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1 lectures accompanying the book: Solid State Physics: An Introduction, by Philip ofmann (2nd edition 2015, ISBN-10: , ISBN-13: , Wiley-VC Berlin. 1

2 Bonds between atoms: contents at the end of this lecture you should understand... bonding in general, attractive and repulsive forces, cohesive energy ionic bonding covalent bonding metallic bonding hydrogen bonding and van der Waals bonding relationship between bonding type and some physical properties of a solid (in particular melting point) 2

3 Bonding in solids: the general idea valence electrons (of the outer shell) achieve bonding (like in chemistry) decrease in total energy stabilises the solid (the solid s energy is lower than that of sum of atoms it is made of) so the energy gain by the bonding must be higher than the energy it costs to promote electrons from the atomic orbitals to the electronic states of the solid. this energy difference is a measure for the strength of the bond. It is called the cohesive energy. cohesive energy = energy of atoms - energy of solid 3

4 Repulsive force interatomic potential (arb. units) 0 attractive part interatomic distance / equilibrium distance (r) = B r m + A r n 4

5 Repulsive force interatomic potential (arb. units) 0 repulsive part attractive part interatomic distance / equilibrium distance (r) = B r m + A r n n>m 4

6 Repulsive force interatomic potential (arb. units) 0 0 resulting potential 1 repulsive part attractive part interatomic distance / equilibrium distance 4 6 (r) = B r m + A r n n>m 4

7 Repulsive force interatomic potential (arb. units) 0 resulting potential repulsive part interatomic force (arb. units) 0 repulsive attractive attractive part interatomic distance / equilibrium distance interatomic distance r (equilibrium distance) (r) = B r m + A r n F(r) = F (r) = grad (r) d dr (r) n>m 4

8 Repulsive force interatomic potential (arb. units) 0 resulting potential repulsive part interatomic force (arb. units) 0 repulsive attractive attractive part interatomic distance / equilibrium distance interatomic distance r (equilibrium distance) (r) = B r m + A r n F(r) = F (r) = grad (r) d dr (r) n>m 4

9 Ionic bonding form positive and negative ions (here Na+ and Cl-) bonding is achieved by electrostatic force and a classical treatment is (partially) meaningful. example NaCl (rock salt): cubic structure Na+ Cl- a 5

10 Turning Atoms in Ions example: NaCl how much energy does it cost? 6

11 Turning Atoms in Ions example: NaCl how much energy does it cost? ionization energy Na: 5.1 ev 6

12 Turning Atoms in Ions example: NaCl how much energy does it cost? ionization energy Na: 5.1 ev electron affinity Cl: 3.6 ev 6

13 Turning Atoms in Ions example: NaCl how much energy does it cost? ionization energy Na: 5.1 ev electron affinity Cl: 3.6 ev net energy cost: (5.1 ev ev) = 1.5 ev per pair 6

14 Ionic bonding example: NaCl what is the energy gain? Na+ Cl - a = 2.8 Å 7

15 Ionic bonding example: NaCl what is the energy gain? potential energy: E Coulomb = e 2 Na+ 4 0 a Cl - this amounts to 5.1 ev per pair a = 2.8 Å 7

16 Ionic bonding example: NaCl what is the energy gain? potential energy: E Coulomb = e 2 Na+ 4 0 a Cl - this amounts to 5.1 ev per pair a = 2.8 Å so the total gain is 5.1 ev ev = 3.6 ev 7

17 Ionic bonding example: NaCl but this was just a molecule: what about the electrostatic energy gain in the solid? consider the centre Na ion energy gain from next 6 Cl: e2 Na+ E = 6 4πϵ 0 a Cl - a 8

18 Ionic bonding example: NaCl but this was just a molecule: what about the electrostatic energy gain in the solid? consider the centre Na ion energy gain from next 6 Cl: e2 E = 6 4πϵ 0 a energy loss from next 12 Na: e 2 E = +12 4πϵ 0 a 2 Na+ Cl - a 8

19 Ionic bonding example: NaCl but this was just a molecule: what about the electrostatic energy gain in the solid? consider the centre Na ion energy gain from next 6 Cl: e2 E = 6 4πϵ 0 a energy loss from next 12 Na: e 2 E = +12 4πϵ 0 a 2 next we get 8 more Cl ions and the total becomes E = e2 4πϵ 0 a ( ) 8 Na+ Cl - a

20 Ionic bonding example: NaCl eventually the series converges and we get (for one ion) E = e2 4 0 a = M d e a Na+ Cl - M d is called the Madelung constant. It is specific for a given structure. a 9

21 E = e2 4 0 a = Ionic bonding M d e a so the total lattice energy for one mole of NaCl because there are Na and Cl ions E tot = 2E Na 2 N a Na + Cl - to count every pair only once This gives 861 kjmol -1. The experiment gives 776 kjmol -1 Note: this is the lattice energy, not the cohesive energy (the lattice energy minus the energy to turn atoms into ions). a

22 Repulsive force interatomic potential (arb. units) 0 resulting potential repulsive part 10% attractive part interatomic distance / equilibrium distance The existence of a (non-classical) repulsive force causes the real cohesive energy to be slightly smaller (10%) than the value calculated by the Coulomb potential. 11

23 Covalent bonding A detailed view on the hydrogen molecule, identical particles (later in connection with magnetism, see online note on A simple view on other covalent bonds. 12

24 The covalent bond: simple picture Cl C The covalent bond: less simple picture (hydrogen molecule) = h2 2 1 h m 2m + e2 { 1 4πϵ 0 R A r 1 1 R B r R + 1 r 1 r 2 1 R A r 2 1 R B r 1 (1) }, Ψ(r 1, r 2 ) = 1 2 (φ A (r 1 )φ B (r 2 ) ± φ A (r 2 )φ B (r 1 )) 13 (see note on

25 The covalent bond: simple picture Cl C 14

26 The covalent bond: simple picture methane hydrocarbons C-based solid 15 C C C C C C C C C C C C C C C C C C

27 The covalent bond: a closer look at C electron configuration: 2 s and 2 p electrons formation of four sp 3 hybrid orbitals as linear combination between the s and three p orbitals directional character of p orbitals is also found in sp 3 orbitals. 16

28 methane The covalent bond: sp 3 bonding diamond 17

29 Bonding in most semiconductors Tetrahedral (sp 3 ) configuration almost ubiquitous: diamond, Si, Ge, III-V (GaAs, AlAs, InP), II-VI (CdS, CdTe) 18

30 Bonding in most semiconductors Tetrahedral (sp 3 ) configuration almost ubiquitous: diamond, Si, Ge, III-V (GaAs, AlAs, InP), II-VI (CdS, CdTe) 19

31 The covalent bond: sp 2 bonding formation of three sp 2 hybrid orbitals as linear combination between the s and two p orbitals. One p-orbital remains 20

32 The covalent bond: sp 2 bonding graphene / graphite bucky-balls carbon nanotubes (rolled-up graphene) 21

33 Covalent bonding Cohesive energies similar to ionic bonding, in the ev range. Very directional bonding. 22

34 Metallic bonding 23

35 metals / non-metals Li Be e B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te l Xe Cs Ba La f Ta W Re Os Ir Pt Au g Tl Pb Bi Po At Rn Fr Ra Ac Rf Db Sg Bh s Mt Ds Rg Uub Ce Pr Nd Pm Sm Eu Gd Tb Dy o Er Tm Yb Lu Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr the boundaries can be disputed simple metals, transition metals, noble metals 24

36 metals / non-metals Li Be e B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te l Xe Cs Ba La f Ta W Re Os Ir Pt Au g Tl Pb Bi Po At Rn Fr Ra Ac Rf Db Sg Bh s Mt Ds Rg Uub Ce Pr Nd Pm Sm Eu Gd Tb Dy o Er Tm Yb Lu Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr the boundaries can be disputed simple metals, transition metals, noble metals 24

37 Metallic bonding (simple metals) outer electrons are delocalized and act as glue between positively charged ion cores generally found for elements with one, two or three valence electrons. cohesive energies in the ev range Na + Na + 25

38 Metallic bonding (simple metals) outer electrons are delocalized and act as glue between positively charged ion cores generally found for elements with one, two or three valence electrons. cohesive energies in the ev range Na + Na + Na + Na + Na + Na + Na + Na + Na + 25

39 Metallic bonding (simple metals): more characteristics smaller cohesive energies than in ionic crystals larger ionic radii, e.g. for Na: 3.82 Å (metal) and 1.94 Å (NaCl) bonding has no directional preference closed-packed atomic configurations are preferred: best possible overlap between the orbitals, no holes in the potential 26

40 Metallic bonding: why is this so favorable? h 2 2 2m (r) + U(r) (r) = E (r) kinetic energy (or amiltonian for a free particle) (negative) average curvature of wave function flatter wave function -> lower energy 27

41 Metallic bonding: why is this so favorable? h 2 2 2m (r) + U(r) (r) = E (r) kinetic energy (or amiltonian for a free particle) (negative) average curvature of wave function flatter wave function -> lower energy x p h/2 less localization -> smaller p variation 27

42 Metallic bonding: why is this so favorable? h 2 2 2m (r) + U(r) (r) = E (r) image source: wikipedia 28

43 Transition metals 4s and 3d have very similar energies 4s electrons form delocalized metallic bonds 3d electrons form more local (covalent-like) bonds higher cohesive energies Li Be e B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te l Xe Cs Ba La f Ta W Re Os Ir Pt Au g Tl Pb Bi Po At Rn Fr Ra Ac Rf Db Sg Bh s Mt Ds Rg Uub Ce Pr Nd Pm Sm Eu Gd Tb Dy o Er Tm Yb Lu Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 29

44 Bonds between molecules molecular solids are very common (but not at RT) ice plastic DNA what makes molecules bond to each other? 30

45 Bonds between molecules: hydrogen bonds O permanent dipole 31

46 Bonds between molecules: hydrogen bonds is positively charged but also very small: another real bond cannot be established without overlap of electron clouds (in this sense it is too big in this drawing). bonding is important in ice, DNA... but not very strong 32

47 Bonds between molecules: van der Waals force E dipole moment caused by fluctuations this is always present as an attractive force (even between e atoms as in this case) it is very weak and depends on the distance as r -6 33

48 Bonds between molecules: van der Waals force E p=αe dipole moment caused by fluctuations this is always present as an attractive force (even between e atoms as in this case) it is very weak and depends on the distance as r -6 33

49 Bonds between molecules: van der Waals force E p=αe attract dipole moment caused by fluctuations this is always present as an attractive force (even between e atoms as in this case) it is very weak and depends on the distance as r -6 33

50 Bonds between molecules: van der Waals force U = E p = E E / 1 r 3 1 r 3 E p=αe attract dipole moment caused by fluctuations this is always present as an attractive force (even between e atoms as in this case) it is very weak and depends on the distance as r -6 33

51 Bond type and physical properties ow does the bond type affect the properties of a solid such as: mechanical strength / melting point electrical conductivity thermal conductivity optical properties 34

52 Bond type and physical properties melting temperature (K) Ne Ar Kr Xe RbK g Ta W Mo Cr B Ti V Pt Fe Be Si Ca Ag Au La Mg Sb Al Zn Te Cd Pb Po Li Se In Sn Na S Ga Br cohesive energy per atom (ev) Cl 35

53 Summary We have looked at different types of bonding: ionic, metallic, covalent, -bonds, van der Waals bonds. In reality, intermediate bonding scenarios are often found. We have some ideas about the relation between between the bonding type and the physical properties (at least for the melting point). 36

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