MICROSCOPE GONIOMETRY J. D. H. DONNAY AND W. A. O’BRIEN Hercules Experiment Station, Hercules Powder Company, Wilmington,
Crystal identification b y microscopical examination has been hampered b y the difficulty of correlating published goniometric data with the angular measurements made on the various views of the crystal, as seen in the microscope. Known graphical methods of crystal drawing and properties of the stereographic projection are here applied quantitatively to interpret microscopical views in terms of published data. Conversely, the axial elements of a crystalline substance can b e derived from microscopical measurements of angles between edges or projected edges.
THEas
usefulness of the microscope in the study of crystalline compounds has long been recognized (10). Optical properties, refractive indices (11, 14), optic axial angle, and dissuch persion of the optic axes (S), and morphological properties, such as crystal system, crystal habit, and outline angles (6, IS), have been employed in characterizing new compounds and identifying unknowns. In the present paper the authors propose to describe methods, here grouped under the name “microscope goniometry”, which enhance the value of the morphological studies that can be made with the microscope. The principles underlying these methods are not new, but their application to microscopical observations has not, to the authors’ knowledge, been discussed in the literature. Morphological observations by means of the microscope are made either on dry crystals, selected from the sample as received provided they are suffiiqiently well formed, or on crystals in contact with the mother liquor, after recrystallization on a glass slide. In either event, the immediate objective of the examination is the determination of the following propertiex crystal system, Miller indices of the faces, and apparent interedge angles seen on different “views”-that is to say, on crystals lying on various faces. With these data, the compound may be recognized again, as long as it is recrystallized in the same manner afid c w tains approximately the same impurities as when the observations were made, so that it exhibits the same habit. In practice, however, such observations often prove inadequate for the identification of an unknown. The chief reason is that published descriptions of microscopical crystal views, needed for comparison, are scanty and scattered in the literature. The microscopist must rely on other morphological data, universally used in crystallographic publications-namely, interfacid angles, coordinate angles, or axial elements. The interfacial angles, which are available for some 12,000 crystalline compounds (7), have been measured by means of the one-circle goniometer. They cannot, in general, be measured with the microscope. [Methods have been proposed in the past for measuring actual interfacial angles by means of the microscope. They have not received wide acceptance. When a zone axis can be placed perpendicularly to the stage of the microscope, the interfacial angle is measured directly (Mallard’s method). The inclination of a face to the vertical may be obtained by means of a calibrated focusing screw and an eyepiece micrometer (@.] The same holds true for coordinate angles of faces (azimuth and polar distance p ) , which have been obtained on the two-circle goniometer. Condensed tables of crystal constants list only the axial elements (axial ratios and interaxial angles), calculated from the measured angles. For the purpose of identification, axial ratios or interfacial angles must be translated in terms of interedge angles as seen on the microscopical views. The angle between two edges that are parallel to the plane of the microscope stage is seen in true value. In general, the true
Del.
interedge angle cannot be measured on the rotating stage whenever one of the edges, or both, are oblique to the plane of the stage. The projection of the true angle onto the stage is called the apparent angle. This is the angle actually measured. When the goal of the examination is the characterization of a new compound, it is desirable to know, in addition to the apparent interedge angles actually observed, the approximate axial elements or the Barker determinative angles (8) of the crystal. These constants are not, in general, readily apparent from the microscopical observations. Two problems, therefore, need to be considered: (1) Given the morphological constants of a crystal (axial elements, interfacial angles, or coordinate angles), to determine the apparent angle between any two edges that will be seen when the crystal lies OD any face. (2) Given the apparent interedge angles measured from microscopical views (one, two, three, or five independent angles, depending on the crystal system), t o determine the morphological constants (axial elements or interfacial angle between any two faces). These problems will be solved by graphical methods. The proposed methods are intended for use by chemical microscopists. For the benefit of those who may not be familiar with crystal projections and terminology, a short section on definitione is included. The reader is referred to standard texts on crystallography for a fuller discussion. DEFINITIONS
A zone is the assemblage of all the faces parallel to a commou direction, called the zone axis. Faces in a zone are said to be tautozonal. Tautozonal faces, produced if necessary, intersect along parallel edges; the direction of these edges is that of the zone axis. A face parallel to a given face is called its counterface. There are two methods of representing the directions of crystal faces on a sphere, drawn around any point inside the crystal witb an arbitrary radius. The first method, which may be called the direct method of spherical representation, is as follows: Through the center of the s here passes a plane parallel to every face (or to every pair of para& faces). his face plane intersects the sphere along a reat circle, called the face circle. The face planes of tautozona? faces intersect in a diameter, which therefore unctures the sphere in a pair of diametrally opposite points. The ratter are called the zone points; they give the direction of the zone axis. The second method, which may be called the polar method (or reciprocal method), is as follows: From the center of the sphere drop a perpendicular on every face. This face normal punctures the sphere in apoint, called the face pole. The face normals of tautozonal faces he in a diametral lane, which therefore intersects the sphere along a great circle. %‘he latter is called a zone circle. The zone axis is given b a line passed through the center perpendicularly to the plane of tge zone circle; such a line punctures the sphere in two diametrally opposite points, called the poles of the zone circle.
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In order to project the spherical representation of the crystal onto a plane, use is made of the stereographic projection, which dates back to Ptolemy. The South Pole of the sphere is taken .athe projection point and the plane of the equator as the projection planer (Geographical parlance is chosen for convenience.) In this system of projection, circles on the sphere are projected as circles (or straight
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
lines), a property whose first complete proof probably is that given by Clavius in 1593. The projection, moreover is mgletrue, BS was shown hy Halley in 1695. It is easily seen that all the p i n t s in the Northern Hemisphere are projected inside the equator (pri+tive circle) m d that points on the equator are themselves their own stereographic projections. Points in the Southern Hemisphere with the above convention, would he projected outside the ri&itive circle I n order to make their projections fall within t i e primitive ci;cle, such points me projected to the North Pole instead of the South Pale. They me represented by circlets in Bterwgraphie projection, whereas the projeotioas of oints in the Roorthern Hemisphere or on the eouator are marked gy points (or hy crarses). I n the established though unfelicitous crystallopphic usage, the term stereographic projection is oonstroed t o mean the stereographic projection of the faee poles (polar method), while the ntereographicprojection of the face circles (direct method) is called eyelographic projection. In the following, these terms are used in the crystdographic sense. The dihedral angle between two faces is given hy its supplement (called the interfacial angle). The plane angle between two edges,
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the interedge eagle, is likewise given by itsmpplement (called the interzonal angle). I n stereographic projection an interfacial angle appears as the BTC of great circle between two faee poles, 89 interzonal mgle as the angle between two sone circle+. I n cyolcgmphie projection an intermnal angle appears as the are of great circle between two zone points, an interfacial mgle as the angle between two faee circles. The projeotion of an interedge mgle on any plane is, of course, the supplement of the projection of the interzonal angle on the same plane. The rotating stereographic net (sc-cdled WuM net) is used for graphic oohstruotions (f). It oomists (Fi1) of a circle, mually of IO-cm. radius, on which two families of circles have been projected: great circles, 2' apart, intersecting in s diameter of the equator, snd small circles likewise 2' a art, perpendicular to the ssme diameter. The ne6 is pasted on {envy oardhomd m d B thumbtack is oushed throueh its center from the hack. [Such new, prepared by >I. A. Pe&,.urk, inn" be purclmcd irom the Vniveniig of Tarontn Pres, Toronto,,Ontario, Canaria (50 CP~IA adozen).i Allconsirurtionsarrmadeonasheei of trarina mer which is b t a t e d around the thumbtack. The center of tKee'sbket is reinforced hy meam of Scotch tape.
s OF THE GRAPHICAL soLunoNs
..
I
.
.
..
.
I
.
.
.
IRI,hit Not
_ \
Figure 9. Edge between Two Facer Projoctod on the Plene of the Stereographic Projection
Fi ure 3. Apparent Angle between two E&or Projected on the Plane oftho Stereographic Projection
A
r
Hguro4. LaseolaLrystal LyingonaPlene Different horn the Plane ofthe Stereographic Projection
^
September, 1945
ANALYTICAL EDITION
zone ab. Pass the zone circle AabB, intersecting the primitive circle along the diameter A B . The pole, p, of the zone circle is located on the diameter, De, perpendicular to A B , a t a stereographic distance, D p , of 90". Call 0 the center of the sphere. The zone axis, Op, is projected in cp. Hence the direction of the projected edge [ab] is perpendicular to the diameter, A B , determined by the zone circle ab. Retaining the assumption that the cr stal lies on the counterfbce of c , wiose pole is the center of the stereographic projection, it is easy to determine the amarent anale between two ed es, in (hjs particub- case. ConFigure 5. Crystal to Be Identified sicker (Fi ure 3) the Stereographic projection oflfour facas, a, b, c , d , o f the crystal. The edges, [ab] and [ad], between the pairs of faces a, b and a, d , res ectivel are oblique to the plane of c. The true interzonal ring[, sugplkientary to the angle between these two edges, is the angle, ad, seen on the stereographic projection. The apparent interzonal angle or the supplement of the an le between the two edges projected onto e, is given by the angle, %cD, between the d i e e + r s A B and A'D, since the angle between the projected ed es 1s e v e n by the perpendiculars to these diameters. Note that &e desired angle is given by the arc, B D , which the zone circles ab and ad interce t OILthe primitive circle. Also note that the primitive circle is t i e cyclographic projection of the face e. Removing our previous assumption, we can now conside; the fenera1 case in which the crystal is lying on a plane different rom the plane of the stereographic projection. Let z (Figure 4) be the stereographic projection of the face on the counterface of which the crystal is lying. The great circle, X B D X whose pole is 2, is the cyclographic projection of the face 2. ?he apparent interzonal angle of the edges [ab] and [ad] (projected onto z) is measured by the arc, B D , intercepted by the sides of the angle bad on the cyclographic projection of z. On Figure 4 this arc is projected stereographically, but its value can be read off in degrees by means of the Wulff net. (By a known theorem, the true value of B D is also iven by the arc B'D', where B' and D' are the intersections of &e straight lines ZB and I D with the primitive circle.) If it is desired to draw a view of the crystal orthor p h i c a l l y projected on the x plane, which is the view that will e seen under the microscope, it is known that the direction of the edge [ab] in such a projection, will be iven by the perpendicular to the radius OB'. The point B' can a t o be located a t the point where the primitive circle is intersected by the small circle BB' the locus of all the points whose spherical distance from X is equai to X B .
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firmation was desired. One of the crystals waa isolated (Figure 6 ) and the following angles were meamrzd on the rotating stage: L1 126.5"; L2 = 41 '; L 3 106 Translated in terms of interzonal angles, the measurements become oonlmr 5t.5'; oco,ml, projected on ml = 139"; O ~ O ~ O Cprojected , on ml = 74 The interference figure showed the optic plane to be parallel to the elongation, with the acute bisectrix inclined about 40" to the left, 89 shown. The suspected compound, &-terpin monohydrate, is described by Groth ( 8 ) ,the following data being especially noted (Figure 6)
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-
.
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m1:rnc = (110):(1~0)= 77O49' ~ 1 : ~=s (011):(011) = 50'57' 01:ml
=
(111):(110) a 52'49'
Orthorhombic bipyramidal, biaxial positive, plane of optic axee parallel to (OlO), acute bisectrix = a axis. From the habit and optical properties it is evident that if the unknown crystal actually is terpin hydrate, it consists of the four m faces, terminated by o faces, and elongated along the c axis. The b and p faces, illustrated by Groth, are missing. The optical properties are in qualitative accord with the dat8 for terpin hydrate, but without a comparison of Groth's goniometric data with the microscopically measured angles, a quantitative check is lacking. (To be sure the refractive index a also fyen by Groth, could be messured on the crystal to provide an a ditional check on the identity of the compound.) The stereographic projection of terpin hydrate (Figure 7) is constructed with the aid of the Wulff net from the goniometric data. The positions of the three pinacoids are first located, with ~(100)at the center of the net, and, since the compound is orthorhombic with b(010) and c(OO1) 90' apart on the primitive circle. &ice the poles of faces ml and mc are 77O 49' apart, each is placed at helf this value from a on zone b'ab. The pole of o1 will lie on zone circle cmlc', 52" 49' from mi, the angular distance being counted off alon the zone circle on the Wulf! net. Since the crystal system is orthorhombic, two independent measurements suffice to locate all faces, and the projection may be completed without additional angular data, the q1:q: angle (or any other) serving as a check on the accuracy of the construction. It remains to determine from t.he projection the angles to be exected for a c stal, of the Figure 6. Crystal of Terpin Hydrate %abit observed, Tying on face m1.
PROBLEM 11. Given the Values oflnterzonal Anglee, OT Apparent Interzonal Angles, Measured on ths Rotating Stags of the Microscope, to Find the Axial Elcmmts. If the axial elements are to be calculated by spherical trigonometry, it is sufficient to sketch an approximate stereographic projection of the crystal, merely to show the general distribution of faces and zones. For graphical solutions, however, it isimperative to plot the stereographic projection, by means of the Wulff net, as accurately as possible. This task is peculiarly awkward when the data are interzonal angles, because the latter appear as angles in the spherical triangles, in contradistinction to interfacial angles, which appear as sides. It is much easier to construct a spherical triangle on thehet when its sides are known than when its angles are known. For this reason, it is advisable either to use the cyclographic projection throughout or at least to plot it as a preliminary step and to derive the stereographic projection from it. An interzonal angle (or its supplement, the interedge angle) can thus be used without difficulty, because it becomes a side in a spherical triangle. APPLICATIONS OF THE GRAPHICAL METHODS
The following examples illustrate the application of these methods to problems confronting the microscopist. EXAMPLE I. Cafirniataon of the Identity of a Compound. An impurity in an organic sRmple consisted of prismatic crystals. These were suspected to be terpin hydrate, and a con-
.
Figure 7.
Stereographic Projection of Terpin Hydrate
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
596
Recrystallization from alcohol on a microscope slide gave the views shown in Figure 9. These were reco nized as monoclinic Class 2 in the hfauguin notation, on morp!ological and opticai evidence, and the faces were assigned the following compatible Miller indices: c(OOl), ml(llO),w!(lll), y(ll1). The following angles were measured on the rotating stage: L 1 = 76.3",L 2 = 68.0", L3 = 116.0", L 4 = 162.0'. Translated in terms of interzonal angles, the measurements become: mlcm4 = 103.7:; mm4c = 112.0'; mamcwl = 64.0";m4mlc,projectedonm4 = 18.0 . When it is known the crystal is monoclinic, the stereogra hic projection may be constructed as follows (Figure 10): Let tie c face be the plane of projection. Zone mlml may be drawn, and zone na,m is constructed so that the interzonal angle, mlm4, marked X on the figure, is equal to 103.7'. The pole of the possible face b(Ol0) lies on the primitive circle. The zone circle. bcb', here a diameter, bisects angle mlcmz. The zone circle am , also a diameter, bisects angle m1cm4; the location of zone circle bmlam4b' is found by a cyclographic determination of ita pole. The cyclographic projection of c is the primitive circle. The pole of zone m4c or edge [mcc] is A ; the pole of zone.mlc or edge [ m ~ c ] is B. Draw the locus of the points whose spherical distance from A is equal to fi = 68' and the locus of the points 68' from B. The intersection, D,of the two loci is the cyclographic projection of the edge [mlm4]. But this is the stereographic rojection of the pole of the zone circle mlw, which may now be Lawn The zone circle m4wl is then located on the WuH net, b utilizing the interzonal angle, m:m4Ul, marked Y OF the figure, wKich is the supplement of mIm4Wl and is equal to 64 The pole 01 is located at the intersection of this zone with zone mtc, and the stereogra hic projection may readily be completed by passing zone circ7es through the appropriate faces. From the stereographic projection, the views observed with the microscope can be accurately constructed, by known methods of
.
M'
Figure 8.
Determination of Microacopically Measured Angler for Terpin Hydrate
The method outlined in Problem I is followed. First the cyclographic projection, MNM', of face ml is drawn, 90"from the pole of ml (Figure 8). The interzonal angle 04mtm4is immediately located on the rojection. Its vertex is at ml; it is marked A (Fi ure 8). T f e arc, N A , which its sides mlo4 and mlm4 intercept on 5 ! f N M ' ,gives the measure of .X in stereographic projection. It will be read off in degrees d o n the great circle of the Wulff net that can be made to coincide w i d MNM'. The second interzonal angle, 0401m1, is likewise easily located on the projection, where it is marked (Figure 8). This angle, however, is the true value of 0401ml-that is, the supplement of the angle that would be measured if the crystal were viewed perpendicularly to face ol. What is wanted is the projection of angle O4Olml on plane ml. As seen above (Problem I), the rpjected angle is measured, in stereographic projection, by arc B h which angle intercepts on the cyclographic projection, MNM', of face ml. The third interzonal angle, 020104, presents a new roblem. Ita true value is marked Y on the projection (Figure 8f One of ita sides, oloz, does not intersect MNM' in the Northern Hemisphere. However, if this side be extended back, throu h the pole of ot', which is the counterface of 02, i t will intersect h N M ' at C. The arc, BC, intercepted on MNM' stereographically measures the projection on ml of the supplement of the interzonal angle o&04. Arc BC is therefore the supplement of the desired projected an le. The vafuea found from these graphical constructions are listed below, together with the observed values:
Figure 9.
Observed Forms of the Lactone of Hydroxytrtrahydroabietic A c i d
Vlslble h e n oblique lo the
plane of Ihe h s e me
hachured
L1 = 127' (graphical) and 126.5' (observed) L 2 42' (graphical) and 41" (observed) L 3 = 107" (graphical) and 106' (observed) The agreement between the two sets of valuos is satisfactory, and the identity of the compound is thereby confirmed. Although three conatanta a p ear to have been determined, in reality these are interrelated. $he morphology of an orthorhombic crystal can yield only two independent constants, corresponding t o the axial ratios a : b a n d c b. The agreement among three angles does, however, serve to check that the crystal is orthorhombic.
EXAMPLE 11. Calculation of Azial Elements. The crystallography of the lactone of hydroxytetrahydroabietic acid ( 4 ) was studied microscopically first, and later checked by two-circle goniometry.
Figure 10. Lectone of Hydroxytetrah droabirtic Acid Stereographically Projected on t l e c Face
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A N A L Y T I C A L EDITION
September, 1945
crystal drawin (1, 6, 18), and compared with the corresponding notebook sket%es. The axial elements are given (6) by the formulas:
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B = 180’ ca a: b = tan mlw c: b = tan wlac Ansle mica is equal to half the measured angle mlcmh-that is, 51.85 The other two angles, ca and WlaC, are read from the stereographic projection, 61.0’ and 34.0O, respectively. The axial elements found graphically are:
.
a : b : c = 1.27:1:0.67, ,9
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119.0”
For comparison, the axial elements calculated from the microscopical data by spherical trigonometry are: a:b:c = 1.272:1:0.670,,9 * 118’27’ The axial elements calculated from twwircle optical goniometry data are:
a : b : c = 1.2875:1:0.6635, ,9 = 117” 17’ The accuracy of the reshlts obtained by microscope oniometry may thus be estimated. It is of the order of 1%(0.5k in favorable cases, 2% in unfavorable cases). The values obtained by Weissenberg x-ray goniometry are: ao:bo:& = 1.285:1:0.670, B = 117.3’
It is interesting t o note that, after the crystal system has been determined as monoclinic, one view alone (Figure 9, c) is sufficient to calculate the axial elements. Including the apparent angle between edq;es [mlmd]and [mlc], which w e not used in the foregoing construction, the mew presents three inde endent angles and is therefore sufficient for the calculation. Tie stereographic projection (Figure 11) may be constructed graphically, as follows: The plane of projection is perpendicular to the c axis. The zone mlmpmsm is projected on the primitive circle. The face pole m4 is arbitrarily located; the zone circle m t m makes an angle, msmac, of 112’ with the primitive circle. The cyclographic projection A B of m, is drawn. On A B the interzonal angle m4mlc intercepts an arc BD, equal t o 18” (Problem I). Point D may so be located. The projection is pivoted on the Wulff net until D comes on the great circle which makes an angle of 112 with the primitive circle. This great circle is the zone circle mlcma. It determines the pole c, a t its intersection with the zone circle m4cm3. The face pole o1 is located by drawing the zone circle mwl, so that the interzonal angle mamcwt equals 64 ’, and determining its intersection with the zone circle mlcma. The. rest of the rojection may then be readil completed. Tie axial elements read from ths graphical construction are: a:b:c = 1.27:1:0.67, ,9 = 118’ in satisfactory ayeement with the elements iven above. I n the monoclinic system, the Barker angfes (d), used for identification purposes, are: cr(001) :(101), ra(lO1) :(loo)!~ ~ $ 1 0 0:) (110), bq(010):(011). These form letters and Miller indices refer to an arbitrary setting, known as the Barker setting, different from the one used above. The values calculated from two-circle goniometry and those obtained graphically from microscope goniometry are compared below: Calculated (2-circle goniometry) Graphical(microacope goniometry)
cr 29’ 9‘ 29O
ra
am
57’4’ 5Q0
48: 51’ 49
bq 56’ 2’
570
Graphically determined Barker an I? will be serviceable, if allowance is made for a possible error of2 FINAL REMARKS
It may not be superfluous, in conclusion, to concede that the rotating stage of a microscope will never supersede the optical reflection goniometer as an instrument for measuring crystal angles. The goniometer measures the angles between face normals (interfacial angles in single-circle goniometry, coordinate angles i n two-circle goniometry) with an accuracy that depends more on the perfection of the faces than on the precision of the instrument. This accuracy is usually a matter of 5 to 10 minutes; in exceptional cases, it may be 1 or even 0.5 minute. Al-
Figure ll. Lactone of Hydrox tetrahydrorbietic A c i d Stereographically Projected on a h a m Normrl to the E Axia
though the rotating stage of the microscope is commonly equipped with a vernier that permits reading to 0.1 ’, the usual error of an angular measurement is 0.5’; the accuracy may be increased somewhat by averaging a large number of readings. The accuracy of microscope goniometry is satisfactory for many purposes, especially in order to exploit the possibility of measuring interzonal angles. Although the microscope is less accurate than the goniometer, the problem of determining axial elements from the measured angles is essentially the same in both cases. The microscope, in a sense, is the equivalent of a single-circle goniometer. There are differences, however: Angles between coplanar edges, instead of angles between tautozonal faces, are measured for each setting of the crystal or, in other words, for each view; and the projections of interzonal angles on the plane of the stage can also be measured. LITERATURE CITED
(1)Barker, T. V., “Graphical and Tabular Methods in Crystallography”, London, Murby, 1922. (2)Barker, T.V.,“Systematic Crystallography”, London, Murby, 1930. (3) Bryant, W.M. D., J . Am. Chem. Soc., 60, 1394-9 (1938); 63, 611-16 (1941); 65, 96-102 (1943);Bryant, W. M. D., and Mitchell, M., Jr., Ibid., 60, 2748-51 (1938); 65, 128-37 (1943). (4) COX,R. F. B., Ibid.,66,765-70 (1944). (6) Dana, E. S., “A Textbook of Mineralogy.”, 4th ed., revised by Ford, W. E., New York. John Wiley & Sons, 1932. (6) Donnay, J. D. H., and Mblon, J., Am. Mineral.. 21, 250-7 (1936). (7) Groth. P.,“Chemische Krystallographie”. 5 vola., Leipsig. Engelmann, 1906-1919. This compendium lists about 8000
substances, a figure which can be increased by an estimated
50% t o cover the compounds investigated to date. (8)Ibid., Vol. 3, p. 658,1910. (9) Hill, E.A.,J . Am. Chem. SOC.,48,651-4 (1926). (10) Jelley, E. E., J . Roy. Microscop. SOC.,62,93-102 (1942). (11) Larsen, E. S., and Berman, H., U. S . Geol. Survey, Bull. 848 (1934). (12) Porter, Mary, Am. Mineral., 5, 89-95 (1920). (13) Shead, A. C.,IND.ENO.CHEM.,ANAL.ED.,9,496-502 (1937); 10,662-5(1938). (14) Winchell, A. N.,“Optical Properties of Organic Compounds”, Madison. University of Wisconsin Psess, 1943.
