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On Homogeneous, Structures and the Symmetrical Partitioning of them, with Application to Crystals

Published online by Cambridge University Press:  14 March 2018

William Barlow
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Summary

The above investigation shows:—

  • 1. The nature of homogeneity of structure, and the properties which distinguish it from structureless homogeneity. The new definition of a homogeneous structure recently put forward by the author in Groth's Zeitschrift is given.

  • 2. A method of realising in a concrete form, and with great generality, the kind of repetition in space which constitutes homogeneity of structure, the models employed for this purpose each consisting of a number of similar plaster hands appropriately arranged in space.

    The total number of types of arrangement, all of which can be represented in this way, is 230, this being the number of typical point systems described by Fedorow and Schönflies, derived by their extension of Sohncke's methods. The various types of homogeneous structure, like the corresponding point-systems, all fall into the 32 classes of crystal symmetry.

  • 3. What property common to all homogeneous structures whatever most nearly coyresponds to Thomson and Tait's definition of homogeneity.

  • 4. Reasons for regarding as untenable the arguments put forward by Fedorow in support of his recent attempt to select from among the types of homogeneous structure those which are possible for crystals, and to determine the shapes of their ultimate units.

  • 5. The possibility of so classifying all conceivable ways of symmetrically partitioning all the types of homogeneous structure as to avoid all reference to the nature of the cell-faces, whether plane or otherwise, and, in other respects also, be perfectly general. Some reasons for undertaking this classification, notwithstanding its complexity, are given, the chief one being the relation of symmetrical partitioning to some stereo-chemical and other experimental facts.

Type
Research Article
Information
Mineralogical magazine and journal of the Mineralogical Society , Volume 11 , Issue 51 , October 1896 , pp. 119 - 136
Copyright
Copyright © The Mineralogical Society of Great Britain and Ireland 1896

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References

page 120 note 1 Fedorow cites some evidences of discontinuity of structure of crystals. See Theorie der Krystallstruktur in Zeitschr. für Kryst. &c. XXV. p. 116. Comp. Thomson and Tait, IL p. 216.

page 120 note 2 Zeitschr. für Kryst. &c. XXIII. p. 1.

page 120 note 3 Sohncke's Entwickelung einer Theorie der Krystallstruktur', p. 28. Those kinds of repetition in space the repeating parts of which have some dimension infinite, are not included in this inquiry.

page 120 note 4 Comp. Wiener's Grundzüge der Weltordnung. Leipzig, 1869. Atomlehre, p. 82.

page 121 note 1 Comp. Fedorow, Zeitschr.für Kryst. XXV. p. 115. The employment of hands was suggested to me by Prof. Miers; its advantages are that the use of so familiar an object greatly facilitates a clear perception of the nature of the arrangement, and that the shape is so exceptional that no one can be led to imagine it to be a necessary feature of any type.

page 121 note 2 Zeitsehr. für Kryst. XXIII. p. 44.

page 121 note 3 Zeitschr. für Kryst. XX. p. 467.

page 121 note 4 This is true of all the types. The same homogeneous structure can, in most cases, be partitioned in a manner compatible with the preservation of its homogeneity and symmetry in an infinite number of ways. The cubic partitioning resorted to in the case under consideration is a very arbitrary one, and a structure of the given type need not have any characteristics corresponding to the exceptional symmetry which this kind of partitioning produces.

As we shall see presently, there are cases of homogeneous structure in which no kind of partitioning into single units (i.e. not of two kinds which are enautiomorphs) is possible which does not impair the symmetry and alter the type of homogeneity. (See p. 132 and 2b1 and 6b2 in table on p. 135).

Some suggestions for the classification of the different kinds of symmetrical par. titioning possible will be made presently.

page 124 note 1 As corrected in some few particulars in an article in the same journal, Vol, XXV. p. 86.

page 124 note 2 By similar parts is mealat parts similarly related to the whole.

page 125 note 1 Fedorow says, "Das Resultat der Beobachtungen kanu auch dahin gedeutet werden, dass die krystalllnisch-homogene Substanz aus gleichen und gleich-orientirteu (d. h. sämmtlich in paralleler Lage geordneten) Theilehen besteht, welche zusammen genommen den Raum lückenlos ausfüllen."--Zeitschr.für Kryst. XXV p. 117.

page 125 note 2 The three necessary properties of a space-unit are--(1) it is continuous; (2) it contains every kind of Point of the structure (i.e. every kind of standpoint from which the structure can be regarded); (3) all the points in it are differently related to the structure as a whole. See Zeitsehr. für Kryst. XXIII. p. 38.

page 125 note 3 The following is Sohncke's definition of a coincidence-movement (Deckbe. wegung):-

Suppose a regular infinite point-system (see p. 120 above) to be made rigid, and moved bodily out of its original position. The positions originally occupied by the points of such a system then collectively furnish a point-system which is congruent with the moved system, and which, to distinguish it from the moveable system, may be called the fixed system.

If now any point of the moveable system be selected, and placed to coincide with any point whatever of those of the fixed system, it is, owing to the congruence of all Linlenbündel, always possible to bring the two systems to coincidence.

The movement which the moveable system executes in order to pass from one position of coincidence with the fixed system to some other position of coincidence, is called a Deckbewegung.

Such Deckbewegungeu can be partly parallel translations, partly rotations or screw movements about certain axes definitely placed in the fixed system.

The Deekbewegungen possessed by any particular system will depend on the properties of the system, and indeed the different kinds of point-systems may be distinguished according to the Deckbewegungen:whieh are proper to them, so that the different sets of Deckbewegungen presented serve as the ground of the classification of the regular point-systems. (See Entwickelung ether Theorie, &c. p. 28. Compare Bravais, "Sur les Systemes formés par des points distribués régulièrement," Journ. de l' Ecole JPolyteehnique, Cahier, XXXIII. p. 57.)

page 126 note 1 See below, p. 127.

page 126 note 2 They may resemble the rhythmically related movements of combined figure skating.

page 126 note 3 This number agrees with the total number of single and double point-systems as ascertained by Fedorow and Schönflies independently.

page 127 note 1 Fedorow's Theorie der Krystallstruktui.. Mögliche Strukturarten. Zeitsehr. für Kryst. XXV. p. 113.

page 127 note 2 Zeitschr. für Kryst. XXV. p. 117.

page 127 note 3 Ibid. See especially p. 218. Cases of this kind are presented by types 3 and 4 in my list. Zeit$ehr. für Kryst. XXIII. p. 11. Compare below, pp. 128 and 135.

page 128 note 1 Fedorow distinguishes "normale" and "anomale" parallelohedra (see below).

page 128 note 2 Cutup. Fedorow, Zeltschr. &c. XXV. Note*, p. 147.

page 128 note 3 Ib. p. 132.

page 128 note 4 Ib. p. 183.

page 128 note 5 Fedorow calls these "extraordinär," those whose contents and outlines are alike sameways-orientated "ordinär" (see p. 135).

The suggestion made by this author (p. 146) that cases of the "extraordinären" kind are unlikely, because it is inconceivable why the contents of the cells should be differently orientated, does not help his contention, because, if there is this difficulty in accounting for various orientations of the composite elements of a mass, there is also the same kind of difficulty in accounting for the similar parts of any such element being put together with their orientations various, and the latter is an essential condition in most homogeneous structures.

page 129 note 1 There does not indeed appear to be any evidence that the divisions of Fedorow's classification are at all traceable by the experimental facts at present known to us regarding crystals.

page 129 note 2 This is true, whether there is any such thing as a crystal molecule distinct from a chemical molecule or not. There does not, it may be remarked, appear to be any adequate evidellee of such a distinotion

page 130 note 1 See Note 3, p. 125.

page 130 note 2 See page 125.

page 130 note 3 Expansion or contraction of the parts of the structure is neglected.

page 130 note 4 If the ideas generally held respecting the nature of matter are substantially correct, this, and not the previous altelmative case, will be that of actual partially liquefied homogeneous matter, i.e. of liquefied crystals the ultimate parts of which retain some kind of stable structure. Cases in which the fragments are not all of the same pattern are probably furnished by crystals containing "water of crystallisation," and by double salts.

page 131 note 1 That is, either identically or enantiomorphically similar. Comp. cases of racemic compounds referred to above, p. 129.

page 131 note 2 Cases are conceivable in which the enantiomorphic property of fragments is manifested only when they are considered with respect to the unbroken structure; they may themselves, considered alone, be destitute of this property. In other words, groupings which are identical with their own mirror-image, and all identically alike, may occupy two different sets of enantiomorphically similar situations in a structure. Indeed, the groupings which survive in the liquid state may, when considered alone, apart from the structure, have any additional elements of symmetry which are compatible with those which they already possess as parts of the structure.

page 131 note 3 Axes are of the same set when their relation to the entire system of axes found in the structure is the same, and that whether they can be brought to coincidence or not.

page 131 note 4 See Zeitsehr.für Kryst. XXIII. p. 59.

page 132 note 1 They may, however, considered apart from the structure, possess the property as a special peculiarity not derived from it. Comp. note 2, p. 131.

page 132 note 2 Comp. Zeitschr. für Kryst. XXV. p. 88.

page 132 note 3 Zeitschr. für Kryst. XXIII. p. 60.

page 132 note 4 The least symmetrical type of all (hemipinakoidal-anorthie), as it possesses no singular points, presents but one type el partitioning for simple structures, i.e. for structures whose ultimate fragments are all similar.

page 133 note 1 See note 3, p. 131.

page 133 note 2 There are some such planes in other types of structure besides tbe two 63c and 64c here referred to, which form a group by themselves. Comp. Zeitschr. XXIII. 59, XXV. p. 90.

page 133 note 3 A fragment may, as we have intimated, have an axis or axes not found in the unbroken structure, but this is outside our province here, as we are not making a classification based on individual peculiarities of the fragments, but one based on the general features of the different types.

page 134 note 1 See Note 3, p. 131.

page 134 note 2 See above, p. 182.