This is a more stringent test than a mere check of strong or weak reflections against large or small amplitudes , and so leads much more directly to the atomic positions by eliminating impossible configurations .sx An example of the comparison of F values is given by the results for iron pyrites quoted at the end of Chapter IV .sx This method is re-served for cases where a structure is to be examined in detail , or is of a type difficult to analyse , because the experimental measurements demand so much labour .sx The agreement of theory and experiment in such cases is evidence of the sound basis of the optical theory .sx In the vast majority of analyses less precise methods can be used .sx An approximate allowance for the factors other than the structure amplitude can be made , and the qualitative comparison is sufficient .sx THE ANALYSIS OF INORGANIC CRYSTALS .sx It is convenient at this point to consider the analysis of inorganic and organic crystals separately , because the further stages of analysis present different features .sx We will suppose that we are analysing a complex inorganic crystal , such as a salt or a silicate .sx The lattice , space-group , and numbers of atoms of each kind in the unit of pattern have been determined .sx We wish to use every possible device to indicate the form of the structure before embarking upon the final test of comparing observed and calculated intensities .sx Certain atoms may be in special positions , lying on symmetry planes and rotation axes or at centres of inversion .sx This must be so if the number of such atoms in the unit cell is less than the number characteristic of a general position .sx A limitation of this kind is of assistance because it reduces the number of parameters , and it is particularly useful when some important atom in the structure is pinned down by a position at a symmetry centre .sx This atom then forms a nucleus around which the rest of the structure can be built up .sx The packing together of the atoms or groups of atoms is a very helpful guide .sx Ions are packed into a structure as if they were spheres of approximately constant size .sx These ionic sizes are described in Chapter VII ; they vary from one atom to another , and it is a general rule that the negative ions are considerably larger than the positive ions .sx The former are therefore themore important in the packing of the structure , and the positive ions may be broadly said to occupy the interstices between the negative ions .sx Acid groups such as , may also be assigned definite dimensions , though their figure is not spherical but resembles a cluster of spheres pressed compactly together .sx In packing the ions together the domains of neighbours must not overlap , and this has an important bearing upon their position relatively to the symmetry elements .sx Purely geometrical reasoning may in this way take us a long step forward in the search for the right structure .sx Let us suppose that an oxygen ion , with a radius of 1.33 A. , is to be incorporated into a structure , and that the space-group has reflection planes .sx Since an oxygen atom must not overlap with its own reflection in the plane , the atomic centre must either lie exactly on the plane or be distant from it by more than the oxygen radius .sx There is therefore a layer on either side of the reflection plane which is forbidden territory , as it were , to the centre of any oxygen atom .sx Similarly , the atom must lie on a twofold axis , or at a centre of inversion , or be more than 1.33 A. distant from either .sx It must be still further away from axes of higher symmetry , if it is not actually on one of them .sx If these forbidden areas are outlined , it is often found that the atom is left with quite a limited range .sx The same holds true for the complex acid groups , which must not overlap symmetry elements .sx These considerations are geometrical , and the physical aspect of the structure may now be considered .sx As the structure is held together by the attraction of positive and negative ions , these ions may be expected to alternate so that a positive is surrounded by negatives , and vice versa .sx Pauling's rule , which is of fundamental importance in the theory of ionic structures , puts this condition in a quantitative form .sx Briefly expressed , it states that there is as far as possible a localised neutralisation of electrical charge in the crystal ; the rule is explained in Chapter VII .sx When regard is paid to the packing together of the ions , and to the satisfying of Pauling's rule , it is often found that only a few alternative structures need be tested .sx The general conditions outlined above are used to simplify as .sx far as possible the final process of testing various atomic arrangements in order to obtain one which explains the X-ray results .sx We can only give a general idea of the methods employed in the final analysis , because they vary so greatly with the type of crystal .sx The spectra with small values of hkl are used to indicate the approximate positions of the atoms , because small variations in the parameters do not affect the structure amplitude greatly when the indices are low .sx The higher orders are then used in a process of fine adjustment .sx If one or more atoms have a high atomic number compared with the others ; their contribution to the is so large that it dominates , and heavy atoms are therefore easy to place .sx This is particularly the case for the higher orders , since , as we shall see later , the scattering of light atoms falls away more quickly with increasing , and so the heavy atoms stand out from the rest .sx The most important guide of all is the knowledge obtained from previous determinations of similar crystals , and the laws of crystal structure which these determinations have revealed .sx All these features enable the investigator to get a first approximation to the atomic arrangement , which is then improved by trying small movements of the atoms and comparing the calculated and observed intensities .sx AN EXAMPLE OF ANALYSIS :sx BERYL .sx The analysis of beryl , Be3Al2Si6O18 , illustrates many of the features which have been described .sx This crystal belongs to the holohedral , or highest class of the hexagonal system .sx It is one of some complexity , containing a large number of atoms in the unit cell whose fixation involves the determination of seven parameters .sx In spite of this , the structure may be found very simply and directly .sx Since all crystals of this class are based on one type of lattice ( the hexagonal lattice ) , it is a simple matter to find the axes from the observed reflections .sx The ratio c :sx a usually adopted as the result of crystallographic measurements is 0.4989 :sx 1 , but the X-ray measurements show that the true ratio is twice as great :sx c = 9.17 A. , a = 9.21 A. , c :sx a = 0.9956: 1 .sx The unit cell contains two molecules of Be3Al2Si6O18 .sx The space-group is found by noting which series of reflections are absent .sx A survey of the possibilities could be made by consulting space-group tables , but it will be carried out from first principles here , because it forms an example of the very simple way in which the various space-groups arise .sx A portion of the hexagonal lattice is shown in fig. 71 , the c axis being normal to the plane of the diagram , or what will be referred to as the `vertical' direction .sx In order that the structure may have the highest form of hexagonal symmetry , there must be twenty-four asymmetrical units around each point of the lattice .sx We need only consider a set of twelve here , because there is in each case a horizontal reflection plane , so that each .sx unit in the diagram actually represents a pair .sx Our problem is that of finding in how many ways the twelve can be arranged so that the full symmetry is represented .sx Fig. 71 shows that this can be done in four ways , the units being represented conventionally as half-arrows .sx We may put all twelve in one plane ( 191) .sx Alternatively six ( drawn in full line ) may be put at top and bottom of the cell , and six ( in dotted line ) half-way up the cell at height c/2 .sx There are three ways of effecting the latter arrangement .sx We may either have six arrows rotating in one sense at the top , and six in the opposite sense at height c/2 ( 192 ) , or two sets of three pairs , a pair being either .sx reflections across a plane parallel to a ( 193 ) , or a plane perpendicular to a ( 194) .sx The figure illustrates the possibilities better than a description .sx Another way of arriving at the result is to recognise that the point group or crystal class has vertical reflection planes 1010 and 1120 parallel and perpendicular to a respectively .sx These must be represented by either reflection or glide planes in the space-group .sx The possibilities are therefore .sx The oxygen atoms are the largest units in the structure , being about 2.7 A. in diameter , and 36 of them must be put into the unit cell without overlapping .sx Suppose an oxygen atom be placed in a general position , such as O1 .sx It must not intersect a symmetry plane or an axis , for this would involve an over-lapping .sx We may picture the twofold axes in the figure as a set of rods , which are at a distance of c/4 or 2.29 A. above a plane .sx A sphere 2.7 A. in diameter must be placed between the .sx Planes perpendicular to ( 1010 ) have indices ( mm2ml ) , and planes perpendicular to ( 1120 ) have indices ( moml) .sx Remembering the rules for the apparent halving of the c axis due to a glide plane , the criteria are seen to be .sx All reflections present .sx l must be even for moml and mm2ml .sx l must be even for moml .sx l must be even for mm2ml .sx In beryl , whereas such reflections as 1231 , 1233 show the true length of the c axis , it is found that 1011 , 1013 , 1015 .sx .. and 1121 , 1123 , 1125 .sx .. are absent .sx Therefore the space-group of beryl is or C6/mcc .sx The framework of symmetry elements has now been found , and it remains to fit the numbers of atoms represented by into the unit cell .sx The framework is complicated , and for simplicity only a part of the unit cell will be considered .sx At the top of the unit cell there is a network of horizontal twofold axes represented by lines in the figure ; vertical axes are denoted by appropriate symbols .sx At a distance c/4 below the top of the cell there is a reflection plane parallel to ( 000l ) , the plane of the diagram .sx We will consider the portion between the horizontal axes and the reflection plane , this being the top quarter of the cell ( fig. 72) .sx rods and the plane .sx Trial shows that only one position is possible , the sphere being wedged between three rods and the plane at O1 .sx The sphere bulges somewhat above the level of the rods , as in fig. 72 ( b) .sx The symmetry operations turn O1 into a group of 6 atoms within this section of depth c/4 , and hence into 24 in the whole unit cell .sx There are 36 oxygen atoms in all , and as an atom in the general position is one of a group of 24 , the remaining 12 must lie on some symmetry element .sx Places on twofold axes are ruled out , because overlapping with the other 24 atoms would result .sx There remains as a possibility .sx a position on the horizontal reflection planes , and it is again found that there is just sufficient room for one at O2 .sx The atoms of the group are shaded in the figure .sx Six appear on the reflection plane considered , and another set of six on the reflection plane at a depth 3c/4 .sx The approximate positions of all the oxygen atoms have now been found .sx Since there are only 4 aluminium atoms in the unit cell , as compared with 24 atoms in a general position , aluminium must be in a highly symmetrical situation .sx A few alternatives present themselves , and the X-ray results immediately decide for a position A where the threefold axis intersects three twofold axes .sx Similarly , the 6 beryllium atoms are found to be at B and its equivalent positions .sx