The Structure of Matter

1. The Structure of Matter

2. §7-1 Introduction • Molecules and bulk matter are aggregates(集合体) of atoms held together by electrical forces. • If the electrons and nuclei of a molecule have a lower energy than that of the separate neutral atoms,the molecule will be stable. • Similarly in metals and semiconductors, the energy of the aggregate is less than that of its isolated constituents.

3. The problem of understanding the structure of matter is one of finding the arrangement that has the lowest energy or greatest binding energy. • Fortunately ,it is not necessary to solve Schroedinger’s equation for all the electrons and nuclei of a system,because this can only be done approximately and is generally difficult. • Instead, we use simplified models based on our knowledge of atoms that are very successful in accounting for the structure and properties of molecules and bulk materials.

4. There are three models of primary interest to us here,which describe ionic,covalent(共价的), and metallic binding(结合),respectively . • Weaker bonds such as the van der Waals and hydrogen bonds are also important where the stronger bonding mechanisms are not effective.

5. §7-2 Ionic Binding Ionic binding • When some molecules and crystals are formed, one or more electrons from one atom are completely transferred to another atom.The atom losing the electrons acquires a net positive charge,and the atom accepting these electrons becomes negatively charged. These two ions are then held together by the electric forces between them.

6. This ionic binding usually involves one atom with one or more loosely bound electrons and a second with a nearly full outer shell. • For example, the alkali metals(碱金属),such as sodium(钠) and potassium(钾) with one valence electron,readily combine with the halides(卤族元素),such as chlorine(氯) and fluorine(氟),that have one vacancy in their outer shells. • This type of binding is responsible both for the formation of single molecules and of bulk crystals.

7. Ionic molecule KCl • Ionic binding is illustrated by the ionic molecule potassium chloride,KCl. In order to understand why it is energetically favorable for potassium and chlorine atoms to form KCl molecules, we break up the formation of the molecule into three stages. • The outer electron of potassium is removed, leaving the K+ ion and an electron. This requires an energy equal to the ionization energy of potassium which is 4.34eV.

8. The extra electron binds to the neutral cholrine atom yielding Cl－. This releases energy called the energy of formation or electron affinity(电子亲和).For chlorine this is 3.82eV.The entire process of transferring one electron from potassium to chorine then requires that (4.34－3.82)=0.52eV of energy be supplied. • The oppositely charged ions approach each other. Because of their mutual attraction,their potential energy initially decreases as they approach.However, when they are so close that their electron clouds begin to overlap,there is a

9. Repulsive force and the potential energy begins to increase(Fig.29.1). The observed equilibrium separation of the ions in KCl is r0 =2.79×10-10 m. The electrical potential energy at this separation is –5.16eV.

10. Ignoring other contributions to the potential energy momentarily, this means that the molecule has an energy of (0.52－5.16)eV=-4.64 eV compared to the separate neutral atoms. • Conversely,4.64eV of energy would be required to break up the molecule into neutral potassium and chlorine.The measured value is 4.40eV.The difference in the two results can be attributed to the repulsion of the inner electron clouds.

11. Ionic crystals • In ionic crystals the ions form a lattice(晶格). • A number of crystal structures are possible, but the common feature of all of them is that ions of one charge have closest neighbors that are ions of the opposite charge.The electric attraction between neighbors is responsible for the binding of the crystals.

12. §7-3 Covalent binding(共价结合) • The vast majority of molecules are formed by sharing the outermost or valence electrons of the constituent atoms.(组成的原子) • In such covalent binding, the distribution of electrons in the separated atoms. • These electronic distributions and other molecular properties are studied by interpreting detailed experimental results with the aid of theoretical models. • These models rely on two important concepts described in the preceding chapter.

13. The square of a wave function |ψ|2determines the relative probability of an electron being at a given place. When two atoms share an electron, it effectively belongs to each atom part of the time. Thus we expect that in molecules the wave functions of the electronic states should overlap so that the shared electron is favorably positioned with respect to both atoms. • When the atoms combine,the Pauli principle requires that each spatial state contains at most two electrons, one with spin up and one with spin down.

14. Hydrogen molecule • We can illustrate these ideas with the diatomic hydrogen molecule, H2. • Neutral hydrogen atoms have one electron in the 1s shell,although this state could hold two electrons if their spins were opposite (Fig.29.2a).

15. When two such atoms move together,one of two things will occur. If the electron spins are parallel,the Pauli principle prevents the electrons from being very close together since they cannot be in the same spatial state.The electrons repel each other and their wave functions are distorted (Fig. 29.2b)

16. On the other hand, if the spins are opposite each electron “fits” into the 1s state of the other atom. The electrons can now be close to each other,and the molecular wave function looks like the sum of the two 1s wave functions of each atom (Fig. 29.2c).

17. The electrons have a high probability of being between the two nuclei where they are attracted by both protons.This reduces the total energy of the molecule below that of the separated atoms and results in a stable molecule

18. The characteristic elongated wave function of Fig .29.2c is called a σ orbital.This bond is said to be a σ bond.

19. Hybridization(杂化) • An approximate approach to the structure of complex covalent molecules that has proved very useful is called hybridization. • This method employs wave functions that are combinations or hybrids of atomic states, such as the 3s and 3p states of the atoms, that have slightly different energies. • These energy differences can usually be ignored in molecules because they are small compared to the molecular binding energy.

20. Magnesium fluorine, MgF2 (氟化镁) • To illustrate the method of hybridization we describe the scheme(构架) for magnesium fluoride, MgF2. • Magnesium (Z=12) has closed n=1 and n=2 shells and two 3s valence electrons; its six 3p levels are empty. • Fluorine (Z=9) is one electron short of having a filled n=2 shell and hence attracts electrons. • Since the outer 3s wave functions of magnesium are spherically symmetric,their overlap with the 2p wave functions of fluorine is small (Fig.29.3a)

21. However, using combinations of magnesium wave functions, the overlap can be greatly increased, resulting in stronger binding (Fig.29.3b,c)

22. We know that the three p wave functions have pairs of lobes(耳垂) oriented along coordinate axes. If a p wave function and an s wave function are added and subtracted, the result is a pair of sp hybridized wave functions, or orbitals, each with a large lobe in one direction and a small lobe opposite (Fig.29.3b).

23. The two magnesium valence electrons can be thought of as being in these states, which overlap well with the single vacant p state of a fluorine atom.Consequently MgF2 is a linear molecule with the magnesium atom between two fluorine atoms. • Note that the sp orbital is a mixture of two atomic orbitals with slightly different energies, so a free magnesium atom would not ordinarily be found in such a state.However, the energy needed to occupy this sp orbital is much smaller than that gained in the formation of the molecule, so in the molecule it is energetically favorable for the electrons to be in this configuration.

24. sp3 hybridization : the water molecule H2O • In sp3 hybridization, we consider combinations of one s state and all three p states. We illustrate this procedure for the water molecule H2O. • Oxygen (Z=8) has two 2s electrons and four 2p electrons in its n=2 shell.Thus it is two electrons short of having a closed n=2. Fig 29.4a shows the four sp3 orbitals that can be formed from suitable combinations of the s and p states of oxygen.These states have a maximum angular separation, 109.50. When occupied by electrons, this will minimize their repulsion.

25. Each sp3orbital can accommodate two electrons with opposite spins.In the water molecule, four of the n=2 oxygen electrons fill two of the sp3 orbits. Each of the other two orbits has one electron from the oxygen and one from a hydrogen atom (Fig.29.4b and 29.4c)

26. A reminder(提醒) that the hybridization model for water is not exact is provided by the experimental bond angle, 104.50,which is slightly less than the predicted angle of 109.50. The difference is due to the effects of electrical forces among the bond electrons and the hydrogen nuclei in the asymmetric(不对称的) water molecule.

27. Carbon • Carbon (Z=6), because it has only four out of a possible eight n=2 electrons, can participate in covalent bonds in a variety of ways. • When the s and p states are combined to form four sp3 orbits, carbon can, for example, bond with four hydrogen atoms forming methane CH4. (甲烷) • Each orbital has an electron from carbon and a second from a hydrogen atom. • The carbon wave functions can also be hybridized to form two sp orbits or three sp2 orbits. The sp2 orbits are used for molecules such as ethylene CH2=CH2(fig.29.6)

29. §7-4 The Metallic Bond The metallic bond • Covalent bonds occur in molecules containing two to several thousand atoms.By contrast, metallic binding holds billions of atoms together. • Usually these substances are solid, with ionized atoms forming a rigid lattice and electrons contributed by the atoms that are free to move

30. over the entire crystal. • The delocalized electrons are the charge and energy carriers in metals and are responsible for their large electrical and thermal conductivities.

31. Conduction band • To see how the electrons become delocalized, consider lithium(锂) atoms, which have one 2s electron outside a filled 1s shell. • When two lithium atoms are brought together, there are only two possible energy levels for the outer electrons. Their wave functions are similar to those for hydrogen, the parallel spin state being higher in energy than the antiparallel state. • Now, when a third atom is brought close to the first two, three separate energy levels become available for the outer electrons (Fig.29.7)

33. The energy levels are clustered near the original energy of the 2s electron in the free atom (Fig29.8)

34. Each time a new atom is added, a new energy level results. However, the net effect of each new atom becomes smaller as the number of atoms already close together increases. • Finally, with N atoms present, we have a band(带) of N closely spaced energy levels,which can accommodate 2N electrons. • The electronic wave functions are spread over large distances, and the electrons in this conduction conduction band are free to move over the entire crystal.

35. There are many unoccupied states in the conduction band into which electrons can move with only a small change in energy.That is the reason why metals are good electrical and thermal conductors. • When a metal is heated at one end, electrons in this region move into higher energy states and transport the heat energy away from this region. • Similarly, a small voltage difference along a metal will lead to an increase in the energy of electrons, which then move to regions of lower potential energy.

36. It is the presence of the closely spaced unfilled energy levels of the band that facilitates(是便利) these conduction processes.In a filled band in a semiconductor or insulator, electrons cannot easily change their states, and conduction is much reduced.

37. A simple model for the normal state of a metal • Consider a metal sample with about 1023 atoms. Suppose that each atom loses one election and settles as a positive ion into a regular position in the lattice structure. • The released electrons occupy states in the conduction band.They move freely in the metal and behave much like noninteracting particles in a box. • Since elections are fermions, each electronic energy state can accommodate a spin-up and a spin-down electron and the conduction

38. electrons tend to fill the lowest electronic energy states first. • In particular, at absolute zero (0 k) the lowest levels are filled up to a maximum energy called the Fermi energy EF(Fig.29.9a).

39. As the temperature is increased, thermal energy may be absorbed by electrons if there is an empty higher-energy state available. • A typical value of the Fermi energy is 5eV, whereas at room temperature thermal energy is 0.026eV,which is about a half percent of the Fermi energy. • This means that even at room temperature, only those electrons in states within about I percent of the Fermi energy can absorb thermal energy and jump into unoccupied states.It is these electrons that participate in electric and thermal phenomena.

40. Meanwhile,the diagram of occupied states is almost unchanged from that at 0 k (Fig.29.9b)

41. Apparently,the above model is just too simple. For example, the electrons are not really free; they interact via electrical forces with each other and with the vibrating ions.That is the reason why this model is not always successful in describing the properties of a metal, especially at low temperatures.

42. §7-5Insulators and semiconductors Energy bands for insulators and semiconductors • In metals the formation of the conduction band plays a vital role in the binding of atoms.In materials such as diamond,germanium(锗),and silicon(硅),bands are also formed when a large number of atoms are assembled. • However, in these cases the valence electrons form a band that is completely filled at zero temperature.

43. Additional bands at higher energies are formed from higher atomic orbitals, but at zero temperature these are completely empty. • There is an energy gap between the fully occupied valence band and the empty conduction band at a higher energy(Fig.29.12)

44. In diamond the energy gap is about 6eV,which is much longer than the thermal energy at room temperature.Thus, virtually(实际上) no electrons will have enough thermal energy to be excited from the filled band into the conduction band at any reasonable temperature. • The motion of electrons due to electrical forces or to temperature gradients requires that they increase their kinetic energy.Since the electrons in the filled band cannot change energy states, conduction is impossible and diamond is an excellent insulator.

45. In germanium (Eg=0.72eV), and in silicon (Eg=1.1eV), the energy gap is much smaller, and a few electrons succeed in being thermally excited from the valence to the conduction band. • Once in the conduction band they behave exactly as the conduction electrons in metals. However, the resistivity is higher in these semiconductors than in metals,because the number of electrons free to move is much smaller than in metals.

46. Semiconductors • A unique feature of semiconductors is the conduction by holes in the filled band. • When an electron is excited to the conduction band, to leaves an empty state or hole in the valence band with an effective positive charge. • If an electric field is applied, electrons from adjacent(邻近的) atoms jump into the hole and the hole moves. Thus, in an electric field, electrons in the conduction band move one way and holes in the valence band the other(Fig.29.13)

48. The conductivity of a semiconductor can be in creased by adding impurities to the sample.For example,if a few parts per million of arsenic are added to germanium, the conductivity increases a thousandfold. • Arsenic has one more electron in its outer shell than does germanium.When added as an impurity, it replaces a germanium atom, and its extra electron is freed for conduction. • Similarly, gallium(録) has one less outer electron than germanium, and when present as an impurity, it takes one electron from germanium, leaving a hole.

49. §7-6 Weaker bonds Introduction • The bonds we have discussed typically have energies of a few electron volts.However, two bonds with energies of at most one-tenth of an electron volt play an important role in nature;the van der Waals bond and the hydrogen bond.

50. These bonds are especially important when the stronger types of bonds are not present. • For example, the inert gases such as helium(氦), neon(氖), argon(氩), and krypton condense to liquids and solids only because of attractive van der Waals forces among their atoms. • Hydrogen bonding is often responsible for important structural features of molecules and solids such as the helical and pleated sheet structures of some organic molecules. • In some instances, both types of bonding are present simultaneously.