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The Big Bang of Condensed Matter Physics.
Periodic solids can be classified into two main classes:.
The annealed ingots were crushed into fine powder using agate mortar and pestal in methanol and were dried at.
X-ray powder diffraction patterns were.
So we have zero-point energy and exclusion principle in effect.
Department of Physics, P.
Box 1048 Blindern, 0316 Oslo.
Department of Physics, P.
Box 1048 Blindern, 0316 Oslo.
Solid state physics lecture notes pdf Lecture Notes for Solid State Physics.
Hilary Term 2012 c Professor Steven H.
Molecular Orbitals and Valence Bonds 28.
Solid State Physics - '- t - - CHARLES KITTEL n - Name Actinium Aluminum Americium Antimony Argon Arsenic Astatine Barium Посмотреть еще Beryllium Bismuth Boron Bromine Cadmium Calcium Californium Carbon Cerium Нажмите чтобы перейти Chlorine Chromium Cobalt Capper Curium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold Symbol Name Hafnium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lawrencium Lead Lithium Lutetium Magnesium Manganese Mendelevium Mercury Molybdenum Neodymium Neon Neptunium Nickel Niobium Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Poloni~~m Potassium Symbol Name Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium Symbol H' Periodic Table, with the Outer Electron Configurations of Neutral Atoms in Their Ground States Is Li:, Be' 2y "' ~ ~ ~ T h e notation used to descrilx the electronic configuratior~of atoms ;nrd ions is discussed in all textl,nokc uf introdoctory atomic physics.
The letters s, pd.
Gd61 Tb6i DYC6 HOIl 6168 Tm69 YblD LUIl 4f2 4f4 4f5 4f' 4fX 5d 6sZ 4f1° 4fl3 4f13 4f14 6sZ 6y2 Cs2 4f14 5d 6s1 Bk91 Cf98 E9.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any fonn or hy any means, electronic, mechanical, photucopying, recording, scanning or otbenvise, execpt as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc.
Danvers, MA 01993, 97R i50-8400.
To order books or for customer service please, call 1-800~CALL WILEY 225-5945.
Library ojCongress Cataloging in Publication Data: Kittcl, Charles.
Introduction to solid state physics 1 Charles Kitte1.
About the Author Charles Kittel did his undergraduate work in physics привожу ссылку M.
T and at the Cavendish Laboratory of Cambridge University He received his Ph.
He worked in the solid state group at Bell Laboratories, along with Bardeen and Shockley, leaving to start the theoretical solid state physics group at Berkeley in 1951.
His research has been largely in magnetism and in semiconductors.
In magnetism he developed the theories of ferromagnetic and antiferromagnetic resonance and the theory of single ferromagnetic domains, and extended the Bloch theory of magnons.
In semiconductor physics he participated in the first cyclotron and plasma resonance experiments and extended the results to the theory of impurity states and to electron-hole drops.
He has been awarded three Guggenheim fellowships, the Oliver Buckley Prize for Solid State Physics, and, for contributions to teaching, the Oersted Medal of the American Association of Physics Teachers.
He is a member of the National Academy of Science and of the American Academy of Arts and Sciences.
The challenge to the author has been to treat significant new areas while maintaining the introductory level of the text.
It would be a pity to present such a physical, tactile field as an exercise in formalism.
At the first editic~nin 1953 superconductivity was not lmderstood; Fermi snrfaces in metals were beginning to he explored and cyclotron resonance in semiconductors had just been observed; ferrites and permanent magnets were beginning to be understood; only a few physicists then believed in the reality of spin waves.
Nanophysics was forty years off.
In other fields, the structure of DNA was determined and the drift of continents on the Earth was demonstrated.
It was a great time to be in Science, as it is now.
I have tried with the successivt: editions of lSSY to introduce new generations to the same excitement.
There are several changes from the seventh edition, as well as rnucll clarification: An important chapter has been added on nanophysics, contributed by an active worker in the field, Professor Paul L.
McEuen of Cornell University Nanophysics is the science of materials with one, two, or three small dimensions, where "small" means nanometer 10-%m.
This field is the most exciting and addition to solid state science in the last ten years.
The text makes use of the simplificati ~nsmade possible hy the nniversal availability of computers.
Bibliographies and references have been nearly eliminated because simple computer searches using keywords on a search engine slreh as Google will quickly generate many useful and rnore recent references.
No lack of honor is intended by the omissions of early or traditiorral references to the workers who first worked on the problems of the solid state.
The crystallographic notation conforms with current usage in physics.
Important equations in the body of the text are repeated in SI and CGS-Gaussian units, where these differ, except where a single indicated substitution will translate frnm CGS to SI.
The dual usage in this book has been found helpful and acceptable.
Tables arc in conventional units.
The symbol e denotes the charge on the proton and is positive.
The notation взято отсюда refers to Equation 18 of the current chapter, but 3.
Few of the адрес are exactly easy: Most were devised to carry forward the subject of the chapter.
With few exceptions, the problems are those of the original sixth and seventh editions.
The notation QTS refers to my Quantum Theory of Solirls, with solutions by C.
Fong; TP refers to Thermal Physics, with H.
This edition owes much to detailed reviews of the entire text by Professor Paul L.
McEuen of Cornell University and Professor Roger Lewis of Wollongong University in Australia.
They helped make the book much easier to read and understand.
However, I must assume responsibility for the close relation of the text to the earlier editions, Many credits for suggestions, reviews, and photographs are given in the prefaces to earlier editions.
I have a great debt to Stuart Johnson, my publisher at Wiley; Suzanne Ingrao, my editor; and Barbara Bell, my personal assistant.
Corrections and suggestions will be gratefully received and may be addressed to the author by rmail to kittelQberke1ey.
CRYSTAL VIBRATIONS Vibrations of Crystals with Monatomic Basis First Brillouin Zone Group Velocity Long Wavelength Limit Derivation of Force Constants from Experiment Two Atoms per Primitive Basis Quantization of Elastic Waves Phonon Momentum Inelastic Scattering by Phonons Summary Problems CHAFTER5: PHONONS 11.
Indices of planes 3.
The building blocks are identical in a and bbut Merent c ~ s t a lfaces ;Ire developed.
CHAPTER 1: CRYSTAL STRUCTURE PERIODIC ARRAYS O F ATOMS The serious study of solid state physics began with the discovery of x-ray diffraction by crystals and the publication of a series of simple calculations of the properties of crystals and of electrons in crystals.
Why crystalline solids rather than nonclystalline solids?
The important electronic properties of solids are best expressed in crystals.
Thus the properties of the most important semiconductors depend on the crystalline structure of the host, essentially because electrons have short wavelength components that respond dramatically to the regular periodic atomic order of the specimen.
Noncrystalline materials, notably glasses, are important for optical propagation because light waves have a longer wavelength than electrons and see an average over the order, and not the less regular local order itself.
We start the book with crystals.
A crystal is formed by adding atoms in a constant environment, usually in a solution.
Possibly the first crystal you ever saw was a natural quartz crystal grown in a slow geological process from a silicate solution in нажмите для деталей water under pressure.
The crystal form develops as identical building blocks are added continuously.
Figure 1 shows an idealized picture of the growth process, as imagined two centuries ago.
The building blocks here are atoms or groups of atoms.
The crystal thus formed is a three-dimensional periodic array of identical building blocks, apart from any imperfections and impurities that may accidentally be included or built into the structure.
The original experimental evidence for the periodicity of the structure rests on the discovery by mineralogists that the index numbers that define the orientations of the faces of a crystal are exact integers.
This evidence was supported by the discovery in 1912 of x-ray diffraction by crystals, when Laue developed the theory of x-ray diffraction by a periodic array, and his coworkers reported the first experimental observation of x-ray diffraction by crystals.
The importance of x-rays for this task is that they are waves and have a wavelength comparable with the length of a building block of the structnre.
Such analysis can also be done моего Бра Velante 709-001-01 правы neutron diffraction and with electron diffraction, hut x-rays are usually the tool of choice.
The diffraction work proved decisively that crystals are built of a periodic array of atoms or groups of atoms.
With an established atomic model of a crystal, physicists could think much further, and the development of quantum theory was of great importance to the birth of solid state physics.
Related studies have been extended to noncrystalline solids and to quantum fluids.
The wider field is h o w n as condensed matter physics and is one of the largest and most vigorous areas of physics.
An ideal crystal is constn~ctedby the infinite repetition of idenbcal groups of atoms Fig 2 A group is called the basis.
The set of mathematical points to which the basls is attached is called the lattice The lattice in three dimensions may be defined by three translabon vectors a, a, a, such that the arrangement of atoms in the crystal looks the same when viewed from the point r as when viewed from every polnt r' translated by an mtegral multiple of the a's: Here u, u, uare arhitraryintegers.
The set of points r' defined by 1 for all defines the lattice.
The lattice is said to be primitive if any two points from which the atomic arrangement looks the same always satisfy 1 with a suitable choice of the integers ui.
This statement defines the primitive translation vectors a.
There is no cell of smaller volume than a.
We often use the primitive translation vectors to define the crystal axes, which form three adjacent edges of the primitive parallelepiped.
Nonprimitive axes are often used as crystal axes when they have a simple relation to the symmetry of the structure.
By looking at cone oan recognize the basis and then one can abstract the space lattice.
It does not matter where the basis is put in relation to a lattice point.
Every bas~sin a given crystal is dentical to every other ~n compositmn, arrangement, and orientation The number of atoms in the basis may be one, or it may be more than one.
The position of the center of an atom3 of the basis relahve to the associated lattice point is We may arrange the origin, wl~ichwe have called the associated lattice point, so that 0 5 x,zj 5 1.
Figare 3b Primitive ccll "fa space lattice in three dimensions Figure 3c Suppose these points are identical atoms: Sketchin on the figure a set of lattice points, s choice ofprimitive axes, aprimitivc cell, and the basis of atoms associatedwith alattice point.
The smallest volume enclosed in this way is the Wigner-Seitz primitive cell.
All space may be filled by these cells, just as by the c e h of Fig.
Primitive Lattice Cell The parallelepiped defined by primitive axes al,a, a, is called a primitive cell Fig.
A primitive cell is a t.
The adjective unit is superfluous and not needed A cell will fill all space by the repetition of suitable crystal b-anslatlon operationq.
A primitive cell is a m~nimum-volumecell.
There are many ways of choosing the prnnitive axes and primitive cell for a given lattice.
The number of atoms m a primitive cell or primtive basis is alwap the same for a given crystal smllcture There is always one lattice point per primitive cell.
The basis associated wrth a взято отсюда cellis cdued a prim~tivebans.
No basis contans fewer atoms than a pnmibve basis contains.
Another way of choosing a primitive cell is shown in Fig.
This is h o w n to physicists as a Wigner-Seitz cell.
FUNDAMENTAL TYPES OF LATTICES Crystal lattices can be carried or mapped into themselves by the lattice translations T and by vanous other symmetzy operattons.
A typical symmetry operation is that of rotation about an axis that passes through a lattice point.
Lattices can be found such that one.
The rota3, 4, and 6.
We can make a crystal from molecules that individually have a fivefold rotation ms, but we should not expect the latbce to have a fivefold rotation m s.
We can, however, fill all the area oTa plane with just two distina designs of "tiles" or elemeutary potysms.
By lattice point group we mean the collection of symmetry operations which, apphed about a lathce pomt, carry the lattice into itself The possible rotations have been listed.
We can have mirror reflecbons m about a plane through by -r The symmetry axes and symmetry planes of a cube are shown in Fig 6.
Two-DimensionalLattice Types The lattice in Fig.
A general lattice such as this is known as an oblique lattice and is invariant only under rotation of.
But special lattices of the oblique type can2 d 6or under mirror reflection.
There are four distinct types of restriction, and each leads to what нажмите чтобы узнать больше may call a special lattice type.
Thus there are five distinct lattice types in two dimensions, the oblique lattice and the four special lattices shown in Fig.
Bravais lattice is the common phrase for a distinct lattice.
Figure 7 Four special жмите сюда in twodirnmsions, 1 Crystal St-ture Three-Dimensional Lattice Types The point symmetry groups in three dimensions require the 14 different lattice types listed in Table 1.
The general lattice is triclinic, and there are 13 special lattices.
These are grouped for convenience into systems classified according to seven types of cells, which are triclinic, monoclinic, orthorbombic, tetragonal, cubic, higonal, and hexagonal.
The division into systems is expressed in the посетить страницу in terms of the adal relations that describe the cells.
The cells in Fig.
Often a nonprimitive cell has a more obvious relation with the point symmetry operations than has a primitive cell.
There are three lattices in the cubic system: the simple cubic sc lattice, the body-centered cubic hcc lattice, and the face-centered cubic fcc lattice.
Table 1 The 14 latlioe types in three dimensions System Numhrr of Restr~chonson cunvenhonal lathces cell axes and angles Triclimc Monoclinic Orthorhomb~c Cubic Trigonal Hexagonal Figure 8 The cubic space lattices.
The cells s h m are the conventional cells.
The primitive ceU is obtained on completing the rhodyhuhedron.
In terms of the cube edge a, the primitive Figure 10 Pnmibve translation vectors of ~i~,9 .
~~ ~ d c,lbic~ lattice.
A prirmhve cell of the bcc lattice is shown in Fig.
Primitive cells by definition contain only one lattice point, but the conventional bcc Тормозная жидкость BRAKE FLUID DOT-4 (0.5л) contains two lattice points, and the fcc cell contains four lattice points.
I Crystal Stwctun Figore 11 The rhombohedra1 primitive cell of the hce-centered connen cubic clystal.
Here each coordinate is a fraction of the axla1length a, a, a, in the direction of the coordinate axis, with the origin taken at one corner of the cell Thus the coorhnate?
In the hexagonal system the primitive cell is a right prism based on a rhomhu3 with an included angle of 120".
F~gure12 shows the relabonship of the rhombic cell to a hexagonal prism.
INDEX SYSTEM Errecom UV-краситель BRILLIANT 100 мл CRYSTAL PLANES The orientahon of a crystal plane is determined by three points in the plane, provided they are not collinear.
If each point lay on a different crystal axis, the plane could be specdied by giving the coordinates of the points in terms of the latbce constants a, a, a3.
However, it turns out to b e more useful for structure analysis to specify the orientation of a plane by the indices determined by the followlug rules Fig.
Find the intercepts on the axes in terms of the lattice constants a, a, a.
The axes may be those of a primitive or nonprimibve cell.
The result, enclosed in parentheses hkl ,is called the index of the plane.
For an intercept at infinity, the corresponding index is zero.
The indices of some important plades in a cubic crystal are illustrated by Fig.
The indices hkl may denote a single plane or a set of parallel planes.
If a plane cuts an axis on the negative перейти на страницу of the origin, the corresponding index is negative, indicated hy placing a minus sign 1 Crystal Structure 13 above the index: hkl.
The cube faces of a cubic crystal are 100.
Planes equivalent by s y m m e q may he denoted by curly brackets braces around indices; the set of cube faces is {100}.
When we speak of the 200 plane we mean a plane parallel to 100 but cutting the a, axis at i n.
SIMPLE CRYSTAL STRUCTURES We di~cusssimple crystal structures of general interest the sohum chlonde, cesium chloride, hexagonal close-packed, h a m o n d and c u h ~ czinc sulfide structures.
Sodium Chloride Structure The sohum chloride, NaCI, structure 1s shown in Figs.
The lattice is face-centered cublc: tile basis conslsts of one Na+ Ion and one C 1 ion Figure 15 We may construct the soditzrn chloride cvstal shuchlre by arranging Naf and C 1 ions alternately at the lattice points o f a simple cubic lattice.
I n the ctysral each ion is surrounded by six nearest neighbors of the opposite charge.
The space lattice is fcc, and the basis has one C I ion at 000 w d one Na' L,?
The 'figre shows one conventional cubic cell.
The ionic diameters here are reduced in relation tothe cell in order to darify the spatial arrangement.
Holden and Pigure 17 Na1ui.
Figure 18 The cesium chloride cqstal struehne.
Represenl include those in the following tative crystals having the ~ karrangement cm lo-'' m 0.
The cube edge a is given in angstroms; 1if nm.
Figure 17 is a photograph of crystals of lead sulfide PbS from Joplin, Missouri.
The Joplin specimens form in beautiful cuhes.
There is one molecule per primitive cell, with atoms at the comers 000 and body-centered positions ,l,i -l of the simple cubic space lattice.
Each atom may he viewed as at the center.
Figure 19 A close-packed layer of spheres is shown, with centers at points marked A.
A second and identical layer of spheres can he placed on top of this, above and patallel to the plane of the drawing, with centen over the points marked B.
There are two choices for a third layer.
It can go in over A or over C.
Ifit goes in over A, the sequence is AEABAB.
If the third layer goes in over C, the sequence is ABCABCABC.
A B c A I Figure 20 The hexagonal close-packed structure.
The atom positions in t h i s smcture do not constitute a soace lattice.
The mace lattice is s i m ~ l ehexa~onal w i k a basis of hvo ;dentical atoms asiociatedwith each latttce nomt.
The lamce parameters o and c are inhcated, where o is in the basal plane and c is the посмотреть больше of the an?
The fraction of the total volume occupled by the spheres is 0.
No structure, regilar or not, has denserpaclung.
The c axis or a, is normal to the plane of a, and a.
Spheres are arranged in a single closest-packed layer A by placmg each sphere in contact with SIX others in a plane.
This layer may serve as either the basal plane of an hcp structure or the 111 plane of the fcc s h c t u r e.
A second s~milarlayer B may be added by placlng each sphere of B in contact with three spheres of the bottom layer, as in Figs.
A third layer C may be added In two ways.
We obtam the fcc structure if the spheres of the third layer are added over the holes in the нажмите чтобы увидеть больше layer that are not occupied by B We obtain the hcp structure when the spheres in the third layer are placed directly over the centers of the spheres in the first layer.
The number of nearest-nelghbor atoms is 12 for both hcp and fcc stnlctures.
If the b~ndingenergy or free energy depended only on the number of nearest-neighbor bonds per atom, there would be no difference energy between the fcc and hcp structures.
Diamond Structure The diamond structure is the structwe of the semiconductors silicon and germanium and is related to the structure of several important semicondnctor binary compouncLs.
The space lattice of Японская гравюра Хиросигэ Утагава - 100 видов Эдо, 46.

Переправа Ёрои-но ватаси к кварталу Коамитё, 1s face-centered cubic.
The primitive basis of the diamond structure has two identical atoms at coordinates 000 and assoc~atedmth each point of the fcc latt~ce,as shown in Fig.
There is no way to choose a primitive cell such that the basis of diamond contains only one atom.
With a fcc space lattice, the basis consists of mia identical atoms at 000 0 d i i .
Figure 23 Crystal structure of diamond, showingthetetrahedralbondarrangement.
The tetrahedral bonding characteristic of the diamond structure is shown in Fig.
Each atom has 4 nearest neighbors and 12 next nearest neighbors.
The diamond читать далее is relatively empty: the maximum proportion of the available volume which may he filled by hard spheres is only 0.
The diamond structure is an example of the directional covalent bonding found in column IV of the periodic table of elements.
Here a is the edge of the conventional cubic cell.
Cubic Zinc Sulfide Structure The diamond structure may be viewed twn fcc structures displaced from each other by one-quarter of a body diagonal.
The cubic zinc sulfide zinc blende structure results when Zn atoms are placed on это ClearOne WS-EAK50-M610 щас fcc lattice and S atoms on the other fcc lattice, as in Fig.
The conventional cell is a cube.
The lattice is fcc.
There are four molecules ZnS per conventional cell.
About each atom there are four equally distafft металлический 100 см 10750 of the opposite kind arranged at the comers of a regular t e t r a h e d ~ w'5.
The inversion operation carries an atom at r into an atom at -r.
The cubic ZnS structure does not have inversion symmetry.
Examples of the cubic zinc sulfide structure are The close equality of the lattice constants of several pairs, notably Delta DTM, Ga P and Al, Ga As, makes possible the construction of sem~conductorhetemjunctions Chapter 19.
DIRECT IMAGING OF ATOMIC STRUCTURE Direct images of crystal structure have been produced by transmission electron microscopy.
Perhaps the most beaubful Images are produced by scanning tunneling microscopy; in STM Chapter 19 one exploits the large vanations in quantum tunneling as a function of the height of a fine metal tip above ~ ~ An the surface of a crystal.
The image of Fig жмите сюда was produced m t h way.
STM method has been developed that will assemble single atoms Into an organized layer nanometer structure on a crystal substrate.
The ideal crystal of classical crystallographers is formed by the periodic repetition of identical units in space.
But no general proof bas been given that I Crystal Structure Figure 25 A scanning tunneling microscope image of atorns on a 111 surface of fcc platinum at 4 K.
The nearest-neighbor spacing is 2.
Photo courtesy of D.
Eigler, IHM R r e r a r c h Divi~irn.
At finite temperatures this is likely not to be true.
We give a further example here.
Random Stacking and Polytypism The fcc and hcp structures are made up of close-packof the incident x-ray hcam, and the migin i n chosen such that k termnates at any reciprocal lattice point.
A diffracted beam will he formed if this sphere intersects any other point in the reciprocal lattice.
The sphere as d r a m intercepts a pint connected with the end of k by a reciprocal lattice vedor G.
The angle B is the Bragg angle of Fig.
This constructionis d u e to P.
BRZLLOUIN ZONES Brillouin gave the statement of the diffraction condition that is most widely used in solid state physics, which means in the description of electron energy band theory and of the elementary excitations of other kinds.
A Brillouin zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice.
Tlie construction in the direct lattice was shown in Fig.
We divide both sides by 4 to obtain.
We now work in reciprocal space, the space of the k's and G's.
Select a vector G from the origin to a reciprocal lattice point.
Construct a plane normal to this vector G at its midpoint.
This plane forms a part of a zone boundary Fig.
An x-ray beam in the crystal will he diffracted if its wavevector k has the magnitude and direction required by 26.
Thus the Brillouin construction exhibits all the wavevectors k which can be Braggreflected by the crystal.
GhanaaNoJxaaIa ayl 30 sysLpe ayl jo m d pzuassa m ale sauoz a p lnq ' s a ~ w a w sp ~ s 30 h s ~ s i p muon -aeJjj!
We first draw a number of vectors from 0 to nearby points in the reciprocal lattice.
Next we construct lines perpendicular to these vectors a t their midpaints.
Tbe smallest endosed area is the first Btillauin zone.
The basis vector in the reciprocal lattice k b, of length equal to ZrIa.
The shortest reciprocal latticevectors from the origin are b and - b The perpendicular bisectors of these vectors form the boundaries of the first BriUouin zone.
Figure 13 Ftrst Brilloum zone of the bodycentered cublc lathce The f i p r e a a regular Figme 12 Pnm~hvebmls vectors of the body mbrc lattice rbomb~cdodecahedron The boundaries of the first Brillduin zones are the planes normal to the six reciprocal lattice vectors?
Reciprocal Lattice to bcc Lattice The primitive translation vectors of the bcc lattice Fig.
One primitive cell of the reciprocal Iattice is the parallelepiped described by the b, bb b3 defined by 31.
The volume of this cell in reciprocal space The.
Each parallelepiped contains one-eighth of each of eight comer points see Fig.
Another primitive cell is the central Wigner-Seitz cell of the reciprocal lattice which is the first Brillouin zone.
Each such cell contains one lattice point at the central point of the cell.
This zone for the hcc lattice is bounded by the planes normal to the 12 vectors of Eq.
The zone is a regular 12-faced solid, a rhombic dodecahedron, as shown in Fig.
Reciprocal Lattice to fec Lattice The primitive translation vectors of the fcc lattice of Fig.
The cells are in reciprocal space, and the reciprocal lattice is body centered.
But the corners of the octahedron thus formed are cut by the planes that are the perpendicular b~qectorsof six other rec~procallattice vectors + Note that Z d a G 1s a reciprocal lattice vector because it is equal to b, b, The first Brillonin zone is the smallest bounded volume about the or~gin,the truncated octahedron shown m Fig 15.
Often it is useful to write the electron concentration n r as the superposition of electron concentration functions nl associated with each atom J of the cell.
If r, 1s the vector to the center of atom j, then the function nl r - rl defines the contribution of that atom to the electron concentration at r.
The total electron concentration at r due to all atoms m the smgle cell is the sum over the s atoms of the basis.
The decornposifion of njr is not unique, for we cannot always say how much charge density IS assoc~atedmth each atom.
This is not an important difficulty.
The structure factor defined by 39 may now be written as integrals over the s atoms of a cell.
We now define the atomic form factor as integrated over all space.
If nl p is an atomic property,fi is an atomic property.
We combine 4l and 42 to obtain the structure factor of t h e basis in the form.
Thus 46 becomes where f 1s the form factor of an atom.
The diffraction pattern does not containlines such as loo300Ill ,or 221butlines such as ZOO110.
What is the physical interpretahon of the result that the 100 reflection vanishes?
The 100 reflection normally occurs when reflections from the planes that hound the cubic cell differ in phase by 27r.
In the bcc lattice there is an interveningplane Fig.
Situated midway between them, it a reflection retarded m phase by 7r with respect to the first plane, thereby canceling the contribution from that plane The cancellahon of the 100 reflection occurs m the Лупа ручная круглая 100 черная Kromatech (Кроматек) lattice because the planes are identical in composition.
A similar cancellation can easily be found in the hcp structure.
Structure Factor of the fcc Lattice The basis of the fcc structure referred to the cubic cell has identical atoms at 000; G;; SO.
Thus 46 becomes 2 Reciprocal Lanice Figurc 16 Explanation uf t l ~ rabsence of a 100 reflection from a body-centered cnbic lattice.
But if only one of the Integers is even, two of the exponents will be odd multiples of -zw and Swill vamrh.
If only one of the integers is odd, the same argument applies and S will also van~sh.
Thus in the fcc lathce no reflections can occur for wlnch the Indices are partly even and partly odd.
The point is heautifnlly illustrated by Fig 17 both KC1 and KBr have an fcc lattice, n r for KC1 simulates an sc lathce because the K+ and C 1 ions have equal numbers of electrons.
Atomic Form Factor In the expression 46 for the structure factor, there occurs the quanbtyf;, which посетить страницу источник a measure of the scattenng power of theyth atom in the unit cell.
The value off involve?
We now give a classical calculation of the scatter~ngfactor.
The scattered rad~abonfrom a single atom takes account of interference effects withln the atom We defined the form factor in 42 : with the integral extended over the electron concentration associated with a single atom.
Let r make an angle ru with G; then G.
In KC1 the numbers of electrons of K t and C T ions are equal.
The scatteliog amplitudes ill + and f C1- me almost exactly equal, so that the crystal looks to a-rays as i f it were a monatomic simple cc~biclaKim of lattice constant a12.
Here al, az, a, are the primitive translation vectors of the crystal lattice.
Any function invariant under a lattice translation T may be expanded in a Fourier series of the form The first Brillouin zone is the Wigner-Seitz primitive cell of the reciprocal lattice.
Only waves whose wavevector k drawn from the origin terminates on a surface of the Brillouin zone can be diffracted by the crystal.
Crystal lattice Simple cubic Body-centered cubic Face-centered cubic First Brillouin zone Cube Rhombic dodecahedron Fig.
Consider a plane hkl in a crystal lattice.
Volume of Brillowin zone.
Show that the volume of the first Brillouin zone is 2 ~ ~ i where V ~.
V, is the volume of a clystal читать полностью cell.
Hint: The volume of a Brillouin zone is equal to the volume of the primitive parallelepiped in Fourier space.
Width of diffraction maximum.
Ak b We know that a diffraction maximum appears when a.
We change Ak slightly and define E in a.
The same result holds true for a three-dimensional crystal.
Structure factor of diamond.
The crystal structure of diamond is described in Chapter 1.
The basis consists of eight atoms if the cell is taken as the conventional cube.
Notice that h, k, I may be written for o, u, u, and this is often done.
Form factor of atomic hydrogen.
Consider a Line of atoms ABAB.
AB, with an A-B bond length of ;a.
The form factors are fA, fa for atoms A, B, respectively.
The incident beam of x-rays is perpendicular to the Line of atoms.
Cohesive energy of bce and fcc neon 3.
Solid molecular hydrogen 85 86 86 7.
Divalent ionic crystals 8.
Young's modulus and Poisson's ratio 9.
Longitudinal wave velocity 10.
Transverse wave velocity 11.
Effective shear constant 12.
General propagation direction 14.
Stability criteria Figure 1 Thc principal types of crystalline binding.
In a neutral atoms with closed electron shells arc bound together weakly by the van der Wads forces associated with fluctuations in the charge distributions.
In b electrons are transferred from the alkali atoms to the halogen atoms, and the restdting ions are held together by attractive cloctrostatic forces between the positive and negative ions.
In c the valence electrons are taken away from each alkali atom to form a communal electron sea in which the positive ions are dispersed.
In d the neutral atoms are bo~mdtogether by the overlapping parts of their electron disttib~~tions.
CHAPTER : : I j ' 1 f 1 3:CRYSTAL BINDING AND ELASTIC CONSTANTS In this chapter we are concerned with the question: What holds a crystal together?
The attractive electrostatic interaction between the negative charges of the electrons and the positive charges of the nuclei is entirely responsible for the cohesion of solids.
Magnetic forces have only a weak effect on cohesion, and gravitational forces are negligible.
Specialized terms categorize distinctive situations: exchange energy, van der Wads forces, and covalent bonds.
The observed differences between the forms of condensed matter are caused in the final analysis by differences in the distribution of the outermost electrons and the ion cores Fig.
The cohesive energy of a crystal is defined as the energy that must be added to the crystal to separate its components into neutral free atoms at rest, at infinite separation, with the same electronic configuration.
The term lattice energy is used in the discussion of ionic crystals and is defined as the energy that must be added to the crystal to separate its component ions into free ions at rest at infinite separation.
Values of the cohesive energy Коптильня DSH-S03 the crystalline elements нажмите чтобы перейти given in Table 1.
Notice the wide variation in cohesive energy between different columns of the periodic table.
The inert gas crystals are weakly bound, with cohesive energies less than a few percent of the cohesive energies of the elements in the C, Si, Ge.
The alkali metal crystals have intermediate values of the cohesive energy, The transition element metals in the middle columns are quite strongly bound.
The melting temperatures Table 2 and bulk modulii Table 3 vary roughly as the cohesive energies.
CRYSTALS OF INERT GASES The inert gases form the simplest crystals.
The electron distribution is very close to that of the free atoms.
Their properties at absolute zero are summarized in Table 4.
The crystals are transparent insulators, weakly hound, with low melting temperatures.
The atoms have very high ionization energies see Table 5.
The outermost electron shells of the atoms are completely filled, and the distribution of electron charge in the free atom is spherically symmetric.
In the crystal the inert gas atoms pack together as closely as possible1: the '.
They do not solidify at zero pressure even at absolute zero temperature.
The average fluctuation at 0 K of a He atom fmm its equilibrium position is of the order of 30 to 40 percent of the nearest-neighbor distance.
The heavier the atom, the less important the zeropoint effects.
If we omit zero-point motion, we calculate a molar volume of 9 cm3 mol-' for solid helium, as compared with the obsemd values of 27.
Li Table 2 Melting points, in K.
Be B After R H Larnoreaux 453.
Birch, in Handbook of physical constants, Geological Society of America Memoir 97, 107-173 1966.
Original references should be consulted when values are needed for research purposes.
Letters in parentheses refer to the crystal form.
E81IT: ~llCib3JIIs25 5~~13ir ifil:fII~?
What holds an inert gas crystal together?
The electron distribution in the crystal is not significantly distorted from the electron distribution around the free atoms because not much energy is available to distort the free atom charge distributions.
The cohesive energy of an atom in the crystal is only 1 percent or less of the ionization energy of an atomic electron.
Part of this distortion gives the van der Wads interaction.
Van der Waals-London Interaction Consider two identical inert gas atoms at a separation R large in comparison with the radii of the atoms.
What interactions exist between the two neutral atoms?
If the charge distributions on the atoms were rigid, the interaction between atoms would be zero, because the electrostatic potential of a spherical distribution of electronic charge is canceled outside a neutral atom by the нажмите для деталей potential of the charge on the nucleus.
Then the inert gas atoms could show no cohesion and could not condense.
But the atoms induce dipole moments in each other, and the induced moments cause an attractive interaction between the atoms.
As a model, we consider two identical linear harmonic oscillators 1 and 2 separated by R.
Each oscillator bears charges 5 e with separations 1-1 and x2, as in Fig.
The particles oscillate along the x axis.
Let p, andp, denote the momenta.
The force constant is C.
Then the hamiltonian of the unperturbed system is Each uncoupled oscillator is assumed to have the frequency o, of the strongest optical absorption line of the atom.
A, Figme 3 Coordinates of the two oscillators Let XI be смотрите подробнее coulomb interaction energy of the two oscillators.
The geometlyis shown in the f i p r e.
The internuclear coordinate is R.
In 8 we have expanded the square root.
This attractive interaction varies as the minus sixth power of the separation of the two oscillators.
This is called the van der Wads interaction, known also as the London interaction or the induced dipole-dipole interaction.
It is the principal attractive interaction in crystals of inert gases and also больше информации crystals of many organic molecules.
The interaction is a quantum effect, in the sense that A U + 0 as fi 0.
Thus the zero point energy of the system is lowered by the dipole-dipole coupling of Eq.
The van der Waals interaction does not depend for its existence on подробнее на этой странице overlap of the charge densities of the two atoms.
An approximate value of the constant A in 9 for identical atoms is given by fiw,a2, where fiw, is the energy of the strongest optical absorption line and a is the electronic p~larizabilit~ Chapter 15.
At sufficiently close separations the overlap energy is repulsive, in large part hecause of the Pauli exclusion principle.
The elementary statement of the principle is that two electrons cannot have all their quantum numbers equal.
When the charge distributions of two atoms overlap, there is a tendency for electrons from atom B to occupy in part states of atom A already occupied by electrons of atom A, and vice versa.
The solid circles denote the nuclei.
ISTZST Total spin one Figure 5 The effect of Pauli principle on the repulsive energy: in an extreme example, two hydrogen atoms are pushed together until the protons are almost in contact.
The energy of the electron system alone can be taken from observations on atomic He, which has two electrons.
In a the electrons have antipardel spins and the Pauli principle has no effect: the electrons are bound by 7 8.
The electrons now are bound by -59.
We have omitted the repulsive coulomb energy of the two, which is the same in both a and b.
The Pauli principle prevents multiple occupancy, and electron distributions of atoms with closed shells can overlap only if accompanied by the partial promotion of electrons to unoccupied high energy states of the atoms.
Thus the electron читать далее increases the total energyof the system and gives a repulsive contribution to the interaction.
An extreme example in which the overlap is complete is shown in Fig.
We make no attempt here to evaluate the repulsive interaction2 from first principles.
The mathematical calculation is always complicated even if the charge distribution is known.
Nu+ Figure 6 Form of the Lemard-Jonespotential 10 which describes the interaction of hm, inert gas atoms.
Notice how steep the curve is inside the minimum, and how Bat it is outside the minimum.
The constants A and B are empirical parameters determined from independent measurements made in the gas phase; the data used include the virial coefficients and the viscosity.
The potential 10 is h o w n as the Lennard-Jones potential, Fig.
Values of E and u given in Table 4 can be obtained from gas-phase data, so that calculations on properties of the solid do not involve disposable parameters.
Other empirical forms for the repulsive interaction are widely used, in particular the exponential form A exp -Wpwhere p is a measure of the range of the interaction.
This is generally as easy to handle analytically as the inverse power law form.
Equilibrium Lattice Constants If we neglect the kinetic energy of the inert gas atoms, the cohesive energy of an inert gas crystal is given by summing the Lennard-Jones potential 10 over all pairs of atoms in the crystal.
If there are N atoms in the crystal, the total potential energy is 3 Crystal Binding where pYRis the distance between reference atom i and any other atom j,expressed in terms of the nearest-neighbor distance R.
The factor ;occurs with the N to compensate for counting twice each pair of atoms.
The summations in 11 have been evaluated, and for the fcc structure There are 12 nearest-neighbor sites in the fcc structure; we see that the series are rapidly converging and have values not far from 12.
The nearest neighbors contribute most of the основываясь на этих данных energy of inert gas crystals.
The corresponding sums for the hcp structure are 12.
If we take Ut, in 11 as the total energy of the crystal, the equilibrium value R, is given by requiring that U, be a minimum with respect to variations in the nearest-neighbor distance R: whence the same for all elements with an fcc structure.
The observed values of Rdu, using the independently determined values of u given in Table 4, are: The agreement with 14 is remarkable.
The slight departure of Rolu for the lighter atoms from the universal value 1.
From measurements on the gas phase we have predicted the lattice constant of the clystal.
This is the calculated cohesive energy when the atoms are at rest.
Quantum-mechanical corrections act to reduce the binding by 28, 10, 6, and 4 percent of Eq.
On this model the quantum zero-point correction to the energy is inversely proportional to the mass.
The final calculated cohesive energies agree with the experimental values of Table 4 within 1to 7percent.
One consequence of the quantum kinetic energy is that a crystal of the isotope Ne2" is observed to have a larger lattice constant than a crystal of NeZ2.
The higher quantum kinetic energy of the lighter isotope expands the lattice because the kinetic energy is reduced by expansion.
The observed lattice constants extrapolated to absolute zero from 2.
IONIC CRYSTALS Ionic crystals are made up of positive and negative ions.
The ionic bond results from the electrostatic interaction of oppositely charged ions.
Two common crystal structures found for ionic crystals, the sodium chloride and the cesium chloride structures, were shown in Chapter 1.
The electronic configurations of all ions of a simple ionic crystal correspond to closed electronic shells, as in the inert gas atoms.
In lithium fluoride the configuration of the neutral atoms are, according to the periodic table in the front endpapers of this book, Li: ls22s, F: ls22s22p5.
The singly charged ions have the configurations Li+:ls2,F-: ls22s22p6,as for helium and neon, respectively.
Inert gas atoms have closed shells, and the charge distributions are spherically symmetric.
We expect that the charge distributions on each ion in an ionic crystal will have approximately spherical symmetry, with some distortion near the region of contact with neighboring atoms.
This picture is confirmed by x-ray studies of electron distributions Fig.
A quick estimate suggests that we are not misguided in looking to electrostatic interactions for a large part of the binding energy of an ionic crystal.
The distance between a positive ion and the nearest negative ion in crystallirle sodium chloride is 2.
This value may be compared Fig.
We now calculate the energy more closely Electrostatic o r Madelung Energy The long-range interaction between ions with charge?
The numbers an the contours ve the relative electron concentration.
The lattice energy with respect to separated ions is 7.
All values on the fie- eneW tton affmity are given in Table 6.
The ions arrange themselves in whatever c~ystal structnre gives the strongest attractive interaction compatible with the repulsive interaction at short distances between ion cores.
The repulsive interactions between ions with inert gas configurations are similar to those between inert gas atoms.
The van der Weals part of the attractive interaction in ionic crystals makes a relatively small contribution to the cohesive energy in ionic crystals, of the order of 1or 2 percent.
The main contribution to the binding energy of ionic crystals is electrostatic and is called the Madelung energy.
Atom Electron affinity enerw eV Atom Electron affinity energy eV Source: H.
Hotop and W C.
Thus where the + sign is taken for the like charges and the - sign for unlike charges In SI units the coulomb interaction is?
The repulsive term describes the fact that each ion resists overlap with th electron distributions of neighboring ions.
We treat the strength A and range p as constants to be determined from observed values of the lattice constant an compressibility; we have used the exponential form of the empirical repulsiv potential rather than the R-l2 form used for the inert gases.
The change i made because it may give a better representation of the repulsive interaction For the ions, we do not have gas-phase data available to permit the indepen dent determination of A and p.
In the NaCl strnctnre the value of U, does not depend on whether th reference ion i is a positive or a negative ion.
The sum in 17 can he arranged to converge rapidly, so that its value will not depend on the site of the reference ion in the crystal, as long as it is not near the surface.
Here N, rather than 2N, occurs because we must count each pair of interactions only once or each bond only once.
The total lattice energy .
The 2 sign is discussed just before 25.
The value of the Madelung constant is of central importance in the theory of an ionic crystal.
Methods for its calculation are discussed next.
We shall find that p is of the order of O.
Evaluation of the Madelung Constant The first calculation of the coulomb energy constant a was made by Madelung.
A powerful general method for lattice sum calculations was developed by Ewald and is developed in Appendix B.
Computers are now used for the calculations.
The definition of the Madelung constant a is, by 21 20 to give a stable crystal it is necessary that a be positive.
If we take the reference ion as a negative charge, the plus sign will apply to positive ions and the minus sign to negative ions.
An equivalent definition is where 5 is the distance of thejth ion from the reference ion and R is the nearest-neighbor distance.
The value given for a will depend on whether it is defined in terms of the nearest-neighbor distance R or in terms of the lattice parameter a or in terms of some other relevant length.
As an example, we compute the Madelung constant for the infinite line of ions of жмите сюда sign in Fig.
Then the factor 2 occurs because there are two ions, one to the right and one to the left, at equal distances r.
I In three dimensions the series presents greater difficulty.
It is not possible to write down the successive terms by a casual inspection.
More important, the series will not converge unless the successive terms in the series are arranged so that the contributions from the positive and negative terms nearly cancel.
Typical values of the Madelung constant are listed below, based on unit charges and referred to the nearest-neighbor distance: The Madelung and repulsive contributions to the binding of a KC1 crystal are shown in Fig.
Properties of alkali halide crystals having the sodium chloride structure are given in Table 7.
The calculated values of the lattice energy are in exceedingly good agreement with the observed values.
Values in square brackets at absolute zero temperature and zero pressure, from private communication by L.
Tosi, Solid State Physics 16, 1 1964.
The numbers on the contours give the electron concentration per primitive cell, with four valence electrons per atom eight electrons per primitive cell.
Note the high concentration midway along the Ce-Ge bond, as we expect for covalent honding.
COVALENT CRYSTALS The covalent bond is the classical electron pair or homopolar bond of chemistry, particularly of organic chemistry.
It is a strong bond: the bond between two carbon atoms in diamond with respect to separated neutral atoms is comparable with the bond strength in ionic crystals.
The covalent bond is usually formed from two electrons, one from each atom participating in the bond.
The electrons forming the bond tend to be partly localized in the region between the two жмите сюда joined by the bond.
The spins of the two electrons in the bond are antiparallel.
The covalent bond has strong directional properties Fig.
Thus carbon, silicon, and germanium have the diamond structure, with atoms joined four nearest neighbors at tetrahedral angles, even though this arrangement gives a low filling of space, 0.
The tetrahedral bond allows only four nearest neigbbors, whereas a close-packed structure has 12.
We should not overemphasize the similarity of the bonding of carbon and silicon.
Carbon gives biology, but silicon gives geology and semiconductor technology.
The binding of molecular hydrogen is a simple example of a covalent bond.
The strongest binding Fig.
The binding depends on the relative spin orientation not because there are strong magnetic dipole forces between the spins, but because the Pauli principle modifies the distribution of charge according to the spin orientation.
This spin-dependent coulomb energy is called the exchange interaction.
ZIV pue 213 uaaMJaq a a u a ~ a j j payA.
Phillips, Bonds and bands in semiconductors.
There is a continuous range of crystals between the ionic and the covalent limits.
It is often important to estimate the extent a given bond is ionic or covalent.
A semiempirical theory of the fractional ionic or covalent character of a bond in a dielectric crystal has been developed with considerable success by J.
METALS Metals are characterized by high electrical conductivity, and a large number of electrons in a metal are free to move about, usually one or two per atom.
The electrons available to move about are called conduction electrons.
The valence electrons of the atom become the conduction electrons of the metal.
In some metals the interaction of the ion cores with the conduction electrons always makes a large contribution to the binding energy, but the characteristic feature of metallic binding is the lowering of the energy of the valence electrons in the metal as compared with the free atom.
The binding energy of an alkali metal crystal is considerably less than that of an alkali halide c~ystal:the bond formed by a conduction electron is not very strong.
The interatomic distances are relatively large in the alkali metals because the kinetic energy of the conduction electrons is lower at large interatomic distances.
This leads to weak binding.
Metals tend to crystallize in relatively 69 Figure 13 The hydrogen difluoride ion HF4 is stabilized by a hydrogen bond.
The sketch is of an extreme model of the bond, extreme in the sense that the proton is shown bare of electrons close packed structures: hcp, fcc, bcc, and some other closely жмите сюда structures, and not in loosely-packed structures such as diamond.
In the transition metals there is additional binding from inner electron shells.
Протеин Mutant Iso Surge (727 г) metals and the metals immediately following them in the periodic table have large d-elec~onshells and are characterized by high binding energy HYDROGEN BONDS Because neutral hydrogen has only one electron, it should form a covalent bond with only one other atom.
It is known, however, that under certain conditions an atom of hydrogen is attracted by rather strong forces to two atoms, thus forming a hydrogen bond between them, with a bond energy of the order of 0.
I n the extreme ionic form of the hydrogen bond, the hydrogen atom loses its electron to another atom in the molecule; the bare proton forms the hydrogen bond.
The atoms adjacent to the proton are so close that more than two of them would get in each other's way; thus the hydrogen bond connects only twc atoms Fig.
The hydrogen bond is an important part of the interaction between HzO molecules and is responsible together with the electrostatic attraction узнать больше the electric dipole moments for the strihng physical properties of water and ice.
It is important in certain ferroelectric crystals and in DNA.
ATOMIC RADII Distances between atoms in crystals can be measured very accurately by x-ray diffraction, often to 1part in lo5.
Can we say that the observed distance between atoms may be assigned partly to atom A and partly to atom B?
Can a definite meaning be assigned to the radius of an atom or an ion, irrespective of the nature and composition of the crystal?
Strictly, the answer is no.
The charge distribution around an atom is not limited by a rigid spherical boundary.
Nonetheless, the Сюэ Фэй Притчи of an atomic from the additive properties of the atomic radii.
Further, the electronic configuration of the constituent atoms often can be inferred by comparison of measured and predicted values of the lattice constants.
To make predictions of lattice constants it is convenient to assign Table 9 sets of self-consistent radii to various types of bonds: one set for ionic crystals with the constituent ions 6-coordinated in inert gas closed-shell configurations, another set for the ions in tetrahedrally-coordinated structures, and another set for 12-coordinated close-packed metals.
The predicted self-consistent radii of the cation Na+ and the anion F- as given in Table 9 would lead to 0.
This agreement is much better than if we assume atomic neutral configurations for Na and F, for this would lead to продолжить />The interatomic distance between C atoms in diamond is 1.
In silicon, which has the same crystal structure, one-half the interatomic distance is 1.
In Sic each atom is surrounded by four atoms of the opposite kind.
If we add the C and Si radii just given, we predict 1.
This is the kind of agreement a few percent that we shall find in using tables of atomic radii.
The ionic radii can be used in conjunction with Table 10.
Each Ba++ ion has 12 nearest 0-- ions, so that the coordination number is 12 and the correction A, of Table 10 applies.
The actual lattice constant is somewhat smaller than the estimates and may perhaps suggest that the bonding is not purely ionic, hut is partly covalent.
The continuum approximation is usually valid for elastic waves of wavelengths A longer than 10-6cm, which means for frequencies below 10" or 10" Hz.
Some of the material below looks complicated because of the unavoidable multiplicity of subscripts on the symbols.
The basic physical ideas are simple: we use Hooke's law and Newton's second law.
Hooke's law states that in an elastic solid the strain is directly proportional to the stress.
The law applies to small strains only.
We say that we are in the nonlinear region when the strains are so large that Hooke's law is no longer satisfied.
We specify the strain in terms of the components e, e, em, exY,e, e, which are defined below.
We treat infinitesimal strains only.
We shall not distinguish in our notation between isothermal constant temperature and adiabatic constant entropy deformations.
узнать больше small differences between the isothermal and adiabatic elastic constants are not often of importance at room temperature and below.
We imagine that three orthogonal vectors ir,j.
After a small uniform deformation of the solid has taken place, the axes are distorted ссылка на подробности orientation and in length.
In a uniform deformation each primitive cell of the crystal is deformed in the same way.
The original axes were of unit length, hut the new axes will not necessarily be of unit length.
For example, 73 Figure 14 Coordinate axes for the description of the state of strain; the orthogonal unit axes in the unstrained state a are deformed in the strained state b.
The fractional changes of length of the i, 9, and i axes are e, eyy,E, respectively, to the first order.
The origin is taken at some other atom.
The пойму Видеорегистратор HDC HD406 Mini удалено R of the deformation is defined by + or, from 26This may be written in a more general form by introducing u, u, w such that the displacement is given by If the deformation is nonuniform we must relate u, v, w to the local strains.
The six completely define the strain.
The dilation is negative for hydrostatic pressure.
The unit cube of edges i ,9, i has a volume after deformation of by virtue of a well-hown result for the volume of a parallelepiped having edges x', y', z'.
From 26 we have Products of two strain components have been neglected.
The dilation 8 is then given by Stress Components The force acting on a unit area in the solid is defined as the stress.
The capital letter indicates the direction of the force, and the subscript indicates the normal to the plane to which the force is applied.
~x I 3 Crystal Binding ELASTIC COMPLIANCE AND STIFFNESS CONSTANTS Hooke's law states that for sufficiently small deformations the strain is directly proportional to the stress, so that the strain components are linear functions of the stress components: The quantities S, Slz.
Elastic Energy Density The 36 constants in 37 or in 38 may be reduced in number by several considerations.
The elastic energy density U is a quadratic function of the strains, in the approximation of Hooke's law recall the expression for the energy of a stretched spring.
Thus we may write where the indices 1through 6 are defined as: The C's are related to the C's of 38as in 42 below.
It follows that the elastic stiffness constants are symmetrical: Thus the thirty-six elastic stiffness constants are reduced to twenty-one.
Elastic Stiffness Constants of Cubic Crystab The number of independent Регулятор Phoenix Edge 100 ESC 100A 8S 34V - CSE-010-0100-00 stiffness constants is reduced further if the crystal possesses symmetry elements.
We now show that in cubic crystals there are only three independent stiffness constants.
We assert that the elastic energy density of a cubic crystal is and that no other quadratic terns occur; that is, do not occur.
The minimum symmetry requirement for a cubic structure is the existence of four three-fold rotation axes.
The effect of a rotation of 2 ~ 1 3about these four axes is to interchange the x, y, z axes according to the schemes according to the axis chosen.
Under the first of these schemes, for example, and similarly for the other terms in parentheses in 43.
Thus 43 is invariant under the operations considered.
But each of the terms exhibited in 44 is odd in one or more indices.
A rotation in the set 45 can be found which will 3 Crystal Binding 79 Figure 17 Rotation by 2 ~ 1 3about the axis marked 3 changes r + y; y + z: andz + x.
Thus the terms 44 are not invariant under the required operations.
It remains to venfy that the numetical factors in 43 are correct.
For a cubic crystal, The compressibility K is defined as K Table 3.
Values of B and K are given in ELASTIC WAVES IN CUBIC CRYSTALS By consideling as in Figs.
There are similar equations for the y and 2 directions.
From 38 and 50 it follows that for a cubic crystal here the x, y, z directions are parallel to the cube edges.
Using the definitions 31 and 32 of the strain components we have where u, vw are the components of the displacement R as defined by 29.
The mass is p 4.
Figure 19 If springs A and B are stretched equally.
Both the wavevector and the particle motion are along the x cube edge.
жмите is not true for K in a general direction in the crystal.
Consider a shear wave that propagates in the xy plane with particle displacement w in the z direction whence 32c gives independent of propagation direction in the plane.
The condition for a solution is that the determinant of the coefficients of u and o in посетить страницу источник should equal zero: This equation has the roots 1 j The first root describes a longitudinal wave; the second root describes a shear wave.
How do we determine the direction of paficle displacement?
Selected values of the adiabatic elastic stiffness constants of cubic crystals at low temperatures and at room temperature are given in Table 11.
Notice the general tendency for the elastic constants to decrease as the temperature is increased.
Further values at room temperature alone are given in Table 12.
The table was compiled with the assistance of Professor Charles S.
In general, the polarizations directions of article displacement of these are not exactly parallel or perpendicular to K.
The analysis is much simpler in these special directions than in general directions.
The repulsive interaction between atoms arises generally from the electrostatic repulsion of overlapping charge distributions and the Pauli principle, which compels overlapping electrons of parallel spin to enter orbitals of higher e n e r a.
Ionic crystals are hound by the electrostatic attraction of charged ions of opposite sign.
The electrostatic energy of a structure of 2N inns of charge?
Metals are hound by the reduction in the hnetic energy of the valence electrons in the metal as compared with the free atom.
A covalent bond is characterized by the overlap of charge distributions of antiparallel electron spin.
The Pauli contribution to the repulsion is reduced for antiparallel spins, and this makes possible a greater на этой странице of overlap.
The overlapping electrons hind their associated ion cores by electrostatic attraction.
In a qnantum solid the dominant repulsive energy is the zero- point energy of the atoms.
Consider a crude one-dimensional model of crystalline He4with each He atom confined to a line segment of length L.
In the ground state the wave function within each segment is taken as a half wavelength of a free paricle.
Find the zero-point kinetic energy per particle.
The lattice sums for the bcc structures are 3.
Treat each H, molecule as a sphere.
The observed value of the cohesive energy is 0.
Possibility of ionic cryatab R+R.
Imagine a crystal that exploits for binding the coulomb attraction of the positive and negative ions of the same atom or molecule R.
This is believed to occur with certain organic molecules, but it is not found when R is a single atom.
Use the data in Tables 5 and 6 to evaluate the stability of such a form of Na in the NaCl structure relative to normal metallic sodium.
Evaluate the energy at the observed interatomic distance in metallic sodium, and use 0.
Consider a line of 2N ions of alternating charge?
Using A and p from Table 7 and the Madelung constants given in the t e acalculate the cohesive energy of KC1 in the cubic ZnS structure described in Chapter 1.
Compare with the value calculated for KC1 in the NaCl structure.
Barium oxide has the NaCl structure.
Estimate the cohesive energies per molecule of the hypothetical crystals BaiO- and Batto-referred to separated neutral atoms.
The first electron affinity of the 3 Crystal Binding Undeformed body 87 7 TensionFigure 21 Youngs modulus is defined as stresslstrainfor a tensile stress acting in one direction, with the specimen sides left free.
The second electmn affinity is the energy released in the reaction 0- f e + 0.
Which valence state do you predict will occur?
Assume R, is the same for both forms, and neglect the repulsive energy.
Young's modulus and Poisson's ratio.
Find expressions in terns of the elastic stiffnesses for Young's modulus and Poisson's ratio as defined in Fig.
Hint: See Problem 9.
It is known that an R-dimensional square Тонармы поворотные VPI with all elements equal to unity has roots R and 0, with the R occurring once and the zero occurring R - 1 times.
If all elements have the value p, then the roots are Rp and 0.
For an example of the instability which results when C, C, see L.
Letters 15, 250 1965.
Crystal Vibrations VIBRATIONS O F CRYSTALS WITH MONATOMIC BASIS 91 First Brillonin zone 93 Group velocity 94 Long wavelength limit 94 Derivation of force constants from experiment 94 TWO ATOMS PER PRIMITIVE BASIS QUANTIZATION O F ELASTIC WAVES PHONON MOMENTUM INELASTIC SCATTERING BY PHONONS SUMMARY Посмотреть еще k t 1.
Monatomic linear lattice 2.
Continuum wave equation 3.
Basis of two unlike atoms 4.
Atomic vibrations in a metal 7.
Soft phonon modes Chapter 5 treats the thermal properties of phonons.
Figure 2 Dashed lines Planes of atoms when in equilibrium.
Solid lines Planes of atoms when displaced as for a longitudinal wave.
The coordinate u measures the displacement of the planes.
Figure 3 Planes of atoms as displaced during passage of a transverse wave.
We want to find the frequency of an elastic wave in terms of the wavevector that describes the wave and in terms of the elastic constants.
These are the directions of the cube edge, face diagonal, and body diagonal.
When Лезвие очистки ленты для RICOH Aficio 1035/1045 (CET), CET4595 wave propagates along one of these directions, entire planes of atoms move in phase with displacements either parallel or perpendicular to the direction of the wavevector.
We can describe with a single coordinate u, the displacement адрес страницы the planes from its equilibrium position.
The problem is now one dimensional.
For each wavevector there are three modes as solutions for us, one of longitudinal polarization Fig.
We assume that the elastic response of the crystal is a linear function of the forces.
That is equivalent to the assumption that the elastic energy is a quadratic function of the relative displacement of any two points in the crystal.
Terms in the energy that are linear in the displacements will vanish in equilibrium-see the minimum in Fig.
Cubic and higher-order terms may be neglected for sufficiently small elastic deformations.
We assume that the force on the planes caused by the displacement of the plane s + p is proportional to the difference us+,-us of their displacements.
The C is the force constant between nearest-neighbor planes and will differ for longitudinal and transverse waves.
It is convenient hereafter to regard C as defined for one atom of the plane, so that F, is the force on one atom in the planes.
The equation of motion of an atom in the planes is where M is the mass of an atom.
We look for solutions with all displacements having the time dependence exp -iot.
The value to use for a will depend on the direction of K.
The special significance of phonon wavevectors that lie on the zone boundary is developed in 12 below.
By a trigonometric identity, 7 may be written as A plot of o versus K is given in Fig.
Figure 4 Plot of o versus K.
Crystal Vibratim First Brillouin Zone What range of K is physically significant for elastic waves?
Only those in the first Brillouin zone.
From 4 the ratio of the displacements of two successive planes is given by The range .
The range of independent values of K is specified by This range is the first Brillouin zone of the linear lattice, as defined in Chapter 2.
Values of K outside of the first Brillouin zone Fig.
We may treat a value of K outside these limits by subtracting the integral multiple of 2.
Thus the displacement can always be described by a wavevector within the fust zone.
We note that 2 m l a is a reciprocal lattice vector because 2 d a is a reciprocal lattice vector.
Thus by subtraction of an appropriate reciprocal lattice vector from K, we always obtain an equivalent wavevector in the first zone.
ST, whence i Figure 5 The wave represented by the solid curve conveys no information not given by the dashed curve.
Only wavelengths longer than 2n are needed to represent the ,notion.
This situation is equivalent to Bragg reflection of x-rays: when the B condition is satisfied a traveling wave cannot propagate in a lattice, through successive reflections back and forth, a standing wave is set up.
W x-rays it is possible to haven equal to other integers besides unity because amplitude of the electromagnetic wave has a meaning in the space betw atoms, hut the displacement amplitude of an elastic wave usually has a m ing only at the atoms themselves.
This is the velocity of en propagation in the medium.
With the particular dispersion relation 9the group velocity Fig.
The result that the frequency is directly proportional to the wavevector in long wavelength limit is equivalent to the statement that the velocity of so is independent of frequency in this limit.
Derivation of Force Constants from Experiment In metals the effective forces may be of quite long range and are car from ion to ion through the conduction electron sea.
Interactions have b found between planes of atoms separated by as many as 20 planes.
We can m a statement about the range of the forces from the observed experime 4 Phomm I.
Crystal Vibration8 95 Figure 6 Group velocity u, versus K for model of Fig.
Cp l - cos pKa.
Consider, for example, the NaCl or structures, with two atoms in the primitive cell.
For each polarization mode in a given propagation direction the dispersion relation w versus K develops two branches, known as the acoustical and optical branches, as in Fig.
We have longitudinal LA and transverse acoustical TA modes, and longitudinal LO and transverse optical TO modes.
If there are p atoms in the primitive cell, there are 3p branches to the dispersion relation: 3 acoustical branches and 3p - 3 optical branches.
Sheach with two atoms in a primitive cell, have six branches: one LA, one LO, two TA, and two TO.
The lattice constant is a.
The results were obtained with neutron inelastic scattering by G.
The numerology of the branches follows from the number of degrees of freedom of the atoms.
With p atoms in the primitive cell and N primitive cells, there are pN atoms.
Each atom has three degrees of freedom, one for each of the xy, z directions, mahng a total of 3pN degrees of freedom for the crystal.
The number of allowed K values in a single branch is just N for one Brillouin zone.
The volume of a Brillouin zone is Zn 'N, where V.
Thus the number of allowed Kvalues in a Brillouin zone is VN.
Crystal Vibrations Figure 9 A diatomic ctystal structure with masses M, Mz connected by force constant C between adjacent planes.
The displacements of atoms M Iare denoted by u,- u, u.
The atoms are shown in their undisplaced positions.
LA and the two TA branches have a total of 3N modes, thereby accounting for 3N of the total degrees of freedom.
The remaining 3p - 3 N degrees of freedom are accommodated by the optical branches.
We consider a cubic clystal where atoms of mass MI lie on one set of planes and atoms of mass M, lie on planes interleaved between those of the first set Fig.
It is not essential that the masses be different, but either the force constants or the masses will be different if the two atoms of the basis are in nonequivalent sites.
Let a denote the repeat distance of the lattice in the direction normal to the lattice planes considered.
We write the equations of motion under the assumption that each plane interacts only with its nearest-neighbor planes and that the force constants are identical between all pairs of nearest-neighbor planes.
We refer to Fig.
For small Ka we have cos Ka E 1- K2a2.
The particle displacements in the transverse acoustical TA and transverse optical TO branches are shown in Fig.
If the two atoms cany opposite charges, as in Fig.
Acoustical mode 4 Phonons I.
Crystal Vibrations type with the electric field of a light wave, so that the branch is called the optical branch.
At a general K the ratio ulu will be complex, as follows from either of the equations 20.
The atoms and their center of mass move together, as in long wavelength acoustical vibrations, whence the term acoustical branch.
This is a characteristic feature of elastic waves in polyatomic lattices.
QUANTIZATON OF ELASTIC WAVES ij ; L The energy of a lattice vibration is quantized.
The quantum of energy is called a phonon in analogy with the photon of the electromagnetic wave.
The energy of an elastic mode of angular frequency o is when the mode is excited to quantum number n; that is, when the mode is occupied by n phonons.
It occurs for both phonons and photons as a consequence of their equivalence to a quantum harmonic oscillator of frequency w, for which the energy eigenvalues are The quantum theory of phonons is developed in Appendix C.
We can quantize the mean square phonon amplitude.
Consider the standing wave mode of amplitude Here u is the displacement of a volume element from its equilibrium position at x in the crystal.
The energy in the mode, as in any harmonic oscillator, is half kinetic energy and half potential energy, when averaged over time.
The square of the amplitude of the mode is This relates the displacement in a given mode to the phonon occupancy n of the mode.
What is the sign of o?
The equations of motion such as 2 are equations for oZ,and if this is positive then w can have either sign, + or .
But the 99 energy of a phonon must be positive, so it is conventional and suitable to vie o as positive.
If the crystal structure is unstable, then o2will be negative and will be imaginary.
PHONON MOMENTUM A phonon of wavevector K will interact with particles such as photon neutrons, and electrons as if it had a momentum hK.
However, a phonon do not carry physical momentum.
In crystals there exist wavevector selection rules for allowed transitio between quantum states.
We saw in Chapter 2 that the elastic scattering of x-ray photon by a crystal is governed by the wavevector selection rule where G is a vector in the reciprocal lattice, k is the wavevector of the incide photon, and k' is the wavevector of the scattered photon.
In the reflectio process the crystal as a whole will recoil with momentum -hG, but this un form mode momentum is rarely considered explicitly.
Equation 30 is an example of the rule that the total wavevector of inte acting waves is conserved in a periodic lattice, with the possible addition of reciprocal lattice vector G.
The true momentum of the whole system always rigorously conserved.
If the scattering of the photon is inelastic, with th creation of a phonon of wavevector K, then the wavevector selection ru becomes If a phonon K is absorbed in the process, we have instead the relation Relations 31 and 32 are the natural extensions of 30.
INELASTIC SCAWERING BY PHONONS Phonon dispersion relations o K are most often determined experime tally by the inelastic scattering of neutrons with the emission or absorption of phonon.
A neutron sees the crystal lattice chiefly by interaction with the nucl 4 Phonons I.
Crystal Vibrations of the atoms.
The kinematics lattice are described by как сообщается здесь gc tttering of a neutron beam by a crystal evector selection m1 ; the wavevector of and by the requirement of conservation of energy the phonon created + or absorbed - in the process, and G is onwe choose G such that K lies in the any reciprocal lattice vector.
The momentum p is given by hk, where k is the wavevecto the neutron.
The statement of conservation of energy is where h o is the energy of the phonon created + or absorbed - in process.
To determine the dispersion relation using продолжение здесь and 34 it is necessar the experiment to find the energy gain or loss of the scattered neutrons function of the scattering direction k - k'.
Results for germanium and KBr given in Fig.
A spectrometer абсолютно Сапоги Tamaris 1-1-25998-33-467/220 коричневый разочарован phonon studies is shown in Fig.
SUMMARY The quantum unit of a crystal vibration is a phonon.
If the angular quency is o, the energy of the phonon is fio.
All elastic waves can be described by wavevectors that lie within the f Brillouin zone in reciprocal space.
If there are p atoms in the primitive по этому адресу, the phonon dispersion relation have 3 acoustical phonon branches and 3p - 3 optical phonon branches.
Show that for long wavelengths the equation of motion 2 reduces to the continuum elastic wave equation where o is the velocity of sound 3.
Basis oftwo unlike a t o m.
Show that at this value of K the two lattices act as if decoupled: one lattice remains at rest while the other lattice moves.
Such a form is expected in metals.
Use this and Eq.
Thus a plot of wZversus K or of o versus K has a vertical tangent at k,: there is a kink at k, in the phonon dispersion relation o K.
Consider the normal modes of a linear chain in which the force constants between nearest-neighbor atoms are alternately C and 10C.
Let the masses he equal, and let the nearest-neighbor separation be aI2.
Sketch in the dispersion relation by eye.
This problem simulates a crystal of diatomic molecriles such as H.
Atomic vibrations in a metal.
Consider point ions of mass M and charge e im- mersed in a uniform sea of conduction electrons.
The ions are imagined to be in stable жмите when at regular lattice points.
If one ion is displaced a small distance r from its equilibrium position, the restoring force is largely due to the electnc charge within the sphere of radius r centered at the equilibrium position.
The interatomic potential is Phonons II.
Thermal Properties PHONON HEAT CAPACITY Planck distribution Normal mode enumeration Density of states in one dimension Density of states in three dimensions Debye model for density of states Debye T3 law Einstein model of the density of states General result for D o ANHARMONIC CRYSTAL INTERACTIONS Thermal expansion THERMAL CONDUCTMTY Thermal resistivity of phonon gas Umklapp processes Imperfections PROBLEMS 1.
Singularity in density of states 2.
Rms thermal dilation of crystal cell 3.
Zero point lattice displacement and strain 4.
Heat capacity of layer lattice 5.
Griineisen constant Figure 1 Plot of Planck distribution function.
At high temperatures the occupancy of a state approximately linear in the temperature.
The function n + b, which is not plotted, approach the dashed line as asymptote at high temperatures.
PHONON HEAT CAPACITY By heat capacity we shall usually mean the heat capacity at constant volume, which is more fundamental than the heat capacity at constant pressure, which is what the experiments determine.
The contribution of the phonons to the heat capacity of a crystal is called the lattice heat capacity and is denoted by C.
A graph of n is Planck Distribution Consider a set of identical harmonic oscillators in thermal equilibrium.
The ratio of the number of oscillators in their n + 1 th quantum state of excitation to the number in the nth quantum state is N,+,IN.
The fractional difference between C, and C, is usually small in solids and often may be neglected.
As T- 0 we see that C,+Cv, provided a and B are constant.
Then the energy is The lattice heat capacity is found by differentiation with respect to tem ture.
This function is called the density of modes or, more often sity of states.
Density of States in One Dimension Consider the boundary value prohlem for vibrations of a one-dimen line Fig.
We su + 5 Phonons 11.
The present figure is in K space.
The dots are not atoms but are the allowed valucs of K.
Of the N + 1 particles on the line, only N - 1 are to move, and their most general motion car1 be expressed in terms of the N - 1 allowed vali~esof K.
This quantization of K has nothing to do with qnantnm mechanics but follows classically from the boondaryconditions that tlre cnd atoms be fixed.
Thus there are N - 1 allowed independent values of K in 12.
This number is equal to the number of particles allowed to Inove.
Each allowed value of K is associated with a sta~ldi~ig wave.
Therc are three polarizations p for each value of K: in one dimension two of these are transverse and one longitudinal.
In three dimensions the polarizations are this simple only for wavevectors in ccrtain special crystal directions.
Another device for enumerating modes is equally valid.
We consider the medium as больше информации, hut require that the solutions be periodic over a large 109 Xq uo!
Fun ad sapour 30 Jaqmnu aya 'suo!
Jjo uoq ~ ~ Xrepunoq 0 3 pua-pax9 aql roj se Xluaa 'apg.
Buuds Dqscla dq paaauuoa j!
The uniform mode is marked with a cross.
We can obtain the group velocity doldK from the dispersion relation o versus K.
There is a singularity in Dl o whenever the dispersion relation w K is horizontal; that is, whenever the group velocity is zero.
Density of States in Three Dimensions We apply periodic boundary conditions over N3 primitive cells within a cube of side L, so that K is determined by the condition whence in K space, or Therefore, there is one allowed value of K per volume 25~lL ~ нажмите сюда values of K per unlt volume of K space, for перейти на страницу polarization and for each branch.
The total number of modes with wavevector less than K is found from 18 to he L125~ ~ times the volume of a sphere of radius K.
читать density of states 2 0 becomes If there are N primitive cells in the specimen, the total number of acousti phonon modes is N.
A cutoff frequency oDis determined by 19 as To this frequency there corresponds a cutoff wavevector in K space: On the Debye model we do not allow modes of wavevector larger than K.
Th number of modes with K 5 K, exhausts the number of degrees of freedom of monatomic lattice.
The thermal energy 9 is given by for each polarization type.
Thermal Properties 113 Figure 7 Heat capacity C, of a solid, according to the Debye approximation.
The vertical scale is in J mol-' K-I.
The holizuntal scale is the temperature normalized to the Debye temperature 0.
The region of the T3 law is below 0.
The asymptotic value at high values of TI0 is 24.
Temperature, K Figure 8 Heat capacity of silicon and germanium.
Note the decrcase at low temperatures.
To convert a value in caVmol-K to Jlmol-K, multiply by 4.
The heat capacity is found most easily by differentiating the middle expression of 26 with respect to temperature.
Then The Debye heat capacity is plotted in Fig.
At T P 0 the heat capacity approaches the classical value of 3Nkn.
Measured values for silicon and germanium are plotted in Fig.
We have where the sum over s-4 is found in standard tables.
Experimental results for argon are plo ted in Fig.
At sufficiently low temperature the T3 approximation is quite good; that when only long wavelength acoustic modes are thermally excited.
These are ju the modes that may be treated as an elastic continuum with macroscopic elas constants.
The energy of the short wavelength modes for which this approxim tion fails is too high for them to he populated significantly at low temperature We understand the T3 result by a simple argument Fig.
Only tho lattice modes having h obecause the velocity sound decreases as the density increases.
Figure 10 To obtain a qualitative explanation of the Debye T3law, we suppose that all phonon modes of wavevector less than Khave the classical thermal energy k,T and that modes between K, and the Debye cutoff K, are not excited at all.
To convert to Jlmol-deg, multiply by 4.
This expresses the Einstein 1907 result for the contribution of N identical oscillators to the heat capacity of a solid.
In three dimensions N is replaced by 3N, there being three modes per oscillator.
The high temperature limit of Это Шарм Эво Оникс 25х75 смотреть!! becomes 3Nk8, which is known as the Dnlong and Petit value.
The Einstein model, however, is often used to approximate the optical phonon part of the phonon spectrum.
General Result for D m We want to find a general expression for D wthe number of states per unit frequency range, given the phonon dispersion relation o K.
BK по этой ссылке where the integral is extended over the volume of the shell in K space hounded by the two surfaces on which the phonon frequency is constant, one surface on which the frequency is w and the other on which the frequency o + dw.
The real problem is to evaluate the volume of this shell.
We let dS, denote an element of area Fig.
~ Here dKL is the perpendicular distance Fig.
The value of dK, will vary from one point to another on the surface.
The specbum for the crystal starts as o2for small o,but discontinuities develop at singular points.
Thc integral is taken over the area of the surface o constant, in K space.
The result refers to a single branch of the dispersion relation.
We can use this result also in electro~lband theory.
There is a special interest in the contribution to D w frorn points at which the group velocity is zero.
Such critical points produce singularitics known as Van Hove singnlarities in the distribution function Fig.
ANHARMONIC CRYSTAL INTERACTIONS The theory of lattice vibrations disciissed thus far has been limited in the potential energy to terms quadratic in the interatomic displacements.
This is the harmonic theory; among its consequences are: Two lattice waves do not interact; a single wave docs not decay or change form with time.
There is no thermal expansion.
Adiabatic and isothermal elastic constants are equal.
The elastic constants are independent of pressure and temperature.
We discuss some of the simpler as pects of anharnionic effects.
The physics of the phonon interaction can be state simply: the presence of one phonon canses a periodic elastic strain whid through the anharmonic interaction modulates in space and time the elasti constant of the crystal.
A second phonon perceives the modulation of the elas tic constant and thereupon is scattered to produce a third phonon, just as from a moving three-dimensional grating.
Thermal Expansion We may understand thermal expansion by considering for a classical osci lator the ellect of anharmonic terms in the potential energy on the mean scpa ration of a pair of atoms at a temperature T.
We take the potential energy of th atoms at a displacement x from their equilibrium separation at absolute zero as with c, g, andf all positive.
The term in x3 represents the asymmetry of th mutual repulsion of the atoms and the term in x4 represents the softening of th vibration at large amplitudes.
T h s m l Properties 121 Figure 15 Lattice constant of solid argon as a Temperature, in K funaion of temperature.
Note that in 38 we have left a2in the exponential, but we have expanded exp pgx3+ pfi4 s 1 pgx3 pfi4.
Measurements of the lattice constant of solid argon are shown in Fig.
The slope of the curve is proportional to the thermal expansion coefficient.
The expansion coefficient vanishes as T+ 0, as we expect from Problem 5.
In lowest order the thermal expansion does not involve the symmetric termfi4 in U x ,but only the antisymmetric term gx3.
This form implies that the process of thermal energy transfer is a random process.
The energy does not simply enter one end of the specimen and proceed directly hallistically in a straight path to the other end, but diffuses through the specimen, suffering frequent collisions.
If the energy were propagated directly through the specimen without deflection, then the expression for the thermal flux would not depend on the temperature gradient, but only on the difference in temperaturc AT between the ends of the specimen, regardless of the Tength of the specimen.
The random nature of the conductivity process brings the temperature qadient and, as we shall see, a mean free path into the expression for the thermal flux.
The e's obtained in this way refer to umklapp processes.
This result was applied first by Debye to describe thermal conductivity in dielectric solids, with C as the heat capacity of the phonons, o the phonon velocity, and e the phonon mean free path.
Several representative values of the mean free path are given in Table 2.
We give the elementary kinetic theory which leads to 42.
The flux of particles in the x direction is in lozl ,where n is the concentration of molec~iles in equilibrium there is a flux of equal magnitnde in the opposite direction.
If c is thc heat capacity of a particle, then in moving frurn a region at local temperatine T + AT to a region at local temperature 2' a particle will give up energy c AT.
Now AT between the ends of a free path of the particle is given hy where T is the average time between collisions.
Thermal Properties Thermal Resistivity of Phonon Gas The phonon mean free path t!
If the forces between atoms were purely harmonic, there would be no mechanism for collisions between different phonons, and the mean free path wolild be limited solely by collisions of a phonon with the crystal boundary, and by lattice imperfections.
There are situations where these effects are dominant.
With anharmonic lattice interactions, there is a coupling between different phonons which limits the value of the mean free path.
The exact states of the anharmonic system are no longer like pure phonons.
We can understand this dependence in terms of the nnmber of phonons with which a given phonon can interact: at high temperature the total number of excited phonons is proportional to T.
To define a thermal conductivity there must exist mechanisms in the crystal whereby the distribution of phonons may be brought locally into thermal equilibrium.
Without such mechanisms we may not speak of the phonons at one end of the crystal as being in thermal equilibrium at a temperature T, and those at the other end in equilibrium at T .
It is not sufficient to have only a way of limiting the mean free path, but there must also be a way of establishing a то, Вращающаяся блесна BLUE FOX VIBRAX FLUORESCENT BFF-1 RT правы thermal equilibrium distribution of phonons.
Phonon collisions with a static imperfection or a crystal boundary will not by themselves establish thermal equilibrium, because such collisions do not change the energy of individual phonons: the frequency o2of the scattered phonon is equal to the frequency o,of the incident phonon.
It is rather remarkable also считаю, Решетка МВ 100 бВС серый восполнить a three-phonon collision process will not establish equilibrium, but for a subtle reason: the total momentum of the phonon gas is not changed by such a collision.
An equilibrium distribution of phonons at a temperature T can move down the crystal with a drift velocity which is not disturbed by three-phonon collisions of the form 45.
Here nK is the number of phonons having wavevector K.
For a distribution with J 0, collisions such as 45 are incapable of establishing complete thermal cquilihrium because they leave J unchanged.
If + Figure 16a Flow of gas molecules in a state of drifting equilibrium down a long open tube with frictionless walls.
Elastic collision processes among the gas molecules do not change the momen tum or energy flux of the gas because in each collision the velocity of the center of mass of the col liding particles and their energy remain unchanged.
Thus energy is transported from left to righ without being driven by a temperature gradient.
Therefore the thermal resistivity is zero and th thermal conductivity is infinite.
Figure 16b The usual definition of thermal conductivity in a gas refers to a situation where n mass flow is permitted.
Here the tube is closed at both ends, preventing the escape or entrance o molecules.
With a temperature gradient the colliding pairs with above-average center of mass ve на этой странице will tend to be directed to the right, those with below-average velocities will tend to he di rected to the left.
A slight concentration gradient, high on the right, will be set up to enable th net mass transport to be zero while allowing a net Скрепки ICO медные, 28 мм, 100 шт.в карт.уп. transport from the hot to the cold end.
Figure 16c In a crystal we may arrange to create phonons chiefly at one end, as by illuminating the left end with a lamp.
From that end there will be a net flux of phonons toward the right end o the crystal.
On arrival o phonons at the right end we can arrange in principle to convert most of their energy to radiation thereby creating a sink for the phonons.
Just as in a the thermal resistivity is zero.
Therefore there is no therma resistance.
The problem as illustrated in Fig.
Thermal Propedies I I I Figure 16d In U processes there is a large net change in phonon momentum in each collision event.
An initial net phonon flu will rapidly decay as we move to the right.
The ends may act as sources and sinks.
The square in each figure represents the first Brillouin zone in the phonon K space; this zone contains all the possible independent values of the phonon wavevector.
Vectors K with arrowheads at the center of the zone represent phonons absorbed in the collision process; those with arrowheads away from the center of the zone represent phonons emitted in the collision.
We see in b that in the umklapp process the direction of the x-component of the phonon flux has been reversed.
The reciprocal lattice vector G as shown is of length 2 d awhere a is the lattice constant of the crystal lattice, and is parallel to the Kaxis.
These processes, discovered by Peierls, are called umklapp processes.
We recall that G may occur in all momentum conservation laws in crystals.
In all allowed processes of the form of 46 and 47energy is conserved.
Such processes are always possible in periodic lattices.
The argument is particularly strong for phonons: the only meaningful phonon K's lie in the first Brillouin zone, so that any longer K produced in a collision must he brought back into the first zone by addition of a G.
A collision of two phonons both with a negative valiie of K, can by an umklapp process G Ocreate a phonon with positive K.
Umklapp processes are also called U processes.
A substantial proportion of all phonon collisions will then he U processes, with the attendant high momentum change in the collision.
In this regime we can estimate the thermal resistivity without particular distinction between Nand U processes; by the earlier argument about nonlinear effects we expect to find a lattice thermal resistivity T at high temperatures.
The energy of phonons K, K, suitable for umklapp to occur is of the order of ikB8, because each of thc phonons 1 and 2 mnst have wavevectors of the order of ;G in order for the collision 47 to be possible.
If both phonons have low K, and therefore low energy, there is no way to get from their collision a phonon of wavevector outside the first zone.
The uniklapp process must conserve energy, just as for the normal process.
The exponential form is in good agreement with experiment.
In summary, the phonon mean free path which enters 42 is the mean free path for urnklapp collisions between phonons and not for all collisions between phonons.
Geometrical effects may also be in~portantin limiting the mean free path.
We must consider scattering by crystal boundaries, the distribution of isotopic masses in natural chemical elements, chemical impurities, lattice imperfections, and amorphous structiires.
When at low temperatures the mean free path t becomes comparable with the width of the test specimen, the value o f t is limited by the width, and the thermal conductivitybecomes a function of the dimensions of the specimen.
This effect was discovered by de Haas and Biermasz.
The abrupt decrease in thermal conductivity of pure crystals at low temperatures is caused by the size effect.
At low tcmperatiires the umklapp process becomes ineffective in limiting the thermal conductivity, and the size effect becomes dominamt, as shown in Fig.
Thermal Properties Figure 18 Thermal conductivity of a highly pr~rifiedcrystal of sodium fluoride, after 11.
The enriched specimen is 96 percent Ge74, natural germanium is 20 percent Ge7", 27 percent Gei2, 8 percent ~ e "37 percent Ge", and 8 percent 6eV6.
The only temperature-dependent term on the right is C, the heat capacity, which varies as T%t low temperatures.
We expect the thermal conductivity to vary as T h t low temperatures.
The size effect enters whenever the phonon mean free pat11 becomes comparahle with the diameter of the specimen.
Dielectric crystals may have thermal conductivities as high as metals.
Synthetic sapphire A1,0, has one of the highest values of Сверло по бетону АМК 6х100 мм 6 x 100 мм conductivity: nearly 200 W c m K-' at 30 K.
The maximum of the thermal conductivity in sapphire is greater than the maximum of 100 W cm-' K-' in copper.
Metallic gallium, however, has a co~lductivityof 845 W cm-' K-' at 1.
T h e electronic contribution to the thermal conductivity of metals is treated in Chaptcr 6.
I n an otherwise perfect crystal, t h e distribution of isotopes of the chemical elements often provides an important mechanism for phonon scattering.
T h e random distribution of isotopic mass disturbs the periodicity of the density as seen by an elastic wave.
I n some substances scattering of phonons by isotopes is comparable in importance t o scattering by other phonons.
Kcsults for gcrmanium are show1 in Fig.
Singularity in density of ntatex.
Here the density oC modes is discontinuous.
Rma thennal dilation of crystal cell.
Take the bulk modulus as 7 X 10" crg ~ m - Note ~.
Zero point lattice displacement and atrain.
Start from the result 4.
We have included a factor of to go from mean square amplitude to mean square displacement.
Heat capacity of layer lattice.
Show that the phonon heat capacity in the Debye approximation in low temperature limit is proportional to 'P.
Thermal Properties b Suppnse instead, as in many layer structures, that adjacent layers are very weakly bound to each other.
What form would you expect the phonon heat capacity to approach at extremely low temperatures?
It is necessary to r e t a i ~the ~ zero-point energy i h o to obtain this result.
Note: Many approximations are involved in this theory: the result a is valid only if o is independent of tcmperature; y may be quite different for differerrt modes.
Kinetic energy of eleclron gas 2.
Pressure and bulk modulus of an elcctron gas 3.
Chemical potential in two dimensions 4.
Fermi gases in astrophysics 5.
Frequency dependence of the electrical conductivity 7.
Dynamic magnetoconductivity tensor for frcc electrons 8.
Cohesive energy of free electron Fermi gas 9.
Static magnetoconductivity tensor 10.
Maximum surface resistance Figure 1 Schematic model of a crystal of sodium metal.
The atomic cores are Na' ions: they are immersed in a Ока Аристократ Венге стекло с узором 900х2000 of electrons.
The conduction electrons are derived from the 3s valence electrons of the free atoms.
The atomic cores contain 10 electrons in the configuration l s 2 2 s 2 2 pIn подробнее на этой странице an alkali metal the atomic cores occupy a relatively small part -15 percent of the total volume of the crystal, hut in a nohle metal Cu, Ag, Au the atomic cores are relatively larger and may he in contact with each other.
The common crystal structure at room temperature is hcc for the alkali metals and fcc for the nohle metals.
In a theory which has given results like these, there must certainly be a great deal of truth.
Lorentz ' We can understand many physical properties of metals, and not only of the simple metals, in terms of the free electron model.
According to this rnodel, die valence electrons of the constituent atoms becorne coriduction electrons and move about freely through the volurrie of the metal.
Even in metals for which the free electron model works best, the charge distribution of thc conduction electrons reflects the strong electrostatic potential of the ion cores.
The utility of the free electron model is greatest for properties that depend essentially on the kinetic properties of the conduction electrons.
The interaction of the conduction clcctrons with the ions of the lattice is treated in the next chapter.
Thc simplest metals are the alkali metals-lithium, sodium, potassium, cesium, and rubidium.
In a free atom of sodium tlie valence electron is in a 3s state; in the metal this electror~becomes a conduction electron in the 3s conduction band.
A r~ionovalentcrystal which contains N atoms will have N conduction electrons and N positive ion cores.
Thc Nat ion core contains 10 electrons that occupy the Is, 29, and 2p shells of the free ion, with a spatial distribution that is csscntially the same when in the metal as in the free ion.
The ion cores fill only about 15 percent of the volume of a sodiurri crystal, as in Fig.
The radius of the free Na+ ion is 0.
The interpretation of metallic properties in terms of the motion of free electrons was developed long before the invention of quantum mechanics.
The classical theory had several conspicuous successes, notably the derivation of tlie form of Ohm's law and the relation between the electrical and thermal conductivity.
The classical theory fails to explain the heat capacity and magnetic susceptibility of the conduction electrons.
These are not failures of the free electron model, but failures of the classical Maxwell distribution function.
There is a further difficulty with the classical model.
From many types of experiments it is clear that a conduction electron in a metal can move freely in a straight path over many atomic distances, undeflected by collisions with other cond~~ction electrons or by collisions with the atom cores.
In a very pure specimen at low temperatures, the mean free path rnay be as long as 10' interatomic spacings rnore tivan 1cm.
Why is condensed matter so transparent to conduction взято отсюда />The answer to the question contains two parts: a A conduction electron is not deflected by ion cores arranged on a periodic lattice because mattcr waves can propagate freely in a periodic structure, Накопительный электрический водонагреватель Gorenje 100 a consequencc of the mathematics treated in thc following chapter.
This property is a consequence of the Pauli exclusion principle.
By a free electron Fermi gas, we shall mean a gas of free electroris subject to thc Pa111iprinciple.
ENERGY LEVELS IN ONE DIMENSION Consider a free electron gas in one dimension, t a k i ~ ~account g of quantum theory and of the Pauli principle.
An electron of maqs m is confined to a length L by infinite harriers Fig.
In quantum theory p may be represented by the operator -i?
We use thc term orbital to denote a solution of the wave equation for a system of only one electron.
The term allows us to distinguish between an exact quantum state of the wave equation of a system of N interacting electrons and an approxirrlate quantum state which we construct by assigning the N electrons to N different orbitals, where each orbital is a solution of a wave equation for one electron.
The orbital model is exact only if there are no interactions between electrons.
They are satisfied if the wavefunction is sir~elike with an integral number n of half-wavelengths between 0 and L: where A is a constant.
The energy levels arc labeled according to the quantum number n which gives thc liu~nberof half-wavelengths in the wavefunction.
The wavelengths are indicated on the wavefunctions.
That is, each orbital can be occupied by at most one electron.
This applies to electrons in atoms, molecules, or solids.
A pair of orbitals labeled by the quantum number n can accomlnodate two electrons, one with spin up and one with spin down.
If there are six electrons, then in the ground state of the system the filled orbitals are thosc given in the table: n Electroll nccupancy n glectron o~rupancy More than one orbital may have the same energy.
The number of orbitals with the saIrle energy is called the degeneracy.
It is convenient to suppose Матрица N116BCA-EB1 N is an even number.
The Fermi energy eF is defined as the energy of the topmost filled level in the ground state of the N electron system.
What happens as the temperature is increased?
This is a standard problem in elementary statistical mechanics, and thc sohition is given by the Fermi-Dirac distribution function Appendix D and TP, Chapter 7.
The kinetic cncrgy of the electron gas increases as the temperature is increased: some energy levels are occupied which were vacant at absolute zero, and some levels are vacant which were occupied at absolute zero Fig.
Thc Fermi-Dirac distribution gives the probability that an orbital at energy E will be occupied in art ideal electron gas in thermal cq~iilihrium: The quantity p is a function of the temperature; p is to be chosen for the in the system particular problcm in siich a way that the total number ofin the comcs out correctly-that is, equal to N.
The results apply to a gas in three di~ne~lsions.
The total number of parti- cles is constant, independent of temperature.
Both contributions to the energy are positive.
The function AU is plotted in Fig.
The heat capacity of the electron gas is f o u ~ ~ond differentiating AU with respect to T.
It is a 6 Free Electron Fenni Gas 143 Figure 6 Temperature dependence of the energy of a noninteracting fermion gas in three dimensions.
The energy is plotted in normalized form as AUINE, where N is the number of electrons.
The temperature is plotted as k B T k p.
Region of degenerate quantum gas Figure 7 Plot of the chemical potential p versus temperature as k,T for a gas of noointeracting fermions m three dimensions.
For convenience in plotting, the units of p and k,T are 0.
These curves were calculated from series expansions of the integral for the number of particles in the system.
The integral in 32 then becomes - whence the heat capacity of an electron gas is From 21 we have for a free electron gas, with kRTF-- eF.
Thus 34 becomes 6 Free Electron Fermi Gas Recall that although Tis called the Fermi temperature, it is not the electron temperature, but only a convenier~treference notation.
The electronic term is linear in T and is dominant at sufficiently low temperatures.
It is convenient to exhibit the experimental values of C as a plot of CIT versus p:1 for then the points should lie on a straight line with slope A and Intercept 7.
Such a plot for potassium is shown In Fig.
Observed values of y, called the Sommerfeld paramctcr, are pven in Table 2.

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