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15 ~Harcourt ~College Publishers

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""VIhere learning

Com s to Life." \vvvvv.harcourtcollege.com )

tate Neil W. Ashcroft N. David Mermin Cornell University

Saunders College Publishing Harcourt College Publishers Fort Worth Philadelphia San Diego New York Orlando Austin San Antonio Toronto Montreal London Sydney Tokyo

l

Copyright© 1976 by Harcourt. Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any fonn or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, lnc., 6277 Sea Harbor Drive, Orlando, FL 32887-6777.

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Library of Congress Cataloging in Publication Data Ashcroft, Neil W. Solid state physics. I. Solids. II. Title.

I. Mermin, N . David. joint author.

QC176.A83 530.4'1 74-9772 ISBN 0-03-083993-9 (College Edition)

Printed in the United States of America SOLJD STATE PHYSICS ISBN# 0-03-083993-9 (College Edition)

0 I 2 3 4 5 6 7 8 9 076 35 34 33 32 31 30 29 28 27

•

for Elizabeth, Jonathan, Robert, and Jan

)

P1e ace We began this project in 1968 to fill a gap we each felt acutely after several years of teaching introductory solid stare physics to Cornell students of physics, chemistry, engineering, and materials science. In both undergraduate and graduate courses we had to resort to a patchwork array of reading assignments, assembled from some half dozen texts and treatises. This was only partly because of the great diversity of the subject ; the main problem lay in i1s dual nature. On the one hand an introduction to solid state physics must describe in some detail the vast range of real solids, with an emphasis on representative data and illustrative examples. On the other hand there is now a weB-established basic theory of solids, with which any seriousl y interested student must become familiar. Rather to our surprise, it has taken us seven years to produce what we needed: a single introductory text presenting both aspects of the subject, descriptive and analytical. Our aim has been to explore the variety of phenomena associated with the major forms of crystalline matter. while laying the foundation for a working understanding of solids through clear, detailed, and elementary treatments of fundamental theoretical concepts. Our book is designed for introductory courses at either the undergraduate or graduate level. 1 Statistical mechanics and the quantum theory lie at the heart of solid state physics. Although these subjects are used as needed, we have tried, especially in the more elementary chapters, to recognize that many readers. particularly undergraduates, will not yet have acquired expertise. When it is natural to do so, we have clearly separated topics based entirely on classical methods from those demanding a quantum treatmenL In the latter case, and in applications of statistical mechanics, we have proceeded carefully from explicitly stated first principles. The book is therefore suitable for an introductory course taken concurrently with first courses in quantu m theory and statistical mechanics. Only in the more advanced chapters and appendices do we assume a more experienced readership. The problems that foil ow each chapter are tied rather closely to the text. and are of three general kinds : (a) routine steps in analytical development are sometimes relegated to problems, partly to avoid burdening the text with formulas of no intrinsic interest, but, more importantly, because such steps are better understood if completed by the reader with the aid of hints and suggestions; (b) extensions of the chapter (which the spectre of a two volume work prevented us from including) are presented as problems when they lend themselves to this type of exposition ; (c) further numerical and analytical applications are given as problems, either to communicate additional

1

Sugf!CStion~

for how to use the text in

I

cou~

of varying length and level ore gh·~n on pp. xviii -xxt.

vii

,·iii

Preface

information or to exercise newly acquired skills. Readers should therefore examine the problems. even if they do not intend to attempt their solution. Although we have respected the adage that one picture i~ wonh a tho usand \\Ords. we are also aware that an uninformative illustration, though decorative, takes up the space that could usefully be filled by several hundred . The reader will rhus cncowuer stretches of expository prose unrelieved by figures, when none are necessary, as well as sections that can profitably be perused entirely by looking at the figures and their captions. We anticipate use of the book at different levels with different areas of major emphasis. A particular course is unlikely to follow the chapters (or even selected chapters) in the order in which they are presented here, and we have written them in a way that permits easy selection and rearrangement. 2 Our particular choice ofsequence follows certain major strands of the subject from their first elementary exposition to their more advanced aspects, with a minimum of digression. We begin the book3 with the elementary classical [I] and quantum [2] aspects of the free electron theory of metals because this requires a minimum of background and immediately introduces, through a particular class of examples, almost all of the phenomena with which thcorie~ of insulators. semiconductors. and metals mu~t come to grips. The reader is thereby spared t he impression that nothing can be understood tlntil a host of arcane definitions (relating to periodic strucll!res) and elaborate quantum mechanical explorations (of periodic systems) have been mastered. Periodic structures are introduced only after a survey (3) of those metallic properties that can and cannot be understood without investigating the consequences of periodicity. We have tried to alleviate the tedium induced by a first exposure to the language of periodic systems by (a) separating the very importan~ consequences of purely translational symmetry [4. 5] from the remaining but rather less essential rotational aspects [7), (b) separating the description in ordinary space [4] from that in the less familiar reciprocal space [5), and (c) separating the abstract a nd descriptive treatment of periodicity from its elementary application to X-ray diffraction [6]. Armed with the terminology of periodic systems, readers can pursue to whatever point seems appropriate the resolution of the difficulties in the free e lectron model of metals or. alternatively, can embark directly upon the investigation of lattice vibrations. The book follows the first line. Bloch ·s theorem is described and its implications examined [8] in general tenns, to emphasize that its consequences transcend the illustrative and very impo rtant practical cases of nearly free electrons (9] and tigh t binding [10]. Much of the content of these two chapters is suitable for a more advanced c:ourse, as is the following survey of methods used to compute real band structures [II]. The remarkable subject of semiclassical mechanics is introduced and given elementary applications [I 2] before being incorporated into the more elaborate semiclassical theory of transport (I 3]. The description of methods by which Fcm1i surfaces arc measured [14] may be more stlitable for advanced readt:rs, but m uch of the survey : The Tabh: o n pp. xi" x>
Preface

Lx

of the ba nd structures of actual metals [15] is readily incorporated into an elementary course. Except for the discussion of screening, an elementary course might also bypass 1he essays on wha t is overlooked by the relaxation-time approximation [16] and by rhe neglect of electron-electron interactions [ 17]. Work fu nctions and other surface properties (18] can be taken up at any time after the discussion of translational symmetry in real space. Our description of the conventiona l classifica tion of solids [19) has been separated from the analysis of cohesive energies [20). Both have been placed after the introduction to band structure, because it i~ in terms of electronic structure that the categories are most clearly distinguished. To motl\atc the study of lattice vibrations (at whatever point after Chapter 5 readers choose to begin the subject) a sttmmary [21] lists those solid propcnics tha t cannot be understood without their consideration. Latttce dynamics is given an elementar) introduction, v. ith the classical [22] and quantum [23] aspects of the harmonic crystal treated separately. The ways in which phonon spectra are measured [24). the consequences of anharmonicity [25], and the special problems associated with phonons in metals [26) and ionic crystals [27] are surveyed at an elementary level, though some parts of these last fottr chapters might well be reserved for a more advanced course. None of the chapters on lattice vi brat ions rely on the usc of normal mode raising and lowering operators; these are described in several appendices for readers wantmg a more advanced treatment. Homogeneous [28] a nd inhomogeneous [29] semiconductors can be examined at any point after the introduction of Bloch's theorem and the elementary discussion of semiclassical mechanics. Crystalline defects [30] can be studied as soon as crystals themselves have been introduced , though parts of earlier c hapters are occasionally referred to . Following a review of atomic magnetism, we examine how it is modified in a solid environment [31], explore exchange and other magnetic interactions [32], and apply the resulting models to magnetic ordering [33]. This brief introduction to magnetism and the co ncluding essay on superconductivity [34] are la rgely self-contained. They are pla4- 27.S All

2

337 - 342

JJ0- 344

2, 4 (6, 8)

354 3(i4

354 3(>4

--

5

23. Quunlum harm(mic crystal

22

-

All AU

396-410

All

.

-

-

All

422 - 4 37

All

452- 464

All

-

l

-

470- 481

All

2, 23

25. Anhurmortie drccl$

23

26. Phooons in metals

17. 23 (16)

27. o .elcctnc properties

19. 22

28. Homc.lgencous semiconductors

2, 8 , ( 12)

562- 580

All

28

590-600

All

-I

499 505

534- 542

31.

DUmlaJPICiism. Paratn.bgnctism

4 (8, 12. 19 . 22. 28. 29)

L

.All 512 519 523 - 526

523 - 526

29. Inhomogeneous

!!.

I

24. Measuring pbonons

30. DefectS

All

/\II

628 - 63(> All

(2, 4. 14)

661 - 665

32. Magnetic inlcrt\t tions

31 (2.8.10. 16, 17)

672

33. Magnetic ordering

4,5.J2

694-700

All

1. 2(26)

726- 736

All

1 34

Superconducllvily

345 - 351

All

19 (17)

22. C la.,;c:nmeta ls one must abo understand metals. for m explaining wh} copper conducts ~o '' d l. ~1ne b.:g111:. to learn why common salt docs not. During the last hundred )C3fS ph~'icists hr some experiments. a nd t he conceptual puzz.les it rat~ed. defined the problems \\llh which 1he theory of metals was to grapple over the ne'\1 q uancr century. These found their resolution only in the rich and ~ubtle structure o f the quantum theory of solid:..

BASIC ASSUMPTIONS OF THE ORL"DE !\lODEL

J. J . Thomson's discovery of the e lectron in 1897 bad a vast :111d unmcdiare impact on theories of the structure of maller. a nd suggested an ob,•ious mechanism fo r conduction in metals. Three years after Thomson's disCO\'ery Urudc constructed hts theory of electrical and therma l conduction by applymg the h1ghly successful kinetic theory of gases to a metal. considered as a gas of electrons. In its simplest form kmettc theory treats the molecules of a gas as identical solid spheres. which move in stra tght lines until they collide \nth one anot her. 2 The time taken up by a sing le colliston IS assumed to be negligtblc. and. e>.cept for the forces commg momentaril} mto pia} during each collision. no o t her forces are assumed to act between the panicles. Altho ugh the re IS only one kind of panicle presem in the stmplest gases. in a metal there must be a t least two. for the electrons are negaurely charged. yet the metal is e lectrically neutral. Drude assumed that the compensating positive charge was at· '

lmiCih•n dlh lh~ ".11'- f the ""'~d cnl.unon!l th~m. af''ing mct.tl< unk ' ), ''-ne i~ 1nh:~tcd 1n \Cr~ f1 nc \\ I h."> thtn "h""t:ts.. c r- c:ff..~ts ut thc 'urr.-.x

Basic .\ s.'m (not to scale). !b) In a metal the nucleu~ and iun core rct;un their conligurauon 111 the free :11om. but the valence electrons leave the a tom to form the electron l!ill>.

-

tached to much heavier particles, wh ich he considered to be immobilt:. At his time, however, there was no precise notion of the origin of the light. mobile electrons a nd the heavier. immobile, positively charged panicles. The solution to this problem is o ne of the fundamental achievements of the modern quantum theory o f solids. ln this discussion of the Drudc model however, we shall simply assume (and in many metals this assumption can be justified) that when atoms o f a metallic element are brought together to form a metal. the valence electro ns become detached and wander freely through the metal. while tbe metallic ions remain intact and play the role of the immobile positive pan ides in Drudc·s theory. This model is indicated schematically in Figure L l. A single isolated atom of the metallic element has a nucleus of charge eZ0 • where Zo is the atomic nu mber and e is the magnitude of the electronic charge 3 : e = 4.80 x 10- 10 electrostatic units (csu) = 1.60 x 10- ' 9 coulombs. Surrounding the nucleus are Za electrons of total charge - eZa. A few of these, Z, are the relatively weakly bound valence electrons. The remainingZa - Z electrons art: relatively tightly bound to the nucleus, play much less of a role in chemical reactions. and are known as the core electrons. When these isolated atoms condense to form a metal, the core electrons remain bound to the nucleus to form the metallic ion, but the valence electrons are allowed to wander far away from their parent atoms. In the meta llic context they are calh:d conduction elecuons.4 • We shall al"'a}'S take(' to be a pos11JVC number. • When. as in the Drude model. the core elect rons play a pa:.-,•ve role and the iun ucts as an indivisible inert entity. o ne often refers to the conduct ion electrons simply as ·· thc electrons." s;.l\ing the fullterm for Lim~'S when the distincllon bel ween conduction and core electrons is to be emphasized.

4

Chapcer 1 The Drude Theory of Metals

Drude applied kinetic theory to this ~gas" of conduction electrons of mass m, which (in contrast to the molecules of an ordinary gas) move against a background of heavy immobile ions. The density of the electron gas can be calculated as follows : A metatlic clement contains 0.6022 x 1024 atoms per mole (Avogadro's number) and p,.,j A moles per cm 3 • where p,., is the mass density (in grams per cubic centimeter} and A is the atomic mass of the clement. Since each atom contributes Z electrons, the number of electrons per cubic centimeter. n = N / V. is , = 0.6022 x 1024

z;,.,.

(1.1)

Table 1.1 shows the conduction electron densities for some selected metals. They are typically o f order IOn conduction electrons per cubic centimeter, varying from 0.91 x 1022 for cesium up to 24.7 x 1022 for beryllium. 5 Also listed in Table 1.1 is a widely used measure of the electronic density. rs- defined as the radius of a sphere whose volume is equal to the volume per conduction electron. Thus

V

I

N = n=

4nr/ 3 ~

- ( 43

rs-

)I 3

7UJ

.

(1.2)

Table 1.1 lists rs both in angstroms (lo- s em) and in units of the Bohr radius a0 = h 2f me2 = 0.529 x 10- 8 em ; the latter length, being a measure of the radius of a hydrogen atom in its g round state, is often used as a scaJe for measuring atomic distances. Note that r jc1 0 is between 2 and 3 in most cases. although it ranges between 3 and 6 in the alkali metals (and can be as large as 10 in some metallic compounds). These densities are typically a thousand times greater than those of a classical gas at normal temperatures and pressures. In spite of this and in spite of the strong electron-electron and electron-ion electromagnetic interactions, the Orude model boldly treats the dense metallic electron gas by the methods of the kinetic theory of a neutral dilute gas, with only slight modifications. The basic assumptions are these: l.

Between collisions the interaction of a given electron, both with the others and with the ions, is neglected. Thus in the absence of externally applied electromagnetic fields each e lectron is taken to move unifom1ly in a stratght line. In the presence of externaUy applied fields each electron is taken to move as determined by Newton's laws of motion in the presenet: of those external fields, but neglecting the additional complicated fields produced by the other electrons and ions. 6 The neglect o f e lectron-electron interactions between collisions is k nown as the independent electron approximation. The correspondjng neglect of electron-ion interactions is knovm as the.free electron appmximation. We s hall find in subsequent chapters that • This is the range for metallic elemen"'~ under nonnal conduions. Htgher densttics can be anajned by application of pressure (which tend~ to favor the metallic shue) Lo wcr densities are found in compounds. " Strictly speaking. the electron-ion interaction is no t enure!) tg no rcd. for the Orude model implicitly assumes that the electrons :ne confined t o the interior of the m.:tal. Evidently th i~ confinement is brought about by their attraill satisfy p,. (1.17) 0 = - e E" - m,p1 - - , T

0 = -e E1

+ w,.p..

Pr

- - . t

where eH

m, = - .

( 1.18)

me

We multiply these equations by

- mn/m and introduce the current densrty com-

ponents through ( 1.4) to find

unE.x = m,rj,.

+ f •.

uo£1 - -m,Tjx

+ j,.,

( 1.19)

where a0 is just the Drude model DC conductivity in the absence of a magnetic field, given by (1.6). " Note that the Lorentz Ioree IS not the same for each electron since it depends on th~ elect rome velocity ' · Therefore the force r in (I 12) is to be wken as the a\·crage force p¢r ck..:tron IM!e Footm>tc 13) llecause. however. the force depends on the electron on which ll acts only throu&h a term linear in the electron's ' 'elocity. the average force IS obUhncd simply by replacing that .elodt.t by the uver~ge velocit~. p,'m.

14

Chapter I The Drude Theory of Metals

The Hall field El is determined by the requi rement that there be no transverse current j 1 • Setting Ir to zero in the second equation of ( 1.19) we find that

£)_= _

(WcT)j " a

=

_ (

0

H

nee

)jx-

(1. 20)

Therefore the HaU coefficient (1.15) is

R,

=

nee

•

(1.21)

This 1s a \'C:ry striking result. for it asserts that the HaU cocmcic:nt depends on no parameters of the metal except the density of carriers. Since we have already calcuLated n assuming that the atomic valence e lectrons become the metallic conduction electrons. a measurement of the Hall constant provides a direct test of the validity of this assumption. r n tr~ ing to extract the electron density 11 from measured Hall coefficients one is faced with the problem t hat. contrary to the prediction of ( 1.211. they generally do depend on magnetic field. Furthermore, they depend on tempct-aturc and on the care with which the sample has been prepared. This result is somewhat unexpected, since the relar.ation time -r. which can depend strongly on temperature and the condition of the sample. docs not appear in ( 1.21 ). However. at very low temperatures in very pure. carefully prepared s amples at very high fields. the measured Hall constants do appear to approach a limiting value. The more elaborate theory of Chapters 12 amll3 predicts that for many (but not a U) metals this limiting value is precisely the simple Drude result ( 1.2 1). Some Hall coefficients at high and moderate fields are listed in Table 1.4·. Note the occurrence of cases in which R 11 is actually positive, apparently corresponding to carriers "ith a positive charge. A striking example of observed field dependence totally unexplained by D r ude theory is shown in Figure 1.4. The Drude result confmns Hall's observation that the resistance does nor depend on field, for when j). = 0 (as is the case in the steady state when the Hall field has been established), tile first equation of (1.19) reduces to ix = a 0 E". the expected result for the conductivity in zero magnetic field. However, more careful experiments on a variety of metals have revealed that there is a magnetic field dependence to the resistance. which can be quite dramatic in some ca tn the charge denSil} .

(1.36) 'anish~.

18

Chapter J The Drude Th~'CJ£) of~ INa I~

then, to a first approximation. Eqs. ( 1.35) and ( 1.29) give (1.37)

.,.,here wp, known as the plasma

frcquenc~ .

i. to some ex tent a piece of good fortune tha t the alkali metals so strikingly display this Drude beha"ior. In o ther metals different comriburions to the dielectric constant compete quite substantially with the ··on1dc term" ( 1.37). A second important consequence of ( 1.37) is that the electron gas can sustain charge density oscillations. By this we mean a disturbance in wbich the electric charge densi ty 22 has an oscillato ry time dependence e ;, '' . From the equation of continuity, ~

V •j = -

c!,

V · j(w) -

((

(1.42)

iwp(w).

and Gauss's law, (1.43)

V · E{w) = 4np(lt>},

we find, in view of Eq. ( 1.30). that icup(w) = 4nu(w)p({IJ).

(1.44)

This has a solution provided tha t 1

+

4nia (w ) = O,

(1.45)

( J)

which is precisely the condition we found above fo r the onset o f propagation o f radiation. In the present context it emerges as the condition the frequency must meet if a charge density wave is to propagate. The n ature o f this charge density wave, known as a plasma oscillation or plasmon, 23 ca11 be understood in terms of a very simple model. Imagine displacing the entire electron gas. as a whole. through a distanced with respect to the fixed positive background o f the ions (Figure 1.5). 24 The resulting surface charge gives rise to an electric field of magnitude 4nu, where u is the charge per unit area 25 at either end of the slab. f igurt' 1.5 Simple model of a plasma oscillation.

o : + ndt'

N clec:tJ'ons A.

+ + + + +

E = 2110 + 2 1JO = 4 nnde

--

y

N Zio ns

o = -nde

The charge density 11 should nol be confused \\ilh the resistiv•ty. a lso generally denoted by p. The conte\t will ah,ays make tl clear which is being referred to . » Since the field of a untform plane o r cbarge ts independem of the di, tance from the plane. th is crude argumem. wtucb places aU of the charge density on 1wo o ppostte surfaces. is not as crude as 11 appears at first glanet:. :~ We obsCTVcd l'ttrlier that the Drudc model dO in "hich th~ can tra nsport thermal energy tthough not electric charge) : the ions can vibmt~ a httlc about their m.:-.m pOst· liOnS. leading tO the l.ranSffii$SIOn of thermal energy in the form Of cl8$t1C \li3VCS pro pagallng through the net" ork or ions. See Cha pter 25 >• Note the analog)> t o the definition of thc: ekctric-.urface. Describe its polanz.ation. This wave is known as a surracc plasmon. (a)

,

R. Bowers et al_ Pl1ys. Rt>t'. l.etrf'TS 7. 339 (1961).

)

2 The Som~nerfeld Theory of Metals Fermi-Dirac Distribution Free Electrons Density of Allowed Wave Vectors Fermi Momentum, Energy, and Temperature Ground-State Energy and Bulk Modulus •

Thermal Properties of a Free Electron Gas Sommerfeld Theory of Conduction Wiedemann-Franz Law

30

Chapter 2 The Sommerrutl> of 1h1• >CO:IIn.soun~ t he n:•lwn.: r
Since the Fermi wave vector is of the o rder of inverse angstroms, the de BrogUe wavelength of t he most energetic elec trons is of the o rder of a ngstroms. The Fermi velocity is 4.20

f r a 5

x 1011 em/sec.

(2.24)

0

This is a substantial velocity (about I percent of the velocity of light). From the viewpo int o f classical s tatistical mechanics this is quite a surprising rf"'" 1 1t, for we are

Ground-State Properties of d!e Dectron Gas 37

describing the ground state (T = 0), and all particles in a classiCi:! I gas have zero velocity at T = 0. Even at room temperature the thermal (i.e., average) velocity for a classical particle with the electronic mass is only of order 107 em/ sec. The Fermi energy is conveniently written in the form (since a0 = 1! 2 fme 2 )

2

f,F =

1i k F 2 2m

=

(

-

e2) (k,Oo)2.

2a0

(2.25)

Here e2 f2a 0 , known as the rydberg (Ry), is the ground-state binding energy of the hydrogen atom, 13.6 electron volts. 13 The rydberg is as convenieut a unit of atomic energies as the Bohr radius is of atomic distances. Since k,a 0 is of the order of unity, Eq. (2.25) demonstrates that the Fermi energy has the magnitude of a typical atomic binding energy. Using (2.23) and a 0 = 0.529 x 10 - 8 em, we find the explicit numerical form:

(2.26)

•

indicating a range of Fermi energies for tbe densities of metallic elements between 1.5 and 15 electron volts. Table 2.1 lists the Fermi e11ergy, velocity, and wave vector for the metals whose conduction electron densities are given in Table l.l. To Ci:llculate the ground-state energy of N electrons in a volume V we must add up the energies of all the one-electron levels inside the Fermi sphere 14 :

E=2

I ~k 2 •

k < lcr

(2.27)

2m

Quite generally, in summing any smomh function F(k) over ali allowed values of k. one may proceed as follows : Because the volume of k-space per allowed k value is ~k = 8n 3 JV(sec Eq. (2.18)) it is convenient to write (2.28) f(k) = : 3 F(k) 8

~

~

~k.

for in the limit as ~k ~ 0 (i.e., V-> oo) the sum :!:F(k) ~k approaches the integral Jdk F(k), provided only that F(k)does not vary appreciably' 5 over distances ink-space or order 2n/L. We may therefore rearrange (2.28) and write lim v- ox:

-vI i F( k) = fdk. 8n k

3

F( k).

(2.29)

Jn applying (2.29) to finite, but macroscopically large, systems one always assumes that ( 1/V) :EF(k) differs negligibly from its infinite volume limit (for example, one u Strictly speaking. the rydberg is the binding energy in the approximation of infinite proton mass. An electron •·olt is the energy gained by anelectroncrossingapotential of I volt: 1 eV = 1.602 x w - u erg = 1.602 x 10- 19 joule. ,.. The factor of 2 is for the two spin levels allowed for each k. 's The most celebrated case in which Ffails to satisfy this condition is the condensation of the ideal Bose gas. In applications to metals the problem never arises.

38 Chapter 2

~

Sommerfeld Theol') of Metals

Table 2.1 FERMI ENERGLES, FER.l\11 TEMPERATURES, FERMI WAVE VECTORS, FERi\fl VEWCITIES FOR REPRESENTATIVE Mt.aALS• ELEMEI'.T

r6Jao

Li Na K Rb Cs Cu Ag Au Be Mg Ca Sr Ba Nb

3.25 3.93 4.86 5.20 5.62 2.67 3.02 3.0 1 1.87 2.66 3.27 3.57 3.71 3.07 2.12 2. 14 2.30 2.59 2.65 2.07 2.19 2.41 2.48 2.22 2.30 2.25 2.14

Fe Mn Zn Cd Hg Al G.~

In 11 Sn Pb Bi Sb

f,F

4.74 eV 3.24 2.12 1.85

1.59 7.00 5.49 5.53 14.3 7.o& 4.69 3.93 3.64 5.32 11.1

10.9 9.47 7.47 7.13 11.7 10.4 8.63 8.15 10.2 9.47 9.90 10.9

kr

TF

5.5 1 x 1o• K

VF

1.12>< 0.92 0.75 0.70 0.65 1.36 1.20 1.21

377

2.46 2.15 1.84 1!.16 6.38 6.42 16.6 8.23 5.44 4.57 423 6.18 13.0 127 11 .0 1!.68 8.29 13.6 12.1 10.0 9.46 11.8 11.0 11.5 12.7

A~D

lo' em

I

1.29 x 108 cmfsec 1.07 0.86 0.81 0.75 157

1.39 1.40

2.25 1.51! 1.28 1.18 1.13 1.37 1.98 1.96 1.83 1.62 1.58 2.03 1.92 1.74 1.69 1.90 1.83 1.87 1.96

1.94

1.36 I. II

1.02 0.98 1.11! 1.71 1.70 1.58 1.40

1.37 1.75 1.66 1.51 1.46 1.64 1.58

1.61 1.70

•

• The table entries are calculated from the values of r,,'a0 given in Table 1.1 using m = 9.1I x 10- 18 grams.

assumes that the electronic energy per unit volume in a 1-cm cube of copper is the same as in a 2-cm cube). Using (2.29) to evaluate (227), we find that the energy density oft he electron gas is:

- =-, f. E

V

I 4n

.l1 kz = 2

Htr

dk

2m

2

I h k/ 1 . n 10m

(2.30)

To find the energy per electron, EJN, in the grow1d state, we must divide this by N/ V = kr 3 /3n 2 , which gives (2.31)

We can also write this result as

E

3

N = 5ki1TF

(2.32)

Ground-State Prop.:rtiL'S uf the Elt-ctrun Gas

39

where T,.., the Fer-mi temperature, is (2.33)

Note, in contrast to this, that the energy per electron in a classical ideal gas, }k8 T, vanishes at T = 0 and achieves a value as large as (2.32) only at T = ~ T,.· ::::: 104 K. Given the ground-state energy E, one can calculate the pressure exerted by the electron gas from the relation P = -(iJEjiJV)N. Since E = !Ne,. and e,.. is proportional to k,.. 2 , which depends on V only through a factor n 20 = (N/ V) 2 ' 3 , it follows that 16 2£

3 v·

P =



One can also calculate the compressibility, K, o r bulk modulus, B = 1/K. defined by: B =

Since E is proportional to

1

K=

-

cP

v av

(2.35)

v- 2' 3, Eq.(2.34) shows that P varies as v- 513, and therefore 5

lOE

B = 3P =

91/ =

2

(2.36}

3 nt.,..

or

6 13 5 B = ( · -) X 10 10 dynes/cm 2 • r.fao

(2.37)

In Table 2.2 we compare the free electron bulk moduli (2.37) calculated from rJa 0 , with the measured bulk moduli, for several metals. The agreement for the heavier alkali metals is fortuitously good, but even when (2.37) is substantially off, as it is in Table 2.2 BULK MOOl."Ll IN 10' 0 OYNES/CM 2 FOR SOME TYPICAL Mt.IALS" MEI"AL

Li Na K Rb Cs Cu

Ag Al

FREE ELECJ"RON

23.9 9.23 3.19 2.28 1.54 63.8 34.5 228

B

MEASURED

0

ll.5 6.42 2.81 1.92 1.43 1343 99.9 76.0

The free electron value is that for a free electron gas at the observed density of the metal, as calculated from Eq. (2.37). Q

'

6

At no

temperatures the pressure and energy dcnstty continue to obey thrs relation. See (2 .101 ).

40

Otapter 2 The Sommerfeld Theory of Metals

the noble metals, it is still o f about the right order ofmagnitude(though it varies from three times too Large to three times too small. through the table). It is absurd to expect that the free electron gas pressure alone should completely determine the resistance of a metal to compression, but T able 2.2 demonstrates that this pressure is at least as important as any other effects.

THERMAL PROPERTIES OF THE FREE ELECfRON GAS: THE FERMI-DIRAC DISTRJBUTION When the temperature is not zero it is necessary to examine the excited states o f the N-electron system as well as its ground state, for according to the basic principles of statistical mechanics, if an N-particle system is in thermal equilibrium at temperature T, then its properties should be calcula ted by averaging over all N-particle stationa ry states, assigning to each state of energy E a weight P,..(E) proportional to e-EiksT:

(2.38)

(H ere Ea"' is the energy of the a th stationary state of the N-electron system, the sum being over a ll such states.) The denominator o f (2.38) is known as the partition function, and is related to the H elmholtz free energy. F = U - TS (whe.re U is the internal energy and S, the entropy) by (2.39) We can therefore write (2.38) more compaclly as: P,..(E)

= e-IE - F,..>t• 8r _

(2.40)

Because o f the exclusion principle, to construct an N-electron state one must fi ll N different one-electron levels. Thus each N-electron stationary state can be specified by listing which o f t he N one-electron levels are fi lled in that sta te. A very useful quantity to know is fi"', the probability of there being a n electron in the particular one-electron level i, when the N-electron system is in thermal equilibrium. 17 This probability is simply the sum of the independent probabilities of finding the Nelectron system in any one ofthose N-electron states in which the ith level is occupied: (summation over a ll N-electron sta tes a in which there is an elee>tron in the one-electron level i).

(2.41)

We can evaluatefi"' by the following th ree observations:

1. Since the probability of an electron being in the level i is just one minus the probability o f no electron being in the level i (those being the only two possibilities , , ln the case we are interested in the level i is sp«ified by the clectron·s wa>-e vector k and the projection s of ats span along some axis.

Derivation of the Fermi-Dirac Distribution

41

allowed by the exclusion principle) we could equally well write (2.41) as

Jt

(summarion over all N-clectron states yin which there is no electron in the one-electron level i) . •

=I - :[P.v(E/)

(2.42)

2. By taking any (N + !)-electron state in which there is an electron in the oneelectron level i, we can construct an N-electr:on state in which there is no electron in the level i, by simply removing the electron in the ith level, leaving the occupation of all the other levels unaltered. Further more, any N-electron state with no electron in the one-electron level i can be so constructed from just one ( N + I )-electron state witlr an electron in the level 1. 18 Evidently the energies of any N-electron state and the corresponding (N + l)-electron state differ by just l:1, the energy of the only one-electron level whose occupation is different in the two states. Thus the set of energies of all N-electron states with the level i unoccupied is the same as the set of energies of all (N + I )-electron states with the level i occupied, provided that each energy in the latter set is reduced by &1• We can therefore rewrite (2.42) in the peculiar fonn (summation over all (N + I )-electron states ex in which there is an electron (2.43) in the one-electron level i).

But Eq. (2.40) permits us to write the summand as r::N + l _ '"·) = PN ( IJa \Jor

•

el&0- p)lksTp (£N + N+ 1 a

J)

'

(2.44)

where 14 known as the chemical potential, is given at temperature T by Jl = FN + l - FN·

(2.45)

Substituting this into (2.43), we find :

Jt

= 1 -

ett ; - p)JksT

L PN 1. 1 (E~·+ 1)

(summation over all (N + I )-electron states ex in which t here is an electron in the one-electron level i).

(2.40)

Comparing the summation in (2.46) with that in (2.41) one finds that (2.46) simply asserts that (2.47) 3. Equation (2.47) gives an exact relation between the probability of the oneelectron level i being occupied at temperature Tin an N-electron system, and in an (N + 1)-electron system. When N is very large (and we are typically interested in N of the o rder of lOu) it is absurd to imagine that by the addition of a single extra electron we could appreciably alter this probability for more than an insignificant handful of one-electron levels. 19 We may therefore replaceff + 1 by J/' in (2.47). which Namely the one obtained by occupying all those IC\>cls occupied in 1he .V-clcclron slate pl11s the itb level. 19 For a 1ypical level. chnnging N by one ahers !he probability of occupat1on by order I N. See Problem 4. 18

42

Lbapter 2 The Sommerfeld Theory of Metals

makes 11 possible to solve for ft :

f,

1

N

=

e~c1 - >J)/ksT + 1·

(2.48)

In subsequent formulas we shaU drop the explicit reference to the N dependence offi. which is. in any event, carried through the chemical potential Jl; see (2.45). The value of N can always be computed, given the};, by noting that}; is the mean number of electrons in the one-electron level 20 i. Since the total number of electrons N is just the sum over all levels of the mean number in each level, (2.49)

which determines N as a function of the temperature Tand chemical potential p.. In many applications, however, it is the temperature and N (or rather the density, n = NJV) that are given. In such cases (2.49) is used to determine the chemical potential 11 as a function of n and T, permitting it to be eliminated from subsequent formulas in favor o f the temperature and density. H owever the chemical potential is of considerable thermodynamic interest in its own right. Some o f its important properties are summarized in Appendix 8 .21

THERMAL PROPERTIES OF THE FREE ELECTRON GAS: APPLICATIONS OF THE FERMI-DffiAC DISTRIBUTION In a gas of free and independent electrons the one-electron levels are specified by the wave vector k a nd spin q uantum number s, with energies that are independent of s (in the absence of a magnetic field) and given by Eq. (2.7)~ i.e., Ji:Zk:Z E(k) =

2m

(2.50)

We first verify that the distribution function (2.49) is consistent with the ground-state (T = 0) properties derived above. In the ground state those and only those levels arc occupied with E(k ) ~ ef., so the ground-state distribution function must be f~u.

= 1, E(k) < St·; = 0, E(k ) > SF.

(2.51)

/'roof: A level can contain either 0 or I de p.. For these to be consistent it is necessary that (2.53)

lim J1. = &FT-0

We shall see shortly that for metals the chemical potential remains equal to the Fem1i energy to a high degree of precision, all the way up to room temperature. As a result. people frequently fail to make any distinction between the two when dealing with metals. This, however, can be dangerously misleading. In precise cakulations it is essential to keep track of the extent to which p., the chemical potential, differs from its zero temperature value. eF. The most important single application of Fermi-Dirac statistics is the calculation of the electronic contribution to the constant-volume specific heat of a metal. T

c,. = V

(cS) (i'u ) cT v

=

II =

fiT v'

u

-

v·

(2.54)

In the independent electron approximation the internal energy U is just the sum over one-electron levels of &(k) times the mean number of electrons in the lcvel 22 :

u

= 2

2: &Ck>J<e

k

We have introduced the Fermijimcrion/(&) to emphasize that fk depends on k only through the electronic energy &{k): I /(&) =

e = h 2

rr .J --,;z·

= 0,

dE;g(E;)F(E;).

(2.60)

- ...

o;

&

>

&

< 0.

(2.61)

Since the integral (2.59) is an evaluation of (1 / V) ~ F(&(k)), the form in (2.60) shows that g(&) d& = (

~)

x [the number of one-electron levels in the energy range from E; to & + de).

(2.62)

For this reason g(&) is known as the density of levels per unit volume (or often simply as the density of Levels). A dimensionally more transparent way o f writing g is g(E;) = 3.!.:. (~)an 2 &F E;F = 0,

e >

0;

e

o,


F, by terms of order T 2 • Thus, correctly to order T 2 , we may write (2.73)

. .

If we apply this expansion to the integrals in (2.71) and (2. n), and replace J1 by f..F in the terms already of order T 2 in these equations, we find

•

(2.74) (2.75)

The temperature-independent first terms on the right sides of (2. 74) and (2.75) are just the values of u and n in the ground state. Since we are calculating the specific beat at constant density,11 is independent of temperature. and (2.75) reduces to

n2

0 = (IL - f..F)g(&F)

+6

(k8 T)2g '(SF),

(2.76)

which determines the deviation of the chemical potential from f..F: 1t

Jl = f,F -

2

6

g'(f..F) (kaT) g(SF) · 2

(2.71)

Thermal Properties of the Free Electron Gas

47

Since for free electrons g(&) varies as &li 2 (see Eq. (263)), this gives

Jl = eF [ 1 - 31

(nk2;F-!\J

2 ]

•

(2.78)

which is, as we asserted above, a shift of the order of T 2 and typically only about 0.01 percent, even at room temperature. Equation (276) sets the term in braces in (2.74) equal to zero, thereby simplifying the form of the thermal energy dem;ity at constant electronic density: 1[2

U = Uo

+ 6 (ksT) 2g(C,F)

(2.79)

where 110 is the energy density in the ground state. The specific heat of the electron gas is therefore (2.80)

or. for free electrons (see (265) ), 2

c, = ' n (ks !\ nk8 • 2 e;)

Comparing this with the classical result for an ideal gas, c,, = 3nk 8 /2., we see that the effect of Fermi-Dirac statistics is to depress the specific heat by a factor of (n 2/ 3) (k 8 T( &F). which is proportional to the temperature. and even at room temperature is only of order 10- 2 • This explains the absence of any observable contribution of the electronic degrees of freedom to the specific heat of a metal at room temperature. If one is willing to dispense with the precise numerical coefficient, one can understand this behavior of the specific heat quite simply from the temperature dependence of the Fermi function itself. The increase in energy of the electrons when the temperature is raised from T = 0 comes about entirely because some electrons with energies within O(kB below &F (the darkly shaded region of Figure 24) have been excited to an energy range of O(ks above f,F (the lightly shaded region of Figure 24). The number of electrons per unit volume that have been so excited is the width, k 8 T. of the energy interval times the density of levels per unit volume g(f.F)· Furthermore, the excitation energy is of order k 8 T, and hence the total thermal energy density is of order g( e,.)(k8 T) 2 above the ground-state energy. Thjs misses the precise result (2.79) by a factor ofrr 2(6, but it gives a simple physical picture, and is useful for rough estimates.

n

•

(2.81)

n

F1gure 2.4

The Fermi function at nonzero T . The distribution differs from its T = 0 form becausesomeelectronsjust below &p (darkly shaded region) have been excited to levels just above f. (lightly shaded region).

f

1 . 0 1 - - - - - - - - - - - -.......... ,

1\.

&

48 Chapter 2 The Sommerfeld Theory of Metals

The prediction of a linear specific heat is one o f the most important consequences of Fermi-Dirac statistics, and provides a further simple test of the electron gas theory of a metal. p rovided one can be sure that degrees of freedom other than the electronic ones do not make comparable o r even bigger contributions. As it happens, the ionic degrees of freedom completely dominate the specific heat at high temperatures. However, well below room temperature their contribution falls off as the cube of t he Table 2.3 SOM E ROUGH EXPERIMENTAL VALUES FOR THE COEFFICIENT OF THE Lll\'EAR TER M IN T OF THE MOLAR SPECrFlC H EATS OF METALS, AND THE VALUES GIVEl'i BY S IMPLE FREE

ELECTRON THEORY ELE..\t.ENT

Li Na K Rb

C.s Cu Ag Au Be Mg Ca

Sr Ba Nb Fe Mn

FREE ELECTRON Y

(in I0- 4 cal-mole- 1 -K- 2 ) 1.8

2.6 4.0 4.6 5.3 l.l

15

u

1.2 2.4 3.6 43 4.7 1.6 1.5

1.5

20 12 40

2.2

24

15

2.9 3 .1

43 35 4.4 7.0

1.8

Cd Hg A1 Ga In T1 Sn

2 .3

Bi Sb

4.2 35 4.7 5.8 7.7 1.6 1.6 1.6 0.5 3.2 6 .5 8.7 6.5

1.4 1.7 5.0 3.0

Zn

Pb

MEASURED Y

2.4

3.3 3.6 4.3 3.9

0.2 1.5

RATI0°

(m•f m)

2.3 1.3

1.2

1.3 1.5

13 l.1 1.1

0.42 1.3 1.8

2.0 1.4 l2

8.0 27 0.78 0.74 2.1 1.4 0.62 1.5 1.1 1.3 1.9

0.047 0.38

• Since the theoret ical value of 'l' is proportional to the density of levels at the Fermi level, which in tum is proportional to the elect ronic mass m., one sometimes defines a specific heat effective mass m• so that m*/m is the ratio o rthe measured 'l' to the free electron}' Beware of identifying this specific beat effective mass with any of the many other effective masses used in solid-state theory. (See, for example, the index entries under "effective mass.~)

•

The Sommerfeld Theory of Conduction in Metals

49

temperature {Chapter 23), and at very low temperatures it drops below the electr onic contribution, which only decreases linearly with T. In order to separate out these two contributions it has become the practice to plot cufT against T 1, for if the electronic and ionic contributions together result in the low-temperature form,

(2.82) then Cv

1

(2.83)

T =y +AT.

One can thus find y by extrapolating the c./T curve linearly down to T 1 = 0, and noting where it intercepts the cvfT-axis. Measured metallic specific heats typically contain a linear term that becomes comparable to the cubic one at a few degrees Kelvin.25 Specific heat data are usually quoted in joules (or calories) per mole per degree Kelvin. Since a mole of free electron metal contains ZNA conduction electrons {where Z is the valence and N Ais Avogadro's number) and occupies a volume ZN Afn, we must multiply the heat capacity per unit volume, c. by ZN,..;,~., in order to get the heat capacity per mole, C:

(2.84)

•

where R = k8 N,., = 8 .314 joules/mole= 1.99 calories/mole-K. Using the free electron density of levels (2.65) and the evaluation (2.33) o f EF/k8 , we find a free e lectron contribution to the heat capacity per mole of C = -yT, where y =

1

-2

rt 1 R~ = TF

0.169z(!:!_) ao

2

X

I0-

4

cal-mole- 1-K- 2 •

(2.85)

Some rough, measured values of yare displayed in Table 2.3, together with the free electron values implied by (2.85) and the values of r./a0 in Table 1.1. Note that the alkali metals continue to be reasonably well described by free electron theory, as do the noble metals {Cu, Ag, Au). f"ote also, however. the striking disparities in Fe and Mn (experiment oft he order of 10 times theory) as well as those in Bi and Sb {experiment of the order of 0.1 times theory). These large deviations are now qualitatively understood on fairly general grounds, and we shall return to them in Chapter 15.

THE SOMMERFELD THEORY OF CONDUCfiON IN METALS To find the velocity distribution for electrons in metals, consider a small 26 volume element of k-space about a point k, of volume dk. Allowing for the twofold spin » Since constant density is hard to arrange expenmentally, one @.tllerally m~sur.s the specific heat at constant pressure, However, one can show (Problem 2) that for the metallic free electron gas at room temperature and below, c,.Jc, = I + O(k 8 T/&,Y. Thus a t temperatures where the eleclronic contribution to the specific heat becomes observable (a few degrees Kelvin) the two specific heats diller by a negligible amount. " Small enough that the Fermi function and other functions of physical interest vary negligably throughout the volume element, but large enough that it contains very ma.ny one-electron levels.

c,.

50

Chnptrr 2 The Sommerfeld Theory of l\trtnls

degeneracy, the number of one-electron levels in this volume element is (see (2.18)} (

~) dk.

(2.86)

The probability of each level being occupied is just /(f.(k)), and there fore the total number of electrons in the k-space volume element is v !Jl k l J(t:.(k)) dk, e(kJ = . (2.87) 4n 3 2rn Since the velocity of a free electron with wave vector k is v = hk/ m (Eq. (2.12)), the number of electrons in an element of volume dv about vis the same as the number in an clement of volume dk = (m/lz)3 dv about k = mvflr. Consequently the total number o f electrons per unit volume of real space in a velocity space element of volume dv about vis (2.88) f(v) dv, where (mj lz)' I (2.89) f(v) = 4n3 exp (Hmt•2 - J.t)lksT] + 1· Sommerfeld reex amined the Drude model, replacing the classical Maxwell-Boltzmann velocity distribution (2. 1) by the Fermi-Dirac distribution (2.89). Using a velocity distribution constructed from quantum-mechanical arguments in an otherwise classical theory requires some j ust ification. 27 One can describe the motion of an electron classically if one can specify its position and momentum as accurately as necessary, without violating the uncertainty principle. 28 • A typical electron in a metal has a momentum oft he order of Ilk F . so the uncertainty in its momentum, llp, must be small compared with ltkF C or a good classical description. Since, from (2.22), kF - l fr., the uncertainty in position must satisfy

II

I llx- - » - -~. !lp k,

~~

where, from (1.2), r, is of the order of the mean interelectronic distance-i.e., angstroms. Thus a classical description is impossible if one has to consider electrons localized to within atomic distances (also of the order of angstroms). However, the conduction electrons in a metal are not bound to particular ions, but can wander freely through the volume of the metal. 1n a macroscopic specimen, for most purposes there is no need to specify their position to an accuracy of 10- 8 em. The Drude model assumes a knowledge of the position of an electron primarily in only the two following contexts : n A detailed analyllcal justification is f1urly complicated to conslruct, just as it is a fairly subtle mauer 10 specify with generality and precision when the quantum theory can be replaced by its classical limit The underlying physics, however, is straightforward. 11 TI1ere is also a somewhat more specializ.ed limitaiionon the use of classical mechanics tn describing conduction electrons. The energy of motion of an electron in the plane perpendicular to a uniform applied magnctu: field is quantized in multiples of ,..(JJ, (Chapter 14). Even for fields as tar~ as 10• gauss. this is a ~ry small energy. but tn suotably prepared samples atttmperaturcs of a few degrees Kelvon.lhesequantum effects become obser,ablc. and are. in £act. of great practical importance.

•

The Sommerfcld Theor) of Conduction in Metals

51

When spatially varying electromagnetic fields or temperature gradients are applied. one must be able to specify the position of an electron on a scale small compared with the distance). over which the fields or temperature gradients vary. For most applications the applied fields or temperature gradients do not vary appreciably on the scale of angstroms, and the necessary precision of definition in the electron's position need not lead to an unacceptably large uncertainty in its momentum. For example, the electric field associated with visible light varies appreciably only over a distance of order 103 A_ If. however. the wave length is very much shorter than this (for example, X rays), one must use quantum mechanics to describe the electronic motion induced by the field. There is also an implicit assumption in the Drude model that one can localize an electron to within substantially less than a mean free path l. and one should therefore be suspicious of classical arguments when mean free paths much shorter than tens of angstroms occur. Fortunately, as we shall see below, mean free paths in metals are of the order of 100 A at room temperature, and become longer still as the temperature drops.

1.

2

There is thus a wide range of phenomena in which the behavior of a metallic electron is well described by classical mechanics. It is not, however, immediately evident from this that the behavior of N such electrons can be described by classical mechanics. Since the Pauli exclusion principle so profoundly affects the statistics of N electrons, why should it not have similarly drastic effects on their dynamics'! That it does not follows from an elementary theorem, which we state without proof, since the proof, though simple, is notationally rather cumbersome : Consider a system of N electrons whose interactions with one another are ignored, and which are exposed to an arbitrary space- and time-dependent electromagnetic field. Let the N-clectron state at time 0 be formed by occupying a particular group of None-electron levels, ljl 1(0), ... , IJ.tN(O). Let ljl1(t) be the level !J,t1(0) would evolve into in time l under the inOuence of the electromagnetic field if there were only a single electron present, which was in the level .;,AO) at time zero. Then the correct N-electron state at time t will be the one formed by occupying the set of N one-electron levels !J.tJ(t), •.. , 1/1,-(l).

Thus the dynamical behavior of N noninteracting electrons is completely determined by considering N independent one-electron problems. In particular, if the classical approximation is valid for each of these one-electron problems, it will also be valid for the whole N-electron system. 29 The use of Fermi-Dirac statistics affe.c ts only those predictions oft he Drude model that require some knowledge of the electronic velocity distribution for their evaluation. If the rate 1/-r at which an electron experiences collisions does not depend on its energy, then only our estimate of the electronic mean free path and our calculation of the thermal conductivity and thermopower are at all affected by a change in the equilibrium distribution function. Note that this implies that any classical configuration consistent with the exclusion princtple at timet = 0 (i.e~ having less than one electron or each spin ~r unit volume, many momentum space region of voltune dp = (2niiJ'J V) will remain consistent with the exclusion principle at aU future times. This result can also be provPrl by purely classical reasoning as a direct coroUary of Liouville's theorem. See Chapter 12. >0

52

Chapter 2 The Sommerfeld Theol') o f Metal.;

Mean Free Path Using vF (Eq. (2.24)) as a measure of the typical electronic speed, we can evaluate the mean free path l = CJFT from Eq. (1.8) as follows~

e = (r.faofZ

92 A.

X

(2.91)

p,_

Since the resistivity io microhm centimeters. p,_, is typically 1 to 100 at room temperature, and since rsfa0 is typically 2 to 6, mean free paths of o rder a hundred angstroms long are possible even at room temperature. 30

Thermal Conductivky We continue to esti rna te the thermal conductivity by Eq. (1.51): (2.92)

The correct specific beat (2.81) is smaller than the classical guess of Drude by a factor of order k 8 Tf&F; the correct estimate of CJ2 is not the classical thermal mean square velocity o f order k8 Tf m, but vF 2 = 26Ffm, which is larger than the classical value by a factor o f order &Ffk 8 T . Inserting these values in (2.92) and eliminating the relaxation time in favor o f the conductivity through ( 1.6), we find

u~ =

2

;

(:sr

= 2.44

X

10-

8

watt-ohm/K.l.

~3)

This is remarkably close to Drude's fort uitously good value, thanks to the two compensating corrections o f order k 8 Tf &F, and in excellent agreement with the data in Table 1.6. We shall see (Chapter 13) that this value of the Lorenz number is a far better one than the very rough derivation of (2.93) would suggest.

Thermopower Drude's overestimate o f the thermopower is also resolved by the use o f Fermi- Dirac statistics. Substituting the specific heat, from Eq. (2.81), i nto Eq. (1.59) we find

k (k T) - 6n e 2

Q =

8

8

f:,F

=

- 1.42

(ks T) eF

X 10

_ 4

VOit/K,

(2.94)

which is smaller than Drudc's estimate (Eq. (1.60)) by O(k 8 T/ SF) ...., 0.01 a t room temperature.

Otlu!r Properties Since the form of the electronic velocity distribution did not play a role in the calculat ion of the DC or AC conductivities, the Hall coefficient, or the magnetoresistance, the estimates given in C hapter 1 remain the same whether one uses Maxwell-Boltzmann or Fermi-Dirac statistics. This is not the case, however, if one uses an energy-dependent relaxation time. If, for example, one thought the electrons collided with fixed scattering centers, then it would be natural to take an energy-independent mean free path, and hence a relaxation time-r = l / v - l / &112 • Shortly after Drude set forth the electron gas model of a It is perhaps just as well that Drudc estimated lusing the very much lower classical thermal velocity. o r he mighL have been sufficiently baffled by such long mean Free paths to abandon furthtt investigation. >()

Problems

53

metal, H. A. Lorentz s howed, using the classical Maxwell-Boltzmann velocity distribution, that an energy-dependent relaxation time would lead to temperature dependence in the DC and AC conductivities, as well as to a nonvanishing magnetoresistaoce and a field- and temperature-dependent Hall coefficient. As one might now expect from the inappropriateness of the classical velocity distribution, none of these corrections were in any way able to bring the discrepancies of the Drude model into better alignment with the observed facts about metals. 31 Furthermore, we shall see (Chapter 13) that when the correct Fermi-Dirac velocity distribution is used, adding an energy dependence to the relaxation time bas little significant effect on most of the quantities of interest in a metal. 32 Tf one calculates the DC or AC conductivities, the magnetoresistance, or the Hall coefficient assuming an energy-dependent •(E), the results one finds are the same as those one would have calculated assuming an energyindependent T, equal to T(EF).ln metals these quantities are determined almost entirely by the way in which electrons near the Fermi level are scattered. 33 This is another very important consequence of the Pauli exclusion principle, the justification of which will be given in Chapter 13.

PROBLEMS •

1.

T!Je Free and Independent Electron Gas in Tf
Those who have not wandered amidst the mineralogical departments of natural history museums are often surprised to learn that meads. like most otl}er solids, are crystalline, for although one is used to the very obvious crystalline features of quartz, diamond, and r ock salt, the characteristic plane faces at sharp angles with one another are absent from metals in their most commonly encountered forms. However, those metals that occur naturally in the metallic state are quite often found in crystalline forms. which are completely disguised in finished metal products by the great malleability of metals, which permits them to be fashioned into whatever macroscopic shape o ne wishes. The true test of crystallinity is not the superficial appearance of a large specimen, but whether on the microscopic level the ions are arranged in a periodic array. 1 This underlying microscopic regularity of crystalline matter was long hypothesized as the obvious way to account for tbe sjmple geometric regularities of macroscopic crystals. in which plane faces make only certain definite angles with each other. It received direct experimental confirmation in 1913 through the work of W . and L. Bragg, who founded the subject of X-ray crystallography and began the investigation of how atoms are arranged in solids. Before we describe how the microscopic structure of solids is determined by X-ray diffraction and how the periodic structures so revealed affect fundamental physical properties, it is useful to survey some o f the most important geometrical properties of periodic arrays in three-dimensional space. These purely geometrical considerations are implicit in almost all the analysis one encounters throughout solid state physics. and shall be pursued in this chapter and in Chapters 5 and 7. The first of many applications of these concepts will be made to X-ray diffraction in Chapter 6.

BRAVAIS LATTICE A fundamental concept in the description of any crystalline solid is that of the Bravais lattice, which specifies the periodic array in which the repeated units of the crystal are arranged. The units themselves may be single atoms, groups of atoms, molecules, ions, etc., but the Bravais lattice summarizes only the geometry of the underlying periodic structure, regardless of what the actual units may be. We give two equivalent definitions of a Bravais lattice 2 : {a)

A Bravais lattice is an infinite array of discrete points with an arrangement and orientation that appears exactly the same, from whichever of the points the array is viewed. A (three-dimensional) Bravais lattice consists of all points with position vectors R of the form

(b)

(4.1} Often a specimen is made. up of many small pieces. e.tch htrgc. on the microscopic scale and contain ing large numbers of periodically arranged ions. This "polycrystalline·· state IS more commonly encountered than a single macroscopic crystal, in which Ute pcriodtcity is perfect. ext.cnding tbrou.g h the entire ~imen.. 1 Why tbe name Bravais appears is explained in Chapter 7 . 1

Bn~i-. Latlitt

65

where a .. a 2 , and a 3 are any three vectors not all in the same plane, and r1" n 2 , and n 3 range through all integral values. 3 Thus the point l:n,a1 is reached by moving n1 steps 4 of le1.1gth a 1 in the direction of a 1 for i = 1, 2, and 3. The vectors a, appearing in definition (b) of a Bravais lattice are caJied primitit•e vectors and are said to gene1·ate or span the lattice. It takes some thought to see that the two definitions of a Bravais lattice are equivalent. That any array satisfying (b) also satisfies (a) becomes evident as soon as both definitions are understood. The argument that atl)' array satisfying definition (a) can be generated by an appropriate set of three vectors is not as obvious. The proof consists of an explicit recipe for constructing three primitive vectors. The construction is given in Problem Sa.

•

•

• •

•

•

Figure 4.1 A general two-dimensional Bravais latlic:e of no particular ~-ymmetry : Lhe obhque net. Primitive vectors a 1 and a 2 are shown. All points in the ne1 are linear combinations of these with integral coefficients; for example. P = a , + 2a 2, and Q = -a, + a2 •

•

•• •

•

•

•

•

•

Figure 4.! shows a portion of a t wo-dimcnsional Bravais lattice.5 C learly definition {a) is satisfied, and the primitive vectors a 1 and a 2 required by definuion (b) are indicated in the figure. Figure 42 shows one of the most familiar of three-dimensional Bravais lattices, the simple cubic. lt owes its special structure to the fact that it can be spanned by three mutually perpendicular primitive vectors of equal length. Figure 4.2 A simple cub1c three-d•mensional Bravais laHice. The three primitive vectors can be taken to bt mutually J>~:rpendicular, and with a common magni tude.

>

We continue wilh 1hc convention that "integer" means a negative tnteger or tero, as well as a

posiuve tnteger • When n 1s negauve. n steps Ul a
By a family oflallice planes we mean a set of parallel, equally spaced lattice planes, which together contain all the points of the three-dimensional Bravais lattice. Any lattice plane is a member of such a family. Evidently the resolution of a Bravais lattice into a family of lattice planes is far from unique (Figure 5.3). The reciprocal lattice provides a very simple way to classify all possible families of lattice planes, which is embodied in the following theorem : For any family of lattice planes separated by a distance tL there arc reciprocal lattice vectors perpendicular to the planes, the shortest of which have a length of 2nfd. Conversely, for any reciprocal lattice vector K. there is a family of lattice planes normal to K and separated by a distance d, where 2n/d is the length of the shortest reciprocal lattice vector parallel to K. The theorem is a straightforward consequence of (a) the definition (5.2) of reciprocal lattice vectors as the wave vectors of plane waves that are unity at all Bravais lattice sites and (b) the fact that a plane wave has the same value at all points lying in a family of planes that are perpendicular to its wave vector and separated by an integral number of wavelengths. To prove the first part of the theorem, given a family oflattice planes, Jet ft be a unit vector normal to the planes. That K = 2nfl/d is a reciprocal lattice vector follows from the fact that the plane wave e'" 'is constant in planes perpendicular to K and has the same value in planes separated by ). = 2rr/ K = d. Since one of the lattice planes contains the Bravais lattice point r = 0, erJ. · r must be unity for any point r in any of the planes. Since the planes comain all Bravais lattice points, e'"K · ' = I for all R, so that K is indeed a reciprocal lattice vector. Furlh,. ~ore, K is the shonest

Miller Indices of Lattice Planes

91

reciprocal lattice vector normal to the planes, for any wave vector shorter than K will give a plane wave with wavelength greater than 2-n/ K = d. Such a plane wave cannot have the same value o n all planes in the family, a nd therefore cannot give a plane wave that is unity at all Bravais lattice points. To prove the converse of the theorem, given a reciprocal lattice vector, let K be the shortest parallel reciprocal lattice vector. Consider the set of real space planes o n which the plane wave eiK · r has the value unity. These planes (one of which contains the point r = 0) are perpendicular to K and separated by a distanced = 2-nf K. Since the Bravais lattice vectors R all satisfy e•K· R = 1 for any reciprocal lattice vector K, they must all lie within these planes; i.e., the family of planes must contain within it a family of lattice planes. Furthermore the spacing between the lattice planes is also d (rather than some integral multiple of d), for if only every nth plane in the family contained Bravais lattice points.. then according to the first part of the theorem, the vector normal to the planes of length 2-nfnd, i.e., the vector K j n, would be a reciprocal la ttice vector. This would contradict our original assumption that no reciprocal lattice vector parallel to K is shorter than K.

MILLER INDICES OF LATTICE PLANES The cor respondence between reciprocal lattice vectors a nd families of lattice planes provides a convenient way to specify the o rientation of a lattice plane. Quite generally one describes the orientation of a plane by giving a vector normal to the plane. Since we know there are reciprocal lattice vectors normal to any fami ly of Janice planes; it is natural to pick a reciprocal lattice vector to represent the normal. To make the choice unique, one uses the shortest such reciprocal lattice vector. In this way one arrives at the Miller indices of the plane: The Miller indices of a lattice plane are the coordina tes of the shortest reciprocal lallice vector normal to that plane, with respect to a specified set of primitive reciprocal lattice vectors. Thus a plane with Miller indices h, k, I, is normal to the reciprocal la11ice vector hb 1 + kb 2 + /b 3 . As so defined, the Miller indices a re integers, since any reciprocal lattice vector is a linear combina tion of three primitive vectors with integral coefficients. Since the normal to the plane is specified by the shortest perpendicular reciprocal lattice vecto r, the integers h, k, I can have no common factor. Note a lso that the Miller indices depend on the particular choice of primitive ve.ctors. In simple cubic Bravais lattices the reciprocal lattice is also simple cubic and the Miller indices are the coordinates of a vector normal to the plane in the obvious cubic coordinate system. As a general rule, face-centered and body-centered cubic Bravais lattice are described in terms o f a conventional cubic cell, i.e., as simple cubic lattices with bases. Since any lattice plane in a fcc or bee lattice is also a lattice plane in the underlying simple cubic lattice, the same elementary cubic indexing can be used to specify lattice planes. In practice, it is only in the description of noncubic crystals that one must remember that the Miller indices are the coordinates o f the normal in a system given by the reciprocal lattice, rather than the direct lattice. The Miller indices of a plane h ave a geometrical interpretation in the direct lattice, which is SOI"T'P.times offered as an alternative way of defining them. Because a lattice

92

Chapter 5 The Reciprocal Lattice

plane with Miller indices h, k, I is perpendicular to the reciprocal lattice vector K hb 1 + kb~ + lb 3 • it will be contained in the continuous plane K · r = A, for suitable choice of the constant A . This plane intersects the axes determined by the direct lattice primitive vectors a, at the points x 1 a~o x 2 a 2 , and x 3 a 3 (Figure 5.4), where the X; are determined by the condition that x,a1 indeed satisfy the equation of the plane: K • (x1a 1) = A . Since K · a 1 = 2-rrh, K · a 2 = 2-rrk, and K • a 3 = 2-rr/, it follows that A A (5.13) 3 x2 = 2-rrk' x = 2nl" Thus the intercepts with the crystaJ axes of a lattice plane are inversely proportional to the Miller indices of the p lane. a3 x 3 a,

Figure 5.4 An illustration of the crysl;.dlographic definition of the Miller .i ndices of a lattice p lane. The shaded p lane can be a portion of the continuous plane in which the points of the lattice plane lie, or any plane parallel to the lattice plane. The Miller indices are inversely proportional to the x, .

Crystallographers put the cart before the horse, defining the Miller indices to be a set of integers with no common factors, inversely proportional to the intercepts of the crystaJ plane along the crystal axes : I l 1 h ·k .. ·l .. = - ·. - · . -. Xa

Xz

(5.14)

XJ

SOME .CONVENTIONS FOR SPECIFYING DIRECTIONS Lattice planes are usually specified by giving their Miller indices in parentheses: (h, k, I). Thus. in a cubic system, a plane with a normal (4, -2, I) (or, from the crystallographic viewpoint, a plane with intercepts (1, -2, 4) along cubic axes) is callc:d a (4, - 2, 1) plane. The commas are eliminated without confusion by writing n instead of -n, simplifying the description to (421). One must know wh at set of axes is being used to interpret these symbols unambiguously. Simple cubic axes are invariably used when the crystal has cubic symmetry. Some examples of planes in cubic crystals are shown in Figure 5.5. A similar convention is used to specify directions in the direct lattice, but to a void confusion with the Miller indices (directions in the reciprocaJ lattice) square brackets arc used instead of parentheses. Thus the body diagonal of a simple cubic lattice lies in the (Ill] direction and, in general the lattice point 11 1 & 1 + n2 a 2 + n 3 a 3 lies in the direction (n 1 n 2 n 3 ) from the origin. There is a lso a notation specifying both a family of lattice planes and a ll those other families that arc equivalent to it by virtue of the symmetry oftl1e crystal. Thus

Problems

(010)

( 110)

93

(Ill)

Figure 5.5 Three latti~-e planes and their Miller indices in a simple cubic Bravais lattice.

the (100), (010), and (001) plane.s are all equivalent in a cubic crystdl. One refers to them collectively as the {100} planes, and in general one uses {hkl} to refer to the {ltkl) planes and all those that are equivalent to them by virtue of the crystal symmetry. A similar convention is used with directions: the [100], [010], [001], [IOO], [OIO], and [OOI] directions in a cubic crystal are referred to, collectively, as the (100) directions. •

This concludes our general geometrical discussion of the reciprocal lattice. ln Chapter 6 we shall see an important example of the utility and the power of the concept in the theory of the diffraction of X rays by a crystal.

PROBLEMS 1. (a)

Prove that the reciprocallallice primitive vectors defined in (5.3) satisfy

(5.15) (Hint : Write b 1 (but not b 2 or b3 ) in tenns of the a~. and usc the orthogonality relations (5.4).) (b) Suppose primitive vectors are constructed from the bi in the same manner (Eq. (5.3)) as the bi are constructed from the a;. Prove that these vectors are just the ai themselves; i.e., show that

b2 x b 3 2n b, . (bz x bl) = a"

etc.

(5.16)

(Hirzt : Write b 3 in the numerator {but not b 2 ) in terms of the· ah use the vector identity A x (B x C) = B{A • C) - QA • B), and appeal to the orthogonality relations (5.4) and the result (5.15) above.) (c) Prove that the volwne of a Bravais lattice primitive cell is (5.17) where the~~; are three primitive vectors. (In conjunction with (5.15) this establishes that the volume of the reciprocal lattice primitive cell is (2n) 3j v.)

94

Chapter 5 The Reciprocal Lattice

2. (a) Using the primitive vectors given in Eq. (4.9) und the construction (5.3) (or by any other method) show that the reciprocal of the simple hexagonal Bravais lauice is also simple hexagonal, with lattice constants 2nfc and 4n!.j3a, rotated through Jo• about the c-axis with respect to the direct lattice. (b) For what value of c'/a does the ratio have the same value in both direct and reciprocal lattices? If cfa is ideal in the direct lattice, what ts its value in the reciprocal lattice? (c) T he Bravais lattice generated by three primitive vectors of equal length a, making equal angles 0 with one another, is known as the trigonal Bravais lauice (see Chapter 7). Show that the reciprocal of a trigonal Bravais lattice is also trigonal, with an angle 0* given by - cos = 111 cos 0/ [ 1 + cos 8], and a primitive vector length a*. given by a* = (2rt/ a)(l + 2 cos cos 0*)- •

e

e•

3. (a) Show that the density oflattice points (per unit area) in a lattice plane is d/v. where vis the primitive cell volume and d the spacing between neighboring planes in the family to which the given plane belongs. (b) Prove that the lattice planes with the greatest densities of points are the {111} planes it1 a face-centered cubic Bravais lattice and the { 110} planes in a body-centered cubic Bravais lattice. (Hint: This is most easily done by exploiting the relation between families oflatticc planes and reciprocal lattice vectors.) 4. Prove rhar any reciprocal lattice vector K is an integral multiple of the shortest parallel reciprocal lattice vector Ko. (Hint: Assume the contrary, and deduce that since the reciprocal lilllice is a Bravais lattice, there must be a reciprocal lattice vector parallel to K shorter than K0 .)

Determination of Crystal Structures by X-ray Di raction Formulation of Bragg and von Laue The Laue Condition and Ewald's Construction Experimental Methods: Laue, Rotating Crystal, Powder Geometrical Structure Factor Atomic Form Factor

96 Chaplcr 6 Determination of Crystal S tructurt..'S b} X-ray Diffra ction Typical interatomic distances in a solid are on the o rder of an angstrom (10- 8 em). An electromagnetic probe of the microscopic structure of a solid must therefore have a wavelength at least this short, corresponding to an energy of order hw =

he

T

=

10

he _ 8 ern ~ 12.3 x 103 eV.

(6.1)

Energies like this, on the order o f several thousands o f electron volts (kilovolts or keY), are characteristic X-ray energies. In this chapter we shall describe how the distribution of X rays scattered by a rigid, 1 periodic 2 array o f ions reveals the locations o f the ions within tha t structure. There are two equivalent ways to view the scattering of X rays by a perfect periodic structure. due to Bragg and to von Laue. Bo th viewpoints are still widely used. The von Laue approach, which exploits the reciproca l lattice, is closer t o the spirit of modem solid sta te physics, but the Bragg approach is still in wide use by X-ray crysta llographers. Both are described below, together with a proof of their equivalence.

BRAGG FORMULATION OF X-RAY DIFFRACTION BY A CRYSTAL In 191 3 W. H . and W . L. Bragg fo und tha t substances whose m acroscopic forms were crystalline gave remarkab ly characteristic p atterns of r eflected X -radiation, quite unlike those produced by liquids. In crystalline materials, for certain sharply defined wavelengths a nd incident directions, intense peaks o f scattered r a dia tion (now known as Bragg peaks) were o bserved. W . L. Bragg accounted fo r this by regarding a crystal as mad e out of parallel planes o f ions, spaced a distance d apart (i.e., the la ttice planes described in Chapter 5). Tbe conditio ns for a sharp peak in the intensity of the scattered radiation were : {1) that the X rays should be specularly reflected 3 by the ions in a ny o ne pla ne and (2) tha t the reflected rays from successive planes should interfere constructively. R ays specularly reflected from adjoining planes are shown in Figure 6. L The path difference between the two rays is just 2d sin 8, whe re 8 is the angle ofincidence.4 For the rays to interfere constructively, this p ath difference must be a n integral number of wavelengths, leading to the celebrated Bragg condition : nl = 2d sin 8. (6.2) The integer tr is known as the order of tbe corresponding reflection. For a beam of X rays containi ng a range of different wavelengths ("white radia tion") many different reflections are observed. Not only can o ne have higher-order reflections from a given set of lattice planes, but in addition one must recognize that there are Actually t he ions vibrate about their ideal equilibrium sites (Cbaptcrs 2 1- 26). This does not affe(:l Ihe conclwdons re-,u:hed in Lhis chapler (though In the early days ofX-roy diffraction it was not clear why such ~1brauons d1d no t obliler:ue the pallem charl!Ctcristic or a penodic structure). II !urns oul Ihat the vibraltons have two lll3in con.~uc:nces (see Appendix N): (a) the intensi ty in the characteristic pe3ks that re,cal the crystal struc:lure is diminished, bm no1 eliminated; and (b) a much weaker cont inuous background o f rtldiation (tbe .. diffuse background.. ) is produced. ~ Amorphous so lids and liquids have 11bout t:bc same density as crystalline solids. and are therefore also susceptible to probing with X rays. However, the discrete, sharp peaks o f scattered radiation charac· teristic o f crystals are no t found. 3 In specular refiecl:ion the angle of incidence equals the angle of reflection. • The angle of incidence in X- ray a;.stalJography is conventionally measure .>ased on the same physical assumptions, and their precise equivalence (see page 99) is to be c~pected.

98

Chapter 6 Detennirultioo of Cr)·stal Structures by X-ray Dilfrao.dOO

Figure 6.4 k

Ulustra ttng that the path di!Teren~:c for rays scattered from two points separated by d is given by Eq. (6.3) or (6.4).

k

•• n k'

d

k'

cos o' =- d. ii'

identical microscopic objects (sets of ions or atoms) placed at the sites R of a Bravais lattice, each of which can reradiate the incident radiation in all directions. Sharp peaks will be observed only in directions and at wavelengths for which the rays scattered from alllallice points interfere constructively. To find the condition for constructive interference, consider first just two scatterers, separated by a ctisplacement vector d (Figure 6.4). Let an X ray be incident from very far away, along a direction 1\, with wavelength )., and wave vector k = 2nfl/).. A scattered ray will be observed in a direction ft' with wavelength 6 ). and wave vector k ' = 2rrfl' /)., provided that the path difference between the rays scattered by each of the two ions is an integral number of wavelengths. From Figure 6.4 it can be seen that this path difference is just

d cos 8 1- d cos 8' = d • (ft - 1\').

(6.3)

The condition for constructive interference is thus d • (fl - fl ') = m.l.,

(6.4)

for integral m. Multiplying both sides of(6.4) by 2rcj).. yields a condition on the incident and scattered wave vectors: (6.5) · d • (k - k') = 211m, for integral m. Next, we consider not just two scatterers, but an array of scatterers, at the sites of a Bravais lattice. Since the lattice sites are displaced from one another by the Bravais lattice vectors R. the condition that all scattered rays interfere constructively is that condition (6.5) hold simultaneously for all values of d that are Bravais lattice vectors: R · (k - k') = 2nm,

for integral m and all Bravais lattice vectors R.

(6.6)

This can be written in the equivalent form eiCk'-kJ· R

= I,

for all Bravais lattice vectors R.

(6.7)

Here (and m the Bragg picture) we assume that the incident and scaucred radiation has the same wavelength. In tenns of photons this means that no energy has been lost in the scattering. i.e. that the scattering is clastic. To a good approAimation the bulk of the scattered radiation is elastically scanercd. though there •s much to be learned from the study of that small component of the radiation that is in· elastically scauered (Chapter 24 and AppendiX N). 6

•

Equh·ai€."11CC e interference will occur provided that the change in wave t>ector, K = k ' - k, is a t·ector of the reciprocal lattice. It is sometimes convenient to have an alternative formulation of the Laue condition, stated entirely in terms of the incident wave vector k. First note that because the reciprocal lattice is a Bravais lattice, if k ' - k is a reciprocal lattice vector, so is k - k '. Camng the latter vector K, the condition that k and k ' have the same magnitude is (6.8) k = lk- Kl. Squaring both sides ?f (6.8) yields the condition (6.9)

k·K =tK;

i.e., the component of the incident wave vector k along the reciprocal lattice vector K must be half the length of K. Thus an incident wave vector k will satisfy the Laue condition if and only if the tip of the vector lies in a plane that is the perpendicular bisector of a hne joining the origin of k-space to a reciprocal lattice point K (Figure 6.5). Such k-space planes are called Bragg planes. Figure6.5 The Laue cortditjon. If the sum ofk and - k' is a vector K . and if I< and k' have the same length, then the Lip of the vector k is cquidistant from the origin 0 and the tip of the vector K, and therefore it lies in the plane bisecting the line joining the origin to the tip ofK.

0

k

It is a consequence of the equivalence of the Bragg and von Laue points of view, demonstrated in the following section, that the k-space Bragg plane associated with a particular diffraction peak in the Laue formulation is parallel to tbe family of direct lattice planes responsible for the peak in the Bragg formulation.