp( G r dv, N s the total umber of lattce pots, ad cell G S G cell ( r exp( G r dv (7 s called the structure factor. It s defed as a tegral over a sgle cell, wth r at oe corer. If there s oly oe lattce pot the bass ad the electro dstrbuto ( r δ ( r, S G. If there are s atoms the bass at postos r j, j,,..,s, the total electro desty ca be expressed as a superposto of electro cocetrato fuctos j at each atom j the bass, so that the structure factor s expressed as tegrals over the s atoms of a cell: S G j s s j ( r r j exp( G r j exp( G r dv j j s j ( ρexp( G ρ dv j ( ρ exp( G ρexp( G r j s j f j exp( G r j dv (8

37 X-ray dffracto 8 where ρ r ad f ( ρ exp( G ρ dv s the atomc form factor, whch depeds r j j j oly o the type of elemet that the atom belogs to. The tegral has to be tae over the electro cocetrato assocated wth a sgle atom. The atomc form factor s a measure of the scatterg power of the jth atom the ut cell. If the charge dstrbuto has a sphercal symmetry, oe ca use sphercal coordates chose such that the polar drecto s alog G. I ths case, dv πρ sϕdρdϕ, G ρ G ρ cosϕ Gρ cosϕ, where ϕ s the agle betwee ρ ad G, ad the atomc form factor becomes f j π ( ρ ρ dρ exp( Gρ cosϕsϕdϕ 4π ( ρ ρ (s Gρ / Gρ dρ. (9 j π j The atomc form factor decreases rapdly wth the dstace ad, the lmt ρ, whe s G ρ / Gρ, f j j 4π ( ρ ρ dρ Z, ( where Z s the umber of electros a atom. Also, whe G Δ (for a dffracted ray collear wth the cdet ray, the phase dfferece vashes ad aga f j ( G Z. f ca be vewed as the rato of the radato ampltude scattered by the electro dstrbuto a atom to that scattered by oe electro localzed at the same pot as the atom. The overall electro dstrbuto a sold, as obtaed from X-ray dffracto expermets, s almost the same as for free atoms,.e. atoms whch the outermost (valece electros are ot redstrbuted formg the sold. X-ray dffracto expermets are thus ot very sestve to small redstrbutos of electros. xample: cosder a bcc lattce as a sc lattce wth a bass cosstg of two atoms at [[]] ad [[/,/,/]]. The prmtve lattce vectors for the ravas ad the recprocal lattces are ths case ax, ay, a az, ad b (π / a, b (π / a, b (π / z, a a x y a respectvely. The dffracto pea of the sc lattce that s labeled by (mp correspods to G m b + b + pb (π / a( mx + y + p ad for ths dffracto pea z

38 X-ray dffracto 9 S mp f + j f f j exp( G r exp[ (π / a( mx + y + pz ( a / ( x + y + z] + f j exp[ π ( m + + p] f exp[ (π / a( mx + y + pz ] ( The bcc dffracto testy s gve by I mp mp + S f + f + Re[ f f exp[ π ( m + p]]. ( If f f f, 4 f, f m + + p eve I mp f [ + exp[ π ( m + + p]] (, f m + + p odd So, for the bcc structure wth the same type of atoms, the (mp dffracto peas of the sc lattce dsappear wheever m + + p s a odd teger. I partcular, t dsappears for a ( reflecto (see the fgure below sce the phase dfferece betwee successve plaes s π, ad the reflected ampltudes from two adjacet plaes are out-of-phase/destructve terferece occurs. π π Observato: for a sc lattce wth oe atom the bass, the dffracto testy would have bee the same, rrespectve of the party (eve or odd of m + + p. Ths example llustrates the effect of the bass o the dffracto testy.

39 Crystal bdg The stablty of sold state materals s assured by the exstg teractos (attractve ad repulsve betwee the atoms the crystal. The crystal tself s deftely more stable tha the collecto of the costtuet atoms. Ths meas that there exst attractve teratomc forces ad that the eergy of the crystal s lower tha the eergy of the free atoms. O the other had, repulsve forces must exst at small dstace order to prevet the collapse of the materal. Oe measure of the stregth of the teratomc forces s the so-called cohesve eergy of the crystal, defed as the dfferece betwee the eergy of free atoms ad the crystal eergy. Smlarly, the cohesve eergy per atom s defed as the rato betwee the cohesve eergy of the crystal ad the umber of atoms. Typcal values of the cohesve eergy per atom rage from to ev/atom, wth the excepto of ert gases, where the cohesve eergy s about. ev/atom. I partcular, the cohesve eergy determes the meltg temperature of sold state materals. Crystals wth <.5 ev have wea crystal bdgs, whle the others are characterzed by strog crystal bdgs. U U As show the fgure above, the potetal/bdg eergy U, whch descrbes the teracto betwee two atoms, approach (or fty for a teratomc dstace R (or to, ad has a mmum at a certa dstace R R. It s composed of a attractve eergy part, domat at R > R, ad a repulsve eergy part that prevals at R < R. The, the most stable state of the system, whch occurs at the lowest possble eergy, s characterzed by the cohesve eergy, the correspodg teratomc dstace, R, beg ow as the U

40 Crystal bdg equlbrum teratomc dstace. The last parameter has typcal values of Å, whch mples that the stablty of the crystal s determed by short-rage forces. The teratomc force, defed as ( R U / R, ( s egatve (attractve for R > R, ad postve (repulsve for R < R. The attractve ad repulsve forces, whch have dfferet orgs, cacel each other at the equlbrum teratomc dstace. The geeral form of the potetal eergy s A U ( r, wth > m. ( m r r The repulsve force betwee atoms the sold has the same org all crystals: Paul excluso prcple, whch forbds two electros to occupy the same orbtal (the same quatum state. The repulsve force s characterzed (see the formula above by the power-law expresso U A / r, wth > 6 or, sometmes, by the expoetal expresso U λ exp( r / ρ, where λ ad ρ are emprcal costats that ca be determed from the lattce parameters ad the compressblty of the materal. Whch expresso s better suted to descrbe the repulsve force depeds o whch oe better fts wth expermetal values. The repulsve potetal s short-raged ad thus t s effectve oly for earest eghbors. The attractve forces create bods betwee atoms/molecules the sold, whch guaratee the crystal stablty ad are of dfferet types depedg o the crystal. Oly the outer (valece electros partcpate the bodg. There are several types of bodg, depedg o the mechasm resposble for crystal coheso: oc, covalet ad metallc, whch gve rse to strog crystal bdgs, ad hydroge bodg ad va der Waals teracto, whch determe wea crystal bdgs. Crystal bdg ert/oble gases. Va der Waals-Lodo teracto The crystals of ert gases have low coheso eergy ad meltg temperature, ad hgh ozato eerges. They are the smplest crystals, wth a electro dstrbuto close to that of free atoms. rom a electrcal pot of vew they are solators, ad from a optcal pot of vew, are trasparet the vsble doma. The wea bdg betwee the costtuet atoms