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12, , ., , d23 V(O 82)2 (0.94)2 32, (07512, 22 014, , =, , d123, , ., , As, , 4, , 692, , 4.692"" =021 nm, planes (246) are double, , Miller indices of the, , d246, , 230.21, , 2, , 2, , R RO, 1., , =, , A, , crystal plane, , has, , = 0-11 nm, , PROBLEMS, , intercepts on, , the three, , indices of the face ?, 2., , 487, , of the (123) planes,, , FOR, , axes, , PRACTICE, , of the crystal, , A face makes intercepts 2a and 3b on the X-axes, and Y-axes, What are the Miller indices of the face ?, , as, , a,b andc. What, , the Miller, , are, , Ans. 9, 4, 51, , respectively, , and does, , not cut the, , Z-axis, , at, , all, , Ans. 3, 2, 01, , (Hint. Refer to Example (ii), page 301), 3., , 3/15, , +4 527 +16, , Calculate the separation of (a) (1, 2, 3) planes and (b) (3, 6, 9) planes of an orthorombic cell with a = 50, pm, b= 100 pm and c= 150 pm., [Ans. 29 pm, 9.7 pm, , 3.11. CRYSTAL SYSTEMS, , single point space, If the various elements of symmetry are distributed theoretically, found that 32 different arrangements are possible. These different, passing through this point), it is, are called crystal classes or, around, , (i.e.., , in, , a, , of symmetry about a single point, have the same axial ratios and, many point groups, Further,, or, groups., point, and all, crystallographic groups, b, : b: c=1:1:1, i.e., a, a, and, B=y=90°, out, =, o, have, of 32, into seven, angles, e.g., five point groups, the 32 point groups have been grouped, this, In, way,, cubic, shape., upon the, these five correspond to the, external geometry of a crystal depends, the, Since, axial ratios and angles., they are, external geometry of the crystal,, types on the basis of, the, basis, the, of, that on, names of these systems, axial ratios and angles, it implies, systems. The, , arrangements, , of the, , elenents, , =, , =c, , called crystallographic, , groups, , which, the number of point, axes of symmetry,, classified into s e v e n types, maximum planes and, chracteristics,, in Table 3.3., along with their axial, each system are given, of, and the examples, Delonging to each system, later in Fig. 3.21, are shown, Information, various, crystal, the, The shapes of, the Related, and, Crystallographic Systems, TABLE 3.3., Examples, are, , System, , Axial, Characteristics, , Maximum, , Symmetry, , Point, Groups, Alums,, NaCI, KCI,, , 1. Cubic, , a, , 2. Tetragonal, , B, , =, =, , a=, , b, , B, , =, , y= 90°, , Planes =9, , Axes = l13, , =c, =, , y= 90°, , a = b# c, , Planes = 5, Axes = 55, , 1, , Sn, SnO.T10;, , Diamond

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3/16, Pradeep's, , 3. Orthorhombic, or Rhombic, , a=B=y=90°, ab#c, , 4. Monoclinic, , Axes = 3, , a B y 90°, abtc, , 7. Hexagonal, , a= B = 90°, , Planes, , Y, , Axes = 7, , 120°, , CuSO45 H,0, K,Cr,0,, , Nil, , NaNO3, ICI, As, Sb, Bi, , Planes= 7, , a=b =C, , B =Y*90°, , PbCO Rhombic sulphur, , Axes = Nil, , 6. Rhombohedral, or Trigonal, , a=, , KNO. KsO4. BaSO4, Monoclinic sulphur, , Axes= 1, , Planes, , Vol., , Na,SO4.10 H0, CasO,2 H,0, , Planes = l1, , abtc, , 5. Triclinic, , 3, , Planes = 3, , PHYSICAL CHEMISTRY, , Axes = 7, , 1, , 7, , HgS, Ice, Graphite, Mg. Zn, Cd, , a = b*c, , 3.12. ISOMORPHISM AND POLYMORPHISM, chemical composition, e.8., sodium phosphate and, Certain crystalline substances which have similar, the same crystal shape. This phenomenon, sodium arsenate (i.e., Na,PO, and NaAsO) are found to possess, substances are said to be isomorphous., is known as isomorphism (Greek: same shape) and the, There are many substances (elements or compounds) which exist in more than one crystalline form, , which are formed under different conditions of temperature and pressure. This phenomenon is known as, , polymorphism. For example, carbon exists in two crystalline forms which are graphite and diamond:, calcium carbonate exists in two crystalline forms called calcite and aragonite. The polynorphism exhibited, , by the elements is commonly called allotropy. The temperature at which one crystalline form changes into, another is called transition temperature., , 3.13. SPACE LATTICE AND UNIT CELL, So far we have been discussing only the external, shape of the crystal. On studying the internal structure, of the, , crystal, it is found that the particles (i.e., ions,, atoms or molecules) constituting the, crystalline, substance are arranged in a regular fashion within, the crystal in the three dimensional, space. For, the, example,, arrangement of particles (represented, by black circles) for a cubic crystal is shown in, Fig., , 3.19, , The, , regular arrangement of points (i.e., ions,, atoms, or molecules), constituting the crystal in the, three dimensional space within the, crystal is called, FIGURE 3.19., the space lattice or, erystal lattice., Space Lattice and Unit cell., Ifa big-sized, crystal is broken more and more, ultimately a, tne, is reached when we, , a, , smallest, , stage, get, possible crystal. If it is broken further, it will break to, give the constituent particles, i.e., ions, ato cube, molecules. For example, if a big cubic, cube,, is, crystal broken, the smallest crystal obtained will de a, as, s, represented by thick lines in Fig. 3.19 above. This, small cubic crystal has all the elements of sy, , mmetry

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SOLID STATE, as, , 3/17, Dossessed by the big cubic crystal. Moreover, it is evident that the complete lattice has been obtained by, , the repetition of this smallest unit in different directions., , The smallest portion of the complete space lattice which has all the elements of symmetry and, which when repeated over and again in different directions produces the complete space lattice is, called the unit cel., , The situation may be compared with that of a thick wall made of bricks where a brick is similar to the, lattice., unit cell and the wall to the complete space, lattice, the, lt may be pointed out that it is the regular arrangement of the points that constitutes, space, them. The lines have been drawn just to indicate the regular repetition more, and not that of the lines joining, of the points. Further, in order that the unit cell may be described, clearly and to describe the position, the unit cell, viz, a, b, c and the angles a, B, y between them, we should know the dimensions of, , completely,, as, , shown in, , Fig., , 3.19., , that in section 3.6, it was, It may be recalled, that we have only 2-fold, 3-fold, 4-fold, , mentioned, rotation and we do not have 5and 5-fold axes of, rotation., or higher fold axes of, fold, 7-fold, 8-fold, due to the fact that the most, This is obviously, should be, a lattice is that it, essential requirement of, cells, the faces of, built up by the repetition ofunit, the faces of the surrounding, each unit cell touching, unit cells without leaving any gap, a, , close, , in-between., , Such, , an object has 2-fold,, packing is possible if, but is not, , GAP, , FIGURE 3.20., llustrating, , of hexagonal objects., (a) the close packing, packing of, (b) a gap created in the, pentagonal objects., , this is, axis of symmetry, of symmetry. As an example,, 3-fold, 4-fold or 6-fold, or higher-fold axis, 8-fold, has 5-fold, 7-fold,, 5-fold axis of symmetry. Confining, possible in case an object, axis of symmetry and, 6-fold, to tile a, having, fact that it is impossible, illustrated in Fig. 3.20 for objects, situation is similar to the, the, sake of simplicity,, to two dimensions for the, in the tiling., etc. without leaving gaps, heptagons, pentagons,, floor with regular, , 3.14. BRAVAIS LATTICES, above, it, , assumed that there are, , points only at, , the, , corners, , of the, , that the points (particles), of the, internal, faces or within the body, unit cell. The detailed study ofthe, the, of, centre, the, unit cells but at, crystallographie, at the c o r n e r s of, to the seven, lattices corresponding, dy be present not only, of, different, types, description of these, n t cells. This gives rise to 14, Bravais lattices. The, called, are, lattices, These 14 types of, The terms simple. jaceon next page., 3.21, ystems already described., in, shown, Fig., are, 1s given in Tale 3.4 and the shapes, considered, In the space lattice, , was, , structure, , by Bravais, , (1848) has shown, , Cs, , as under:, end-centred and body-centred imply, unit cell., the corners of the, are present only at, at, the, When, points, )Simple., are points present, , centred,, , points at the, , corners,, , there, , the, When in addition to the points at, , corners,, , there, , in, ace-Ccentred. When, , addition to the, , of each face of the unit cell., Or the, , body, body, , d-centred., , end faces of the unit cel., , Giv) B 0uy-centred., , of each unit cell., , When in addition to the points at the, , corners,, , there, , are, , is, , points present, , one, , i, , uit, , at the, , point present, , of the unit, , at the corners, cell, i.e., when the points are present only, i ecell. The, prmitive unit, other types of unit cells are called non-primitive unit cells., celld, , the centre, , centre, , witnin, , cell,, , un, , 1s c a l l e d

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3/18, , Pradeep's PHYSICAL CHEMISTRY, [Vol., , 1. CUBIC, , a B y =90°|, a b=cC, , SIMPLE, , FACE-CENTRED, , BODY-CENTRED, , 2. TETRAGONAL, , a, , By 90°|, , SIMPLE, , BODY-CENTRED, , 3. ORTHORHOMBIC, , a, , 90°, c, , a, , SIMPLE, , FACE-CENTRED, , END-CENTRED, , 4. MONOCLINIC, , BODY-CENTRED, , a =y= 90° #, la#b#c, , SIMPLE, , BODY-CENTRED, , SIMPLE, , SIMPLE, , 5. TRICLINIC, , a # pty # 90, , ab#C, , 6. RHOMBOHEDRAL, , a, , a, , p=y+90°1, , b c, , SIMPLE, 7. HEXAGONAL, , 90°,y =120°, ap=, la =b* c, , FIGURE 3.21., Bravais lattices, , corresponding to seven crystallographic systems

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SOLID STATE, , 3/19, , TABLE 3.4. Bravais lattiees corresponding to different crystal systems, Number of, , System, , 3, , 1. Cubic, , or, , 4, , Rhombic, , Simple, Face-centred, End-centred, Body-centred., Simple, End-centred., , 4. Monoclinic, , Simple, , 5. Triclinic, , 6., , Simple, Face-centred, Body-centred, , Simple, Body-centred., , 2. Tetragonal, , 3. Orthoromic, , Description of the lattices, , Barvais Lattices, , Rhombohedral or, , Simple, , trigonal, 1, , 7. Hexagonal, , Simple, , unit cell (N) is given by the expression, The number of points per, , number of points present on, , the corners of the unit cell., , N., unit cell., on the faces of the, N= number of points present, the interior of the unit cell., number of points present in, N,, shared by 8 unit cells,, on the corner is, =, , where, , =, , each point present, follows from the fact that, the interior is shared by, cells and the point present in, unit, two, shared, by, on the face is, , This expression, , each point present, no, , other unit cell., , For example, for any simple, , lattice,, , so that, N=8, N=0, N, =0,, , so that, N =8, N=6, N, 0,, =, , For, , a face-centred, , For body-centred, , For, , cubic lattice,, , orthorhombic, , end-centred monoclinic, , 3.15. SPACE GROUPS, , lattice,, , lattice,, , so, N =8,Np=0, N, =1,, , N, , =8, N,=2, N,, , =0,, , so, , N= +0+0, N=, , 1, , +0=4, , that N=+0+1=2, that, , N=++0=2, , there are 32 different ways ofarranging, theoretically, that, 3.11, section, in, However,, lthas already been discussed, called point groups., arangements are, different, These, ne elements of symmetry about a point., dimensional space,, of the pattern in the three, repetition, the, of, regular, detailed, d, complete space lattice, because, become possible. The, serew axes, and, elements called glide planes, 1s, book. However, it, u o n a l symmetry, the, of, present, is beyond the scope, of arrangeneus, ICuSSIon of these new elements of symmetry, of symmetry, the number, elements, additional, these, of, to 250., CCsting to mention that because, becomes manifold, viz, equal, lattice, a, of, space, about a point, are callea spuce, n e elements of symmetry, T, a lattice point, about, symmetry, elements, of, the, labi, is given in, O pOssible arrangements of, systems, different, crystal, the, corresponding to, or, Pup these space groups, , 3.5 On, on next, , page.

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3/20, Pradeep's, TABLE 3.5., , PHYSICAL CHEMISTRY, , [Vol. n, , Distribution of space groups among crystal systen, , Systems, , No. of space groups, , 1. Cubic, , 36, , 2., , 68, , Tetragonal, , 59, , 3. Orthorhombie, , 13, , 4. Monoclinic, , 2, , 5., , Triclinic, , 6., , Trigonal or Rhombohedral, , 30, 22, , 7. Hexagonal, , To Sum up:, , Number of Crystallographic Systems, Number of Point Groups, Number of Bravais Lattices, , Number of Space groups, 3.16. X-RAYS, , DIFFRACTION, , BY, , =7, , = 32, = 14, , 230, , CRYSTALS: STUDY, , OF THE INTERNAL, , STRUCTURE, , On this basis,, of an optical microscope., seen with the help, be, could, a, of, crystal, The external shape, internal structure, i.e.,, discussed. However the, seven systems as already, into, classified, were, the crystals, of X-rays (by, not be studied until the discovery, atom within the unit cell could, each, of, exact, the, position, the internal, have been carried out to understand, X-rays, the different studies that, in, , 1895). Using, discribed, structure of crystals are briefly, Roentgen, , below:, , 3.16.1. Laue's Method, , that if a crystal consists of a large number of, Laue in 1912 made a valuable suggestion, atomic planes in the crystals were separated by, ions arranged in an ordered manner and the, , Max, atoms or, , von, , the wavelength of the X-rays, then if these X-rays are, distances which are of nearly the same magnitude as, and will be scattered by the electrons*, allowed to strike the crystal, the rays will penetrate into the crystal, The rays reflected from different layers of atoms, due to wave, atoms or the ions of the, , crystal., , of the, , a diffraction pattern, just, nature, will then undergo interference (constructive and destructive) to produce, a large number of closely spaced lines., as it happens in case of light passing through a grating containing, In other words, crystals should act as a three-dimensional grating for X-rays. These ideas of von Laue, , were put to experimental test and found to be true., In the Laue method, continuous spectrum of X-rays (called white radiation) having a wide range of, , wavelengths is used (e.g. from a tungsten target). The purpose to do so is that at least some of theradiatio, would have the proper wavelength to undergo interference. The method used is shown in Fig. 3.22 (a) and, , the Laue diagram obtained for NaCl is shown in Fig. 3.22 (b)., , Such diagrams confirm the definite arrangement of the atoms in a crystal. However, their interpretation, to know the exact arrangement, ofatoms within the crystal is difficult. A much simple method has beenpu, , forward by W.H. Bragg as discussed below, , *X-rays are scattered by electrons. In neutron diffraction, neutrons are scattered by the nuclei of the atoms or, , ions of the, , crystal.

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EFL CTED BEAM, , X, OF, BEAM

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3/22, Pradeep's, , respectively. The reflect, d, reflected rays, , , they, , PHYSICAL, , CHEMISTRY [Vol. I), with each, , other. It these, , interference, Deams like BC, EF, etc., then undergo over the crests and troughs over the, are in, On the other, the reflected ravs, crests fall, pnase (i.e., in tne, is maximum., re the intensity of, reinforce into each other and, of the reflected rays, beam, , phase, , ntensity, , ofthe reflected, , renethe intensitypattern, oisis obtained., nand, if the reflected rays are out of phase (i.e., crests fall over the troug.action, obtained., Yay 1OW. If a photographic plate is placed to receive the reflected rays, dijfrachon pans, , BC and, , reflected, , EF may be in, , phase, the, , extra distance, , rays,, the A-raya,, DViOusthatin order that the, wavelength A of, the, of, travercod, multiple, d v I S e d by the, ray DEF should be an integral, , =na, , Distance JEK, , CTC, , 7 1s an, , d is, integer, i.e., 1, 2, 3, 4, etc. If, , the, , distance between, , the, , JEK, , l.e,, , successive atomic, , 1), planes, it is, , obvious from Fig. 3.23 that, JE, So that, , =, , d sin 0, , JEK =, , Putting this value in equation, , (),, , 2), , EK=, , 2 d sin 6, , we, , get, , 3), , 2 dsin 0= na, , This equation is called, , Bragg's, , us the, equation. It gives, monochromatic, , which, conditions under, , X-rays, A is, , the reflected beam, , particular crystal, certain values of, be integer only for, , constant., , Also for a, , Using, will, will have maximum intensity., constant. Hence n, observed corresponding, the X-rays, d is, is, face, facing, positions are, when a particular, value of 8, a number of, the, increasing, For other values of elyingin, the angle 6. Thus by gradually, have maximum intensity., beam, will be, will, reflected, where the, Thus a diffraction pattern, ton= 1,2,3,4 etc.,, than the maximnum., less, called diffraction, beam will have intensity, between, the reflected, 3 etc. These are respectively, 1,2,, n, to, correspond, various maxima, obtainedin which the, order etc., second order, third, maximum intensity is observed), maxima of the first order,, maximum occurs (i.e., first, the, first, which, at, value of d can be calculated, using, Measuring the angle 0, A the X-rays used, the, wavelength, the, and knowing, so that n=1,, =, , of, , for, equation (3)., the internal structure of crystals (ie.,, for, studying, used by Bragg, values of the angle 8) is, Apparatus. The apparatus, beam corresponding to different, reflected, , of the intensities of the, and is shown in, called Bragg's X-rays spectrometer, measurement, , Fig. 3.24., , uLLllul, , TARGET, X-RAYS, , FIGURE 3.24., Bragg's X-ray spectrometer.

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D, , INTENSITY OF THE, REFLECTED X-RAYS

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CHEMISTRY, , (Vol., , PHYSICAL, , 3/24, , Pradeep's, , The length (a) of the edge of the cubic unit cell as, , ohtained, , X-ray, from Bragg's, , alternative, , based on, method, o, the example, , the, , studies, , knowledge, , of NaCl crystal, , calculation from, using, can be checked, checked by calcunA, it cell. Taking, Taking, g eqns. (2) and (3), can, unit cell., , an, , dittraction, , er inin section 3.18). ie., , later, of the density of the crystal and the typeof lattice of theenit cell, (explained, unit cell (explaine, , section3, , Nat and 4 C ions per, , S Tace-centred cubic, it has 4, , it has 4 NaCl units per unit cell, Mass of unit cell, , =, , Mass of 4 NaCl units, 4x Mass, , -, , ofone NaCI, , unit, , M, , 4X No, mass ofNaC1(-, , M=molar, , where, , ltp is the density of NaCI, , crystals, , 585 g, , mol), , to, (which is equal, , Volume, , ofthe unit, , cell, , number, number, , cm), , 2 165 g, 4, , Avogadro's, , N ==A v o g a d r o ' s, , No, , and, , M, , px No, 4M, , Edge ofthe unitcell=p N,, The value, , Bragg's, , of the edge (a), , thus, , determined, , diffraction, , to, further gives support, method. This, , the, support to, , structure, , of NaCl, , as, , conceived, , wavelength, , Example 1. Using X-rays of, , Ag, , atoms, , Bragg's X-ray calculations above., , and used in the, , PROBLEMS ON, , from silver, , to occur, crystal wasfound, , that gave rise, , Solution. Here,, , to, , Using Bragg's equation,, , BRAG'S, , 154.1 pm, , at, , 6, , n, , =, , 1,, , EQUATION, , the glancing angle, the reflection, of, spacing between the planes, , andstartingfromthe, , 22:20. Calculate, , =, , the above reflection., , 154-1 pm,, , =, , with that calculated from, very well, to, agree, found, is, lends, method. The method also, , 22-20°, , (sin, , =, , 0:3778)., , 0=22-20°, , viz.,, 2 dsin 0=na, , na, , d2 sin, , 1x154 1, , 2x0.3778 P=, , 204 pm., , of, was observed at a glancing angle, cubic, crystal, ofa, planes, (11l), Example 2. A reflection from, unit cell, What is the length of the side of the, 1 2 when X-rays of wavelength 154 pm were used., , Ar what angle the reflection will take place from (123) planes ? (sin 11-2° = 0.1944, , =, , Solution. Here,, , =, , 154 pm,, , n, , =, , 1,, , 8= 112°, , Applying Bragg's equation, 2 d sin 6 = na, , na, , Ix154, , 396 pm., 2sin 6 2x0.1944 P=, , I.e.,, , Further, the separation between the planes of a cubic crystal is given by, , dl, d, or, , P++12 +12 396 pm, a, , =, , 396, , x, , 3, , =, , 686 pm, , (calculated above)

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3/27, , SOLID STATE, , the reflections from the different planes are, To see to which type of cubic lattice the crystal belongs,, at which these reflections take place. The interplanar distances for, studied and the angles are determined, na, d, viz., calculated, equation, sin A: The ratio of the interplanar, using Bragg's, ach set of planes is, ratios for, calculated and compared with the theoretically calculated, distances of the three planes is then, above., the different types of cubic lattices given, CsCI. From the external geometry, as found, Determination of the structure of NaCI, KCl and, found to belongtocubic system., all the three, i.e., NaCI, KCI and CsCI have been, =, , using optical microscope,, In, at, , case, , of NaCl erystal, the first order reflections, , from the three faces of its crystal are, , equation, i.e., 2d sin 9, 5.9, 8.4° and 5.2° respectively. Applying Bragg's, , nid, , =, , will be in the ratio of 1/sin, evident that the ratio of the three interplanar spacings, will be in the ratio, constant for the given experiments). Thus, they, 1, 1, 1, 1, 0-0919:71:6-65: 10.99, , or, , found to occur, , d, , values (as, , =1:0-705:, , 0-146, , 0103, sin5.9° sin8.4 sin5.2, NaCl has, hence the given crystal of, face-centred, cubic,, of, that, to, close, ratio, is, As the, , cubic lattice., , Proceeding exactly, , n, , found that KCl also, in the same way as above, it is, , n, , inA, , it is, , and A. are, , 1-132, , a face-centred, , has face-centred cubic, , structure, , cubic structure., whereas CsCI has body-centred, , a given cubic crystal, 100, 110 and 111 planes of Determine the type of, 8.4° and 5.2° respectively., to occur at angles 5.9,, , order reflections from the, , Example. The first, (NaCl crystal) werefound, , belongs., cubic lattice to which the crystal, 1 so that, n, For first order reflections,, =, , Solution., , we, , have, , nN, , d, , 2sin 6 2sin, = 4-854 A, , d100 2sin 5.9°, 110, , ., , d100d110:d, , Obviously,, , 2x0.103, , 2sin 84, , 2x0-146, , 1 2sin 5.2, , 2x0-091, , 1:0-706:, =4-854:3.425:5495 =, , = 3-425, =, , 5-495, , N, , 1-132., , the, face-centred cubic. Hence,, to that of the, close, is, ratio, very, this, , centred cubic., , given crystal is, , PROBLEMS FOR PRACTICE, , 23:7,, , crystal occurs at angles, that sin &,, crystal belong ? Given, , cubic, , 6,, , =, , 111 planes of a given, from the 100, 110 and, The first order reflection, cubic lattice does the, To what type of, respectively., 34-70 and 0 20-4, 0-5693 and sin 0, 0-3486., =, , 0-4019, sin 0,, , face-, , =, , [Ans. Face-centred cubic], , =, , =, , developed by Debye, for studying the X-ray, This is the simplest method, a single large crystal is, 1917. In the Bragg's method,, in, Hull, by, rotated about a sua, and Scherrer in 1916 and independently, m o r e o v e r it has to be, tube, and, obtain), to, difficult, thin walled glass, (which may be sometimes, small, a, in, form, eguired, is taken in the powdered, Wnereas in this method the sample, , 3.16.3. The, , dS, , Powder, , Method, , diffraction, , by crystals. It was

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3/28, , (capillary tube) and, , Pradeep's PHYSICAL CHEMISTRY [Vol.I, , (small crystals) in the required. Ofcourse, the tube is rotated to ensure that the crystallites, are randomly oriented. A, powder, r are, sample and a circulare, randomly oriented. A Debeam of monochromatic X-rays is focussed onthe, notographic, the, hown inin Fig., photog film is placed around it as shown, 3.27a Since in the powder,, Fig. 3.27(a)., crystal planes0T, of difterent, different small, smal crystals are oriented at all possible angles to the beam, there wil be, always some, y s tcrystals, a l s which, will, have the proper orientation to produce diffraction maxima trom.a tne, wh, r, , originating, , cach set of (hkl) planes will give rise a diffraction, prnciple,, POWder. Thus, a large number of concentric cones will be produced due to the retlections from, A:, alferent sets of planes and, satisfying Bragg's law Fig. 3.2/[0)., to, , cone, , SPECIMEN, , CYLINDRICAL, , POWDER, , CAMERA, , SPECIMEN, , 29, FILM, X-RAY, BEAM, , SLIT, SYSTEM, , X-RAY X-RAY, FILM BEAM, , FIGURE 3.27., of a monochromatic, Debye-Scherrer method for diffraction, X-ray beam by powder sample., two lines (or arcs) at equaEach cone will meet the photographic film at two points producing, undiffracted beam). Thus, a large number of lines, distance from the bright centre spot (produced due to the, each pair of lines being equidistant on the right and le, (or arcs) will be obtained on the photographic film,, reflected beam makes an angle 20 with the, from the centre spot. Further, if the incident angle is 0, the, idea of the general diffractiordirection of the incident beam, as shown in Fig. 3.27. An approximate, of the, may be shown as given in fig. 3.28., , powder, , pattern, , - INTERSECTION OF THE, , FIRST CONE ON THE FILM, , FIGURE 3.28., X-ray diffraction pattern a powder sample, 1.e, After getting the X-ray diffraction pattern of the powder, the next step is the indexing of lines,, assigning Miller indices to the planes responsible for producing these lines. For this purpose, the distan, , rof each line from the central spot is measured usually by taking half of the distance between the two in-, , (of, , the, , circumference,, cone) on either side of the centre. If the radius of the film is r, the, to a scattering angle of 360°. This implies that, , same, , corresponds, , 20, , x, , 360, , 27Tr, , Thus, 0 can be calculated. Knowing 0, the interplanar spacing can be, viz, 24 sin 6= nh., , calculated using Bragg's equau, , i

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3/29, , SOLID STATE, , diffraction patterns obtained by powder method are characteristic of the solid substances, i.e.,, the crystal planes and hence, the lines on the photograph depend upon the spacing between, the positions of, for, used, qualitative as well quantitative, this reason, this method is quite often, an the type of the crystal. Forestimate, this method is much easier, and, pure substances and mixtures. Moreover,, to, identify, i.e.,, analvsis,, The, , than the single, , crystal method., , 3.17. X-RAYS, , DIFFRACTION PATTERNS, , OF CUBIC LATTICES, , and each, because all lengths of the unit cell are equal, distance can be, with Miller indices (hkl), the interplanar, , of all the systems, Cubic system is the simplest, anole is, , equal, , of, to 90°. For any set, , calculated using, , planes, , the expression, , .1), , a, , dnlh+k? +12, Also according, , to, , Bragg Sequation,, , for the, , for first order diffraction,, glancing reflection, i.e.,, , 2 dhe sin9=, 2), , sin -, , 2dh, , or, from eqn., value of dh, Substituting the, sin, , (1), we get, , .3), , - x v + k + 1 2, , a, , .4), sin, , 4a2 ( + + A, , or, , For a, constant., , and A are constant, X-rays used, a, , is, , and for the particular, becomes,, to A, eqn. (4), constant equal, , given cubic system, , Putting this, , and hence, , +, sin 0 (h2 +k2 P)A, , values, Substituting the, , of, , ofsin, the value of 0 in, This is explained, photographic film., terms, , lattices,, , three types of cubic, present in the, the, planes, for, be obtained, the indices (hkl), the ines that would, which, predict, calculated, lattices., 0 can be, the three cubic, below for each of, and 1,, , Using eqn. (1), , and the integral, , indices h,, values for the, , on, , the, , dha may, , k, , obtained from, 7 cannot be, because, missing, Note that alV7 is, arealso given inthe same, Table 3.6 below., in, values forsin 8, diffraction, given, corresponding, values, The, have, integers., conclude that the powder, k, we, are, and then again, possible. Thus,, +*+ when h, and I, to 7A is also not, followed by a gap, 0, lines, equal, of, value, The, 1able., sin, of six equally spaced, lattice will consist, , 1. Primitive cubic, , pattem of primitive, , lattice., , cubic, , a set of lines and so on., TABLE, , hkl, , 100 110, , daua, sin, , A, , a, , 111, , 3.6., , 200, , and, , 210, , sin, , 211, , 220, , 3A, , 4A, , SA, , 300, 21, 221, , 310, , 311, , 222, , 320, , a, , 9, , 6, 2A, , cubic lattice, , a, , a, , a, , 6A, , 8A, , 9A, , V10, 10A, , II 12, 11A, , 12A, , 13A, , take, ak place, reflections in phase, that, found, is, abov, For this type of lattice, it, calculations as, out the, red cubic lattice., Carrying, even., odd or all, tor which hkl values are either all, , fom, es, the, thevol, values of the dha and sin? 0, only, , dh, , 0 values for primitive, , are given in Table 3.7 below :

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3/31, , SOLID STATE, , Thus, the type ofcubic lattice can be determined by observing the X-ray diffraction pattern produced, the type of lattice, missing lines extend a great help in ascertaining, from which the reflection takes place (i.e., corresponding to, Knowing the (hk/) value of the plane, value of in' 9 and hence sin 9., , the, , from, , hich, the, , a, , edge, , The, , sample., , is obtained) and the correeponding, line in the diffraction pattern, (3) which can, cubic unit cell can be calculated using eqn., , ofthe, , sin Vh, , a, from any (hk/), For reflection, come, , plane, , be, , written as, , +, , value ofa will always, and the corresponding sin 9 value, the, , be constant, , out to, , KI AND CsCI, STRUCTURES OF NaC1,, CRYSTAL, 3.18., NaCl crystal, diffraction studies of, Nray, as, , lattice, f a c e - c e n t r e d cubic, it has, that, show, stnucture consists, 3.30. In fact, this, shown in Fig., face-centred cubic, interpenetrating, , of, , two, , lattices,, , one composed, , of, the other, , Cl, , unit cell, , edge, its, by the fact that, is further supported, , contains, , 4, , Na, , and 4, , ions (i.e.,, , C, , in mind the, , contributions, , C, , Na, , face-centres (1/2), edge, , FIGURE 3.30., , by, , made, , (1)., (1/4) and body-centre, , centres, , KCI also has, similar to, , face-centred, , cubic lattice, , which Na, that of NaCI in, , replaced by K, , Fig., , shown in, , latuces,, atoms, , one, , atoms, , (black circles, , Cr ions), circles represent, , have, , atoms., , 3.31. In, , of cesiun, , which, , fact, it has, , atoms, , are interlocked, , body-centred lattice. This, , two, , simple cubic, , chlorine, and the other of, , together, , structure, , to, , CI, , cubic lattice, , C, , give pseudo-, , is further supported, , one Cs* and, the fact that its unit cell contains, , one, , CT, , by, ion (i.e., one CsCI unit)., t 1s, , interesting, , NaCl, cubic lattice of, c o n s and open, Nat, represent, , Face-centred, , have a body-centred, CsCl is found to, as, , and, , any, , keeping, 4 NaCl units)., c o r n e r s (1/8),, , been, , ions, , chlorine lattice is, ions. The, of the cube. This, , displaced al2 aong, structure, , entirely of Na", , to, , CI, , point out that through, , KCI have similar structure, their diffraction, , remarkably different., , This is explained, , on, , /C, , NaCI and, , patterns are, the basis, , of, , the fact that X-rays are scattered by the electrons and, different atoms or ions contain different number o, , electrons. Hence, their scattering strengths are difterent., , CI, , FIGURE 3.31., Body-centred, , unit, , cell of CsCl

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3/32, , CHEMISTRY, , Pradeep's, , Greater the number of, , PHYSICAL, , Moreover, the, , electrons, the wavesis the, scattering strength., Scattered by anions interfere with greater, scattered by cations and, and depending u, 01IS, Une reflections, Is, , may be in phase, , less than that in CF ions, , or, , out of phase. In case of, , of the X-rays, orientation of the, , waves, , dependinEelectrons, he, , a, , number, , Vol. I, , of, , electrons, , in, different but, , case, , in Nat ion, , of KCI, both, Thus, when, , are, , equal., and, therefore, their scattering strengths, strengths, of, in, scattering, and Cions contain the same number ofelectronsand their, completely, instead, out complecmilar, cancel out, and instead, similar, similar and, Insteau, and, ions do not, very, Cl, look, Or phase, the, and, of, scattering strengths Na, Out, therefore, all, that, of primitive, similar to, K and Cions, the cancellationis complete. he ions in KCI,, obtained is, pattern, are, , whereas, , ppearing to be face-centred cubic, the, , powder, , case, , diffraction, , cubic unit cell (Fig. 3.32)., , K, Na, , C, , b., , FIGURE3.32., Difference, , KCI shown in, in NaCI and, , atoms, , a, , octant), portion (one, , is, or ions ina crystal, , determined, , positions of, In fact, the relative, diffraction photograph., the X-ray powder, the lines obtained in, of cubic, different, , types, , in, Coordinates of lattice points, as that of NaCl,, face-centred cubic such, , (1) For a, , that its coordinates are, , ions present in the face, , (b) the coordinates of the Cl, , ions present at the, , by studying the, , centres, , intensities of, , crystals, , taking the Nat ions at, , (0, 0, 0),, , Nat, (a) the coordinates of, , of the unit cell., , the, , corner as, , the origin so, , will be, , edge centres will be, , oaoa.a, (c) the coordinates of the Cl ion present at the body centre will be |, , (ti) For simple cubic unit cell, the coordinates of lattice points, , 9, , at the cormers are, , (00), (001),, , (010), (100), (011). (101), (110) and (1)., , (in) For a body-centred cubic, lattice points, while that of the point at the, , at the, , body-centre are (1 51, 22, , corners, , have the coordinates, , as, , in (i) abOve, , For a cubic lattice, distance between any two points with coordinates ( . y, 21) and (2 Y2 2)a, be calculated using the formula

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d=, , x - X2) +(y ¥2) + (7-2,), where a = length of the edge of the unit cell, 19,, , 3/33, , -, , xa, , CRYSTAL STRUCTURE OF CALCIUM FLORIDE, (FLUORITE), , In fluorite, each Ca** ion is surrounded by cight, Fions, giving a body centred cubie arrangement of, Fions around Ca** ion. Since there are twice as many, , Fions as Ca* ions, the coordination number of both, , ions is different, and four Ca** ions are tetrahedrally, ach F ion. The coordination, arranged around, , numbers are therefore 8 and 4, so this is called an8:, The fluorite structure is found when, 4 arrangement., 0.73, the radius ratio is, , or, , above., , An alternative description of the structure is that, face-centred cubic arrangement., the Ca2t ions form a, small to touch each other, so, The Ca2* ions are too, , J, , Ca, , oONS, , O- F ONSs, FIGURE 3.33., , close packed. However, the, the structure is not, a close-packed arrangement,, structure is related to, the same relative positions, since the Ca" ions occupy, structure, and the F- ions, as for a cubic close-packed, tetrahedral holes., occupy all the, , Fluorite (CaF2) structure., , SUAMANAIRY FOR REVIEW, 1. Solid. A Solid is defined, , as, , that form of matter which possesses, , rigidity, , and hence, , a, , definite, , shape, , definite volume., 2. Classification of Solids., , (atoms., , various constituent particles, said to be crystalline if the, is, solid, A, solids., elements and comp, i) Crystalline, within the solid, e.g., all, pattern, definite, geometrie, molecules) are arranged in a, and undergo a clean cleavage., , Tney, , have, , sharp melting points. They, , are, , is said to, 1) Amorphous solids. A solid, rubber, Tegular fashion, e.g., glass, pitch,, undergo an irregular cut., , anisotropic, not arranged, constituent particles are, be amorphous if the, They are isotros, a temperature range., etc. They melt over, , 3. Classification of crystalline Solids, , wewww.oon, , Examples, , www.wwwwnewwwwevwwowwwwww, , *****, , Crystal type, , Constituent particles, , i) lonic, (ii) Molecular, (iii) Covalent, (iv) Metallic, , +ve and-ve 1ons, , ecoxsa, , na*******, , Molecules, , Law, , van, , foreces, der Waal's, , +ve ions and, , 1s, , lodine, Ice., , CO, (s», , Diamond, Silicon e=, , and alloAll metals, , Atoms, , of crystalP, , n, , NaCl. KNO, BaSC, Electrostatic forces, , Covalent bonds, , mobile electrons, , 5. S, , Binding forces, , Electrostatic, , (Metallic bond), , deals with the, that branch of science which, , of constancy of Interfacial angles. The angles, , a, , study of geometry,, , between, u, , a, , y, , s, , the, , properucs, , Taces,, the corresponding of, the, independent, same, , s