Chapter 3 Crystal Binding. Phys 175A Dr. Ray Kwok SJSU

Transcription

1 Chapter 3 Crystal Binding Phys 175A Dr. Ray Kwok SJSU

2 4 basis categories

3 Molecular Bonds Introduction To understand the crystal binding, one should understand how molecules bind together The bonding mechanisms in a molecule are fundamentally due to electric forces The forces are related to a potential energy function A stable molecule would be expected at a configuration for which the potential energy function has its minimum value

4 Features of Molecular Bonds The force between atoms is repulsive at very small separation distances This repulsion is partially electrostatic and partially due to the exclusion principle Due to the exclusion principle, some electrons in overlapping shells are forced into higher energy states The energy of the system increases as if a repulsive force existed between the atoms The force between the atoms is attractive at larger distances (e.g. due to shifted charge distribution, induced dipole-dipole interaction)

5 Potential Energy Function The potential energy for a system of two atoms can be expressed in the form A B U( r ) = + n m r r r is the internuclear separation distance m and n are small integers (usually) A is associated with the attractive force B is associated with the repulsive force

6 Graph U(x) At large separations, the slope of the curve is positive Corresponds to a net attractive force (F = du/dr) At the equilibrium separation distance, the attractive and repulsive forces just balance At this point the potential energy is a minimum The slope is zero (F=0)

7 Molecular Bonds Types Simplified models of molecular bonding include Ionic Covalent van der Waals Hydrogen

8 Ionic Bonding Ionic bonding occurs when two atoms combine in such a way that one or more outer electrons are transferred from one atom to the other Ionic bonds are fundamentally caused by the Coulomb attraction between oppositely charged ions (e.g. Na + Cl )

9 Ionic Bonding, cont. The energy required to remove an electron from an atom is called Ionization Energy. (Na + ) The amount of energy gained by adding an electron (from far away E=0) to an atom (E<0) is called the Electron Affinity of the atom. (Cl ) The Binding Energy (or Dissociation Energy) is the amount of energy needed to break the molecular bonds and produce neutral atoms

10 Ionic Bonding, NaCl Example The graph shows the total energy of the molecule vs the internuclear distance The minimum energy is at the equilibrium separation distance Binding energy = 4.2 ev

11 Ionic Bonding,final The energy of the molecule is lower than the energy of the system of two neutral atoms It is said that it is energetically favorable for the molecule to form The system of two atoms can reduce its energy by transferring energy out of the system and forming a molecule

12 Covalent Bonding A covalent bond between two atoms is one in which electrons supplied by either one or both atoms are shared by the two atoms Covalent bonds can be described in terms of atomic wave functions The example will be two hydrogen atoms forming H 2 (bonding energy 4.48 ev)

13 Covalent bonds Bonding can occur without outright removal or addition of an electron. In these types of bonds, the connection occurs through orbital overlap.

14 Wave Function Two Atoms Far Apart Each atom has a wave function (1s 1 ) 1 r ao ψ1 s( r ) = e πa There is little overlap between the wave functions of the two atoms when they are far away from each other 3 o

15 Wave Function Molecule The two atoms are brought close together The wave functions overlap and form the compound wave shown The probability amplitude is larger between the atoms than on either side

16 Covalent Bonding, Final The probability is higher that the electrons associated with the atoms will be located between them This can be modeled as if there were a fixed negative charge between the atoms, exerting attractive Coulomb forces on both nuclei The result is an overall attractive force between the atoms, resulting in the covalent bond

17 Van der Waals Bonding Two neutral molecules are attracted to each other by weak electrostatic forces called van der Waals forces (typically 0.1 ev) Atoms that do not form ionic or covalent bonds are also attracted to each other by van der Waals forces The van der Waals force is due to the fact that the molecule has a charge distribution with positive and negative centers at different positions in the molecule

18 Van der Waals Bonding, cont. As a result of this charge distribution, the molecule may act as an electric dipole Because of the dipole electric fields, two molecules can interact such that there is an attractive force between them Remember, this occurs even though the molecules are electrically neutral e.g. Liquid nitrogen molecules N 2

19 Types of Van der Waals Forces Dipole-dipole force An interaction between two molecules each having a permanent electric dipole moment Dipole-induced dipole force A polar molecule having a permanent dipole moment induces a dipole moment in a nonpolar molecule

20 Types of Van der Waals Forces, cont. Dispersion force An attractive force occurs between two nonpolar molecules The interaction results from the fact that, although the average dipole moment of a nonpolar molecule is zero, the average of the square of the dipole moment is nonzero because of charge fluctuations The two nonpolar molecules tend to have dipole moments that are correlated in time so as to produce van der Waals forces

21 Hydrogen Bonding In addition to covalent bonds, a hydrogen atom in a molecule can also form a hydrogen bond (weak 0.5 ev) Using water (H 2 O) as an example There are two covalent bonds in the molecule The electrons from the hydrogen atoms are more likely to be found near the oxygen atom than the hydrogen atoms

22 Hydrogen Bonding H 2 O cont. This leaves essentially bare protons at the positions of the hydrogen atoms The negative end of another molecule can come very close to the proton This bond is strong enough to form a solid crystalline structure

23 Hydrogen Bonding, Final The hydrogen bond is relatively weak compared with other electrical bonds Hydrogen bonding is a critical mechanism for the linking of biological molecules and polymers DNA is an example

24 Bonding in Solids Bonds in solids can be of the following types Ionic Covalent Metallic

25 Ionic Bonds in Solids The dominant interaction between ions is through the Coulomb force Many crystals are formed by ionic bonding (I-VII, II-VI) e.g. NaCl Ions are closed electronic shells. [e.g. LiF, Li (1s 2 2s) becomes Li + (1s 2 ), F (1s 2 2s 2 2p 5 ) becomes F (1s 2 2s 2 2p 6 )]

26 Electrostatic Energy The net effect of all the interactions is a negative electric potential energy U attractive = e αke r 2 α is a dimensionless number known as the Madelung constant The value of α depends only on the crystalline structure of the solid

27 Total Energy in a Crystalline Solid As the constituent ions of a crystal are brought close together, a repulsive force exists The potential energy term B/r m accounts for this repulsive force This repulsive force is a result of electrostatic forces and the exclusion principle

28 Total Energy in a Crystalline Solid, cont The total potential energy of the crystal is U 2 e = α k + r B r t o t a l e m The minimum value, U o, is called the ionic cohesive energy of the solid It represents the energy needed to separate the solid into a collection of isolated positive and negative ions

29 Ionic Bonds, NaCl Example The crystalline structure is shown (a) Each positive sodium ion is surrounded by six negative chlorine ions (b) Each chlorine ion is surrounded by six sodium ions (c)

30 Na-Cl Na ev Na + + e - (ionization energy = 5.14 ev) Cl + e - Cl ev (electron affinity = 3.61 ev) Na + + Cl NaCl ev (cohesive energy = 7.9 ev) i.e. the energy per molecule of NaCl is ( ) = 6.4 ev lower than the energy of separated neutral atoms.

31 More properties of Ionic Crystals They form relatively stable, hard crystals They are poor electrical conductors They contain no free electrons (filled shells) Each electron is bound tightly to one of the ions They have high melting points

32 Properties of Solids with Covalent Bonds Properties include Usually very hard Due to the large atomic cohesive energies High bond energies High melting points Good electrical conductors

33 More about Covalent Bonds Share electrons (usually 2 e - ) Directional (along orbital that share e - ) Electrons tend to localize between atoms Prefer anti-parallel spins (Pauli) Act as glue to atoms No clear cut range to be ionic or covalent

34 Cohesive Energies for Some Covalent Solids

35 Covalent Bond Example Diamond Each carbon atom in a diamond crystal is covalently bonded to four other carbon atoms This forms a tetrahedral structure

36 Metallic Solids Metallic bonds are generally weaker than ionic or covalent bonds The outer electrons in the atoms of a metal are relatively free to move through the material (high mobility) The number of such mobile electrons in a metal is large (high carrier density) High electrical conductivity (& thermal)

37 Metallic Solids, cont. The metallic structure can be viewed as a sea or gas of nearly free electrons surrounding a lattice of positive ions The bonding mechanism is the attractive force between the entire collection of positive ions and the electron gas

38 Properties of Metallic Solids Light interacts strongly with the free electrons in metals Visible light is absorbed and re-emitted quite close to the surface (reflective) This accounts for the shiny nature of metal surfaces (screening, plasma frequency) High electrical conductivity More discussions on metal later.