{"id":146,"date":"2018-11-30T06:45:43","date_gmt":"2018-11-30T06:45:43","guid":{"rendered":"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=146"},"modified":"2018-12-05T05:35:15","modified_gmt":"2018-12-05T05:35:15","slug":"application-of-x-ray-diffraction-in-the-study-of-crystal-defects-ii","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/chapter\/application-of-x-ray-diffraction-in-the-study-of-crystal-defects-ii\/","title":{"rendered":"Application of X-ray diffraction in the study of crystal defects-II"},"content":{"raw":"
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\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n10.1 Assessment of crystal defects - Experimental techniques of x-ray diffraction.<\/strong>\r\n\r\n \r\n

As explained above x-ray diffraction has the potential of being an exploratory technique for the assessment of various types of defects in crystals. Its penetration through the matter makes it a bulk method as it can lead to investigations into the observation and characterization of defects within the body of the crystal. There are a number of experimental arrangements for recording x-ray topographs. We may describe here two types of experimental techniques that are widely used. These two techniques are:<\/p>\r\n \r\n\r\n1.\u00a0\u00a0 Berg-Barrett technique. Also known as Newkirk technique.\r\n\r\n2.\u00a0\u00a0 Lang technique.\r\n\r\n \r\n\r\nThese two techniques may be described in the sections that follow.\r\n\r\n \r\n\r\n \r\n\r\n10.1.1 Berg \u2500 Barrett (Newkirk) technique.<\/strong>\r\n\r\n \r\n

In 1931 Berg gave an idea about a method wherein x-ray could be used as probes for the study of imperfections in crystals. Barrett further developed the method in 1945. X-ray topography requires point to point correlation between the x-ray intensity in the topograph and the crystalline perfection in the crystal. It means that any single spot in the crystal should be imaged as such in the topograph. That is what is actually meant by one to one correlation between what exists in the crystal and what gets imaged and registered in the topograph. To enable this happen certain experimental conditions are required to be met. Let us describe those conditions which will have to be realized in order to obtain one to one correspondence between what exists in the imperfect region and what is detected in terms of contrast of the topograph.<\/p>\r\n \r\n\r\n\"\"\r\n

Figure 10.1 : Schematic diagram showing the formation of image of point O of the crystal on photographic film at P<\/em><\/strong><\/p>\r\n \r\n

Figure 10.1 shows x-ray source of finite size and diffraction of x-rays is taking place from a point O on the crystal specimen. In order that the Bragg condition for diffraction is satisfied, the angle between the incident x-ray and the normal ON at O should be (90 \u2500 \u03b8B), where \u03b8B is the Bragg angle. All the x-rays that are\u00a0incident and lie within the cone having its axis ON and semi apex angle (90 \u2500 \u03b8B) will satisfy Bragg\u2019s law. Let us consider that the source XS is at a distance L from the crystal and that the diffracted beam from O is registered on a photographic film P which is placed at a distance X from the crystal. XS being a source of finite size the cone intersects the source along a curve XI1<\/sub> XI2<\/sub> XI3<\/sub> on the photographic plate P. The curve XS1<\/sub> XS2<\/sub> XS3<\/sub> is recorded as image which appears as XI1<\/sub> XI2<\/sub> XI3<\/sub> on P. The angle between XS1<\/sub>O and XS3<\/sub>O is equal to the angle between XI1<\/sub>O and XI3<\/sub>O.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nXI1<\/sub> and XI3<\/sub> could be treated as almost straight lines. Same treatment applies to XS1<\/sub> and XS3<\/sub>. Therefore, one may say that:\r\n\r\n \r\n\r\n\u2206XI = \u2206XS. X\/L ,\r\n\r\n \r\n\r\nWhere, \u2206XI = XI1 XI3 and \u2206XS = XS1XS3\r\n\r\n \r\n\r\n\u2206XI\u00a0 should be as small as possible and for that:\r\n\r\n \r\n\r\n(i) the source size \u2206XS should be as small as possible, and\r\n\r\n \r\n\r\n(ii) the distance X from the crystal specimen to the photographic film should be as small as possible and\r\n\r\n \r\n\r\n(iii) The distance L from the source to the crystal specimen should be as large as possible.\r\n\r\n \r\n

It is not possible to meet all these requirements for this experiment. For example, it is not possible to reduce \u2206XS or X to zero or to make L as large as infinity. However, what best can be done is to make compromise with these requirements and meet these conditions as best as one can and as practically feasible. This point is kept in mind while designing the x-ray topography set-up. Use is made of x-ray source of as small size as possible while maintaining the brilliance of the source high. The brilliance is defined as X-ray output per unit area which is required to be kept as high as possible.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 10.2: Experimental arrangement of Berg-Barrett technique in which Bragg diffraction in symmetric geometry is shown<\/em><\/strong><\/p>\r\n \r\n

The experimental arrangement used in Barrett technique is schematically shown in figure 10.2. S is an X-ray source. The crystal to be examined is placed at a distance of about 30 cm from the X-ray source S. The photographic film is positioned very close to the crystal specimen. When the diffracting planes of the crystal are parallel to the crystal surface, it is known as symmetrical Bragg diffraction and in this condition the distance between the crystal specimen and the photographic plate is extremely small, may be hardly a millimetre or so.The source of x-rays is a sealed focus tube and is hardly of the dimensions 1mm x 1mm. Because of the closeness between the specimen and the photographic film, only a few millimetre square area of the crystal specimen can be scanned. The technique is used in Bragg geometry for observation of the crystal surface. However, this technique is not useful in the detection of defects within the volume of the crystal.<\/p>\r\n\"\"\r\n

Figure 10.3 : Experimental set-up of Berg-Barrett method using asymmetrical Bragg reflections; the diffracting planes are not parallel to the surface of the crystal<\/em><\/strong><\/p>\r\n\r\n<\/div>\r\n \r\n

\r\n

Berg-Barrette technique can be used for covering greater areas by using asymmetrical Bragg condition. This happens when the lattice planes are not parallel to the surface as shown in figure 10.3. In this experiment diffraction is allowed to happen by lattice planes which are not parallel to the surface. When these lattice planes are aligned such as to satisfy the Bragg condition, the incident x-ray beam is set at small angle with the crystal surface but the diffracted x-ray beam are at large angle with it.<\/p>\r\n \r\n

The equipment used in this experiment is simple and is widely used for rapid assessment of perfection of single crystals. However, it has certain limitations too which were overcome by improving the experimental technique by Lang. The technique introduced by Lang is described in the section that follows:<\/p>\r\n \r\n\r\n10.1.2 Lang Technique of x-ray topography.<\/strong>\r\n\r\n \r\n\r\n\"\"\r\n

Figure 10.4: Schematic diagram of an experimental set-up of Lang technique<\/strong><\/p>\r\n \r\n

Figure 10.4 is a schematic diagram of an experimental set-up, popularly known by the inventor\u2019s name as Lang technique. It uses a micro-focus x-ray source to produce x-ray beam for probing perfection of a given crystal. The x-ray beam from this source is collimated by a 50 cm. long collimator. The emerging x-ray beam from the collimator has a divergence of nearly 40\u2070 in the horizontal plane. In order to set the crystal specimen in the correct position there is a provision for the required orientation of lattice planes. The crystal can be suitably rotated around two axes. One of the axes is in the vertical direction and the other is in the horizontal direction. The turntable enables rotation of the specimen about the vertical axis. The vertical circle goniometer enables rotation around a horizontal axis. So, by appropriate positioning and orienting the specimen one can achieve setting of the lattice planes for Bragg diffraction. To receive the x-rays as diffracted by the lattice planes of the crystal specimen at an angle 2\u03b8B with respect to the direct beam, the x-ray detector is set as shown in the figure. The area irradiated by the x-rays sends out a diffracted beam which gets recorded as a thin line. In order to record diffracted x-rays from the whole volume of the crystal specimen, the same (crystal specimen) is traversed across the x-ray beam. In order to ensure that the diffracted x-rays are properly recorded, the\u00a0photographic film is rigidly coupled to the crystal specimen. It enables it to move with the specimen while new regions of the crystal specimen and the photographic film are presented for each setting. A slit is provided between the photographic film \/ plate and the crystal specimen so that the direct x -ray beam does not fall on the photographic film\/plate. The positioning of the slit is adjusted in a way that it permits the diffracted x-ray beam to pass through but stop the direct x-ray beam. This technique allows scanning of the whole volume of the crystal specimen and so has the potential of exposing defects\/imperfections in the entire volume of the crystal. The whole volume topographs recorded by this technique are called projection topographs.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

In this technique, it is desirable that a diffraction curve is first obtained. This diffraction curve is also known as rocking curve. In this curve , intensity of the diffracted x-ray beam say I is plotted against angle of orientation of the crystal specimen ( \u03b8 ) around the Bragg angle \u03b8B setting , keeping the slit of the x-ray detector suitably wide. The shape of the rocking curve or diffraction curve is indicative of the overall perfection of the crystal specimen. Single crystals which are perfect give well \u2013 resolved peaks due to wavelength components of the characteristic radiation. However, if the curve of a crystal specimen shows several peaks separated by a few minutes of arc from each other, it can be attributed to be as a result of low angle grain boundaries.<\/p>\r\n \r\n\r\nAdvances have been made in recent years to improve the efficiency and sensitivity of X-ray topographic and experimental techniques. Using high brilliance and a high power source the exposure time is greatly reduced. X-ray generators of the rotating anode type have been used with great success. This type of generator with electrical power of 30 KW is available. Synchrotron radiations at a wavelength of 1\u00c5 have the potential for a great future in the field of x-ray diffraction topography. This radiation is highly collimated with divergence of around 10\u25004 radians and is able to irradiate large areas of the crystal specimen at a time. Use of synchrotron radiation has several advantages which may be summarily put as under: (i) It is a well collimated beam\r\n\r\n \r\n\r\n(ii)\u00a0 It offers broad spectrum of wavelengths in the synchrotron radiation which enables one to record\r\n\r\nseveral reflections at a time as in Laue experiment. All reflections can be recorded in just few seconds.\r\n\r\n(iii) Photographic film\/plate need not be placed very close to the crystal specimen.\r\n\r\n \r\n

It is a criterion for good resolution in ordinary Laue technique to keep the two (photographic film and crystal specimen) very close. In the case of synchrotron radiation photographic film placed at a distance of 10 cm. from the crystal specimen without affecting the resolution.\u00a0<\/span>may be<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n10.2 Example of topographs and diffraction curves.<\/strong>\r\n\r\n \r\n

X-ray topography is applied in transmission or reflection geometry in order to make assessment of perfection and investigate defects in crystals. Generally, diffraction curve or a rocking curve is also recorded before taking topographs. X-ray topography is a preferred technique for studying the crystal defects in bulk. The advantages of this method are that it is relatively simple to adjust, its resolution can be good, it does not require very expensive equipment and exposure times are short. What makes the technique more interesting is the fact that the distribution of defects in a section of large crystal can be investigated without having to cut the crystal.<\/p>\r\n \r\n

The resolved defects generally observed by topography are extended defects like dislocations, inclusions or precipitates, surface defects, long range strains, growth striations etc.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

We describe here a few examples that would reveal the potential of x-ray diffraction topography in characterizing the crystal defects and also in studying sources that may generate defects in otherwise perfect crystals. The topographs are recorded by a Lang camera, using a collimated x-ray beam with a horizontal divergence of about 03 min.of arc, the geometrical resolution of the apparatus having been estimated to about 03 \u00b5m. The topographs were recorded in a reflection (RXRT) and transmission (TXRT) scanning geometry. In certain cases, the Agk\u03b1 radiation was employed after having thinned down the samples to about 150 \u00b5m. In the other cases, Cuk\u03b1 x-ray wavelength was used as the probe of investigation.<\/p>\r\n\"\"\r\n

Figure 10.5: RXRT of Strontium Hexaferrite crystal exhibiting near perfection but for some absorbing particles from the surface<\/em><\/strong><\/p>\r\n \r\n

Figure 10.5 is a topograph of strontium hexaferrite (SrFe12O19) taken in reflection (RXRT) scanning geometry with 0.0.18 reflection. The crystal was grown using fluxed-melt technique. The sample is nearly perfect but for some small white contrast areas and some cleavage like patterns seen in the upper right hand region, which may be due to presence of some absorbing particles on the sample surface.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 10.6 : TXRT showing precipitate like defects inside Strontium Hexaferrite crystal<\/strong><\/p>\r\n \r\n\r\nFigure 10.6 is a topograph of another crystal taken in transmission (TXRT) scanning geometry using 3.0.0.reflection. The topograph shows precipitate like defects inside the crystal which could be flux inclusions. No growth bands are revealed.\r\n\r\n\"\"\r\n

Figure 10.7: RXRT showing practically defect free Strontium Hexaferrite crystal<\/strong><\/p>\r\n \r\n

Figure 10.7 is a topograph taken in RXRT geometry using 0.0.18 reflection which reveals that the sample is a reasonably good single crystal; practically defect free within 70% of its area. One observes fringes in the\u00a0bottom part which probably are stress related features. The white contrast area could be due to some misoriented grains, which most probably could be flux particles or due to some surface damage. Figure 10.8 is RXRT scanned topograph employing 0.0.18 reflection. One finds growth bands, a few dislocations on the basal plane, and a large misoriented region corresponding to a grain emerging from a sample surface. The large misoriented grain attached with the main crystal indicates a coalescence phenomenon due to crystals getting nucleated at sites near to each other during flux growth.<\/span><\/p>\r\n \r\n

In order to illustrate the potential of x-ray diffraction for characterization of crystal defects and assessment of perfection of a given crystal, we may take up an example of recording diffraction curve of a crystal of strontium hexaferrite (SrFe12 O19) grown by fluxed-melt technique. In this experiment direct comparison of the diffraction patterns of irradiated and unirradiated crystal (the same sample) was made.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 10.8: RXRT of SrGa<\/strong>5<\/strong>In<\/strong>0.8<\/strong>Fe<\/strong>6.2<\/strong>O<\/strong>19<\/strong> (Gallium Indium substituted Strontium Hexaferrite) crystal revealing growth bands, dislocations and a misoriented grain<\/strong><\/p>\r\n \r\n

First, the x-ray diffraction pattern of a crystal specimen was recorded by using high resolution x-ray diffractometer facility available at the National Physical Laboratory, New Delhi. The x-ray diffraction curve of strontium hexaferrite crystal with (008) diffraction planes is shown in figure 10.9<\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

Figure 10.9: X-ray diffraction curve of Strontium Hexaferrite crystal showing sharp and smooth peak suggestive of good quality crystal<\/em><\/strong><\/p>\r\n \r\n

It shows sharp and smooth peak with half width \u02dc<\/strong> 5 arc sec. which is suggestive of good quality crystal. The crystal may be declared as nearly perfect. The same crystal was irradiated at room temperature with 50 MeV Li3+ ions beam delivered by 15 UD Pelletron Accellerator (facility available at Nuclear Science Centre, New Delhi) at a fluence of 1 X 1014 ions cm-2 and its diffraction curve was recorded as is shown in figure 10.10. It is interesting to find sharp decline in the diffracted x-ray intensity and smoothening of the curve disappears. Diffused peaks also appear. The curve has developed kinks at intervals. The variation in the shape of the diffracted curve suggests a possibility of low angle grain boundaries in the irradiated crystal of strontium Hexaferrite. Half width increases from 5.6 arc sec. in the case of pristine (un \u2013 irradiated) to 31 arc sec. in the case of irradiated crystal.<\/p>\r\n \r\n\r\nThe results suggest presence of point\/clusters of defects causing amorphization in the crystal on irradiation.\r\n\r\n<\/div>\r\n\"\"\r\n

Figure 10.10: Diffraction curve of irradiated Strontium Hexaferrite crystal showing kinks and half width of 31 arc sec , suggestive of creation of defects on irradiation<\/em><\/strong><\/p>\r\n \r\n

The direct comparison of diffraction curves of un-irradiated (pristine) and irradiated crystal is best done by figure 10.11. The diffraction curve gets almost flattened on irradiation. Disappearance of smoothness in the irradiated crystal implies that a stress\/strain is generated in the Hexaferrite crystal by swift heavy ion irradiation. This experiment not only offers an example of effectiveness of x-ray diffraction in delineating imperfections in crystals but also its role as a powerful tool to study radiation damage in crystalline solids. There are several problems on assessment, characterization and defects related investigations which can be undertaken by employing x-ray diffraction topography and rocking curve as a powerful technique.<\/p>\r\n \r\n\r\n\"\"\r\n

Figure 10.11: A plot showing comparison between the x-ray diffraction curves o un-irradiated and Irradiated strontium Hexaferrite crystal<\/em><\/strong><\/p>\r\n \r\n\r\n \r\n\r\n\r\n\r\n
you can view video on Application of X-ray diffraction in the study of crystal defects-II<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nReferences.<\/strong>\r\n
    \r\n \t
  1. Newkirk, J.B. and Wernick,J.H. (Eds) \u201c Direct observation of imperfections in crystals,Interscience., N.Y.,1962.<\/li>\r\n \t
  2. Tanner,B.K. \u201c X-ray diffraction topography\u201d, Pergamon, N.Y. ,1976.<\/li>\r\n \t
  3. Nuffield, E.W. \u201c x-ray diffraction \u201c, Wiley, N.Y.1966.<\/li>\r\n \t
  4. Verma, A.R., Srivastava, O.N.\u201d Crystallography for Solid State Physics\u201d, Wiley Eastern Ltd.,N.Delhi 1982.<\/li>\r\n \t
  5. Lal, K. \u201c X-ray diffraction topography\u201d,J.Phys.Edu.,4 (1-5)<\/strong> 1976.<\/span><\/li>\r\n \t
  6. Kittel,C. \u201c Introduction to Solid state physics\u201d.IV edition,Wiley,N.Y.,1971.<\/li>\r\n \t
  7. Sherwood, J.N.\u201d Lattice Defects and the Plasticity of Plastic Crystals\u201d, The Plastically Crystalline State ( Sherwood , J.N. ed.) , Wiley, London.1979.<\/li>\r\n \t
  8. Hooper,R.M. and Sherwood,J.N. \u201c Dislocations in Pyrene crystals \u201c, J.Chem.Soc. Faraday, 1,72, 2872.<\/li>\r\n \t
  9. Azaroff,L.V. \u201c Elements of x-ray Crystallography\u201d, MacGraw-Hill, N.Y.,1968<\/li>\r\n \t
  10. Raina,U.,Bhat.S, ,FranzosiP,Licci,F., Kotru,P.N. \u201c X-Ray Topographic Assessment of Defects in Pure & substituted Hexaferrite crystals\u201d, Krist. Und Tech. 31<\/strong> , 1995.<\/span><\/li>\r\n \t
  11. Raina,U.: Ph.D. Thesis, Jammu university,1999.<\/li>\r\n<\/ol>\r\n \r\n\r\n \r\n\r\nGeneral References For detailed Information Particularly Related To Defects And Their Revelations<\/strong>\r\n\r\n \r\n\r\n1.\u00a0 Lang,A.R. \u201c Recent Application of X-Ray Topography, Modern Diffraction And Imaging\r\n\r\n2.\u00a0\u00a0\u00a0 Lang, A.R.\u201dThe Properties and Observation of Dislocations, Crystal Growth : An Introduction (Hartman,P.\r\n\r\nEd.),North Holland, London.,1973.\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0 Holt, D.B. \u201c Assessment of Crystalline Perfection \u201c in \u201c Crystal Growth\u201d (Ed. Brian R. Pamplin),\r\n\r\nInternational Series in the Science of the Solid State,Vol.16,Pergamon press, II edition,1980.\r\n\r\n4. Authier,A. in \u201c Current Topics In Materials Science\u201d vol II( E.Kaldis&H.J.Scheel ,Eds.)North\u00a0<\/span>Holland <\/span>\r\n\r\nPublishing Co,, Amsterdam, 1977.<\/span>\r\n\r\n5. Lang, A.R. \u201c Advances in X-Ray Analysis\u201d,<\/span>10,<\/strong> Plenum Press ,N.Y., 1967.<\/span>\r\n\r\n6. Laudise,R.A. in \u201c Crystal Growth & Characterization\u201d eds.R.Ueda and J.B. Mullin, North<\/span>\r\n\r\n7. Amelinckx,S,\u201d Surface Methods,Solid State Physics, <\/span>6<\/strong>(supplement), Seitz,F. ed.,Academic Press N.Y.,1969.<\/span>\r\n\r\n<\/div>\r\n ","rendered":"
    \"epgp<\/a>
    \n<\/span><\/div>\n
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     <\/p>\n

    10.1 Assessment of crystal defects – Experimental techniques of x-ray diffraction.<\/strong><\/p>\n

     <\/p>\n

    As explained above x-ray diffraction has the potential of being an exploratory technique for the assessment of various types of defects in crystals. Its penetration through the matter makes it a bulk method as it can lead to investigations into the observation and characterization of defects within the body of the crystal. There are a number of experimental arrangements for recording x-ray topographs. We may describe here two types of experimental techniques that are widely used. These two techniques are:<\/p>\n

     <\/p>\n

    1.\u00a0\u00a0 Berg-Barrett technique. Also known as Newkirk technique.<\/p>\n

    2.\u00a0\u00a0 Lang technique.<\/p>\n

     <\/p>\n

    These two techniques may be described in the sections that follow.<\/p>\n

     <\/p>\n

     <\/p>\n

    10.1.1 Berg \u2500 Barrett (Newkirk) technique.<\/strong><\/p>\n

     <\/p>\n

    In 1931 Berg gave an idea about a method wherein x-ray could be used as probes for the study of imperfections in crystals. Barrett further developed the method in 1945. X-ray topography requires point to point correlation between the x-ray intensity in the topograph and the crystalline perfection in the crystal. It means that any single spot in the crystal should be imaged as such in the topograph. That is what is actually meant by one to one correlation between what exists in the crystal and what gets imaged and registered in the topograph. To enable this happen certain experimental conditions are required to be met. Let us describe those conditions which will have to be realized in order to obtain one to one correspondence between what exists in the imperfect region and what is detected in terms of contrast of the topograph.<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    Figure 10.1 : Schematic diagram showing the formation of image of point O of the crystal on photographic film at P<\/em><\/strong><\/p>\n

     <\/p>\n

    Figure 10.1 shows x-ray source of finite size and diffraction of x-rays is taking place from a point O on the crystal specimen. In order that the Bragg condition for diffraction is satisfied, the angle between the incident x-ray and the normal ON at O should be (90 \u2500 \u03b8B), where \u03b8B is the Bragg angle. All the x-rays that are\u00a0incident and lie within the cone having its axis ON and semi apex angle (90 \u2500 \u03b8B) will satisfy Bragg\u2019s law. Let us consider that the source XS is at a distance L from the crystal and that the diffracted beam from O is registered on a photographic film P which is placed at a distance X from the crystal. XS being a source of finite size the cone intersects the source along a curve XI1<\/sub> XI2<\/sub> XI3<\/sub> on the photographic plate P. The curve XS1<\/sub> XS2<\/sub> XS3<\/sub> is recorded as image which appears as XI1<\/sub> XI2<\/sub> XI3<\/sub> on P. The angle between XS1<\/sub>O and XS3<\/sub>O is equal to the angle between XI1<\/sub>O and XI3<\/sub>O.<\/span><\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    XI1<\/sub> and XI3<\/sub> could be treated as almost straight lines. Same treatment applies to XS1<\/sub> and XS3<\/sub>. Therefore, one may say that:<\/p>\n

     <\/p>\n

    \u2206XI = \u2206XS. X\/L ,<\/p>\n

     <\/p>\n

    Where, \u2206XI = XI1 XI3 and \u2206XS = XS1XS3<\/p>\n

     <\/p>\n

    \u2206XI\u00a0 should be as small as possible and for that:<\/p>\n

     <\/p>\n

    (i) the source size \u2206XS should be as small as possible, and<\/p>\n

     <\/p>\n

    (ii) the distance X from the crystal specimen to the photographic film should be as small as possible and<\/p>\n

     <\/p>\n

    (iii) The distance L from the source to the crystal specimen should be as large as possible.<\/p>\n

     <\/p>\n

    It is not possible to meet all these requirements for this experiment. For example, it is not possible to reduce \u2206XS or X to zero or to make L as large as infinity. However, what best can be done is to make compromise with these requirements and meet these conditions as best as one can and as practically feasible. This point is kept in mind while designing the x-ray topography set-up. Use is made of x-ray source of as small size as possible while maintaining the brilliance of the source high. The brilliance is defined as X-ray output per unit area which is required to be kept as high as possible.<\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    Figure 10.2: Experimental arrangement of Berg-Barrett technique in which Bragg diffraction in symmetric geometry is shown<\/em><\/strong><\/p>\n

     <\/p>\n

    The experimental arrangement used in Barrett technique is schematically shown in figure 10.2. S is an X-ray source. The crystal to be examined is placed at a distance of about 30 cm from the X-ray source S. The photographic film is positioned very close to the crystal specimen. When the diffracting planes of the crystal are parallel to the crystal surface, it is known as symmetrical Bragg diffraction and in this condition the distance between the crystal specimen and the photographic plate is extremely small, may be hardly a millimetre or so.The source of x-rays is a sealed focus tube and is hardly of the dimensions 1mm x 1mm. Because of the closeness between the specimen and the photographic film, only a few millimetre square area of the crystal specimen can be scanned. The technique is used in Bragg geometry for observation of the crystal surface. However, this technique is not useful in the detection of defects within the volume of the crystal.<\/p>\n

    \"\"<\/p>\n

    Figure 10.3 : Experimental set-up of Berg-Barrett method using asymmetrical Bragg reflections; the diffracting planes are not parallel to the surface of the crystal<\/em><\/strong><\/p>\n<\/div>\n

     <\/p>\n

    \n

    Berg-Barrette technique can be used for covering greater areas by using asymmetrical Bragg condition. This happens when the lattice planes are not parallel to the surface as shown in figure 10.3. In this experiment diffraction is allowed to happen by lattice planes which are not parallel to the surface. When these lattice planes are aligned such as to satisfy the Bragg condition, the incident x-ray beam is set at small angle with the crystal surface but the diffracted x-ray beam are at large angle with it.<\/p>\n

     <\/p>\n

    The equipment used in this experiment is simple and is widely used for rapid assessment of perfection of single crystals. However, it has certain limitations too which were overcome by improving the experimental technique by Lang. The technique introduced by Lang is described in the section that follows:<\/p>\n

     <\/p>\n

    10.1.2 Lang Technique of x-ray topography.<\/strong><\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    Figure 10.4: Schematic diagram of an experimental set-up of Lang technique<\/strong><\/p>\n

     <\/p>\n

    Figure 10.4 is a schematic diagram of an experimental set-up, popularly known by the inventor\u2019s name as Lang technique. It uses a micro-focus x-ray source to produce x-ray beam for probing perfection of a given crystal. The x-ray beam from this source is collimated by a 50 cm. long collimator. The emerging x-ray beam from the collimator has a divergence of nearly 40\u2070 in the horizontal plane. In order to set the crystal specimen in the correct position there is a provision for the required orientation of lattice planes. The crystal can be suitably rotated around two axes. One of the axes is in the vertical direction and the other is in the horizontal direction. The turntable enables rotation of the specimen about the vertical axis. The vertical circle goniometer enables rotation around a horizontal axis. So, by appropriate positioning and orienting the specimen one can achieve setting of the lattice planes for Bragg diffraction. To receive the x-rays as diffracted by the lattice planes of the crystal specimen at an angle 2\u03b8B with respect to the direct beam, the x-ray detector is set as shown in the figure. The area irradiated by the x-rays sends out a diffracted beam which gets recorded as a thin line. In order to record diffracted x-rays from the whole volume of the crystal specimen, the same (crystal specimen) is traversed across the x-ray beam. In order to ensure that the diffracted x-rays are properly recorded, the\u00a0photographic film is rigidly coupled to the crystal specimen. It enables it to move with the specimen while new regions of the crystal specimen and the photographic film are presented for each setting. A slit is provided between the photographic film \/ plate and the crystal specimen so that the direct x -ray beam does not fall on the photographic film\/plate. The positioning of the slit is adjusted in a way that it permits the diffracted x-ray beam to pass through but stop the direct x-ray beam. This technique allows scanning of the whole volume of the crystal specimen and so has the potential of exposing defects\/imperfections in the entire volume of the crystal. The whole volume topographs recorded by this technique are called projection topographs.<\/span><\/p>\n<\/div>\n

    \n

     <\/p>\n

    In this technique, it is desirable that a diffraction curve is first obtained. This diffraction curve is also known as rocking curve. In this curve , intensity of the diffracted x-ray beam say I is plotted against angle of orientation of the crystal specimen ( \u03b8 ) around the Bragg angle \u03b8B setting , keeping the slit of the x-ray detector suitably wide. The shape of the rocking curve or diffraction curve is indicative of the overall perfection of the crystal specimen. Single crystals which are perfect give well \u2013 resolved peaks due to wavelength components of the characteristic radiation. However, if the curve of a crystal specimen shows several peaks separated by a few minutes of arc from each other, it can be attributed to be as a result of low angle grain boundaries.<\/p>\n

     <\/p>\n

    Advances have been made in recent years to improve the efficiency and sensitivity of X-ray topographic and experimental techniques. Using high brilliance and a high power source the exposure time is greatly reduced. X-ray generators of the rotating anode type have been used with great success. This type of generator with electrical power of 30 KW is available. Synchrotron radiations at a wavelength of 1\u00c5 have the potential for a great future in the field of x-ray diffraction topography. This radiation is highly collimated with divergence of around 10\u25004 radians and is able to irradiate large areas of the crystal specimen at a time. Use of synchrotron radiation has several advantages which may be summarily put as under: (i) It is a well collimated beam<\/p>\n

     <\/p>\n

    (ii)\u00a0 It offers broad spectrum of wavelengths in the synchrotron radiation which enables one to record<\/p>\n

    several reflections at a time as in Laue experiment. All reflections can be recorded in just few seconds.<\/p>\n

    (iii) Photographic film\/plate need not be placed very close to the crystal specimen.<\/p>\n

     <\/p>\n

    It is a criterion for good resolution in ordinary Laue technique to keep the two (photographic film and crystal specimen) very close. In the case of synchrotron radiation photographic film placed at a distance of 10 cm. from the crystal specimen without affecting the resolution.\u00a0<\/span>may be<\/span><\/p>\n<\/div>\n

    \n

     <\/p>\n

    10.2 Example of topographs and diffraction curves.<\/strong><\/p>\n

     <\/p>\n

    X-ray topography is applied in transmission or reflection geometry in order to make assessment of perfection and investigate defects in crystals. Generally, diffraction curve or a rocking curve is also recorded before taking topographs. X-ray topography is a preferred technique for studying the crystal defects in bulk. The advantages of this method are that it is relatively simple to adjust, its resolution can be good, it does not require very expensive equipment and exposure times are short. What makes the technique more interesting is the fact that the distribution of defects in a section of large crystal can be investigated without having to cut the crystal.<\/p>\n

     <\/p>\n

    The resolved defects generally observed by topography are extended defects like dislocations, inclusions or precipitates, surface defects, long range strains, growth striations etc.<\/span><\/p>\n<\/div>\n

    \n

     <\/p>\n

    We describe here a few examples that would reveal the potential of x-ray diffraction topography in characterizing the crystal defects and also in studying sources that may generate defects in otherwise perfect crystals. The topographs are recorded by a Lang camera, using a collimated x-ray beam with a horizontal divergence of about 03 min.of arc, the geometrical resolution of the apparatus having been estimated to about 03 \u00b5m. The topographs were recorded in a reflection (RXRT) and transmission (TXRT) scanning geometry. In certain cases, the Agk\u03b1 radiation was employed after having thinned down the samples to about 150 \u00b5m. In the other cases, Cuk\u03b1 x-ray wavelength was used as the probe of investigation.<\/p>\n

    \"\"<\/p>\n

    Figure 10.5: RXRT of Strontium Hexaferrite crystal exhibiting near perfection but for some absorbing particles from the surface<\/em><\/strong><\/p>\n

     <\/p>\n

    Figure 10.5 is a topograph of strontium hexaferrite (SrFe12O19) taken in reflection (RXRT) scanning geometry with 0.0.18 reflection. The crystal was grown using fluxed-melt technique. The sample is nearly perfect but for some small white contrast areas and some cleavage like patterns seen in the upper right hand region, which may be due to presence of some absorbing particles on the sample surface.<\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    Figure 10.6 : TXRT showing precipitate like defects inside Strontium Hexaferrite crystal<\/strong><\/p>\n

     <\/p>\n

    Figure 10.6 is a topograph of another crystal taken in transmission (TXRT) scanning geometry using 3.0.0.reflection. The topograph shows precipitate like defects inside the crystal which could be flux inclusions. No growth bands are revealed.<\/p>\n

    \"\"<\/p>\n

    Figure 10.7: RXRT showing practically defect free Strontium Hexaferrite crystal<\/strong><\/p>\n

     <\/p>\n

    Figure 10.7 is a topograph taken in RXRT geometry using 0.0.18 reflection which reveals that the sample is a reasonably good single crystal; practically defect free within 70% of its area. One observes fringes in the\u00a0bottom part which probably are stress related features. The white contrast area could be due to some misoriented grains, which most probably could be flux particles or due to some surface damage. Figure 10.8 is RXRT scanned topograph employing 0.0.18 reflection. One finds growth bands, a few dislocations on the basal plane, and a large misoriented region corresponding to a grain emerging from a sample surface. The large misoriented grain attached with the main crystal indicates a coalescence phenomenon due to crystals getting nucleated at sites near to each other during flux growth.<\/span><\/p>\n

     <\/p>\n

    In order to illustrate the potential of x-ray diffraction for characterization of crystal defects and assessment of perfection of a given crystal, we may take up an example of recording diffraction curve of a crystal of strontium hexaferrite (SrFe12 O19) grown by fluxed-melt technique. In this experiment direct comparison of the diffraction patterns of irradiated and unirradiated crystal (the same sample) was made.<\/span><\/p>\n<\/div>\n

    \n

    \"\"<\/p>\n

    Figure 10.8: RXRT of SrGa<\/strong>5<\/strong>In<\/strong>0.8<\/strong>Fe<\/strong>6.2<\/strong>O<\/strong>19<\/strong> (Gallium Indium substituted Strontium Hexaferrite) crystal revealing growth bands, dislocations and a misoriented grain<\/strong><\/p>\n

     <\/p>\n

    First, the x-ray diffraction pattern of a crystal specimen was recorded by using high resolution x-ray diffractometer facility available at the National Physical Laboratory, New Delhi. The x-ray diffraction curve of strontium hexaferrite crystal with (008) diffraction planes is shown in figure 10.9<\/p>\n

     <\/p>\n

    \"\"<\/p>\n<\/div>\n

    \n

    Figure 10.9: X-ray diffraction curve of Strontium Hexaferrite crystal showing sharp and smooth peak suggestive of good quality crystal<\/em><\/strong><\/p>\n

     <\/p>\n

    It shows sharp and smooth peak with half width \u02dc<\/strong> 5 arc sec. which is suggestive of good quality crystal. The crystal may be declared as nearly perfect. The same crystal was irradiated at room temperature with 50 MeV Li3+ ions beam delivered by 15 UD Pelletron Accellerator (facility available at Nuclear Science Centre, New Delhi) at a fluence of 1 X 1014 ions cm-2 and its diffraction curve was recorded as is shown in figure 10.10. It is interesting to find sharp decline in the diffracted x-ray intensity and smoothening of the curve disappears. Diffused peaks also appear. The curve has developed kinks at intervals. The variation in the shape of the diffracted curve suggests a possibility of low angle grain boundaries in the irradiated crystal of strontium Hexaferrite. Half width increases from 5.6 arc sec. in the case of pristine (un \u2013 irradiated) to 31 arc sec. in the case of irradiated crystal.<\/p>\n

     <\/p>\n

    The results suggest presence of point\/clusters of defects causing amorphization in the crystal on irradiation.<\/p>\n<\/div>\n

    \"\"<\/p>\n

    Figure 10.10: Diffraction curve of irradiated Strontium Hexaferrite crystal showing kinks and half width of 31 arc sec , suggestive of creation of defects on irradiation<\/em><\/strong><\/p>\n

     <\/p>\n

    The direct comparison of diffraction curves of un-irradiated (pristine) and irradiated crystal is best done by figure 10.11. The diffraction curve gets almost flattened on irradiation. Disappearance of smoothness in the irradiated crystal implies that a stress\/strain is generated in the Hexaferrite crystal by swift heavy ion irradiation. This experiment not only offers an example of effectiveness of x-ray diffraction in delineating imperfections in crystals but also its role as a powerful tool to study radiation damage in crystalline solids. There are several problems on assessment, characterization and defects related investigations which can be undertaken by employing x-ray diffraction topography and rocking curve as a powerful technique.<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

    Figure 10.11: A plot showing comparison between the x-ray diffraction curves o un-irradiated and Irradiated strontium Hexaferrite crystal<\/em><\/strong><\/p>\n

     <\/p>\n

     <\/p>\n\n\n\n
    you can view video on Application of X-ray diffraction in the study of crystal defects-II<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    References.<\/strong><\/p>\n

      \n
    1. Newkirk, J.B. and Wernick,J.H. (Eds) \u201c Direct observation of imperfections in crystals,Interscience., N.Y.,1962.<\/li>\n
    2. Tanner,B.K. \u201c X-ray diffraction topography\u201d, Pergamon, N.Y. ,1976.<\/li>\n
    3. Nuffield, E.W. \u201c x-ray diffraction \u201c, Wiley, N.Y.1966.<\/li>\n
    4. Verma, A.R., Srivastava, O.N.\u201d Crystallography for Solid State Physics\u201d, Wiley Eastern Ltd.,N.Delhi 1982.<\/li>\n
    5. Lal, K. \u201c X-ray diffraction topography\u201d,J.Phys.Edu.,4 (1-5)<\/strong> 1976.<\/span><\/li>\n
    6. Kittel,C. \u201c Introduction to Solid state physics\u201d.IV edition,Wiley,N.Y.,1971.<\/li>\n
    7. Sherwood, J.N.\u201d Lattice Defects and the Plasticity of Plastic Crystals\u201d, The Plastically Crystalline State ( Sherwood , J.N. ed.) , Wiley, London.1979.<\/li>\n
    8. Hooper,R.M. and Sherwood,J.N. \u201c Dislocations in Pyrene crystals \u201c, J.Chem.Soc. Faraday, 1,72, 2872.<\/li>\n
    9. Azaroff,L.V. \u201c Elements of x-ray Crystallography\u201d, MacGraw-Hill, N.Y.,1968<\/li>\n
    10. Raina,U.,Bhat.S, ,FranzosiP,Licci,F., Kotru,P.N. \u201c X-Ray Topographic Assessment of Defects in Pure & substituted Hexaferrite crystals\u201d, Krist. Und Tech. 31<\/strong> , 1995.<\/span><\/li>\n
    11. Raina,U.: Ph.D. Thesis, Jammu university,1999.<\/li>\n<\/ol>\n

       <\/p>\n

       <\/p>\n

      General References For detailed Information Particularly Related To Defects And Their Revelations<\/strong><\/p>\n

       <\/p>\n

      1.\u00a0 Lang,A.R. \u201c Recent Application of X-Ray Topography, Modern Diffraction And Imaging<\/p>\n

      2.\u00a0\u00a0\u00a0 Lang, A.R.\u201dThe Properties and Observation of Dislocations, Crystal Growth : An Introduction (Hartman,P.<\/p>\n

      Ed.),North Holland, London.,1973.<\/p>\n

      3.\u00a0\u00a0\u00a0\u00a0 Holt, D.B. \u201c Assessment of Crystalline Perfection \u201c in \u201c Crystal Growth\u201d (Ed. Brian R. Pamplin),<\/p>\n

      International Series in the Science of the Solid State,Vol.16,Pergamon press, II edition,1980.<\/p>\n

      4. Authier,A. in \u201c Current Topics In Materials Science\u201d vol II( E.Kaldis&H.J.Scheel ,Eds.)North\u00a0<\/span>Holland <\/span><\/p>\n

      Publishing Co,, Amsterdam, 1977.<\/span><\/p>\n

      5. Lang, A.R. \u201c Advances in X-Ray Analysis\u201d,<\/span>10,<\/strong> Plenum Press ,N.Y., 1967.<\/span><\/p>\n

      6. Laudise,R.A. in \u201c Crystal Growth & Characterization\u201d eds.R.Ueda and J.B. Mullin, North<\/span><\/p>\n

      7. Amelinckx,S,\u201d Surface Methods,Solid State Physics, <\/span>6<\/strong>(supplement), Seitz,F. ed.,Academic Press N.Y.,1969.<\/span><\/p>\n<\/div>\n

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