ORIENTATION ARD STRUCTURE OF 'SELECTRON CRYSTALS BY XARAY DWCTEON 11m. for m. m of M. s. MtCHiGAN sure com David em. M: Conmfl 1953 ILL\LLLL\|LLLLLLLLLLLLLLLLLLLLLLLI L LIBRARY Michigan State _ \ university ‘”“ ' “ "_ ' ‘ ' ‘ This is to certify that the thesis entitled Orientation and Structure of Silectron Er fstas X-ray D f rac iony presented by David Ellis KcConnell b has been accepted towards fulfillment of the requirements for M. .S..__degree mm '. \ .761“ Major professor Date W 0-169 -u—A—_—.~ —----—A—- - - _.--- .-_. —— —_- PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 1/98 cJClRCJDateDuepGS—p.“ ORIEN$AEION'LND STRUCTURE OF SILECTRON CRYSTALS BY X-RAI DIFFRACTION 13: David Ellie McConnell A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1953 ACKNOWLEDGICENT The author wishes to express his sincere appreciation to Drs. J. C. Lee and B. D. Spence for suggesting this problem.and assisting in its execution. He is also indebted to Mr. J. A. Coven for his valuable help in , the operation of the xhray unit and his encouraging suggestions, and to Mr. C. L. Kingston for his help and suggestions in the construction and.maintenance of the equipment. 300056 I. II. III. IV. V. VI. VII. TABLE OF CONTENTS II‘JTRODUCTION . O O O O O C C I O O O O O O O O . INTERACTION OF XhRAYS AND CRYSTALS . . . . . . . PRELIMINARY INVESTIGATIONS AND CHOICE OF METHOD CONSTRUCTION OF A.SHEET METAL GONIOMETER . . . . DETERMINATION OF CRYSTAL ORIENTATION . . . . . . CRYSTAL SYSTEM DETERMINATION . . . . . . . . . . ADDITIONAL CRYSTAL ANALYSIS . . . . . . . . . . APPENDIX - REPAIR AND MAINTENANCE OF XFRAI UNIT BIBLIOGRAPHY . O O O O O O O O O O O O O O O O 0 RACE 10 12 15 13 23 35 I . INTRODUCTION The study of crystals is no doubt one of the oldest of the scien- tific studies. Surely the men who lived on the earth thousands of years ago must have noticed that rocks usually broke along definite geometrical lines and planes and wondered why they did so. By the end of the seventeenth century some men were measuring the angles between these cleavage planes. Huygens published his Treatise _o_n_ Mg; in 1690 and described.the shape of calcite crystal, which is one of the more complex forms. However, Huygens was primarily interested in op- tics, and it was not until nearly a hundred years later that Hauy laid the foundation of the science of crystallography by the publication of his book, Essai d'une theorie sur la structure? des cristaux. He recog- nised that the external form of a crystal was based on a sermon under- lying structure; that is, a large crystal is formed by stacking together many small identical elementary unit cells. Advancement during the next 120 years was largely geometrical, based on the visual study of the external faces and angles of crystals. Considering the different ways in which space can be divided into iden- tical unit cells, Bravais showed in lens that all types of crystalline symmetry can be grouped into the seven systems shown in Figural. Bessel had shown in 1830 that as many as 32 classes of symmetry were possible for crystals based on space lattices, but his work remained unnoticed until Sohnke recognized its significance sixty years later. Fedorov, Schoenflies, and Barlow, all working independently between 1885 and 18911:, showed that, considering all possible symmetry axes, reflection and glide planes of symmetry, axes of rotary inversion, and centers of in— version, the seven systems and 32 classes could be extended to 230 ac— tual types of crystals. About a century ago the theory that matter was composed of numer- ous small particles, or atoms, was becoming widely accepted. In a crys- talline solid where the binding forces are comparatively large, these atoms assume a regular and periodic arrangement. This does not mean that they always extend in an unbroken line from one edge of such.a solid to the other, but that the length of the unbroken rows must be large as compared to the distances between adjacent atoms. This regular three- dimensional arrangement of unbrdken rows of atoms is called a space lat- tice. .If the rows of atoms are parallel to three mutually perpendicular axes and the distances between adjacent atoms along the three axes are the same, we call the arrangement a cubic space lattice. We can imagine planes normal to each of the axes and passing through the atoms in such a way as to divide the crystal into small cubes with an atom at every corner. Such an atomic arrangement which is repeated at regular inter- vals along all three axes is called.a unit cell. Sometimes the atoms have the above cubic arrangement except that there are additional atoms at the center of each of the six faces of the unit cube, or there may be one atom.in the center of the unit cube. Such arrangements are called face-centered or body-centered cubic lat- tices. Of course the other six systems also may have face-centered or bodybcentered variations. o 0 RHOMBCHEDRAL TRICLINIC MONOGLINIC HEXAGONAL UNIT CELLS OF THE SEVEN SYSTEMS Figure l IOO PLANES HO PLANES Ill PLANES 3 It would be easy to imagine planes through the atoms of the cubic cell at angles other than normal to the three axes. In general such families of parallel planes will not be as densely populated as the planes mentioned.above, but the distance between two adjacent members of a family will be less. Figure 2 (boss three types of planes which may be passed through a cubic lattice. The planes are designated at the top of the figure by their Miller indices. To find the Miller in- dex of a plane in.a crystal lattice we must first choose a definite orientation for the crystal with respect to three (in the case of the hexagonal unit cell - four) axes, x, y; and g, and then determine there the plane would intercept each of the three axes. When we take the reciprocals of the intercepts, write them in order, and nmltiply by an appropriate number so that no fractions remain, we have the Miller indices of the plane or any plane parallel to this plans. If the three axes are not mutually perpendicular, the angles between them must also be specified. In writing negative Miller indices, the minus sign is ‘usually placed above the integer. In Figure l the three top unit cells have three mutually perpen- dicular axes, but the four bottom unit cells do not. The rhombohedral unit cell has three equal but not 90° angles and three equal sides. The monoclinic unit cell has two 900 angles and no equal sides; ihile the triclinic unit cell has no equal angles or sides. Also indicated in Figure l are some of the two-fold, three-fold, fourbfold, and Illb fold axes of symmetry. These axes of symmetry are very important in the xpray analysis of a crystal lattice. h The use of the word ”atom? to describe the element of'matter which appears at the corners of the unit cells does not mean that this dis- cussion is being restricted to the structure of the elements. It has been found that the crystal lattices of the majority of chemical com. pounds are composed of atoms rather than molecules. For example the 'unit cell of rock salt can be considered.a simple cube with chlorine atoms at the diagonal corners of every side and sodium.atome the other corners. However, to follow strictly our definition of a unit cell, we must take a cell with twice the dimensions of the above cube (eight times the volume) and consider the structure to be that of an inter- penetrating face-centered cubic lattice. Figure 3 illustrates this model and the models of several other compounds. Only in the case of carbon dioxide do actual molecules appear at the corners of the unit cell. Hexagonal Close-packed Figure 5 II. INTERACTION 0F XERAIS AND CRYSTALS When Laue discovered in 1912 that spot patterns could be obtained on a photographic film by passing x-rays through a crystal in an ar- rangement similar to Figure 1}, a new and very fruitful period of crys- tal analysis began. At the time scientists were still debating whether x-rays were of corpuscular or wave nature, and H. L. Bragg soon showed that when the crystal was rotated in the x-ray beam, the spots moved Just as if they were spots of light reflected from the surface of a mirror. By assuming that the reflecting surface for any one spot was a family of parallel planes, he was able to derive the fundamental. law of x-ray diffraction: n A: 2d sin a where )1 is the wave length of the x-radiation, d is the distance be. tween two adjacent planes, and e is the angle the incident beam nukes with the parallel planes. Il'he law can be derived as follows. Let p, p' , and p" be three parallel, equally-spaced, crystal planes and x the incident x-ray beam at an angle 9 with these planes. At A part of the beam will be reflected along [8' and part will continue on to 0 to be reflected along 61). In general the beams that arrive at 3' and D will not be in phase, but if the angle of incidence is such that the distance BOD is any-integral multiple, :1, of wave lengths, they will be in phase. In this case the reflected beams from all the crystal planes will reinforce each other in such a way as to form a diffraction spot on the film. But BC is Just d sin 9 and Bragg's law follows directly. PRODUCTION OF LAUE PHOTOGRAPH Figure 4 Fab . . . .. . L a . -g... a..-” a ”.5-.- ......~s. - a _. a. a - a . - ., T '3; #2 3H? 9 IZf , J I ll 1 ‘ i a lo? t 9 ‘ ‘ 33 " i E 7 l 2 j TUNGSTEN -———-~ ! :‘3 5' ' ‘ z 4 g 53. -. 1 MOLYBDENUM . 4 . P 3 ' CHROMIUM . 2, I : .9 4 3 - a.-.._..._.......,_._...~,.-;_,. . '.I° .‘2‘""'.'3 4 .5 .6 .7 .s .9' ANGSTROM UNITS X-RAY SPECTRUM OF THREE TARGETS AT 35000 VOLTS Figure 5 While W. L. Bragg was determining how the Lane spot patterns were formed, his father, H; H. Bragg, constructed an ionisation spectrometer with which the position and intensity of x-rays reflected from crystal planes could be accurately measured. This instrument allowed a study of the xpray spectrum.in.much greater detail, and intense bands of 1b rays were discovered at definite wave lengths which depended on the material of the target in addition to the white radiation whose wave length is independent of the target‘hut depends on the drop in potential across the tube. In Figure 5 the intensity cf the radiation is plotted against the wave length for three different targets at 35000 volts potential drop. The characteristic spectrum lines show up only for molybdenum because the voltage is not sufficient to produce character- istic lines for a tungsten target, and the characteristic lines of 7 chromium fall to the right of the portion of spectrum shown, at about 2.29 A0. It was at about this time that Bohr advanced his orbital electron model for the atom.in an attempt to explain visible spectra, and it was not too great a step to assume that the characteristic 1— radiation was the result of electrons ”dropping" to an orbit closer to the nucleus of the atom to fill up a vacancy caused in the orbit by the collision of the original electron with a high—energy incident electron. The most important of these lines are the K“ and,KF lines shown in the figure for molybdenum. All three lines approach zero intensity at the same wave length on the left, that is, the short wave—length limit is independent of the material of the target. The incident electrons that produce xsrays of this wave length give up all of their energy in the process so that we can write: from whi ch, = 102:” x 1.0”)“.I Cm. )\ min vmax In the production of the Lane photograph the crystal is stationary and the requirement for diffraction as stated by Bragg's Law is met in the general case only because a wide variety of wave lengths of radia- tion is present. Often it is better to know the wave length of the radiation being used than to know its angle of incidence on a crystal plane. For this purpose Debye and Scherrer developed the powder photo- graphic method in 1916. It is especially useful for substances which can only be obtained in a finely divided crystalline form, and has 8 found many applications in the field of metals and alloys. The powder is usually mixed with some adhesive material such as gum tragacanth and made into a thick paste which is formed into a long cylinder about the size of a pencil lead. When this cylinder is dry it is mounted at the center of the camera, where the x-ray beam will fall on it, and rotated during the exposure. Thefilm usually takes the form of a narrow band surrounding the sample at a constant distance from the axis of the sample. In such a finely divided sample some of the crystals are always at the correct angle with the beam to cause diffraction of the lines of characteristic radiation, and these diffracted rays can" a pattern of black lines, which is characteristic of the type of crystal being exam- ined, to appear across the developed film. The first use of the rotating single crystal for analysis purposes seems to have been by Shiebold in 1919. This method is quite similar to the powder method except that a single crystal is rotated at the center of the camera and spots are obtained on the film instead of lines. The pattern is not of much use unless the film is wide enough to con- tain a large number of spots and the crystal is rotated around one of its axes of symmetry. Often a flat film is used, and it is sometimes placed between the crystal and the x-ray tube in order to obtain a back- reflection photograph. Other variations are often used for the task of interpretating the patterns obtained, Often the crystal is only oscil- lated through a small angle rather than being rotated through 360°. 1 method first used by weissenberg in 1921!- is to have the cylindrical film move in the direction of the cylinder axis while the crystal is rotating. The rotating crystal method with its variations can give very good results for someone with enough experience to interpret its patterns correctly, but it is easy for the beginner to make errors in this method. III. PRELIIIRKEY INVESTIGATIONS AND CHOICE OF METHOD Silectron is Allegheny Ludlum Steel Corporation's trade name for one of their polycrystalline sheet metal products often used for trans- former cores. The composition is about 97 percent iron and 3 percent silicon, and it is claimed that the crystals are quite well oriented with respect to each other. The particular sample which was chosen for examination was a sheet about .35 millimeters thick and.measuring about three by five centimeters. It was found that at a peak potential difference of forty kilo- volts, an exposure of about fifteen minutes with a filament current of about fifteen mdlliamperes gave quite dense diffraction spots on a film.five centimeters from.the crystal' sample. The first Laue spot photographs obtained were not symmetric but appeared somewhat like the ones shown in Figure 7 . On one or two of the photographs the spots were smaller and more abundant indicating that the xprey beam.had passed through portions of two or more crystals. The sheet was etched in a 30 percent nitric acid solution in an attempt to determine the size and shape of the crystals visually. After etching the sheet appeared similar to Figure 11, although some of the boundary lines shown in the sketch were very difficult to detect until the sheet was rotated through wide angles under a bright light. .A person looking at the etched polycrystalline sheet might well ask the following questions: How are the various crystals oriented with respect to each other? How valid is the etched surface in indicating 5 94‘5me 11 the size and shape of the crystals, or are there other crystal bound— aries not evident to the eye? Do all the crystals have the same struc- ture, and if so what is this structure? Symmetric Laue photographs of the various crystals could very easily answer the first question if the sheet could be mounted in such a way that angles for the various symmetric patterns could be recorded. Laue spot photographs could also answer the second question if the sheet could be translated without rotation between exposures. Laue photographs are not nearly as good as the various monochromatic xbray methods in determining crystal structure, but they can be used successfully if the crystal structure is not too complex. In the present case a powder photograph seemed impossible since powdering the crystals. would most certainly introduce strains in the crystal lattice, which would cause streaking of the powder photograph. Since there was no way of knowing how the sample was treated when the crystals were originally formed, one could not be sure that any treatment given the powder to rid it of strains would preserve -the original crystal structure. These and etha- factors made it seem desirable to employ Laue photographs in all phases of the study of the Silectron sheet. IV. CONSTRUCTION OF A. SHEET METAL GONIOMETER Laue photographs are not very valuable for interpretation purposes unless the crystal is orientated in such.a way that the xpray beam is incident along one of the axes of symmetry. This required some method of mounting the silectron sheet so that it could be rotated around each of two axes perpendicular to each other and to the xhray beam. Small single crystals are usually mounted at the center of a two-circle goniometer, but such goniometers are difficult to construct and expen- sive to purchase, and one was not available for the present investiga- tions. If one had been available it is doubtful that it would have been found suitable for as large a sample as was used in this case, and it would have been difficult to have mounted the silectron crystal so that it could have undergone translation without rotation. The size and shape of the metal sheet suggested a slightly different method of mounting; therefore, the sheet metal goniometer which is pictured in Figures 8 and 9 was constructed. The metal sheet.A to be examined is held against the back of a square copper sheet B, which has a.hole in its center, by two spring steel bands 0. The face of the copper sheet is marked off in vertical and horizontal lines one millimeter apart and indexed in such a way that the position of the sample can be recorded and it can be returned to the same position at any subsequent time. At the top and bottom the copper sheet is held between two copper channels D, in such a way that one side or the whole sheet can be moved in a di- rection normal to the sheet by adjusting one or more of the four screws 3. m seamen ‘ure 9 Fe L K D 13 The two copper channels are bolted to the top and bottom of a square frame constructed of copper strap F in such a way that they can rotate through angles of at least 60° both sides of the center (parallel) po- sition. The square copper frame is in turn bolted through its two sides to either of two copper strap uprights thastened to the base H of the goniometer, in such a way that it too can rotate through angles of at least 60° both sides of the vertical. Protractors J are mounted to show how many degrees the copper sheet or the square frame is rotated around either of these two axes which are perpendicular to each other. By means of the four screws 1, the copper sheet can be adjusted so that both protractors will read 900 when the plane of the sample is normal to the xPray beam. The collimator K, which was constructed by drilling small holes in each of two pieces of lead in either end of a copper tube, is adjusted so that the x-ray beam falls on the sample at the intersec- tion of the two perpendicular axes. Thus when the goniometer is cor- rectly adjusted the sample may be rotated through wide angles around either or both.axes without moving laterally with respect to the xaray beam. The film L is placed in a holder M which is set at any desired distance behind the sample; and the entire goniometer is placed in the correct position before the xpray tube by means of the three leveling screws N which fit into appropriate depressions in a previously adjusted camera stand 0. After construction the goniometer operated exactly as had been planned except that because it was built large enough to examine large sheets of metal the corners of the copper sheet prevented bringing the 'ure 10 ["d J- .4- .. ~_/ 15% ~ ' 4 \‘.L I L...‘ (Ni » . e'ttg‘ v . $335 $2 91’ a ‘ (it; it a " “WM? Egg “V522,? ESE-Egg A". O. ah. attsa‘tt' st " 11+ film closer to the sample than eight or nine centimeters when the sample is turned at a large angle from the perpendicular position. This means that some of the exposures must be three or four times as long as would be required if it was possible to employ the standard crystal-film.dis- tance of five centimeters. This goniometer can be used to examine any part of a sheet which is not more than six centimeters square, and will accomodate sheets whose greatest dimension is not more than twelve cen— timeters. V. DETERMINATION OF CRYSTAL ORIENTATION A number of the larger crystals from all parts of the sample were selected and given arbitrary numbers as shown in Figure 11 so that they could be easily identified. Crystal number 1 was chosen for the first real attempt to secure a symmetric Laue photograph. About ten ex- posures were taken at various angles before symmetry was recognized around the vertical axis; and it required another dozen exposures at various angles around this axis before the symmetric pattern of Figure 10 was obtained. The protractor readings were recorded and the crystal moved so that the x-ray beam now passed through crystal number 2. The search for the same symmetric pattern of this crystal followed much the same procedure as before except that fewer pictures were required now that it was known what pattern would finally appear. A.similar process was used to obtain the same symmetric pattern for 28 more of the numbered crystals and the protractor readings for the symmetric settings of the thirty crystals are given in the columns at the left of Figure 11. After some experience it was often possible to tell which way the sheet had to be rotated in order to give the desired pattern, and several times the amount of rotation needed was correctly estimated. At other times, however, the search approached to within three or four degrees of the symmetric pattern and still the proximity of the pattern was not recognized. The pattern at the left of Figure 7' is eight degrees in one direction and two degrees in the other from perfect symmetry; the pattern on the right is only three degrees in one direction and five 16 degrees in the other from the desired orientation. Complete orientation of the thirty crystals with.possible error of one degree required over 150 exposures. It will be noticed from the list that only for crystal number 30 was it necessary to search far from the perpendicular position for the symmetric pattern. The averages of the other 29 crystals comes out to be very nearly ninety degrees around all three of the axes. Five of the 29 crystals show a difference of somewhat more than ten degrees from the average around one axis but in no case does the difference ex- ceed fifteen degrees. On the basis of this investigation it would seem safe to assume the majority of the crystals are aligned within fifteen degrees. Only crystal number 30 of those examined had an error greater than this. Also the investigation seemed to indicate that it is possible to detect visually the boundary line between two crystals whose orientations differ by more than two or three degrees. 'At only two places on the sheet was a crystal boundary found that had not been seen previously on the etched sample, and in each case it was possible to see the boundary when it was known to be there. The two cases were the boundary between crystals 6 and “0, whose orientations differ only around the axis normal to the sheet; and the boundary between crystal number 1 and the small crystal to its upper left, which was found only when a portion of each crystal happened to lie in the x-ray beam for an exposure and the pattern turned out to be a double one with corresponding spots less than a cenp timmter apart. It was thought at first that perhaps the filmthad.been I 1.‘ ill: lalllll.‘ l1|1l| 17 moved slightly during the exposure but a repeat exposure gave the same double pattern. Thus one can find the majority of the crystal boundaries by merely looking at the sheet and rotating it through large angles under a light, but if differences of less than two or three degrees cannot be tolerated, it is much safer to inspect for these differences of orientation by means of x-ray diffraction. CROSS SECTION OF CUBIC CRYSTAL LATTICE SHOWING INTERACTION OF HETENOCNNONATIC X‘RAY BEAN AND THE LATTICE PLANES . C I ' s. ..... 0 g a ..... F E ' a r m est 03 «1 air on sea on “33' VI. CRYSTAL SYSTEM DETERMINATION The first crystal analysis was made from Laue photographs but it ‘was soon found that other methods using monochromatic x—radiation were Inore direct and powerful. However, in the present case there were dif— ficulties connected with using any of the other methods, as mentioned above, so that an attempt to use Laue photograph analysis was made. First it will be helpful if the method of formation of Laue photographs is examined in somewhat greater detail. Figure 12 shows how an x-ray beam is diffracted by the various lat- tice planes of a cubic crystal lattice. Of course in this cross-sec- tional diagram the formation of only a few of the spots (namely those lying in one straight line) of the Lane photograph can be shown. This particular cross-section is normal to the x-axis, but a cross-section normal to the y-axis would show the formation of a similar line of spots. Other cross-sections parallel to the z-axis would show the position of other spots, and when all the cross-sections were added together the Laue photograph would be formed as is shown on.Figure h. In the case of the cubic lattice all spots lie on ellipses which pass through the intersection of two perpendicular lines, and the foci of all of the ellipses lie on one or the other of the lines. The first step in a Laue photograph analysis is to index the spots; that is, to determine what plane of the crystal lattice caused each of the many spots. This would appear to be a formidable task, but it can be accomplished quite readily by using one of two common methods of 19 projection illustrated below. In stereographic projection the incident ray falls on a crystal at O, and a reflecting plane 02 forms a spot at R. If a diagram is made in which the point B' is substituted for every point B, by means of the construction shown in the figure, the ellipses on which the points lie are transformed into circles. Figure lI-I shows such stereographic projections for NaCl (left) and 101. Figure 13 In the gnomonic projection, which is usually simpler and more sat- isfactory, the point R in the diagram transforms to the point G, where the normal OG to the reflecting plane meets the tangent to the circle. If the radius of the circle, which is always taken to be the crystal- film distance of the Laue photograph, is 5 centimeters, the relation. ship between a Laue spot 3 and its gnomonic projection G will be: @201! cote =5cot9 and RN=ONtan29=5tan29. fl ash 14%;: x .7" \ V ‘ "o:¢”%/A\%‘\\t ‘ .."'e e“... .‘e 0’ ~ - Q A -——-— Hug! Figure 14 ma enemaa In» I . 42.4 41: 4, 5:} 2443.4 N n .w 0 0% ~ 0 w n 0 h 20 Such a projection will transform the points lying on any one ellipse into points lying on a straight line. The simplest and.most rapid way to carry out a gnomonic projection is to construct a protractor such as shown in.Figure 15. The left side of this ruler, divided into millimeters fron.a.central point 0', measures the distance RN of a Laue spot from the central spot of the pattern; the right side is graduated in accordance with the requirements of the expression above to read the corresponding projected distances NG. In making the projection, the Laue photograph, or a reproduction.made from it, is fastened near the center of the piece of paper upon which the projection is to be made. The ruler is fastened to the Laue pattern by means of a thumb tack.passed through.the point C“ of the ruler and the center of the pattern. The distance from the center to a spot of the pattern is read off on the short end of the ruler and a dot placed on the paper at the corresponding number of the long end. In this way spots near the center of the pattern appear as dots exactly on the opposite side of the center and near the outside of the gnomonic projection. Figb ures l6, l7, and 18 are gnomonic projections of three symmetric Laue photographs obtained from.a silectron crystal., Notice that in Figure 16, which is a gnomonic projection of the two- fold symmetric pattern of Figure 10, the lines formed by the rows of dots divide the pattern into rectangles. This pattern is typical of either a tetragonal or an orthorhombic space lattice. However, Figure 1 showed that we could obtain a two-fold symmetric pattern by letting the xpray beam enter the edge of a staple cubic cell, therefore, that pos- a g .2 l,° 3‘; I’ a. all e 8 A. d: ‘3 , u‘ , 3's: 2.30 0 0° 0 e e e - . '.. Q. bike an -o sees m ”I t. )3 O C e o o e he s 0 ° ° 0 M e e 3|, s I O . v91 ‘03 . SA. 33 . I . . . I.‘ be . . . . . . ‘1! . ' 3. - ' 0 0 ° 9- I: A, ,5, L", 3' I, at u 3.: es: 2’ a: s . ' :s .u 3: w n: ('s . 0 e .a'. as! or is see ' I I:5 0" 2, I20 ”' . 1‘ O O 9. «I no no GNOMONIC PROJECTION OF LAUE PHOTOGRAPH OF SILECTRON CRYSTAL PRAY BIA“ NORMAL TO “0” PLANE Figure 115 “I 0h 0. .ss 0!! 0“ all e]. obs do w ‘00 0. out}; eh e!!! 0b the O” eh 00¢ out slit of“ as: C"! e”: .us 01‘! on. .m- site 0!. Old! GNONONIO PROJECTION OF LAUE PHOTOGRAPH OF SILECTRON CRYSTAL 0U 0" ens .013 “I Can on 0714 Oh ole eoSI obs sis OH Ow eat out on! we“ .33! “HI 0!" o“! b es! 0 0‘” es“ 0“! edl 0'! Oz! om X-RAY BEAN NORMAL TO (00') PLANE Figure 17 «I: On CHI no: «In .sls he on“. .‘fl. 5. “NT «no. n9. wzja E; or 44880! 140. >4¢ux mass .1 . .rru _ Adhmrco zozkouma to 113505.085 9.2.. to vac—housoxa 0.201029 «£0 .‘fl. 3. Jae ace no.0 «E. 3e 3. new. «as. .30 an . eon. Q’s mnNe 3n. en's nuns tune a... use 3. 3' use I. as .30 31 o o o . .3. as. Sue See moo. mi. 2.. nae 0.“. in. i. '0. 3.. add. e... ' CRYSTAL X Y Z TARGET KY MA. TIME OF DATE | .. 3.‘:F°-39.53--.9_9.§Z4é§ .9? 2975M612fio-52 I , , ‘“""* I I .’ . I .. e ' I. I O Q. I e ' O . e 0 e I .l O O. ‘ o O \ o - To \ Q o \ \ ‘ ‘ - ' Figure 19 21 sibility must be examined in this case. Also it is possible to obtain a two-fold symmetric pattern from a monoclinic lattice. By means of simple geometry it was determined.that the Laue spot pattern of Figure 10 would be formed by reflection from three mutually perpendicular families of planes whose inter-planar distances were in the ratio of about 7:7:5. The incident x-ray beam was parallel to a long edge of the cell. ‘ 4A simple way to test for a cubic lattice was to lookaor a four- fold symmetric pattern about N5° on either of two opposite sides of this symmetric pattern. .A previous search for other symmetric patterns had proved fruitless, but now the four-fold symmetric pattern shown in Figure 19 was found at the two places mentioned above. Figure 20 shows the angles around the y—axis that the crystal had to assume in order to obtain these three patterns. The four-fold symmetric patterns are all more dense on one side than on the other because the diffracted rays had to penetrate more of the crystal in one direction than in the other when the sheet was at an angle of about “5° with the xbray beam. The gnomonic projection of this four-fold symmetric Laue photo- graph is given in Figure 17. Notice that the lines formed by the rows of dots divide the pattern into squares. In assigning Miller indices to these spots, a right-hand orthogonal systemthas been assumed with the xpaxis to the right, the ybaxis up (along the rolling direction of the sheet) and the z-axis normal to the papen "me first integer of the Miller indices on the vertical lines are all the same, and.the second integer of all indices on the horizontal lines are the same. The third cm mazmflm m.x<..>. m1». 50¢ .2ng 805.03“... us... 0.2.2.39. >n aux-(boo ”32505.02 35 oflhg>m >¢hu22>m “Scat-:9; E35 as 53.9.8; aboaacaoa ___-r--_-‘-‘ am esenaa 9x4 4403mm) m1» b.3004 4mo zomhowdm m1» Ozfldhom rm 0mz_d._.mo mxddmoohbxd wbdq 0Ehm22>m 35125 30...-«32 5:225 30..-:th 351:5 Solos» \ A O a \ A e O 0 0 I 0 I O . 0 e t O I O . a O O Q o e 0 O O o O . . . e . e o o . O s O N... on «3.1:? no a. ... 5 om en own—m n90? o re 3 me a 380.88.“ ingot. can. 0‘ finsn.0ms§ 3390.3»? {Lona ~ L x .4393 ugtorhgyxhg N > x 45.20 u 240 5.0 I» 4! >1 >83 N > 1955 22 integer of all indices on the lines is one. This simple arrangement is what makes it so easy to index Laue photographs when using gnomonic projections. It will be remembered that crystal number 30 in.the silectron sheet was rotated about “5° from the average around one axis and about 53° from the average about another axis. ‘Use was now made of this “5° rotation about the x—axis to obtain a series of three symmetric patterns with rotation around one axis as shown in Figure 21. On the left is the two-fold symmetric pattern of Figure 10 except that here it is laid on its side. On the right is the four-fold symmetric pattern of Figure 19, and the rotation about the vertical axis is exactly 90° from.the two-fold pattern. In between these two patterns is a three-fold symn metric pattern formed when the xpray beam is incident normal to the (111) plane of a cubic cell. This pattern is found about 370 from.the two-fold pattern and 53° from the four-fold pattern, which is exactly where we would expect to find it for a cubic crystal lattice. The gnomonic projection of this three-fold pattern is shown in Figure 18. From the gnomonic projection of the two-fold symmetric pattern the spots of the Lane pattern were indexed in Figure 22, and the same was done for the three—fold pattern in Figure 23. OOII . . euo em eI63 e36! ° ' 0031 .I51 .25. ONO-5 .053 .230 . eh» DWI .350 .5”. - '55,, .on one .31 . .I13 .. .61 , _ en» ‘ ‘M .215 3 .03: .no out on] . . , , as i’ . , e"?- .132 “3' 3 .3“ em Oafi . ' . _ .515 ' .103 .q't‘z- out e3l2 _ - m . " °T . 0:215 . '3” .2 e307 - e312 . ."3 - _ "5‘ .237 .2“ .5”. . . __ .3; on :30 -- " " ..23 3 '3‘, .531 .3 ' e251! " -- - .13» Con _ - .720 .151 .311 ’ ."st 0053 .m' .350 .6; 0033 - _ - - - “7 em e25: 023° .3 Om e36? em a . w I . ’ eITo INDEXED LAUE SPOT PATTERN OF SILECTRON CRYSTAL x—mw sees NORMAL TO (too PLANE , in 71': "9'7 e — LLIT' :4 ‘J‘V cam "”2 an: e out [=26 e311 , , _ ‘3'» 'z. .331 ' .35. 'm .x m ’0’: O”: ‘ elt .> ON! 2 . .4 I15 203 l3 .3!» on! .373 .03: can our Qua .2“- .I10 ens. .913 flux» .26] .m 0'11 0233 one .1" e1...I e052. , I52 m, . '”' em 0332 ""3 .032. 0053 QB: «u .3 20 “2' m .‘It .~:. 05'th “'3 0530 CM em .392 .3” .35» .211 '51: us: .25. .93 .512 ezao 053: e223 .‘9/3 an} .325 'OZI e334 e516 .20! “,1 NJEXED LATE SPOT PATTERN or SILEOTRON CRYSTAL x-ssv seAN NosNAL TO «m PL»: *4": elm. 01' L. #5144. I .’,.,L) VII. ADDITIONAL CRYSTAL ANALYSIS The next step after determining the crystal system is to determine if the lattice is simple, face-centered, or body-centered. In the sim. ple geometrical analysis of Figure 10 mentioned in the last chapter, it was determined that the tetragonal cell formed by the intersections of three pairs of mutually perpendicular planes had.the third, or odd, side shorter than the two sides of equal length. Such a tetragonal cell could only be formed in a body-centered cubic lattice, as the tet- ragonal cells formed by the planes of simple cubic lattice or face- centered cubic lattice would.have the third side longer than the two equal sides. This phase of the analysis is usually accomplished on the basis of spots present or spots absent on the Laue photograph. It can be shown that for a simple cubic lattice spots will appear which have any combination of liller indices. For a face—centered cubic lattice only those spots will appear in which the Miller indices are either all odd or all even. For a body-centered cubic lattice only those spots will appear in which the three integers of the Miller index add.up to an even number. Of the 82 spots which appear on the four-fold.Laue photo- graph of Figure 17, only six:Miller indices have integers which add up to an odd number. Perhaps these six spots appear only because sil- ectron is not a pure element but has a few atoms of silicon which are much lighter than the iron atoms. It is likely that from a powder dif- fraction pattern of the silectron some information could be learned 2% about the location of the silicon atoms with respect to the unit cubes. Of course the alloy may be of such a nature that the location of these atoms is entirely at random. ,At any rate such information cannot be obtained from a Laue photograph because it is almost impossible to take due consideration of all the factors which contribute to the density of any one Laue spot. It is often possible, however, to distinguish between two chemical compounds having a similar crystal structure by using Laue photographs. For instance in Figure 1“ some of the spots that are present for K61 are absent for NaCl. Potassium.has about the same atomic weight as chlorine so that it does not make much difference which atom is at any given place in the lattice; however, sodium is much lighter than chlor- ine so that a plane composed entirely of sodium atoms will give a less dense diffraction spot than a plane composed entirely of chlorine atoms. Thus certain families of planes are composed of alternate chlorine and sodium planes and.there is destructive interference of the xpray'bmam in some directions and constructive interference in other directions. Such effect can tell much about the location of the individual atoms. ,Assuming that the silicon atoms do assume an orderly arrangement in the silectron crystal, just what could this arrangement be? The fact that iron has a much greater atomic weight than silicon would.mean that in a sample which contained about three percent silicon by weight, about six percent or one in sixteen of the atoms would be silicon atoms. However, there are two atoms in every unit cube of a bodyacen- tered cubic lattice, and this would give about one silicon atom for 25 every eight unit cubes. The entire cubic lattice would have an orderly arrangement if each of these silicon atoms were shared by the eight unit cubes that come together at any corner of a unit cube in a cubic lattice. This would mean that we could consider the unit cubes to be combined into larger cubes of twice the dimensions, and there would be a silicon atom at the corners of every one of these larger cubes. It would be interesting to determine if this is the actual arrangement. To determine the exact size of the unit cells in a crystal it is necessary to employ one of the monochromatic x—ray methods; however, it is possible to gain a fair estimation of the lattice constants of a crystal from a Laue photograph. When only the first order diffraction by Bragg's law is considered and the angle of incidence is made smaller and smaller, the wave length of the radiation used in the diffraction must become shorter and shorter until finally a point is reached beyond which there is no radiation available to produce diffraction. Thus around the center of the Laue photograph there will be an area where no spots will be formed because the short wave length limit is determined by the potential difference across the x-ray tube. Figure 2A shows two Laue photographs which were taken under ident- ical conditions except that one was exposed for two hours at forty kilovolts and the other was exposed for fifteen minutes at fifty kilo- volts. Several of the intense spots closest to the center of the latter appear only very faintly on the former. These spots must have been formed by radiation which was very close to the short wave length limit for a potential difference of forty kilovolts. The distance of the Figure 24 26 closest of these spots to the center is 2.05 centimeters, and the crys- tal-film distance for this photograph was five centimeters. Using these values the angle of incidence 9 and the short wave length limit Moan be computed and substituted into Bragg's law to allow the determination of the interplanar distance d. Ian 2 9 = @5521: .I-Il 29 = arctan .1+1 = 22° 18' 9 = 11° 9' tan 9 = .196 =1??? sin 6 = .193 2 sin 9 = .386 J; Ann": = W = .309 Angstroms d = $51-6— = %= .798 Angstroms This d, however, is not the length of the side of a unit cube but is only the interplanar distance. Figure 25 shows that when the radiation is incident on the edge of the unit cube, as it is in the present case, the angle 9 is the angle between the diagonals of the unit cube faces and the reflecting planes. It is evident from the geometry of the fig- ure that to obtain the lattice constant a,‘ the interplanar distance d must be multiplied by: 2 cos (1+5Tr + 11:6 97V this gives: a = cos Ego 9. = 1-3;??— = 2.87 Angstroms. CROSS SECTION OF CUBIC LATTICE SHOWING INTERACTION OF X‘RAY BEAM INCIDENT ON EDGE OF UNIT CUBE WITH (3I2) PLANES n; f :-rure fiS 37 This compares very favorably with the value of 2.86 Angstroms which is given for iron at room.temperature. APPENDIX REPAIR AND MAINTENANCE OF X—RAY UNIT 29 REPAIR AND MAINTENANCE OF X-RAY UN IT One of the first requirements of a modern crystallographer is a suitable source of x-radiation. In the present investigation the first phase of the problem was leak hunting in the high vacuum system of a Hilger max X—ray Diffraction Unit (Figure 26) that had not been oper- ated for several months because sufficient vacuum could not be obtained. In the cross section of the vacuum system.pictured in Figure 27 it will be noticed that there are 22 openings, including the four win. dows of the tube, where high vacuum.seals from the atmospheric pressure must be made. The majority of these seals are made with either flat neoprene gaskets or neoprene O-rings, but two seals are metal-to-metal cone joints, and.the windows use flat lead washers and discs of aluminum or beryllium.foil. The magnetic valve at the bottom of the reservoir automatically isolates the system from.the rotary pump then the latter is not running. Through the top of the reservoir a dish of P205 is in. sorted to act as a drying agent for the system. The two discharge tubes are used to indicate the quality of vacuum.in the system; the upper tube is connected into a vacuum.trip relay circuit which disconnects the high voltage, the tube filament, and the diffusion pump if the vacuum in the system drops below a certain predetermined value. The annular recess around the top of the diffusion pwmp is kept filled with the same kind of silicone oil as is used in the pump in order to improve the vacuum- holding qualities of the cone joint. Cold water is circulated behind the nose cap of the target when the tube is in operation in order to carry away the heat produced when the target absorbs the energy of the Figure 26 am mssmfln 2m...m>m §8<> mo Erroumnmmoco t2: zo....o<¢nx amen-.1 case ; fl :5. mSuS>nxPNZQSzI e azam \\\\\\\\\ \. 205w taut—o monk >