For other angles of incidence this reinforcement is prevented by interference .sx The necessary condition for the reinforced beam is that , where is the wave length , d the distance apart of the planes , the angle of incidence , and n an integer .sx Hence , knowing and , d can be found .sx The first experiments were made by v. Laue , but the method has been largely developed by Sir W. H. Bragg and W. L. Bragg ( 13 ) , Debye and Scherrer ( 15 ) , and Hull ( 16 ) ; for the technique of the experiments reference should be made to these authorities .sx It must suffice here to remark that not only the dimensions of the space lattice , the arrangement of the atoms , and the orientation of the crystals , can be deter-mined by this method , but internal strains and lattice distortion can also be detected .sx The Crystal Boundaries .sx The Amorphous Theory .sx It will be readily realised that given a number of differently oriented crystals , all growing on a regular plan until stopped by adjacent crystals , there must be a discontinuity between them where they meet .sx Some difference of opinion exists as to what form this discontinuity takes .sx Rosenhain has suggested that it probably consists of a layer of irregularly arranged atoms , which , though subject to the play of inter-atomic force , exhibits many of the properties of an amorphous material ( amorphous material = material which has solidified without crystallising , i.e. in the manner in which glass , pitch , and amorphous sulphur solidifies) .sx Whatever be its nature , at ordinary temperatures the inter-crystalline structure is harder and stronger than the crystalline grains themselves , for fractures prefer to pass through the grains rather than between them , see 319 .sx At high temperatures the reverse appears to be the case ( 320) .sx Slip Bands .sx If a specimen of ductile material be polished and etched , so as to bring out the crystal boundaries , and is then stressed sufficiently to cause a slight permanent set , microscopic examination with direct illumination will show that the polished surfaces of the crystals are crossed by a series of lines .sx In any one crystal the lines are usually , though not always , straight and parallel , but their direction is different in different crystals .sx These lines are minute steps on the polished surface , which , under oblique illumination catch the light and appear as bright lines on a dark ground .sx They are due .sx to slip along certain crystallographic planes , and were called slip bands by Ewing and Rosenhain ( 8 ) who first discovered them .sx ( i ) Fig. 380 is a section representing the junction between two crystals , the parallel lines representing the traces of the planes in question .sx When slip occurs along these planes the material deforms , much as a pack of cards would deform , and the steps described above appear on the polished surface as shown at ( ii) .sx Recent research ( 22 ) appears to show that in iron crystals , " rod " instead of " plane " deformation occurs ; that is to say , the deformation resembles that which would occur if a bundle of wood , rather than a pack of cards , were thus deformed , and the slip bands are jagged or curved .sx It may be inferred , therefore , that plastic deformation is the result of slip in the interior of the crystals .sx Slip appears to take place first in crystals with a particular orientation , and on certain definite crystallographic planes .sx While the deformation is small , therefore , slip bands appear only in a few of the crystals .sx As the deformation increases the lines get closer together , .sx due to slip on fresh planes , and further slip takes place in more and more of the crystals .sx In very severe straining other series of lines may appear in the same crystals , crossing the first series .sx Slip bands first become visible when the slip along the planes is of the order 0.00001 inch , i.e. about 1000 atom diameters ; in severe straining the slip may be five times as great as this , or more .sx Slip , having started along one series of slip planes , does not continue along these planes until fracture occurs .sx The motion is arrested , and the deformation continues along other planes and in other crystals .sx The extra resistance which prevents further motion is accompanied by the loss of plasticity and increase in strength and hardness known as strain- or work-hardening .sx During the process of cold working another phenomenon accompanies the slipping during the deformation .sx X-ray analysis has shown that the crystals tend to re-orient themselves , so that as far as possible one of the crystal faces lies in , or parallel to , the surface perpendicular to the pressure .sx In very severely cold-worked metals the fragments of the broken-up crystals all tend to assume the preferred orientation .sx 317 .sx Slip and the Space Lattice .sx Strain Hardening .sx It was pointed out in 314 that a metallic crystal is built up of atoms , placed in regular order on a space lattice , and held in position by the mutual attractions or repulsions between adjacent atoms .sx When a small .sx stress is applied to the material its effect is to modify these attractions , with the result that the relative positions of the atoms are altered to suit the new distribution of inter-atomic force ; in other words , the space lattice will be distorted .sx When the stress reaches a certain magnitude , the attraction between certain of the atoms will be entirely overcome , and slip will occur .sx As the slip proceeds , atoms , or groups of atoms , will be torn from their places , and when the slip comes to an end there will be a layer of displaced atoms along the slip plane which will not register properly with the crystal space lattice ; the severer the deformation , the greater the thickness of this layer .sx Its presence appears to increase the resistance to motion along the slip plane , and finally , unless the distortion be too great , to stop the sliding .sx It is not improbable that in a crystal aggregate ( a structure composed of many minute crystals , as in an ordinary metal , see 313 ) , a large part of the resistance stopping the motion comes from adjacent crystals ; but the fact that failure does not occur on the plane of original slip in specimens composed of a single crystal t proves that the distorted lattice on the slip planes offers a greater resistance to sliding than does the regular crystalline structure .sx The fact that , at ordinary temperatures , sliding takes place on the interior planes of the crystal , instead of through the distorted structure at the junction between the separate crystals , also indicates the superior resistance to sliding of a distorted structure .sx It has been further suggested that the distortion of the lattice extends inwards from the actual slip plane ( 46 ) , and that this " rumpling " of the adjacent space lattice increases the resistance to slip .sx A similar rumpling must occur at the crystal boundaries .sx Before strain takes place , there must be considerable distortion of the space lattice at these boundaries , for the regular distribution of atomic force must be upset , which distortion and consequent extra resistance to slip will increase under stress , and the accompanying tendency of the crystals to re-orient themselves .sx This conception will explain not only the extra resistance to sliding which the grain boundaries possess , but the fact that crystal aggregates harden more quickly under strain the smaller the size of the grain ; and also the hardening effect of cold working , since the latter much distorts the internal structure of the material .sx According to the above reasoning , therefore , strain hardening is the effect of lattice distortion due to slip during plastic deformation , and any circumstance which disturbs the regular arrangement of the atoms increases the hardness of the material .sx 318 .sx Lders' Lines .sx Strain Etching .sx The effects of slip manifest themselves in a number of different ways .sx If a specimen of ductile material such as mild steel be polished and then tested in tension , a series of parallel lines ( i , Fig. 381 ) will appear on the polished surface immediately the yield point is passed .sx These lines were first observed .sx by Lders ( 31 ) and arc called Lders' lines .sx They may be seen in many different conditions of stress .sx In a tension specimen they make an angle of about 60 with the axis , and have on this account been held to support the internal friction theory , ( 228) .sx Lders' lines must not be confused with slip bands .sx The latter make their appearance in all metals as soon as the proportional limit is overstepped , and are only visible under a microscope .sx They can be removed by etching , but reappear if the material be again placed under stress .sx Lueders' lines are visible to the naked eye , but only appear in metals with a decided yield point .sx They cannot be removed by etching .sx Strain Etching .sx If a specimen of mild steel , C 0.3 per cent .sx , which has been strained slightly beyond the yield point , be heated for about half an hour to a temperature of 200 C. , then ground and polished and etched with a strongly acid solution of copper chloride Fry etching ( 36 ) a series of dark markings appear on the polished surface .sx ( ii ) Fig. 381 Jevons ( 39 ) , shows these markings on a tension member which has just passed the yield point , and Fig. 282 shows the markings on a short compression member .sx These markings are found to correspond exactly with Lders' lines ( 37 ) , and are the surface indications of wedges of permanently distorted metal .sx The wedges appear to start from the most highly strained locality , and broaden slowly at the base as the apexes rapidly advance across the section .sx The material in the wedge .sx is found to be strain hardened , and under the microscope exhibits slip , crystal breakdown , or grain boundary failure .sx The wedges may be straight or curved depending on the direction of the strain , but the distortion is not a mass slip along a plane right through the specimen .sx Rupture only takes place after the distortion wedges have intersected one another all over the area of failure .sx Such wedges appear to be a fundamental characteristic of permanent deformation in mild steel ( cf .sx Modern Yield-point Studies , 216) .sx Some steels cannot be strain etched .sx 319 .sx Mechanism of Fracture .sx It was explained in 316 that deformation takes place by slip on inter-crystalline planes , and manifests itself by the appearance of slip bands on the crystal surfaces .sx As the intensity of the stress increases , more and more slip bands appear in more than one direction , crossing each other , and the bands broaden out .sx Dark patches or markings appear on the crystals , made up of a profusion of slip bands .sx Fracture occurs when the inter-atomic forces on the slip planes are finally overcome by the applied stress .sx With certain exceptions ( 320 , 321 ) , and in all kinds of tests , the fracture .sx invariably passes through the crystals and not along the crystal boundaries ( Fig. 382) .sx In a tensile test , the crystals in the neighbourhood of the local extension are much elongated , and near the fracture are drawn out into mere threads .sx This thread-like structure is the explanation of the silky appearance of the fracture in ductile steels .sx Nevertheless , X-ray analysis has shown that the drawn-out material retains its crystalline character , though there appears to be evidence of lattice distortion .sx The development of fatigue fractures has been studied by Ewing and Humfrey ( 41 ) , Gough and Hanson ( 46 ) , and others ( 43 ) ; also at the University of Illinois Engineering Experimental Station ( 50) .sx In a specimen exposed to an unsafe range of stress a few slip lines at first appear , then more and more , which gradually broaden out , until , after many cycles , they become changed into relatively wide bands and lose their original sharp definition .sx At a later stage of the test , parts of the crystals are found to be covered with dark markings made up of groups of broadened slip bands .sx These develop into a crack which gradually extends through crystal after crystal until fracture results .sx It is to be observed ( 46 ) that slip bands appear at ranges of stress much less than the limiting range ; and , prior to the formation of the crack , the micro-structure presents the same appearance , with broad , dark markings formed of masses of fine slip bands , whether the range of stress be safe or unsafe .sx