Structure and Bonding in Crystalline Materials: A Textbook...

Structure and Bonding in Crystalline Materials: A Textbook for

Materials Science and Engineering Students

Gregory S. Rohrer

Department of Materials Science and Engineering

Carnegie Mellon University

Pittsburgh, Pennsylvania 15213-3890

The Journal of Materials Education, 20 (1998) 135.

correspondence should be addressed to G.S. Rohrer at:

phone 412-268-2696

fax 412-268-7599

e-mail [email protected]

www http://neon.mems.cmu.edu/rohrer.html

G.S. Rohrer p. 2

ABSTRACT

A textbook has been prepared for an entry level Materials Science and Engineering graduate

course on structure and bonding in crystalline materials. The course is intended to serve as

a foundation for other graduate courses by providing techniques to understand crystal

structures and the cohesive forces that stabilize them. The intrinsically connected topics of

structure and bonding are presented in the same course because without knowledge in both

of these areas, it is not possible to address the fundamental relationship between the

composition, structure, and properties of solid matter that motivates nearly all materials

research. The purpose of this paper is to describe the rationale used during the

development of the course and the textbook that has been prepared to assist Materials

Science and Engineering students and instructors.

Keywords:

Textbook, Crystal Structures, Bonding in Crystals

G.S. Rohrer p. 3

INTRODUCTION

The interdisciplinary nature and continual evolution of the field of Materials Science

and Engineering (MSE) make the choice of a core graduate curriculum a difficult issue.

Nevertheless, most departments select a small subset of courses that are required for all

graduate students. Ideally, the material taught in the core part of the curriculum will

provide all students (even those with undergraduate training in other disciplines) with the

tools necessary to address the central issue that motivates nearly all materials research: how

does the composition of solid matter and the way in which it is processed influence its

structure and properties? To explore this fundamental question in a systematic manner, one

needs to understand how the structure of matter is described, how to experimentally probe

structure, and the cohesive forces in solids that determine the structure. The scope of the

course described here is limited to descriptions of ideal atomic structure (the defect structure

and microstructure being course material unto themselves), the theory of diffraction,

bonding in crystals, and the relationship between structure and bonding.

The intellectual match between the topics of crystal structure and bonding is difficult

to dispute. This structure-bonding relationship was realized in the early part of the century

and discussed extensively through the decades by Pauling [1], Hume-Rothery [2], Pearson

[3], Phillips [4], and Villars [5-7]. However, the graduate curricula in most MSE

departments continue to greatly emphasize structure over bonding. In many cases, students

are exposed to advanced models for crystal cohesion only during courses on electrical

properties or solid state physics. Clearly, there is a potential advantage to teaching the topic

of crystal cohesion at the same time as the topic of crystal structures so that the connections

can be emphasized.

There are numerous existing books on crystallography [8-11], structural solid state

chemistry [12,13], metallurgy [14], and solid state physics [15-17] that can be used as

resources when teaching a course on structure and bonding. However, none of these

books is wholly appropriate and this means that one must select sections of each. In my

G.S. Rohrer p. 4

experience, this complicates the instructor's task of educating and the student's task of

learning. The motivation for preparing this textbook was to provide a self-contained,

coherent resource for both students and instructors. I should mention that another book

targeted for MSE curricula, "Structure and Bonding in Condensed Matter", by C.S.

Nichols [18], appeared three years ago. The textbook described here differs from this

mainly in the balance between the two topics. The book by Nichols, which is no longer in

print, has greater detail in the quantum mechanical models for bonding, while the book

described here has a more extensive description of crystal structures, diffraction, and

phenomenological models for bonding.

THE SUBJECT MATTER AND ITS ORGANIZATION

The most important (and possibly contentious) organizational decision in the

structure of the textbook was whether to present structure or bonding first. There are

reasonable philosophical and pragmatic arguments for both possible arrangements. For

example, it has been argued that the distribution of electrons in the solid (the bonding)

determines the structure and, therefore, bonding must be presented before structure. While

this view can be supported, it leads to two problems. The first is the implication that the

relationship between bonding and structure is a one way street. In fact, the atomic

configuration in a crystal and the electronic distribution (spatial and energetic) seek a joint

equilibrium. Consider, for example, what happens when a phase transformation is initiated

by the application of hydrostatic pressure. In this case, the interatomic distances are

externally altered and the bonds respond to the structure by changing to a new

configuration. The second problem with presenting the bonding section before the

structure section is that it betrays the scientific process. Models to explain phenomena are

developed based on observations. Models for chemical bonds in solids were developed

only after extensive crystallographic studies. When examining a model, it is essential that

G.S. Rohrer p. 5

one has a strong working knowledge of the phenomenon being explained. Since crystal

structure data are the observations on which bonding models have been formulated, I have

chosen to present this section in advance of the bonding models. A complete table of

contents for the text is presented in Table 1.

The case for presenting structure before bonding can also be argued from a

pragmatic standpoint. The development of every analytical model for bonding is based on

an understanding of the periodicity of the Bravais lattice and symmetry. Furthermore, the

concepts of the reciprocal lattices, plane wave scattering, and systematic absences are

easiest to introduce when describing diffraction, where the relationships between atomic

structure and the reciprocal lattice (diffraction pattern) are concrete. It is then much easier

to use these tools to describe the nearly free electron theory, the stability of electron

compounds, and the reduced zone scheme for the presentation of band structures.

The decision to present the structure section before the bonding section is hedged

somewhat by the first, introductory chapter. In Chapter 1, simple models for crystal

structures and bonding are introduced. It is assumed that for most students in the course,

this is a review. However, it does provide a foundation in bonding models before the

beginning of the structure section. There are two points of emphasis in this chapter: the

periodic nature of bonding, structure, and properties and the weakness of the simple

models that most students were exposed to as undergraduates. This is intended to motivate

the more detailed sections that follow.

Chapters 2 though 5 comprise the structural section of the course. Chapter 2 is a

review of undergraduate crystallography with all of the terminology and mechanics needed

for any advanced discussion. For example, the chapter includes sections on the real space

lattice, crystallographic notation for locations, directions and planes, the reciprocal lattice,

methods for calculating geometric parameters of the lattice, and methods for representing

crystal structures. I have found that this is not only a useful review for MSE students, but

it also provides an adequate foundation for students from other disciplines.

G.S. Rohrer p. 6

The third chapter describes the formal methods used to specify the symmetry of a

crystal. The symmetry groups and operators used by crystallographers are a concise set of

instructions for generating the positions of the atoms in a crystal. The goal of the chapter is

not to explain the theory of groups, but to provide students with the information necessary

to interpret crystallographic data written using conventional notation. This chapter contains

sections on symmetry operators, the 32 crystallographic point groups, the 230

crystallographic space groups, and the interpretation of standard crystallographic data.

The goals of the fourth chapter are to acquaint the student with typical prototype

crystal structures and to present a less formal, but more convenient way of describing

structures. Briefly, simple structures are described as eutactic arrays with an ordered

occupation of close-packed and/or interstitial sites. While this method lacks the precision

of the one presented in Chapter 3, it is valuable because it leads to easy visualization and

provides a means for comparison. More complex ternary and quaternary structures are

described in comparison to the simple ones and noncrystalline structures (quasicrystalline,

incommensurate, liquid crystalline, and amorphous structures) are described briefly in the

last section.

The following example illustrates the methods emphasized in the first several

chapters of the book. Chapters 3 and 4 include data for more 50 structure types in the

format illustrated by Tables 2 and 3. In a worked problem, the reader is asked to describe

the packing in the scheelite structure, compare this ternary structure to a more well known

binary prototype structure, and to develop a geometric criterion that can be used to

differentiate the ABO4 compounds that take the scheelite structure from those that take the

rutile structure. Based on the data in Table 3 and the information provided in Chapters 2

and 3, the schematic in Fig. 1 is constructed. From this sketch, we recognize that the metal

atoms occupy the sites of a cubic close packed lattice and that the oxygen occupy the

tetrahedral interstices. Based on this packing description, we note a similarity to the

fluorite structure. However, scheelite differs from fluorite because the ordering of the two

G.S. Rohrer p. 7

metal atoms on the close packed sites doubles the length of the unit cell along [001]. This

ordering of the two metal atoms also causes the O to occupy a position that is distorted

form the ideal tetrahedral coordination. To develop a geometric criterion to separate those

ABO4 compounds that take the rutile structure from those that take the scheelite structure,

we recall that in rutile all of the metal atoms are 6-coordinate. In scheelite, the A cations are

8-coordinate while the B cations are 4-coordinate. This suggests an A/B size differential.

To test this, the data in Tables 2 and 3 are used to make the plot in Fig. 2, which clearly

illustrates that if rA/rB < 1.3, the rutile structure is preferred over scheelite.

The goal of the fifth chapter is to provide students with enough of a foundation in

the theory of diffraction that they understand how structures are determined and what

typical data mean. Rather than surveying crystallographic techniques or methods (most

MSE departments devote an entire course to this subject), the chapter concentrates on the

theory that accounts for diffraction. An emphasis is placed on understanding how the

lattice and the basis of a structure can be determined from the angular positions and

intensities of diffracted peaks, respectively. Intensity calculations are used to determine

relative site occupations and specific position parameters. This particular section is placed

after the chapters on basic crystallography and symmetry because it is not possible to have

a detailed discussion of diffraction without the mechanical skills for dealing with real space

and reciprocal space lattices or knowledge of the symmetry operations that lead to

systematic absences.

The section on bonding is found in Chapters 6 through 9. Chapter 6 is about

secondary bonding, Chapter 7 is about ionic bonding, Chapter 8 is about metallic bonding,

and Chapter 9 describes the linear combination of atomic orbitals (LCAO) method that can

be used for the whole range of polar covalent bonding. While the subject is divided under

the traditional headings, the book always emphasizes that the first three classifications

(secondary, ionic, metallic) are simply limiting cases of a single phenomenon that should

simply be called chemical bonding. In each case, the emphasis is placed on making

G.S. Rohrer p. 8

quantitative calculations of structural parameters and stabilities. The order of presentation

was selected for the simple reason that there is an escalation in the skills needed for each

model. Chapter 6, on secondary bonding, introduces the Lennard-Jones pair potential and

the methods used to account for long range forces.

In Chapter 7 on ionic bonding, the Madelung potential is added to the Lennard-

Jones pair potential. The fundamental steps involved in ionic bond formation are also

described in terms of the Born-Haber cycle. Effects not included in the quantitative model,

including the crystal field stabilization energy and ionic polarization, are assessed. The

concept of electronegativity (first introduced in Chapter 1) is revisited and different

quantitative scales are described. Finally, the ability to predict structure based on these

models is discussed.

The eighth chapter is about metallic bonding and begins with the development of the

free electron theory. After pointing out that this provides no cohesion for the crystal,

electrostatic terms are introduced so that the stability of metallic crystals can be explained.

Next, the nearly free electron theory and dispersion in the reduced zone scheme are

described to introduce the concepts of energy bands, zone boundaries, and energy gaps.

The objective of the ninth chapter is to describe the polar covalent bond within the

framework of the LCAO model. In this final chapter on bonding, I note that the LCAO

model can, in its limits, also be used to describe materials that are considered ionic or

metallic in the simple models. This fact emphasizes the idea that chemical bonding is a

single phenomenon that can be described by a single model. After a brief description of

how the LCAO method is applied to small molecules, it is applied to two dimensional

homopolar crystals, polar crystals, and three dimensional crystals. The Freidel model is

used to explain the presence of covalent bonds in transition metal crystals. The factors that

lead to hybridization and tetrahedral geometries are also described. While this chapter will

not make the student an expert in electronic structure methods, it will provide the

foundation needed to understand the electronic structure of a crystal well enough to

G.S. Rohrer p. 9

interpret the band structure diagrams and density of states plots that are the common result

of modern theoretical studies.

The methods emphasized in Chapter 9 are illustrated by the following example.

The reader is asked to consider a 2 dimensional binary compound (AB) with a square lattice

structure where each site in the lattice is alternately occupied by an A or B atom. Assuming

that each atom has a single s valence orbital, the reader is asked to apply the nearest

neighbor LCAO model and determine the dispersion along the [100] direction, plot the

dispersion assuming that the hypothetical compound is LiCu, LiAl, and LiCl, and, for each

case, determine the width of the s-derived bands and the gap between the occupied and

unoccupied states. Although the situation is clearly fictitious and the approximations

unwarranted, the results are instructive (see Fig. 3). The quantitative calculations, which

can be done by hand, illustrate that increasing the polarity in a compound narrows the

widths of the bands and increases the gap between occupied and unoccupied states.

The tenth and final chapter returns to the question that is central to materials

research: How do the composition of solid matter and the way in which it is processed

influence its structure and properties? The scope of this question is much greater than that

of this course, but two parts can be addressed. First, when a given set of elements are

combined, will they react to form a compound, will they dissolve in one another, or will

they be immiscible? Second, what will the ideal atomic scale structure of the combination

be? In this chapter, phenomenological approaches to these questions are reviewed because

they have been more effective than the quantitative physical models. In the first section,

Miedema's [19] and Villars' [20] methods for predicting the existence of stable compounds

are reviewed. In the next section, special electron concentration and radius ratio rules are

reviewed. Laves phases, interstitial compounds, and Hume-Rothery rules [2] are

described. The next section describes the factors that determine structure in polar covalent

crystals. Among the topics described in this section are Pauling's rules [1] and the bond

G.S. Rohrer p. 10

valence method [21]. In the final section, structure stability maps developed by Mooser

and Pearson [3], Phillips [4], and Villars [5-7] are described.

Throughout the book, an emphasis is always placed on quantitative data, numerical

examples, and problem solving. For example, the extensive data allow students to examine

how chemical periodicity, electronic structure, and size factors influence structure selection.

In the chapter on diffraction, all of the data needed for the calculation of diffraction peak

intensity are included and sample calculations are carried out. The bonding models in

Chapters 6-9 are all presented in such a way that quantitative solutions can be determined

using simple calculations. In this regard, the methods and parameters established by

Harrison [17] are followed. While these models are not at the state-of-the-art, they are

qualitatively accurate and have the advantage of being simple enough that numerical results

for specific compounds and elements are readily obtained and compared by the student.

Whenever appropriate, the text includes worked numerical examples and each chapter

concludes with an extensive set of problems.

THE COURSE'S TIMING AND TRACK RECORD

In the MSE department at Carnegie Mellon University, this course is taught to all

first semester graduate students. A smaller number of senior MSE undergraduates and

graduate students from Chemical Engineering, Electrical and Computer Engineering,

Mechanical Engineering, and Physics also enroll. The class is held on a normal semester

schedule and meets for 28, 110 minute lectures. The chapter structure of the text book is

by no means evenly divided. The approximate schedule that I have used is illustrated in

Table 4. This is covers 26 of 28 lecture periods and leaves two exam and/or review

classes.

I have taught this course annually since 1992 and each class has consisted of 17 to

27 students; the total number of students that have taken this course is 130. At the end of

G.S. Rohrer p. 11

each semester, students are asked the question, "What was the overall quality of this

course?" The response is an integer rating, from 1 to 5, with 5 being the best. The results

for the first 5 years (the 1997 results are not yet available) are summarized in Table 5. As a

baseline for the rating system, consider the fact that this question is asked of every student

in every class in the college and the mean for all courses taught during the same five years

was 3.98.

In the first year, there was no text for the course. Students were asked to read from

one of about 10 different existing books before each lecture. During this period, the course

was rated as slightly above average. The second year was much the same, but I made a

transcribed version of my lecture notes available to students a few days before each lecture.

During this year, the course was rated well below the college average. Before the third

year, I organized, corrected, and greatly expanded the transcription of the notes and made a

draft version of the book available to the class before the semester. This draft version was

the primary source for the class. At this point, the students' opinions of the class improved

significantly. During subsequent years, the content and the organization of the book were

continuously refined and the ratings of the course have also improved.

CONCLUSION

A textbook has been prepared from which it is possible to teach the topics of crystal

structures and crystal cohesion in an integrated way. Student approval of a course based

on this text has been high. Currently, there are no firm plans for publication. However, I

anticipate that in the near future, this book will be available for use by others; students and

instructors interested in this text prior to publication are encouraged to contact the author.

ACKNOWLEDGMENT

G.S. Rohrer p. 12

I thank the National Science Foundation for support under a YIA grant DMR-9458005.

REFERENCES

[1] L. Pauling, J. Am. Chem. Soc., 51 1010 (1929).

[2] W. Hume-Rothery, G. Raynor, "The Structure of Metals and Alloys" (Institute of

Metals, London 1962).

[3] E. Mooser and W. B. Pearson, Acta Cryst, 12 1015 (1959).

[4] J.C. Phillips, Rev. Mod. Phys., 42 317 (1970).

[5] P. Villars, J. Less-Common Metals, 92 215 (1983).



[8] D.E. Sands, "Introduction to Crystallography" (W. A. Benjamin, New York 1969).

[9] M.J. Buerger, "Contemporary Crystallography" (McGraw Hill, New York 1970).

[10] H. Megaw, "Crystal Structures: A Working Approach" (W. B. Saunders,

Philadelphia 1973).

[11] B.D. Cullity, "Elements of X-ray Diffraction, 2nd edition" (Addison-Wesley,

Reading, MA 1978).

[12] A.F. Wells, "Structural Inorganic Chemistry, 5th edition" (Clarendon Press, Oxford,

1984).

[13] A.R. West, "Solid State Chemistry and its Applications" (J. Wiley & Sons,

Chichester 1984).

[14] C.S Barrett and T.B. Massalski, "The Structure of Metals" (McGraw Hill, New York

1966).

[15] C. Kittel, "Introduction to Solid State Physics 5th edition" (J. Wiley & Sons, New

York 1976).

[16] N.W. Ashcroft, N.D. Mermin, "Solid State Physics" (Holt Rinehart and Winston,

New York 1976).

G.S. Rohrer p. 13

[17] W.A. Harrison,"Electronic Structure and the Properties of Solids: The Physics of the

Chemical Bond" (Dover Publications, Inc., New York 1989).

[18] C.S. Nichols, "Structure and Bonding in Condensed Matter" (Cambridge University

Press, Cambridge, 1995).

[19] D.G. Pettifor, Solid State Physics 40 43 (1987).

[20] P. Villars, J. Less-Common Metals 109 93 (1985).

[21] M. O'Keeffe and N.E. Brese, J. Am. Chem. Soc., 113 3226 (1991).

G.S. Rohrer p. 14

Figure Captions

Figure 1. A sketch of the atomic positions in the scheelite structure. The projection is

shown along the [100] direction.

Figure 2. The radius ratios for the rutile and scheelite structured ABO4 compounds listed

in Tables 2 and 3, respectively. For a consistent comparison, the octahedral radii are used.

Figure 3. Dispersion relations for three hypothetical Li compounds. (a) The two different

atoms alternately occupy the vertices of a square lattice. (b) The dispersion of electron

energy levels along [100] for three combinations of atoms. The dispersion relation, or

energy (E) as a function of wave vector, (k), is shown below. EA is the term state for the

valence s electron of the A atom and EB is the term state for the valence s electron of B

atoms. Vssσ is the covalent energy for the overlap of the s states, and r1 and r3 are the

vectors defined in (a).

G.S. Rohrer p. 15

0 0

0

0.5

0.5 0.5

0 00.5

0

00

0.5

0.5

0.5

0.25 0.25

0.25

0.75 0.75

0.75

0.11 0.110.89 0.89

0.89 0.11

0.75 0.25

0.25 0.75

0.39 0.61

0.61 0.39

0.890.11

b

c

= Ca

= W

= O

b

c

Figure 1. G.S. Rohrer

G.S. Rohrer p. 16

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Smal

ler

Cat

ion

Rad

ius,

Å 0.65

0.7

0.3 0.6 0.9 1.2 1.5

Larger Cation Radius, Å

scheelite

rutile

1.8

ra = rb


G.S. Rohrer p. 17

r1

r3

E(k) = 1/2(EA+EB) ± 1/4(EA-EB)2 + {2Vssσ[cosk•r1+cosk•r3]}

vacuum level (unbound)

-5

0

-10

-15

-20

-25

-30

Ene

rgy,

eV

0 [kx,0,0] π/a 0 [kx,0,0] π/a 0 [kx,0,0] π/a

LiCu LiAl LiCl


G.S. Rohrer p. 18

Table 1. Table of Contents for Structure an Bonding in Crystalline Materials

1. Chapter 1, Introduction 1A. Introduction 2B. The Importance of the Periodic Table: Mendeleev Predicts Ge in 1871 2C. Periodic Trends in Atomic Properties 3D. Bonding Generalizations Based on Periodic Trends in the Electronegativity 6

(i) Classification of the elements 6(ii) Simple bonding models and typical properties 7(iii) Ketelaar's triangle 10(iv) Examples of trends in bonding 11

E. Generalizations About Crystal Structures Based on Periodicity 14(i) Close-packing in metallic solids 14(ii) Radius ratios in ionic structures 17(iii) Orbital hybridization in covalent structures 21(iv) Examples of periodic trends in structures 24

F. The Limitations of Simple Models 25(i) Polymorphism 25(ii) Systems with mixed properties and mixed bonding 27

G. Problems 30H. References and Sources for Further Study 31

2. Chapter 2, Basic Structural Concepts 33B. The Bravais Lattice 34

(i) Definition 34(ii) Geometry of the 14 three dimensional Bravais lattices 36

C. The Unit Cell 47(i) The primitive cell 47(ii) The non-primitive conventional cell 47(iii) The Wigner-Seitz cell 50

D. The Crystal Structure. A Bravais Lattice Plus a Basis 50E. Specifying Locations, Planes and Directions in a Crystal 53

(i) Locations 53(ii) Directions 53(iii) Indices for planes 54

F. The Reciprocal Lattice 57(i) Mathematical formalism 57(ii) The relationship between the direct lattice and the reciprocal lattice 60(iii) The first Brillouin zone 62

G. Quantitative Calculations Involving the Geometry of the Lattice 64(i) The distances and angles in the direct lattice 64(ii) Interplanar spacings, angles between planes, and cell volume 67

H. Beyond the Ball and Stick Representation of Crystals 71(i) Polyhedral models 71(ii) Block representations 74(iii) The stereographic projection 78

I. Problems 81J. References and Sources for Further Study 84

3. Chapter 3, Symmetry in Crystals 85A. Introduction 86B. The Symmetry Operators 86C. The First Five Symmetry Operators 87

(i) Definitions 87(ii) The distinction between proper and improper operations 91

G.S. Rohrer p. 19

Table 1, continued. Table of Contents for Structure an Bonding in Crystalline Materials

D. Combining Operators to Form the 32 Distinct Crystallographic Point Groups 91(i) Compatibility of rotations and translations 91(ii) Group notation 93(iii) Combinations of proper rotations 93(iv) Groups formed by improper rotations 95

E. The 32 Point Groups Classified by Bravais Lattice 97(i) Point group nomenclature 97(ii) The importance of an inversion center 99(iv) Dealing with cubic point groups 103(v) General and special positions 105(vi) Notes on determining the crystallographic point group of an object 108

F. The 230 Space Groups 109(i) The screw and glide operators 112(ii) The origin of the space groups and their symbols 113(iii) The space group symbols 114(iv) Space group representations 115(v) Trigonal and hexagonal groups 124(vi) Information in the International Tables 128

G. The Interpretation of Conventional Crystal Structure Data 129H. Problems 145I. References and Sources for Further Study 150

Chapter 4. Crystal Structures 153A. Introduction 154B. Close-Packed Arrangements 154

(i) Types of close-packing 154(ii) The ccp and hcp arrangements 157

C. The Interstitial Sites 160D. Classifying and Naming Crystal Structures. 163E. Close-Packed Metallic Structures 165

(i) The B2 or CsCl structure 165(ii) The L10 or CuAu structure 166(iii) The L12 structure 167(iv) The DO3 and L21 structures 168(v) The DO19 and DO24 structures: hcp superlattice structures. 170(vi) Filling the interstices of the bcc lattice: the A15 structure 172(vii) The C14, C15, C16 structures 173

F. Close-Packing in Ionic Compounds 176(i) The rock salt and NiAs structures 176(ii) The spinel and olivine structures 179(iii) The rutile and CdI2 structures 183(iv) Other AB2 structures, the CdCl2 and fluorite structures 185(v) The perovskite structure 187(vi) Superlattice structures 192(vii) Extensions of the close-packing description to more complex structures 193

G. Structures Derived From ReO3 193(i) Bronzes 194(ii) Structures derived by crystallographic shear 195

H. Tetrahedral Structures 196(i) The sphalerite and wurtzite structures 196(ii) Complex structures based on sphalerite and wurtzite 198

G.S. Rohrer p. 20


I. More Complex Packing Sequences 202(i) Notation for stacking sequences 202(ii) The SiC polytypes 202

J. Van der Waals Solids 204K. Noncrystalline Solid Structures 204

(i) Incommensurate structures 204(ii) Quasicrystalline structures 206(iii) Liquid crystalline structures 208(iv) Amorphous materials 209

L. Problems 211M. References and Sources for Further Study 217

5. Chapter 5, Diffraction 219A. Introduction 220B. Bragg's Formulation of the Diffraction Condition 220C. The Scattering of X-rays From a Periodic Electron Density 222

(i) The periodic electron density 222(ii) Elastic scattering 225(iii) Scattering from many centers 230(iv) Equivalence of the Bragg condition and the elastic scattering condition 233(v) The Ewald construction 234(vi) Concluding remarks 236

D. The Relationship Between Diffracted Peak Intensities and Atomic Positions 237(i) The structure factor 237(ii) Systematic absences due to lattice centering 242(iii) Systematic absences due to glide operators 244(iv) Systematic absences due to the presence of screw axes 244(v) Systematic absences and the reciprocal lattice 247

E. Factors Affecting the Intensity of Diffracted Peaks 252(i) Measuring diffracted beam intensities 252(ii) Example of a structure factor calculation 254(iii) The atomic scattering factor 255(iv) The temperature factor 256(v) The Lorentz - polarization factor 258(vi) Multiplicity 259(vii) Computing the intensities 260(viii) Preferred orientation 262(ix) The absorption correction 263(x) The extinction correction 263

F. Some Diffraction Techniques and Their Uses 264(i) Laue back-reflection 264(ii) Powder diffraction 265(iii) Single crystal methods 268(iv) Electron diffraction 270(v) Neutron diffraction 270

G. Supplementary Information 272(i) Basics of the Fourier series 272(ii) Coefficients for scattering factors 275

H. Problems 276I. Review Problems 283J. References and Sources for Further Study 286

G.S. Rohrer p. 21


6. Chapter 6, Van der Waals Bonding 287A. Introduction 288

(i) Substances held together by van der Waals bonds 288(ii) The effect of molecular weight 289(iii) The effect of molecular shape 291

B. A Physical Model for the van der Waals Bond 292(i) The origin of molecular attraction and repulsion 292(ii) The Lennard-Jones potential energy model 294(iii) Calculating the lattice constant, the cohesive energy, and the bulk modulus 296(iv) Accounting for longer range interactions 300(v) Connecting the generalizations with the physical models 303

C. Interactions of molecules with permanent dipole moments 306D. Problems 309E. References and Sources for Further Study 311

7. Chapter 7, Ionic Bonding 312A. Introduction 313

(i) Solids that are held together by ionic bonds 313(ii) The effect of charge and separation on bond strength 313(iii) The effect of size on the structure 314

B. A Physical Model for the Ionic Bond 316(i) The formation of an ionic bond 316(ii) The ionization energy and electron affinity 317(iii) Estimating lattice energies and stability using the Born-Haber cycle 320(iv) The Coulombic bonding force and the ionic pair potential 322(v) Ionic bonding in the crystal 326(vi) The Madelung constant 328(vii) Evaluation of the Madelung constant 330(viii) Calculating the equilibrium separation, lattice energy, and compressibility 333

C. Refinements to the Model 336(i) The van der Waals attraction 336(ii) The zero-point energy 337(iii) The crystal-field stabilization energy 337(iv) Polarization 341

D. Predicting the Structures of Ionic Compounds 343(i) The minimum energy structure 343(ii) Ionic radius and coordination 345(iii) The influence of polarization on structure 347

E. Electronegativity Scales 349F. The Correlation of Physical Models with the Generally Observed Trends 353G. Problems 355H. References and Sources for Further Study 358I. Pauling Electronegativities . 360J. Table of Ionic Radii 361

8. Chapter 8, Metallic Bonding 364A. Introduction 365

(i) Materials that are held together by metallic bonds 366(ii) Variations in the bond strength 366(iii) Qualitative free electron theory 367

G.S. Rohrer p. 22


B. A Physical Model for the Metallic Bond Free Electron Theory 368(i) Wave functions, operators, and expectation values 368(ii) Formulation of the free electron theory 371(iii) The energy density of states 379(iv) The free electron energy 381

C. Failures of the Free Electron Theory 394D. Electrons in a Periodic Lattice 394

(i) The relationship between reciprocal space and wave vector space 394(ii) Plotting dispersion in the reduced zone scheme 396(iii) The nearly free electron theory: the origin and magnitude of the energy gap 398(iv) Metals and insulators 402

E. Correlation of the Physical Models with General Trends 405F. Problems 406G. Further Reading 408

9. Chapter 9, LCAO Model for the Polar Covalent Bond 409A. Introduction 410

(i) Which solids have covalent bonds 410(ii) The crystal structures of covalent solids 411(iii) The effect of polarity and separation on the bond strength and band gap 412

B. A Physical Model For the Covalent Bond in a Molecule 413(i) Atomic orbitals 414(ii) The homopolar bond in a diatomic molecule 417(iii) The polar covalent bond 422(iv) Bond polarity 425(v) The general case 426

C. Physical Model for the Covalent Bond in a Crystal 428(i) Introduction 428(ii) A homopolar crystal with a square lattice 428(iii) The effect of polarity 435(iv) The ccp structure 442(v) The sphalerite structure 455

D. Transition Metals 458E. Orbital Hybridization 462F. The Cohesive Energy of a Covalently Bonded Solid 465G. Overview of Results From the LCAO Theory 468H. Structural Implications of the LCAO Model 470I. Problems 472J. Further Reading 476

10. Chapter 10, Models for Predicting Phase Stability and Structure 477A. Introduction 478B. Compound formation 479

(i) Miedema's Rules for Alloy Formation 479(ii) Villars' method for predicting compound formation 486

C. Electron Concentration and Radius Ratio rules for Intermetallic Phases 491(i) Hume-Rothery rules and electron compounds 491(ii) Laves Phases 496(iii) Interstitial Compounds 499

G.S. Rohrer p. 23


D. Factors that Determine Structure in Polar-Covalent Crystals 501(i) Pauling's rules 501(ii) The bond valence method 505(iii) The maximum volume principle 511(iv) Cation eutaxy 513(v) Sanderson's description of the polar covalent bond 516

E. Structure Stability Diagrams 526(i) Mooser-Pearson Plots 526(ii) Phillips ionicity 531(iii) Villars' structure stability diagrams 534

F. Problems 539G. References and Sources for Further Study 540

G.S. Rohrer p. 24

Table 2. The structure of rutile (C4), titanium dioxide

Formula Unit TiO2, titanium dioxideSpace group: P42/mnm (no. 136)Cell dimensions: a = 4.594Å, c = 2.958ÅCell contents: 2 formula unitsAtomic positions: Ti in (2a) mmm 0 0 0; 1

21

21

2O in (4f) m2m x x 0; x x 0

( 1

2+ x ,

1

2− x ,

1

2) ,( 1

2− x ,

1

2+ x ,

1

2)

x ~ 0.3Examplescompound a(Å) c(Å) x compound a(Å) c(Å) x

TiO2 4.5937 2.9581 0.305 CrSbO4 4.57 3.042CrO2 4.41 2.91 CrTaO4 4.626 3.009GeO2 4.395 2.859 0.307 FeNbO4 4.68 3.05IrO2 4.49 3.14 FeSbO4 6.623 3.011

β-MnO24.396 2.871 0.302 FeTaO4 4.672 3.042

MoO2 4.86 2.79 GaSbO4 4.59 3.03NbO2 4.77 2.96 RhSbO4 4.601 3.100OsO2 4.51 3.19 RhVO4 6.607 2.923PbO2 4.946 3.379 CoF2 4.6951 3.1796 0.306RuO2 4.51 3.11 FeF2 4.6966 3.3091 0.300SnO2 4.7373 3.1864 0.307 MgF2 4.623 3.052 0.303TaO2 4.709 3.065 MnF2 4.8734 3.3099 0.305WO2 4.86 2.77 NiF2 4.6506 3.0836 0.302

AlSbO4 4.510 2.961 0.305 PdF2 4.931 3.367CrNbO4 4.635 3.005 ZnF2 4.7034 3.1335 0.303

G.S. Rohrer p. 25

Table 3. The structure of scheelite, CaWO4

Formula unit CaWO4Space group: I41/a (no. 88)Cell dimensions: a = 5.24 Å, c =11.38 ÅCell contents: 4 formula unitsAtomic positions:

Ca (4b) 0,0,1/2; 1/2,0,1/4; + IW (4a) 0,0,0; 0,1/2,1/4; + I

O (16f)x,y,z; x, y,z; x,y + 1 / 2,1 / 4 − z;x,1 / 2 − y,1 / 4 − z;

y,x, z; y,x, z; y,x + 1 / 2,z + 1 / 4;y,1 / 2 − x,z + 1 / 4; + Ix=0.25, y=0.15, z=0.075

Examplescompound a(Å) c(Å) compound a(Å) c(Å)AgReO4 5.378 11.805 NaReO4 5.362 11.718AgTcO4 5.319 11.875 NaTcO4 5.339 11.869BaMoO4 5.56 12.76 PbMoO4 5.47 12.18BaWO4 5.64 12.70 PbWO4 5.44 12.01BiAsO4 5.08 11.70 RbReO4 5.803 13.167CaMoO4 5.23 11.44 SrMoO4 5.36 11.94CdMoO4 5.14 11.17 SrWO4 5.40 11.90CeGeO4 5.045 11.167 ThGeO4 5.14 11.54HfGeO4 4.849 10.50 TlReO4 5.761 13.33KReO4 5.675 12.70 UGeO4 5.084 11.226KRuO4 5.609 12.991 YNbO4 5.16 10.91KTcO4 5.654 13.03 ZrGeO4 4.871 10.570

G.S. Rohrer p. 26

Table 4 Approximate Schedule for Structure and Bonding in Crystalline Materials

Chapter Number and Title Number of Lectures

1. Introduction 1.5

2. Basic Structural Concepts 1.5

3. Symmetry in Crystals 3

4. Crystal Structures 3

5. Diffraction 4

6. Secondary Bonding 1

7. Ionic Bonding 2.5

8. Metallic bonding 2.5

9. LCAO Model for the Polar Covalent Bond 4

10. Models for Predicting Phase Stability and Structure 3

G.S. Rohrer p. 27

Table 5. Student Evaluations of the Course

Year Mean Course Rating Number of Students

1992 4.07 17

1993 3.65 27

1994 4.45 22

1995 4.52 23

1996 4.65 20

1997 NA 21

Structure and Bonding in Crystalline Materials: A Textbook...

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