January 19651 BARKER INDEX OF CRYSTALS AS AN ANALYTICAL TOOL 3 The Barker Index of Crystals as an Analytical Tool BY L. W. CODD (Beatlaads A ddlestone Surrey) THE publication of Volume I11 of the Barker Index of Crystals brings to completion work started in Oxford as long ago as 1931. Volume I published in 1951 dealt with crystals of the tetragonal hexagonal trigonal and orthorhombic systems Volume 11 published in 1956 with monoclinic crystals and the present Volume deals with anorthic crystals. The substances dealt with are essentially (with some additions) those described in the five volumes of Groth’s Clzemische Krystallographie ; apart from its more specialised application as a method for identification the Barker Index thus provides a reference source to the data on some 8000 substances presented in the earlier work now out of print.The idea of using crystal angles as a means of identification naturally attracted crystal- lographers long ago but in spite of several attempts in the past no generally useful system emerged. It was in fact Barker’s own unsuccessful attempt to put to practical use the system developed by his teacher Fedorov in his monumental work Das KrystaZlreich that led him to develop his own method. The basis of the Barker method was first set out in his book “Systematic Crystallo- graphy” (Thos. Murby and Co. London 1930). The long history of this project since Barker’s death in 1931 is set out in the prefaces to the three volumes of the Index. Although consider- able modification in detail has been found necessary in the course of the work it remains firmly based on the Barker “principle of simplest indices” described in the book just men- tioned.Since the Index was started developments in the X-ray field have to some extent overshadowed the earlier morphological methods and it therefore seems appropriate at the present time to attempt a revaluation of the Barker method of identification as a possible tool for the analyst. For this purpose it will be necessary firstly to explain several technical terms but by studying the two practical examples given below it is thought that the reader will understand the principles of the method and will at the same time be convinced that it is possible to make use of it without extensive preliminary training in specialised crystallography. BARKER’S PRINCIPLE OF SIMPLEST INDICES- As shown in elementary text books of physical chemistry the position of any face of a crystal can be described in terms of its intercepts on three arbitrarily chosen crystallographic axes a b and c whose unit lengths are defined by the intercepts of an arbitrarily chosen “para- metral plane.’’ The choice of a particular set of axes and parametral plane is known as a “setting,” and the object of Barker’s principle of simplest indices is to give a single setting from all the possible alternatives.When a setting has been chosen the positions of all the faces are expressed by the crystallographic “indices,” which consist of three digits depending on the intercepts of the face on the three axes. The face (111) for example is one that cuts off unit length on each axis (this is of course the parametral plane).The face (121) cuts off intercepts 1/1 1/2 and 1/1 on the a- b- and c-axes respectively and so on. (It should be noted that in forming the indices the denomiaators of the fractions representing intercepts are taken). Where 0 occurs as in ( l O l ) the corresponding intercept is l/O which equals co i.e. in this instance the face is parallel to the b-axis. The Barker principle limits the choice of setting by specifying that the setting that results in assigning the maximum number of simple indices to faces on the crystal is to be preferred.* In many instances the principle of simplest indices leads at once to a single setting but in circumstances in which more than one setting satisfying the principle can be found simple “auxiliary setting rules” are provided that remove the ambiguity.CLASSIFICATIOS AXGLES- When the Barker setting has been found the angles by which the crystal is classified in the The parameters used to characterise a crystal (those quoted in *. Simple indices are those containing only the digits 1 or 0; e . g . (111) (101) and (001) are simple but Index can readily be found. (221) is not. 4 [Proc. SOC. Anal. Chem. Groth for instance) are usually the axial lengths a b and c and the axial angles a ,8 and y . Barker uses instead the angles between certain of the faces-in the tetragonal system for example the angle between the faces (001) and (101). This not only gives a better spread for classification purposes but also has the advantage that the Barker angles are frequently angles that have already been measured or are otherwise known so that further calculation is un- necessary; the calculation of the axial lengths and angles on the other handy is relatively involved.The classification angles used in the various systems will be defined when practical examples are considered. GRAPHICAL METHODS THE SPHERICAL PROJECTION- Before a practical example is discussed it is necessary to say something about the graphical methods used for representing the crystal angles diagrammatically. This will only be done briefly since these methods are described in detail in the Introductions to the Index itself. The method of expressing a crystal face by means of indices is useful in discussing crystal symmetry and for bringing out the relation of one face to another but for present purposes we need a method of expression more closely related to the actual measurements carried out on the crystal.Such a method is the “spherical projection.’’ In this the crystal is imagined to be placed inside a sphere and radii are then drawn from the centre of the sphere perpendicular to each face. The points at which these radii cut the surface of the sphere are known as the “face poles” of the corresponding faces. When a crystal has been plotted in this way we have a system of face poles on the surface of the sphere representing the relative positions of all the faces. These face poles can be regarded as analogous to positions on the earth’s surface; they can be particularised by measurements corresponding to latitude and longitude and all the normal formulae of terrestrial navigation can be applied to express and calculate the angular distances between them.This method has the further advantage that the angles of latitude and longitude as measured on the spherical projection are directly related to the angles obtained by measuring a crystal on the two-circle goniometer. In this instrument a crystal is rotated about two mutually perpendicular axes and the positions of normals to the faces are determined by an optical-reflection method the locations of the faces then being expressed in terms of an angle + corresponding to degrees of longitude and p representing the co-latitude or polar distance which is used instead of the latitude measured from the equator. THE GNOMONIC PROJECTION p-VALUES- plane surface. BARKER INDEX OF CRYSTALS AS AN ANALYTICAL TOOL As with navigational problems it is more convenient to make graphical constructions on a Several types of projection are used by crystallographers but for the purpose P’ Fig.1. Gnomonic projection vertical section through sphere and projection plane Fig. 2. Gnomonic projection #- and p- values of the present discussion the “gnomonic projection’’ will suffice. In Fig. 1 the circle is a vertical section through the centre of the sphere the north and south poles being at P and P‘ respectively. X is a face pole of co-latitude p which it is desired to represent in the gnomonic January 19651 BARKER INDEX OF CRYSTALS AS AN ANALYTICAL TOOL 5 projection. The projection plane is placed horizontally at the top of the sphere with its central point touching the sphere at P ; the projection plane is shown in the diagram as PX’Y’.To find the gnomonic projection of the face pole X a radius OX is drawn and produced to cut the projection plane at X’. X’ is then the required gnomonic face pole and if r is the radius of the sphere then PX’ == rtanp. Knowing the p-value of any point on the sphere it is thus possible to find its position on the gnomonic projection by measuring off the distance rtanp from the centre. This can be done either with an ordinary measuring scale and a table of tangents or more simply with a special scale in which the lengths of the angular graduations are proportional to rtanp; such a scale is provided on the “Barker protractor,” obtainable through the publishers of the Barker Index. Fig. 3 is a reproduction of such a scale; the reader can if he wishes use it in conjunction with a pair of dividers to make his own projection from the data to be given later.OO 10 20 30 40 50 60 70 75 I I I I l l l l l u u L 1 l I I I I l I I I 1 I I I I I I I I 1 I I 1 1 Fig. 3. Gnomonic scale One special instance must be mentioned namely that of a face pole lying at B (see Fig. 1) on the equator of the sphere. The projection of such a pole lies at an infinite distance from the centre of the gnomonic projection; it cannot therefore be represented by a point in the projection but its longitude can be indicated by means of an arrow as shown in the next paragraph. THE GNOMONIC PROJECTION #-VALUES- In Fig. 2 the projection plane is the plane of the paper and we are therefore looking down on the sphere from a point above the north pole. The circle of radius Y is the locus of points having p = 45” (since tan 45” = 1).The position of a gnomonic face pole of longitude # and co-latitude p is found as described below- Taking the point €3 as the zero point for the longitude measurement the line OP is drawn with an ordinary angular protractor to make the angle BOP = #. On OP a distance OX (equal to rtanp) is then marked off as was done in Fig. 1 and the desired gnomonic face pole is thus obtained. An arrow head added to any such line as OP (whether it contains another face pole or not) will indicate the presence of an equatorial pole in that direction. PRACTICAL EXAMPLE SETTING- projection. Chemische Krystallogra$hie Volume 3 p. 475. We can now reproduce the measurements of an actual crystal in the form of a gnomonic The following #- and p-values are obtained from the data given in Groth’s Face No.. . .. . . .. 1 2 3 4 5 6 7 +-value . . .. .. .. __ 0” 0” 90” 90” 56’1’ 56”l’ p-value . . * . * . . . 0” 58’58‘ 39’44‘ 67’55‘ 50”57’ 71’30’ 90” These measurements are plotted in a gnomonic projection as described above; the reader is recommended to do this for himself with an angular protractor for the #-values and marking off with dividers the p-values from the scale of Fig. 3. For the purposes of the present discussion it will be sufficient to work to the nearest degree. Fig. 4 shows the resulting pro- jection. It should be noted that all the measured poles lie in one octant of the spherical pro- jection of which the diagram is the gnomonic version. The crystal in question is actually orthorhombic and we shall assume this fact to be known; in practice of course more faces would have been measured both as a check on the symmetry and also to provide more results for averaging purposes.Since an orthorhombic crystal has three “planes of symmetry,” the pattern of face poles in one octant of the sphere will be repeated by “reflection” in the other seven. The effect of this is shown so far as the upper hemisphere is concerned by the unnumbered dots in Fig. 4. It will be seen that all the face poles lie in a regular pattern through which straight lines can be drawn to form a rectangular network symmetrical about the centre of the projection. (The rectangular net and the fact that it is symmetrical about the centre are expressions of the orthorhombic symmetry of the crystal). The axial system of the crystal is represented in the 6 [I-’roc.SOC. Anal. Chem. gnomonic projection by the axes A’OA B’OR and the perpendicular through the plane of the paper at 0. BARKER INDEX OF CRYSTALS AS AN ANALYTICAL TOOL I A’ Fig. 4. Gnomonic projection of an orthortiombic crystal obtained by using 6- and p- values Taking the rectangle 1-2-6-4 as a kind of “unit cell” in the network and considering the distances 1-4 and 1-2 as unit lengths along OA and OR respectively we may now assign co- ordinates to each of the numbered face poles taking the x co-ordinate along OA and the y co-ordinate along OR. (The equatorial pole face No. 7 will be omitted for the moment.) We have then- Face No. . . . . .. .. 1 2 3 4 5 6 7 Co-ordinates . . . . . . 0,0 0,1 0,h 1,0 ;,0 1,1 It can be shown (although it will not be proved here) that from a gnomonic projec- tion of this kind the crystallographic indices of the various faces may be obtained simply by adding a third digit 1 to the two-figure co-ordinates.Indices for the equatorial faces which have no co-ordinates of their own being merely indicated by directional arrows are indexed by taking the co-ordinates of any point on the direction line and adding 0; thus the co-ordinates of the point 6 are 1,1 and the indices of 7 are therefore (110). I_ We thus have- Face No. . . .. .. .. 1 2 3 4 5 6 7 Indices . . .. .. . . (001) (011) (061) (101) (401) (111) (210) or (012) or (102) The indices thus obtained are of course those corresponding to the particular (arbitrary) setting in which the axes are A‘OA R’OR and the perpendicular through 0 with the para- metral plane (1 11) given by face 6.The axial system is more or less fixed bv the ortho- rhombic symmetry of the crystal but it will be appreciated that another worker might perhaps have taken the rectangle 1-3-8-5 or some other as the unit cell. Barker’s principle of simplest indices provides a criterion to assist in the choice of a single setting. January 19651 BARKER INDEX OF CRYSTALS AS AN ANALYTICAL TOOL 7 In counting the number of simple indices given by a setting it is assumed that a face such as (001) is always accompanied by its antipode (OOi) on the spherical projection and the same for all other faces. Such a pair of parallel faces is termed by Barker a “plane,” and in making the count we take the number of simple planes not of faces. In the orthorhombic system a face (101) with its antipode (TOT) is “reflected” to (TC41) and (101) and thus counts as two planes.The face (111) appears in each octant of the sphere and counts as four planes. The same applies to (01 1) and (1 10). Q7e thus have- (loo) (OlO) (001) = 1 plane each (101) (Oll) (110) = 2 planes each (111) = 4 planes. For the simple planes in the crystal (001) = 1 plane (011) (101) and (110) = 6 planes and (1 11) = 4 planes giving 11 simple planes in all. It will be found that any other setting gives fewer simple planes (e.g. with 1-2-8-5 as the unit cell there are seven) and the setting already obtained is therefore the preferred Barker setting. Each face is given a standard letter and we may therefore proceed to fill in the appropriate letters in the projection as given below- c = (001),r = (lo]) a = (loo),% = (110) b = (010),q = (011) Until the “orientation” has been determined (see below) the setting is only provisional and to indicate this the standard letters are placed in brackets.0 RIENTA4TION- In order to obtain the provisional Barker setting the co-ordinates were taken in the order x y with 1 or 0 as the third index N. This was equivalent to assuming that A‘OA B’OR and the perpendicular through 0 represented the three crystallographic axes a b and c respect- ively. There are however five other ways of choosing the axes depending on the permuta- tions of the three digits which would be NXY xNy yxN Nyx and yNx. Each of these choices o f axes corresponds to a different orientation. It will be seen that change of orientation is equivalent to turning the crystal over inside the sphere and the orientation as it appears in the gnomonic projection will depend on the position in which the crystal was set up for measurement on the goniometer.Of course a fresh projection could be made for each orienta- tion but fortunately this is not necessary and the effect of changes of orientation on the indices can easily be studied by taking x y and N in the six sequences in turn. The values of the six angles corresponding to (cr) (am) (bq) and their complements remain unchanged and it is only the letters attached to them that vary with change of orientation. In our measured crystal the values of (cr) (am) (bq) and their complements were as given below- (cr) 67’55’ (am) 33’59’ (bq) 31’2’ (ar) 22’5’ (mb) 56’1’ (qc) 58’58’ (These are all measured angles or the Complements of measured angles so that no calculations are required).The Barker orientation rule lays down that the orientation is to be so chosen that the final classification angle am (heavy type is used to distinguish the final from the provisional angles) is less than 45’ and nearer to 45’ than either cr or bq. In the six angles given above the provisional angle (am) = 33’59’ already fulfils this condition and the provisional setting can be adopted as the final one. The final classification angles are thus- cr == 67’55’ am = 33’59’ bq = 31’2‘ Had it been necessary to investigate other orientations further sets of indices would have been derived by using the other permutations until a set satisfying the orientation rule had been found. With a little practice it is quite easv to pick out the correct orientation by mere inspection of the projection.IDENTIFICATION FROM THE CLASSIFICATION ANGLES- The three classification angles for the orthorhombic system are listed in the tables in the order cr am bq with cr in the first column as the principal classification angle. The identi- fication is made by listing from. the tables all substances having values of the three angles lying not more than 30’ on either side of the measured angles. The results are set out on p. 8. 8 BARKER INDEX OF CRYSTALS AS AN ANALYTICAL TOOL [PYOC. S O C . A.naZ. Chem. The figures Acr Aam and Abq are the amounts by which the angles of each substance differ from the measured angles. cr am bq Acr Aam A b q Substance 67'39' 34'24' 30'59' 16' 25' 3 0.589 Diammonium dihydrogen 67'55' 33'59' 31'2' 0' 0' 0' 0.921 Citric acid monohydrate 68"O' 33"30' 31"O' 5' 29' 2' 0 .6 9 Phosphorous pentabromide hypophosphate The correspondence of all the classification angles of 0.921 with those of the measured crystal make it highly probable that the crystal is citric acid monohydrate. The necessity fas confirmatory tests will not be discussed here since the main point is to show the use of the classification angles but under the entry 0.921 in the Crystal Descriptions will be found further data on specific gravity refractive indices and other physical properties. It will be noticed that under this entry three settings A B and C are given; this fact deserves comment since it illustrates an important feature of the Index. I t often happens that the presence or absence of a face alters the Barker setting.The reader may care for example to find the setting for the above crystal on the assumption that the face numbered 6 is absent. The fact that other settings are given reduces the likelihood of failure to find a setting owing to the absence of a face. OTHER CRYSTAL SYSTEMS- The principles involved in dealing with crystals of the tetragonal and hexagonal systems are essentially the same as those already discussed. In these two systems there is only one classification angle cr. The monoclinic system does not introduce any special problems and will not be discussed here. It has four classification angles cr ra am and bq of which am is taken as the principal angle and is listed first. The anorthic system on the other hand is complicated by its low symmetry which gives little guidance as to the choice of axes; the fact that all three axes have positive and negative ends increases the number of possible orientations.It is therefore not surprising that it has been necessary to formulate several (24 in all) auxiliary setting and orientation rules. The application of these rules to each individual crystal would be intolerably laborious but it has been found possible to lighten the task by reducing the multiplicity of possible patterns of the 13 possible simple planes in a crystal to 764 projectively distinct configurations. The setting rules have been applied once and for all to these 764 configurations and recorded in an Atlas and in Tables of Configurations. The finding of a setting is then reduced to the matching of the observed pattern with one in the Atlas or the Tables which also provide instructions on the setting and orientation.ANORTHIC SYSTEM A PRACTICAL EXAMPLE- In this system each face is accompanied by only one parallel face and there are no duplica- tions by reflection. The words "plane" and "face," as used in counting are therefore synonymous. For reasons that need not be explained here the classification angles for this system are those lying in the upper left-hand quadrant of the projection cR Ra' a'm' mLb' b'Q and Qc i.e. (OOl) (101); ( T O l ) (TOO); (TOO) (110); (TTO) (010); (OTO) (011) and (Oll) (001). As an example let us assume that the following measurements have been made- Face . . . . . . b m a n G 0 w I' 4 P . . .. . . 0" 61" 89' 117" 61" 61" 115" 176" . . . . . . 90" 90" 90" 90" 4" 44" 43" 40" (The above two-circle measurements have been roughly calculated to the nearest degree from data given by Groth in the form of single-circle data in Chemische KrystaZZographie Volume 1 p.373. In spite of the low precision they will be sufficient for the present demonstration.) Fig. 5 is the gnomonic projection drawn from these data. When straight lines are drawn through the poles it will be seen that once again a regular pattern is obtained of which c-0-04' is one cell. For an anorthic crystal the net is no longer rectangular and the face pole c is not at the centre of the projection; nor are the direction lines of the equatorial poles always coincident with the net lines so that On and Ob do not coincide with cw and cl' although they are parallel to them. January 19651 BARKER INDEX OF CRYSTALS AS AN ANALYTICAL TOOL 9 We now have all the information needed for determining the classification angles.Remembering that in the anorthic system each face represents one plane we find eight planes present b m a ‘Y c 0 w and Z’. The number of zones* containing four planes (four-plane zones) are now counted and it is found that bgaa.tz = 1 four-plane zone. There are 5 three-plane zones bcE’ bow moc m ~ l ’ and ‘YWC. The figures 8.1.5 are therefore noted and the small dia- grams given under this heading in the Atlas consulted. It will be seen that the first diagram under 8.1.5~1 corresponds exactly with the present pattern. (To get exact correspondence the little figure must be viewed through the back of the paper but such reversal does not affect the topological relations between the various zones which are all that matter at the present stage).By copying the figure the letters a p y 6 b Q and c can now be inserted in the projection. To find the position of a in the Barker setting we are told to compare the angles ca and c’p,? i.e. the angles between the crystal faces MW and n’c. Since in the present instance n’c is the greater the Barker a is placed at the crystal face originally marked c. \ / n Fig. 5 . Gnomonic projection of an anorthic crystal The required angles can now be found by simple geometrical constructions on the projection or calculated$ from the known data. We therefore have (to the nearest degree)- In terms of the old crystal angles cR = nw Ra’ = u c a’m’ = co m’b’ = om b‘Q = ma and Qc == an. cR = 47” Ra‘ = 41” a’m‘ = 40” m’b’ = 46” b’Q = 28” and Qc == 28” If we now consult the table of angles in which the principal angle a’m’ appears in the first column for any substance having all its angles within 30’ on either side of the measured angles we find only one namely A.24 which has a’m‘ = 40”17’ m’b’ == 45”50’ b’Q = 27”42’ Qc = 38”9’ cR = 47”7’ and Ra’ = 40”37’.A.24 is tetraethylammonium mercury trichloride. * A “zone” is a set of faces in the same straight line in the gnomonic projection. Poles on such a line represent a set of faces all of which are parallel to a “zone axis.” They may be visualised as laces all of which are tangential to the sphere on a great circle. In the gnomonic projection counting in the equatorial poles must not be forgotten. Thc two parallel faces making up a Barker “plane” bear the same letter one being distinguished from the other by a dash thus C c‘ n n’.$ Angles cannot be read off directly from a gnomonic projection except in zones passing through the centre but there is a simple construction involving the drawing of only three additional lines that permits this to be done (see P. Terpstra and I,. W. Codd “Crystallometry,” Longmans Green & Co. London 1961 p. 5 ) . 5 The angle between two faces P and Q having +IpI and +2p2 is given by- cos PQ = cos pI. cos p2 + sin p,,sin pz.cos (C#I~-+~) 10 [Proc. SOC. Anal. Chew. The Tables of Configuration give the same information in a different form in which the various patterns are indexed under a series of “code numbers.” These can easily be derived for any pattern met with in a projection.Depending on circumstances either the Atlas or the Tables may be more convenient for a particular example but the end result is of course the same. BARKER INDEX OF CRYSTALS AS AN ANALYTICAL TOOL In the above example the Atlas was used. ADVANTAGES AND LIMITATIONS- The examples given above have it is hoped been sufficient to show that the Barker method is not difficult to understand and that it is capable of being used by workers without specialised crystallographic training. Those who are interested should read the Introductions in the first volume of the Index one of which was specially written with the non-specialist in mind. Further examples of the identification of unknown substances with detailed accounts of working methods including the actual process of measurement are given in a paper by the present writer,” and examples of the use of the method by students at Groningen University in Holland are to be found in a workt mentioned previously.The fact that it is necessary to prepare a crystal specially for measurement has sometimes been cited as a disadvantage of the method. It is in fact possible to make identifications with relatively imperfect crystals such as might be available without special preparation provided that the end faces have not been too badly damaged. Moreover the fact that an existing crystal can be examined without destruction is an im- portant advantage in some instances. In any event recrystallisation need not be an elaborate process. For example the crystal of sodium thiosulphate described in the paper on crystallo- chemical analysis was taken straight from a crop obtained by allowing a strong solution to cool undisturbed overnight.Quite small crystals can be measured; one of those described in the paper mentioned above measured 1.5 x 0.5 x 0.5 mm and even smaller specimens can be dealt with. Indeed the faces on small crystals usually provide the best reflections. Another possible objection to the method is that identification is naturally restricted to substances appearing in the Index. It must be remembered on the other hand that the method is generally applicable and that one of its important uses may be for instance to demonstrate the identity of two substances obtained by different synthetic routes. Moreover increasing use of the Barker method would itself lead to an increase in the available data and if analysts became familiar with the method they would doubtless also compile lists of the compounds in which they were most interested. To sum up it is hoped that this discussion may encourage analysts to examine the method and to test how far it meets their particular needs. Only in this way can the ultimate value of this work be decided. This is only partially valid. * 1,. W. Codd “Crystallochemical analysis and the Barker Index of Crystals.” t 1’. Terpstra and I,. W. Codd “Crystallometry,” Longmans Green & Co. London 1961 p. 218. Proceedings of the 15th International Congress of Pure and Applied Chemistry Volume 111 Lisbon 2956.