Chapter Three: Crystal Binding

Transcription

1 Fouth Edition SI Vesion Chapte Thee: Cystal Binding Cengage Leaning, Engineeing. All Rights Reseved. 1

2 Classification of cystalline solid: (a) ionic, (b) van de Waals, (c) covalent, and (d) metallic 2

3 3.1 Intenal enegy of a cystal includes (1)lattice enegy (cohesive enegy): electostatic attactions and epulsions. Most of the time, the book does not conside epulsive foce, except deiving B and n. Shot ange: epulsive foce: ion esists ovelap with the electon distibutions of neighboing ions. (when a solid is compessed, less than 0 ) 3

4 Pauli exclusion pinciple (no two electons could be at the same quantum state) Hund's ule: 4

5 (2) themal enegy (lattice vibation): vibations of the atoms about thei equilibium. The lowest allowed enegy is not zeo. A paticle theefoe cannot be at est in a system, but, must have a finite zeo-point kinetic enegy. 5

6 3.2 Ionic cystal: NaCl, CsCl, ZnS, CaF 2, TiO 2 Combination of a highly electopositive metallic element with a highly electonegative element (halogens, O o sulfu). NaCl: CN6 CsCl: CN8 Zincblend (two kind of atoms): CN4 Na + Cl - Cl - Cs + S 2- Zn 2+ 6

7 Inteaction: long ange and shot ange. Long ange: attaction foce: Coulomb (electostatic) foce e 1 12 e 2 Coulomb potential enegy (J): ke 1 e 2 / 12 Coulomb Foce (nt): f / ke 1 e 2 /( 12 ) 2 e 1, e 2 : the chages on the ions 12 : cente to cente distance between ions k: constant 7

8 3.3 The Bon Theoy of Ionic Cystals Potential: M + R ; Columb and epulsive enegy Empiical potential fo epulsive tem: B n Fo example: one NaCl Potential: -Az 2 e 2 / + Be 2 / n z: the chages on the ions : distance between ions A, B: constants n: usually 9 8

9 Lattice enegy vs distance, assume n 9 R M R : detemine the shape of the total enegy cuve at small due to lage n. M : contol facto at lage U 0 : at 0 : the equilibium sepaation at 0 K. 9

10 Fo NaCl stuctue, 6 neaest neighbos with distance ; 12 second neaest neighbos with distance 2; 8 thid neaest neighbos with distance 3;... M -6e 2 /1 + 12e 2 /2-8e 2 /3 + 6e 2 /4 24e 2 /5 M -Ae 2 / -e 2 / [ ] A (Madelung numbe) [ ] (Electostatic/Coulomb only)

11 1. It is not possible to wite down the successive tems by a casual inspection. 2. The seies will not convege unless the successive tems in the seies ae aanged so that the contibutions fom the positive and negative tems nealy cancel. Thee ae numeically moe poweful, but moe complex, ways of computing such Coulomb lattice sums, which ae, howeve, all guided by the same physical citeion. The most famous is due to Ewald. 11

12 C. Kittel, Intoduction to Solid State Physics 7th ed., Chapte 3., Appendix B, John Wiley & Sons, Inc., Singapoe, Data fo Geneal, Oganic, and Physical Chemisty Copyight 1989 by C.D. Schaeffe, J. (Elizabethtown College) C.A. Stausse, M.W. Thomsen, and C.H. Yode (Fanklin & Mashall College) 12

13 1 st method: measue lattice enegy (intenal enegy) epulsive tem: B n The paamete used in epulsive tem must be evaluated; Two expeimental quantities could be used (1) equilibium inteionic sepaation 0 at 0K: net foce 0 d tot d 0 d d (- Ae 2 Be + 2 ) 0 n A is known, so n and B is elated. 13

14 d tot d 0 d d (- Ae 2 Be + 2 ) 0 n B n A 0 n-1 Ae - 2 Be + 2 n - 0 Ae 2 0 A n-1 ( 0 ) n + 0 n e 2 -Ae 2 (1-1 n ) n can be deduced then. 0 0 can be found fom x-ay diffaction, A is known. Compaing with the expeimental data, n can be deduced. 14

15 15 The pecise value of the lattice enegy may not be detemined expeimentally, because of the impossibility of pepaing an adequate amount of gaseous ions and cations and measuing the enegy eleased duing thei condensation to fom the solid. Howeve, the value of the lattice enegy may eithe be deived theoetically fom electostatics o fom a themodynamic cycling eaction, the so-called Bon Habe cycle Bon-Habe Cycle fo NaCl kj/mol 5 Lattice enegy

16 2 d s i n n Jounal of Physics and Chemisty of Solids Vol. 70, Iss. 1, Jan. 2009, P

17 2 nd method: measue lattice enegy (intenal enegy) (2) A bette pocedue to get n: By measuing the compessibility (K 0 ) of the solid at 0 K. Solids and liquids, the volume changes in metallugical eactions ae usually vey small. K 0 To make sue V V p T K 0 : compessibility V: volume of the cystal Relative ate of change of volume with pessue at constant temp. 17

18 Fom themodynamics G U + PV -TS At 0 K T 0 G U + PV G: Gibbs fee enegy U: intenal enegy S: entopy At themal equilibium G 0 G U + PV TS 0 U + PV U -PV 0 du -(PdV + VdP) G > 0: nonspontaneous G < 0: spontaneous Most metallugical pocesses of inteest, such as feezing, occu at constant pessue. 18

19 K 0-1 V V p T P -(du/dv) -1 V V -U V T B 0 (bulk modulus) 1 K 0 V U V V T Na + Cl - Fo NaCl V (2) 3 4 (NaCl) 2 3 (pe NaCl) dv 6 2 d 19

20 B 0 (bulk modulus) 1 K 0 V U V V T V U 6 2 d 6 2 d T V U 6 2 d d d dx T dv uv u + v dx du dx dv 6 2 d V d 0 2 T V U d 0 2 T 20

21 U () - Ae 2 Be + 2 n B 0 du () Ae 2 - d U d 0 2 T nbe 2 n+1 du () when d 0 B n A 0 n-1 U() 0-2 Ae 2 (n+1)nbe ( + 2 ) 3 n+2 0 Ae 2 (-2+n+1) 3 0 Ae 2 (n-1)

22 U d 2 B Ae 2 Ae 2 (n-1) (n-1) n 18B Ae 2 0 : detemined by x-ay diffaction A: Madelung numbe, known B 0 : bulk modulus, can be measued To get n To get B B n A 0 n-1 22

23 3.4 Anothe type of bonding: Van de Waals foces (cystals) Electically neutal and possess electon configuation chaacteistic of inet gases. Van de Waals-London Inteaction: small and shot ange Atoms: inet gases: He, Ne, A, Molecules: H 2, N 2, O 2, Cl 2, CH 4 N 2 (s) 63K N 2 (l) 77K N 2 (g) The van de Waal bonds occu to some extent in all mateials but ae paticulaly impotant in plastics and polymes. These mateials ae made up of a long sting molecules consisting of cabon atoms covalently bonded with H, N, O, o F. 23

24 3.6 Inet Gases Inet gas (Ne, A ) is the pototype fo the van de Waals cystal; closed shell; no aveage dipole moment; only have instantaneous dipole moment (induced dipoles). They cystallize at low temp. in the FCC system. 24

25 3.12 Molecula Cystal Many molecules fom cystals which ae held togethe by van de Waals foces. 25

26 1. Non-pola molecules N 2, H 2, CH 4, ae typical covalent molecules in which the atoms shae valence electons to effectively obtain closed shells fo each atom in the molecule. non-pola molecules, no pemanent dipole moments. 2. Pola molecules pola molecules (like H 2 O) possess pemanent dipoles > much stonge binding (van de Waals binding) in thei espective cystals 26

27 - -e 3.5 Dipoles A pai of oppositely chaged paticles (+e & -e) sepaated by a small distance a. a Electic potential at the point p Θ + +e l 1 +(a/2)cosθ V e l 2 -(a/2)cosθ p k e e e - e V - ( a / 2 ) cos ( a / 2 ) cos ( a / 2)cos 1 ( a / 2)cos V V e ( a / )cos 1- [( a / 2)cos ] 2 Fo >> a, (a/2)cos <<1 V ea cos 2 27

28 F F The electic field in the adial and tansvese diection: E The foce which acts on a chage e at point p is ee - V 2e cos ; F 3 2ea cos ; E 3 μea: dipole moment. ee - V e sin 3 ea sin 3 ea cos 2 Note: the E due to a dipole vaies as the invese 3, V invese 2 the E ( ke/ 2 ) due to a single chage vaies as 2, V invese Using k 9x10 9 Nm 2 /coul 2 to get mks units 2ke cos ke sin kee ; F kee V cgs units k 1 dyne(cm 2 )/statcoul 2

29 3.7 Induced dipoles When an atom is placed in an extenal electical field, its electons displaced fom thei nomal positions elative to the nucleus. > induced dipole This chage edistibution may be consideed equivalent to the fomation of a dipole inside the atom. 29

30 (A) A μ I α E α: polaizability (A) B induced dipole moment μea Due to the movement of the electons aound the nucleus. This dipole moment at A will poduce E at B. Induced dipoles e and +e. Let E be the field intensity due to A at e, The coesponding field at +e: E 2 E+ (de/d)a 30

31 The total foce on the induced dipole due to the field of the othe dipole f f -ee e E μ I αe de d a ea de d I de d Dipole moment de d ae ; f a ( ) a 3 3 d d The enegy of a pai of inet-gas atoms due to dipole inteaction is now 2 2 a d a 7 6 van de Waals between a pai of inet gas atoms due to the dipole inteactions: invese sixth powe of thei distance

32 E f Dipole - V 2 cos 3 ee 2e cos 3 V cos 2 Dipole-dipole a 2 7 a a d 7 2 a 6 32

33 3.8 the Lattice Enegy of an Inet-gas Solid Cohesive enegy of an inet-gas solid A B Repulsive foce U n A, B, and n: constants Dipole dipole inteation (attaction) Ionic NaCl: U -Az 2 e 2 / + Be 2 / n It was shown, ove 50 yeas ago, that if n 12, coelating well with the obseved popeties of ae-gas solid. Attactive tem: total enegy fo one mole of the cystal caused by the dipole-dipole inteactions between all the atoms of the solid. 33

34 3.11 Dipole-Quadupole and Quadupole-Quadupole Tems Dipole-Quadupole and Quadupole-Quadupole tems: Synchonization of the motion of the electons on the vaious atoms of a solid > van de Waals attactive enegy (lowe enegy). Moden QM > van de Waals attactive enegy of an ion ψ()-(c 1 / 6 +c 2 / 8 +c 3 / 10 ) ~83%, ~16%, <1.3% Dipole-Quadupole inteaction Double dipoles: fou chages Quadupole-Quadupole inteaction 34

35 An elementay quadupole can be epesented as two dipoles oiented antipaallel. A double dipole consisting of fou chages. 35

36 Lattice vibations and themal popeties Heat capacities (J/K) Examining how enegy is taken up by the vibations of atoms. Classical theoy of lattice heat capacity Dulong and Petit s Law Mean enegy of each atom oscillating in 1-D in a solid would be kt. U 3N A kt 3 RT U C V V 3R (J/molK) T independent of temp. Fails when small masses, small moments of inetia and small tansfes of enegy ae involved: quantum effect. 36

37 37 Einstein model (1907) The discepancies between classical theoy and expeiment in the specific heat of solids at low temp. wee esolved by Albet Einstein. The Einstein solid is a model of a solid based on two assumptions: 1. Each atom in the lattice is an independent 3D quantum hamonic oscillato. 2. All atoms oscillate with the same fequency (contast with the Debye model). He applied quantum theoy fo a hamonic oscillato. The assumption that a solid has independent oscillations is accuate. In Einstein's model, the heat capacity appoaches zeo exponentially fast at low tempeatues. This is because all the oscillations have one common fequency. not accuate At high temp. C v ~ 3Nk B

38 Debye model (1912) The coect behavio is found by quantizing the nomal modes of the solid. Then the fequencies of the waves ae not all the same, and the heat capacity goes to zeo as a powe law, which matches expeiment. This modification is called the Debye Model, which appeaed in In themodynamics and solid state physics, the Debye model is a method developed by Pete Debye in 1912 fo estimating the phonon contibution to the specific heat (heat capacity) in a solid. It teats the vibations of the atomic lattice (heat) as phonons in a box (diffeent fequency). The Debye model coectly pedicts the low tempeatue dependence of the heat capacity, which is popotional to the Debye T 3 law. At high temp. Cv ~ 3Nk B 38

39 But due to simplifying assumptions, its accuacy suffes at intemediate temp. (lowe) 39

40 3.9 The Debye Fequency: The zeo point enegy of a cystal is its themal (vibating) enegy when the atoms ae vibating in thei lowest enegy state (at 0K). In a cystal solid (3-D), thee vibational degees (tansvese (othogonal) of feedom pe atom > N atoms > 3N oscillatos. 3D 40

41 1D 41

42 The vibations of these atoms have constains: thee is no eal physical tanslation of the objects. > Bounday condition > Standing waves 2a Shotest wavelength (maximum fequency) allowed (neighboing atoms a vibate against each othe): min 2a > m v/ min ; v: sound velocity; v ~ 5x10 3 m/s ; assume a 0.25 nm > m Hz; Simple calculations to epesent the v of an atom in a cystal m fo fou masses IR (10 12 ~10 14 Hz): vibation fequency of an atom in a cystal. 42

43 The speed of sound Ai wate Ion m/s 1,484 m/s 5,120 m/s 1200 km/h 43

44 3.10 The Zeo point Enegy Density of states (DOS) In solid state and condensed mattes physics, the density of states (DOS) of a system descibes the numbe of states pe inteval of enegy at each enegy level that ae available to be occupied by electons. f DOS f d Total numbe of states Fo lattice vibation Enegy fequency Electonic levels vibational modes 44

45 m f ( ) d 0 3N f() Aea 3N m Fequency spectum of a cystal accoding to Debye. 45

46 # of modes f(k) dk 4 k 2 dk (volume) 2 ( ) 3 L L 3 ( 2 ) 4 3 k2 dk (single mode volume) V ( 2 ) 4 3 k2 dk In k space V: volume 46

47 Fo phonons # of modes f(k) dk V ( 2 ) 4 3 k2 dk Fo the density of state pe unit fequency ange E h h v hvk vk k f() d V ( 2 ) 4 3 Total # of states (modes) ( ) 2 v d ( ) v v v V ( 2 ) 3 v 3 d d dk 47

48 m f d 3N m V (2 ) v 2 d V ( 2 ) v 3 3 m 3 V ( 2 ) v 3 3 m 9N [ ] f() d f() V ( 2 ) v 3 2 9N m 3 [ ] d 2 d 9N m 3 2 E Z 0 m f h d 2 48

49 The enegy levels of the quantum hamonic oscillato ae Zeo-point enegy (1/2)h (1/2)(h/2)(2) (h/2) 49

50 Classical theoy: Because the classical gound state completely specifies both the paticle's speed (zeo) and position (at the minimum). The enegy is zeo. Quantum theoy: It violates the famous Heisenbeg Uncetainty Pinciple. Quantum physics, via the Uncetainty Pinciple, foces the paticle to spead out both in position and velocity and so causes it to have an enegy somewhat highe than the classical minimum. The ZPE is defined as this shift: E(ZPE) E(quantum minimum) - E(classical minimum) > 0 Scientific Ameican Aug 18,

51 In 1-D, the density of vibational modes (density of states) is the same in any fequency inteval d; in 3-D, the density of vibational mode in the ange of to + d. The total vibational enegy of the cystal at 0K E Z 0 m 9N h 9hN 1 9hN ( 0 m f h d v )( ) d v m m 2 2 m 4 8 m f() 9 8 Nhv m m f ( ) d 0 m 3N Aea 3N 51

52 3.13 Refinements to the Bon Theoy of Ionic Cystals U M + R + van de Waals + zeo-point enegy ionic cystals also have some potions of van de Waals. Ae Be - 2 Ce Ee - 2 De U { ( ) }+ ( Nh n m) 8 Quadupole-Quadupole

53 3.14 Covalent and Metallic Bonding: ionic and inet-gas cystals: electons ae consideed to be tightly bound to thei espective atoms. easie to intepet in tems of the law of classical physics. Moe quantum mechanics is not necessay unless geat accuacy is needed. 53

54 e.g. diamond; C: 1S 2, 2S 2, 2P 2 > 4 hybid (valence e - s) bonds, to shae electon with the 4 neaest neighbo > achieve the electon configuation of 1S 2, 2S 2, 2P 6. These valence electons belong to the cystal as a whole Cengage Leaning, Engineeing. All Rights Reseved

55 The simplest covalent bond system is Hydogen molecule. (see. Fig.3.12) (1) Two hydogen atoms with the same spin ae bought togethe 55

56 (2) Two hydogen atoms with the opposite spin (Pauli exclusion pinciple) ae bought togethe a. each nucleus can accommodate two electons > +,- chage pai (moe time in the neighbohood of one nuclei, exteme case) 56

57 b. The electon pai could move back and foth fom one nuclei to anothe > esonance effect at a vey fast ate > ~80% of the total binding enegy Fom QM > electons spend moe of thei time is the egion between the two potons. c. Thee ae othe moe complicated inteactions: emaining 15%. 57

58 Covalent cystals: the electons ae shaed between neighboing atoms (not fee) and fom diected bonds. Covalent cystals fom complicated stuctues to have a closed-shell configuation. ¼ ¼ ¼ associates with each point of FCC Tetahedal angles:

59 Metallic cystals: Lattice of positively chaged nuclei in a nea fee electon sea. Valence electons move in all diections. Unifom distibution can be thought of electon gas. Valence electons ae also shaed between atoms in metallic cystal. > metallic cystal tends to be closepacked (most close-packed lattice (fcc & hcp)), in which the diectionality of the bonds between atoms is of seconday impotance. 59