246 PHILIPS TECYNICAL REVIEW VOL." 15, No. 8-9

LATTICE IMPERFECTIONS AND PLASTIC DEFORMATION IN METALS

1. NATURE AND CHARACTERISTICS OF LATTICE IMPERFECTIONS,

NOTABLY DISLOCATIONS.

by H. ,G. van BUEREN

. Not more than 30 or 40 years ago our knowledge of the physical properties of metals was basedalmost entirely on experience. After the first world war a change come.brought about by agradual expansion in the s,tudy of the physics of metals, the object of which was to justifytheoreticallv the observed characteristics. Recently, much progress has been made in the fieldof plastic properties of metals. The new conception, that the plastic d~for1]lationof materials isintimately connected with the occurrence and concentration of imperfections in the regularstructure ofthe atoms, that is, oflattice imperfections, has beenfound to be very fruitful. It canbe applied not only to metals, but also to other materials.

548.4: 539.374:669

Introduction

At the beginning of this century it was still thegeneral belief that the more important physicalproperties of crystalline substances could heexplained exclusively in terms of the periodicarrangement of the component atoms. It wasgenerally thought that any defects in that structure(irregularities in the crystal lattice) had but littlebearing on the characteristics, and these irregu-larities were accordingly disregarded; the crystallattices were held to he perfect.

Although this standpoint originally met with con-siderable success, e.g. in the explanation of X-raydiffraction patterns and in the theories of cohesionbetween the atoms in crystals, it was soon foundthat there were many phenomena which woulddefinitely not answer to theoretical considerationson that basis. Particularly in the study of transportphenomena such as the conduction of heat andelectricity and diffusion, insurmountable diffi-culties were encountered.The first and most important step towards a

better understanding of, these phenomena can besaid to have been the recognition of thermallatticevibrations. The atoms at the lattice points of a,crystal vibrate, practically as harmonic oscillators,about their position of equilibrium, and the energywith which they do this, and hence the ~mplitudes,increase considerably with the temperature.

Although this conception led to important addit-ions to the theory of certain transport phenomena,many effects were still without any satisfactoryexplanation, in particular, those which relate tothe mechanical properties of crystals. Ultimately, 'therefore, it was found necessary to take into con-sideration other imperfections in the crystalstructure. To enter into a general discussion on the

influences of lattice imperfections upon the phy-sical and chemical properties of crystals, would notbe practicable within the scope of this article. Weshall therefore limit our discussion to two articlesdealing with some of the phenomena connectedwith the plastic deformation of metals. For a reviewof the many other domains in which lattice defectsare of interest, the reader may refer to a recentarticle in this Review by G. W. H.athenau 1), andto the comprehensive survey by F. Seitz 2).

Some phenomena related to the plastic deformationofmetals

Microscopie examination of the surface of metalswhich have been polished and subsequently de-formed, usually reveals a pattern of fine, more orless straight lines (fig. la). The more the metal isdeformed, the more clearly these so-called slip linesbecome visible and the greater is their number. Theorientation of these lines appears almost invariablyto correspond to the planes in the crystal in whichthe atoms are most closely packed. This relationshipis found to be the most marked in metals with theclosest packed crystal structure. In metals havingbody-centred cubic lattices which are not packedas closely as possible, the slip lines are somewhatirregular, . but the relationship between the sliplines and the crystal planes still remains.

Examination ofthe slip lines under high magnifica-tion by means of the electron microscope, shows thatwhat appear to he single lines when seen with lowermagnification, are in many cases groups of some

1) G. W. Rathenau, Philips tech. Rev. 15, 105-113, 1953(No. 4).

2) F. Seitz. Imperfections in nearly perfect crystals. JohnWiley, New York, 1952, p. 3-76.

FEBRUARY-MARCH 1954 LATTICE IMPERFECTIONS IN METALS 247

Fig.1.a) Micrograph of the polished surface of a metal single crystal subjected to 7%deformation. Magnification approx. 200 X (R. \V. Cahn, J. Inst. Metals 79,129-158,1951).b) Slip band on the surface of a polished, deformed crystal as seen under the electronmicroscope. Magnification 25,000 X (A. F. Brown, Advances in Physics, 1, 421-479, 1952,No.4).

a

tens to hundreds of adjacent lines (fig. Ib), whichmay be termed slip bands. Single lines are seen aswell, however.A study of the slip lines or bands has revealed the

fact that these are in effect "steps" in the surfaceof the metal, of which the height may vary fromsome tens to some thousands of times the atomicspacing. The conclusion to be drawn is clear, viz.that in the deformed metal, translations take placealong certain crystallographic planes whose orien-tation corresponds to the direction of the slip line.

_1 77977

Fig. 2. Diagrammatic representation, on a highly exaggeratedscale, of slip in a metal crystal. g slip planes; T shear stress.

This mechanism is sketched in jig. 2, which refersto simple slip lines and is, of course, drawn on anexaggerated scale. This deformation mechanism,which appears to be of a very universal nature, isknown as gliding or slip; the lattice plane alongwhich the slip takes place is termed the slip plane,

b

and the direction of the translation, the slip direction.The slip plane and direction together constitute theslip system.The theoretical critical shear stress Tcr, that is, theminimum shear stress necessary to produce a sliptranslation of this kind, is of the order of 0.1 G,where G is the modulus of rigidity, or torsionalmodulus of the material.

This result is obtained in the following manner (videFren keI 3».

Take the case of two neighbouring rows of atoms in a simplecrystal. Let a denote the spacing of the atoms in theundistorted condition. The force required to move the oncrowan infinitesimal distance over the other depends upon thedisplacement that both rows have already undergone. In theundeformed state, which is the state of equilibrium, this forceis zero. Each time the rows are displaced an integral numberof times a/2 from the undeformed state, an equilibrium stateis restored, as a symmetrical configuration of the lattice isagain reached. It would appear to be a reasonable assumptionthat the relationship between the shear .stress (T) and therelative displacement (x) might be approximated by a simpleperiodic function:

T = k sin 2nx .a

With only a small displacement this becomes T = k.2nx/a.In this case also Hooke's law applies, which states that:

T = Cx]«,

From this, k = G/2n; hence:

G . 2nxT = 2n S111 -;;- •

If T is larger than G/2n, it follows from this formula that thedisplacement x of the atoms is unrestricted and slip will occur.Closer investigation yields a value for the theoretical criticalshear stress slightly lower than G/2n, although still of the orderof 0.1 G.

3) .J. Frenkel, Z. Physik. 37, 572, 1926.

24,8 PHILIPS' TECHNICAL REVIEW VOL. 15, No. 8-9

For most metals, G lies between some thousandsand some ten-thousands of kilograms/mmê, fromwhich a theoretical critical shear stress of a fewhundreds or a few thousands of kilograms/mm''follows. It is a well-known fact, however, that metalscan be very much more easily deformed than wouldappear from these values. The ,observed criticalshear stress of well-annealed single crystal occursbetween 0.1 and 10 kgfmm2;' annealed polycrystall-ine metals yield somewhat higher values but stillvery much less than the theoretical shear stress.

To explain this ease of deformation, it was postu-lated that a particular kind of lattice defect, namelydislocations, occur very frequently in every crystal.It was very soon apparent that such dislocationsdo play a, dominant rêle in the process of deforma-tion of a metal.

We have referred above to the critical shear stressof well-annealed materials. The annealing mini-:mizes the, consequences of any previous deforms-tions which would otherwise manifest themselves,for example, by the increase which occurs in thestress required to deform a metal as the deformationitself increases (fig. 3). This effect is known as workhardening (it does' not occur in all materials;substances such as pitch, for example, undergo nowork hardening). A severely deformed and un-annealed metal may have a critical shear 'stressmany tens of times higher than that in the annealedstate.

77978 _ê

Fig. 3. Diagram showing the relationship between the defor-mation 6 and the shear stress -c in a pure metal. The straightpart at the commencement of the curve represents the elasticdeformation. When the critical shear stress -Ccris exceeded,plastic déformation sets in, this being accompanied by strainhardening, as. a result of which the characteristic assumes aparabolic form (in cubic metals). In practiee, irregular curvesare often encountered, but the general form remains the same.

As we shall see presently, work hardening inmetals is attributable mainly to the mutual inter-action of the dislocations associated with the defor-mation.The extent to which a metal can be deformed is

dependent not only on the previous deformation;

variations in the temperature, will also affectthe ductility, albeit to a relatively small degree.Thus, materials which are brittle when cold arenearly always ductile at elevated temperatures.Foreign atoms or conglomerates in a metal also affectthe ductility appreciably; the various hardeningprocesses for metals are based on this fact 4).

Lastly, it appears that the plastic deformationof a metal affects its general physical properties invarying degrees. Amongst other things, the elec-trical conductivity will change, and it is found thatthe study of this and other subsidiary consequencesof plastic deformation can reveal valuable informa-tion about the behaviour of lattice imperfections..

In this section we have surveyed only the groundon which our further considerations are to be based.Before entering into a detailed discussion of therelationship between the mechanical properties ofmetals and lattice imperfections in crystals,however, it will be necessary for us to say a littlemore about present-day conceptions of the natureand behaviour of possible kinds of lattice imper-fections. The rest of the present article is accor-dingly devoted to this subject.

Possible kinds of lattice imperfections

Imperfections in a crystal lattice may be one-,two- or three-dimensional. Also, singularities mayoccur at the ~attice points themselves, in which casewe might speak of zero-dimensional, or pointdefects. The three-dimensional defects such as macro-scopic holes or inclusions, precipitates etc, and two-dimensional faults which may be taken to includethe crystal boundaries and surface layers, can bedealt with quite briefly. In recent years, severalarticles have appeared in this Review on the subjectof three-dimensional imperfections 4) 5); as to thet~o-dimensio~al defects, these can often be succes-fully interpreted as more or less ordered associa-tions of linear and point defects.

Dislocations belong to the category of linearimperfections. Point defects may be taken toinclude vacancies, i.e. lattice points where an atomi~ missing, and further, interstitial atoms (atoms atintermediate points in the lattice) and impurities(foreign atoms). Whereas the concept of dislocationswas originally introduced to provide an explana-tion of the mechanical properties of crystals,vacancies etc. have been postulated to promote adeeper insight into diffusion effects and other trans-

4) J. L. Meijering, Hardening of metals, Philips tech. Rev.14, 203-211, 1953 (No. 7).

5) J. D. Fast, Ageing phenomena in iron and steel after rapidcooling, Philips tech. Rev. 13, 165-171, 1951.

FEBRUARY-MARCH 195't LATTICE IMPERFECTIONS IN METALS 249

port phenomena. It is only in recent years that aclose relationship has been found to exist betweenthe different kinds of lattice imperfections.The conception of linear lattice irregularities, i.e.

dislocations, was introduced in the theory of metalsindependently by Taylor, Orowan and Polanyi 6)in 1934. Since 1939, when Burgers made the firstdetailed theoretical study of the behaviour ofdislocations 7), the work of N. F. Mott and hiseo-workers at Bristol has been mainly responsiblefor the almost general acceptance of the hypothesisof dislocations. The work of the British scientistsgave the impetus to the first direct experimentalpointers to the existence of dislocations in crystals,

r -- - ~-hI III IIL__ - -

r+r+ -ohI II IIL - -

DislocationsThe dislocation concept links up very closely with

the most important deformation mechanism inmetals, namely the above-mentioned "slip". Fig. 4illustrates the present conception of this mechanism.The two halves of the crystal do not move over eachother as a whole: this,' as already mentioned, wouldrequip~ a much higher shear stress than that whichis indicated by experiment.

The 'displacement commences at one side (left,in fig. 4) of the crystal and is propagated veryrapidly to the other side. A situation whereby allthe atoms simultaneously assume non-equilibriumpositions never arises; the deformation at a given

9

Ir" --1-- -hI I

L_ \_.-

I

-9

r _._-- 1i I

II 11">L_ -p

Fig. 4. Diagram representing on an atomic scale the occurrence of slip in crystals, accord-ing to the dislocation mechanism. The full lines reprosent the lattice planes. The twoparts of the crystal do not slip simultaneously as a whole along the slip plane g; thedeformation is localized within a small zone (shown in dotted-line rectangles). Only thecross-section in one atomic plane is illustrated, but it should be visualised as extendedwithout limit perpendicular to the plane of the drawing. Movement of the dislocationfrom left to right (in the sequence of figures a to f) completes the slip.

9

r1

viz. the spiral growth of crystals from super-saturated vapour 8), as well as the recently dis-covered sub-structures of photo-sensitive silverbromide crystals 9). Further direct evidence for theexistence of dislocations in metals has been gatheredby workers in the United States, from the study ofcrystal boundaries 10).

G) G. I. Taylor, Proc. Roy. Soc. A 145, 362 1934; E. Oro-wan, Z. Phys. 89, 634, 1934; M. Polanyi, Z. Phys. 89, 660,1934.

7) J. M. Burgers, Proc. Kon. Ned. Akad. Wet. Amst. 42,293 and 377, 1939.

8) F. C. Frank, Advances in Physics. 1, 91, 1952. See alsofig..2 in the article referred to in footnote 1.

9) J. M. Hedges and J. W. Mitchell. Phil. Mag. 44, 223-224,1953.

10) See, e.g. W. T. Read, Dislocations in crystals, McGrawHill, New York, 1953.

--'--t-t--t-t-+-t--t-t--I-+-rg

f 78018

instant is always localized within a zone of some 3or 4 times the atom spacing. This zone, of whichfig. 4 shows only a cross section, i.e. one 'atomiclayer (this layer should be regarded as arbitrarilyextended in a direction perpendicular to the plane ofthe drawing), contains a linear lattice defect, and itis this that has come to be known as a dislocation.

The displacement of the two halves of the crystalin relation to each other has already taken place tothe left of the dislocation in fig. 4d, but not on theright-hand side. The deformation of the crystal asa whole approaches completion as the dislocationmoves to the right along the slip plane; once ithas' arrived at the opposite side of the crystal.(fig. 4f)' the shear is complete. In this example it is

250 PHILlPS TECHNICAL REVIEW VOL. 15, No. 8-9

seen that plastic deformation due tó slip is simplythe movement of a dislocation.

The critical shear stress is now the force requiredto initiate a dislocation and cause it to be propagated.To produce this effect, only a [mction. of the totalnumber of atoms in the region of the slip planeneed be simultaneously in a condition of increasedenergy, so that it follows that this force is verymuch less than in the case of a perfect lattice.

Dislocations occur not only in metal crystals; in principlethey may arise in any crystalline substance. Whether or notthey play an important part in the behavier of the material,depends very largelyon the crystal structure and the cohesionbetween atoms. Recent investigations have indicated that inionic crystals and in non-polar semiconductors, dislocationsprobably play quite an important role, especially with regardto the optical and electrical properties of these materials. Thestudy of dislocations in non-metals is still in an early stage, how-ever, and we shall therefore not concern ourselves with it here.

~ ~

I1

------------ ----1- -.:::---------1:-----t_"., "b 'i.. -,

1',

~///

the Burgers vector with respect to the dislocationaxis. The dislocation referred to in our discussionon the mechanism of slip, is the edge dislocation, ascharacterized by a Burgers vector perpendicular tothe dislocation axis. Fig. 5a illustrates an edge dis-location in perspective, and it will be seen that, inorder to complete the translation of the- one part ofthe crystal over the other, the edge dislocation mustbe propagated in a direction at right angles to itsaxis, that is, parallel to its Burgers vector. Fig. Sbdepicts a cross-section of an edge dislocationperpendicular to the dislocation axis. On closeinspection of this diagram (and also of fig. 4), it willbe seen that an edge dislocation can be consideredas being initiated by the introduetion of an addit-ional plane of atoms perpendicular to the slip planein one of the halves (in this case the upper half) ofthe crystal. As a result of this, variations in density

hII• • • • • • • • •II• • • • • • • ••II.... 'e.~.I--~-~-~-~-*-~-~-~~---g

• • • • • • • e

9 · .' . . . . ...• • • • • • • •• e , • • • • • •

711019

Fig. 5. a) Perspective diagram of an edge dislocation which has progressed half way througha crystal. The chain-dot line d is the disloontion axis; the light lines reprosent the atomicplanes. The Burgers vector b is oriented in the direction of the shear stress .. which strivesto complete the slip. To do this, the dislocation must move in the direction of its Burgersvector, along the slip plane g. b) Cross-section of an edge dislocation perpendicular to thedislocation axis as indicated by X; g is the slip plane, b the Burgers vector; h. extra atomichalf-plane.

Types of dislocationLet us now consider the concept of dislocations

in a broader sense. The axis of the linear latticeimperfection '(which need not necessarily be astraight line) is called the dislocation axis. Thedislocation separates the part of the lattice whichhas already slipped, from the part which has notyet undergone displacement. The degree and direc-tion of the translation in the displaced zone isrepresented hya vector b, known as the Burgersvector. Usually, the extent of the translation ata dislocation will be equal to the spacing of thelattice plane in the corresponding crystallographicdirection, and the magnitude of the Burgers vector'is then equal to this distance. The direction of thevector corresponds to that of the slip direction." It is thus possible to differentiate between varioustypes of .dislocation according to the orientation of

of the crystal lattice are produced around the axis,of an edge dislocation; ~ fig. Sb the upper half ofthe crystal in the region of the dislocation is moreclosely packed than normally, the lower half less so.

An exactly similar distortion of the crystal,having the same slip plane as in fig. 5, and alsoproduced by the movement of an edge dislocation,can be thought of as being initiated by the intro-duetion of an extra plane of atoms in the lower halfof the crystal. The direction in which this disloca-tion would have to be propagated in order to com-plete the movement is then reversed in sign, and themore densely and less densely packed regions aboutthe dislocation axis change places. We then speakof a dislocation of opposite sign to that of the original.

A second kind of dislocation is the so-calledscrew-dislocation, the Burgers vector of which isparallel to the dislocation axis. A dislocation of

FEBRUARY-MARCH 1954 LATTICE IMPERFECTIONS IN METALS 251

/

1 T /1 / '//yJ-----------------.:-~:t':..---- ----

)',/' i vr-K)/ -- V---""" )- - - - f-t - - - - g1/ If -- -----,

" 1\[/ L77/ / r>" I.~ll

/~/ 1/

~'~~~//~~~LU~I~/rcl

Fig. 6. a) Perspective diagram of a screw dislocation displaced half way along its slipplaneg. The chain-dot line d is the dislocation axis... is the shear stress which tends to movethe dislocation along the slip plane, perpendicular to the Burgers vector b and to the shearstress itself, in order to complete the slip. The curved arrow round the dislocation axisindicates that a complete revolution about the dislocation axis within the same atomicplane does not result in a return to the point of origin; the atomic planes describe a helicalsurface about the dislocation axis. b) Atom positions in the lattice planes above and belowthe dislocation axis d and parallel to that axis. The open circles reptesent atoms in the upperplane and the dots those in the lower plane. b is the Burgers vector.

this kind is shown in perspective in jig. 6a. Theregions in which slip has occurred (right-hand side)and has not occurred (left-hand side) are nowseparated by a linear lattice imperfection parallel~to the direction of slip, in contrast to the edgedislocation, which is oriented perpendicular to theslip direction. Once more, in order to complete thetranslation, the dislocation must move in a directionperpendicular to its axis, but here this meansperpendicular to the Burgers vector. In consequenceof the peculiar mutual orientations of the Burgersvector and the dislocation axis, the 'screwdislocationhas no specific slip plane, as this plane is definedas the plane through the dislocation axis and theBurgers vector. Whereas the edge dislocation, inorder to accomplish slip, must move over a very

definite slip plane, the screw dislocation is free toselect its own direction of propagation (providedit is perpendicular to its axis).

This is not the only point of difference betweenthe two kinds of dislocation. From fig. 6a it will beseen that a screw dislocation can be imagined asformed by making an incision of some depth (inthe perfect crystal), this being followed by displace-ment of the part above the incision, parallel to thebottom of the cut, through a distance equal to theatomic spacing. The bottom of the cut is the' dislo-cation axis.' If we now describe a path round thedislocation axis, keeping within the same atomicplane, it will be seen that on completion of onewhole turn we do not return to the point of originas in .the undistorted crystal, but that a translation

has been. accomplished equal in extent to theBurgers vector in the direction of the. dislocationaxis. In other words, a set of parallel atomicplanes (perpendicular to the dislocation axis) inthe undistorted crystal, forms, on the introduetionof a dislocation, a helical surface having the dislo-cation line as axis. The atomic structure around ascrew dislocation is depicted in fig. 6b.·It is notpossible to visualise the screw dislocation as beingproduced by introducing' an additional atomicplane, and there is accordingly no question of anyappreciable variation in density within the crystal,as occurs around edge dislocations.A sign can also be attributed to the screw disloca-

tion, since here again an equivalent translationcan be brought about in two different ways: the

screw dislocation can transform the atomic planesinto either right-hand or left-hand helical surfaces.Fig. 6 shows a right-hand screw. Right and left-handscrew dislocations must move in opposite senses inorder to produce the same translation.Edge and screw dislocation represent only the

extreme instances of the general conception ofdislocations in which the angle between the Burgersvector and the dislocation axis is arbitrary.The character of a dislocation need not neces-

sarily be the same throughout its length: it mayvary between one point and another. Moreover, adislocation element ¥lay change its character inthe course of propagation through the crystal. Aprobably common form of dislocation is thedislocation loop depicted in jig. 7, the feature of

252 PHILIl'S TECHNICAL REVIEW VOL. 15, No. 8-9

which is that within the loop the crystal hasslipped and that outside it no slip has occurred,since by definition, a dislocation separates a zonein which slip has taken place from another in whichit has not. Eveqwhere within the loop, the ~'elativeamount of slip must be of the same magnitude andin the same direction; otherwise the loop wouldreveal branches. In other words, around a closeddislocation loop without any branch points, theBurgers vector -- which determines the trans-lation - is at all points constant in magnitude anddirection. Only certain parts of the loop, as indicatedby E and S in fig. 7, conform to the requirement ofa purely edge Ol' screw type of dislocation; at allother points the character of the dislocation iscomposite. The dislocation therefore nearly every-where includes elements with at least some edge-character, i.e. differences in density are present,and the movement will be confined to a definiteslip plane.

77979

Fig. 7. Dislocation loop. b is the Burgers vector which isconstant along the whole loop. Elements E, perpendicular tothe Burgers vector are purely edge-type elements; elements S,parallel to the Burgers vector, arc purely screw type.

Dislocation loops are probably of frequent occur-rence, since dislocations cannot be terminatedsomewhere in the middle of a perfect crystal.Along the dislocation there are invariably two zoneswhich are displaced a definite amount with respectto each other. Where the dislocation is terminated,this relative displacement must disappear, and thiscan only happen on the surface of a crystal, at acrystal boundary, or at points where anotherdislocation is present. In the absence of any of theseessentials, that is in the centre of a perfect crystal,only continuous, e.g. loop dislocations are possible.

Mechanical model of a dislocation

To study the characteristics of dislocations, it isusually sufficient in principle and without sacrificeof generality to confine our investigation to pureedge and screw dislocations. Using a mechanical

Fig. 8. Mechanical model for reproducing dislocations.Magnets N are arranged in a square pattern on a baseplate P. A rack capable of movement to the left and right (notshown in the illustration) carries a number of pendnla withiron balls B which, in the condition of eqnilibrium, hang justabove the magnets. Springs S are fitted between the pendula.The illustration shows an edge dislocation with axis L Theextra atomic plane is clearly seen. Slight movement ofthe boardcarrying the pendula causes the dislocation to progress throughthe model in a direction perpendicular to its axis, i.e. to theright or left hand side, thus producing a displacement,

rnodel !"] such as that shown in jig. 8, it is possibleto illustrate the occurrence and movement of thedislocations.A large number of permanent magnets are arran-

ged on a board in a square pattern, to represent asingle lattice plane of the crystal (which for con-venience is supposed to be simple-cubic). Abovethis board a rack is mounted which is movablein either of the square-directions and which carriesa number of pendula with "bobs" of soft iron.These iron balls also represent a lattice plane. Inthe state of equilibrium, each pendulum hangs overa magnet.

Whereas the "atoms" in the lower plane havefixed positions, those in the upper plane are capableof movement. To represent the inter-atomic forces,all the pendula are interconnected by small helicalsprings which exert no tension in the state ofequilibrium, but which somewhat impede theapproach Ol' separation of any two of the pendula.When the rack of pendula is pushed to one side

by hand, the magnetic forces between the magnetsand iron balls is at first sufficient to resist the resul-tant "shear stress". When the stress reaches acertain appreciable value, first one row of ballswill be seen to jump over a distance equal to onelattice spacing in the direction of the stress; theapplication of a very slightly larger stress causesneighbouring rows in succession to carry out thesame movement until finally the whole system of

11) H. G. van Bueren, Brit. J. App!. Phys 4,144-145,1953.

FEBRUARY-MARCH 1954 LATTICE IMPERFECTIONS IN METALS 253

pendula has moved up a distance equal to thelattice space. Thus a dislocation is seen to migratethrough the lattice.It is possible that the first row of balls to jump

over will be perpendicular to the direction in whichthe stress is applied, as shown in fig. 8; in this casewe have an edge dislocation. Again, the first row of"atoms" executing the movement may be parallelto the direction of the applied stress; in this caseit is a screw dislocation which is seen to developand move through the lattice (fig. 9). Whichparticular kind of dislocation will occur dependsentirely on accidental factors.

Fig. 9. As fig. 8, but showing a screw dislocation. Movement ofthe pendulum rack causes the dislocation to move backwardsor forwards through the model.

Movement of dislocations

Although the model provides only a very roughapproximation to the real conditions, it is possibleto note at once two important properties of disloca-tions. Firstly, the stress needed to initiate a disloca-tion is very much greater than that required toset it in motion. Secondly, under the influence ofthe stress, the dislocations are indeed propagatedin such a way that the deformation of the crystalby slip is completed. Edge dislocations move inthe direction of the stress and screw dislocationsperpendicular to it.It can be said, then, that a force operates on the

dislocation. The magnitude of this force can becomputed in the following manner (vide Mott andNabarro 12)).

Let us assume a dislocation of which the Burgersvector is b, running from one side of a cube-shapedcrystal of dimension L to the other side (cf. fig. 5).When it reaches the opposite side, one part of thecrystal will have been displaced in relation to theother through a distance b. Let the components of the

12) N. F. Matt and F. R. N. Naharro, Report on strength ofsolids, The Physical Soc. London, 1948, p. 1.

shear stress in the slip direction be Tg; the crystalarea on which this operates is L2, so that the workperformed by the external force III producingthe slip is:

The length of the dislocation line is L, as is alsothe distance travelled by the dislocation. If we nowdenote the force acting on a unit length of thedislocation by F, it may be said that the work donemust be equal to:

w= FV.

(This is actually the definition of a "force acting ona dislocation".)The result of a shear stress whose component in

the direction of the Burgers vector is Tg (and it canbe shown that this is universally valid) is thereforea force F per unit length of the dislocation equal to

. (1)

which tends to move the dislocation III the senseof completing the slip movement.

Only a slight force is needed for the propagationof the dislocation along its slip plane. If the wholedislocation is to be displaced by atomic distance,it is only necessary for the atoms themselves tomove a fraction of this distance (fig. la). In doingthis, half of the atoms move under the influenceof an attracting force and the other half under arepelling force, so that, to a first approximation,the forces counterbalance each other and the

77980

Fig. 10. Diagrammatic representation of the movement ofan edge dislocation in its slip plane. The upper row containsone more atom than the lower. In the first instance, thedislocation axis lies at the cross marked A, and tbe atomsassume the positions shown by the dots. In order to displacethe dislocation axis by one atomic spacing to the point B, theatoms themselves need undergo only slight displacement, vizfrom the dots to the adjacent circles. In doing so, the atoms inthe upper row to the left of A move so as to approach moreclosely to the position of equilibrium, i.e. directly above theatoms in the lower row. These atoms move under the influenceof attracting forces. The atoms to the right of A move furtherfrom the position of equilibrium against the action of repellingforces. As a first approxirnation, the effects of the two forces arecounterbalanced and the dislocation can move freely in itsslip plane. (A. H. Cottrell, Progress in Metal Physics 1,77-125, 1949).

dislocation moves easily. It is owing to the discon-tinuous structure of the crystal that a dislocationcannot move in its slip plane entirely withouteffort, as might be inferred by the above remarks.Different positions of a dislocation axis betweentwo lattice planes are not exactly equivalent

254 PHIL1PS TECHNICAL REVIEW VOL. 15, No. 8-9

energetically. In metals of cubic and hexagonalstructure, this results in only a slight reductionin the mobility of the dislocation in its slip plane.The mobility is in any case quite high and, as theatomic arrangement and hence the lattice vibra-tions in the crystal affect it but little, it is notvery dependent on temperature. It is for thisreason that dislocations play such an importantpart among lattice defects.

In the foregoing we have referred to motionwithin the slip plane of the dislocation. For a screwdislocation, every plane of movement can beconsidered as a slip plane. For dislocations notbearing the character of purely screw dislocations,however, we should also consider movements whichdo not meet this condition. .

Dislocations with some edge-character are alwaysassociated, in the manner described for pure edgedislocations, with an extra half plane of atoms.

77981 8 8'

VA'A

Dc

Fig. 11. Illustrating non-conservative movement of an edgedislocation (vide A. H. Cottrell. Dislocations and plastic Howin crystals. Oxford Univ. Press, 1953).

Fig.ll illustrates this situation diagrammatically.The plane of the drawing represents a cross-sectionperpendicular to the dislocation axis A. The extrahalf plane is represented by the line AB, and theBurgers vector lies in the slip plane of which theline CAD is the cross-section. With movement outof the slip plane, for example towards the point A I,the length of the extra half plane changes from ABto A'B'. This means that a number of extra atomshave been removed (or added), and this can happenonly if accompanied by a simultaneous transportof atoms to or from other parts of the crystal, or,if this is impossible, at the expense of considerablelocal distortion. Both processes necessitate activa-tion energy (e.g. for diffusion); so also, therefore,does the movement of a dislocation beyond its slipplane.

Accordingly, whereas movement within the slipplane needs very little energy and is hardly depen-

dent on temper~ture (conservative movement),movement outside the slip plane (non-conservativemovement) demands very much more energy, whichcannot in general be derived from the externallyapplied forces; furthermore, owing to the fact thatan activation energy is involved, the latter is highlydependent on the temperature.

Clearly, a screw dislocation performs only con-servative movements, since each crystallographicplane containing its axis can serve as slip plane.Needless to say, screw dislocations in crystals aswell as other kinds of dislocation can be propagatedonly over crystallographic planes. Chalmers andMartius 13) have pointed out that it makes adifference which planes are involved. They showedthat ,during its propagation, a dislocation will reveala preference for planes containing the most closelypacked layers of atoms. The slipping movementtherefore occurs mainly along such planes.

Since the slip lines in a distorted crystal are themarkings of the slip planes, this explains why, asmentioned above, the slip lines in many metalcrystals run, along the most closely packed crystalplanes.

Origin of dislocations

Dislocations represent a considerable amount ofenergy by reason of the distortion in the crystallattice that accompanies such lattice defects. Theenergy of a dislocation is defined by its stress field,the nature of which is rather intricate and wouldtake us too far afield to consider here. It appearsthat the stresses decrease in inverse proportionto the distance from the dislocation axis, in conse-quence of which, in a continuous medium, theenergy of a dislocation would be infinitely high.This is not the case with crystals because of theirdiscrete atomic structure; it is found that theenergy of a dislocation per atomic plane, is severaltimes Cb3, i.e. in most metals, several times lO-12erg.

A proper dislocation is at least some tens of timesthe atomic spacing in length. The energy of suchdislocations, that is, the energy required to producethem, is thus always 10-10 erg or more.

The presence of a dislocation increases theentropy of the crystal. It can be shoron that thiseffect is negligible compared to the energy offormation, which means that dislocations cannotpossibly be initiated .thermally, or exist in thermalequilibrium with the crystal lattice. In a metalcrystal plastically elongated about 1%, we mustnevertheless accept the occurrence of dislocations

13) B. Chalmers and U. N. Martius, Nature 167, 681, 1951.

FEBRUARY-MARCH 1954. LATTICE IMPERFECTIONS IN METALS 255

with a totallength in the crystal of at least 107 cmper cm'' (i.e. a dislocation density of 107• cm-2),as demonstrated by experiments which will bediscussed in the second part of this article.This apparent contradietien has :been solved by .

adopting the view that dislocations occur as a resultof the actual process under consideration, viz. theplastic deformation. In principle this can beexplained in the following manner.If inhomogeneities occur somewhere within the

crystal or at its surface, e.g. small cracks or inclusions,applying a stress will result in local stress coneen-trations, i.e. zones in which the stresses are muchhigher than elsewhere in the crystal. It is plausiblethat the formation of one or more dislocations insuch zones might lead to a reduction in energy.If this is so, such zones will be capable of acting assources of dislocations.Which particular kind of inhomogeneities are

likely to act best as sources, is net- yet known withcertainty. It is very probable that the most likelysource of dislocations will he found at the crystalboundaries.

Dislocations mayalso he formed during thegrowth of the crystal. According to a theory putforward by Frank 8), screw dislocations can veryappreciably accelerate the growth of a crystal. Thispoint has already been mentioned in the articlereferred to in footnote 1), so that there is no need todwell further on this very remarkable and interest-ing point here. So far as we are concerned here, weare interested only in the fact that a crystal grownin accordance with the mechanism described byFrank will invariably contain dislocations.Although the two formative mechanisms described

throw some light on the origin of certain dislocationsdistributed at random throughout the crystal, theyby no means explain the observed phenomenonthat, in a given slip plane, slip often occurs over anumber of atomic spacings simultaneously, whereasin other crystallographic planes no slip takes placeat all. This phenomenon can be due only to themovement oflarge numbers of dislocations alllyingin the same slip plane. A possible way out of the dif-ficulty is offered by the' hypothesis of a dynamicmultiplication mechanism for the dislocations ini-tiated in one of the ways outlined above. A conditionfor the occurrence of this mechanism (a Frank-Read source, so named after the originators ofthis conception 14) is the presence of dislocationlines in the crystal, anchored at two points, so thatthey can deflect as a result of a shear stress without

14) F. C. Frank and W. T. Read, Phys. Rev. 79,722, 1950.

being able to m~ve as àwhole. Mot~ 15) has shownthe plausibility of the existence of such anchoreddislocation elements in metallic crystals. He pointedout that the dislocations in non-distorted metals.with cubic symmetry very probably form a space

Fig. 12. Spatial network of dislocations in a crystal (videMottI5».network, contammg many nodes in which threedislocation lines, not located in the same plane, meet(fig. 12). If the orientation of the applied stress issuch that a considerable force acts on one of thethree converging dislocations, this usually meansthat the other two dislocations are subjected to onlya slight force, or even to a force in the oppositedirection. The nodes (e.g. A and B in fig. 12) willthen co?stitute more or less fixed points in thedislocation network, and: hence act as anchoringpoints for those dislocations which would otherwisetend to move under the inflence of one force oranother.An anchored dislocation element (1) is illustrated

infig. 13. For convenience we will suppose that this

Fig. 13. Diagram demonstrating the multiplication mecha-nism of dislocations according to Frank and Read.

15) N. F. Mott, Phil. Mag. 43, 1151, 1952.

256 PHILlPS TECHNICAL REVIEW VOL. ] 5, No. 8-9

is a purely edge-type disloéation. If a' shear stress 7:

is applied, the dislocation will be deflected; it thusbecomes longer, and the dislocation energy accumu-lated within it increases. Deflection is continueduntil a position is reached (e.g. position 2 in fig. 13)whereby the gain in potential energy by the disloca-tion in the stress field 7: is balanced by this increasein dislocation energy. If the stress increases, position3 will ultimately be reached, when the dislocationwill have assumed the form of a semi-circle. Thisis a critical form. It can be shown that if the stressis increased gradually from this point, the energyrequired to produce further deflection of the disloca-. tion (and hence to increase its length) is alwaysless than the gain in potential energy that accom-panies this further deflection. The stress requiredto produce the critical semi-circular configurationcan be computed, being approximately:

Gb7:0 = -1-' . . . . . . . (2)

where b is the Burgers vector of the dislocation andI the length of the original straight anchored element,If the critical value 7:0 of the shear stress is exceeded,the dislocation expands ad lib without any furtherincrease in the stress. Those parts of the semi-circlewhich are now practically perpendicular to theoriginal edge dislocation now have the character ofa screw dislocation (the Burgers vector, whichremains constant, is at those points almost parallelto the dislocation axis). These screw-like parts ofthe dislocation thus move laterally outwards, per-pendicular to the direction of the strain. Thoseparts which are practically parallel to the originaldislocation, progress as an edge dislocation parallelto the direction of the strain. This being so, positions4 and 5 are traversed, the strain being the same allthe time, until position 6 is reached, Here, two partsof the screw of opposite sign face each other and,the movement being continued, will meet. Theythen cancel each other, since two dislocations ofopposite sign coming together result once more ina perfect lattice. Finallyposition 7 is assumed, thisbeing none other than the condition at 1, plus adislocation loop enclosed by the "source" andmoving outwards under the influence of the appliedstress.

In principle, an unlimited number of loops couldemanate from the source, assuming that the appliedstress is at least equal to the critical stress 7:0,

This would provide a simple explanation forthe observed translations of several hundreds oftimes the atomic spacing along a slip plane.

It is an obvious step to identify (in order ofmagnitude) the observed critical shear stress withthe critical stress .0' If this is done, it will befound from formula (2) that in most metals thelength I of the source in fig. 13 is several times10-4 cm. Of course, longer sources will start workingat lower stress levels than short ones, but it mustapparently be accepted that the length of mostsources is about 10-4 cm; this is of the same orderof magnitude as the estimated average length ofthe network· elements in Mott's model.

Formatlon of vacancies and interstitial atoms'

If the energy of formation of a lattice imperfec-tion and the change in entropy of, the crystalassociated with it are known, it is possible toestimate the thermal equilibrium concentratio,n ofsuch imperfections as a function of remperature.This concentration is proporrional to the Boltzmannfactor ,

where U is the (free) energy of formation, T theabsolute temperature and k Boltzmann's constant.At room temperature, the value ofthe product kT is4 X 10-10 erg.

The formation energy of vacancies and inter-stitial atoms in'metals is not known with any degreeof accuracy. Huntingdon and Seitz 16) have com-puted the energy for a vacancy in a metal such ascopper to be roughly 10-12 erg. The formationenergy of irrterstitial atoms will probably be severaltimes higher. These values agree fairly well withthe conclusions drawn from experimental work,and with their aid it can be computed that a con-centration of vacancies or interstitial atoms ofsome appreciable magnitude can exist in equili-brium with the lattice only at temperatures nearthe melting point of a metal. If any indications arefound of the formation of such lattice defects atnormal temperatures, it must be concluded that

, their origin is not thermal. In fact, there are indi-cations that during the process of deformation of ametal at ordinary temperatures, very large numbersof vacancies and interstitial atoms do occur, andwe shall refer to this again in our next article.

The obvious course is to seek a mechanism thatis responsible for the' occurrence of lattice defectsduring deformation, i.e. during the formation andpropagation of dislocations. Several mechanismshave been proposed, of which only one will bediscussed here.

16) H. B, Huntingdon and F. Seitz, Phys. Rev. 61, 315 and325, 1942.

FEBRUARY-MARCH 1954 LATTICE IMPERFECTIONS IN METALS 257

We have already seen above that the non-conserv-ative movement of a dislocation (i.e. of an edgetype of dislocation outside its slip plane) is neces-sarily accompanied by a variation in the length ofthe "extra half plane" of atoms involved in thedislocation. A number of extra atoms is withdrawnor added. If the temperature is high enough, or ifthe movement takes place very slowly, these atomscan be transferred to other parts of the crystal bydiffusion. If these conditions are not fulfilled,however, a shortage, or surplus, of atoms occurs inthe neighbourhood of the dislocation; in other words,there will be a number of vacancies or interstitialatoms. The question is, then: can non-conservativemovements of dislocations occur during plasticdeformation at low temperatures? Seitz and Mott 17)have shown that such movements do indeed takeplace on a large scale.Dislocations emanating from a Frank-Read source

travel through the crystal, and it is inevitable thatthey encounter parts of the original and still existentnetwork. The further movement, as a result of theapplied stress, of a dislocation loop straight acrossthe network element, results in the formation ofdiscontinuities in both dislocation elements at thepoint where they intersect. Consider the simplecase of two screw dislocations at right angles toeach other (fig. 14). If dislocation 1 is,intersectedbydislocation 2, the relative translation within thecrystal produced by the latter will result in a

b,,..- ,"'-,..-'

,I /'2 ba/ ............

/ ,// 1/

..... / 2 ao

Ê. !l. 77983

Fig. 14-,a) Two mutually perpendicular screw dislocations 1and 2 with Burgers vectors bI and b2 approach each otheralong the dotted line. b) When they interseet each other, eachdislocation reveals a kink equal in magnitude to the Burgersvector of the other dislocation. These kinks behave as edgedislocations, with a Burgers vector which is the same as thatof the screw dislocationin ,-::hichthey occur.Further movementof the screw dislocations in the original sense results in non-conservative movement of the kinks.

17) N. F. Mott, Proc. Phys. Soc. B 64, 729, 1951; F. Seitz,Advances in Physics 1, 43,1952.

displacement of the two parts of dislocation 1 onboth sides ofdislocation 2. A "j og" is thus formed indislocation 1 equal in length to the Burgers vector'of 2, and in the same direction as this vector. Con-versely, a jog occurs in dislocation 2 in accordancewith the Burgers vector of dislocation 1. These jogsbehave like small sections of dislocation, with thesame Burgers vector as that of the original dislo-cation, but of course with a different orientation ofaxis. In this particular case, the jogs constitutesmall edge dislocations, since their Burgers vectors(those of the original screw dislocations) are per-pendicular to their axes. Further movement of thescrew dislocation is in effect non-conservativemovement of these edge dislocations. The movementof the jogs would be conservative only if it tookplace in their own slip planes, that is, the planesthrough the jog and the axis of the screw disloca-tion. Actually, however, these jogs move togetherwith the screw dislocation, i.e. they move in a planeperpendicular to that of the axis of the screw disloca-tion. Consequently, with further rapid movement asproduced by the applied stress, the jogs leavebehind them a row of vacancies or interstitial atoms.

Not only will the perpendicular intersection ofscrew dislocations almost always produce a jog witha non-conservative movement impressed on it:in general, the intersection of any' two dislocationswill also do this. In this way it is possible for verylarge numbers of vacancies and interstitial atomsto occur during plastic deformation.

We have now discussed both the formation andthe principal characteristics of dislocations and otherlattice defects in metals. In the next article, theeffect of such lattice defects on the deformation andother related phenomena will be investigated.

Summary. The theoretical necessity for the existence oflattice imperfections in crystals, such as vacancies, interstitialatoms and dislocations, is clarified in the light of a number ofphenomena related to the plastic deformation of metals, e.g.the occurrence of slip ,lines, work hardening and variationsin the electrical conductivity. This article is devoted mainlyto a discussion of the more important characteristics ofdislocations, and deals successively with the geometricalaspects-of various kinds of dislocation, their behaviour undershear stresses in the crystal as illustrated by means of a model,and the movement of dislocations, taking into account thedifference between conservative and non-conservative move-ments. Lastly, the formation of lattice defects of differentkinds as the outcome of interaction between existing defectsis examined. The effects of these imperfections on certainmechanical properties willbe dealt with in a subsequent article.