上海交通大学：《材料与文明》课程教学资源（参考资料）Understanding Mater_Chapter 3 - Mechanisms

3 Mechanisms 3.1.The Atomic Structure of Condensed Matter This chapter strives to explain the fundamental mechanical prop- erties of materials by relating them to the atomistic structure of solids and by discussing the interactions which atoms have with each other. Atoms!that make up condensed matter are often arranged in a three-dimensional periodic array,that is,in an ordered man- ner called a crystal.The periodic arrangement may differ from case to case,leading to different crystal structures.This fact has been known since the mid-nineteenth century through the work of A.Bravais(1811-1863),a French physicist at the Ecole Poly- technique.His concepts were confirmed (1912),particularly af- ter X-ray diffraction techniques were invented,by Laue and Bragg,and routinely utilized by Debye-Scherrer,Guinier,and others.(Periodic arrays of atoms act as three-dimensional grat- ings which cause X-rays to undergo interference and thus pro- duce highly symmetric diffraction patterns from which the peri- odic arrangement of atoms can be inferred.)Since about a decade ago,the arrangement of atoms also can be made visible utilizing high-resolution electron microscopy (or other analytical instru- ments).Specifically,a very thin foil of the material under inves- tigation is irradiated by electrons,yielding a two-dimensional projection pattern of the three-dimensional atomic arrangement, as seen in the lower part of Figure 3.1. Atomos (Greek)=indivisible.This term is based on a philosophical con- cept promulgated by Democritus (about 460-370 B.c.)postulating that matter is composed of small particles that cannot be further divided

3 This chapter strives to explain the fundamental mechanical prop￾erties of materials by relating them to the atomistic structure of solids and by discussing the interactions which atoms have with each other. Atoms1 that make up condensed matter are often arranged in a three-dimensional periodic array, that is, in an ordered man￾ner called a crystal. The periodic arrangement may differ from case to case, leading to different crystal structures. This fact has been known since the mid-nineteenth century through the work of A. Bravais (1811–1863), a French physicist at the École Poly￾technique. His concepts were confirmed (1912), particularly af￾ter X-ray diffraction techniques were invented, by Laue and Bragg, and routinely utilized by Debye–Scherrer, Guinier, and others. (Periodic arrays of atoms act as three-dimensional grat￾ings which cause X-rays to undergo interference and thus pro￾duce highly symmetric diffraction patterns from which the peri￾odic arrangement of atoms can be inferred.) Since about a decade ago, the arrangement of atoms also can be made visible utilizing high-resolution electron microscopy (or other analytical instru￾ments). Specifically, a very thin foil of the material under inves￾tigation is irradiated by electrons, yielding a two-dimensional projection pattern of the three-dimensional atomic arrangement, as seen in the lower part of Figure 3.1. Mechanisms 3.1 • The Atomic Structure of Condensed Matter 1Atomos (Greek)
indivisible. This term is based on a philosophical con￾cept promulgated by Democritus (about 460–370 B.C.) postulating that matter is composed of small particles that cannot be further divided

3.1.The Atomic Structure of Condensed Matter 25 FIGURE 3.1.High-reso- 0.o1m lution electron micro- graph of silicon in which a boundary be- tween a crystalline (lower part)and an amorphous region (upper part)can be observed.(Courtesy of S.W.Feng and A.A. Morrone,University of Florida.) A perfect crystal in which all atoms are equidistantly spaced from each other is,however,seldom found.Instead we know from indirect observations that some atoms in an otherwise or- dered structure are missing.These defects are called vacancies. Their number increases with increasing temperature and may reach a concentration of about 1 per 10,000 atoms close to the melting point.In other cases,some extra atoms are squeezed in between the regularly arranged atoms.They are termed self- interstitials (or sometimes interstitialcies)if they are of the same species as the host crystal.These defects are very rare because of the large distortion which they cause in the surrounding lat- tice.They may be introduced by intense plastic deformation or by irradiation effects,for example,in nuclear reactors.On the other hand,if relatively small atoms are crowded between regu- lar lattice sites which are of a different species(impurity elements such as carbon,hydrogen,nitrogen,etc.),the term interstitial is used instead.Moreover,parts of planes of atoms may be absent, the remaining part ending at a line.These line imperfections are referred to as dislocations.(Dislocations are involved when ma- terials are plastically deformed,as we will discuss in detail later on.) In still other solids,one detects an almost random arrange- ment of atoms,as observed in the upper part of Figure 3.1.This random distribution constitutes an amorphous structure. The mechanical properties of solids depend,to a large extent, on the arrangement of atoms,as just briefly described.Thus,we need to study in this chapter the microstructure and the crystal- lography of solids.The crystal structures,however,depend on

A perfect crystal in which all atoms are equidistantly spaced from each other is, however, seldom found. Instead we know from indirect observations that some atoms in an otherwise or￾dered structure are missing. These defects are called vacancies. Their number increases with increasing temperature and may reach a concentration of about 1 per 10,000 atoms close to the melting point. In other cases, some extra atoms are squeezed in between the regularly arranged atoms. They are termed self￾interstitials (or sometimes interstitialcies) if they are of the same species as the host crystal. These defects are very rare because of the large distortion which they cause in the surrounding lat￾tice. They may be introduced by intense plastic deformation or by irradiation effects, for example, in nuclear reactors. On the other hand, if relatively small atoms are crowded between regu￾lar lattice sites which are of a different species (impurity elements such as carbon, hydrogen, nitrogen, etc.), the term interstitial is used instead. Moreover, parts of planes of atoms may be absent, the remaining part ending at a line. These line imperfections are referred to as dislocations. (Dislocations are involved when ma￾terials are plastically deformed, as we will discuss in detail later on.) In still other solids, one detects an almost random arrange￾ment of atoms, as observed in the upper part of Figure 3.1. This random distribution constitutes an amorphous structure. The mechanical properties of solids depend, to a large extent, on the arrangement of atoms, as just briefly described. Thus, we need to study in this chapter the microstructure and the crystal￾lography of solids. The crystal structures, however, depend on 3.1 • The Atomic Structure of Condensed Matter 25 FIGURE 3.1. High-reso￾lution electron micro￾graph of silicon in which a boundary be￾tween a crystalline (lower part) and an amorphous region (upper part) can be observed. (Courtesy of S.W. Feng and A.A. Morrone, University of Florida.)

26 3·Mechanisms the type of interatomic forces that hold the atoms in their posi- tion.These interatomic bonds,which result from electrostatic in- teractions,may be extremely strong (such as for ionic or cova- lent bonds;see below).Materials whose atoms are held together in this way are mostly hard and brittle.In contrast,metallic and particularly van der Waals bonds are comparatively weak and are thus responsible,among others,for the ductility of materials.In brief,there is not one single physical mechanism that determines the mechanical properties of materials,but instead,a rather com- plex interplay between interatomic forces,atomic arrangements, and defects that causes the multiplicity of observations described in Chapter 2.In other words,the interactions between atoms play a major role in the explanation of mechanical properties.We shall endeavor to describe the mechanisms leading to plasticity or brittleness in the sections to come. In contrast to this,the electrical,optical,magnetic,and some thermal properties can be explained essentially by employing the electron theory of condensed matter.This will be the major theme of Part II of the present book,in which the electronic properties of materials will be discussed. 3.2.Binding Forces Between Atoms Ionic Bond Atoms consist of a positively charged nucleus and of negatively charged electrons which,expressed in simplified terms,orbit around this nucleus.Each orbit (or "shell")can accommodate only a maximum number of electrons,which is determined by quantum mechanics;see Appendix I.In brief,the most inner "K- shell"can accommodate only two electrons,called s-electrons. The next higher "L-shell"can accommodate a total of eight electrons,that is,two s-electrons and six p-electrons.The fol- lowing "M-shell"can host two s-electrons,six p-electrons,and ten d-electrons;and so on. Filled outermost s+p shells constitute a particularly stable (nonreactive)configuration,as demonstrated by the noble (in- ert)gases in Group VIII of the Periodic Table (see Appendix). Chemical compounds strive to reach this noble gas configuration for maximal stability.If,for example,an element of Group I of the Periodic Table,such as sodium,is reacted with an element of Group VII,such as chlorine,to form sodium chloride (NaCl), the sodium gives up its only electron in the M-shell,which is transferred to the chlorine atom to fill the p-orbit in its M-shell; see Figure 3.2.The sodium atom that gave up one electron is now positively charged (and is therefore called a sodium ion)

the type of interatomic forces that hold the atoms in their posi￾tion. These interatomic bonds, which result from electrostatic in￾teractions, may be extremely strong (such as for ionic or cova￾lent bonds; see below). Materials whose atoms are held together in this way are mostly hard and brittle. In contrast, metallic and particularly van der Waals bonds are comparatively weak and are thus responsible, among others, for the ductility of materials. In brief, there is not one single physical mechanism that determines the mechanical properties of materials, but instead, a rather com￾plex interplay between interatomic forces, atomic arrangements, and defects that causes the multiplicity of observations described in Chapter 2. In other words, the interactions between atoms play a major role in the explanation of mechanical properties. We shall endeavor to describe the mechanisms leading to plasticity or brittleness in the sections to come. In contrast to this, the electrical, optical, magnetic, and some thermal properties can be explained essentially by employing the electron theory of condensed matter. This will be the major theme of Part II of the present book, in which the electronic properties of materials will be discussed. Atoms consist of a positively charged nucleus and of negatively charged electrons which, expressed in simplified terms, orbit around this nucleus. Each orbit (or “shell”) can accommodate only a maximum number of electrons, which is determined by quantum mechanics; see Appendix I. In brief, the most inner “K￾shell” can accommodate only two electrons, called s-electrons. The next higher “L-shell” can accommodate a total of eight electrons, that is, two s-electrons and six p-electrons. The fol￾lowing “M-shell” can host two s-electrons, six p-electrons, and ten d-electrons; and so on. Filled outermost s p shells constitute a particularly stable (nonreactive) configuration, as demonstrated by the noble (in￾ert) gases in Group VIII of the Periodic Table (see Appendix). Chemical compounds strive to reach this noble gas configuration for maximal stability. If, for example, an element of Group I of the Periodic Table, such as sodium, is reacted with an element of Group VII, such as chlorine, to form sodium chloride (NaCl), the sodium gives up its only electron in the M-shell, which is transferred to the chlorine atom to fill the p-orbit in its M-shell; see Figure 3.2. The sodium atom that gave up one electron is now positively charged (and is therefore called a sodium ion), Ionic Bond 26 3 • Mechanisms 3.2 • Binding Forces Between Atoms

3.2.Binding Forces Between Atoms 27 M M FIGURE 3.2.Schematic representation of the formation of ionically bound Nacl by transferring one electron from Na to Cl Na CI to yield Nat and Cl-ions. whereas a negatively charged chlorine ion is formed by gaining one electron.The net effect is that the two oppositely charged ions attract each other electrostatically and thus produce the ionic bond.Ionic bonds are extremely strong(see Table 3.1).Any mechanical force that tries to disturb this bond would upset the electrical balance.It is partly for this reason that ionically bound materials are,in general,strong and mostly brittle.(Still,in par- ticular cases for which the atomic structure is quite simple,plas- tic deformation may be observed,such as in NaCl single crystals, which can be bent by hand under running water. The question arises why the oppositely charged ions do not ap- proach each other to the extent that fusion of the two nuclei would occur.To answer this,one needs to realize that the elec- trons which surround the nuclei exert strong repulsive forces upon each other that become exceedingly stronger the closer the two ions approach.Specifically,the orbits of the electrons of both ions start to mutually overlap.It is this interplay between the electrostatic attraction of the ions and the electrostatic repulsion (caused by the overlapping electron charge distributions)that brings about an equilibrium distance,do,between the ions,as shown in Figure 3.3. Characteristic examples of materials which are held together in part or entirely by ionic bonds are the alkali halides,many ox- ides,and the constituents of concrete.The strength and the ten- dency for ionic bonding depend on the difference in "electroneg- TABLE 3.1 Bonding energies for various atomic bonding mechanisms Bonding mechanism Bonding energy [kJ.mol-1] Ionic 340-800 Covalent 270-610 Metallic 20-240 Van der Waals <40

whereas a negatively charged chlorine ion is formed by gaining one electron. The net effect is that the two oppositely charged ions attract each other electrostatically and thus produce the ionic bond. Ionic bonds are extremely strong (see Table 3.1). Any mechanical force that tries to disturb this bond would upset the electrical balance. It is partly for this reason that ionically bound materials are, in general, strong and mostly brittle. (Still, in par￾ticular cases for which the atomic structure is quite simple, plas￾tic deformation may be observed, such as in NaCl single crystals, which can be bent by hand under running water.) The question arises why the oppositely charged ions do not ap￾proach each other to the extent that fusion of the two nuclei would occur. To answer this, one needs to realize that the elec￾trons which surround the nuclei exert strong repulsive forces upon each other that become exceedingly stronger the closer the two ions approach. Specifically, the orbits of the electrons of both ions start to mutually overlap. It is this interplay between the electrostatic attraction of the ions and the electrostatic repulsion (caused by the overlapping electron charge distributions) that brings about an equilibrium distance, d0, between the ions, as shown in Figure 3.3. Characteristic examples of materials which are held together in part or entirely by ionic bonds are the alkali halides, many ox￾ides, and the constituents of concrete. The strength and the ten￾dency for ionic bonding depend on the difference in “electroneg- 3.2 • Binding Forces Between Atoms 27 + 11 + 17 K L M K L M Na Cl FIGURE 3.2. Schematic representation of the formation of ionically bound NaCl by transferring one electron from Na to Cl to yield Na and Cl ions. TABLE 3.1 Bonding energies for various atomic bonding mechanisms Bonding mechanism Bonding energy [kJmol1] Ionic 340–800 Covalent 270–610 Metallic 20–240 Van der Waals 40

28 3·Mechanisms Repulsive energy 0 Net energy FIGURE 3.3.Schematic represen- tation of the potential energy, Attractive Epot,of an Na+and a Cl-ion energy as a function of the internu- clear distance,d.The equilib- rium distance,do,between the Na do two ions is the position of smallest potential energy. Repulsion→ Attraction ativity"between the elements involved,that is,the likelihood of an atom to accept one or more extra electrons.Chlorine,for ex- ample,is strongly electronegative because its outer shell is al- most completely filled with electrons (Figure 3.2).Sodium,on the other hand,is said to be weakly electronegative (actually it is electropositive:Group I of the Periodic Table!)and,therefore, readily gives up its valencel electron. Covalent Covalently bound solids such as diamond or silicon are typically Bond from Group IV of the Periodic Table.Consequently,each atom has four valence electrons.Since all atoms are identical,no elec- trons are transferred to form ions.Instead,in order to achieve the noble gas configuration,double electron bonds are formed by electron sharing [see Figure 3.4(a)].In other words,each Group IV atom,when in the solid state,is "surrounded"by eight elec- trons,which are depicted in Figure 3.4(a)as dots. The two-dimensional representation shown in Figure 3.4(a)is certainly convenient but does not fully describe the characteris- tics of such solids.Indeed,in three dimensions,silicon atoms, for example,are arranged in the form of a tetrahedron2 around a center atom having 10928'angles between the bond axes,as depicted in Figure 3.4(b).Because of this directionality and be- Valentia (Latin)=capacity,strength 2Tetraedros (Greek)=four-faced

ativity” between the elements involved, that is, the likelihood of an atom to accept one or more extra electrons. Chlorine, for ex￾ample, is strongly electronegative because its outer shell is al￾most completely filled with electrons (Figure 3.2). Sodium, on the other hand, is said to be weakly electronegative (actually it is electropositive: Group I of the Periodic Table!) and, therefore, readily gives up its valence1 electron. Covalently bound solids such as diamond or silicon are typically from Group IV of the Periodic Table. Consequently, each atom has four valence electrons. Since all atoms are identical, no elec￾trons are transferred to form ions. Instead, in order to achieve the noble gas configuration, double electron bonds are formed by electron sharing [see Figure 3.4(a)]. In other words, each Group IV atom, when in the solid state, is “surrounded” by eight elec￾trons, which are depicted in Figure 3.4(a) as dots. The two-dimensional representation shown in Figure 3.4(a) is certainly convenient but does not fully describe the characteris￾tics of such solids. Indeed, in three dimensions, silicon atoms, for example, are arranged in the form of a tetrahedron2 around a center atom having 109°28 angles between the bond axes, as depicted in Figure 3.4(b). Because of this directionality and be￾FIGURE 3.3. Schematic represen￾tation of the potential energy, Epot, of an Na and a Cl ion as a function of the internu￾clear distance, d. The equilib￾rium distance, d0, between the two ions is the position of smallest potential energy. 28 3 • Mechanisms Na+ Cl– d0 d Repulsion Attraction 0 Epot Repulsive energy Net energy Attractive energy 1Valentia (Latin)
capacity, strength. 2Tetraedros (Greek)
four-faced. Covalent Bond

3.2.Binding Forces Between Atoms 29 4 :: Si : FIGURE 3.4.(a)Two-dimensional and (b)three-dimensional representations (a) of the covalent bond as for silicon or carbon(diamond cubic structure). The charge distribution between the individual atoms is not uniform but cone-shaped.The angle between the bond axes,called the valence angle,is (b) 10928'.(See also Figure 16.4(a).) cause of the filled electron shells,covalently bound materials are hard and brittle.Typical representatives are diamond,silicon, germanium,silicate ceramics,glasses,stone,and pottery con- stituents. In many materials a mixture of covalent and ionic bonds ex- ists.As an example,in GaAs,an average of 46%of the bonds oc- cur through electron transfer between Ga and As,whereas the remainder is by electron sharing.A range of binding energies is given in Table 3.1. Metallic Bond The outermost (that is,the valence)electrons for most metals are only loosely bound to their nuclei because of their relative re- moteness from their positively charged cores.All valence elec- trons of a given metal combine to form a "sea"of electrons that move freely between the atom cores.The positively charged cores are held together by these negatively charged electrons.In other words,the free electrons act as the bond (or,as it is often said, as a "glue")between the positively charged ions;see Figure 3.5. Metallic bonds are nondirectional.As a consequence,the bonds do not break when a metal is deformed.This is one of the rea- sons for the high ductility of metals. Examples for materials having metallic bonds are most metals such as Cu,Al,Au,Ag,etc.Transition metals(Fe,Ni,etc.)form mixed bonds that are comprised of covalent bonds (involving their 3d-electrons;see Appendix I)and metallic bonds.This is one of the reasons why they are less ductile than Cu,Ag,and Au. A range of binding energies is listed in Table 3.1

cause of the filled electron shells, covalently bound materials are hard and brittle. Typical representatives are diamond, silicon, germanium, silicate ceramics, glasses, stone, and pottery con￾stituents. In many materials a mixture of covalent and ionic bonds ex￾ists. As an example, in GaAs, an average of 46% of the bonds oc￾cur through electron transfer between Ga and As, whereas the remainder is by electron sharing. A range of binding energies is given in Table 3.1. The outermost (that is, the valence) electrons for most metals are only loosely bound to their nuclei because of their relative re￾moteness from their positively charged cores. All valence elec￾trons of a given metal combine to form a “sea” of electrons that move freely between the atom cores. The positively charged cores are held together by these negatively charged electrons. In other words, the free electrons act as the bond (or, as it is often said, as a “glue”) between the positively charged ions; see Figure 3.5. Metallic bonds are nondirectional. As a consequence, the bonds do not break when a metal is deformed. This is one of the rea￾sons for the high ductility of metals. Examples for materials having metallic bonds are most metals such as Cu, Al, Au, Ag, etc. Transition metals (Fe, Ni, etc.) form mixed bonds that are comprised of covalent bonds (involving their 3d-electrons; see Appendix I) and metallic bonds. This is one of the reasons why they are less ductile than Cu, Ag, and Au. A range of binding energies is listed in Table 3.1. Metallic Bond 3.2 • Binding Forces Between Atoms 29 (b) (a) Si Si Si Si Si Si Si Si Si FIGURE 3.4. (a) Two-dimensional and (b) three-dimensional representations of the covalent bond as for silicon or carbon (diamond cubic structure). The charge distribution between the individual atoms is not uniform but cone-shaped. The angle between the bond axes, called the valence angle, is 109°28. (See also Figure 16.4(a).)

30 3·Mechanisms FIGURE 3.5.Schematic representation of metallic bonding.The valence electrons become disassociated with "their" atomic core and form an electron "sea" that acts as the binding medium be- tween the positively charged ions. Van der Compared to the three above-mentioned bonding mechanisms, Waals Bond van der Waals bonds!are quite weak and are therefore called secondary bonds.They involve the mutual attraction of dipoles. This needs some explanation.An atom can be represented by a positively charged core and a surrounding negatively charged electron cloud [Figure 3.6(a)].Statistically,it is conceivable that the nucleus and its electron cloud are momentarily displaced with respect to each other.This configuration constitutes an electric dipole,as schematically depicted in Figure 3.6(b).A neighboring atom senses this electric dipole and responds to it with a simi- lar charge redistribution.The two adjacent dipoles then attract each other. FiGURE 3.6.(a)An atom is represented by a (a) positively charged core and a surrounding B negatively charged electron cloud.(b)The electron cloud of atom'A'is thought to be displaced,thus forming a dipole.This in- duces a similar dipole in a second atom,'B'.Both dipoles are then mutually attracted,as proposed by van der Waals. (b) IJohannes Diederik van der Waals (1837-1923),Dutch physicist,re- ceived in 1910 the Nobel Prize in physics for his research on the math- ematical equation describing the gaseous and liquid states of matter.He postulated in 1873 weak intermolecular forces that were subsequently named after him

Compared to the three above-mentioned bonding mechanisms, van der Waals bonds1 are quite weak and are therefore called secondary bonds. They involve the mutual attraction of dipoles. This needs some explanation. An atom can be represented by a positively charged core and a surrounding negatively charged electron cloud [Figure 3.6(a)]. Statistically, it is conceivable that the nucleus and its electron cloud are momentarily displaced with respect to each other. This configuration constitutes an electric dipole, as schematically depicted in Figure 3.6(b). A neighboring atom senses this electric dipole and responds to it with a simi￾lar charge redistribution. The two adjacent dipoles then attract each other. FIGURE 3.5. Schematic representation of metallic bonding. The valence electrons become disassociated with “their” atomic core and form an electron “sea” that acts as the binding medium be￾tween the positively charged ions. 30 3 • Mechanisms 1Johannes Diederik van der Waals (1837–1923), Dutch physicist, re￾ceived in 1910 the Nobel Prize in physics for his research on the math￾ematical equation describing the gaseous and liquid states of matter. He postulated in 1873 weak intermolecular forces that were subsequently named after him. + – – – – – – – – A B (a) (b) + + FIGURE 3.6. (a) An atom is represented by a positively charged core and a surrounding negatively charged electron cloud. (b) The electron cloud of atom ‘A’ is thought to be displaced, thus forming a dipole. This in￾duces a similar dipole in a second atom,‘B’. Both dipoles are then mutually attracted, as proposed by van der Waals. Van der Waals Bond

3.3.Arrangement of Atoms(Crystallography) 31 FIGURE 3.7.Two polymer chains are mutually at- tracted by van der Waals forces.An applied exter- nal stress can easily slide the chains past each other. Many polymeric chains which consist of covalently bonded atoms contain areas that are permanently polarized.The cova- lent bonding within the chains is quite strong.In contrast to this, the individual chains are mutually attracted by weak van der Waals forces (Figure 3.7).As a consequence,many polymers can be deformed permanently since the chains slide effortlessly past each other when a force is applied.(We will return to this topic in Section 16.4.) One more example may be given.Ice crystals consist of strongly bonded H2O molecules that are electrostatically attracted to each other by weak van der Waals forces.At the melting point of ice, or under pressure,the van der Waals bonds break and water is formed. Mixed As already mentioned above,many materials possess atomic Bonding bonding involving more than one type.This is,for example,true in compound semiconductors (e.g.,GaAs),which are bonded by a mixture of covalent and ionic bonds,or in some transition met- als,such as iron or nickel,which form metallic and covalent bonds. 3.3.Arrangement of Atoms (Crystallography) The strength and ductility of materials depend not only on the binding forces between the atoms,as discussed in Section 3.2, but also on the arrangements of the atoms in relationship to each other.This needs some extensive explanations. The atoms in crystalline materials are positioned in a periodic

Many polymeric chains which consist of covalently bonded atoms contain areas that are permanently polarized. The cova￾lent bonding within the chains is quite strong. In contrast to this, the individual chains are mutually attracted by weak van der Waals forces (Figure 3.7). As a consequence, many polymers can be deformed permanently since the chains slide effortlessly past each other when a force is applied. (We will return to this topic in Section 16.4.) One more example may be given. Ice crystals consist of strongly bonded H2O molecules that are electrostatically attracted to each other by weak van der Waals forces. At the melting point of ice, or under pressure, the van der Waals bonds break and water is formed. As already mentioned above, many materials possess atomic bonding involving more than one type. This is, for example, true in compound semiconductors (e.g., GaAs), which are bonded by a mixture of covalent and ionic bonds, or in some transition met￾als, such as iron or nickel, which form metallic and covalent bonds. 3.3 • Arrangement of Atoms (Crystallography) 31 FIGURE 3.7. Two polymer chains are mutually at￾tracted by van der Waals forces. An applied exter￾nal stress can easily slide the chains past each other. 3.3 • Arrangement of Atoms (Crystallography) Mixed Bonding The strength and ductility of materials depend not only on the binding forces between the atoms, as discussed in Section 3.2, but also on the arrangements of the atoms in relationship to each other. This needs some extensive explanations. The atoms in crystalline materials are positioned in a periodic

32 3·Mechanisms that is,repetitive,pattern which forms a three-dimensional grid called a lattice.The smallest unit of such a lattice that still pos- sesses the characteristic symmetry of the entire lattice is called a conventional unit cell.(Occasionally smaller or larger unit cells are used to better demonstrate the particular symmetry of a unit.) The entire lattice can be generated by translating the unit cell into three-dimensional space. Bravais Bravais!has identified 14 fundamental unit cells,often referred Lattice to as Bravais lattices or translation lattices,as depicted in Figure 3.8.They vary in the lengths of their sides (called lattice constants, a,b,and c)and the angles between the axes (a,B,y).The char- acteristic lengths and angles of a unit cell are termed lattice pa- rameters.The arrangement of atoms into a regular,repeatable lat- tice is called a crystal structure. The most important crystal structures for metals are the face- centered cubic(FCC)structure,which is typically found in the case of soft (ductile)materials,the body-centered cubic (BCC) structure,which is common for strong materials,and the hexag- onal close-packed (HCP)structure,which often is found in brit- tle materials.It should be emphasized at this point that the HCP structure is not identical with the simple hexagonal structure shown in Figure 3.8 and is not one of the 14 Bravais lattices since HCP has three extra atoms inside the hexagon.The unit cell for HCP is the shaded portion of the conventional cell shown in Fig- ure 3.9.It contains another "base"atom within the cell in con- trast to the hexagonal cell shown in Figure 3.8. The lattice points shown as filled circles in Figure 3.8 are not necessarily occupied by only one atom.Indeed,in some materi- als,several atoms may be associated with a given lattice point; this is particularly true in the case of ceramics,polymers,and chemical compounds.Each lattice point is equivalent.For ex- ample,the center atom in a BCC structure may serve as the cor- ner of another cube. We now need to define a few parameters that are linked to the mechanical properties of solids. cla Ratio The separation between the basal planes,co,divided by the length of the lattice parameter,ao,in HCP metals(Figure 3.9),is theoret- ically V8/3 =1.633,assuming that the atoms are completely spherical in shape.(See Problem 3.6.)Deviations from this ideal ratio result from mixed bondings and from nonspherical atom shapes.The c/a ratio influences the hardness and ductility of ma- terials;see Section 3.4. See Section 3.1

that is, repetitive, pattern which forms a three-dimensional grid called a lattice. The smallest unit of such a lattice that still pos￾sesses the characteristic symmetry of the entire lattice is called a conventional unit cell. (Occasionally smaller or larger unit cells are used to better demonstrate the particular symmetry of a unit.) The entire lattice can be generated by translating the unit cell into three-dimensional space. Bravais1 has identified 14 fundamental unit cells, often referred to as Bravais lattices or translation lattices, as depicted in Figure 3.8. They vary in the lengths of their sides (called lattice constants, a, b, and c) and the angles between the axes (, , ). The char￾acteristic lengths and angles of a unit cell are termed lattice pa￾rameters. The arrangement of atoms into a regular, repeatable lat￾tice is called a crystal structure. The most important crystal structures for metals are the face￾centered cubic (FCC) structure, which is typically found in the case of soft (ductile) materials, the body-centered cubic (BCC) structure, which is common for strong materials, and the hexag￾onal close-packed (HCP) structure, which often is found in brit￾tle materials. It should be emphasized at this point that the HCP structure is not identical with the simple hexagonal structure shown in Figure 3.8 and is not one of the 14 Bravais lattices since HCP has three extra atoms inside the hexagon. The unit cell for HCP is the shaded portion of the conventional cell shown in Fig￾ure 3.9. It contains another “base” atom within the cell in con￾trast to the hexagonal cell shown in Figure 3.8. The lattice points shown as filled circles in Figure 3.8 are not necessarily occupied by only one atom. Indeed, in some materi￾als, several atoms may be associated with a given lattice point; this is particularly true in the case of ceramics, polymers, and chemical compounds. Each lattice point is equivalent. For ex￾ample, the center atom in a BCC structure may serve as the cor￾ner of another cube. We now need to define a few parameters that are linked to the mechanical properties of solids. The separation between the basal planes, c0, divided by the length of the lattice parameter, a0, in HCP metals (Figure 3.9), is theoret￾ically 8/3
1.633, assuming that the atoms are completely spherical in shape. (See Problem 3.6.) Deviations from this ideal ratio result from mixed bondings and from nonspherical atom shapes. The c/a ratio influences the hardness and ductility of ma￾terials; see Section 3.4. Bravais Lattice 32 3 • Mechanisms 1See Section 3.1. c/a Ratio

3.3.Arrangement of Atoms (Crystallography) 33 Simple Cubic Face-Centered Body-Centered (SC) Cubic (FCC) Cubic (BCC) Simple Body-Centered Tetragonal Tetragonal Simple Body-Centered Base-Centered Face-Centered Orthorhombic Orthorhombic Orthorhombic Orthorhombic Simple Base-Centered Hexagonal Rhombohedral Monoclinic Triclinic Monoclinic FIGURE 3.8.The 14 Bravais lattices grouped into seven crystal systems: First row:a=b=c,a=B=y=90(cubic); Second row:a=b≠c,a=B=y=90°(tetragonal); Third row:a≠b≠c,a=B=y=90°(orthorhombic: Fourth row:at least one angle is 90.Specifically: Hexagonal:a=B=90°，y=120°a=b≠c(the unit cell is the shaded part of the structure); Rhombohedral:a=b=c,a=B=y≠90°≠60°≠109.5°； Monoclinic:a=y=90°，B≠90°，a≠b≠c; Triclinic:a≠B≠y≠90°，a≠b≠c

FIGURE 3.8. The 14 Bravais lattices grouped into seven crystal systems: First row: a
b
c, 


90° (cubic); Second row: a
b  c, 


90° (tetragonal); Third row: a  b  c, 


90° (orthorhombic); Fourth row: at least one angle is  90°. Specifically: Hexagonal: 

90°, 
120° a
b  c (the unit cell is the shaded part of the structure); Rhombohedral: a
b
c, 

  90°  60°  109.5°; Monoclinic: 

90°,   90°, a  b  c; Triclinic:       90°, a  b  c. 3.3 • Arrangement of Atoms (Crystallography) 33 Simple Cubic (SC) Face-Centered Cubic (FCC) Body-Centered Cubic (BCC) Simple Tetragonal Body-Centered Tetragonal Simple Orthorhombic Body-Centered Orthorhombic Base-Centered Orthorhombic Face-Centered Orthorhombic Hexagonal Rhombohedral Simple Monoclinic Base-Centered Monoclinic Triclinic