 | We now generalize the present view ofdislocations as follows: |
|  | 1. Dislocation lines may be - never mind that we cannot, atthe present, easily imagine the atomic picture to that. |
| | 2. can be Burgers vectors, and as we will see later, evenvectors that are lattice vectors arepossible. A general definition that encloses all cases is needed. |
| As ever so often, the basic ingredients needed for"making" dislocations existed before dislocations in crystals wereconceived., coming from the mechanics of thecontinuum (even crystals haven't been discovered yet), had defined all possiblebasic deformation cases of a continuum (including crystals) and in thoseelementary deformation cases the basic definition for dislocations was alreadycontained! |
| | The link showsVolterra' basic deformationmodes - three can be seen to produce dislocations in crystals, one generates a dislocation. |
| | Three more cases produce defects called "disclinations". While of theoreticalinterest, disclinations do not really occur in "normal" crystals, butin more unusual circumstances (e.g. in the two-dimensional lattice of fluxlines in superconductors) and we will not treat them here. |
| Volterra's insight gives us the tool to definedislocations in a very general way. For this we invent a little contraptionthat helps to imagine things: the "Volterraknife", which has the property that you can make any conceivablecut into a crystal with ease (in your mind). So lets produce dislocations withthe Volterra knife: |
| 1., any cut, into the crystal using the Volterra knife. |
| | The cut is always defined by some inside the crystal (here the plane indicatedby he red lines). |
| | The cut does not have to be on a flat plane, butwe also do not gain much by making it "warped". The picture shows aflat cut, mainly just because it is easier to draw. |
| | The cut is by necessity completely containedwithin a, the (most of it on the outside of the crystal). |
| | That part of the cut that is the crystal will define the line vectort of the dislocation to be formed. |
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| 2. separated by the cut relative to each other bya;allowing elastic deformation of the lattice in the region around thedislocation line. |
| | The translation vector chosen will be theb of thedislocation to be formed. The sign will depend on the convention used. Shownaremovements leading to anedge dislocations(left) and a screw dislocation (right). |
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| 3. or take some out, ifnecessary. |
| | This will be necessary for obvious reasons wheneveryour chosen translation vector has a component perpendicular to the plane ofthe cut. |
| | Shown is the case where you have to fill inmaterial - always preserving the structure of the crystal that was cut, ofcourse. |
| | : Aftercut and movement.: After filling upthe gap with crystal material. |
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| 4. by "welding" together the surfaces of the cut. |
| | Since the displacement vector was a, the surfaces willfit together everywhere - exceptin the region around the dislocation line defined as by the cut line. |
| A one-dimensional defect wasproduced, defined by the (= linevectort of the dislocation) and the which we callBurgers vectorb. |
| | It is rather obvious (but not yet proven) thatthe Burgers vector defined in this way is identical to the onedefined before. This will becometotally clear in the following paragraphs. |
| From the Volterra construction of adislocation, we can not only obtain the simple edge and screw dislocation thatwe already know, but dislocation.Moreover, from the Volterra construction we can immediately deduce a new listwith more properties of dislocations: |
| | 1. The for a given dislocation is always the same, i.e. it does notchange with coordinates, because there is only displacement for every cut. On the other hand,the may be different at everypoint because we can make the cut as complicated as we like. |
| | 2. Edge- and screwdislocations (with an angle of90° or0°, resp.,between the Burgers- and the line vector) are just of the general case of a mixed dislocation, which has anarbitrary angle betweenb andt that may evenchange along the dislocation line. The illustration shows the case of a curveddislocation that changes from a pure edge dislcation to a pure screwdislocation. |
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| |  | We are looking at the plane of the cut (sort of a semicirclecentered
in the lower left corner). Blue circles denote atoms just below,
red circles atoms just above the cut. Up on the right the dislocation is
a pure edge dislocation, on the lower left it is pure screw. In between
it is mixed. In the link thisdislocation is shown moving in an
animated illustration. |
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| | 3. The Burgers vector must be independentfrom the precise way the Burgers circuit is done since the Volterraconstruction does not contain any specific rules for a circuit. This is easy tosee, of course: |
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| | | | | Old circuit | Two arbitrary alternative Burgers circuits.
The colors serve to make it easier
to keep track of the steps. |
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| | 4. A dislocation in the interior of an otherwise perfectcrystal (try to make a cut that ends internally with your Volterra knife), butonly at |
| | 5. If you do not have to add matter or totake matter away (i.e. involve interstitials or vacancies), the Burgers vectorbwhich has two consequences:- The cut plane must be planar; it is defined by the line vector and theBurgers vector.
- The cut plane is the of thedislocation; only in this plane can it move without the help of interstitialsor vacancies.
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| | The glide plane is thus the planespread out by the Burgers vectorbt. |
| | 6. Plastic deformation is promoted by themovement of dislocations in glide planes. This is easy to see: Extending yourcut produces more deformation and this is identical to moving the dislocation! |
| | 7.The magnitude ofb (=b) is a measure for the"strength" of the dislocation, or the amount of elasticdeformation in the core of the dislocation. |
| A not so obvious, but very importantconsequence of the Voltaterra definition is |
| | 8. At adislocation knot the,Σb = 0, provided all line vectors pointinto the knot or out of it. A dislocation knot is simply a point where three ormore dislocations meet. A knot can be constructed with the Volterra knife asshown below. |
| Statement8. can be proved intwo ways: Doing Burgers circuits or using the Voltaterra construction twice. Atthe same time we prove the equivalence of obtainingb froma Burgers circuit or from a Voltaterra construction. |
| | Lets look at a dislocation knot formed by threearbitrary dislocations and do the Burgers circuit - always taking the directionof the Burgers circuit from a "right hand" rule |
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| | Since the sum of the two individual circuits mustgive the same result as the single "big" circuit, it follows: |
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| | Or, more generally, after reorienting allt -vectors so that they point into the knot: |
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| Now lets look at the same situationin the Voltaterra construction: |
| | We make a first cut with a Burgers vectorb1 (the green one in the illustration below). |
| | Now we make a second cut in the same plane thatextends partially beyond the first one with Burgers vectorb2 (the red line). We have three differentkinds of boundary lines: red and green where the cut lines are distinguishable,and black where they are on top of each other. And we have also produced adislocation knot! |
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| | Obviously the displacement vector for the blackline, which is the Burgers vector of that dislocation, must be the sum of thetwo Burgers vectors defined by the two cuts:b =b1 +b2. So weget the same result, because our line vectors all had the same "flow"direction (which, in this case, is actually tied to which part of the crystalwe move and which one we keep "at rest"). |
| If we produce a dislocation knot by two cuts thatare coplanar but keep the Burgersvector on the cut plane, we produce a knot between dislocations that do nothave the same glide plane. As an immediate consequence we realize that thisknot might be - it cannot move. |
| | A simple example is shown below (consider that theBurgers vector of the red dislocation may have a glide plane different from thetwo cut planes because it is given by the (vector) sum of the two originalBurgers vectors!). |
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| We can now draw some conclusion abouthow dislocations must behave in circumstances not so easy to see directly: |
| | Lets look at the glide plane of a. We can easily produce a loop withthe Volterra knife by keeping the cut totally inside the crystal (with a knife that could not be done). In theexample the dislocation is an edge dislocation. |
| | The glide plane, always defined by Burgers andline vector, becomes aglide cylinder! Thedislocation loop can move up or down on it, but no lateral movement ispossible. |
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| | What would the glide plane of a screw dislocationloop look like? Well there is no such thing as a - you figure that one out for yourself! |
| | A pure (straight) dislocation has no particular glide planesinceb andt are parallel and thusdo not define a plane. A screw dislocation could therefore (in principle) moveon any plane. We will see later why there are still some restrictions. |
| This leaves the touchy issue of thesign convention for the line vectort.! The sign of the line vectordetermines the sign of the Burgers vector, and the Burgers vector, includingsign, is what you will use for many calculations. This is so because for aBurgers circuit you must define if you go clockwise or counter-clockwise aroundthe line vector, using the right-hand convention.! |
| | The easiest way of dealing with this is toremember that the sum of the Burgers vectors must be zero if all line vectorseither point into the knot or away from it. |
| | As long as only three dislocations meet at onepoint, there is no big problem in being consistent in the choice of line vectorand Burgers vectors, once you started assigning signs for the line vectors, youcan throw in the Burgers vector. There is however no principal restriction toonly three dislocations meeting at one point; in this case the situation is notalways unambiguous; we will deal with that later. This is not as easy as itseems. We will do a little exercise for that. |
| | Last we define: The circuit is to close around thedislocation; the circuit in the reference crystal then defines the Burgersvector. |
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© H. Föll
