THE DETERMINATION OF THE GEOMETRICAL CONSTANTS OF A CRYSTAL FROM ITS INTERFACIAL ANGLES.

BY ALFRED J. MOSES.

The face symbols and elements of a crystal result either from angles between faces obtained with one-circle goniometers or from angles analogous to latitude and longitude obtained with two-circle goniometers.

For the purposes of instruction and especially in a course designed principally to rivet loose conceptions obtained in an elementary study of models and crystals it is better to use the onecircle goniometer and measure the very zones and angles actually seen on the crystal. It may also be said that neither the solution. of the problems of the crystallographer * nor the obtaining of clear ideas as to structure, symmetry, indices and other crystal relations are necessarily connected with the deduction of the trigonometrical or other formulæ used. All this can be eliminated and a course presented which can be followed understandingly by any student able to use logarithmic tables.

The purpose of this paper is to present such a course under the principal divisions of:

I. Measurement of the Interfacial Angles.

II. Stereographic Projection.

III. Graphic Determination of Symmetry, Indices and Elements. IV. Calculation of Axial Elements and Interfacial Angles.

I. THE MEASUREMENT OF THE INTERFACIAL ANGLES.

1. SELECTION OF THE CRYSTAL.

Before any measuring is done all available crystals are carefully examined to find:

(a) The most brilliant flawless crystals.

"The chief problems of the crystallographer are: the determination of the face symbols and elements of the crystal when the angles are measured; or from a knowledge of the elements and the face indices to determine the true values of the angles." W. J. Lewis, "Crystallography," p. 161.

(b) Crystals which show rare faces or unusual combinations of faces or other peculiarities.

During and after selection the crystals are handled by either a pencil of wax or by the forceps, never by the fingers. If in any way blurred they are cleaned with chamois skin and, unless soluble therein, by alcohol.

2. SKETCHING THE CRYSTAL.

Each selected crystal is carefully studied with a hand-glass especially for:

I. Recurrence of similar groupings of faces, indicating symmetry. 2. Parallelism of edges, indicating zones.

3. Similarity of lustre or markings, indicating faces that belong to the same form.

4. Cleavages, parting and etchings.

A sketch is made, usually a horizontal projection of the crystal with some selected zone vertical. The bounding lines a, b, c, d, e, f, g, h, Fig. 2, which represent the vertical faces are drawn approximately at their true angles and apparently parallel edges between faces are drawn parallel. A letter or number is assigned to each face and thereafter represents it.

3. THE GONIOMETER.

Among "one-circle" goniometers preference is usually given to those in which the crystal is revolved about a vertical axis, and one of the best and simplest is the Fuess (4, 4) shown in Fig. 1.

The axes of the two telescopes, C and T, are in the same horizontal plane and intersect in the axis of rotation. If a crystal is attached by wax at p, and adjusted, as later described, so that an edge between two faces coincides with the axis of rotation, then a ray of light sent through C will be reflected into T for just one position of each face and the difference between the two readings of the graduated circle for these positions will be the supplement angle between the faces.

4. THE ADJUSTMENT.

Before the objective of Tswings a lens which, when down, brings the crystal into focus. The crystal is attached by wax to the plate pas nearly as possible in the correct position. One of the faces of the angle to be measured is placed approximately parallel to one of the sliding screws, for instance », and that screw is placed at right angles to the telescope.

*R. Fuess, Steglitz, near Berlin, marks 260, or about 65 dollars.

The lens on T is lowered and the crystal raised or lowered into the field after loosening the screw d. The crystal is then moved by n and tipped by the screw k of the corresponding circular arc until the edge appears to coincide with the vertical cross hair. The screw a is then loosened, the crystal turned 90° by the wheel f and the edge moved by o and tipped by the other circular arc

until again in coincidence with the vertical cross hair. This is repeated until through a rotation the edge and cross hair appear to coincide.

The lens is then raised and if the images of the signal from the faces can be bisected by the cross hairs the adjustment is satisfactory.

With small crystals it is not necessary to recenter on each edge, but is sufficient by a slight motion of the centering screws to make the vertical hair coincide with an imaginary central axis within the zone.

5. THE MEASURING.

The screw G is loosened and the telescope set at 100 to 120 degrees to the collimator C. The screw a is then loosened, the graduated circle and crystal are turned together by the wheel ƒ,

until the reflected signal is seen through the telescope, then a is tightened and the fine adjustment made by the tangent screw F. The vernier is then read and recorded.

The screw a is again loosened and the rotation continued until the signal is received from a second face; this is centered by F and a and recorded as before. The difference between the two readings is the supplement angle between the faces.

6. THE ORDER OF MEASUREMENT.

The zone first measured is usually one showing numerous or very well-developed faces. The sketch and the projection usually suggest other prominent zones. For instance, Fig. 2 suggests that

the first zone be followed by one of the two zones [c, v, x, y, p, g], and [a, s, n, e]. There are also zones [u, t, s, r, q] and [w, n, o].

7. CONNECTING THE ZONES.

Care must be taken to avoid assumptions as to the relative positions of two measured zones. In general a zone is determined by two non-parallel faces and a face by its angles with two known faces.

8. RECORDING THE MEASUREMENTS.

Two records should be kept, one in a note book and the other on circles of convenient size, one for each zone. For instance, for the zone [a, b, etc.] of the colemanite crystal, Fig. 2, draw a circle,

* After bringing the central line of the signal into coincidence with the vertical cross hair it is well to have the entire apparatus stand with the lamp lighted for 15 to 20 minutes. If there is a notable displacement of the signal from unequal heating of the outer and inner axles, readjust and let stand again until the centering can be maintained.

Fig. 3, and let the point a, represent the face a, the other faces b, c, may then be represented by other points b, c, on the circumference found by laying off arcs corresponding to the angles of the note book. Distortions will thereby be eliminated and the relations between the faces judged from these points. Opposite points will mean parallel faces, symmetrically related points will mean corresponding faces and so on.

The note book should provide columns for the readings and angles of three separate rotations, for instance:

9. AVERAGING CORRESPONDING ANGLES.

It will usually happen that the reflections from the different faces concerned are of unequal distinctness. The brighter images can be more exactly centered and should have greater weight in the conclusions reached. The method of least squares is sometimes used but a simpler method is to assign a quality mark to each image of the signal. For instance, a perfect image may be 2, a good image 1, a poor image o, and the proportionate value of any measurement may then be taken as the sum of the quality marks of the readings. These may be recorded in tabular shape, as in the following example:

The average angles obtained from the three readings by measuring the zone [a, b] are given in the last column. The sketch, Fig. 2, and these angles indicate that bc, cd, fg and gh are corresponding angles. The process would be: