Fedorov’s or Universal Optical Method

this displacement is. Bringing into focus, in the objective, points lying on the upper surface of the thin section generally corresponds to setting the given mineral into clear view; as points located on the lower surface, various specks may serve, which are usually found between the Canada balsam and the mineral plate.

To obtain more accurate results, several observations must be made, and on various points. Moreover, in order to eliminate the harmful influence of the backlash of the screw, each setting is determined as the arithmetical mean of readings obtained when the microscope tube is brought to this position from both sides.

In order to make the focusing more sensitive, it is necessary to use the strongest objectives, and even with immersion of the latter in strongly refracting liquids.

If the described method of determining the thickness of a thin section could be applied in a completely parallel light, then the displacement of the microscope tube would exactly answer to the required magnitude. But we have just seen that very strong objectives must be used, and this is connected with the introduction of a condenser, i.e. the observation has to be conducted in conoscopic light. In order to free ourselves from various accidental circumstances, we shall always lower the somewhat illuminating apparatus so as to have divergent light during the measurements. Under such conditions the following correction must be introduced into the measured thickness of the thin section.

Let a bundle of diverging rays of light pass through the point \(P\), lying on the lower surface of the plate (Fig. 30). Obviously, this point will appear to occupy the position \(P'\), which is the point brought into focus in the objective. Consequently, in order then to bring into focus the upper point \(O\), the microscope tube will have to be moved by the distance \(OP'\), and not \(OP\), and the measured thickness of the thin section will prove less than the true one.

It is easy to see that

\[ OR = OP \cdot \operatorname{tg} r = OP' \cdot \operatorname{tg} i, \]

whence

\[ OP = OP' \cdot \frac{\operatorname{tg} i}{\operatorname{tg} r}. \]

Since usually, in this method, strong objectives with a narrow aperture are used, and the particles being selected have an insignificant size and are placed in the center of the field of view, the angles \(r\) and \(i\) have a very small magnitude. Therefore the formula given may be represented in the following form:

\[ OP = OP' \cdot \frac{\operatorname{tg} i}{\operatorname{tg} r} = OP' \cdot \frac{\sin i}{\sin r} = OP' \cdot \frac{n}{\mu}, \]

where \(n\) and \(\mu\) denote the refractive indices of the mineral and of the surrounding medium; moreover, if no immersion system is used, then \(\mu = 1\).

Since the mineral under investigation can always be approximately determined, the numerical value of \(n\) is usually more or less known. From the formula it is evident that an error in determining the refractive index of the mineral will affect the measurement result the less, the greater the refractive index of the surrounding medium.

When strongly refracting liquids are used, errors of \(0.1\) no longer have any effect at all on the results of the measurements.

If the mineral has very high birefringence, then the refractive index of that ray is inserted into the formula which vibrates in the direction coinciding with the principal section of the polarizer.

The described method for determining the thickness of a thin section is not applicable to the Fedorov stage proper, where comparatively weak objectives have to be used.

§ 37. Nikitin’s method.

On the other hand, the universal optical method for investigating rock-forming minerals provides a new method for determining the thickness of a thin section, requiring, it is true, that in the grain under investigation there be some crack, or an intergrowth of a thin plate of another mineral, passing through the entire thickness of the thin section.

The measurement of the latter is based on the following¹. Let \(e\) (Fig. 31) be the thickness of the thin section, \(AB\) the crack cutting through it, \(BZ\) a line parallel to the axis of the microscope, and \(BN\) the normal to the plane of the thin section. In the position of the latter shown in the figure, the crack will appear as a band bounded by the traces \(A\) and \(B\) of the crack on the upper and lower surfaces of the thin section. Let the angle formed by the plane of the crack with the normal to the thin section be \(\angle DBA = \rho\), and the angle between this normal and the axis of the microscope be \(\angle NBZ = \varphi\).

From triangle \(ABC\) we have that

\[ \frac{d}{s} = \sin \angle BAC. \quad \text{But} \]

\[ \angle BAC = 90^\circ - \angle CBA,\ \text{and} \]

\[ \angle EBC + \angle ABD = 90^\circ - \angle CBA. \]

Consequently,

\[ \angle BAC = \angle EBC + \angle ABD = \varphi + \rho. \]

Thus,

\[ \frac{d}{s} = \sin(\varphi + \rho). \]

But if, for generality, different signs are assigned to the angles corresponding to the displacement of the schliere in different directions from the horizontal position, then the last formula must be represented as follows:

\[ \frac{d}{s}=\sin(\varphi-\rho). \]

Finally, from the triangle \(ABD\), the thickness of the schliere is

\[ e=s\cos\rho=\frac{d\cdot \cos\rho}{\sin(\varphi-\rho)}. \]

The quantities entering into the last expression are determined in the following way. First of all, the angle \(\rho\) is measured by the method described for the setting of the cleavage planes.

As for the angle \(\varphi\), it is determined just as simply by a direct reading on the vertical limb, at that position of the stage at which the visible projection of the fissure \(d\) is measured. In the numerical value of the angles, one or the other sign is put in depending on the direction of rotation of the stage about the axis \(J\). The determination of the width of the fissure \(s\) is the more exact, the smaller the angle \(OBA\), i.e. the closer the fissure is brought to the horizontal plane, for then the angle \(BAC\), entering into the formula, reaches its greatest value, so that its sinus varies within wider limits; but with excessively large inclinations about the axis \(J\) the edges of the fissure become difficult to distinguish. In general, the limit of the magnitude of the angle between the axis of the microscope and the plane of the fissure, at which measurement of its width is still possible, is sometimes not more than \(20^\circ\)—\(30^\circ\), and only very rarely reaches \(60^\circ\). This limit depends on the transparency of the schliere and the sharpness of the fissure.

The actual measurement of \(d\) is carried out with the aid of a screw ocular micrometer or the previously described Compensator-ocular. In the first instrument, one system of parallel threads is displaced in its plane by turning a special screw provided with a limb. If objective No. 2 is used, then usually the displacement of the thread of the ocular micrometer corresponding to a turn of the latter’s screw by one division measures \(0.00036\)—\(40\ mm\). This constant of the instrument is readily established by direct measurement of an objekt-mikrometer placed on the stage. Thus, in order to determine the length \(d\), we bring the thread of the ocular micrometer into coincidence with one edge of the fissure, noting at the same time the division of the micrometer limb. Then we move the thread to the other edge of the fissure, or of its projection, and the difference of the readings, multiplied by the “number” of the instrument, gives the required quantity.

In computing the formula given above it is necessary to make certain corrections. Thus, if the refractive index of the mineral differs from that of the glass segments, then the magnitude of the angles \(\rho\) and \(\varphi\) must be corrected according to the diagram of Fig. 6.

However, the angle remains unchanged if the crack is gaping. But in the latter case the measured quantity \(d\) must be corrected. In Fig. 32 it is seen that \(CB\), the projection of the crack onto the plane perpendicular to the axis of the microscope, is not equal to its measured apparent width \(d\). If, with the preparation in a horizontal position, the gaping of the crack \(\sigma\) is measured with an ocular micrometer, then \(CB = d - \sigma \cos \varphi\).

Hence the true thickness of the thin section is:

\[ e = \frac{(d - \sigma \cos \varphi)\cos \rho}{\sin(\varphi - \rho)} . \]

It is evident from the formula that, of all the quantities entering into the latter, chief attention must be paid to the measurement of \(d\). But even with perfectly correct determinations of all the elements, an incorrect result may easily be obtained. This occurs either because the crack does not pass through the entire thickness of the thin section, or because the edges of the walls of the crack have been somewhat broken off during the preparation of the specimen, as a result of which the measured width \(\sigma\) proves to be exaggerated. However, if cracks with regular walls and with a sufficient distance between them are selected, one can be more or less guaranteed against errors caused by these latter circumstances.

In any case, the described method of determining the thickness of a thin section cannot have general significance, since its application depends on the chance occurrence, in the grain under investigation, of a suitable crack or inclusion.

Determination of the magnitude of birefringence.

§ 38. a) By the direct method.

Thus, we have described separate procedures of optical investigation used in determining the magnitude of birefringence. In the general case the latter problem consists in determining the retardation of rays vibrating along selected directions, and the thickness of the thin section. Usually the magnitude of birefringence is referred to the axes of greatest and least elasticity, but in the case of a biaxial crystal it is sometimes necessary to find also two other characteristic quantities of birefringence: \((n_g - n_m)\) and \((n_m - n_p)\). Before determining the retardation, the selected axes of elasticity must be brought into the plane perpendicular to the axis of the microscope. For this purpose, in the usual manner, the plane of elasticity is set, the pole of which is one of these axes, thereby coinciding with the axis of the stage \(J\). Then, according to the rules set forth in the chapter on the determination of the axes of elasticity, the second axis is brought...

...is brought into the horizontal plane. Now, when measuring the retardation by means of a compensator, it is necessary to turn the microscope stage exactly by \(45^\circ\), so that the greater axis of elasticity lies in the direction of the slot, since in all compensators the smaller axis coincides with this direction.

When comparators are used, exact orientation of the microscope stage is not required; it is necessary only to obtain equal intensity of illumination. By one means or another the required retardation \(p\) is determined. Finally, by one of the methods described above, the thickness of the section \(e\) is measured.

According to the definition of the magnitude of birefringence, the latter is

\[ = \frac{p}{e_1}, \]

where \(e_1\) denotes the thickness of that layer of the mineral through which the light rays pass. Obviously, \(e_1\) depends on the inclination of the preparation about the axes \(H\) and \(J\) and is equal to

\[ \frac{e}{\cos H . \operatorname{Cos} J}. \]

Thus the required quantity is

\[ x = p \cdot \frac{\operatorname{Cos} H . \operatorname{Cos} J}{e}. \]

§ 39. c) By means of an auxiliary mineral.

If in the section, next to the mineral being investigated, there is a grain of another mineral whose birefringence is known exactly, then, taking the thickness of the plates of both minerals to be the same, we can derive the required birefringence of the given mineral from a comparison of the retardations of the latter.

Denoting the latter by \(x\), the corresponding retardations of the given and auxiliary minerals by \(p_1\) and \(p_2\), and the thickness of the section by \(e\), and taking into account the inclination of the preparation about the axes \(H\) and \(J\), we have

\[ p_1 = x \cdot \frac{e}{\cos H_1 . \operatorname{Cos} J_1} \quad \text{and} \quad p_2 = (n_g - n_p) \cdot \frac{e}{\cos H_2 . \operatorname{Cos} J_2}. \]

Hence

\[ \frac{x}{p_2} = \frac{x}{(n_g - n_p)} \cdot \frac{\operatorname{Cos} H_2 . \operatorname{Cos} J_1} {\operatorname{Cos} H_1 . \operatorname{Cos} J_2}, \quad \text{and} \]

\[ x = \frac{p_1}{p_2} (n_g - n_p) \frac{\operatorname{Cos} H_1 \operatorname{Cos} J_2} {\operatorname{Cos} H_2 \operatorname{Cos} J_1}. \]

As an auxiliary mineral one can in most cases take only quartz, which has a constant composition and in general does not change. In order to substitute into the formula the value of the birefringence of quartz, \(n_g - n_p = 0.009\), it is necessary to lay the optical axis of this mineral in the horizontal plane; but this, as we saw above, is not always possible. In this case one must proceed as follows—

in the following manner. We bring into coincidence with the plane of symmetry of the microscope the principal section of quartz, shown in Fig. 33 by the corresponding ellipse of elasticity \(A_1B_1A_2B_2\).

Let us suppose that, when the preparation is tilted about the axis of the stage \(J\), the optical axis \(A_1A_2\), occupying an extreme position, forms with the horizontal plane an angle \(\varphi\), which is easily determined on the working net.

For this position of the preparation there will be measured the path difference of the rays vibrating in the section \(B_1B_2\) along the vector \(OB_2\) and along the axis \(n_p\).

Starting from the equation of the ellipse, we may write that

\[ OB_2 = \frac{n_g n_p} {\sqrt{\,n_g^2 \sin^2 \varphi + n_p^2 \cos^2 \varphi\,}} . \]

If, moreover, we take into account that, in setting up the principal section of the quartz, it is not necessary to make a tilt about the axis \(H\), then the formula given above assumes the form:

\[ x = \frac{p_1}{p_2}\, n_p \cdot \left( \frac{n_g}{\sqrt{\,n_g^2 \sin^2 \varphi + n_p^2 \cos^2 \varphi\,}} - 1 \right) \cdot \frac{\cos H_1 \cos J_1}{\cos J_2}, \]

where for quartz

\[ \begin{aligned} n_g &= 1.556,\\ n_p &= 1.547. \end{aligned} \]

In deriving the quantity \(OB_2\) we assumed that the extraordinary ray remains normal to the section \(B_1B_2\), which is in general incorrect. Let us therefore consider what error follows from this assumption of ours.

If the extraordinary ray deviates from its original direction, so that the corresponding vector of elasticity \(OC\) makes an angle \(\xi\) with the optical axis, then, according to the formula of Liebisch¹,

\[ \tan \xi = \frac{n_g^2}{n_p^2}\tan \varphi . \]

There is no doubt that the angle \(\varphi\) cannot exceed \(30^\circ\), because otherwise the optical axis can be brought into coincidence with the plane perpendicular to the axis of the microscope. If this is so, then, substituting the values of the quantities entering into the right-hand side of the equation, we obtain \(\xi = 30^\circ 17'\).

Thus the error does not exceed the accuracy of the instrument, and therefore it may be completely neglected.

Turning now to the method itself for determining the magnitude of birefringence, it must be said that it has an even more limited application than the direct method described above, but in comparison with the latter it is more accurate, since in it errors arising from the measurement of the thickness of the thin section are avoided.

§ 40. Determination of the magnitude of birefringence by the method of M. Lévy.

The following method makes it possible to determine the magnitude of birefringence rather accurately and quickly, without the aid of a compensator, but on the condition that the thickness of the thin section is known. The method consists in observing the interference color of the mineral under investigation, corresponding to the path difference of rays vibrating in the selected directions, and in comparing it with a special chromatic table constructed by M. Lévy. In this table¹, whose scheme is shown in Fig. 34, along the ordinate axis there is laid off a series of commonly occurring values of the thickness of a thin section, and along the abscissa axis—path differences expressed in millionths of a millimeter; Newtonian colors corresponding to these path differences follow in increasing order from left to right in the form of vertical columns. The use of the table is as follows. Knowing the thickness of the mineral plate \(e\), we follow the corresponding horizontal straight line to the place where the interference color observed in the mineral is shown, and obtain a definite abscissa corresponding to the actual path difference \(p\). Then the required magnitude of birefringence is

\[ x=\frac{e}{p}. \]

Of course, when working on the Fedorov stage, the thickness of the thin section must be corrected for the inclination about the axes \(H\) and \(J\), i.e., one must take not the true value \(e_0\), but

\[ e=\frac{e_0}{\cos H \cdot \cos J}, \]

so that

\[ x=\frac{p \cdot \cos H \cdot \cos J}{e_0}. \]

In order not to make calculations each time, certain additions are provided on the table.

Giving the above formula the form

\[ \frac{e}{p}=\frac{1}{x}, \]

we see that different values of the coordinates corresponding to one and the same magnitude of birefringence give points lying on one straight line passing through the origin of the coordinate axes. Therefore, having obtained a point on the table, we connect it with the origin of the coordinates by a straight line, which we continue until it intersects the upper or right side of the table, where the corresponding magnitudes of birefringence are entered.

It is evident from the table that, with a small path difference and with a small thickness of the thin section, the determination of the magnitude of birefringence by this method will be very inaccurate, because in this region the indicated straight lines crowd together strongly. And in general this method can be used only in preliminary investigations, when the thickness of the thin section is known only approximately.

In conclusion it should be noted that M. Lévy’s table makes it possible to determine the thickness of a mineral plate if the magnitude of its birefringence is known, since this problem is the inverse of the preceding one.

Chapter III

Determining the relation between the optical and crystallographic elements of minerals.

Investigation of twins.

Until now we have considered the optical constants characterizing the properties of minerals as physical bodies. We shall now pass to those constants which express the spatial relations of the elements of minerals. In order that these relations may be determined, the grain under investigation must possess at least one face, or one plane of definite cleavage; and this circumstance greatly narrows the sphere of microscopic investigations of this kind.

The creator of the universal method, E. S. Fedorov, provides a means of finding the indicated relations between the optical and crystallographic elements of minerals even in the absence of the above-mentioned planes. This means consists in the study of twinned formations, the presence of which in the mineral grain under investigation is the only limiting condition.

Since the determination of twinned formations is one of the most essential derivatives of the universal-optical method of investigation, it is necessary to dwell on the laws of twinning in greater detail.

§ 41. Formation and types of twins.

A twin is a crystal¹ consisting of two individuals having in common some plane and, at the very least, one direction lying in the latter. This condition will be fulfilled only in the event that one individual is derived from the other by one of the three following procedures:

1) by rotation through 180° about a line called the twin axis;
2) by mirror reflection in a certain plane called the twin plane; and
3) by both, or by another of the preceding methods, leading to the same result.

These three laws differ only under the condition that all the physical properties of the mineral in general are taken into account, among which many depend on the direction of the vector representing the corresponding property. If, however, the crystal is considered from the point of view only of optical properties, or of other properties homogeneous with the latter, which find their external expression in the very general case—

... in the form of an ellipsoid with three axes, then all three laws of the formation of twins are, obviously, reduced to one, characterized by the simultaneous presence of a twinning plane and a twinning axis perpendicular to it. Therefore, among other things, in the microscopic study of crystals many varieties of twins formed according to the first two laws cannot in general be distinguished.

The determination of minerals by means of observing twinned formation is based on those spatial relations which exist between the axes of elasticity of the crystal and the elements of the twin, chiefly the twinning axis, and on the crystallographic significance of the latter. In order to understand what position the elements of the twin may occupy in the crystal, it is necessary to dwell on the theory of twinned formation, which in brief outline may be set forth as follows.

Proceeding from the theory of the molecular structure of crystalline substance, we may imagine a homogeneous crystal as consisting of the smallest physical, or crystalline, particles, arranged parallel to one another and forming in the aggregate the so-called space lattice. The form of the latter is determined by the forces of cohesion acting between the individual particles. In the general case the magnitude of the forces of cohesion varies with direction, and there may be several of their maxima and minima.

Let, in Fig. 35, \(OA\), \(OB\), and \(OC\) represent the vectors of three such special values of the forces of cohesion, of which \(OA\) and \(OB\) correspond to two maxima, defining a plane which will have the greatest crystallographic significance. During the formation of a mineral, each newly deposited particle strives to assume such a position that its axes of greatest cohesion are parallel to the corresponding directions in the crystal already formed. In this there may be several different cases.

1) All three special directions remain parallel. In this case, obviously, the particles are arranged quite parallel to one another, and an ordinary homogeneous crystal is formed.

The appearance and development of the faces of the latter will depend chiefly on the relation of the forces of cohesion acting in the various directions. Thus we may expect that the plane \(AOB\), in which the molecular network possesses the greatest density, will attain the greatest development.

If along the direction \(OC\) the force of cohesion is not great, then the indicated plane may manifest itself in the form of a cleavage plane.

2) Under certain conditions of crystallization, another state of equilibrium may arise, in which the directions of the principal forces of cohesion \(OA\) and \(OB\) remain coincident, but the newly deposited particles themselves will be rotated through \(180^\circ\) about the axis \(OR\), perpendicular to the plane formed by these directions.

If the changing conditions of crystallization prove to be constant, then these particles will act upon the newly deposited ones in such a way that the latter assume a position entirely parallel to them, and as a result a crystal will be obtained consisting of two individuals intergrown with one another according to the twin law, determined by the twin axis \(OR\). Since in the most general case, relating chiefly to crystals of the triclinic system, only the plane \(OAB\), as containing the directions of the greatest forces of cohesion, will have crystallographic significance, the twin formations just derived define the 1st type, in which the twin axis is the normal to a possible face of the crystal.

In crystals of higher degrees of symmetry, characterized by the presence of right angles between crystallographic elements, the twin axis may be a possible edge of the crystal, but the crystal itself for the most part loses its twin character for physical properties.

3) Finally, the relations between the forces of cohesion under given conditions of crystallization may be such that, for the newly deposited particles, there will be only one direction of the greatest force of cohesion in common with the particles already deposited. In this case there may be several special cases.

a) The axis of rotation will prove to be that same direction of the greatest force of cohesion, for example \(OA\). If the angle of rotation is arbitrary, then there results only a parallel intergrowth of two individuals, essentially representing a deviation in the growth of a homogeneous crystal. Such intergrowth, theoretically possible, is often encountered in reality, and in this respect feldspars can provide many interesting examples¹.

Of the mass of cases of such intergrowth, the one of greatest importance for us is that in which the angle of rotation about the axis mentioned equals \(180^\circ\), for only then can we speak of a twin formation. This second type of twins differs from the one considered above also in that, in it, the real realization in the most general case is already found by the twin axis, as coinciding with the direction of the greatest force of cohesion. Thus, in the 2nd type the twin axis is a possible edge of the crystal.

Of course, in crystals of more or less high degrees of symmetry, mutually perpendicular edges and faces are often encountered, so that there this type of twins is not infrequently reduced to the first type.

b) During deposition of the particles of the second individual of the crystal, the coincidence of the greatest axis of cohesion \(OA\) may also be attained by rotating the particles by \(180^\circ\) about a straight line perpendicular to this axis. There exist—

there is a whole series of such lines, whose geometrical locus is a plane perpendicular to the axis \(OA\), and many of these lines may serve for the formation of twins, provided only that the plane in which this line and the axis \(OA\) lie has crystallographic significance. Obviously, among the mass of such twins the one whose individuals have in common a plane containing the greatest axes of cohesion will have the greatest chance of existence, for under this condition equilibrium is most readily established in the formation of the crystal.

Strictly speaking, only one line, \(OS\), satisfies the condition derived; thus in the 3rd type the twin axis is normal to a possible edge and lies in a possible plane of the crystal. This type is found chiefly in the triclinic system, since in crystals of a higher degree of symmetry it reduces to cases already known to us.

The three derived types of twin formations are characterized by the mutual perpendicularity of the twin axes belonging to them, as a result of which each of them can be derived from the other two, if one admits that the second individual of the twin is subjected to a successive rotation through \(180^\circ\) about axes corresponding to these two types. Of course, this representation has a purely geometrical significance, since a crystalline particle, in its orientation, can hardly have the possibility of undergoing such a complex displacement.

Strictly speaking, what has been said exhausts all cases of twin formations theoretically admissible in the construction of crystals from perfectly homogeneous physical particles, on the assumption that the latter are arranged either parallel to one another or—at least—so that the principal, under the given conditions of crystallization, plane of the forces of cohesion and one direction lying in this plane remain parallel.

But observations¹ have shown that a fourth type of twin formations is also possible. If we imagine that two principal directions of the forces of cohesion lying in a given plane have almost the same value, then equilibrium will also occur in the case where, in the formation of a double crystalline accretion, these differently named directions of the individuals of the latter coincide. This coincidence will evidently take place on the condition that the axis of rotation is the line bisecting the angle between the indicated directions. From a geometrical point of view, this formation may justly be called twinned, and thus in the 4th type the twin axis lies in a possible face and bisects the angles between possible edges of the crystal situated in this face.

Thus, in the most general case, all twins can be formed with the aid of only one twin axis, which is the fundamental element of these formations.

The twin plane perpendicular to this axis is added either in crystals of higher degrees of symmetry, or when considering physical properties whose functions have a simple form.

But there may also be a special type of crystalline formation, determined by a single twin plane that is a plane of symmetry of the entire crystal. In this case, obviously, the individuals of the twin consist of enantiomorphous particles possessing somewhat different properties, and therefore the given formation cannot be assigned to true twins¹.

It must be said that among rock-forming minerals this kind of crystalline intergrowth occurs very rarely and is of no significance for the petrographer.

Which of the derived types of twins, then, should occur more often?

If homogeneous crystals consisting of parallel particles occur more often than twins, then among the latter, in the struggle for existence, that one will prevail whose individuals have more common elements. Therefore the order in which the types are designated will correspond to the degree of their prevalence. It is true that the 2nd and 3rd types seem to have equal rights to existence, but in the latter the principal axis of cohesion of one individual of the twin, although parallel to the corresponding axis of the other individual, has the opposite direction, owing to rotation through 180° about a straight line perpendicular to it, as a result of which the twin formation itself is less stable.

As regards the forms in which twin formations manifest themselves, they can be very diverse.

In the case analyzed above, it is assumed that, toward the end of the formation of one individual of the twin, the conditions of crystallization changed somewhat, and another individual began to crystallize out.

But the formation of twins may also be conditioned by other circumstances, for example, by the surface tension of the crystal², and then this process may take place under entirely constant conditions, so that both individuals will develop simultaneously.

The latter circumstance, in many minerals, especially the plagioclases, must in all probability have a dominant significance, since these minerals are generally known only in the form of twins. The condi-

the effects of crystal formation may change repeatedly, and this is reflected in the fact that the latter will consist of several individuals, intergrown with one another alternately or according to one and the same, or according to different, twin laws, and forming so-called polysynthetic and polygenetic twins.

Depending on the state of the medium in which crystallization takes place, the intergrowth of the separate individuals of twins may proceed in a very different manner and along different planes, even along surfaces of the most irregular kind. But there usually exist, characteristic of one or another kind of twin formation, special planes along which the individuals of the twin preferentially intergrow or penetrate one another.

These planes coincide with the planes of the greatest cohesive forces common to both individuals, and therefore attain the greatest development, thereby determining the general habitus of the crystal.

Thus, in the first type of twins, such a plane will be the plane perpendicular to the twin axis.

In the second type, the plane common to both individuals has no great crystallographic significance; therefore the intergrowth is for the most part indefinite, and the crystal itself develops along the direction of the axis of greatest cohesive force common to both individuals and coinciding with an edge of the latter.

In the third type, both the axis of greatest cohesion and the twin axis lie in the possible plane, which will also be the plane of intergrowth and of greatest development of the individuals of the twin.

The arrangement of the elements of twins of the fourth type approaches that of the preceding type, but, owing to the almost identical significance of the two axes of cohesion, the crystal itself is more or less isometric.

The absolute and relative dimensions of the separate individuals of a twin fluctuate within wide limits and not infrequently descend to a microscopic magnitude. Many polysynthetic twins, especially of the first and third types, often represent a packet of the thinnest plates, distinguishable sometimes only under high magnifications; some minerals, such as leucite and microcline, consist of almost submicroscopic little plates intergrown according to various laws, etc. On the other hand, minerals are known that give only simple twins with uniformly developed individuals; these twins for the most part belong to the second and fourth types.

In general, apart from the chemical nature of the mineral and the conditions of crystallization, the outward appearance of twin formation is greatly influenced by the law of the latter, since with it is connected the development of some or other planes and directions. These twin elements, corresponding to the axes of the greatest cohesive forces, usually coincide with the principal crystallographic elements of the mineral, but sometimes

depending on the purely local conditions of crystallization, and are secondary elements having a more complex index.

From what has been said it follows that in every mineral species, besides the twinning formations usual for the latter, one may always encounter twins of more or less accidental origin, the investigation of which may yield interesting results in the sense of elucidating the molecular structure of the substance and the conditions of crystallization.

There is no doubt that such twins can most likely, and in the greatest number, be encountered among the constituent parts of rocks; therefore the exact and rapid determination of a twinning formation in rock thin sections is of great importance.

The universal method, like no other, makes it possible to cope with this problem.

§ 42. Determination of the elements of a twinning formation.

As was indicated above, every twinning formation, when optically investigated, appears to possess both a twinning plane and a twinning axis perpendicular to it. Both of these elements can easily be determined on the Fedorov stage. Indeed, the twinning plane is the plane of symmetry for the parts of the optical indicatrix belonging to each of the two individuals of the twin; therefore, if the indicated plane is brought into coincidence with the plane of symmetry of the microscope, then, upon rotating the preparation about the axis $J$, both individuals will brighten and darken simultaneously, i.e. the twin will as it were disappear.

The actual setting is carried out as follows. The thin section is cemented so that the seam line separating one individual of the twin from the other passes through the intersection of the threads of the ocular cross. Since the plane of intergrowth of the twin is for the most part also the twinning plane, first of all the seam line is brought into coincidence with the vertical thread of the cross, and by inclining the preparation about the axis $H$ and rotating the stage around the axis $J$ one tests whether the phenomenon described above will occur. If the disappearance of the twin does not occur under all possible inclinations about the axis $H$, then the twinning plane, in all probability, occupies a position different from that of the plane of intergrowth, and it is necessary to find it by trial, by rotating the preparation about the axis $N$ and inclining it along the axis $H$.

The readings of the coordinates of the twinning plane thus found, and the plotting of it on the net, are carried out in the known manner. In the example of Table VIII, the twinning plane, represented by the arc $DD^{1}$, was found from the coordinates: $2^\circ$, to the right $50^\circ$.

If this plane occupies a position rather close to the section of the thin section, it cannot be observed; and therefore one must at once proceed to finding the twinning axis. The latter is characteri-

it is found that, when it is brought into coincidence with the axis of the microscope, the parts of the ellipsoid of elasticity belonging to the individuals of the twin are superposed one upon the other along the plane of symmetry perpendicular to the axis of the microscope. Therefore, if after such a coincidence the stage is rotated arbitrarily, the twin will behave toward polarized light as a single individual.

The indicated coincidence is, of course, effected by trial; the determination of the coordinates of the axis found and the plotting of the projection of the latter have already been described in the corresponding place (§ 25).

In Table VIII the projection of the twin axis \(B_{1.2}\) is found from the coordinates: \(272^\circ\), toward oneself \(40^\circ\).

Through this point, as the pole of the twin plane, one could at once draw the projection of the latter—the arc \(DD^1\).

The direct path described for finding the elements of the twin formation is in many respects inconvenient and does not always give precise results, not to mention that in some cases these elements are not at all amenable to observation.

Much more fruitful is another method of indirect determination of the elements of a twin, chiefly of the twin axis, based on the following. By the very definition of the concept of a twin, all unambiguous elements of the individuals of the latter form equal angles with the twin axis, or, conversely, the twin axis bisects the angles formed by the unambiguous elements of the individuals of the twin. Among the latter, the principal axes of elasticity \(n_g\), \(n_m\), and \(n_p\), which, as was shown earlier, can readily be found in any grain of a mineral, are of the greatest significance for us.

Thus, if the twin axis bisects the angles between like-named axes of elasticity, then it represents the line of intersection of the planes passing through the latter. This circumstance makes it possible quickly to find the position of the twin elements from the axes of elasticity found and, at the same time, to verify the accuracy of the determination of the latter, since, according to the preceding, the planes passing through like-named axes of elasticity must intersect exactly in one line—the twin axis, which can then already be found from the known coordinates in the instrument.

Let us consider, using Table VIII as an example, how this is done. Without changing in any way the position of the thin section on the inner circle of the stage, we determine the position of the axes of elasticity for the first individual (I), denoting them by \(n_g^1\), \(n_m^1\), \(n_p^1\), and then for the second (II), with the corresponding designation of the axes of elasticity as \(n_g^2\), \(n_m^2\), \(n_p^2\). The elements found are already sufficient to determine the position of the twin axis, and the further work is limited to constructions on the net.

For this purpose we draw arcs of great circles through like-named poles, which is done according to Problem 4 (§ 12). In the present case the arcs drawn intersect exactly at the point \(B_{1.2}\), which will be the pre—

constitute the pole of the twin axis; from the latter one may at once construct also the projection of the twin plane.

It remains now to verify whether, in the given twin, the elements found by the indicated construction are actually observed. Turning to the diagram, we see that the twin plane occupies a position fairly close to the section of the thin section; therefore it will be rather difficult to bring it, with the requisite accuracy, into the plane of symmetry of the microscope. On the contrary, the twin axis can much more easily be made to coincide with the axis of the latter. For this purpose, in accordance with the principles set forth in § 25 of this chapter, we draw a diameter through the point \(B_{1\cdot 2}\) and bring the index of the inner circle of the stage into coincidence with the division corresponding to the reading on the circumference of the network at one of the ends of this diameter, for example \(272^\circ\).

Next, along the polar parallels, we count off the angular distance of the point \(B_{1\cdot 2}\) from the center of the network, corresponding to the projection of the microscope axis, and by the same number of degrees we deflect the stage about the axis \(J\) in the direction opposite to that observed on the network, i.e. away from ourselves (the marked end of the diameter must face toward us). Then the twin axis should coincide with the microscope axis, which is checked by rotating the stage of the latter.

In other cases it is more convenient to verify the twin plane, which is found from the coordinates read off on the diagram.

Thus, the position of the twin axis is determined quite easily by one of the methods indicated above.

But what is of interest to us is not, strictly speaking, the twin axis itself, but the angles which it forms with the various elements of the crystal, among which the greatest significance is attached to the same axes of elasticity. Since the twin axis is the bisector of the angles between like-named directions of the two individuals, it becomes wholly immaterial to us with the elements of which individual we have to deal.

The measurement of the angles formed by the twin axis with the axes of elasticity, or the determination of the coordinates of this axis relative to the latter, is of course most readily carried out with the aid of a three-legged compass. In our example the indicated coordinates have the following values:

\[ \angle B_{1\cdot 2} \begin{cases} n_g = 37^\circ \\ n_m = 59^\circ, \\ n_p = 72^\circ \end{cases} \]

As is seen, the coordinates represent the angular distance of the pole of the twin axis to the corresponding nearest axis of elasticity.

The described method of determining the coordinates of the twin axis gives the most reliable results, but it also requires rather much time. In order to avoid the latter, the given problem can be considerably—

Fedorov’s or universal-optical method.

considerably simplified. Indeed, the twin axis bisects the angles between the axes of elasticity; therefore it is only necessary to measure these angles, divide them in half,—and we obtain the required coordinates, without knowing precisely the position of the twin axis. In doing this it is only necessary to pay attention to whether the twin axis lies in the chosen angle (for the intersecting lines form two different angles, of which only one can be fully represented in stereographic projection). To resolve this question, we mentally draw arcs of great circles through the like-named poles and find, at their intersection, the approximate position of the twin axis, which also facilitates our choice of angles.

Let us turn to our example. Assuming that the position of the twin axis has not yet been exactly found, by the method indicated we at once determine it near the point \(B_{1\cdot 2}\). Of all three angles between the axes of elasticity that can be measured directly on the diagram, only the angle \(n_g^{1} n_g^{2}\) contains the twin axis and therefore at once gives the corresponding coordinate if it is divided in half; the two other coordinates are obtained by halving the supplements to \(180^\circ\) of the angles \(\angle n_m^{1} n_m^{2}\) and \(\angle n_p^{1} n_p^{2}\).

All that has been said is written in the margin of the net in the following manner:

\[ \begin{array}{rcl|rcl} \angle n_g^{1}\, n_g^{2} &=& 74^\circ & & 74^\circ & \\ \angle n_m^{1}\, n_m^{2} &=& 61^\circ & & 119^\circ & \text{and } \angle B_{1\cdot 2}\left\{ \begin{array}{l} n_g = 37^\circ\\ n_m = 59.5^\circ\\ n_p = 72^\circ \end{array} \right.\\ \angle n_p^{1}\, n_p^{2} &=& 36^\circ & & 144^\circ & \end{array} \]

Depending on how the section of the thin section is oriented with respect to the elements of the twin, several different cases of distribution of the angles between the axes of elasticity may arise.

In distinguishing these angles, one may also take into account that in the final result one of them must be acute, and the other two obtuse; moreover, it is not difficult to see that this acute angle is the largest of all the acute angles formed by the three pairs of like-named axes of elasticity.

Bearing this last circumstance in mind, one need not even look for the approximate position of the twin axis, since it is sufficient only to measure the angles and, taking, if necessary, the supplement to \(180^\circ\), see which acute angle is the largest; then, converting the other two angles into obtuse ones and dividing all the angles in half, we find the coordinates. However, it is necessary not only to find, at least approximately, the position of the twin axis, but also to verify the latter in the specimen, since sometimes, even with an incorrect determination of the axes of elasticity, an arrangement of the latter may result that is close to the conditions of twin formation.

In general, verification in the specimen of the twin axis found by construction on the diagram is of decisive importance; only in the following particular case can it sometimes lead to misunderstandings.

Let the axes of elasticity of the twin be shown in Fig. 36, and suppose that its axis coincides with \(OB^1\).

Since in the given twin the axes \(n^1\) and \(n^2\) merge, two other pairs of axes of elasticity lie in one plane; therefore the position of the twin axis, representing the line of intersection of the planes passing through the like-named axes of elasticity, will be indeterminate, i.e. it may coincide with the two directions \(OB^1\) and \(OB^{11}\). Turning now to the testing of these presumed twin axes, we observe that both of them satisfy the conditions required of them, as is clearly seen from the drawing. Therefore, if we stopped at testing only the direction \(OB^{11}\), we might take it for the true twin axis and thus fall into error. The actual axis will be the one that has a possible crystallographic significance. Approximately the same phenomenon is obtained if one pair of axes of elasticity is close to coincidence, which, as we shall see below, is often encountered among twins of feldspars.

In determining the coordinates of the twin axis we could find only their absolute magnitudes. Meanwhile, like all other coordinates, they will fully characterize the direction measured by them only when it is known in which of the eight octants formed by the coordinate axes this direction is situated. But the coordinate axes, which in this case are the axes of elasticity of the mineral, can be established only when the principal crystallographic elements of the latter are known. It must be said that in rock-forming minerals it is almost never possible to determine a sufficient number of these elements, and thus the use of the coordinates of the twin axis inevitably leads to certain misunderstandings; the latter, however, are for the most part easily eliminated by some collateral observations.

§ 43. The significance of twin elements for the determination of minerals.

How, then, should one make use of the coordinates of the twin axis found by means of the method described when determining minerals?

We know that twin elements always have crystallographic significance; therefore the above-mentioned coordinates express that relation between the optical and crystallographic elements which for the most part fully characterizes the given mineral species. Depending on the availability of certain other data, the determination of the coordinates of the twin axis has a threefold significance:

1) If the nature of the mineral is known, then the coordinates found may indicate the law of twinning, i.e. the crystallographic significance of the elements of the latter. Indeed, knowing with what mineral we are dealing, we have an idea...

information as to how the optical ellipsoid is situated in it. If all the elements of the mineral are represented in stereographic projection, then the solution of the problem posed can be carried out by a purely graphical method.

Let, for example, Fig. 37 represent a mixed projection of the elements of orthoclase, in which the circles denote poles, or gnomostereographic projections of planes, and the crosses denote grammo-stereographic projections of directions.

Suppose that, in studying some twinned formation of this mineral, we have found the coordinates of the twin axis:

\[ \angle B \left\{ \begin{array}{l} \mathfrak{p}_{g}=45^\circ\\ \mathfrak{p}_{m}=45^\circ\\ \mathfrak{p}_{r}=86^\circ \end{array} \right. \]

In order to find from these coordinates on the orthoclase diagram the point representing the pole of the twin axis, it is necessary to draw, about the poles of the axes of elasticity, the loci of points distant from these poles by the amount of the corresponding coordinate. Such a construction is made according to problem 7 (§ 14).

It is evident that the pole of the twin axis will be represented by the point of intersection of all three arcs of a small circle, which are the indicated loci. It is not difficult to see that, if such circles are drawn about both poles of each axis of elasticity, eight points will be obtained satisfying the stated condition.

In our example, of these eight points only two, representing the poles of the planes \((0\bar{2}1)\) and \((021)\), correspond to rational elements of the mineral. Therefore it must be accepted that the twin axis is the normal to the plane \((0\bar{2}1)\) or \((021)\), i.e. the orthoclase twin is formed according to the so-called Baveno law.

From this example we see that, in order to determine the character of the twin law of minerals, it is necessary to have diagrams representing all the elements of the latter.

In manuals of mineralogy that have appeared in recent times, such diagrams are appended for many of the more frequently occurring mineral species (see H. Rosenbusch, Mikroskopische Physiographie. I₂, 1905).

But usually these diagrams have the inconvenience that no stereographic net is drawn on them. In order to be able to make the necessary constructions, it is necessary to have copies of transparent nets of the corresponding diameter, the use of which was described in § 15. It is still better each time to transfer all the data of the diagram onto Nikitin’s hemisphere, where the drawing of arcs of a small circle is extremely simple.

2) Conversely, if the character of the twinned forma-

of the mineral under study, then we may make use of the coordinates of the twin axis for the determination of the latter.

True, this problem is considerably more difficult than the preceding one, since we do not yet have tables in which all the numerical data characterizing the possible twinned formations of minerals would be collected in a known order. But in any case, by means of certain signs we can assign the mineral under study to some class or species and thereby considerably narrow the field of our searches. Since the given problem is the inverse of the preceding one, in these searches it is likewise necessary to make use of the diagrams indicated above.

Let us take the following schematic case as an example. Suppose that in a thin section of a rock there is a twin of feldspar formed according to the Baveno law, as may be judged from the characteristic diagonal arrangement of the seam line, and that the coordinates of the twin axis have the following value:

\[ \angle B \begin{cases} n_g = 45^\circ \\ n_m = 45^\circ, \\ n_p = 86^\circ \end{cases} \]

Let us then suppose that, among the representatives of the feldspar group, only orthoclase and unstriated microcline can enter into the composition of the twin under study.

Turning to the diagrams of these minerals and plotting on them, as before, the coordinates of the twin axis, we find that the latter, only in the case of orthoclase, coincides with the pole of the plane \((021)\) or, in general, has a rational value. On this basis we are entitled to conclude that the given mineral is orthoclase.

3) Finally, in the case when neither the character of the given twinned formation nor the chemical nature of the mineral composing it is known to us, the coordinates of the twin axis by themselves cannot in general serve for the determination of the mineral, since this is equivalent to solving an equation with two unknowns. Only in the case when we have to deal with a definite mineral group, the separate representatives of which are connected with one another in a regular manner, can the coordinates of the twin axis simultaneously solve both problems posed.

Up to the present time this method has so far been applied to the determination of the isomorphous group of plagioclases, where it has attained remarkable development, thanks to the labors of the creator of the universal method of investigation—E. S. Fedorov.

Determination of Minerals by Means of Cleavage Planes

§ 44. Significance of the coordinates of the pole of cleavage.

The axes of elasticity are sometimes the only elements of a mineral that can be determined in any grain of the latter.

It is obvious that every mineral species is characterized by a special position of these axes relative to its crystallographic elements.

Above we saw how simply this relation may be determined in the case of twin formation, the axis of which, being found indirectly, always has crystallographic significance. But this method of determining minerals requires that the grains under investigation be found in the form of twins, which occurs only in certain mineral species. Far more often, perhaps, real crystallographic planes are met with in rock-forming minerals, appearing chiefly in the form of cleavage cracks.

How, then, are the latter to be used for determining minerals?

Clearly, here one may apply the same principle that was taken as the basis for determining minerals by means of investigations of twin formations. Indeed, having established the cleavage plane, we measure the coordinates of its pole, which will have the same significance as the coordinates of the pole of the twin axis. Likewise, using the diagram of one mineral or another, we can determine either the mineral itself, or the crystallographic significance of the cleavage of the latter, or both together.

It must be said that here, too, in the general case, with one system of cleavage cracks, as regards crystals of triclinic system, it is insufficient for avoiding ambiguities in solving the question; and for the complete determination of a mineral, additional observations and data are still needed. But in crystals of higher degrees of symmetry the planes of the axes of elasticity already acquire the significance of principal crystallographic planes, so that the coordinates of the pole of the cleavage plane can often fully characterize the given mineral. However, in the latter case it is sometimes more convenient to use the coordinates not of the cleavage plane, but of some other derived crystallographic elements.

Let us consider, by one example, how the cleavage cracks present are to be used for determining the relation between the optical and crystallographic axes of a mineral. Let us take monoclinic pyroxene, which, together with amphibole, forms two analogous groups of very important rock-forming minerals.

Let Fig. 38 represent the gram-stereographic projection of the results of measurements of the elements of one augite. In setting up

corrections for the angles of inclination about the axes \(H\) and \(J\), respectively, were taken into account for the different planes of this mineral, for the refractive index of augite \(n = 1.71\).

As is seen from the diagram, in the grain investigated it was possible to establish precisely the cleavage plane, which in augites coincides with the plane of the vertical prism. In addition, it is known that in these minerals the plane of the optical axes \(n_g n_p\) lies in the second pinacoid \((010)\), which, in its intersection with the prism plane, gives an edge parallel to the vertical axis \(C\).

Thus, at the point of intersection with the projections of the planes just mentioned, we find the projection of this crystallographic axis. It remains now to measure the angular distance of the latter from the axes of elasticity, and we shall obtain data fully characterizing the position of these axes in the crystal under investigation.

Among the coordinates of the pole of the vertical axis it is customary to use only the angle \(C n_g\), since the two other coordinates are easily derived from the latter when a plane of symmetry exists in a monoclinic mineral. In the old terminology this coordinate is called the extinction angle of augite. It seems to us that, when using the universal method, such a name is quite superfluous, if only because this extinction angle is determined graphically and for the most part cannot be checked in the section, for, as in our example, the well-established plane of the optical axes occupies an almost vertical position.

Making the measurement on the diagram, we find the angle \(C n_g = 38^\circ\). The coordinate obtained indicates that the given pyroxene belongs to the ordinary augites, with which the angle between the optical axes \(2V = +66^\circ\) also agrees.

The accuracy of the determination of the angle \(C n_g\) depends chiefly on the accuracy with which the cleavage plane is set, because the latter intersects the pinacoid \((010)\) at a rather acute angle.

This setting is easily checked in the following way. As is known, in monoclinic pyroxene the plane \((010)\) bisects the angle between the cleavage planes along the prism; moreover, it is known that this angle \(= 87^\circ\).

Thus, if the angle measured on the diagram between the plane \(n_g n_p\) and the cleavage plane is equal to

\[ \frac{87}{2} = 43.5^\circ, \]

then this means that the setting of the latter was made correctly. But the angle between planes is equal to the angle, or to the supplement to \(180^\circ\) of the angle, formed by the poles of these planes. Measuring the required angle in the indicated way, we have:

\[ \angle B_{m} n_m = 180^\circ - 136^\circ 5' = 43^\circ 5'. \]

Thus, the setting of the cleavage plane was made correctly, po-

to which one can trust also the obtained magnitude of the coordinate of the angle \(C n_g\).

From the example just given of finding the relation between the optical and crystallographic axes of a mineral by means of observations of the cleavage plane, we see that this method is a very flexible one, since in each particular case the combinations of elements present in a given grain of a mineral may be most diverse, not only in number but also in the quality of their manifestation. Thus there is no possibility of giving an exact and definite scheme for such investigations—a scheme which would reduce the separate stages of observation to a series of mechanical procedures. Here it was possible only to indicate the general character of the work, as has been done in the form of the analysis of a particular case of determining augite.

§ 45. On extinction angles.

We have seen that the presence of a single system of cleavage cracks in rock-forming minerals, belonging chiefly to the triclinic system, is insufficient for orienting the optical ellipsoid of elasticity. Therefore, for the closer determination of such minerals it is necessary, in addition to the coordinates of the pole of cleavage, to establish some other constants dependent on this cleavage and thus having a somewhat conditional character.

Among such constants, attention has long been paid to the so-called extinction angles, obtained in the following manner.

Usually in every mineral species there is some characteristic crystallographic element, manifested in each individual of such a mineral. Thanks to this circumstance it sometimes becomes possible to prepare sections oriented according to a known plane of the crystal. Let us observe such a preparation under the microscope; if in the section of the thin section there is a known direction—for example, the trace of a definite cleavage—then the angle formed by the latter with the axis of the ellipse of elasticity of the section, and appearing upon rotation of the microscope stage as the extinction angle relative to this direction, will characterize the given mineral. But to prepare oriented preparations from specimens of rocks is extremely difficult and often impossible; and in sections of arbitrary cut the constant derived can be used only in rare cases, when by some signs it is possible to establish the character of the section of the given mineral grain. Therefore, in the universal-optical investigation of rock-forming minerals, one must completely abandon the extinction angle on definite faces as the guiding optical constant. If by chance we should have at hand the conditions necessary for measuring the indicated extinction angles, then these conditions are still better used for another purpose. Indeed, the extinction angle can be measured only in

in the case when, in the mineral grain under investigation, there are two crystallographic planes. But, having established the latter with the aid of the universal stage, we find the axis of their zone and then determine the coordinates of this axis relative to the axes of elasticity of the mineral, i.e., we learn directly the orientation of the optical ellipsoid in the given mineral, which is of far greater significance for determining the latter than measuring the angle formed by secondary directions in the crystal.

If such conventional constants as extinction angles are to be introduced into the universal method at all, they must be established on somewhat different grounds. It is true that even here one cannot dispense with at least one system of cleavage cracks, but the latter may be found in almost every grain of a mineral possessing good cleavage. Since, in order to obtain an angle, besides the cleavage plane another geometrical element is also necessary, as the latter we may take one or another plane of the axes of elasticity.

Then in each mineral grain we can find three extinction angles which, taken together, determine the sought relation with sufficient precision. Let in Fig. 39 there be given the projections of the planes of elasticity and of the cleavage plane of some mineral. Finding the points of intersection of the latter with the projections of the planes of elasticity, we at once measure the required angles: the angle \(X n_p\), the angle \(Y n_p\), and the angle \(Z n_g\). If it were possible to bring the planes of the axes of elasticity into a position perpendicular to the axis of the microscope, then the indicated angles could be verified by direct observation as extinction angles relative to the cleavage trace.

Constants of this kind were introduced already by M. Fouqué for the determination of feldspars, in the discussion of which they will be analyzed in detail below.

PART THREE.

Determination of Feldspars.

Feldspars are the most widespread rock-forming minerals. Classifications of rocks based on mineralogical composition reflect above all the character of the feldspars present. But, apart from their practical significance, feldspars are of enormous theoretical interest: suffice it to say that all advances in the study of feldspars have at the same time been stages in the development of crystallography and chemical mineralogy. Hence the understandable desire to expand the sphere of more or less precise investigation of these minerals.

Of all methods for determining feldspars in thin sections of rocks, Fedorov’s universal method is the most suitable, since it makes it possible to find almost all the most important elements of these minerals. And it must be said that up to the present time this method has been applied chiefly to the determination of feldspars, since the latter are colorless minerals with a comparatively low refractive index and an almost imperceptible dispersion of the axes, i.e., they possess such properties under which investigations on the Fedorov stage give the most reliable results.

Thus, the description of methods for determining feldspars fully illustrates the practical application of the universal-optical method of investigation to the study of rock-forming minerals.

The history of the development of this method, which is still far from having spoken its last word, compels us to begin with the consideration of plagioclases.

Lime-soda feldspars.

§ 46. Isomorphism of the plagioclases.

Plagioclases constitute a sharply defined group of lime-soda feldspars forming an isomorphous series, at one end of which stands pure sodium feldspar, or albite, whose composition is expressed by the formula $Na_2Al_2Si_6O_{16}$, while the other terminal member of the series is anorthite, having the formula $CaAl_2Si_2O_8$.

If the molecules of albite and anorthite are denoted by \(Ab\) and \(An\), then the composition of any member of the plagioclase series may be expressed by the formula: \(Ab_mAn_n\). For convenience in designating the individual members of this numerous series, it is more convenient to take \(m+n=100\), and then the above formula will have the form: \(Ab_{100-n}An_n\). At the suggestion of Prof. Fedorov, the designation of plagioclases may be still further abbreviated if, instead of the name, one puts a number indicating the percentage content of anorthite molecules, equal to \(n\) in the last formula. Thus, plagioclase No. 35 contains 35 molecular percent of anorthite and 65% of albite.

It must be said that even after G. Tschermak’s discovery of the isomorphism of the plagioclases, for a very long time only a few, more frequently occurring representatives of the latter were known, which had long since received special names.

Although at the present time there is no special need to preserve these names, nevertheless they are still often used, meaning by them broader categories of these minerals. Therefore below is given a table of the various designations of the best-known members of the plagioclase series.

Name of plagioclase	Formula	No.
Albite	\(Ab\)	0
Albite-oligoclase	\(Ab_7An_1\)	13
Oligoclase	\(Ab_3An_1\)	25
Oligoclase-andesine	\(Ab_2An_1\)	33
Andesine	\(Ab_3An_2\)	40
Labrador	\(Ab_1An_1\)	50
Labrador-bytownite	\(Ab_3An_5\)	63
Bytownite	\(Ab_1An_3\)	75
Anorthite	\(An\)	100

Of course, this table may be changed depending on what composition is assigned to a plagioclase bearing one or another name.

The belonging of the plagioclases to a single isomorphous series is proved by the gradual and regular change of their physical properties depending on the relative quantity of both end-members of the series entering into the composition of each intermediate member of the latter. Among such properties, whose regular change is usually not subject to doubt, are: specific gravity, volume, melting point, and crystalline forms¹.

With respect to the optical properties of the plagioclases, the opinions of various scholars differ somewhat. Although almost all these properties change in one direction with the change in the composition of the plagioclases, nevertheless no strict regularity is observed in this phenomenon. Thus E. S. Fedorov showed¹ that the curves of his diagram, representing the geometrical loci of the projections of the twin axis obtained for different members of the plagioclase series, do not have the simple form that would have been expected under the isomorphism of the latter. Likewise C. Viola found² that in one and the same specimen of albite the extinction angles on the planes (010) and (001) gave fluctuations of up to \(6^\circ\), and that, in consequence of this, there is hardly any practical possibility of proving the affiliation of plagioclases to one isomorphous series, at least from the point of view of their optical properties.

But the well-known investigator of the feldspars, F. Becke, in the work cited above, after a thorough analysis of the optical properties of the plagioclases, comes to the conclusion that, in the aggregate of these properties, the plagioclases must be assigned to one isomorphous series, and that if certain comparatively insignificant deviations from the regularity in the change of constants required by theory are observed, this is explained in the following way.

The end members of the series under consideration have somewhat different crystalline forms; therefore, upon entering into an isomorphous compound, they must adapt themselves to one another, and this will certainly be reflected in the most sensitive properties of minerals, to which the optical properties belong.

For this reason the existing theories of Mallard, Pockels, and M. Lévy, whose aim is to derive the optical properties of isomorphous compounds from the known properties of the end members of the series, often give results inconsistent with the actually observed values, since in all these theories it is assumed that the properties of the end members of the series remain unchanged when the latter enter into compounds.

Thus the determination of the optical constants of the individual members of the plagioclase group must for the time being proceed by the experimental route, while the isomorphism of this series can serve only as a guiding idea that checks the scattered observations and links them into a single whole.

Chapter I.

Fedorov’s Method for Determining Plagioclases.

§ 47. General foundation of the method.

The universal-optical method is based on the use of the Fedorov stage, which makes it possible, chiefly, to determine the spatial relations between the optical and crystallographic elements of minerals. But it is precisely these relations that, for the most part, characterize well the nature of the latter. We have seen above that, in elucidating the indicated relations, the study of twin intergrowths is of enormous importance. Indeed, the twin axis, found by simple construction, must be either an edge or a perpendicular to a possible face of the crystal; therefore, by measuring the angles between this direction and the axes of elasticity of one or another individual of the twin, we obtain the relation sought. This convenient method for determining minerals is applicable only in the case when the latter always occur in the form of twins.

Among such minerals the plagioclases in particular belong. Since they are of exceptional importance in rocks, it is understandable that the universal method of investigation was, from the very beginning of its origin, applied to these minerals; with their study its own development was connected, and it is with them, chiefly, that it is concerned at the present time. As a consequence of this, quite often by the universal-optical, or Fedorov, method people incorrectly understand only the method for determining plagioclases.

Let us set forth, in brief outline, wherein lies the essence of this Fedorov method understood in the narrow sense.

Suppose that we have material representing crystals of various members of the plagioclase series, and that the latter form twins according to one and the same law. Having made chemical analyses of each of the specimens, we accurately establish the composition of the corresponding plagioclase and thus determine the place which it occupies in the series. On the other hand, having prepared thin sections from the given material, we subject the twin intergrowths of these plagioclases to investigation on the Fedorov stage, and find the coordinates of the twin axis, i.e. the angles which it forms with the axes of elasticity of one or another individual of the twin. Let us now depict the obtained results in the form of a diagram. For this we shall take a stereographic net (Table X) and assume that its center coincides with the axis \(n_m\), and with its horizontal and vertical diameters—the axes \(n_g\) and \(n_p\). In order to find on this net the projection of the twin axis, it is necessary to lay off its coordinates from the projections of the corresponding axes of elasticity, using polar or equatorial parallels. It is not difficult

we see that, in the general case, four points are obtained on the net corresponding to these coordinates, since the projection of the axis \(n_g\), from which the corresponding coordinate is measured, may be taken both on the right and on the left end of the horizontal diameter, and the projection of the axis \(n_p\)—on both ends of the vertical diameter. To avoid this indeterminacy, the coordinates obtained for the twin axis must be assigned definite signs.

Proceeding in this way, we shall obtain on the net a series of points representing the projection of the twin axis for various plagioclases that form twins according to one and the same law. If it were possible somehow to carry out such an investigation for all possible members of the plagioclase series, then the points obtained on the net would yield a continuous line of more or less regular character, owing to the fact that plagioclases belong to an isomorphous series. For this reason, however, it is possible to limit oneself to finding only a few points corresponding to various representatives of these minerals; by joining these points with a smooth curve, we shall obtain, with a sufficient degree of accuracy, the required diagram. It remains then, for the convenience of using the latter, to mark on it, by interpolation, points corresponding, for example, to every tenth No. of plagioclase.

Obviously, in exactly the same way one may construct curves corresponding also to other known and more frequently encountered twin laws of plagioclases.

These curves, taken together, form what is called the Fedorov diagram, represented in Table X.

It must be said that no special work of the character described above has yet been done, since it is hardly likely that material sufficient for this purpose can be found anywhere. Therefore the twin diagram was compiled in a somewhat different manner: it was necessary to use those generally insufficient data on the distribution of the axes of elasticity in individual plagioclases that are available in the literature.

The diagram obtained in one way or another can already serve for determining plagioclases in ordinary petrographic work. In the selected grain of plagioclase, the twinning is determined in the usual manner and the coordinates of the twin axis are found. Since, in the general case, there is no criterion by which the signs could be assigned to the obtained coordinates one way or the other, when the projection of this axis is plotted on the net of our diagram four points are obtained. In the event that all measurements have been made correctly and that the diagram contains a curve corresponding to the law by which the twin of the plagioclase under investigation is formed, one of these four points will fall on the corresponding curve or close to it. Obviously, this curve will indicate to us the law of twinning, and the point marked on it the No. of the given plagioclase.

Thus, by means of comparatively simple measurements and constructions, Fedorov’s method makes it possible to determine not only the chemical composition of plagioclases, but also the character of the twins that they form.

§ 48. Laws of twin formations of plagioclases.

Thus, the twin diagram is a necessary accessory in the method of determining plagioclases described here. On it, each curve represents the geometric locus of the projections of the twin axis of some one law. It is obvious that, both in compiling the diagram and in using it, one must know which twin formations are generally characteristic of plagioclases, and what crystallographic significance the elements of the twins have.

In the chapter on twin formations it was indicated that the most general twin laws belong to crystals of triclinic syngony. Since plagioclases belong to the latter, very diverse twin formations are theoretically possible among them; and if comparatively few of them are known, this is explained by the circumstance that, until recent times, twins were determined chiefly on macroscopic crystals with well-developed faces, while in general there is an insufficient amount of material satisfying this condition. Therefore many of the twin laws derived below still require experimental verification, which can for the most part be successfully carried out only with the aid of the universal method.

As was shown earlier, the character of a twin formation is determined by the directions of the principal forces of cohesion in the crystal. These forces of cohesion are manifested externally in the directions of preferential growth of crystals and in the presence of cleavage planes, formed by the directions of the greatest forces of cohesion. If this criterion is applied to plagioclases, then in the latter the directions of the most sharply expressed forces of cohesion must coincide with the principal crystallographic axes of the mineral, since feldspars are almost always elongated along these axes and have very good cleavage along the pinacoids $(001)$ and $(010)$.

Thus the most important and most frequently encountered twin formations of plagioclases must be connected with their principal crystallographic axes.

But under certain conditions of crystal formation, the character of a twin may be influenced by cohesive forces of the second or even third order, if the directions of these forces lie in the plane most strongly developed under the given conditions. Thus, feldspars also have not especially good cleavage along the planes $(110)$ and $(1\bar{1}0)$, and therefore the latter may sometimes also be twin planes.

Then in the literature there are indications that faces of so complex a symbol as (454) can serve as a twin plane, or that such edges as (001)(340) or (001)(130) can be twin axes¹.

Strictly speaking, it is difficult even to foresee all possible twin formations, since the conditions of crystallization vary in the most diverse manner, and since the vectorial character of the forces of cohesion has as yet been little studied. Therefore, below there will be set forth those twin laws which theoretically are the most possible, and for which there are at least some indications in the literature.

These twin laws are distributed, according to the types derived earlier, in the following manner.

I. The twin axis is perpendicular to a possible face of the crystal.

1) Albite law. The second pinacoid (010) represents the most important face and the plane of perfect cleavage of the plagioclases. Therefore the perpendicular to this plane is the axis of the most widespread twin law, called the albite law. The individuals of this twin grow together, evidently, for the most part along the same plane (010), although observations show that sometimes the composition plane may also have an irregular form.

2) Manebach law. On the same basis, the perpendicular to the plane of another good cleavage of the plagioclases, (001), serves as the twin axis of the so-called Manebach law. The latter, in general investigation, is often confused with the better-known Carlsbad law, and therefore, before the application of the universal method, it was mentioned in the literature comparatively rarely; but already the first detailed works² employing the latter method showed that the law under consideration enjoys considerable prevalence, as should have been expected from the theoretical point of view. The composition plane of the individuals of the twin most often coincides with the twin plane (001), but sometimes it may also be the second pinacoid, in view of the greater crystallographic weight of the latter, or may even represent an irregular surface, depending on the conditions of crystallization.

3) In the preceding laws the twin planes were determined by the axes \(a, c\) and \(a, b\). Obviously, the plane determined by the axes \(b\) and \(c\), and coinciding with the first pinacoid (100), will likewise have a special significance and therefore may serve as a twin plane. Since this plane is usually rarely observed in crystals of feldspars, the corresponding twin law, too, must occur only in exceptional cases. And indeed, con—

there are only vague indications of this twin law in the mineralogical literature¹.

4) Along the plane of the vertical prism \((\bar{1}10)\), a cohesion is not infrequently manifested, indicating that directions of forces of the second order lie in it. Therefore, under certain circumstances twins may arise with an axis perpendicular to the face \((\bar{1}10)\). Hintze¹ mentions such twins, which have no special name.

5) The face \((110)\) has approximately the same property as the plane \((110)\); therefore everything said concerning the latter is also applicable to the former.

6) As observations show, the plane \((021)\) has special significance, since along it twins are not infrequently formed that recall the Baveno twins of orthoclases.

7) The same applies also to the plane \((0\bar{2}1)\).

Many other twins are also possible that belong to the first type and have, as twin planes, other possible faces of the feldspars, for example \((111)\), along which imperfect cohesion sometimes occurs, \((201)\), \((130)\), etc. But the probability of such twins is so slight that it is better not to discuss them, so as not to make the diagram very motley. Meanwhile the latter must and can satisfy only the practical aims of determining plagioclases.

II. The twin axis is a possible edge of the crystal.

8) The law de l’Esterel. The cohesion of feldspars along \((001)\) and \((010)\) corresponds to the planes of greatest density of the crystallographic lattice. Evidently, of all three axes \(a\), \(b\), and \(c\), which determine these planes, the axis \(a\), corresponding to the line of intersection of the latter, coincides with the direction of the greatest axis of cohesion. Therefore twins of the second type, having this direction as the twin axis, should be encountered most frequently.

Although this twin law, which received the name de l’Esterel, was known already from the time of Des Cloizeaux², nevertheless it is rarely mentioned in the literature. This is explained partly by the fact that it can easily be mistaken for the better-known “complex” law, partly by the fact that it is most often encountered among microlites, since the latter are usually elongated along the axis \(a\), corresponding to the direction of greatest rate of growth; therefore this law is in general little accessible to investigation.

9) The Carlsbad law. Of the two other crystallographic axes, the axis \(c\) has the next significance after the axis \(a\), since it lies

in the most developed plane of the crystal \((010)\). Twins whose axis is this direction, and which are called Carlsbad twins, are encountered rather often, although they are for the most part confined to large crystals, slightly elongated along the vertical axis. The plane of intergrowth of the individuals of the twin often has an irregular appearance, but belongs predominantly to the vertical zone, being either \((010)\) or \((100)\).

10) Pericline law. This twin law, whose axis is the second crystallographic axis \(b\), corresponding to the direction of the weakest of the three principal cohesive forces, is encountered much more rarely than the preceding law. It must be said, however, that owing to the proximity of the axis \(b\) to the perpendicular to the plane \((010)\), this law almost coincides with the albite law and therefore may easily be taken for the latter in microscopic investigations. It is distinguished chiefly by the circumstance that the plane of intergrowth of the individuals of the twin is constant and belongs to the so-called rhombic section, generally close to the plane \((001)\).

The three laws given exhaust the practically possible twin formations of the 2nd type. It is true that there exist¹ twins of this kind connected with directions of the 2nd and 3rd orders, lying in the above-mentioned planes \((110)\), \((0\bar{2}1)\), etc.; but such twins must occur extremely rarely, since even twins of the 1st type depending on these directions are not widely distributed.

III. The twin axis lies in a possible plane and is perpendicular to a possible edge of the crystal.

11) The most widespread twin formations of this type must be connected with the second pinacoid, since this plane contains the directions of the greatest cohesive forces.

First of all it is necessary to dwell on twins whose axis is perpendicular to the principal axis of cohesion \(a\). The law according to which such twins are constructed, and which has the formula: twin axis—perpendicular to \([100]\) in \((010)\), has no special name and was only recently established²—in all probability because it is close to the Manebach law and in general occurs rarely. The plane of intergrowth of the individuals of the twin must, for the most part, coincide with \((010)\).

12) La Roc Tourné law. In this twin law the axis is a perpendicular to \([001]\) in \((010)\) and almost coincides with the perpendicular to \((100)\), for which reason the law itself is difficult to separate from the law described in point 3.

Although, from the theoretical point of view, this law has no special advantages over the latter, nevertheless even Des Cloizeaux¹ had occasion to describe it, explaining, however, the manner of its formation somewhat differently from what is accepted here. True, this law was encountered very rarely, so that in recent times E. S. Fedorov and C. Viola², in their investigations of feldspars, as it were rediscovered it and each gave it a special name: E. S. Fedorov called this law compound, and C. Viola—the 2nd Carlsbad. It seems to us that there are no sufficient grounds for departing from the original name—la Roc Tourné. Why this law is better known than others analogous to it will be said below.

13) Let us now pass to the next principal plane of the crystal, $(001)$. As before, let us first direct attention to the twinning law determined by the axis—the perpendicular to $[100]$ in $(001)$, which almost coincides with the second crystallographic axis $b$, and for this reason the law itself is very close to the pericline law. Despite this, it had already been noticed on crystals³, although it had not yet received a special name.

14) The Scopi law. This law, characterized by an axis representing the perpendicular to $[010]$ in $(001)$, almost coincides with the de l’Esterel law and was discovered by C. Viola.

15 and 16) In view of the fact that the plane $(100)$ has a special crystallographic significance, two more twinning laws of the 3rd type are possible, determined by the axes: the perpendicular to $[001]$ in $(100)$ and the perpendicular to $[010]$ in $(100)$.

The first of them almost coincides with the albite law, and the second is rather close to the Carlsbad law. It must be said that neither of these laws has yet been observed up to the present time, and indeed they should occur very rarely.

It goes without saying that, with respect to directions of cohesion forces of the 2nd, etc., orders, the principle of twinning formations of the 3rd type is almost inapplicable.

IV. The twin axis lies in a possible plane and bisects the angle between possible edges of the crystal.

17) Since the cleavage planes $(010)$ and $(001)$ are often approximately equally developed, the cohesion forces coinciding with the crystallographic axes $b$ and $c$ have almost the same magnitude. Therefore the fourth type of twinning formations is applicable first of all to the plane $(100)$, so that the axis of the corresponding twin will bisect the angle between the crystallographic

planes \([0\bar{1}0]\) and \([001]\). This law is more or less close to the Baveno law and, properly speaking, has not yet been observed.

18 and 19) The same may also be said with regard to possible theoretical twins whose axes are the bisectors of the angles between the axes \(a\)—\(c\) and \(a\)—\(b\). Since the difference between the cohesion forces directed along the two latter axes is comparatively great, the corresponding twin law too must occur in exceptional cases.

In constructing the twin diagram we shall confine ourselves to the twin laws cited, as being those most likely to be widespread.

§ 49. Analysis of Twins.

In enumerating the twin laws it was indicated that different laws occur with unequal frequency. If a crystal develops regularly, then only those twins should be formed which are connected with the directions of the greatest cohesion forces. And indeed, according to observations, the most widespread are those twin laws of the 1st and 2nd types whose elements relate to the most prominent planes \((010)\) and \((001)\). To these belong the laws: albite, Manebach, de l’Esterel, Carlsbad, and pericline. The law perpendicular to \((100)\), belonging to the 1st type, must apparently have an insignificant distribution, as was explained above. As for twin laws of the 3rd type, connected with the crystallographic axes, they should generally occur much more rarely; moreover, establishing these laws is rather difficult, since for the most part they either closely or almost coincide with the laws enumerated above. Consequently, very little is known about twins of the 3rd type as separate crystals, and if some of these twins have nevertheless been more or less studied, this is a consequence of the following circumstance.

In its proper place it was mentioned that three twins with mutually perpendicular axes are closely connected with one another by the manner of their formation. V. V. Nikitin¹ showed that if a crystal consists of a pair of such twins having common individuals, then some of the latter must be related according to a third law.

In such complex twinned crystals (polysynthetic twins) it has been possible best of all to detect the twins of the 3rd type under consideration. The prevalence of the latter depends on the significance of those twins with which they form these complex crystals.

Guided by this, one may compose the following triads of such

complex twins with mutually perpendicular twin axes. These triads are arranged according to the degree of their probability.

a) Laws: albite, de l’Esterel, and the law with the twin axis perpendicular to \([100]\) in \((010)\).

Theoretically such a triad should occur most often, since both \((010)\) and \([100]\) are the strongest crystallographic elements; but in reality it was discovered only recently¹, in all probability because the last two laws must occur chiefly among microlites.

b) Albite, Carlsbad, and la Roc Tourné. Since the albite and Carlsbad laws occur most often among crystals in mineralogical collections, the law la Roc Tourné (“complex” Fedorov) was already known long ago, and the probability of finding it under the same conditions will be the greatest.

c) Manebach, de l’Esterel, and the law with the twin axis perpendicular to \([100]\) in \((001)\).

Apparently, by fortunate chance, the last law has also proved to have been studied, though on only a few specimens. One may think that this triad, too, is found more often among microlitic formations.

d) Manebach, pericline, and Scopi. For the reasons set forth in point (b), this triad has likewise been rather well studied.

e, f) The remaining two triads are connected with the law: perpendicular to \((100)\); but since the latter is still little known, the probability of finding them will likewise be insignificant.

Although the indicated twins of the 3rd type are formed with the participation of principal crystallographic elements, nevertheless they are in general encountered very rarely.

It is therefore highly probable that twins of the 1st type, connected with cleavage axes of the 2nd order, will be encountered no less, if not more, often. And indeed, twin laws with axes: perpendicular to \((110)\), perpendicular to \((021)\), etc., have already been repeatedly established and more or less studied.

As for the twin laws of the 4th type, they will be encountered only as exceptions under special conditions of crystallization. It was mentioned above that among these laws the greatest significance will attach to the law with a twin axis bisecting the angle between \([010]\) and \([001]\).

§ 50. Construction of the twin diagram.

After examining the possible twin laws of the plagioclases, one may proceed to the compilation of the twin diagram mentioned above, which will make it possible to determine these minerals from their twin formations.

For this it is necessary to know the position of the optical ellipsoid, at least in typical representatives of the plagioclases.

In preparing his diagram, the creator of the method, E. S. Fedorov, was guided¹ chiefly by the works of M. Lévy, M. Schuster, and Fouqué, who gave a series of constants for certain types of plagioclases. These constants are, for the most part, extinction angles in definite sections of minerals; and in order to compile from them the required diagram, it was necessary to carry out a whole series of constructions and calculations. Naturally, in doing so it was difficult to avoid inevitable errors, all the more because the constants themselves had been determined with an insufficient degree of accuracy. On the other hand, Fedorov’s diagram² contains curves for only four, though the most widespread, twin laws.

Consequently it appears necessary not only to correct, but also to supplement this diagram; this has been done, as far as possible, in the present work.

The diagram appended hereto (Table X) is constructed for eight members of the plagioclase series, corresponding to Nos. 5, 13, 20, 25, 37, 52, 75, and 100. The material of these plagioclases comes from the classic localities of Western Europe and was subjected to chemical and optical investigations by different methods and at different times. In 1906 F. Becke, in his work “Die optischen Eigenschaften der Plagioklase” (Tschermack’s Petr. u. Min. Mittheil. XXV, 1906; 1–42), brought together all the known data on these plagioclases in the form of a special diagram showing the position of the axes of elasticity relative to the crystallographic elements. These data of Becke, which are the most accurate up to the present time, were used as the basis for constructing the twin diagram. However, No. 63 of Becke’s diagram was entirely discarded, and the constants of No. 37 were taken from the analogous diagram of Rosenbusch (Mikroskopische Physiographie, 1905, I.; Taf. XVII), since these plagioclases, according to Becke himself, had in general been investigated by him least accurately.

In this somewhat modified form, Becke’s diagram is presented in Table IX. On it, in stereographic projection, are plotted the axes of elasticity of the indicated types of plagioclases; the second pinacoid (010) is taken as the plane of projection, the pole of which thus coincides with the center of the net, while the third crystallographic axis \(c\) is made to coincide with the vertical diameter of the latter. In addition, the diagram marks and labels the poles of other planes and directions needed for the subsequent constructions.

Here it is necessary to draw attention to the fact that the crystalline forms

of the individual members of the plagioclase series are not entirely identical; however, the variations in the angles between definite elements are, in general, insignificant, and only the angle between the crystallographic axes \(a\) and \(b\) varies within limits of about \(3^\circ\), so that the position of the axis \(b\) on the diagram must be found separately for each type of plagioclase, using the following data\(^1\).

No. of plagioclase	\(\angle ab\)
0	\(88^\circ 9'\)
20	\(90^\circ 4'\)
37	\(89^\circ 59'\)
52	\(89^\circ 54'\)
100	\(91^\circ 13'\)

With the aid of the Becke diagram it is very easy to obtain all the quantities required for constructing the twin diagram. Indeed, the curves of the latter represent the geometrical loci of the projections of the twin axis, referred to the axes of elasticity as coordinate axes. But the crystallographic significance of the twin axis for each law is already known to us; therefore, in order to find the required coordinates, it is necessary to measure on the Becke diagram the angles between the corresponding crystallographic direction and the axes of elasticity of the given type of plagioclase, which can be successfully done by means of a three-legged divider or an overlaid transparent net.

In order, when constructing the diagram, to avoid ambiguity, it is necessary to agree to assign definite signs to the coordinates obtained. In determining the latter it is best to proceed from the accepted orientation of the crystals of the feldspars. These crystals are usually set in such a way that the obtuse angle between \((001)\) and \((010)\) is to the right of the observer, and the face \((001)\) is inclined forward. Since in the majority of types of feldspars the axes of elasticity occupy approximately the position shown in Fig. 40, then, according to the convention adopted in mathematics, the directions of the coordinate axes \(n_g\), \(n_m\), and \(n_p\), respectively to the right, upward, and forward, must be regarded as positive. Accordingly, on the Becke diagram all projections of the axis \(n_g\), as well as the projections of the axes \(n_m\) and \(n_p\) lying respectively in the upper and left halves of the net, determine positive coordinates. Therefore, for example, the coordinates of the twin axis according to the albite law for No. 13 will have the following form:

\(^1\) A. Lacroix, Minéralogie de la France et des ses colonies. II. 1897; p. 131.

Fedorov’s or Universal Optical Method

\[ \angle \perp (010) \begin{cases} n_g = +10^\circ,\\ n_m = +80^\circ,\\ n_p = -89^\circ. \end{cases} \]

It has already been indicated earlier how it is customary to arrange the coordinate axes in a twinning diagram. Let us now assume that the projections of the positive directions of the axes \(n_g\) and \(n_p\) are located respectively at the right-hand end of the horizontal diameter and at the lower end of the vertical diameter of the net. As for the projection of the axis \(n_m\), which coincides with the center of the net, it can have only one value, which in the present case will be positive. Therefore, if the corresponding coordinate has a negative sign, the signs of all three coordinates must be changed to the opposite ones. Similarly, if the coordinate for the same axis \(n_m\) exceeds \(90^\circ\), one must take its supplement to \(180^\circ\) with the opposite sign and, if necessary, then carry out the above-mentioned transformation.

In the diagram constructed in this way, the curves identical with the curves of Fedorov’s diagram are arranged in the same way as in the latter, but the coordinate axes, as is clear from the article by E. Stratanovich¹, who popularized the method, have the opposite meanings. In the cited article it is not stated what considerations guided the author in determining the sign of these axes; later it will be explained that this is the result of an arbitrary assumption.

The new diagram contains curves for all the twinning laws of plagioclases derived above, with some curves corresponding to two laws so close to one another that, at the scale of the diagram, the difference in the positions of both curves is almost imperceptible.

According to the prevalence of one or another of the laws, the latter are divided into three categories, marked by different thicknesses of the lines of the corresponding curves. The first category includes the twinning laws of the 1st and 2nd types, associated with the crystal planes \((001)\) and \((010)\); these laws are the ones commonly encountered in practice, and the curves corresponding to them stand out by their thickness. To the 2nd category are assigned the first four twinning laws of the 3rd type and the laws of the 1st type associated with the planes \((1\bar{1}1)\), \((0\bar{2}1)\), \((02\bar{1})\), and \((100)\).

Finally, the remaining laws, as occurring in general very rarely, are marked by dashed curves.

Turning to an examination of the character of the curves of the diagram, one may note that in the general case they consist of a combination of three arcs, as though indicating that the entire plagioclase series is divided—

are divided into three unequal parts, within the limits of which these minerals represent true isomorphous compounds. As may be seen from the diagram, the end members of these partial isomorphous series are approximately Nos. 0, 20, 50, and 100.

On the Fedorov diagram the position of the twin axis is marked for every tenth No. of plagioclase; this was done by interpolation between points corresponding to more or less precisely established chemical species of these minerals. Of course, the calibration of the diagram in precisely this manner is somewhat arbitrary, and none of the points on it can lay claim to special accuracy. Therefore, on the curves of the new diagram only those points are marked which correspond to the adopted types of plagioclases, according to which the diagram itself was constructed; and only the presumed positions of the twin axis for No. 0 are marked.

By means of the points present on the diagram, the curves are divided into seven parts, which may be considered sufficient for determining intermediate plagioclases with an accuracy of up to 2 numbers.

It goes without saying that, for greater convenience, but not accuracy, these curves may be subdivided by interpolation into smaller segments; in doing so, however, it is necessary to leave marked all the original points, as those found most accurately.

The construction of the diagram may be checked in the following manner.

Above, a series of twin laws was indicated that have mutually perpendicular axes. Obviously, the points of the corresponding curves of the diagram, belonging to one and the same plagioclase number, must be 90° apart from one another. This check is carried out by ordinary methods.

Then the twin laws may be divided into groups characterized by the fact that their axes lie in one plane, or that the twin planes belong to one zone. For example, one such group is formed by the laws: albite, Manebach, perpendicular to (021), perpendicular to (021), perpendicular to [100] in (001), and perpendicular to [100] in (010).

It is not difficult to see that the points of the corresponding curves, belonging to one and the same plagioclase number, must lie on one arc of a great circle, whose pole will coincide with the corresponding point of the curve [100].

By all these checks the correctness of the constructions from the data of Becke’s diagram is determined. The actual verification of the curves themselves belongs, of course, to future mineralogical work, the task of which is to determine the twin formations of chemically established types of feldspars.

§ 51. Analysis of the twin diagram.

The constructed twin diagram may serve for determining plagioclases from the found coordinates of the twin axes of the latter. If it were possible in each particular case to assign a definite sign to these coordinates, then, by plotting them on the diagram, we would obtain a single point determining both the number and the twin law of the given mineral grain. Thus, in solving the problem there would be no uncertainty. But, as we know, signs can be determined only on the condition that the given grain is brought into a definite setting, for which the presence of at least two principal planes of the crystal, \((001)\) and \((010)\), is necessary. Meanwhile, in grains of feldspars determined in rock sections, at best only one such plane can be found, chiefly in the form of cleavage. Consequently one must limit oneself to finding only the absolute values of the coordinates of the twin axis, and in this case the solution of the problem is very often accompanied by uncertainty. Indeed, when the found coordinates are plotted on the diagram, four points are obtained, which may indicate not only different laws of twinning, but also different numbers of plagioclases.

Let us turn to the consideration of the diagram, in order to determine when uncertain solutions will be encountered. As is known, the most widespread twin law is the albite law; meanwhile, when determining it one must precisely encounter ambiguous solutions. In fact, the curve of the albite law is divided into two symmetrical parts belonging to different poles of the \(n_g\) axis; therefore, in the part of the plagioclase series between Nos. 0–36, determining the plagioclase number from one twin formation alone is completely impossible. The matter is further complicated by the circumstance that, alongside the albite curve, at a very small distance from it, lies the curve of the pericline law, for which reason, if even a small inaccuracy in determining the coordinates is admitted, one may not only confuse these two laws, but also obtain a plagioclase whose number differs by four from the true one.

The same must also be said with regard to the law de l’Esterel, whose curve is likewise split into two parts. This case is still more complicated, since over a certain extent the curve of the law la Roc Tourné almost coincides with this curve, and the plagioclases corresponding to parallel points of these curves differ by more than 25 numbers.

The situation is somewhat better with the Manebach and Carlsbad curves, but they too are accompanied by parallel curves of other laws—

...laws, and many of their points have their analogues on the curves $\perp(110)$, la Roc Tourné and some others.

As for laws whose curves are very close to, or almost coincide with, the curves of more widespread twin formations, their determination is for the most part possible only when one or several known crystallographic elements are present that have a quite definite relation to the given twin axis. Thus, the law: perpendicular to $[100]$ in $(010)$, is easily distinguished from the Manebach law close to it by the fact that its twin axis lies in the cleavage plane $(010)$, etc.

Limiting ourselves to this general survey of the diagram, we already see that it is comparatively rare that a proper determination of plagioclase can be made by studying only one twin formation. To this it must also be added that sometimes the twin axis deviates from its proper position in the crystal, and then, obviously, its coordinates will give on the diagram a point which either does not lie at all on the existing curves, or corresponds to an entirely different plagioclase.

Such a deviation of the twin axis must be due chiefly to the influence of admixtures $K_2O$ and $BaO$, often noted in analyses of plagioclases; but how strong the action of these elements is, and in what direction they produce the deviation, remains still entirely unclear.

On the other hand, the noted anomaly in the coordinates of the twin axis may be the consequence of a regular intergrowth of individuals belonging to different numbers of plagioclases. E. S. Fedorov mentions one such twin.^1)

Thus, not only is it impossible to confine oneself to consideration of a single twin formation of plagioclase, but it is necessary to subject each individual of the twin to possible study.

Finally, an arbitrary position of the projection of the twin axis on the diagram may correspond to some twin law not taken into account, connected, for example, with the faces $(111)$ or $(\bar{2}01)$. The exact establishment of such a law requires, of course, a careful mineralogical study of the given crystal.

Of course, in the mass determination of minerals in thin sections of rocks, it is often possible to make do with the coordinates of the twin axis alone, since both the general character of the rock and the existing paragenesis of the constituent parts of the latter indicate which of the two, or of the larger number, of plagioclase numbers corresponding to the given coordinates is the most probable. Moreover, in such cases one can always find a grain of the given mineral, represent—

^1) L. v. Fedorow. Universalmethode etc. III, 648.

complex twins formed according to several laws. From comparison of the results of the investigation of each twinned formation of such a crystal, one can often obtain a direct solution of the question.

§ 52. On a method for determining plagioclases by means of the coordinates of the pole of the cleavage trace.

Be that as it may, the determination of plagioclases by the coordinates of the pole of the twin axis is for the most part insufficient, since various ambiguities usually arise.

The author of the universal method has proposed the following way of resolving the arising ambiguity. The method requires that, in the grain being studied, there be cleavage cracks along one of the two planes (010) and (001).

Marking on the working net the position of this plane, they measure the coordinates of the pole of the cleavage trace, represented by the corresponding diameter of the net, and, assigning known signs to the coordinates obtained, they find on the twinning diagram the projection of this trace. The further construction is based on the following: if a plane is drawn perpendicular to the given cleavage trace, then it will contain the normal to the latter; but the loci of the projections of these normals for the whole series of plagioclases are represented on the twinning diagram by the curves of the albite and Manebach laws.

Thus, if the projection of the cleavage trace is taken as the pole of an arc and the latter is drawn, then at its intersection with the corresponding curve there will be obtained a point indicating the No. of the given plagioclase. Sometimes this arc intersects both curves, but the general character of the cleavage and its relation to the axes of elasticity determines to which plane it belongs.

In this way, this method of investigating plagioclases is analogous to the basic method and is only more complicated than the latter. But perhaps it has the advantage that it permits definite signs to be assigned to the coordinates of the pole of the cleavage trace?

E. S. Fedorov¹ arrived empirically at the conclusion that the coordinates obtained must be assigned the values of one of three combinations:

\[ \begin{array}{ccc} -\,a & +\,a & -\,a \\ +\,b & +\,b & +\,b \\ +\,c & -\,c & -\,c \end{array} \]

depending on the position of the pole of the trace in one or another of the regions of the net into which the latter is divided by the projections of the principal planes of symmetry of the optical ellipsoid.

But even at a glance at the coordinate values given above it is evident that they satisfy only the special case when—namely—the pole of the given cleavage plane lies in the first octant formed by the coordinate axes \(N_g\), \(N_m\), and \(N_p\). Meanwhile, the trace of the cleavage, depending on the position in the crystal of the plane of the latter and of the section of the preparation, may occupy any position, and therefore the signs of the coordinates of the pole of the trace of this cleavage must present eight, or, provided the coordinate along the axis \(N_m\) is always positive, four possible combinations, and not three, as was said above. It is not surprising, therefore, that the determination of plagioclases by the method described often gives uncertain results.

E. S. Fedorov*) explains this latter circumstance by the existence of sections unfavorable for the given mineral. We have just seen that the probability of favorable sections is not greater than \(1/4\). In general, definite signs can be assigned to the coordinates of the pole of the cleavage trace, as also to the coordinates of the pole of the twin axis, only in the case when at least two planes \((001)\) and \((010)\) are known, by means of which the crystal under investigation can be subjected to a known setting.

Therefore the method under consideration, being rather complicated and in general entirely analogous to the method of determining plagioclases from the coordinates of the twin axis, must be abandoned.

This method is all the more inapplicable if, instead of the cleavage trace, one uses the trace of the twin intergrowth, as is recommended especially by E. Stratanovich. Indeed, the plane of intergrowth of twins does not always have a constant crystallographic significance, and to base more or less exact investigations upon it is entirely impossible. For this reason, among others, orienting the preparation by the twin seam or by the cleavage trace, usually adopted for the convenience of measuring the corresponding coordinates, is superfluous and only introduces delay into the work.

*) Ibidem, p. 647.

Chapter II.

Additional methods for determining plagioclases.

Above we have seen that the determination of plagioclases from the coordinates of the twin axis, despite the exceptional importance of the method itself, is in general insufficient in the investigation of rocks.

In order to remove the ambiguity that arises in deciding to which number a given plagioclase belongs, or according to what law its twin is formed, it is necessary to resort to additional investigations, which, together with the former, often characterize the various properties of these minerals. A description of these investigations follows in the order given below.

§ 53. Angle between the optical axes.

The angle between the optical axes, the determination of which is possible in almost any grain of plagioclase, sometimes helps to resolve the uncertainty that is obtained when finding the number of this feldspar by means of the coordinates of the twin axis.

This angle, in the types of plagioclases that have been subjected to exact investigation, has the following values¹.

Nos. of plagioclases	2V.
5	+78°5
13	+85°5
20	−86°
25	−81°
37	90°
52	+75°
63	+82°
75	−86°
100	−76°5

As is seen from the table, the angle between the optical axes in different representatives of the plagioclase series changes in different directions; but, undoubtedly, in view of the isomorphism of this series, there cannot be an abrupt jump between the given constants belonging to two adjacent varieties of plagioclases. Therefore the change in the angle between the optical axes of the latter must

obey some regularity, which is easy to obtain graphically in the form of the diagram shown in Fig. 41.

From this diagram it is evident that \(2V\) varies according to a law expressed approximately by a sinusoidal curve. If, for the entire plagioclase series, the latter represented only half a wave, then the determination of plagioclases could be based on the magnitude of the angle between the optical axes. But the diagram consists of \(1 \tfrac{1}{2}\) waves, as a result of which this constant can fully characterize the plagioclase under investigation only in particular cases.

Thus, we have seen that in the case of the albite law one always obtains an ambiguous solution within the limits between Nos. 0 and 36, since plagioclases equally distant from No. 21 have approximately identical coordinates of the twin axis.

In this case, the magnitude \(2V\), as is evident from the diagram just cited, can often resolve the ambiguity, and the more so the farther apart the compared numbers of these minerals are.

The same also applies to other curves having points with coordinates equal in absolute value. Obviously, in each particular case it is easy, with the aid of the diagram, to determine the value of the constant \(2V\) for the purpose of resolving the ambiguity that arises in determining plagioclases. In any case, it is always necessary to find the angle between the optical axes, since quite often this constant, as we shall see below, makes it possible to distinguish certain plagioclases from other varieties of feldspars resembling them.

§ 54. Magnitude of the birefringence.

Like the angle between the optical axes, the magnitude of the birefringence of plagioclases, on the basis of the formula

\[ \tg \frac{v_p}{2} = \sqrt{\frac{\mathrm{n}_g - \mathrm{n}_m}{\mathrm{n}_m - \mathrm{n}_p}}, \]

constitutes a constant whose variation must obey a rather complex law. To determine the latter, we shall derive the magnitude of the birefringence of plagioclases, using the principal refractive indices of the latter.

The following table is taken from the above-mentioned work of F. Becke (§ 30) and only supplemented by M. Lévy’s data for plagioclase No. 40.

Nos. of plagioclases	Refractive indices	Refractive indices	Refractive indices
	\(\mathrm{n}_g\)	\(\mathrm{n}_m\)	\(\mathrm{n}_p\)
5	1.5387	1.5321	1.5285
13	1.5431	1.5381	1.5341
20	1.5463	1.5428	1.5388
25	1.5490	1.5458	1.5417

Plagioclase Nos.	Refractive indices	Refractive indices	Refractive indices
	\(n_g\)	\(n_m\)	\(n_p\)
40	1.5560	1.5530	1.5490
52	1.5632	1.5583	1.5553
75	1.5730	1.5690	1.5640
100	1.5885	1.5835	1.5756

Using this table, we construct a diagram showing the dependence between the magnitude of the birefringence along the principal sections of the plagioclases and the composition of the latter (Fig. 42). The curves of this diagram have a wave-like form, and therefore in the general case the constants derived cannot be used for the determination of plagioclases; but, just like the angles between the optical axes, they help one to find one’s way in certain special cases.

Thus, the quantities \((n_g—n_p)\) and \((n_g—n_m)\) make it possible, quite well, to distinguish andesine from albite, which very often cannot be separated when determined by means of the coordinates of the pole of the twin axis.

Then the quantities \((n_g—n_p)\) and \((n_m—n_p)\) quite clearly separate bytownite from anorthite, which in some cases, for example in the formation of twins according to the Carlsbad law, are difficult to distinguish from one another.

In general, determination of the magnitude of the birefringence of plagioclases is of substantial benefit only in certain combinations of those results which are obtained in the investigation of these minerals. To this must also be added that, owing to the imperfection of the methods of measuring the magnitudes of birefringence, the application of the latter in determining plagioclases must be still more limited.

However, in the second part of the present work it was mentioned that the indirect method of determining birefringence by comparing the mineral under investigation with a completely determined mineral lying alongside it in the thin section is rather convenient and gives, for the most part, good results, and that quartz may be used exceptionally well as such a standard. The latter mineral occurs chiefly in acid rocks, the plagioclases of which, in their basicity, rarely exceed No. 40. But the acid half of the plagioclase series presents precisely the greatest difficulties for the principal method of determining these minerals, whereas in terms of the magnitude of birefringence it is characterized rather distinctly. Thus, in granites and rocks related to them, measurement of the birefringence of plagioclases may render great assistance in the determination of the latter. As for the choice of sections of the mineral for this measurement, preference should be given to those which give the greatest absolute values of the birefringence, namely \((n_g—n_p)\).

§ 55. Determination of Plagioclases by Fouqué’s Principle.

The extinction angles on the planes (001) and (010), which until recently were regarded as the most important optical constants of plagioclases and served for the practical determination of the latter, cannot have universal significance, since their use requires preparations whose sections are oriented according to the above-mentioned planes. True, when working on Fedorov’s stage there is no need to have a section with an oriented cut, but on the other hand the presence, in the grain under investigation, of both cleavage planes (001) and (010) is required, making it possible to determine these extinction angles. But if a grain of plagioclase possesses two systems of cleavage cracks amenable to setting, then there is no reason to resort to the aid of such constants, since in this case the coordinates of the zone axis make it possible to make an exact determination.

It is another matter when there is only one system of cleavage cracks; here, too, in the universal-optical method of investigation, we can introduce the concept of extinction angles, using for this purpose only the planes of elasticity of the mineral.

It has already been mentioned above (§ 45) that such extinction angles were first applied by M. Fouqué specifically for the determination of plagioclases. Although it represented a step forward in the study of the latter, Fouqué’s method nevertheless did not have general significance, because only sections more or less close to the planes \(n_g n_m\) and \(n_m n_p\) could be subjected to investigation, whereas by no means every grain of these minerals in rock thin sections satisfies such a condition.

But when working on Fedorov’s stage, any individual of plagioclase possessing clearly expressed cleavage can be subjected to the indicated test, and the extinction angles on the plane \(n_g n_p\) are then quite determinable. Indeed, by plotting on the net all the planes of elasticity and the cleavage plane, we determine by direct measurement with the aid of a three-legged compass the angles formed by the line of intersection of the cleavage plane with each principal section of the mineral and with some axis of elasticity lying in the latter.

These extinction angles can also be checked on the preparation, provided only that the axis of elasticity perpendicular to the chosen principal section of the mineral coincides with the axis of the microscope.

M. Fouqué¹ gave incomplete tables for the extinction angles on planes perpendicular only to the axes \(n_g\) and \(n_p\). In order to make possible a detailed investigation of plagioclases by the method described, we must compile such tables for all three principal—

...of these minerals, and for both principal cleavage planes, (010) and (001).

These data are easy to find, at least on the twin diagram, where the coordinate planes represent sections perpendicular to the axes of elasticity, and the curves of the albite and Manebach laws are the geometric loci of the poles of the planes of both cleavages for the entire plagioclase series. Having drawn the projections of these planes for different representatives of the plagioclases, we measure the required angles in the usual way. Since the present method should be used only in the case when orientation of the grain under investigation is altogether impossible, only the absolute magnitudes of the angles may be taken in this measurement.

Having found the latter, we shall express graphically the dependence between the composition of the plagioclases and the corresponding extinction angles; for this purpose, on the abscissa axis of the orthogonal projection we shall plot the former, and on the ordinate axis the latter.

The diagram constructed in this way (Fig. 43) is divided into three parts, corresponding to sections perpendicular to the axes \(n_g\), \(n_m\), and \(n_p\), and each part consists of two curves corresponding to extinction angles referred to the cleavage planes (001) and (010). In the name of each curve the sides of the measured angles are indicated: thus, the curve \(n_g\) (001) corresponds to extinction angles of which one side is the axis \(n_g\), and the other is the trace of the cleavage (001) on the chosen section of the mineral, perpendicular to the axis \(n_m\). The curves are not everywhere drawn with a solid line: the dotted line corresponds to the case when the cleavage plane forms an acute angle with the chosen section, owing to which the determination of the angle becomes indeterminate, since a small error in finding the line of intersection of the indicated planes entails a large change in the measured angle. Such parts of the curves should be used only when the corresponding extinction angles can be observed directly in the preparation.

The derived diagram may be used for a twofold purpose:

1) If the value of the cleavage plane is known, then the extinction angles will serve for determining the plagioclase. It is true that we see that only the second halves of the curves of the diagram will give quite definite solutions; however, if the extinction angles are found on all three planes, then, by comparing them with one another, we can very often avoid indeterminacy.

For illustration let us take the following example. Let the extinction angles in the plagioclase grain under investigation, referred to the cleavage plane (010), be expressed by the following magnitudes:

\[ \begin{aligned} \angle\, n_m\ (010) &= 15^\circ,\\ \angle\, n_p\ (001) &= 6^\circ,\\ \angle\, n_g\ (010) &= 88^\circ. \end{aligned} \]

Turning to the diagram, we see that plagioclases Nos. 5 and 42 correspond to the first angle, while the remaining angles characterize in general the acid members of the plagioclase series, not exceeding No. 33. Thus it must be accepted that the given plagioclase belongs to No. 5.

2) Above we assumed that the crystallographic value of the cleavage plane is known to us. But, strictly speaking, cleavage cannot be distinguished by its external appearance—especially since sometimes even secondary cleavage along \((110)\) or \((111)\) attains considerable perfection. Thus one must first determine the character of the cleavage, and only then proceed to finding the number of the plagioclase.

The character of the cleavage is also learned by comparing on the diagram the measured extinction angles.

Let—for example—the extinction angles be equal to:

on a plane perpendicular to the axis \(n_p\) . . . . . . . . . . . . . . . . \(65^\circ\)
” ” ” ” \(n_g\) . . . . . . . . . . . . . . . . \(32^\circ\)
” ” ” ” \(n_m\) . . . . . . . . . . . . . . . . \(50^\circ\).

On the diagram we see that the cited extinction angles correspond to different curves and to different numbers of plagioclases, which, of course, is impossible. Therefore it is necessary to suppose that this discrepancy is the result either of incorrect observations or of the circumstance that the given cleavage does not belong to the planes \((001)\) or \((010)\). In the present case we can no longer use the diagram to determine the plagioclase itself.

It is also necessary to note that if the plagioclase under investigation belongs to the acid half of the series, and the pole of the cleavage lies near the axis \(n_g\) or \(n_m\), then the cleavage belongs respectively to \((010)\) or \((001)\).

§ 56. Determination of plagioclases by means of the pole of the cleavage plane.

We have seen that the presence of at least one system of cleavage cracks is necessary for an unconditionally exact determination of a plagioclase. Thus, by means of it the above-described extinction angles are found, and often only in its presence is it possible to determine the magnitude of the birefringence.

We shall now show that cleavage can fully replace twinning in the sense of determining the number of a plagioclase from a twin diagram. Indeed, the normal to the cleavage planes \((001)\) and \((010)\) represents the twin axis of the Manebach and albite laws, and therefore the coordinates of this normal may be taken as the coordinates of the corresponding twin axis and from them the number of the plagioclase may be determined on the twin diagram.

It has already been indicated above how to determine the crystallographic value of the cleavage plane. Concurrently with the determination of the coordinates

of the latter pole, one can also measure the extinction angles on the principal sections of the mineral, and thus make the determination of the plagioclase more or less complete.

To this it must be added that if the grain under investigation represents a twin formation according to one of the two above-mentioned laws and possesses a cleavage whose plane coincides with the corresponding twin plane, then it is very convenient to verify the correctness of the determination of these and other elements. Thus, according to the albite law, the pole of the twin axis must coincide with the pole of cleavage on \((010)\).

Finally, only in the presence of a cleavage plane can one distinguish twin laws whose axes, for certain plagioclase numbers, have almost identical coordinates. Turning to the twin diagram, we see, for example, that the curves of the laws \([001]\) and perpendicular to \((110)\) almost coincide in the absolute value of the coordinates of the pole of the twin axis.

The ambiguity is easily resolved if there is cleavage on \((010)\), since in the case of a Carlsbad-law twin the twin axis must lie in the plane of this cleavage, whereas the twin axis—perpendicular to \((110)\)—will be situated far from the latter. If, under the same conditions, the investigated grain has cleavage on \((001)\), then the problem of distinguishing the laws is also solved quite satisfactorily; for this it is necessary to measure the angular distance between the poles of cleavage and the twin axis on the working net and to compare it with the corresponding value found, for example, on Becke’s diagram.

In the same way one can investigate other combinations of twin laws having analogous coordinates of their axes.

However, with the degree of accuracy that is generally characteristic of measurements on the Fedorov stage, some laws cannot be separated from one another; for example, the laws la Roc Tourné and perpendicular to \((100)\), whose curves almost completely merge with one another.

§ 57. Determination of new twin laws.

The twin diagram, which serves as the basis in the determination of plagioclases, includes only those twin laws that occur most frequently; therefore it has a provisional character, satisfying only the principal needs of petrographic practice.

How, then, should one proceed in those cases when the projection of the pole of the twin axis does not fall on any of the curves present on the diagram, or when the constants of the mineral give contradictory indications?

It must be said that, owing to the insufficiency of the data that we can obtain when studying the rock-forming minerals, such a question in the general case cannot be resolved. Only occasionally is it possible to determine the character of the twinning of the plagioclase grain that has interested us, and this can be done only by an indirect route. The latter consists in this: on the basis of certain indications we set ourselves the composition of this plagioclase and, using the coordinates found for the twin axis, determine by construction the crystallographic significance of the latter. Then, comparing the results of the constructions with other data, we verify the correctness of our assumptions. This method was described in brief outline in the second part (§ 43), and here we shall give a detailed example of its application.

Let us suppose that, while studying the twinning of one plagioclase in a thin section of a typical gabbroid rock, we found the following coordinates of the twin axis:

\[ \angle B \begin{cases} n_g = 84^\circ,\\ n_m = 71^\circ 5,\\ n_p = 20^\circ \end{cases} \]

Turning to the twin diagram, we see that these coordinates correspond most closely to plagioclase twin No. 5, constructed according to the law of de l’Esterel. The angle between the optic axes \(2V = +77^\circ\) agrees with this determination of the composition of the plagioclase, but the coordinates of the pole of the existing cleavage along \((001)\), the angle between the normal to this cleavage and the twin axis, the extinction angles in sections perpendicular to the axes of elasticity, and the general character of the rock indicate that this determination is hardly correct.

Indeed, when studying other grains of the same mineral in the given thin section of the rock, we everywhere find plagioclase No. 52. It is therefore natural to suppose that the first grain of this mineral also has the same composition, and that its twin is formed according to some new law.

In order to determine the latter, we draw on the Becke diagram, around the poles of the axes of elasticity corresponding to plagioclase No. 52, the geometrical loci of the above-given coordinates. All three arcs of small circles constructed in this way intersect at one point, close to the pole of the plane \((\bar{1}11)\), which may, in a first approximation, be taken as the sought twin plane. But we know that, if such a construction is made on a sphere, eight points of intersection of the indicated circles must be obtained. Thus it is necessary to verify whether \((\bar{1}11)\) is indeed a twin plane. For this one can use the angle between the pole of the cleavage \((001)\) and the twin axis of the grain under investigation; this angle,

according to measurements on the net \(= 82^\circ\). Measuring the same angle on Wülfing’s diagram, we obtain \(121^\circ\) or \(59^\circ\). Hence one may conclude that \((\overline{1}\overline{1}1)\) cannot be the twin plane of our twin. In order to find another point of intersection of the arcs of the small circle, we take the opposite pole of some axis of elasticity, for example, \(n_p\).

This pole will obviously lie outside the circumference of the net and will occupy the position of point \(52\). Carrying out the corresponding construction, we find one more point of intersection of the arcs of the small circle; this point coincides with the perpendicular to \((\overline{2}01)\), and, since the angle between it and the perpendicular to \((001)\) is equal to \(81 \tfrac{1}{2}^\circ\), the plane \((\overline{2}01)\) is indeed the true twin plane of the plagioclase grain under investigation. There is no need to find the remaining six points of intersection of the arcs of the small circle, since there can be no other answer to the question posed. It must be said that sometimes certain points cannot be obtained by construction—for example, when one of the poles of the axes of elasticity lies close to the projection point.

Thus, all the data concerning the plagioclase under consideration now agree, and the latter does indeed belong to No. 52.

Obviously, in exactly the same way one may determine the significance of other crystallographic elements, different from \((010)\) and \((001)\).

The example given shows that Fedorov’s stage makes it possible to make full use of all the numerous data furnished to us by rock-forming minerals, and thereby to achieve very significant results in the investigation of the latter. These results not only have practical value in the sense of the nearest determination of the composition of rocks, but at times are also very interesting from the theoretical point of view, for example, in the question of the character of twin formations.

For this reason the universal stage is fully applicable also to purely mineralogical investigations, at least to those concerning material which earlier had been the subject only of more narrowly petrographic study. When it comes into use in mineralogical laboratories, what a wealth of manifestations of minerals and their combinations will be opened up for precise investigation!

Chapter III.

Potassium–sodium feldspars.

§ 58. Isodimorphism of the potassium–sodium feldspars.

Above we examined the most important group of feldspars—the plagioclases, which form a clearly expressed isomorphous series, the individual members of which are characterized by gradual and rapid changes in various physical properties. Thanks to this latter circumstance, these minerals have been well studied, so that their true nature is more or less known to us; but this cannot be said of another group of feldspars, distinguished from the plagioclases by the fact that they are aluminosilicates of \(K_2O\) and \(Na_2O\).

Judging by the general habitus of the minerals of both groups, they must be recognized as simmorphous, so that the potassium–sodium feldspars should likewise yield a more or less isomorphous series.

But the known members of this presumed series differ from one another in such an insignificant way and often possess such a finely lamellar structure that, up to the present time, not only has the connection between them not been satisfactorily established, but the true nature of some species of these minerals is still unknown.

The latter are usually divided as follows:

1) Orthoclases, representing a potassium feldspar with an admixture, sometimes of a rather considerable amount, of \(Na_2O\), belong to the monoclinic system. In view of the fact that the albite and pericline twin laws, according to which finely lamellar polysynthetic twins are formed in feldspars, disappear completely in orthoclases, the latter always have a homogeneous structure.

For the characterization of these minerals it must be noted that in them the plane of the optical axes \(n_g n_p\) is perpendicular to the plane of symmetry of the crystal, while the angle \(n_p . a = 3^\circ—10^\circ\), and that the angle between the optical axes for varieties richest in \(K_2O\) is approximately \(-70^\circ\).

2) Anorthoclases represent an isomorphous mixture of \(K_2Al_2Si_6O_{16}\) and \(Na_2Al_2Si_6O_{16}\), with the latter predominating, and already belong to the triclinic system; but their forms are for the most part so close to the corresponding forms of orthoclases that it is often possible to establish the difference between these minerals only by means of especially precise methods. In general, the greatest difference in the angles between corresponding faces of the two mineral species does not exceed \(2^\circ\).

And the optical properties of anorthoclases are rather close to the analogous properties of orthoclases. For example, the axes of elasticity are arranged in them almost identically, so that the extinction angle on the plane \((001)\)

rarely reaches a value of \(4^\circ\), while the extinction angle on the plane \((010)\), which is almost the measure of the angle between the axes \(a\) and \(n_p\), is for the most part \(9^\circ\)¹).

As for the angle between the optical axes, it is subject to rather strong fluctuations, apparently depending on the relative amounts of \(Na_2O\) and \(K_2O\), varying from \(-32^\circ\) to \(-54^\circ\). Moreover, these minerals have not yet been studied sufficiently fully, so that the above figures must be regarded as only approximate. If one also takes into account the refractive indices and the magnitudes of birefringence, we shall see that the orthoclases and anorthoclases are rather difficult to distinguish from one another by most of their physical properties.

But there is one morphological feature that makes it quite easy to distinguish these minerals. The point is that anorthoclase belongs to the triclinic system and—consequently—twin formations according to the albite and pericline laws are possible in it; these always have the appearance of thin lamellar crystals, and the individuals of the twins, in their optical orientation, almost merge with one another, owing to the closeness of the crystal forms of the mineral to the monoclinic system. Sometimes anorthoclases, even with crossed nicols, seem homogeneous; but when the preparation is rotated about the axes of the Fedorov stage, it is almost always possible to find a section in which the twin striation nevertheless appears. In a section close to the \((010)\) plane—the composition plane of polysynthetic twins according to the albite law—patchy extinction is often observed, due to the irregular superposition of twin lamellae whose thickness is less than the thickness of the thin section²).

The same is observed in sections close to the \((001)\) plane, when the polysynthetic twins are formed according to the pericline law. Sometimes twin formations according to both laws appear in one grain, and then, in a section perpendicular to the first axis, a twin lattice can be observed.

3) Among the fairly numerous analyses of feldspars belonging to monoclinic minerals, there are some in which the amount of \(Na_2O\) many times exceeds the content of \(K_2O\)³).

Since microscopic investigations show that such a predominance of \(Na_2O\) is not due to mechanical admixtures, it is obvious that the molecules of sodium feldspar must give a monoclinic lattice, i.e., that there must exist a monoclinic modification of this

¹) H. Rosenbusch. Mikroskopische Physiographie I 2, 1905; 327.
²) M. Fouqué. Contribution à l’étude des feldspaths des roches volcaniques. Bull. soc. min. fr. 1894, p. 397—419.
³) M. M. Barbier et Prost. Sur l’existence d’un feldspath sodique monoclinique isomorphe de l’orthose.
Bull. de la Société chimique de France. 4 série, III, 1908; p. 894.

of the substance. In the feldspar studied by M. Barbier, the ratio \(K_2O : Na_2O = 1 : 10\); such a mineral may be considered an almost pure sodium feldspar. Thus, the monoclinic modification of the latter has apparently already been proved, which confirms the predictions of Des Cloizeaux¹.

If this is so, then sodium feldspar has three modifications: albite, anorthoclase, and a monoclinic mineral, which as yet has no special name.

Since this mineral forms an isomorphous group with orthoclase, it is very convenient to call it natronorthoclase, and the whole series—the natronorthoclase series.

4) Like sodium feldspar, potassium feldspar is also polymorphous: a triclinic variety of this substance is known, called microcline. In its crystalline forms the latter is very close to orthoclase, and its belonging to the triclinic syngony is proved chiefly by the fact that the axes of elasticity of this mineral occupy a sharply asymmetric position in the crystal; thus, the extinction angle on the plane \((001)\) is approximately \(+15^\circ\). Then, another property distinguishing microcline from the other potassium-sodium feldspars is the magnitude of the angle between the optical axes, equal on average to \(83^\circ\).

Finally, one morphological feature must be noted which is very characteristic of microcline. It consists in the fact that, owing to the mineral’s ability to form very thin-platy polysynthetic twins simultaneously according to two planes almost perpendicular to one another, an original grating is visible in sections, normal to the zones of the latter, under crossed nicols. These twin formations belong to the albite and pericline laws. The latter law, however, has in microcline the peculiarity that the trace of the plane of intergrowth of the individuals of the twin on the plane \((010)\) forms an angle of \(99^\circ\) with the trace of the cleavage \((001)\), for which reason some call the law itself microcline².

The grating of microcline is visible only in sections more or less close to \((001)\); in the other sections the extinction is five-bladed, owing to the irregular superposition of the twin lamellae one upon another.

It must be said that it is by no means necessary that both systems of twin formations be present in every individual of microcline; twins according to the less developed pericline law may also be absent, and then the mineral becomes similar to certain finely—

striated oligoclases, or else the twin microstructure is not manifested at all.

Despite the fact that there is a whole series of features distinguishing microcline from orthoclase, many regard¹ these minerals as identical, supposing that orthoclase is a submicroscopically twinned microcline. Against such an assumption there already speaks the circumstance that both in the individual rods of the microcline lattice and in the apparently homogeneous mass of a crystal of the latter the extinction angle and the angle between the optical axes remain the same.²

And there is nothing impossible in potassium feldspar, like the majority of minerals, possessing dimorphism. If, as some have noted,³ orthoclase, under the influence of the pressure of mountain-forming processes, passes into manifest microcline, then this only confirms the proposition that the character of the manifestation of a dimorphous substance depends on physical conditions.

It is interesting that recently P. Barbier³, by means of spectral analysis, demonstrated the different composition of the feldspars under consideration by us: namely, in orthoclase there is always a known quantity of lithium or rubidium, whereas in microcline the latter are not observed at all. However insignificant in general the content of these rare elements may be, nevertheless they may—according to the author—give an impetus to the formation of the corresponding dimorphous variety of the mineral, like certain catalysts. Of course, the influence of these elements by no means denies the importance of pressure or other physical conditions: they may act now in one direction, now in different directions.

In general, the independence of microcline as a mineral species can scarcely be subjected to special doubt.

5) Besides microcline, which has the composition of more or less pure potassium-sodium feldspar, there are several further types of feldspars resembling, in their properties, now anorthoclases and now microclines. True, these types have as yet been studied very inadequately, owing to the difficulty connected with this study because of the finely lamellar structure of the minerals.

Thus, M. Fouqué⁴ described two such feldspars, displaying here and there a lattice characteristic of microcline, but having angles between the optical axes \(2V = -56^\circ\) and \(-62^\circ\), close to the corresponding angles of anorthoclases. Unfortunately, it was not possible to measure the extinction angles on the plane \((001)\), in order more closely to determine

relation of these minerals to microcline. But the data cited were sufficient to recognize these minerals as representatives of the isomorphous series microcline—anorthoclase.

Actually, a priori one might have thought that such a series should exist among the triclinic potassium–sodium feldspars, because in the monoclinic syngony, as we have seen, minerals of analogous composition are isomorphous.

But if this is so, then the potassium–sodium feldspars constitute isodimorphous compounds.

This conclusion is exceedingly important, since the view we have adopted of the feldspars under consideration provides a criterion that will help to make sense of the scattered data concerning these minerals and will make it possible to define the latter more or less consistently.

Indeed, the physical properties of isomorphous compounds, as they vary, obey a certain regularity¹. If, as shown in Figure 46, the ratios of the alkali molecules entering into the composition of the feldspars of our isomorphous series are plotted on the axis of abscissas, and the magnitudes characterizing the corresponding properties on the axis of ordinates, then the resulting diagram will consist of two lines parallel to one another.

Although our series have been very poorly studied, nevertheless even an approximately constructed diagram can indicate the character of the necessary work and will, so to speak, guide it.

It goes without saying that such a diagram can also be used in determining the data of minerals. True, almost every ordinate corresponds to two abscissas belonging to different lines of the diagram, but for the most part the series differ rather sharply in certain simple features, so that ambiguous solutions will occur very rarely.

In constructing the diagram described, it is necessary to pay attention to the circumstance that its curves will turn out parallel only for such constants as characterize simple properties of a purely additive character. To these should be assigned the refractive index, the specific gravity, the angle between the optical axes, and the relation between the crystallographic axes and the axes of elasticity of the mineral. As for extinction angles, here, in view of a certain conditionality of the constants themselves, the diagram may also have an irregular appearance.

6) The classification of the potassium–sodium feldspars outlined above is grossly disturbed by the existence of modifications of these minerals called sanidines. The latter belong to the monoclinic—

… in syngony and in their chemical composition are completely identical with representatives of the sodium-orthoclase series. Nevertheless, they differ rather sharply in certain optical properties. Thus, the angle between their optical axes is usually small and even falls to zero; at the same time the axes are sometimes situated in the plane of symmetry of the crystal.

Observations show that these strange minerals are characteristic exclusively of the newest rocks, where they completely replace the orthoclases of more ancient rocks.

When comparing the properties of orthoclases and sanidines it seems incomprehensible why these two minerals, having the same chemical composition and crystalline forms, differ so sharply from one another in certain optical properties. It seems still more strange that sanidines have been encountered only in the newest eruptive rocks—and precisely where the boundary lies between the latter and more ancient rocks, in which only orthoclase is found.

The resolution of these questions requires, above all, a detailed study of the potassium-sodium feldspars among rocks of different ages, something that until now almost no one has undertaken. It is possible that among ancient rocks as well, under such study, feldspars analogous to the newest sanidines will be discovered.

Experiments on heating these minerals shed some light on the relation of the latter to the orthoclases. It turns out that the angle between the optical axes of orthoclases decreases with increasing temperature, while the plane of these axes may take a position coinciding with the plane of symmetry of the crystal, so that in the end a mineral is obtained that acquires the properties of sanidines. On the other hand, when the latter are heated, their properties change in the opposite direction.

Thus one gains the impression that sanidine represents only a pathological case of orthoclase, caused by the latter being found under special circumstances.

This is also consistent with the fact that almost all well-studied specimens of sanidines are volcanic ejecta or inclusions that have been subjected to the action of intense heat and then rapid cooling.

If this is so, then sanidines can be found only in the most superficial volcanic formations, which among ancient deposits have almost nowhere been preserved. It is possible, therefore, that the sanidines of ancient rocks, owing to diagenetic processes, have managed to pass into the original state of orthoclase. All these suppositions require substantial experimental verification, which, with the spread of the universal optical method of investigation, will probably not be long in coming.

Principal constants and methods for determining the potassium-sodium feldspars.

§ 59. General.

In considering the potassium-sodium feldspars and in deriving methods for determining the latter, we shall base ourselves on such data as come from wholly reliable sources and represent the arithmetic mean of several investigations. For some varieties of these feldspars, especially orthoclase, there is, as it were, a fairly extensive literature, but in some cases only the crystalline forms or optical properties are analyzed in detail, and the chemical composition of the minerals investigated is not given; in other cases, a mass of analyses is cited, and very little is said about the physical properties.

Insofar as we have been able to acquaint ourselves with the literature on the question of the potassium-sodium feldspars, all the more or less detailed, complete, and exact data concerning the latter can be accommodated in the following table (see p. 125).

In comparison with the isomorphous series of plagioclases, the potassium-sodium feldspars are known in relatively few representatives. But if one takes into account that, despite the insignificant quantity of available data, these representatives are scattered over the entire extent of the series we have outlined, and that their optical properties in general change in proportion to the change in chemical composition, then one must acknowledge the existence of these series.

Indeed, if we turn to the history of the study of the plagioclases, with respect to which there has never been any lack of material and which are well characterized by their constants, varying by a very considerable amount among the different representatives of the series, then even among these minerals comparatively few species were known for a long time, and only with the introduction of the universal-optical method of investigation did almost all the practically determinable representatives of the isomorphous series emerge.

The same should also be the case with the potassium-sodium feldspars, the closer study of which, as we shall see below, presents no small difficulties for the universal method as well.

§ 60. Morphological properties

The morphological properties of the potassium-sodium feldspars in many cases are so characteristic that they can render substantial assistance in determining these minerals. We have said above that, owing to the small difference in the constants ...

Monoclinic series

No. in order	Deposit	Sources	% by weight K₂O	% by weight Na₂O	% molecule Na₂O/K₂O	2V	Extinction angle: ⟂ np, ng	Extinction angle: \< nm (001)	Extinction angle: \< np (001)	Refractive index ng	Refractive index nm	Refractive index np	Notes
1	Celonite	Des Cloiseaux (Manuel de Minéralogie I, p. 338)	14.15	0	0	69°45′	0°	−5°	—	—	—	—	Analyses taken from Des Cloiseaux (ibid., p. 337).
2	From Oligoclase	Kohlrausch (Z. f. Kr. II, 102)	14.17	1.44	13	−66°5′	0°	—	1.5246	1.5230	1.5192	—	—
3	Brazil	C. Hintze (Handbuch d. Miner., p. 1400)	12.89	3.08	26	−54°	0°	—	—	—	—	—	—
4	Aranavast	C. Hintze (Handbuch d. Miner., p. 1401)	9.09	6.03	50	−48°	0°	—	—	—	—	—	—
5	Island of Pantelleria	Förstner (Z. f. Kr. VIII, 1933)	5.40	7.57	68	−40°30′	0°	+9°	—	—	1.5556	—	—
6	”	”	5.45	7.63	67	−41°	0°	+9°	—	—	1.5556	—	—

Triclinic series

No. in order	Deposit	Sources	% by weight K₂O	% by weight Na₂O	% molecule Na₂O/K₂O	2V	Extinction angle: ⟂ np, ng	Extinction angle: \< nm (001)	Extinction angle: \< np (001)	Refractive index ng	Refractive index nm	Refractive index np	Notes
7	Ural	M. Lévy (Étude sur la détermin. des feldspaths 2, 102)	—	—	—	−84°	3°	7°	—	—	—	—	Without characterization of the mineral.
8	Ilmen Lake	H. Rosenbusch, I₂, 316	13.50	1.56	15	−78°37′	—	—	—	1.5248	1.5228	1.5186	Analyses taken from Des Cloiseaux (Annales chim. et phys. 1876, p. 403).
9	Mursinka	”	13.90	1.66	15	−77°7′	—	—	—	1.5263	1.5238	1.5199	Analyses taken from Des Cloiseaux (Annales chim. et phys. 1876, p. 403).
10	Kilimanjaro	C. Hintze (Handbuch d. Min. II, 1424)	5.34	4.09	52	−60°45′	2°	—	—	—	—	—	From the analysis, apparently \((\mathrm{CaO}+\mathrm{Na_2O})\); hornblende is visible.
11	Island of Pantelleria	Förstner (average of 7 analyses)	4.25	7.34	72	−50°	—	9°	—	—	1.5262	—	Hornblende and riebeckite inclusions \(\parallel [100]\).
12	Central France	Fouqué (average of 7 analyses)	2.80	9.10	83	−45°	2°	10°	—	1.5289	1.5281	1.5225	Hornblende and riebeckite inclusions \(\parallel [100]\).

representatives of the monoclinic and triclinic series of these feldspars, their separation often meets with great difficulties.

In this case, the investigation of morphological features is an almost decisive factor. Indeed, monoclinic natronorthoclases always exhibit a homogeneous structure, whereas among representatives of the microcline-anorthose series a polysynthetic-twin composition is usually encountered; moreover, in microcline the latter develops almost always in two directions, whereby a characteristic lattice is formed. As some observations show, the latter is also found in other representatives of this series that are richer in $Na_2O$, having, however, distinctive peculiarities. Thus, in microcline this lattice is visible in sections more or less close to $(001)$, whereas in anorthoclases it can be noticed only in sections normal to the first axis $[100]$. This is due to the fact that in the first mineral species the plane of intergrowth of the polysynthetic twins according to the pericline law forms an angle of $99^\circ$ with the basal pinacoid, whereas in anorthoclases it is more or less close to the latter. In view of the closeness of the forms of the triclinic potassium-sodium feldspars to the monoclinic syngony, the two laws according to which the polysynthetic twins develop merge optically with one another. Therefore it is very possible that the twin lattice of these minerals is a consequence of the intergrowth of polysynthetic twins according to a single albite law, since among the plagioclases, whose twin individuals attain a more considerable size, such formations are sometimes observed quite distinctly.

In sections differing from those indicated above, the microstructure of microcline-anorthoses appears as a fibrous mottling. Sometimes, however, the twin formations of these minerals develop only in one direction, and then the latter become similar to thin-lamellar plagioclases.

Finally, it should be noted that the described microstructure may appear only in separate areas in a given feldspar grain.

§ 61. Extinction angles.

Observation of morphological properties often makes it possible to separate monoclinic feldspars from triclinic ones, but sometimes this is insufficient, in view of the fact that the twin striation of some representatives of the triclinic series reaches submicroscopic dimensions, or under certain conditions does not appear at all.

In this case the question can be resolved, for the most part, only in the presence of some crystallographic element, [[unclear: continuation cut off on page]].

for example, the cleavage plane, which we can use for measuring the extinction angles referred to the principal sections of the mineral, as in the plagioclases.

On the basis of the data in the table given above, let us construct a diagram indicating the variation of these extinction angles as a function of the composition of the feldspar.

From this diagram (Fig. 44), on which curve I refers to the monoclinic series, and II to the triclinic series, it is not difficult to see that the question of distinguishing the syngony of the minerals under consideration is most conveniently resolved by means of the cleavage plane (010), but that this very method of investigation is not very reliable, since the difference in the extinction angles not only for the individual series, but often also for different representatives of the latter, does not exceed \(2^\circ\), which is precisely only the accuracy of setting of the greater part of cleavage cracks.

For a complete characterization of the given minerals it is also necessary to know the extinction angles on the plane perpendicular to the axis \(n_m\). In monoclinic feldspars this angle is, obviously, \(0^\circ\), since the plane \(n_m n_p\) lies in the plane of symmetry of the crystal, but in triclinic feldspars it is variable. Meanwhile, we have no corresponding data; nor are there diagrams showing the relation of the optical and crystallographic elements of these minerals, and therefore we cannot find the above-mentioned angles by means of constructions, as we did in the case of the plagioclases.

True, M. Lévy¹ gave such a diagram for one variety of microcline, but its indications disagree with the data of such a scholar as M. Fouqué², and therefore it still requires verification. In any case, it will be appropriate to mention here that, according to this diagram, the extinction angles on the plane perpendicular to

\[ \begin{aligned} n_g &\ .\ .\ .\ .\ \angle n_p(001) = 5^\circ_{5} \\ n_p &\ .\ .\ .\ .\ \angle n_m(010) = 8^\circ \\ n_m &\ .\ .\ .\ .\ \angle n_p(010) = 16^\circ_{5} \end{aligned} \]

Be that as it may, the extinction angles measured on all three planes of elasticity make it possible to construct diagrams of the feldspars under discussion. This construction, based on the principle of notches and in the general case rather complex, requires, as an indispensable condition, that the direction be known, or the sign of the extinction angle. This condition is fulfilled only when an exact orientation of the feldspar grain under investigation is possible, which is, of course, achieved in petrographic work only very rarely. But if such a case occurs, then signs must be assigned to the extinction angles as follows—

...their generally accepted bases. If the crystal is set in the manner described in § 50, then, when considering planes perpendicular to the positive directions of the axes of elasticity, the extinction angles have the values indicated in Fig. 45.

In distinguishing the character of the cleavage in calcareous feldspars, one must be guided by the circumstance that, if the pole of the cleavage lies near \(n_g\), then the latter is \((010)\), whereas if this pole is close to \(n_m\), then the cleavage belongs to \((001)\). Leaving aside the imperfect cleavages along \((110)\) and \((111)\), which are very easily distinguished by the rule just given, we must draw attention to the frequently occurring—especially in monoclinic feldspars—frequent and fine parting along \((801)\), the pole of which lies not far from \(n_p\).

Finally, it should also be noted that in the case of the sanidines, in which the plane of the optic axes for the most part coincides with \((010)\), the axes of elasticity \(n_g\) and \(n_m\) change places. Even in this case, however, the character of the cleavage is very easy to determine.

Indeed, in monoclinic feldspars the two axes of elasticity lying in the plane of symmetry of the crystal \((010)\) are characterized by the circumstance that the extinction angles on planes perpendicular to these axes are equal to \(0^\circ\).

Thus we can always mark the axis of elasticity perpendicular to the second pinacoid \((010)\). If this is the axis \(n_g\), then before us is a feldspar with the normal position of the plane of the optic axes; but if the plane in which oblique extinction is observed is perpendicular to the axis \(n_m\), then the mineral under investigation has the character proper to sanidine.

To facilitate the construction of the diagram of a feldspar, in addition to determining the extinction angles it is also necessary to take the coordinates of the pole of cleavage, which, like the coordinates of the pole of the twin axis, may sometimes characterize the mineral under investigation.

§ 62. Twin formations.

From the consideration of extinction angles it may be noticed that the axes of elasticity are situated approximately the same in many representatives of the calcareous feldspars. Therefore the principle underlying the determination of plagioclases by means of the coordinates of the twin axis is not applicable here—the more so because twins, at least those accessible to practical determination, are not encountered especially often in these feldspars.

But the study of these twin formations can nevertheless be used for the solution of certain rather important questions.

Thus, above all, this method makes it possible to find bo-

…more or less exactly the position of certain crystallographic elements of the mineral and thus determine their relation to the axes of elasticity of the latter.

Then, knowing the composition of the feldspar, we can determine the character of the twinning formation, which, to be sure, is of greater theoretical interest. In the latter case it is necessary to have a diagram for the corresponding mineral, which for the present is possible in the case of monoclinic feldspars (such, for example, is the diagram in Fig. 37).

In order not to make constructions every time in carrying out the determinations indicated above, there is given below a table of the coordinates of the twin axis for the possible and known laws of one species of orthoclase, having \(n_p [100] = 6^\circ\).

All the necessary data were obtained by simple measurement on the diagram of Fig. 37.

No. in order	Name of the law	Twin axis	Orthoclase \(< n_p [100] = 6^\circ\): coordinates of the twin axis, \(n_g\)	Orthoclase \(< n_p [100] = 6^\circ\): coordinates of the twin axis, \(n_m\)	Orthoclase \(< n_p [100] = 6^\circ\): coordinates of the twin axis, \(n_p\)	Microcline according to M. Lévy: coordinates of the twin axis, \(n_g\)	Microcline according to M. Lévy: coordinates of the twin axis, \(n_m\)	Microcline according to M. Lévy: coordinates of the twin axis, \(n_p\)
1	Albite	\(\perp (010)\)	none	none	none	\(-18^\circ\)	\(82.5^\circ\)	\(+74^\circ\)
2	Manebach	\(\perp (001)\)	\(90^\circ\)	\(6^\circ\)	\(+84^\circ\)	\(+81^\circ\)	\(11.5^\circ\)	\(+84^\circ\)
3		\(\perp (100)\)	\(90^\circ\)	\(71^\circ\)	\(+19^\circ\)	\(+72^\circ\)	\(72.5^\circ\)	\(+25^\circ\)
4	De l’Esterel	\([100]\)	\(90^\circ\)	\(84^\circ\)	\(-6^\circ\)	\(-75.5^\circ\)	\(81.5^\circ\)	\(-16.5^\circ\)
5	Carlsbad	\([001]\)	\(90^\circ\)	\(19^\circ\)	\(-71^\circ\)	\(+87.5^\circ\)	\(18.5^\circ\)	\(-70.5^\circ\)
6	Pericline	\([010]\)	none	none	none	= albite	= albite	= albite
7		\(\perp [100]\ \parallel (010)\)	= Manebach	= Manebach	= Manebach
8	La Roc Tourné	\(\perp [001]\ \parallel (010)\)	= No. 3	= No. 3	= No. 3
9	Scopi	\(\perp [010]\ \parallel (001)\)	= De l’Esterel	= De l’Esterel	= De l’Esterel
10		\(\perp [100]\ \parallel (001)\)	none	none	none	= albite	= albite	= albite
11		\(\perp [001]\ \parallel (100)\)	none	none	none	= albite	= albite	= albite
12		\(\perp [010]\ \parallel (100)\)	= Carlsbad	= Carlsbad	= Carlsbad
13
14	Bisectrix	\(\angle [100]\,[001]\)	\(90^\circ\)	\(74.5^\circ\)	\(-15.5^\circ\)	\(-76.5^\circ\)	\(73^\circ\)	\(-22^\circ\)
15	Bisectrix	\(\angle [001]\,[010]\)	\(90^\circ\)	\(38.5^\circ\)	\(+51.5^\circ\)	\(+73^\circ\)	\(41.5^\circ\)	\(+53^\circ\)
16	Bisectrix	\(\angle [010]\,[100]\)	\(+45^\circ\)	\(48^\circ\)	\(-76.5^\circ\)	\(+45^\circ\)	\(55^\circ\)	\(-65^\circ\)
17	Baveno	\(\perp (0\bar{2}1)\)	\(+45^\circ\)	\(85^\circ\)	\(+45.5^\circ\)	\(+31^\circ\)	\(80^\circ\)	\(+61^\circ\)
18	„	\(\perp (021)\)	\(45^\circ\)	\(45.5^\circ\)	\(86^\circ\)	\(-56.5^\circ\)	\(38.5^\circ\)	\(+75^\circ\)
19		\(\perp (1\bar{1}0)\)	\(59.5^\circ\)	\(74^\circ\)	\(+36^\circ\)	\(-77^\circ\)	\(71^\circ\)	\(+24^\circ\)

The coordinates of the twin axis for other members of the natron-orthoclase series will differ more or less strongly from those given only in those cases when this axis lies not far from the plane \((010)\), in which the axes \(n_m\) and \(n_p\), shifting with a change in the chemical composition of the feldspar, are located.

In the same way, the coordinates given are fairly characteristic also for triclinic feldspars rich in \(Na_2O\)—with only those …

a substantial difference, namely that in the latter there can be no coordinates at all equal to \(90^\circ\).

If these coordinates are plotted on a twin diagram of plagioclases, then everywhere we obtain, on the corresponding curves, points lying near No. 20; this is just what was to be expected in view of the closeness of the forms and the almost identical arrangement of the axes of elasticity in these and other feldspar species.

But in the first half of the triclinic series, rich in \(K_2O\), the axes of elasticity, judging by the above-mentioned diagram of M. Lévy, occupy a more original position. Thus, if we plot, in the manner known to us, the twin-axis coordinates of microcline given above on this diagram, we obtain points far removed from the corresponding plagioclase curves.

§ 63. Angle between the optical axes.

Thus, none of the above-described methods for determining the feldspars under consideration is distinguished by any particularly notable sensitivity; by means of them, for the most part, we can only distinguish these minerals according to their belonging to one series or another, but we are not able to determine what place they occupy within the latter.

Incomparably better results in this respect are given by finding the angle between the optical axes. Indeed, using the table given in § 59, we can express the dependence between this angle and the chemical composition of the corresponding feldspar in the form of the diagram shown in Fig. 46.

From it it is evident that, from the angles between the optical axes, one can judge, with a sufficient degree of accuracy, the chemical nature of the minerals in question. Of course, we cannot determine the latter with an accuracy to one or two numbers, as is the case in the investigation of twin formations of plagioclases. The principal errors in this determination depend on the following causes.

1) As has already been said more than once above, the setting of the optical axes is usually not made with sufficient accuracy; moreover, if the angle \(2V\) is obtained by constructions while observing only one axis, then the error is doubled.

2) Further, we know that this constant is in general readily subject to fluctuations depending on external conditions, and that in sanidines it has an entirely different value; meanwhile, we as yet have no criterion for assigning a given feldspar to one species or another.

3) Sometimes it is impossible to determine to which series the investigated grain of potash feldspar belongs, and in this

case, obviously, one value of the angle \(2V\) corresponds to two feldspars differing in composition.

4) Finally, it must be noted that the diagram presented is only preliminary and needs verification.

Thus we see that even the angle between the optical axes, which represents a regular and rapidly varying function of the chemical composition of the feldspars under consideration, still cannot yet serve as a fully reliable basis in determining the latter.

§ 64. Birefringence.

To finish the consideration of methods for determining the potash-soda feldspars, let us also turn our attention to the constants connected with birefringence.

Of these, practically only the magnitude of the birefringence can be determined more or less accurately; but the latter varies by a very small amount in individual representatives of the given feldspars, having on average the following values:

\[ \begin{aligned} n_g - n_p &= 0.006\text{--}8,\\ n_g - n_m &= 0.001\text{--}2,\\ n_m - n_p &= 0.005\text{--}6. \end{aligned} \]

Thus this constant is not characteristic of the minerals in question, and by means of it one cannot distinguish not only the individual members of isodimorphous series, but even identify the latter.

Only sanidines are determined fairly well by means of the magnitude of the birefringence, which for these minerals has the following values:

\[ \begin{aligned} n_g - n_p &= 0.003\text{--}5,\\ n_g - n_m &= 0.000\text{--}2,\\ n_m - n_p &= 0.003\text{--}5. \end{aligned} \]

The situation is somewhat better with the refractive index, but the latter does not belong to those constants by means of which one can determine rock-forming minerals. Moreover, even this constant, on passing from one series to another, changes only to an insignificant degree, so that it cannot be used to distinguish the kind of syngony to which the given potash-soda feldspar belongs.

Only in one case can the constants given be of substantial use—namely, in the determination of microcline. The point is that the latter sometimes has a polysynthetically twinned structure in only one direction, and since the angle \(2V\) in it sometimes

rises to \(-88^\circ\), if one judges from the data of M. Fouqué; in this case it is extremely difficult to distinguish this mineral from oligoclase, which has the same outward appearance and approximately the same angle between the optical axes, and only the magnitude of the birefringence or—still better—the refractive index distinguishes these two minerals rather well.

Potassium-barium feldspars.

There is one more type of feldspar, called celsian and representing an aluminosilicate of \(BaO\) with the formula \(Ba Al_2 Si_2 O_8\).

This mineral, together with orthoclase, forms an isomorphous group of monoclinic feldspars. The latter apparently has no especially wide distribution, occurring chiefly in rocks of foyaitic and theralitic magmas¹. Nevertheless, thanks to the works of Strandmark, Sjögren, and some others, the optical constants of these minerals have already been studied rather well and may be set out in the following table, taken from the cited work of Rosenbusch.

% celsian content	Sp. wt.	\(2V\)	\(\langle n_p\rangle [100]\)	\(n_g\)	\(n_m\)	\(n_p\)	\(n_g-n_p\)	\(n_g-n_m\)	\(n_m-n_p\)	Investigator
5	2.593	\(-71^\circ\)	\(+2^\circ\)	1.5257	1.5240	1.5201	0.0056	0.0017	0.0039	Strandmark.
9	2.645	—	\(+5^\circ\)	1.5335	—	—	—	—	—	Baumhauer.
20	2.725	\(-75^\circ 5\)	\(+2^\circ\)	—	—	—	—	—	—	Strandmark.
21	2.733	\(-74^\circ\)	\(-6^\circ\)	1.5416	1.5395	1.5373	0.0043	0.0021	0.0022	—
24	2.756	\(-77^\circ\)	\(-11^\circ\)	1.5426	—	—	—	—	—	—
30	2.818	\(-78^\circ 5\)	\(-18^\circ\)	1.5469	1.5451	1.5419	0.0050	0.0018	0.0032	—
100	3.384	\(+86^\circ 5\)	\(62^\circ\)	1.5940	1.5886	1.5837	0.0103	0.0054	0.0049	—

By their morphological characters, potassium-barium feldspars can be taken only for natron-orthoclases, from which they are distinguished by a whole series of properties.

Thus, first of all, the large angle between the optical axes is conspicuous, close to the corresponding angles of the plagioclases. Then the refractive index is also a rather characteristic constant, especially for varieties rich in \(BaO\). As for the extinction angles, the latter can help in determining these feldspars only in the case when the latter are close to celsian, or when oriented sections are available, from which one can determine what position in the crystal is occupied by the axes of elasticity.

The position of the latter changes greatly throughout the series. Figure 47 shows how the elements of the optical ellipsoid are arranged in orthoclase (1) and celsian (2). From it one can see that, in the representatives of this series, all the axes of elasticity change their places. For this reason, determining the spatial relations between the elements of the minerals under consideration in grains in which they cannot be precisely established encounters rather serious difficulties.

In general, the methods for determining potassium–barium feldspars are analogous to those for soda orthoclases.

Perthitic intergrowths of feldspars

In studying the chemical composition of feldspars, results are often obtained that are inconsistent with certain properties of these minerals. This discrepancy is for the most part caused by the regular intergrowth, in a single crystal, of two feldspar species, one of which often appears in the form of such small lamellae that, in ordinary observations, the latter may remain entirely unnoticed.

Regular intergrowths of two feldspar species are generally called perthitic and are divided into the following varieties.

1) Microperthites represent the commonest case of such formations. In them the principal mineral is some potassium–sodium feldspar, enclosing portions of usually very acid plagioclase, close to albite.

These portions appear for the most part in the form of very thin bands, arranged along the plane \((\overline{8}01)\) of the mineral enclosing them and sometimes having so regular an appearance that the whole crystal seems to be a polysynthetic twin (Fig. 48a). In other cases the bands wedge out, change their width, or occur as spindle-shaped inclusions, almost coinciding with the vertical axis of the mineral and having an elliptical or round cross-section (Fig. 48b).

In all these more or less regular bands of albite it is usually difficult to discern the twin striation of the latter.

But there are also such intergrowths in which the portions of albite already attain considerable dimensions, having for the most part an irregular appearance. When these perthites are examined, one gets the impression that two crystals, consisting of potassium–sodium feldspar and thin-platy plagioclase, have grown through one another in the most accidental manner, while the relative quantity of these mineral species may be very varied, up to the complete displacement of one mineral by the other (Fig. 48c).

In these cases it is sometimes possible to determine that both minerals intergrow in such a way that the planes $(010)$ are parallel, and this determines the regular character of these formations.

The so-called Schachbrett-Albit (checkerboard albite)¹ is a microperthite in which the substance of the potassium-sodium feldspar has been completely displaced by albite. This substitution is reflected in the albite crystal in that the twin lamellae of the latter are often interrupted, their length usually being no greater than their width. Owing to such a microstructure, an albite section cut normal to $(001)$ and $(010)$ resembles a chessboard between crossed nicols.

2) When the width of the microperthitic intergrowths is reduced, transitions are obtained from the formations just described to the so-called cryptoperthites. Since these intergrowths are arranged along the plane $(\overline{8}01)$, which represents a frequent and fine parting of the potassium-sodium feldspars, it is often difficult to decide whether one is dealing with the parting alone or also with cryptoperthitic bands deposited along the latter.

Here chemical analysis of such formations, taken to be homogeneous minerals, may give quite erroneous results.

To distinguish such formations it is sometimes useful to subject the section under investigation to rotations about the axes of the universal stage, since in certain positions the latter intergrowths sometimes show themselves rather distinctly. But for the most part one must resort to the method of F. Becke,² which consists in this: by simple procedures the sharpness of the outlines and the relief of adjacent parts of the section, possessing different refraction, is brought out, whereby they are easily separated from one another.

Finally, to attain the stated aim it is necessary to pay attention to the following circumstance. In those places where the cryptoperthitic intergrowths are strongly crowded together, a certain amount of $Na_2O$ is taken up by the enclosing feldspar, as a result of which the extinction angle of the latter on a plane perpendicular to $n_g$ is sometimes considerably increased. This phenomenon, however, may be explained by the combined action on the transmitted light of both minerals, the oriented lamellae of which are superposed upon one another.

3) Further, there exists a type of perthitic intergrowths which differs from the preceding ones in that the enclosing mineral is plagioclase, while the intergrowths consist of potassium-sodium feldspar.³ These intergrowths usually have the form of more or less short

¹ F. Becke. Zur Physiographie der Gemengtheile der krystallinen Schiefer. Wien. 1906, 28.
² F. Becke. Ueber die Bestimmbarkeit der Gesteinsgemengtheile etc. Wien. Ak. 102 (1) 1893; 358.
³ F. Suess. Ueber Perthitfeldspäthe aus krystallinischen Schiefergesteine. Jahrbuch. d. g. Reichsanstalt. LIV, 1904; s. 417.

six-sided inclusions of rectangular or trapezoidal form. With their long direction they are arranged either along the vertical axis or along the edges of intersection of the prism faces with the principal pinacoid of the plagioclase. Figure 49 gives an idea of such formations, called antiperthites.

The described perthitic formations explain the content of \(CaO\) in analyses of many potassic-sodic feldspars and \(K_2O\) in analyses of plagioclases. But it is possible that sometimes the aluminosilicates of these oxides enter into the composition of the corresponding minerals in the form of isomorphous admixtures, and that, consequently, there must exist a monoclinic modification of the calcium feldspar and a second triclinic modification of the potassium feldspar. The latter is all the more probable, since the sodium feldspar is dimorphous in the triclinic system.

This question has not yet been completely clarified; meanwhile, if such modifications do indeed exist, they must influence the properties of those feldspars into whose composition they enter.

4) Finally, in the literature one may find indications of regular intergrowths, of perthitic character, of two feldspar species belonging to one and the same isomorphous series. Such formations, by analogy with the preceding ones, might be called isoperthites. However, we still have an insufficient number of observations on the basis of which these formations could be regarded as fully proved.

The most definite data concerning the latter are given in the recently published work of A. N. Zavaritsky: “Petrographic observations in the vicinity of the Miass factory” (Transactions of the Mining Institute, III, 1910; 66—67), where the author points to the existence of perthitic intergrowths of orthoclase, microcline, and other representatives of the potassic-sodic feldspars.

Assuming that all the observations were made quite correctly, we must pay attention to the circumstance that the striped bands observed by the author in some grains of microcline, and possessing a considerably smaller angle \(2V\) than this mineral may have, represent twin lamellae of the same microcline, lying irregularly one upon another in the given section of the thin section and therefore producing a deviation in the optical properties of the whole system.

Such an assumption can easily be verified by rotating the apparatus about the axes of the universal stage to such a position at which the vertical axis of the mineral, lying not far from \(\eta_m\), coincides with the axis of the microscope. In such a position the striping of homogeneous microcline disappears, and in its place there appears an original lattice.

In general, certain optical constants of triclinic feldspars are not only dependent on their composition, but also on the structure of the crystal; and in different sections of the latter, and sometimes even in different portions of one and the same section, they will have different values. This proposition has been demonstrated by the investigations of C. Viola¹ on polysynthetic twinned albite and, of course, may be applied still more fully to the feldspars here under consideration.

The probability of finding isoperthites must be insignificant, for, as observations show, different representatives of an isomorphous series, when occurring in a single crystal, intergrow in a perfectly parallel manner with one another, and in most cases grains with so-called zonal structure are formed. Such formations are especially common among the plagioclases, but are also characteristic of the potash-soda feldspars.

The indicated tendency to produce perfectly parallel intergrowths is so great among members of isomorphous series that one may sometimes observe some layers of zonal plagioclase being deposited on strongly resorbed surfaces of an earlier-formed crystal,² or a crack in the crystal of some mineral being healed by another mineral isomorphous with the former.³

In any case, such intergrowths must have a regular zonal character, depending on the very conditions of their formation, and therefore they differ from perthites, which properly represent the results of the unmixing (Entmischung) of two heterogeneous substances.

APPENDIX.

A detailed example of the determination of a plagioclase.

Table VIII.

The thin section is cemented on so that the twin seam between the two widest, differently oriented individuals of the polysynthetic plagioclase twin passes through the center of the ocular cross (§ 4).

The microscope stage is fixed in its initial position (§ 5, item 3 c).

Examination of the first individual.

Loosening screw $i$, we tilt the stage about axis $J$ away from ourselves and, by rotating the inner circle with the aid of the screws of the upper segment, bring the selected individual into extinction.

We rotate the stage about the same axis toward ourselves and attain extinction by tilting the inner circle about axis $H$, fixing the position of the latter with screw $h$.

We repeat this procedure several times (sometimes in the reverse order), until, when the stage is rotated about axis $J$, extinction is preserved (§§ 18 and 19).

We take readings of the coordinates of the established plane of the axes of elasticity. The index of the inner circle standing opposite the division $344^\circ$ of the inner limb indicates the longitude of the plane.

The second coordinate is determined in the following way. Since the inner circle is inclined to the right, then, having loosened screw $r$, we turn ring $R$ (holding only the serrated edge $m$) counterclockwise through $90^\circ$. We insert the Bertrand lens and, tilting the stage about axis $J$ away from ourselves, bring the shining edge of the inner circle into coincidence with the horizontal thread of the ocular cross.

In this position a reading of $82^\circ$ is obtained on the vertical limb; hence the height of the plane is equal to $90^\circ - 82^\circ = 8^\circ$ (if the zero of the vernier does not coincide with $90^\circ$ of the vertical circle with the same setting of the shining edge of ring $R$, then one must take into account the correction for the given instrument, determined exactly on the basis of § 5, item 8 a). The coordinates obtained are written down in the margins of the mesh, reckoning the inclination

plane opposite to that observed on the instrument: 1) 344°, to the left 8°.

We seek the optical axes in the plane that has been set.

Loosening the microscope screw, we turn the stage of the latter approximately 45° counterclockwise and, by rotating the instrument about the axis $J$, try to bring the section to complete extinction, preserved as the microscope stage is rotated (§ 22). In the present case this is achieved, and with exact setting to complete extinction (§§ 19 and 27) the mean reading on the vertical circle is equal to 355°. Thus the altitude of the optical axis found is equal to $360^\circ - 355^\circ = 5^\circ$ (the correction for the 360° graduation of the vertical circle is, obviously, the same as for the 90° graduation). For a graphical representation of the optical axis we read the inclination of the section in the opposite direction and write on the margins of the net: $A^1_1$—away from oneself 5°.

The second optical axis, lying in our plane, is inaccessible to observation.

We plot the results of the measurements on the stereographic net. First we mark the circumference of the latter from 0° to 360° counterclockwise (this may also be done mentally, using the existing divisions). We turn the net toward ourselves to division 344° and mark the diametrically opposite point of the circumference. The arc of a great circle representing the projection of the plane found must rest on these two points, being inclined to the left. Having set off on the circumference of the net 90° from one of the marked points, we draw the diameter $ab$. Counting by means of the concentric parallels along this diameter 8° from the right-hand end of the latter and from the center of the net, we obtain, respectively, the pole of the arc $p$ and the third point of the latter $c$. The pole found must be denoted by $n_m^{1}$, as perpendicular to the plane of the optical axes, and the arc is easily drawn through the three points with the aid of a circular ruler (§§ 9 and 11).

To find the projection of the optical axis, we set off on the circumference of the net, from the left-hand end of the above-mentioned diameter $ab$, 5° away from ourselves and, connecting the point obtained with the pole $n_m^{1}$, we have, at the intersection with the arc, the projection of the axis $A^1_1$ (§ 27).

We set the second plane of the axes of elasticity. Having brought the microscope stage to its initial position, we turn the inner circle approximately 90° and here seek, as before, the plane of elasticity mentioned. We obtain the coordinates: 2) 80°, to the left 34°.

The graphical construction of this plane can be considerably simplified, since we need to find the axes of elasticity themselves. We turn the net toward ourselves to division 80° and, having counted from the latter 90°, draw through the division obtained (170°) a diameter on which will lie the pole of the plane, distant from the right-hand end of this ...

of the diameter by 34°. We obtain the point \(f_1\), which in the present case happens exactly to coincide with the arc found earlier. If there were no such coincidence, it would be necessary to adjust the results of all the measurements, according to § 21.

To find the third axis of elasticity we project the point \(f_1\) from \(n_m^1\) onto the circumference of the net, mark off 90° on the latter and, joining the point thus obtained with \(n_m^1\), at its intersection with the arc we have the projection of the required axis \(f_2\).

We determine the magnitudes of the axes of elasticity. We take the second plane of elasticity and set the axis \(f_2\) vertically; for this purpose we rotate the instrument about the axis \(J\) toward ourselves (if the net is kept with division 80° toward oneself) through the angle \(f_2 f_1 O\), equal to approximately 8°, and fasten the screw \(i\). We rotate the microscope stage by 45° counterclockwise and insert the mica compensator into the slot of the microscope tube: a weakening of the interference color is obtained. Therefore (§ 23, table), the axis lying in the direction of the slot, i.e. \(n_m^1\), will be greater than the axis \(f_1\) compared with it, which must be called \(n_p^1\), and the axis \(f_2\)—of course—\(n_g^1\).

It is now possible to determine the angle between the optical axes and the sign of the plagioclase. From the pole of the plane of the optical axes \(n_m^1\) we project onto the circumference of the net the points \(n_g^1\) and \(A_1^1\), and mark off the angle between the projections of the latter, equal to 42.5°.

Obviously, \(2V = +85^\circ\) (§ 27).

Study of the second individual.

In the manner described above we study the second individual, finding the axes of elasticity \(n_g^2\), \(n_m^2\), and \(n_p^2\). The optical axes in this individual are not directly determinable. Although in the present case there is no pressing necessity to find them, since the angle \(2V\) of the plagioclase is already known to us, nevertheless, for the sake of illustrating the procedures set forth in § 28, we shall do this.

We set the axis \(n_m^2\) vertically, without tilting the specimen about the axis \(J\). For this we draw a diameter through \(n_m^2\), mark off from the end of the latter on the circumference of the net 90°, and place the index of the inner circle of the stage opposite the corresponding division, for example, 318°. Then we rotate the outer ring \(R\) by 90° clockwise and tilt the specimen about the axis \(J\) away from ourselves by the spherical distance of the point \(n_m^2\) from the center of the net, i.e. we bring the zero of the vernier of the vertical circle into coincidence with 66° (90°–24°) of the latter. Finally, we move in the Bertrand lens and, by tilting the inner circle around the axis \(H\), bring the shining edge of the latter into coincidence with the horizontal thread of the ocular cross. If we now set the outer ring \(R\) and the vertical circle in their initial position, the axis \(n_m^2\) will prove to be in coincidence with the axis of the microscope, while the axes of elasticity \(n_g^2\) and \(n_p^2\) will occupy the positions \(n_1\) and \(n_2\).

Since the angle \(2V\) of the second individual of our twin must in general be the same as that of the first individual, its optical axes will, in all probability, in the present setting occupy the positions \(OE\) and \(OF\), with the arc \(En_1=-n_1F=42.5^\circ\).

Having made this supposition, we turn the outer ring \(R\) by \(53^\circ\) clockwise, and then the optical axis \(OE\) should come into the plane of symmetry of the microscope. To verify this position we incline the preparation about the axis \(J\) toward ourselves—positively—by \(40^\circ\). Then the projection of the axis \(n_m^2\) will move to the point \(I\), and the projection of the optical axis \(OF\) will be the point \(F^1\). According to § 28, we draw a diameter through the latter point and bisect the arc \(EF_1\).

If the microscope stage is now turned clockwise by half of this arc, equal to \(43^\circ\), extinction should occur, which was indeed observed.

From this it may be concluded that the position of the optical axes of the second individual was accepted by us correctly, and that thus

\[ 2V=+85^\circ . \]

Determination of the twinning formation.

Drawing arcs of great circles through the like poles of the axes of elasticity, we find at their intersection the projection of the twin axis \(B_{1,2}\).

To verify the latter, we set the index of the inner circle opposite the division \(272^\circ\) and incline the instrument about the axis \(J\) by \(40^\circ\), away from ourselves: in this case, when the microscope stage is rotated, the twin appears to be a single individual.

We measure, with a three-legged compass (§ 13), the coordinates of the twin axis:

\[ \angle B_{1,2} \left\{ \begin{array}{rcl} n_g^1 \ \text{or}\ n_g^2 &=& 37^\circ,\\ n_m^2 &=& 59.5^\circ,\\ n_p^1 &=& 72^\circ . \end{array} \right. \]

For the abbreviated method of determining these coordinates, see § 42. Plotting on the twin diagram (Table X) \(37^\circ\) from the right or left end of the horizontal diameter, \(72^\circ\) from the upper or lower end of the vertical diameter—along the equatorial parallels—and \(59.5^\circ\) from the center of the net along the concentric circles, we find at the mutual intersection of the parallels four points, of which only two fall on the existing curves: one point indicates No. 63 of the twin according to the pericline law, and the other point—approximately No. 26 of the twin according to the law: the twin axis is the bisector of the angle between \([010]\) and \([001]\).

To eliminate this ambiguous solution we turn first of all to the angle between the optical axes, which in the diagram of Figure 41 definitely indicates No. 63.

Since there are good cleavage cracks in the selected grain, we make a further check of our determination by means of the latter.

First we establish the plane of this cleavage (§ 24) and, after several readings, find its coordinates with an accuracy to \(1/2^\circ\): \(114^\circ\), to the right \(17.5^\circ\). From these coordinates we construct the projection of the cleavage plane with its pole at the point \(B_{001}\).

Extinction angles.

Having drawn the projections of all three planes of the axes of elasticity of the 2nd individual, in which the established cleavage crack is situated, we measure the angular magnitudes \(X n_m^2\), \(Y n_p^2\), and \(Z n_g^2\), as the extinction angles on the planes perpendicular to the axes \(n_p^2\), \(n_g^2\), and \(n_m^2\) (§ 55):

\[ \begin{aligned} \angle X n_m^2 &= 45^\circ,\\ \angle Y n_p^2 &= 37^\circ,\\ \angle Z n_g^2 &= 53^\circ. \end{aligned} \]

The last angle can also be checked in the preparation. For this, as before, we bring the axis \(n_m^2\) into coincidence with the microscope axis and, rotating the stage of the latter, directly measure the extinction angle referred to the trace of the cleavage crack.

Plotting the values of the quantities found on the diagram of Figure 43, we see that, except for one abscissa corresponding approximately to No. 63, they determine the cleavage \((001)\).

Thus, by the extinction angles as well, the plagioclase belongs to No. 63.

Coordinates of the pole of cleavage.

We measure the coordinates of the pole of cleavage:

\[ \angle B_{(001)} \begin{cases} n_g^2 = 52^\circ,\\ n_m^2 = 51^\circ,\\ n_p^2 = 62^\circ. \end{cases} \]

Since the perpendicular to \((001)\) corresponds to the twin axis according to the Manebach law, then, plotting the obtained coordinates on the twin diagram, we take the projection of this axis near the Manebach curve: the marked point is close to No. 63.

Thus, several determinations consistently indicate that the given plagioclase belongs to No. 63. Therefore the twin law according to which the investigated twin is formed is the pericline law.

Having the crystallographic plane \((001)\), established with an accuracy to \(1/2^\circ\), we check the crystallographic value of the twin axis. From the stereogram of the given plagioclase we see that this axis lies in the plane \((001)\) and therefore may in fact be \([010]\).

In addition, the appended figure shows that the plane of intergrowth—

the extinction positions of the individuals of the twin are more or less parallel to the indicated cleavage, which likewise characterizes the pericline law, making it possible to distinguish it from the albite law, which resembles it.

In conclusion it should be noted that, with some skill, the described detailed determination of plagioclase can be carried out within one hour; for determination by a single twin law, which is what is usually encountered in the treatment of petrographic material, even ten minutes is often sufficient.

Thus the universal method is a quite convenient petrographic method, and the Fedorov stage, in combination with the microscope, should be an indispensable accessory of every petrographer.

Misprints.

Page	Line	Printed	Should be
1.	21 from bottom	caps	caps
4.	1 ”	Feldspathstudien,	Feldspathstudien I.
5.	16 from top	equal	equal
6.	1 ”	a—	$\alpha$—
”	16 ”	concentric	concentric
”	2 from bottom	Traité pètrographique	Traité pétrographique
”	” ”	and—the chief—segments,	and segments,
7.	22 ”	objective	objective
9.	25 ”	O =	OQ =
16.	2 ”	$90^\circ\ 2\alpha$	$90^\circ - 2\alpha$
17.	5 from top	Further	Further,
”	8 ”	there was not,	there was not;
19.	26 ”	Dreispitrzzirkels	Dreispitzzirkels
19.	2 from bottom	Thus	Thus,
20.	18 from top	light-colored	light-colored
21.	7 from bottom	coincide	coincide.
25.	21 from top	tube	tube
”	22 ”	$336^\circ$	$336^\circ$,
32.	2 ”	Now	Now,
”	6 ”	H	H,
39.	7 ”	sion	sion
45.	9 from bottom	Univers lmethodе	Universalmethode
48.	1 ”	manipulations,	manipulations
51.	20 ”	$\varphi^0$	$\varphi_0$
59.	11 ”	Kr.	f. Kr.
”	1 ”
60.	8 from top. The right-hand side of the formula is to be multiplied by 0.009.
62.	8 ”	Bl.	Bl,
65.	7 ”	obtaining	obtaining
”	21 ”	thus	thus,
66.	4 ”	division	division
68.	1 ”	angle	angle $\rho$
69.	14 from bottom	H and J	H and J,
”	9 ”	$\dfrac{\mathrm{Cos}\ H_2 \cdot \mathrm{Cos}\ J_1}{\mathrm{Coc}\ H_1 \cdot \mathrm{Cos}\ J_2}$	$\dfrac{\mathrm{Cos}\ H_2 \cdot \mathrm{Cos}\ J_2}{\mathrm{Cos}\ H_1 \cdot \mathrm{Cos}\ J_1}$
”	8 ”	$\dfrac{\mathrm{Cos}\ H_1 \cdot \mathrm{Cos}\ J_2}{\mathrm{Cos}\ H_2 \cdot \mathrm{Cos}\ J_1}$	$\dfrac{\mathrm{Cos}\ H_1 \cdot \mathrm{Cos}\ J_1}{\mathrm{Cos}\ H_2 \cdot \mathrm{Cos}\ J_2}$

M. A. Usov.

Page	Line	Printed	Should be
70.	6 from top	joints	joint.
71.	17 from top	equality of path \(p\)	difference of path \(p\).
71.	18 from top	\(x=\dfrac{e}{p}\).	\(x=\dfrac{p}{e}\).
83.	13 from top	\(n_r\)	\(n_p\)
83.	15 from bottom	From	Because of
87.	1 from top	coordinates	coordinates —
87.	4 from top	plane	planes
91.	5 from top	Thus	Thus,
96.	1 from bottom	Minéralogie	minéralogie
96.	1 from bottom	\(^{3})\)	\(^{2})\)
96.	1 from bottom	110.	310.
98.	4 from top	hoving	zoning
98.	3 from bottom	F ldspathstudien	Feldspathstudien
100.	4 from top	axis	axis:
100.	9 from top	laws	law
102.	8 from bottom	then	then,
102.	1 from bottom	des	de
105.	15 from top	to be limited	to limit oneself
106.	1 from top	\((110)\)	\((\bar{1}10)\)
110.	13 from top	identical	identical
110.	12 from bottom	from	is
111.	11 from bottom	apply	to apply
112.	2 from bottom	d s	des
119.	3 from bottom	m noclinique	monoclinique
120.	1 from bottom	\(^{1})\)	\(^{2})\)
122.	1 from bottom	Min ralogie	Mineralogie
125.	2 from top	refractions	refraction

Table VII

No. 307e
1.906a
Epidote — Kars collection

I

\(A_1\), optic axes: \(5^\circ\)

1) \(34.4\), measured \(8^\circ\)
2) \(80^\circ\), measured \(34^\circ\)

\[ 2V = +85^\circ \]

II

1) \(55^\circ\), measured \(2.5^\circ\)
2) \(146^\circ\), measured \(24^\circ\)

\[ B_{1,2} \begin{cases} n_g = 37^\circ,\\ n_m = -59.5^\circ,\\ n_p = -72^\circ. \end{cases} \]

Orientation in the II interference figure:
\(114^\circ\) to the right; \(17.5^\circ\).

\[ \begin{aligned} n_m^2(001) &= 45^\circ,\\ n_p^2(001) &= 37^\circ,\\ n_g^2(001) &= 53^\circ. \end{aligned} \]

Annular No. 63.

Brachypinacoidal grain.

Table IX.

(The visible page consists of a full-page crystallographic projection diagram with numerical and crystallographic labels, including forms such as \([100]\), \([010]\), \([001]\), \((010)\), \((100)\), and indices such as \(101\), \(201\), \(301\), \(401\), \(100\), \(110\), etc. No running prose or equations are visible on this page.)

Table X

No.	Name of the equator	Twin axis
I	[[unclear: handwritten mineral name]]	\(\perp (010)\)
II	[[unclear: handwritten mineral name]]	\(\perp [001] \parallel (100)\)
III	[[unclear: handwritten mineral name]]	\([100]\)
IV	[[unclear: handwritten mineral name]]	\(\perp [010] \parallel (001)\)
V	[[unclear: handwritten mineral name]]	\([010]\)
VI		\(\perp [001] \parallel (001)\)
VII	[[unclear: handwritten mineral name]]	\(\perp [001] \parallel (010)\)
VIII		\(\perp (100)\)
IX		\(\perp (110)\)
X		\(\perp [010] \parallel (100)\)
XI		\(\perp (021)\)
XII		\(\perp (021)\)
XIII	[[unclear: handwritten mineral name]]	\(\perp [100][001]\)
XIV		\(\angle [010][001]\)
XV		\(\angle [010][100]\)

1. C. Viola, “Ueber die optische Orientierung des Albits und das Tschermack’sche Gesetz.” T. M. P. M. 20, 1901; 199. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩

2. J. Schmutzer, “Ueber Zonarstruktur, Rekurrenz und Resorption.” Centralblatt f. Min. etc. 1910, No. 13, p. 389. ↩↩↩↩↩↩↩↩↩↩↩

3. V. V. Nikitin, “A case of secondary overgrowth of primary feldspars.” Proceedings of the Mining Institute 1, 1907; 236. ↩↩↩↩

4. M. Fouqué. Contribution à l’étude des feldspaths etc. p. 420. ↩