Transcript

Solid State Chemistry

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Solid State Chemistry

byR.C. Ropp WarrenNew Jersey USA

2OO3

ELSEVIER AMSTERDAM - BOSTON- HEIDELBERG- LONDON- NEW YORK- OXFORD PA R IS - S A N D I E G O - S A N F R A N C I S C O - S I N G A P O R E - S Y D N E Y - T O K Y O

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

9 2003 Elsevier Science B.V. All rights reserved.

This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-maih [email protected]. You may also complete your request on-line via the Elsevier Science homepage (http://www.elsevier.com), by selecting 'Customer Support' and then 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Science & Technology Rights Department, at the phone, fax and email addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2003 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.

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Preface

Most of the

books

concerning

"Solid

State

Chemistry"

that

I have

e xamin ed have been w r i t t e n for the specialist or for the advanced s t u d e n t . I have c o m p o s e d this book with the novice in mind. That is, those w h o have little b a c k g r o u n d but w i s h to begin to learn about the solid state and w h a t it entails can start at the be gi nn i ng and build u p o n their knowledge and experience. This includes those working in biotechnology, p h a r m a c e u t i c a l a n d p r o t e o m i c s own the

fields. At this point in

time, they have s e q u e n c e d the h u m a n g e n o m e and are trying to define t h e s t r u c t u r e of proteins. It is m y opinion t h a t the material in this book will be helpful to t h e m as well. With this in mind, I have p r e s e n t e d the information in this book in a f o r m t h at can be easily u n d e r s t o o d . I t h i n k that it is quite i m p o r t a n t t hat any s t u d e n t of the body of knowledge t h a t we call "science" needs to be cognizant of the history a nd effort t hat has been m a d e by those w h o p r e c e d e d us. It was Newton who said: "If I have seen further, it is because I have stood on the s h o u l d e r s of giants". Thus, I have tried to give a s h o r t history of each particular s e g m e n t of solid state theory and technology. I have enjoyed c o m p o s i n g the material in this book and t r u s t t h a t t h e reader can use it as a self-study to build u p o n his (her) knowledge. I t h a n k m y wife, Francisca Margarita, profusely for her s u p p o r t during t h e time t h a t it has t a k e n to finish this c o m p o s i t i o n . R.C. R o p p March 2 0 0 3 store of usable

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Table off contentsPreface Table of Contents CHAPTER 1 - The Phase Chemistry of Solids 1 . 1 . - Phase Changes of Solids, Liquids and Gases 1.2.- Differences Between the Three States of Matter a. The Gaseous State b. The Liquid State c. The Solid State 1.3.- The Close-Packed Solid 1.4. Phase Relations Between Individual Solids References Cited Problems for Chapter 1 CHAPTER 2- Determining the Structure of Solids 2.1.- Scientific Basis for Determining the Structure of Solids 2.2.- Solid State Structure Conventions and Protocols 2.3.- How to Determine the Structure of Compounds 2.4.- Symmetry Distribution of Crystals 2.5.- Phase Relationships Among Two or More Solids SUGGESTED READING Problems for Chapter 2 CHAPTER 3- Defects in Solids 3.1.- The Defect Solid a. The Point Defect in Homogeneous Solids b. The Point Defect in Heterogeneous Solids c. The Line Defect d. The Volume Defect 3.2.- Mathematics and Equations of the Point Defect a. The Plane Net 3.3.- Non-Stoichiometric Solids 3.4.- Defect Equation Symbolism

PaBei iii

2 9 12 15 17 23 27 28 31 31 45 55 61 64 69 69 71 73 74 72 82 85 88 88 95 98

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3.5.- S o m e Applications For Defect C h e m i s t r y a. P h o s p h o r s 3.6.- Defect Equilibria a n d Their E n e r g y 3.7. Defect Equilibria in Various Types of C o m p o u n d s a. S t o i c h i o m e t r i c Binary C o m p o u n d s of MXs b. Defect C o n c e n t r a t i o n s in MXs C o m p o u n d s 3.8.- The Effects of Purity (And I m p u r i t i e s ) Suggested Reading P r o b l e m s for C h a p t e r 3 A p p e n d i x 1. C o n c e n t r a t i o n s of Defects in Non-Ionized a n d Non-Stoichiometric Compounds A. Defects in N o n - S t o i c h i o m e t r i c MXs + d C o m p o u n d s A p p e n d i x I I. Analysis of a Real Crystal Using the Thermodynamic Method A. The AgBr Crystal with a Divalent Impurity, Cd 2+ B. Defect Disorder in AgBr- A T h e r m o d y n a m i c A p p r o a c h AoDendix III. Statistical M e c h a n i c s a n d the Point Defect CHAIrI'ER 4- M e c h a n i s m s a n d Reactions in the Solid S t a t e 4. I.- P h a s e C h a n g e s 4.2.- The Role of P h a s e B o u n d a r i e s in Solid State R e a c t i o n s 4.3.- Reaction Rate P r o c e s s e s in Solids 4.4.- Defining H e t e r o g e n e o u s Nucleation P r o c e s s e s 4.5.- The T a r n i s h i n g R e a c t i o n 4.6.- Fick's Laws of Diffusion 4.7.- Diffusion M e c h a n i s m s 4.8. - Analysis of Diffusion R e a c t i o n s 4.9. - Diffusion in Silicates 4. I0.- Diffusion M e c h a n i s m s W h e n the Cation C h a n g e s V a l e n c e P r o b l e m s for C h a p t e r 4 A p p e n d i x I- Math Associated with Nucleation Models of 4 . 4 . 1 . A p p e n d i x II- Math Associated w i t h Incipient Nuclei G r o w t h A p p e n d i x III- H o m o g e n o u s Nucleation P r o c e s s e s A p p e n d i x IV- Diffusion E q u a t i o n s Relating to F u n d a m e n t a l Vibrations of the Lattice

99 100 101 103 104 107 110 112 113 115 115 118 118 120 124 129 130 132 138 140 146 148 151 156 161 171 175 177 182 184 188

CHAPTER 5- Particles and Particle Technology 5.1.- Sequences in Particle Growth 5.2.- Sintering, Sintering Processes and Grain Growth 5.3.- Particle Size 5.4.5.5.5.6.5.7.Particle Distributions Particle Distributions and the Binomial T h e o r e m Measuring Particle Distributions Analysis of Particle Distribution P a r a m e t e r s A. The Histogram B. Frequency Plots C. Cumulative F r e q u e n c y D. Log Normal Probability M e t h o d 5.8.- Types of Log Normal Particle Distributions A. Unlimited Particle Distributions B. Limited Particle Distributions C. Particle Distributions with Discontinuous Limits D. Multiple Particle Distributions Case I: Fluorescent Lamp Phosphor Particles Case II: T u n g s t e n Metal Povcder 5.9.- A Typical PSD Calculation 5.10.- Methods of Measuring Particle Distributions A. The Microscope- Visual Counting of Particles B. Sedimentation Methods C. Electrical Resistivity-The Coulter Counter D. Other Methods of Measuring Particle Size Permeability Gas adsorption Particle size by laser r e f r a c t o m e t r y Suggested Reading Problems for Chapter 5 Chapter 6. - Growth of Crystals 6. I.- Methods of Growth of Crystals 6.2.- Furnace Construction 6.3.- Steps in Growing a Single Crystal

191 192 193 203 207 209 213 217 218 218 219 220 222 223 223 224 225 226 228 229 232 233 237 241 245 245 245 247 248 249 251 252 253 258

Czochralski Growth of Single Crystals 6.5.- The Bridgeman-Stockbarger Method for Crystal Growth 6.6.- Zone Melting as a Means for Forming Single Crystals 6.7.- Zone Refining 6.8.- The Impurity Leveling Factor 6.9.- The Verneufl Method of Crystal Growth 6. I0.- Molten Flux Growth of Crystals 6.11.- Hydrothermal Growth 6.12.- Vapor Methods Used for Single Crystal Growth 6.13.- Edge Defined Crystal Growth 6.14.- Melting and S t o i c h i o m e t r y.4. -

260 270 274 275 278 282 285 288 292 294 296 299 302 303 308 308 310 311 313 315 318 328 328 329 330 330 334 337 338 343 345 350 351 352

6.15.- Actual Imperfections in Crystals 6.16.- Electronic Properties of Crystals A. Conductivity in Ionic Compounds 6.17.- Silicon Single Crystals and Integrated Circuits A. Silicon B. Silicon as a S e m i - C o n d u c t o r C. Semiconductor Devices D. Integrated Circuits E. Manufacture of Integrated Circuits F. Steps in the Manufacture of Integrated Circuits G. Film Deposition H. Impurity Doping I. Lithographic P a t t e r n i n g J. E t c h i n g K. A Final Look at the IC Manufacturing Process L. Crystal Growth and Crystal Defects Affecting IC's 6.18.- Future Methods for Manufacture of Integrated Circuits 6.19.- Pushing the Limits of Semi-Conductor Technology 6.20.- The Solar Cell A. Crystalline, PolycrystaUine and Amorphous Solar Cells B. Thin Film Solar Cells 6.21.- Piezo-electric materials and Their Uses A. Applications Sonar

xi

Medical U l t r a s o u n d Micro-positioning and M i c r o - m o t o r s Piezoelectric T r a n s f o r m e r s Active Noise a n d Vibration D a m p i n g SUGGESTED READING References on Silicon Devices Problems for C h a p t e r 6 Chapter 7- M e a s u r e m e n t of Solid State P h e n o m e n a 7.1.- Methods of M e a s u r e m e n t of Solid State Reactions A. Differential T h e r m a l Analysis (DTA) B. Differential S c a n n i n g Calorimetry 7.2.- Utilization of DTA and DSC A. Applications of DTA B. Uses of DSC 7.3.- T h e r m o g r a v i m e t r y 7.4.- D e t e r m i n a t i o n of Rate Processes in Solid State Reactions A. Types of Solid State R eact i ons B. The Freeman-Carroll Method applied to DTA Data C. The Freeman-Carroll Method Applied to TGA Data 7.5.- Dilatometry 7.6.- T h e r m o m e t r y 7.7.- Application of Dilatometry to Plastic Materials 7.8.- Optical M e a s u r e m e n t s of Solids A. Defining Light B. M e a s u r e m e n t of Color C. The Nature of Light D.- Absorbance, Reflectivity and T r a n s m i t t a n c e E. M e a s u r e m e n t of Color I. The S t a n d a r d Observer II. The Nature of C h r o m a III. Intensity and S c a t t e r i n g IV. Color Processes and Color Matching S y s t e m s I. T r i s t i m u l u s Coefficients

353 353 353 353 353 354 354 357 357 358 374 376 376 380 381 389 389 392 393 394 401 403 405 405 409 410 411 415 417 417 418 420 426

F. The S t a n d a r d Observer and The First Color-Comparator 4 2 1

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II. Chromaticity Coordinate Diagrams 7.9.- Color Spaces A. The Munsell Color T r e e B. Color Matching and MacAdam Space R e c o m m e n d e d Reading Problems for Chapter 7

42 43 43 43 43 44

Chapter 1The P h a s e C h e m i s t r y of Solids

ii

....

[]

To u n d e r s t a n d the solid state, we m u s t first u n d e r s t a n d how m a t t e r exists. T h a t is, m a t t e r w a s originally defined as "anything t h a t occupies s p a c e a n d h a s weight". This definition w a s m a d e in the e i g h t e e n t h c e n t u r y w h e n little w a s k n o w n a b o u t "matter" . T h u s , it w a s defined in t e r m s of its general s h a p e a n d physical c h a r a c t e r i s t i c s . We will delve into the "ouflding-blocks" of m a t t e r a n d how solids originate. It is a s s u m e d t h a t you have some initial knowledge c o n c e r n i n g p h y s i c s a n d c h e m i s t r y as well as s o m e ability in m a t h e m a t i c s . The m o r e difficult m a t h e m a t i c a l t r e a t m e n t s are reserved to the a p p e n d i c e s of certain c h a p t e r s . Originally, in the 18th century, w h e n the "Scientific Revolution" w a s j u s t getting started, science w a s referred to as "Natural Philosophy". R e s e a r c h e r s were i n t e r e s t e d in n a t u r a l science, including w h a t we call today: Physics, Chemistry, Biology a n d Biochemistry. All of t h e s e scientific disciplines deal with molecules having various physical aspects and properties (even proteins). The earliest w o r k dealt with air, w a t e r a n d solids since t h e s e were the easiest to m a n i p u l a t e in t h o s e times. Note t h a t we r e g a r d all of their scientific a p p a r a t u s to be c r u d e by o u r s t a n d a r d s . Yet, even the m o s t s o p h i s t i c a t e d i n s t r u m e n t s (by their criteria) h a d to be built b y h a n d . Even so, early w o r k e r s k n e w t h a t m a t t e r existed in three forms, i.e.- solids, liquids a n d gases. W a t e r w a s s t u d i e d extensively b e c a u s e it w a s e a s y to freeze a n d boil. As we will show, m a n y of o u r basic m e a s u r e m e n t s s t a n d a r d s r e s u l t e d from s u c h studies. Reversible t r a n s f o r m a t i o n s b e t w e e n the t h r e e forms (phases) of m a t t e r , are called "Changes of State". The science related to t h e s e p h a s e c h a n g e s lies in the r e a l m of "Physical Chemistry" while their a c t u a l "chemistry", or "reactiveproperties", lies either in the r e a l m of "Inorganic" or "Organic" chemistry. Inorganic materials include both elements and compounds. Organic m a t e r i a l s have a c a r b o n b a c k b o n e . Nearly all inorganics are solids at a n d scientific

a m b i e n t t e m p e r a t u r e . At elevated t e m p e r a t u r e s , t h e y t r a n s f o r m or melt to form a liquid a n d t h e n a gas. Most of t h e s e p h a s e c h a n g e s are reversible. A

few are g a s e s at r o o m t e m p e r a t u r e , b u t liquids are rare. In contrast, organic compounds can be either solid, liquid, or gaseous. Although organic c o m p o u n d s do u n d e r g o some C h a n g e s - o f - S t a t e easily a n d reversibly, their t h e r m a l stability is a problem. If one tries to i n d u c e a c h a n g e of s t a t e in an organic c o m p o u n d , one u s u a l l y finds t h a t liquid organic c o m p o u n d s will freeze a n d / o r sublime. Notwithstanding, m o s t organic solids d e c o m p o s e if the g a s e o u s state is sought. This is due to their relatively low t h e r m a l stability, i.e.- "they burn". Organic h y d r o c a r b o n c o m p o u n d s u s u a l l y r e a c t with oxygen and convert to c a r b o n to 600 ~ because to 2 0 0 0 ~ of dioxide and water at r a t h e r low t e m p e r a t u r e s of 200 ~ interested in while m a n y inorganic c o m p o u n d s are a n d even higher. We will be m o r e their greater thermal stability.

stable at r a n g e s of 1500 ~ inorganics

Additionally, inorganic c o m p o u n d s have physical properties t h a t c a n n o t be d u p l i c a t e d in the organic d o m a i n of chemistry, a n d vice-versa. 1.1.- PHASE CHANGES OF SOLIDS, LIQUIDS AND GASES The b e s t w a y to u n d e r s t a n d p h a s e c h a n g e s is to e x a m i n e those observed with water, i.e.- t h o s e involving the molecule, H20. We are, of course, familiar with the three s t a t e s of water, n a m e l y ice, w a t e r a n d s t e a m , since these forms are e n c o u n t e r e d daily. The a c t u a l difference b e t w e e n t h e m is a m a t t e r of energy. To illustrate this fact, let u s calculate how m a n y calories are required to c h a n g e ice into w a t e r into s t e a m . This p r o b l e m is typical of those c o n c e r n i n g Changes-of-State. In solving s u c h a problem, we will also define certain c o n c e p t s a n d c o n s t a n t s relating to its solution. For example, we k n o w t h a t w a t e r (a liquid) will c h a n g e to ice (a solid) ff its internal temperature falls below a certain t e m p e r a t u r e . Likewise, ff its i n t e r n a l t e m p e r a t u r e rises above a certain point, w a t e r c h a n g e s to s t e a m (a gas). B e c a u s e w a t e r is so a b u n d a n t on the Earth, it w a s u s e d in the p a s t to define C h a n g e s of S t a t e a n d even to define T e m l ~ r a t u r e Scales. However, the concept of "heat" is also involved, a n d we need to also define the perception of h e a t as it is u s e d in this context. Note t h a t defining h e a t implies t h a t we have a reproducible w a y to m e a s u r e t e m p e r a t u r e . A great deal of w o r k w a s r e q u i r e d in the p a s t to r e a c h t h a t stage. First, you have to e s t a b l i s h t h a t certain liquids e x p a n d w h e n heated. T h e n you m u s t e s t a b l i s h

how m u c h t h e y expand. You c a n do this b y malting a glass capillary {a glass t u b e with a u n i f o r m small-bore hole in it} a n d t h e n m e a s u r i n g how far the liquid m o v e s from a given t e m p e r a t u r e point to a n o t h e r . The easiest w a y to do this is to establish the freezing point of w a t e r a n d its boiling point on y o u r "thermo" meter. Lastly, you t h e n d e t e r m i n e if y o u r c h o s e n liquid e x p a n d s in a linear m a n n e r b e t w e e n t h e s e two t h e r m a l points (For a history, see p. 401}. There are two h e a t factors involved in a n y p h a s e change. T h e y are: "heat capacity", i.e.- Cp or C v , a n d "heat of transformation", u s u a l l y d e n o t e d as H. The former factor is c o n c e r n e d with i n t e r n a l t e m p e r a t u r e c h a n g e within a n y given m a t e r i a l (solid, liquid or gas) w h e r e a s t h e latter is involved in c h a n g e s of s t a t e of t h a t material. The a c t u a l n a m e s we u s e to describe H d e p e n d u p o n the direction in w h i c h the p h a s e c h a n g e occurs, vis: 1.1.1.C h a n g e s of S t a t e in W a t e r TEMPERATURE CHANGE OF STA.TE ice to w a t e r w a t e r to ice w a t e r to s t e a m s t e a m to w a t e r HLHEAT OF: Fusion Solidification Vaporization Condensation ~ 0 0 100 100 OF 32 32 212 212

The terms, heat, h e a t c a p a c i t y a n d h e a t of t r a n s f o r m a t i o n were originally defined in t e r m s of water. The earliest concept w a s t h a t of "heat", b u t it w a s soon f o u n d t h a t every m a t e r i a l h a d its own "heat capacity". T h a t is, the u n i t of "heat" (or i n t e r n a l energy) itself w a s originally defined as the a m o u n t of energy r e q u i r e d to raise the t e m p e r a t u r e of 1.00 g r a m of w a t e r b y 1.00 degree Celsius, the u n i t of h e a t energy being set equal to one {1.0) calorie. Heat c a p a c i t y itself w a s originally defined as the a m o u n t of h e a t r e q u i r e d to raise the t e m p e r a t u r e of one {1) cubic c e n t i m e t e r (cc.) of w a t e r (whose

density w a s later defined as 0.9999 @ 3.98 ~ by one (1.00) degree. However, after tbJs concept h a d b e e n defined, it w a s f o u n d t h a t the s a m ea m o u n t of h e a t d i d n o t raise the i n t e r n a l t e m p e r a t u r e of other m a t e r i a l s to the same degree. This led to the concept of h e a t capacity. Heat of t r a n s f o r m a t i o n w a s discovered s o m e w h a t later. How t h e s e c o n c e p t s arose c a n be illustrated as follows.

S u p p o s e you decide to solve the p r o b l e m of defining the c o n c e p t s of h e a t c a p a c i t y a n d h e a t of t r a n s f o r m a t i o n . The former is hhe a m o u n t of energy required to c h a n g e the t e m p e r a t u r e of a m a t e r i a l while the latter relates to how m u c h energy is n e e d e d to c h a n g e from one p h a s e to another. Defining t h e s e c o n c e p t s is a n intellectual r e s e a r c h exercise in itself. In doing this work, it is clear t h a t you m u s t be able to define s o m e sort of m e a s u r e of energy, or heat, itself (even t h o u g h you have not yet defined a u s a b l e scale of heat). Since w a t e r is plentiful (it covers a b o u t 75% of the E a r t h ' s surface), you decide to u s e w a t e r as a s t a n d a r d . You begin by distilling w a t e r to obtain a p u r e p h a s e . You will find t h a t this is difficult since even the a p p a r a t u s you u s e will c o n t a m i n a t e the distilled w a t e r ff you are not careful. After observing its behavior as its t e m p e r a t u r e is changed, you decide to set u p a t e m p e r a t u r e scale in t e r m s of water. In this work, you define the meltW_g point of ice as 0 . 0 0 0 ~ a n d the boiling point of w a t e r as 100.000 ~ (C is "Celcius", n a m e d after the r e s e a r c h e r who did the original w o r k in 1734). You do this b e c a u s e t h e s e are easily observable points in a n a r b i t r a r y scale of t e m p e r a t u r e as related to h e a t (and more i m p o r t a n t l y are reproducible a n d reversible points). As we said, the m o s t i m p o r t a n t p a r t of this w o r k is to obtain a s a m p l e of p u r e water. M u c h effort w a s e x p e n d e d in the p a s t in a c c o m p l i s h i n g this goal. T a k i n g y o u r p u r e p h a s e of water, you t h e n m e a s u r e its d e n s i t y as a function of t e m p e r a t u r e . Note t h a t you have already defmed a "specific volume" for u s e in y o u r work, ie- cubic c e n t i m e t e r = cc. You d e t e r m i n e t h a t the point of m a x i m u m density of w a t e r o c c u r s at 3 . 9 8 0 ~ d e c r e a s e s b o t h as you a p p r o a c h 0 . 0 0 ~ a n d t h a t the d e n s i t y of w a t e r a n d its boiling point of 100.00 ~

You t h e n assign the a r b i t r a r y value of 1.000 g r a m / 1 . 0 0 0 ml. of w a t e r (or 0 . 9 9 9 8 g r a m / c c , as later more precise m e a s u r e m e n t s have revealed) as the d e n s i t y of w a t e r at its t e m p e r a t u r e of m a x i m u m density. By t h e n m e a s u r i n g the a m o u n t 1.000 ~ of h e a t r e q u i r e d to raise the t e m p e r a t u r e of 1.000 cubic by c e n t i m e t e r of w a t e r b y 100th of y o u r a r b i t r a r y t e m p e r a t u r e range, i.e. 1.000 calorie). You have now defined "heat" itself as the a m o u n t of energy r e q u i r e d to raise 1 cc. of w a t e r by one degree as one calorie. However, the t e m p e r a t u r e at

you have also defined y o u r s t a n d a r d u n i t of h e a t (and called it

w h i c h you m e a s u r e d y o u r s t a n d a r d calorie w a s not specified. For this reason, you choose the t e m p e r a t u r e r a n g e of 14.5 ~ to 15.5 ~ b e c a u s e you have d e t e r m i n e d t h a t this r a n g e is one w h e r e the d e n s i t y of w a t e r r e m a i n s relatively uniform. T h u s , the s t a n d a r d calorie w a s originally defined as the a m o u n t of h e a t r e q u i r e d to raise the t e m p e r a t u r e of o n e g r a m of w a t e r from 14.5 ~ to 15.5 ~ at a c o n s t a n t p r e s s u r e of one (1) a t m o s p h e r e . Note t h a t in y o u r r e s e a r c h to define heat, h e a t c a p a c i t y a n d h e a t of t r a n s f o r m a t i o n , you have discovered several a n o m a l i e s of w a t e r t h a t h a d not b e e n k n o w n before y o u r w o r k w a s accomplished. Next, you now m e a s u r e o t h e r m a t e r i a l s to deterrrdne how m u c h energy is required to raise their t e m p e r a t u r e by one degree a n d find t h a t the a m o u n t of energy ,caries from m a t e r i a l to material. In this way, you e s t a b l i s h the notion of h e a t capacity. However, you also note t h a t e a c h m a t e r i a l h a s its own i n t e r n a l energy (heat capacity) at a given t e m p e r a t u r e in relation to t h a t of water. The next step t h a t you t a k e is to s h o w t h a t the t e m p e r a t u r e of ice does n o t c h a n g e as it m e l t s (or t h a t the t e m p e r a t u r e of w a t e r does not c h a n g e as it freezes). This t a k e s several very careful m e a s u r e m e n t s to e s t a b l i s h this fact. It is this concept t h a t e s t a b l i s h e d t h a t there are two k i n d s of "heat" involved in Changes-of-State, n a m e l y t h a t of h e a t of t r a n f o r m a t i o n a n d h e a t capacity. T h u s , the i n t e r n a l t e m p e r a t u r e of a m a t e r i a l c h a n g e s in a linear m a n n e r as energy is a d d e d until the point of t r a n s f o r m a t i o n (phase change) occurs. Then, the i n t e r n a l t e m p e r a t u r e does n o t c h a n g e u n t i l the p h a s e c h a n g e is complete. Reiterating, h e a t of t r a n s f o r m a t i o n , or H, is involved in c h a n g e of form of the m a t e r i a l while h e a t c a p a c i t y relates to its i n t e r n a l c h a n g e in t e m p e r a t u r e as it a p p r o a c h e s a n o t h e r point of change. Both of t h e s e c o n s t a n t s are b a s e d u p o n the s t a n d a r d of energy, or heat, of one (1.00) calorie. Heat c a p a c i t y is also k n o w n as t h e r m a l capacity. We t h u s label h e a t c a p a c i t y as Cp, m e a n i n g the t h e r m a l c a p a c i t y at c o n s t a n tpressure.

Sometimes, it is also called specific heat, m e a n i n g the ratio of

t h e r m a l c a p a c i t y of a n y given m a t e r i a l to t h a t of water, defined as 1.000.

This is the a m o u n t of h e a t in calories t h a t it t a k e s to raise a n y given material 1.00 ~ Originally, C v , the t h e r m a l capacity at c o n s t a n t v o l u m e w a s also used, b u t its u s e is rare n o w a d a y s . The r e a s o n for this is t h a t m o s t m a t e r i a l s e x p a n d over a given t e m p e r a t u r e range, a n d this complicates a n y a c c u r a t e m e a s u r e m e n t (since you have to c o m p e n s a t e for this c h a n g e in volume). We have now d e m o n s t r a t e d t h a t there are two types of "heat" involved in the physical c h a n g e s of a n y given material, one involved in c h a n g e of internal t e m p e r a t u r e of the material a n d the other d u r i n g its t r a n f o r m a t i o n . We specify its internal energy as the "heat" of a material, u s i n g HS,L,G. where S, L, or G refer to either solid, liquid or gas. Note t h a t this internal energy differs from the h e a t of t r a n s f o r m a t i o n , It. (This concept is p e r h a p s one of the m o r e difficult o n e s for one to grasp. However, the internal energy is t h a t energy the s u b s t a n c e h a s b e t w e e n HS,L,G a n d Cp is: I.I.2.at a given temperature). Thus, the relation

{ H = Cp. T}S,L,G, or

ikI-Is,L,G = Cp(S,L,G) ATs,L,G

where the a c t u a l state of m a t t e r is either S, L, or G. (Note t h a t Cp will differ d e p e n d i n g u p o n the state of m a t t e r involved). In contrast, t h e i n t e r n a l t e m p e r a t u r ethe material undergoes of a material does not change as

a c h a n g e o f s t a t e . (Thus, its internal energy does

not c h a n g e at t h a t point). Therefore, for a c h a n g e of state between solid a n d liquid, we w o u l d have: 1.1.3{ I ~ - HI.} = Cp (T(s) - T(L) ) or tdtI = Cp AT

Note t h a t AIt here is a h e a t of t r a n s f o r m a t i o n involved in a c h a n g e o f s t a t e w h e r e a s zkHs,L,G refers to c h a n g e in ~ t e r ~ d h e a t for a given state of matter. In the case of water, w e have defined its heat capacity, whereas its

heats of transformation were measured. These c o n s t a n t s r e p r e s e n t therelative a m o u n t s of energy n e e d e d to c a u s e c h a n g e s in form of the molecule, H20, from one p h a s e to a n o t h e r state. Their values are given in the following values as s h o w n in 1.1.4. on the next page:

1.1.4.Ice: Water: Steam: 0.5 c a l / g m / ~ 80 c a l . / g m . (S ~ L) 1.00 c a l / g m / ~ 5 4 0 c a l . / g i n . {L ~ G) 0.5 c a l . / g i n / ~

We c a n n o w c a l c u l a t e the a m o u n t of energy r e q u i r e d to raise o n e g r a m o f i c e at - 10 ~ following: 1.1.5.- CALORIES REOUIRED TO CHANGE 1 G r a m OF ICE TO STEAM FORM Ice Ice Water Water to S t e a m 0 AT -10 to 0 0 0 to 100 Ca 0.5 ...... 1.0 100 cal. . . . . . . C ~ 5.0 cal. &H_K --80 cal.(fusion) --540 cal. (vaporization) Total = 730.0 cal. ADDED 5.0 cal. 85.0 cal. 100.0 cal. 5 4 0 . 0 cal. to form o n e g r m n o f s t e a m at 110 ~ This is s h o w n in the

In all cases, AItx is specified in t e r m s of the type of c h a n g e of s t a t e occurring, while Cp AT is the c h m a g e i n i n t e r n a l h e a t w h i c h o c c u r s as the t e m p e r a t u r e rises (or falls). At a given c h a n g e of state, all of t h e energy goes to the c h a n g e of s t a t e a n d t h e t e m p e r a t u r e does not c h a n g e until t h e t r a n s f o r m a t i o n is complete. In t h e n e x t section, we shall see w h e r e the energy goes. The s a m e t r a n s f o r m a t i o n s t a k e place r e g a r d l e s s of the n a t u r e of t h e a t o m s composing the compound, unless, as we said above, the compound d e c o m p o s e s to s o m e o t h e r form. Let u s now e x a m i n e t h e t h e r m a l c h a n g e s we have d e s c r i b e d above in t e r m s of a g r a p h s h o w i n g t h e c h a n g e in a c t u a l t e m p e r a t u r e . This is i l l u s t r a t e d in the follo,~mg d i a g r a m , given on the n e x t page as 1.1.6. Note t h a t we have plotted the calories a d d e d on a s h o r t e n e d scale so as to

get the total calories a d d e d to the s y s t e m on one page. The s a m e n u m b e r s given in 1.1.5. are u s e d in 1.1.6. b u t it is easier to see t h a t the calories involved in C h a n g e - o f - S t a t e p r e d o m i n a t e over t h o s e w h i c h m e r e l y c h a n g e the i n t e r n a l energy, or t h e t e m p e r a t u r e , of the m a t e r i a l (here, ice or steam). The next c o n c e p t w h i c h we w i s h to e x a m i n e is t h a t of the differences b e t w e e n the three s t a t e s of m a t t e r , gases, liquids a n d solids. In t h i s case, we will fred very significant differences in their e n e r g y content, n a m e l y t h a t t h e g a s e o u s form is t h e m o s t energetic while the solid h a s t h e l e a s t energy.

1,2.- DIFFERENCES B ~ E N

THE THREE STATES OF MATTER

In order to define s u c h differences, we m u s t first s h o w how t h e s e s t a t e s differ physically from one another. We will s t a r t with gases, t h e n liquids a n d finally solids. As we shall see, the m a j o r difference b e t w e e n these s t a t e s is a m a t t e r of energy, the solid having the least energy of all. G a s e o u s molecules are free to roam, w h e r e a s m o l e c u l e s in the liquid s t a t e are b o u n d together a n d molecules in the solid state are b o u n d a n d ordered into a tight-knit structure. a. The G a s e o u s State The g a s e o u s s t a t e h a s b e e n defined as "a state of m a t t e r in w h i c h the s u b s t a n c e e x p a n d s readily to fill a n y c o n t a i n i n g vessel" (1). W h a t this m e a n s is t h a t a n y collection of molecules in the g a s e o u s state is free to move in all directions a n d t h a t the g a s e o u s molecules will fill a n y c o n t a i n e r in w h i c h t h e y are confined. For the H20 molecule in a g a s e o u s state (which h a s 3 a t o m s per molecule), there will be 9 degrees of freedom (since there are 3 d i m e n s i o n s in w h i c h it c a n move). These c a n be divided into 3-translational, 3-vibrational a n d 3-rotational degrees of freedom. These axe s h o w n in the following diagram, given as 1.2.1. on the next page. Note t h a t the drawings are exaggerated from the a c t u a l condition to illustrate the c h a n g e s in vibrational a n d rotational states. T h e s e molecules are free to move in a n y of three directons, a n d c a n rotate a n d vibrate in v a r i o u s m o d e s in a n y of the three directions as shown. Their energy s t a t e s are quantized, b u t well s e p a r a t e d in energy as s h o w n in the following diagram, also given on the next page as 1.2.2. Note t h a t the drawings are exaggerated from the actual condition to illustrate the c h a n g e s in vibrational a n d rotational states. These molecules are free to move in a n y of three directons, a n d c a n rotate a n d vibrate in v a r i o u s m o d e s in a n y of the three directions as shown. In this diagram, we have s h o w n 3 s e p a r a t e vibrational s t a t e s with rotational s t a t e s s u p e r i m p o s e d u p o n them. For o u r g a s e o u s w a t e r molecule w h i c h h a s three a t o m s per molecule, (1 oxygen a n d two h y d r o g e n atoms), the 3 vibrational degrees of freedom will have (2J+ 1 = 7) rotational s t a t e s s u p e r i m p o s e d u p o n

10

t h e m (J is the n u m b e r of a t o m s in the molecule, having q u a n t i z e d vibrational states).

If the

molecule

happened

to be

NH3,

then

the

expected

number

of

vibrational s t a t e s would be nine. If we m e a s u r e the a b s o r p t i o n s p e c t r a of a n y molecule, M X 2 , in the infra-red region of the s p e c t r u m , we will obtain

11

r e s u l t s similar to t h o s e s h o w n in the d i a g r a m given above, the exact region of the s p e c t r u m d e p e n d i n g u p o n the type of molecule p r e s e n t (i.e.- m o l e c u l a r vceight). It t h u s s h o u l d be clear t h a t e a c h g a s e o u s molecule is free to move in a n y of the three d i m e n s i o n s until it collides with either the walls of the container, or with s o m e o t h e r molecule. The average d i s t a n c e t h a t e a c h molecule m o v e s before collision is called the "mean free path". The m e a n free p a t h will be a function of b o t h the t e m p e r a t u r e a n d the p r e s s u r e of the gas. This concept arose from the Kinetic T h e o r y of G a s e s w h i c h in t u r n arose from Avagadro's Hypothesis. In 1811, Avogadro p o s t u l a t e d t h a t equal v o l u m e s of g a s e s c o n t a i n equal n u m b e r s of molecules (at a given t e m p e r a t u r e a n d pressure). Following this, J o u l e explained in 1843 t h a t the p r e s s u r e of a gas is c a u s e d b y the i n t e n s e m o t i o n of the molecules w h i c h b o m b a r d the walls of the container. The exact p r e s s u r e w a s p r o p o s e d to be proportional to the speed a n d m o m e n t u m of these molecules. Both Avogadro's a n d J o u l e ' s theories were d i s p u t e d over a n u m b e r of y e a r s on several g r o u n d s until 1906 w h e n J e a n Perrin directly observed, a n d Einstein explained, the Brownian m o v e m e n t in gases. These arguments, coupled with direct scientific observation, finally served to e s t a b l i s h t h e s e theories as Laws. However, it w a s Maxwell in 1848 who s h o w e d t h a t molecules have a

distribution of velocities a n d t h a t they do not travel in a direct line. One experimental m e t h o d u s e d to s h o w this w a s t h a t a m m o n i a molecules are not detected in the time expected, as derived from their calculated velocity, b u t arrive m u c h later. This arises t "ore the fact t h a t the a m m o n i a molecules

interdiffuse a m o n g the air moi~.cules b y i n t e r m o l e c u l a r collisions. The m o l e c u l a r velocity calculated for N:i3 molecules from the w o r k done b y J o u l ein 1843 w a s 5.0 x102 m e t e r s / s e c , at r o o m t e m p e r a t u r e . This implied t h a t the odor of a m m o n i a o u g h t to be detected in 4 millisec at a d i s t a n c e of 2.0 m e t e r s from the source. Since Maxwell observed t h a t it took several m i n u t e s , it w a s fully obvious t h a t the m o l e c u l e s did not travel in a direct p a t h . Analysis b y C l a u s i u s in 1849 s h o w e d t h a t the mmmonia molecules travel only s o m e 0.001 cm. b e t w e e n collisions with air molecules at a t m o s p h e r i c p r e s s u r e , in time intervals of a b o u t 10 -10 sec. between collisions. This m e a n t

12

t h a t t h e y d e s c r i b e a long a n d intricate p a t h in the p r o c e s s of a c q u i r i n g a d i s p l a c e m e n t of several m e t e r s . The m a t h e m a t i c s involved are intricate a n d we will not p r e s e n t t h e m here. However, t h e y a r e available for t h o s e who w i s h to s t u d y t h e m (2). Nevertheless, t h e fact t h a t t h e r e is a n average d i s t a n c e t h a t m o l e c u l e s travel b e t w e e n collisions h a s given u s the c o n c e p t of the Mean Free P a t h (which is the average d i s t a n c e b e t w e e n collisions). A b y - p r o d u c t of the w o r k b y Perrin a n d Einstein w a s the first reliable e v a l u a t i o n of Avogadro's n u m b e r , the n u m b e r of m o l e c u l e s in a mole. The b e s t c u r r e n t value is believed to be: 6.02204531 x 1023 m o l e c u l e s per k i l o g r a m mole. It is well to note, at this point, t h a t all of t h e s e o b s e r v a t i o n s are the r e s u l t of m a n y h o u r s of w o r k by prior investigators from the p a s t . T h u s , we have their experience a n d i n t u i t i o n to d r a w u p o n for a n y w o r k t h a t we m a y c a r r y forward to t h e benefit of m a n k i n d . b. The Liquid S t a t e In the liquid state, the m o l e c u l e s are still free to move in t h r e e d i m e n s i o n s b u t still have to be confined in a c o n t a i n e r in the s a m e m a n n e r as the g a s e o u s s t a t e if we expect to be able to m e a s u r e t h e m . However, there are i m p o r t a n t differences. Since the m o l e c u l e s in the liquid s t a t e have h a d energy removed from them in order to get them to condense, the t r a n s l a t i o n a l degrees of freedom are found to be restricted. This is d u e to the fact t h a t t h e m o l e c u l e s are m u c h closer together a n d c a n i n t e r a c t with one a n o t h e r . It is this i n t e r a c t i o n t h a t gives the liquid s t a t e its u n i q u e properties. T h u s , the m o l e c u l e s of a liquid are not free to flow in a n y of the three directions, b u t are b o u n d by i n t e r m o l e c u l a r forces. T h e s e forces d e p e n d u p o n the electronic s t r u c t u r e of the molecule. In the c a s e of water, w h i c h h a s two electrons on the oxygen a t o m w h i c h do n o t p a r t i c i p a t e in the b o n d i n g s t r u c t u r e , the m o l e c u l e h a s a n electronic m o m e n t , i.e.- is a "dipole". This r e s u l t s in a p r o p e r t y w h i c h we call fluid viscosity since the m o m e n t of e a c h m o l e c u l e i n t e r a c t s with all of its n e a r e s t neighbors. Yet, the s a m e vibrational a n d r o t a t i o n a l s t a t e s are still p r e s e n t b u t in a different form. T h a t is, t h e y are m u t a t e d forms of the s a m e energy levels t h a t we found in the g a s e o u s state. This is i l l u s t r a t e d in t h e following diagram:

13

Note t h a t in the above d i a g r a m there axe still vibrational s t a t e s b u t t h a t the rotational s t a t e s are "smeared" one into the other. There is little t r a n s l a t i o n a l m o t i o n for the w a t e r molecules within the interior of the liquid u n l e s s t h e y escape from the liquid p h a s e . If t h e y do so, we call this "evaporation" (This m a y be c o n t r a s t e d to the escape of molecules from a solid w h i c h we call "sublimation"). :Most liquids do have a def'med v a p o r p r e s s u r e w h i c h m e a n s t h a t molecules c a n a n d do escape from the surface of the liquid to form a gas. This is a n o t h e r r e a s o n t h a t the properties of a liquid vm3r from those of the g a s e o u s state. Hence, we still have the vibrational a n d rotational degrees of freedom left in the liquid, b u t not t h o s e of the t r a n s l a t i o n a l mode. A r e p r e s e n t a t i o n of 'water molecules in the liquid state is p r e s e n t e d in the s h o w n as 1.2.4. on the next page. (Ignore the o p e n s p a c e s since this is merely a simulation). As s h o w n in this diagram, the w a t e r molecules flu the c o n t a i n e r a n d also have a free surface from w h i c h they c a n escape. T h u s , we conclude t h a t the molecules of a liquid are free to slide p a s t one a n o t h e r b u t the overall a s s e m b l a g e of molecules does n o t have a definitive form, except t h a t of the c o n t a i n e r u s e d to hold it. For this reason, a liquid h a s b e e n defined as "a s u b s t a n c e or state of m a t t e r w h i c h h a s the capacity to flow u n d e r extremely small s h e a r s t e s s e s to conform to the s h a p e of a n y following diagram,

14

confining vessel, b u t is relatively incompressible and lacks the capacity to expand without limit" (I).1.2.4.-

Therefore, as we change the state of matter, the translational degrees of freedom in liquids become severely restricted in relation to those of the gaseous state. And, the vibrational a n d rotational degrees of freedom appear to be s o m e w h a t restricted, even t h o u g h m a n y of the liquid vibrational and rotational states have been found to be quite similar to those of the gaseous state.

15

c. The Solid S t a t e If we n o w r e m o v e m o r e ener~r from t h e liquid, it finally r e a c h e s a

t e m p e r a t u r e w h e r e it "freezes", t h a t is - it converts to a solid. W h a t h a p p e n s , in a m o l e c u l a r sense, is t h a t the m o l e c u l e s b e c o m e o r d e r e d . A n o t h e r w a y to s a y this is t h a t t h e y form a lattice-like framework. A r e p r e s e n t a t i o n of the solid s t a t e is s h o w n in ~ e foUowing diagram:

In this case, the m o l e c u l e s are f o u n d to have eu-rmnged t h e m s e l v e s in orderly rows. A l t h o u g h we c a n see only t h e top layer, t h e r e are several layers below with the exact s a m e a r r a n g e m e n t (This is not the exact a r r a n g e m e n t f o u n d in "ice" b u t is a stylized r e p r e s e n t a t i o n of the solid s t a t e of water). It s h o u l d thus be clear t h a t as we c h a n g e the state both of m a t t e r , ~ibrational the and

t r a n s l a t i o n a l degrees of f r e e d o m p r e s e n t in g a s e s b e c o m e r e s t r i c t e d in liquids ~md d i s a p p e a r in solids. For g a s e o u s molecules, r o t a t i o n a l degrees of freedorn are p r e s e n t while t h o s e of the liquid s t a t e are modified to the point w h e r e only ~ibrational s t a t e s c a n be said to truly-free s t a t e s . The s a m e c a n n o t be said for m o l e c u l e s in the solid state. In t h e solid,

16

only the vibrational s t a t e s remain, b u t t h e y do not resemble t h o s e of the g a s e o u s a n d liquid states. This is illustrated as follows:

Note t h a t t h e s e vibrational s t a t e s in the solid are not recognizable in t e r m s of those of the g a s e o u s or liquid states. And, the rotational s t a t e s a p p e a r to be completely absent. It h a s b e e n d e t e r m i n e d t h a t solids have quite different vibrational s t a t e s w h i c h are called "phonon modes". These vibrational s t a t e s are

quantized

vibrational m o d e s within the solid s t r u c t u r e w h e r e i n the

a t o m s all vibrate

together

in a specific pattern. T h a t is, the vibrations have

clearly defined energy m o d e s in the solid. The n u m b e r of p h o n o n m o d e s are limited a n d have b e e n described as " p h o n o n b r a n c h e s " w h e r e two types are present, "optical " a n d "acoustical". (These n a m e s arose due to the original m e t h o d s u s e d to s t u d y t h e m in solids). For the solid state, there will be a specific n u m b e r of p h o n o n b r a n c h e s f o u n d in the vibrational s p e c t r u m of a n y given solid, which d e p e n d s u p o n the n u m b e r of a t o m s c o m p o s i n g the solid. The n u m b e r of b r a n c h e s f o u n d in the p h o n o n m o d e s can be f o u n d from the following equations, given in 1.2.7. on the next page. C o n t r a s t this s i t u a t i o n with those of b o t h the liquid state a n d the g a s e o u s state. (What do we m e a n b y "quantized p h o n o n m o d e s in the solid?- the vibrations have specific a m o u n t s of energy a n d these m o d e s a p p e a r only as r e s o n a n t vibrations- i.e.- the molecules or a t o m s vibrate together only- at certain frequencies, d e p e n d i n g u p o n the m a s s or the

17

molecules (atoms or ions) a n d the chemical b o n d s holding the s t r u c t u r e together). 1.2.7.- N u m b e r of B r a n c h e s of P h o n o n Dispersi0.n: Equations: Acoustical = y a t o m s / m o l e c u l e Optical = 3 y -3

P h o n o n States:

For w a t e r with 3 a t o m s per mole c.ule: Acoustical Optical =3 = 6 (i.e.- [3x31 -3)

The m a j o r difference, then, between the 3 p h a s e s we have d i s c u s s e d is t h a t the solid c o n s i s t s of a n a s s e m b l a g e of

close-packed

molecules

w h i c h we have s h o w n to have a r i s e n - w h e n we r e m o v e d e n o u g h energy from the molecules so as to c a u s e t h e m to c o n d e n s e a n d to form the solid state. Let u s n o w examine the properties of a t o m s or molecules w h e n t h e y are crowded together to form a "close-packed" solid. 1.3.- THE CLOSE PACKED SOLID 'We have already said t h a t the solid differs from the o t h e r s t a t e s of m a t t e r in t h a t a long range ordering of a t o m s or molecules h a s appeared. To achieve long r a n g e order in a n y solid, one m u s t s t a c k a t o m s or molecules in a s y m m e t r i c a l w a y t o c o m p l e t e l y fill all of the space available. This is not a trivial m a t t e r since solids require t h a t all of the a t o m s be a r r a n g e d in a synm]etical p a t t e r n hi t h r e e d i m e n s i o n s . T h u s , if we could actually see these a t o m s in a solid, we w o u l d find t h a t t h e y are c o m p o s e d of specific "building blocks", w h i c h we shall call "propagation models" or "Units". (Actually, it is now possible to directly observe the p a c k i n g of a t o m s in solids, b u t t h a t is a n o t h e r story, t h a t of the atomic-force microscope). S u c h m o d e l s m u s t be entirely synm]etrical in t h r e e - d i m e n s i o n a l space so t h a t we c a n a r r a n g e t h e m properly to form a 3 - d i m e n s i o n a l solid. B e c a u s e of this limitation, we find t h a t only certain types of p r o p a g a t i o n m o d e l s will work. And, in doing so, we c a n gain f u r t h e r insight into the properties of a solid. To u n d e r s t a n d tt'ds, consider the following discussion.

18

Of t h e t h r e e - d i m e n s i o n a l m o d e l s available to u s , only c e r t a i n s h a p e s c a n b e u s e d to f o r m a s y m m e t r i c a l solid. T h e s e are e v e n - n u m b e r e d s e t s of a t o m s , a r r a n g e d in e i t h e r of t h e following forms1.3. I.- Atomic F o r m s S u i t a b l e for A s s e m b l i n g Long D i s t a n c e A r r a n g e m e n t s T e t r a g o n a l = 4 a t o m s per Unit H e x a g o n a l = 6 a t o m s p e r Unit Cubic = 8 a t o m s p e r Unit

T h e s e specific s h a p e s are s h o w n in t h e following d i a g r a m :

Note t h a t t h e r e are 4, 6 or 8 a t o m s p e r Unit, b u t o d d - n u m b e r e d u n i t s of 5, 7, or 9 a t o m s p e r Unit are n o t u s e d . If s u c h U n i t s are tried, o n e finds t h a t t h e y c a n n o t be fit t o g e t h e r in a t h r e e - d i m e n s i o n a l p a t t e r n w h i c h h a s long r a n g e o r d e r a n d s y m m e t r y (If y o u do n o t believe this, t r y it yourself. You will Fund t h a t a five-atom Unit, w h i c h is t h e h e x a g o n a l Unit given above m i n u s one atom, cannot be stacked together without losing part of t h e threed i m e n s i o n a l space. In o t h e r w o r d s , t h e r e will b e "holes" in t h e s t r u c t u r e ) . You m i g h t w o n d e r w h y we did n o t specify e i t h e r 1, 2 or 3 a t o m s p e r Unit. T h e r e a s o n lies in t h e fact t h a t 1 a t o m , or 2 a t o m s are n o t t h r e e - d i m e n s i o n a l

19

b u t are t w o - d i m e n s i o n a l (a fact t h a t you c a n a s c e r t a i n b y glueing s o m e pingpong balls together to m a k e individual m o d e l s - t h e s e c o m p o n e n t s c a n be u s e d to form t h e three Units given in 1.3.2.). Note t h a t we have now e s t a b l i s h e d t h a t only specific s h a p e s c a n be u s e d to a s s e m b l e a solid. The next q u e s t i o n t h a t n e e d s to be a n s w e r e d is how the solid w o u l d a p p e a r if we could see the a t o m s directly. There are, in general, two kinds of solids,

homogeneous

and

heterogeneous.

The former is c o m p o s e d from a t o m s t h a t are all the s a m e

a n d the latter from a t o m s not the s a m e . If the a t o m s are all of one kind, i.e.one of t h e e l e m e n t s , t h e p r o b l e m is s t r a i g h t forward. S e t s of eight a t o m s , e a c h set a r r a n g e d as a cube, will g e n e r a t e a cubic s t r u c t u r e . Two sets of three a t o m s , e a c h set of t h r e e a r r a n g e d in a triangle, will p r o p a g a t e a hexagonal pattern with three dimensional symmetry. E l e m e n t a l solids h a v i n g a t e t r a g o n a l s t r u c t u r e are very few a n d it is e a s y to a s c e r t a i n t h a t m o s t of the e l e m e n t s form s t r u c t u r e s t h a t are either cubic or hexagonal, b u t rarely tetragonal. The r e a s o n for this is t h a t t e t r a g o n a l u n i t s are m o r e conducive for the c a s e w h e r e not all of t h e a t o m s are the s a m e , i.e.the h e t e r o g e n e o u s case. One e x a m p l e of this is t h e c a s e w h e r e we have 1 p h o s p h o r o u s atom, c o m b i n e d w i t h 4 oxygen a t o m s to form the p h o s p h a t e Unit, i.e.- P04. A n o t h e r case m i g h t be w h e r e we c o m b i n e 1 c a r b o n a t o m w i t h 3 oxygen a t o m s to form a c a r b o n a t e Unit. In t h e first case, we have a t e t r a g o n a l Unit, w i t h the p h o s p h o r o u s a t o m at t h e c e n t e r of the 4 a t o m s c o m p o s i n g a tet_rahedron. In the s e c o n d case, the c a r b o n a t o m aligned w i t h 3 oxygen a t o m s w h i c h lie in t h e form of a t e t r a h e d r o n . In the first case, the t e t r a h e d r o n arises b e c a u s e of s p a t i a l preferences, w h e r e a s in the s e c o n d case, we k n o w t h a t t h e c a r b o n a t o m prefers to form t e t r a h e d r a l b o n d s . T h u s , it s h o u l d be clear t h a t the specific s t r u c t u r e f o u n d in a solid arises either b e c a u s e of s p a t i a l c o n s i d e r a t i o n s , or b e c a u s e of b o n d i n g p r e f e r e n c e s of certain atoms comprising the structure. In forming a solid from a t o m s or molecules, p a r t of t h e p r o b l e m lies in t h e fact t h a t we m u s t h a n d l e a very large n u m b e r of a t o m s or molecules. For example, 6.023 x 1023 (60.23 septiUion) a t o m s c o m p r i s e one mole a n d we m u s t s t a c k e a c h of t h e s e in a s y m m e t r i c a l m a n n e r to form the c l o s e - p a c k e d

20

solid. (As a m a t t e r of c o m p a r i s o n ,

calcium carbonate, whose molecular

weight is a b o u t 100 g r a m s / m o l e , c a n be held in b o t h of y o u r c u p p e d h a n d s . This a m o u n t c o n t a i n s the 6.0 septillion (1028) molecules). There is also a n o t h e r i m p o r t a n t factor. T h a t is, in building a solid s t r u c t u r e one finds t h a t s o l i d s t r u c t u r e s are b a s e d o n t h e l a r g e s t a t o m p r e s e n t , as well as how it s t a c k s together (its valence) in space filling-form. For m o s t inorganic c o m p o u n d s , this is the o x y g e n atom, e.g.- oxides, silicates, p h o s p h a t e s , sulfates, borates, t u n g s t a t e s , v a n a d a t e s , etc. The few exceptions involve chalcogenides, halides, hydrides, etc., b u t even in t h o s e c o m p o u n d s , the s t r u c t u r e is b a s e d u p o n aggregation of the largest atom, e.g..- the sulfur a t o m in ZnS. Zinc sulfide exhibits two s t r u c t u r e s , the sulfide atoms. Divalent zinc atoms have sphaleritea cubic same a r r a n g e m e n t of the sulfide atoms, a n d wurtzite- a h e x a g o n a l a r r a n g e m e n t of essentially the coordination in b o t h s t r u c t u r e s . The following d i a g r a m s h o w s how s u c h a solid w o u l d look on a n atomistic scale:

W h a t we have s h o w n is the surface of several rows of a t o m s c o m p o s i n g the

21

solid s t r u c t u r e . We see the o u t e r m o s t layer of the s t r u c t u r e w h i c h is likely to be the oxygen a t o m s c o m p o s i n g the c o m p o u n d . Let u s n o w consider p r o p a g a t i o n u n i t s in their space-filling aspects. As we defined t h e m above, they are solid state building blocks t h a t we c a n s t a c k in a symmeh--ical form to infinity. Thus, 4 - a t o m s will form a 3 - d i m e n s i o n a l t e t r a h e d r o n (half a c u b e is oniv 2-dkmensional) w h i c h is a valid p r o p a g a t i o n lunit. This m e a n s t h a t we c a n take tetrahech-ons a n d fit t h e m together 3dimensionally to form a s y m m e t r i c a l s t r u c t u r e w h i c h extends to infini~-. However, the s a m e is not true for 5-atoms, which foiTns a four-sided pyramid. This s h a p e c a n n o t be completely fitted together in a s y m m e t r i c a l a n d space-filling mmnner, b e c a u s e w h e n we s t a c k these p}Tamids containing 5 atoms, we find t h a t their t r a n s l a t i o n a l properties preclude formation of a s}Tnmetrical s t r u c t u r e b e c a u s e there is lost space between the p y r a m i d s which r e s u l t s in holes in hhe long-range s t r u c t u r e . However, if one m o r e a t o m is a d d e d to the pyramid, we t h e n have a n o c t a h e d r o l i w h i c h is space-filling with t r a n s l a t i o n a l properties. This r e s u l t s in a hexagonal s t r u c t u r e . Going further, c o m b i n a t i o n s of seven a t o m s are asymmetrical, b u t eight a t o m s form a cube w h i c h c a n be p r o p a g a t e d to infu-'dty to form a cubic s t r u c t u r e . Note t h a t b y t u r n i n g the top layer of four a t o m s b y 45 ~ (see 1.3.1.), we have a hexagonal u n i t w h i c h is related to the hexagonal u n i t c o m p o s e d of 6 atoms, two triangles atop of each other. By t~g ping-pong balls and" gluing t h e m together to form the p r o p a g a t i o n u n i t s s h o w n above, one c a n get a b e t t e r perspective between cubic a n d hexagonal close-packing. Although we c a n c o n t i n u e with more a t o m s per p r o p a g a t i o n unit, it is easy to s h o w t h a t all of those are related to the four basic p r o p a g a t i o n u n i t s found in the solid state, to wit: 1.3.4.Propagation Units Usually F o u n d ~ So!i'ds Tet_rahedron Octahedron Hexagon Cube (4) (6) (6 or 8) (8)

22

We t h u s conclude t h a t s t r u c t u r e s of solids are based, in general, u p o n t h e s e four (4) b a s i c p r o p a g a t i o n units, w h i c h c a n be s t a c k e d in a syrranetrical a n d space-filling form to n e a r infinity. The synm~etry will be t h a t of the largest a t o m in the s t r u c t u r e , u s u a l l y oxygen in inorganic solids. Variation of s t r u c t u r e d e p e n d s u p o n w h e t h e r the o t h e r a t o m s forming the s t r u c t u r e are larger or smaller t h a n the basic p r o p a g a t i o n u n i t s c o m p o s i n g the s t r u c t u r e . In m a n y c a s e s t h e y are smaller a n d will fit into the i n t e r s t i c e of the p r o p a g a t i o n unit, illustrated b y the PO4 - t e t r a h e d r o n m e n t i o n e d above. In this case, the 1~+ a t o m is small e n o u g h to fit into the center (interstice) of the t e t r a h e d r o n formed b y the four oxygen atoms. If we c o m b i n e it with Er 3+ (which is slightly smaller t h a n PO4), we obtain a tetragonal s t r u c t u r e , a sort of elongated c u b e of high s y m m e t r y . But if we combine it with La 3+ , w h i c h is larger t h a n PO4 3- , a monoclinic s t r u c t u r e with low s y m m e t r y results. There still is one a s p e c t of p h a s e c h e m i s t r y t h a t we have not yet a d d r e s s e d . T h a t is the case w h e r e m o r e t h a n one solid p h a s e exists. The basic properties of a solid include two factors, n a m e l y composition a n d s t r u c t u r e . We will a d d r e s s s t r u c t u r e s of solids in the next chapter. The composition of solids is one w h e r e the individual c o n s t i t u e n t s will vary ff the solid is h e t e r o g e n o u s . T h a t is, the two types of inorganic solids v a r y according to w h e t h e r t h e y are h o m o g e n e o u s or h e t e r o g e n e o u s . This is s h o w n in the following: 1.3.4.- Properties of Solids Variation In H o m o g e n e o u s Solids: H e t e r o g e n e o u s Solids: Elemental Compounds S t r u c t u r e only S t r u c t u r e a n d Composition

W h a t this m e a n s is t h a t elements (e.g.- metals) c a n have m o r e t h a n one s t r u c t u r e . For example, Fe exists in 4 s t r u c t u r e s , i.e.- Fe is t e t r a m o r p h o u s . a-Fe is the one stable at r o o m t e m p e r a t u r e . But, it t r a n s f o r m s to ~-Fe, t h e n y-Fe a n d finally ~ -Fe as the t e m p e r a t u r e is raised. These c h a n g e s are all reorganizations in packing d e n s i t y before the melting temperature is reached. In c o n s t r a s t , h e t e r o g e n e o u s solids u s u a l l y exist in one s t r u c u r a l

23

modification (a few c h a n g e s t r u c t u r e as t e m p e r a t u r e is increased) b u t t h e y exist in m a n y compositions, d e p e n d i n g u p o n how t h e y were formed. For example, a large n u m b e r of c a l c i u m silicates are known, including: 1.3.5.- Known C a l c i u m Silicates Ca2SiO 3 Ca 3 Si20v CaSi205 Ca2SiO4 CaSiO 3 of varying Ca4(ShO 17)(OH)~.

Ca4Si2Ov(OH)2

Ca9Si6021 (OH)

This is only a partial list (I c o u n t e d 58 k n o w n c o m p o u n d s

composition). E a c h h a s its own composition a n d sWacture. In order to differentiate a m o n g s u c h complicated s y s t e m s , i.e.- o x y g e n a t e d c o m p o u n d s of c a l c i u m a n d silicon, we r e s o r t to w h a t is called a "phase-diagram". A p h a s e d i a g r a m s h o w s t h o s e c o m p o u n d s w h i c h are formed w h e n varying m o l a r ratios of CaO a n d SiO 2 are r e a c t e d together. 1.-4. P h a s e Relations Between Individual Solids To differentiate a n d to be able to d e t e r m i n e the differences b e t w e e n the p h a s e s t h a t m a y arise w h e n two c o m p o u n d s are p r e s e n t (or are m a d e to r e a c t together), we u s e w h a t are t e r m e d " p h a s e - d i a g r a m s " to illustrate the n a t u r e of the i n t e r a c t i o n s b e t w e e n two solid p h a s e compositions. Consider the following. S u p p o s e we have two solids, "A" a n d "B". It does not m a t t e r w h a t the exact composition of e a c h m a y be. A will have a specifc melting point (ff it is stable a n d does not d e c o m p o s e at the M.P.) a n d likewise for the c o m p o u n d , B. We f u r t h e r s u p p o s e t h a t the M.P. of B is h i g h e r t h a n t h a t of A. F u r t h e r m o r e , we s u p p o s e t h a t A a n d B form a solid solution at all v a r i a t i o n s of composition. W h a t this m e a n s is t h a t from 100% A- 0% B to 50% A-50% B to 0% A-100% B, the two solids dissolve in one a n o t h e r to form a completely h o m o g e n o u s single p h a s e . If we plot the M.P. of the system, we find t h a t t h r e e different c u r v e s could result, as s h o w n in t h e following diagram, given as 1.4.1. on the n e x t page. In this case, the m e l t i n g point of the ideal solid solution s h o u l d i n c r e a s e linearly as the ration of B / A increases. However, it u s u a l l y does not.

24

Either a negative deviation or a positive deviation is regularly observed. In any p h a s e diagram, composition is plotted against temperature. In this way, we can see how the interactions between p h a s e s change as the t e m p e r a t u r e changes a n d the behavior as each solid p h a s e then melts. Either two-phase or three p h a s e sys t em s can be illustrated. This is shown in the following:

Here, the two-phase diagram is simplified to show a hypothetical p h a s e involving "A" and "B" c o m p o u n d s which form a solid solution from 100% A to 100% B. The solid solution is labelled as "r The melting t e m p e r a t u r e of A is higher t h a n t h a t of B. Therefore, the melting t e m p e r a t u r e of a drops as the composition becomes richer in B. At specific t e m p e r a t u r e s on the diagram (see 1. & 2.), a two-phase s y s t e m appears, t h a t of a liquid plus t h a t of a. Finally, as the t e m p e r a t u r e rises, the melt is h o m o g e n o u s a n d the solid, a , h a s melted. In the t h r e e - p h a s e system, only the relationship between A, B

25

a n d C c a n be i l l u s t r a t e d on a t w o - d i m e n s i o n a l drawing. A t h r e e - d i m e n s i o n a l d i a g r a m w o u l d be r e q u i r e d to s h o w the effect of t e m p e r a t u r e as well. The p h a s e d i a g r a m s we have s h o w n are b a s e d u p o n the fact t h a t A a n d B form solid m u t u a l l y soluble solid s t a t e solutions. If t h e y do not, i.e.- t h e y are not m u t u a l l y soluble in the solid state, t h e n the p h a s e d i a g r a m b e c o m e s m o r e complicated. As a n example, consider the foUowing, w h i c h is the c a s e of limited solid solubility b e t w e e n A a n d B. (N.B.- s t u d y the following d i a g r a m s carefully)"

Here, we have the c a s e w h e r e A & B form two slightly different c o m p o n d s , a a n d ~. The composition of a is AxBy while t h a t of ~ is AuBv , w h e r e the s u b s c r i p t s indicate the ratio of A to B (these n u m b e r s m a y be whole n u m b e r s or t h e y m a y be fractional n u m b e r s ) . At low B c o n c e n t r a t i o n s , a exists as a solid (the left side of the diagram). As the t e m p e r a t u r e increases, a m e l t is o b t a i n e d a n d a r e m a i n s a s a solid in the melt (L) {This is indicated by the details given j u s t below the s t r a i g h t line in the diagram}. As the t e m p e r a t u r e is increased, t h e n a m e l t s to form a u n i f o r m liquid. In the middle c o n c e n t r a t i o n s , r a n d g exist as two s e p a r a t e p h a s e s . At 75% A-25% B, ~ m e l t s to form a liquid p l u s solid a w h e r e a s j u s t the opposite o c c u r s at 25% A a n d 75% B. W h e n A e q u a l s B, t h e n b o t h a a n d ~ m e l t together to form the liquid melt, i.e.- the "eutectic point".

26

In two similar c a s e s b u t differing cases, i.e.- a s y s t e m with a melting point m i n i m u m a n d a n o t h e r with a different type of limited solid solubility, the behavior differs as s h o w n in the following diagram:

In the case on the left, a composition exists w h i c h in w h i c h a m i n i m u m melting point exists. Again, A a n d B form a w h i c h partially melts to form a + L. However, two forms of a also exist, a compositional a r e a k n o w n as a "miscibility gap", i.e.- " a l a n d a2". But at higher t e m p e r a t u r e s , b o t h of t h e s e melt into the single p h a s e , a. Finally, we obtain the melt p l u s a. Note t h a t at a b o u t 80% a i - 2 0 % a ~ , a melting point m i n i m u m is seen w h e r e it directly i n s t e a d of forming the t w o - p h a s e system, a + L. In the case of limited solid solubility, the p h a s e b e h a v i o r b e c o m e s m o r e complicated. Here, b o t h a a n d ~ form (where the a c t u a l composition of a is AxBy while t h a t of ~ is A u B v , as given before. Note t h a t the values of x a n d y change, b u t t h a t we still have the a phase. The s a m e holds for u a n d v of the phase). At low A c o n c e n t r a t i o n s , a exists alone while ~ exists alone at higher B c o n c e n t r a t i o n s . A region exists w h e r e the two p h a s e system, + exists. We notew t h a t as the t e m p e r a t u r e is raised, a two p h a s e s y s t e m is also seen consisting of the melt liquid p h u s a solid p h a s e . The p h a s e behavior s h o w n on the right side of the d i a g r a m arises b e c a u s e the two p h a s e s , A a n d B, have limited solubility in each other. melts

27

Now, let u s consider the case w h e r e three (3) s e p a r a t e p h a s e s a p p e a r in the p h a s e diagram, s h o w n as follows:

In this case, we have three (3) s e p a r a t e p h a s e s t h a t a p p e a r in the p h a s e diagram. T h e s e p h a s e s are a , ~, a n d o , w h o s e c o m p o s i t i o n s are: ct = AxBy, = AuBv a n d ~ - AcBd, respectively (the v a l u e s of x, y, u, v, c, a n d d all differ from e a c h other so t h a t AxBy is a specific c o m p o u n d as are the others). By s t u d y i n g this p h a s e d i a g r a m carefully, you c a n see how t h e individual p h a s e s relate to e a c h other. As you c a n see, a p h a s e d i a g r a m c a n b e c o m e quite complicated. However, in m o s t c a s e s involving real c o m p o u n d s , the p h a s e d i a g r a m s are u s u a l l y simple. Those involving c o m p o u n d s like silicates c a n be complex, b u t those involving alloys of m e t a l s s h o w simple behavior like limited solubility. R E F E R E N C E S CITED I. "Dictionary of Scientific a n d Technical Terms" - D.N. Lapedes- Editor in Chief, McGraw-Hill, New York (1978)

28

(2) "Enclyclopedia of Physics- 2 n d Ed." - Edited b y R.G. Lerner & G.L. Trigg, VCH Publishers, NY (1990). PROBLEMS FOR CHAPTER 1 i. Given I 0 . 0 g r a m s of ice at - 65 ~ required to c h a n g e it to s t e a m at 240 ~ 2. Do the s a m e for 24.0 g r a m s of ice at -11 ~ converted to boiling water. 3. Look u p the following h e a t capacities: Br2 CO2 PCI3 CH4 NH3 C2H4 NH3 CO2 - gases - liquids - solids calculate the n u m b e r of calories

4. List aU of the metallic e l e m e n t s a n d their crystal s t r u c t u r e s . 5. List the n u m b e r of expected vibrational a n d rotational m o d e s for the following g a s e o u s molecules: Br2

co2PCI3

cI-14C2H4 C2H6 6. Draw the following p l a n e s for the cubic lattice (see C h a p t e r 2 for help): {101} {222} {101} {301}.

29

7. Given the foUowing phase diagram, label the individual phases present, assu.~ that three phases are present. Use a , 13, and o as symbols for the three phases:

This Page Intentionally Left Blank

31

Chapter 2D e t e r m i n i n g t h e S t r u c t u r e of Solids This c h a p t e r will p r e s e n t m o r e a d v a n c e d topics t h a n t h o s e of the first c h a p t e r in t e r m s of d e t e r m i n i n g the s t r u c t u r e of solids. C o n s e q u e n t l y , you will gain s o m e knowledge of h o w one goes about determining the s t r u c t u r e of a solid, even if you n e v e r have to do it. 2 . 1 - S C I E N T I F I C DETERMINATION OF THE STRUCTURE OF SOLIDS In this section, we will p r e s e n t the basis developed to explain the

s t r u c t u r e of solids. T h a t is, t h e c o n c e p t s t h a t w e r e p e r f e c t e d in o r d e r to a c c u r a t e l y d e s c r i b e h o w a t o m s or ions fit t o g e t h e r to form a solid p h a s e . This w o r k was a c c o m p l i s h e d by m a n y p r i o r w o r k e r s w h o e s t a b l i s h e d t h e r a t i o n a l e u s e d to define the s t r u c t u r e of a s y m m e t r i c a l solid. As you will recall, we said t h a t the basic difference b e t w e e n a gas, liquid a n d t h a t of a solid lay in the o r d e r l i n e s s of the solid, c o m p a r e d to t h e o t h e r p h a s e s of the s a m e m a t e r i a l .

We

have

already

indicated

that

solids

can

have

several

forms

or

s y m m e t r i e s . To e l u c i d a t e the s t r u c t u r e of solids in m o r e detail, at l e a s t t h r e e p o s t u l a t e s apply: First, the f o r m a t i o n of a solid r e s u l t s from a s y m m e t r i c a l "stacking" of a t o m s to n e a r infinity from a t o m s or m o l e c u l e s with s p a c i n g s is m u c h s m a l l e r t h a n t h o s e f o u n d in the liquid or g a s e o u s state. S e c o n d l y , if we w i s h to gain an i n s i g h t into h o w t h e s e a t o m s are a r r a n g e d in the solid, we n e e d to d e t e r m i n e w h a t k i n d of p a t t e r n t h e y form w h i l e in close p r o x i m i t y . T h i r d l y , we c a n t h e n relate o u r p a t t e r n define the s y m m e t r y of solids in g e n e r a l . One w a y to a p p r o a c h a solution of the last two p o s t u l a t e s is to define t h e to o t h e r structures and thus

32

s t r u c t u r e of any given solid in t e r m s of its l a t t i c e p o i n t s , W h a t this m e a n s is t h a t by s u b s t i t u t i n g a p o i n t for e a c h atom(ion) c o m p o s i n g the s t r u c t u r e , we find t h a t t h e s e p o i n t s c o n s t i t u t e a latticework, i.e.- t h r e e - d i m e n s i o n a l solid, having c e r t a i n s y m m e t r i e s (Examples of the s y m m e t r i e s to w h i c h we refer are given in 1.3.2. of C h a p t e r 1). A lattice is not a s t r u c t u r e per se. A lattice is d e f i n e d as a s e t of t h r e e d i m e n s i o n a l p o i n t s , having a c e r t a i n s y m m e t r y . T h e s e p o i n t s may, or m a y

not, be totally o c c u p i e d by the a t o m s c o m p o s i n g the s t r u c t u r e . C o n s i d e r a cubic s t r u c t u r e s u c h as t h a t given in the following diagram:

Here, we have a set of p o i n t s o c c u p i e d by a t o m s (ions) a r r a n g e d in a simple three-dimensional cubic pattern. The latticedirections

are

defined, b y c o n v e n t i o n , as x , y & z. Note t h a t t h e r e are eight (8) c u b e s in our example. In o r d e r to f u r t h e r define o u r cubic p a t t e r n , we n e e d to analyze b o t h t h e s m a l l e s t Unit a n d h o w it fits into the lattice. We call the s m a l l e s t Unit a "unit-cell" a n d find t h a t t h e u n i t - c e l l is the s m a l l e s t c u b e in the d i a g r a m .

33

We also n e e d to define h o w large t h e u n i t - c e l l is in t e r m s of b o t h t h e l e n g t h of its s i d e s a n d its v o l u m e . We do so b y d e f i n i n g t e r m s of t h e x, y, & z d i r e c t i o n s h a v i n g d i r e c t i o n s c o r r e s p o n d i n g to: the unit-cell d i r e c t i o n s in t e r m s of its "lattice u n i t - v e c t o r s " . T h a t is, we define it in of t h e u n i t cell w i t h specific v e c t o r s

X~X

Z-~ ~

w i t h t h e l e n g t h of e a c h u n i t - v e c t o r b e i n g e q u a l to 1.0. (Our n o t a t i o n for a v e c t o r h e n c e f o r t h is a l e t t e r w h i c h is "outlined", e.g.- ~ is d e f i n e d h e r e as t h e u n i t - c e l l v e c t o r in t h e "x" d i r e c t i o n ) . given p o i n t . We c a n n o w define a set of v e c t o r s called " t r a n s l a t i o n " v e c t o r s , i.e.- a , ~, a n d ~ in t e r m s of t h e following: 2.1.2.a. = a x ~ + a y :y + a z g As you p r o b a b l y r e m e m b e r , a "vector" is specified as a line w h i c h h a s b o t h d i r e c t i o n a n d d u r a t i o n f r o m a

b=bxz= Cx :~

+t~]] +bz~+ Cy ] + Cz

Here, t h e a , b, a n d ~ v e c t o r s are n o w d e f i n e d in t e r m s of t h e a, b, a n d c lattice c o n s t a n t s in e a c h of t h e x, y, a n d z d i r e c t i o n s of 2.1.1., w h e r e a, b, a n d c are lattice c o n s t a n t s . T h e y are real n u m b e r s ~ n g s t r o m (A) d i s t a n c e u n i t s , i.e.- 10- 8 c m . corresponding to

Note t h a t we n o w have: a as a v e c t o r in t e r m s of x, y a n d z v e c t o r s as a f u n c t i o n of t h e lattice d i s t a n c e s in t h e t h r e e d i r e c t i o n s , t h r e e d i r e c t i o n s . T h e s a m e h o l d s for b, a n d ~ v e c t o r s . ax, ay & az. Here, a = ax, ay & az, i.e.- a is c o m p o s e d of c o m p o n e n t s in e a c h of t h e

34

In g e n e r a l , we u s e o n l y t h e lattice c o n s t a n t s to define t h e solid s t r u c t u r e ( u n l e s s we are a t t e m p t i n g to d e t e r m i n e its synm~etry). We c a n then define a s t r u c t u r e factor k n o w n as t h e t r a n s l a t i o n vector. It is a e l e m e n t r e l a t e d to t h e u n i t cell a n d d e f i n e s t h e b a s i c u n i t of t h e s t r u c t u r e . We will call it ~ . It is d e f i n e d a c c o r d i n g to t h e following e q u a t i o n : 2.1.3.-

=

n l & + n2 5

+ n3

w h e r e n l , n2 , a n d n3 is t h e n d e f i n e d as: 2.1.4.Vector V = { ~]

are i n t e r c e p t s of t h e u n i t - v e c t o r s , ~ , y , a , o n

t h e x, y, & z - d i r e c t i o n s in t h e lattice, r e s p e c t i v e l y . T h e u n i t cell vohxme

x a }

(This is a " d o t - c r o s s " v e c t o r p r o d u c t ) . e a s i e s t w a y to specify unit-cell of t h e

n o t a t i o n is b e i n g u s e d h e r e b e c a u s e t h i s is t h e lies in t h e fact t h a t we c a n n o w

define t h e unit-cell. T h e r e a s o n for u s i n g b o t h u n i t lattice v e c t o r s a n d translation vectors parameters in t e r m s of a, b, a n d c (which are t h e i n t e r c e p t s

t r a n s l a t i o n v e c t o r s o n t h e lattice). T h e s e ceil p a r a m e t e r s are v e r y useful since t h e y specify t h e actual l e n g t h a n d size of t h e u n i t cell, usually in A., as we shall see. A l t h o u g h t h e cubic s t r u c t u r e looks like t h e s i m p l e s t o n e of t h o s e p o s s i b l e , it actually is t h e m o s t c o m p l i c a t e d in t e r m s of s y m m e t r y . W h a t t h i s m e a n s is t h a t we c a n "spin" t h e lattice by h o l d i n g it at a c e r t a i n point and r o t a t i n g it a r o u n d t h i s axis w h i l e still m a i n t a i n i n g t h e s a m e a r r a n g e m e n t of a t o m s in space. (Take a c u b e a n d do t h i s for yourself). T h e c u b i c l a t t i c e gives rise to a great many (As we symmetry will see, elements if t h e in contrast to less and synm~etrical l a t t i c e s lattice is e l o n g a t e d

d i s t o r t e d f r o m cubic, s o m e of t h e s e synm~etry e l e m e n t s do n o t arise). In 1 8 9 5 , R 6 n t g e n e x p e r i m e n t a l l y d i s c o v e r e d "x-rays" a n d p r o d u c e d the

first p i c t u r e of t h e b o n e s of t h e h u m a n h a n d . This w a s followed b y w o r k by von Laue in 1912 w h o s h o w e d t h a t solid c r y s t a l s c o u l d act as d i f f r a c t i o n gratings depended to f o r m symmetrical patterns of "dots" whose arrangement u p o n h o w t h e a t o m s w e r e a r r a n g e d in t h e solid. It w a s s o o n

35

realized t h a t the a t o m s f o r m e d

"planes" w i t h i n the solid. In 1913,

Sir

William H e n r y Bragg a n d his son, William L a w r e n c e Bragg, analyzed t h e m a n n e r in w h i c h s u c h x-rays w e r e reflected by p l a n e s of a t o m s in t h e solid. T h e y s h o w e d t h a t t h e s e reflections w o u l d be m o s t i n t e n s e at c e r t a i n angles, a n d t h a t the values w o u l d d e p e n d u p o n the d i s t a n c e b e t w e e n t h e p l a n e s of a t o m s in the crystal a n d u p o n the w a v e l e n g t h of t h e x-ray. T h i s r e s u l t e d in t h e Bragg equation:

2.1.5.-

n~. = 2d sin 0

w h e r e d is t h e d i s t a n c e in a n g s t r o m s (A = 10 -8 cm) b e t w e e n p l a n e s a n d 0 is the angle in d e g r e e s of the reflection. In Bragg's x - r a y diffraction equation, i.e.- the a n g l e , 0 , is actually t h e angle b e t w e e n a given p l a n e of a t o m s in the s t r u c t u r e a n d the l m t h of the x - r a y b e a m . The unit, "d", is defined as the d i s t a n c e b e t w e e n p l a n e s of the lattice a n d k is t h e wavelength of the radiation. Itwas

Georges

Friedel

who

in

1913

d e t e r m i n e d t h a t the i n t e n s i t i e s of reflections of x - r a y s from t h e s e p l a n e s in the solid could be u s e d to d e t e r m i n e the s y m m e t r y of the solid. T h u s , by convention, we usually define planes, not points, in t h e lattice. We have i n t r o d u c e d the t e r m "symmetry" in the solid. Let u s n o w e x a m i n e exactly w h a t is m e a n t by t h a t t e r m . To illustrate this c o n c e p t , on the n e x t page as 2 . 1 . 6 . In this case, we c a n s h o w t h a t the p l a n e s of the cubic lattice are d e f i n e d by m o v i n g various d i s t a n c e s in t h e lattice from t h e origin. W h a t is m e a n t by this t e r m i n o l o g y is t h a t as we move from the (0,0,0) origin j u s t 1 . 0 0 unit-cell d i s t a n c e in the x" d i r e c t i o n to the (1,0,0) point in the cubic lattice, we have defined t h e {100} p l a n e (Note t h a t ,are are u s i n g t h e n l i n t e r c e p t of the unit-cell). In a like m a n n e r , if we move in t h e "y" d i r e c t i o n a d i s t a n c e of j u s t 1 unitcell, we have defined the {010} plane. Using the the "z" d i r e c t i o n s gets us a set of o n e - d i m e n s i o n a l p l a n e s (See line A in 2 . 1 . 6 . ) . examine the s y m m e t r y e l e m e n t s of o u r cube, given in the follovcing d i a g r a m , given

36

By m o v i n g I unit-cell d i s t a n c e in the b o t h the x- a n d y - d i r e c t i o n s , t h e {110) h a s b e e n defined, etc. (Line B a b o v e - n o t e t h a t we have n o t i l l u s t r a t e d the {101} plane). Moving 1 unit-cell in all t h r e e d i r e c t i o n s t h e n gives u s t h e {111} plane. In a l i k e m a n n e r , we c a n o b t a i n the {200}, {020}

37

a n d {002} planes. A set of p l a n e s s u c h as the {003} a n d {004} series a r e not as c o m m o n , b u t do exist in s o m e solids. Also s h o w n are the r o t a t i o n a l axes of the cubic lattice, s h o w n as the diad, t r i a d a n d t e t r a d axes. "What this m e a n s is ff ,are spin t h e c u b e on its face, we u s e the t e t r a d axis to d e t e r m i n e t h a t it t a k e s a total of 4 t u r n s to b r i n g a n y specific c o r n e r b a c k to its original position. Likewise, the t r i a d axis u s e s t h e c o r n e r of the c u b e where the lattice is m o v e d a total of t h r e e times, a n d t h e diad axis e m p l o y s the diagonal a c r o s s t h e cube to p e r f o r m the synm~ehur o p e r a t i o n . T h u s , t h e p l a n e s of the lattice are f o u n d to be i m p o r t a n t and can be

defined by m o v i n g along one or m o r e of t h e lattice d i r e c t i o n s of the unitcell to define t h e m . Also i m p o r t a n t are the s y m m e t r y o p e r a t i o n s t h a t c a n be p e r f o r m e d w i t h i n the unit-cell, as we have i l l u s t r a t e d in the p r e c e d i n g diagram. T h e s e give rise to a total of 14 different lattices as we will s h o w below. But first, let us confine o u r d i s c u s s i o n to j u s t the simple cubic lattice. It t u r n s out t h a t t h e m e t h o d u s e d to decribe the p l a n e s given above for the cubic lattice c a n also be u s e d to define the p l a n e s of all of the k n o w n lattices, by u s e of t h e so-called "Miller Indices", w h i c h are r e p r e s e n t e d by: 2.1.7.{h,k,l}

T h e w a y t h a t Miller I n d i c e s a r o s e c a n be u n d e r s t o o d by c o n s i d e r i n g t h e h i s t o r y of crystal s t r u c t u r e work, a c c o m p l i s h e d by m a n y i n v e s t i g a t o r s . In 1921, E~wald d e v e l o p e d a m e t h o d of c a l c u l a t i n g the s u m s of diffraction i n t e n s i t i e s from different p l a n e s in the lattice by c o n s i d e r i n g called t h e "Reciprocal Lattice". The reciprocal lattice w h a t is by is o b t a i n e d

d r a w i n g p e r p e n d i c u l a r s to e a c h p l a n e in the lattice, so t h a t the axes of t h e r e c i p r o c a l lattice are p e r p e n d i c u l a r to t h o s e of the c r y s t a l lattice. This has t h e r e s u l t t h a t t h e p l a n e s of the r e c i p r o c a l lattice are at right angles (90 ~) to the real p l a n e s in the unit-cell. For o u r cubic lattice, t h e r e c i p r o c a l lattice -would look as s h o w n in t h e following d i a g r a m , given as 2.1.8 on t h e n e x t page.

38

In this diagram, a series of h e x a g o n - s h a p e d p l a n e s are s h o w n w h i c h are orthogonal, or 90 degrees, to each of the c o r n e r s of the cubic cell. E a c h plane c o n n e c t s to a n o t h e r plane (here s h o w n as a rectangle) on e a c h face of the unit-ceU. T h u s , the faces of the lattice unit-cell a n d t h o s e of t h e r e c i p r o c a l unit-cell c a n be s e e n to lie on the s a m e plane while t h o s e at the c o r n e r s lie at right angles to the c o r n e r s . The n o t i o n of a r e c i p r o c a l lattice arose from Ewald who u s e d a s p h e r e to represent how the x-rays i n t e r a c t with any given lattice plane in t h r e e d i m e n s i o n a l space. He e m p l o y e d w h a t is now called the Ewald Sphere to s h o w how reciprocal space could be utilized to r e p r e s e n t diffractions of xrays by lattice planes. Ewald originally r e w r o t e the Bragg e q u a t i o n as: 2.1.9.sin 0 = n k/ 2d {hkl} = 1 / d {hkl} 2/X

Using this equation, Ewald applied it to the case of the diffraction s p h e r e w h i c h we s h o w in the following d i a g r a m as 2.1.10. on the n e x t page. the S t u d y this d i a g r a m carefully. In this case, the x-ray b e a m e n t e r s

s p h e r e e n t e r s from the left a n d e n c o u n t e r s a lattice plane, L. It is t h e n diffracted by the angle 20 to the p o i n t on the s p h e r e , P, w h e r e it

39

is r e g i s t e r e d as a diffaction p o i n t on the r e c i p r o c a l lattice. The d i s t a n c e b e t w e e n p l a n e s in the r e c i p r o c a l lattice is given as 1/dhkl w h i c h is r e a d i l y o b t a i n e d from the diagram. It is for t h e s e r e a s o n s , we c a n u s e the Miller Indices to indicate p l a n e s in the real lattice.

The

reciprocal

lattice

is useful in

defining

some

of the

electronic

p r o p e r t i e s of solids. T h a t is, w h e n we have a s e m i - c o n d u c t o r (or even a c o n d u c t o r like a metal), we find t h a t the e l e c t r o n s are confined in a b a n d , defined by the reciprocal lattice. This h a s i m p o r t a n t effects u p o n t h e c o n d u c t i v i t y of any solid a n d is k n o w n as the "oand theory" of solids. It t u r n s out t h a t the r e c i p r o c a l lattice is also the site of the Brfllouin zones, i.e.- the "allowed" electron e n e r g y b a n d s in the solid. How this o r i g i n a t e s is e x p l a i n e d as follows. The free electron r e s i d e s in a q u a n t i z e d e n e r g y well, defined by k (in w a v e - n u m b e r s ) . This r e s u l t c a n be derived from the S c h r o e d i n g e r waveequation. However, in the p r e s e n c e of a periodic a r r a y of e l e c t r o m a g n e t i c p o t e n t i a l s arising f r o m the a t o m s c o n f i n e d in a crystalline lattice, orbit a n d are r e s t r i c t e d the energies of the e l e c t r o n s f r o m all of the a t o m s are severely limited in to specific allowed e n e r g y b a n d s . T h i s p o t e n t i a l o r i g i n a t e s from a t t r a c t i o n a n d r e p u l s i o n of the e l e c t r o n c l o u d s from t h e periodic array of a t o m s in the s t r u c t u r e . S o l u t i o n s to this p r o b l e m w e r e

40

made

b y B l o c h in

1930

who

showed

they

had

the

form

(for

a one-

dimensional lattice). 2.1.11.= e ikx u ( x ) - o n e d i m e n s i o n a l e ika u k (X) - t h r e e d i m e n s i o n a l

~ k (a) =

where

k is t h e w a v e n u m b e r

of the

allowed

band

as m o d i f i e d

by the

l a t t i c e , a m a y b e x, y o r z, a n d uk (x) is a p e r i o d i c a l f u n c t i o n w i t h t h e s a m e p e r i o d i c i t y as t h e p o t e n t i a l . O n e r e p r e s e n t a t i o n is s h o w n in t h e f o l l o w i n g :

41

We have s h o w n the least c o m p l i c a t e d have said, are the allowed energy

one w h i c h bands

t u r n s o u t to be t h e as w e in any given when one

simple cubic lattice. S u c h b a n d s are called "BriUuoin" zones a n d , of e l e c t r o n s

crystalline latttice. A n u m b e r of m e t a l s a n d simple c o m p o u n d s have b e e n s t u d i e d a n d t h e i r Brilluoin s t r u c t u r e determined. However, gives a r e p r e s e n t a t i o n of t h e e n e r g y b a n d s in a solid, a "band-model" is usually p r e s e n t e d . T h e following d i a g r a m s h o w s t h r e e b a n d m o d e l s : 2.1.13.- E n e r g y B a n d M o d e l s

In the solid, e l e c t r o n s r e s i d e in the valence b a n d b u t c a n be excited i n t o the c o n d u c t i o n b a n d by a b s o r p t i o n of energy. The e n e r g y gap of various solids d e p e n d s u p o n the n a t u r e of the a t o m s c o m p r i s i n g the solid. S e m i c o n d u c t o r s have a r a t h e r n a r r o w e n e r g y gap (forbidden zone) vchereas t h a t of i n s u l a t o r s is wide (metals have little or no gap). Note t h a t e n e r g y levels of the a t o m s "A" are s h o w n in the valence b a n d . T h e s e will vary d e p e n d i n g u p o n the n a t u r e a t o m s p r e s e n t . We will n o t delve f u r t h e r into this a s p e c t h e r e since it is the s u b j e c t of m o r e advanced studies of electronic a n d optical m a t e r i a l s .

42

R e t u r n i n g to t h e u n i t - c e l l , we c a n also utilize t h e v e c t o r m e t h o d to d e r i v e t h e origin of t h e Miller I n d i c e s . T h e g e n e r a l e q u a t i o n for a p l


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