Effect of Rotational Symmetry on Colored LatticesAuthor(s): David HarkerSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 75, No. 12 (Dec., 1978), pp. 5751-5754Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/68833 .

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Proc. Natl. Acad. Sci. USA Vol. 75, No. 12, pp. 5751-5754, December 1978 Chemistry

Effect of rotational symmetry on c( (generalized symmetry/space-group theory/crystallography/mag

DAVID HARKER

Medical Foundation of Buffalo, Inc., 73 High Street, Buffalo, New York 14203

Contributed by David Harker, August 30, 1978

ABSTRACT A colored lattice L C has a geometrical lattice L. A subgroup lattice L' of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L' in L is given by A, the determinant of the matrix (tik) that converts L into L'. This is the order of the factor group {L/L'}, and is also the number n of colors present. The crystal systems-i.e., the combinations of rotational symmetry axes- of L ,% L, and L' are all the same, but L and L' may have differ- ent centerings; this results in 39 combinations of centerings, called here the 39 symmetry types of colored lattices. These types are tabulated here, together with the special forms taken by (t'k) and the formulas for A. In only 7 of the 39 types can the numter of colors be arbitrary; in most types certain numbers of colors are impossible.

The presence of axes of rotational symmetry in three-dimen- sional lattices places restrictions on the relative orientations of the lattice rows and nets in them. These restrictions lead to the classification of geometrical lattices into the 14 well-known Bravais types. These types in turn are grouped into crystal systems characterized by certain combinations of rotational symmetry axes. In this paper the simplest combination of axes necessary to establish a crystal system will be used to charac- terize that system. (Additional axes of rotation or rotation- inversion, where present, can specialize some of the mathe- matical expressions to be derived here, but do not change their important properties. These effects will be discussed where appropriate.) Table 1 displays the classification to be used here.

A colored lattice Lc can be characterized by a "color-blind" (or geometrical) lattice L and by a subgroup lattice L' with nodes all of one color. Each of the cosets of L' consists of like- colored nodes, so that the index of L' in L is the number of different colors present.

The relationship between L and L' can be expressed, as fol- lows:

Let L = ma + nb + pc, and [1]

L' = ua' + vb' + wc' [2]

where m, n, p, u, v, and w range over all the integers. The three non-coplanar vectors a, b, and c define a primitive unit cell in the L lattice and a', b', and c' define a primitive unit cell in the L' lattice. Define

a' = tlla + t12b + t1c,

b= t2la + t22b + t2c, and [3]

c' = t31a + t32b + t33c.

The matrix (tjk) establishes the group of color permutations {P} that is connected with the group of lattice translations {L} to

The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "ad- vertisement" in accordance with 18 U. S. C. ?1734 solely to indicate this fact.

57

ilored lattices netic structures/order-disorder reactions)

form the group of translational colored-symmetry operations of the colored lattice Lc. The number n of different colors is given by the magnitude of the determinant I tjk I = A-i.e., n = I A I . The order of {P} is also n. The abstract structure of {P} is the same as that of the quotient group {L/L'}-i.e., {P} {L/L'}. The details are discussed in ref. 1.

L and L' cannot belong to different crystal systems. For, suppose that the crystal system of the geometrical, "color-blind" lattice L contains rotational axes not present in the "'one-col- ored" subgroup lattice L'. Then these cannot be present in the colored lattice Lc, because the arrangement of colors cannot have them. Conversely, if L', considered alone, appears to have axes not present in L, they cannot survive in Lc, which can possess only such axes as exist in L. Therefore, in examining the crystal system of a colored lattice Lc, we need only investigate cases for which L and L' belong to the same system, which will be called the crystal system of the colored lattice Lc.

Coordinate axes for primitive unit cells of Bravais lattices In order that the matrix (tjk) consist of integers, it is necessary that both L and L' be referred to axes that define primitive unit cells, whether or not these cells display the full symmetry of the lattices they define. Table 2 sets forth definitions of the axes to be used in this work to define each of the Bravais lattices, and it relates them to the minimal collection of symmetry axes re- quired to establish the crystal system in each case.

The three-dimensional colored lattices Lc will be denoted by a sequence of symbols, as follows: first, a capital letter de- noting the crystal system (G = triclinic, M = monoclinic, 0 =

orthorhombic, S = tetragonal, H = hexagonal, K = cubic); next, a capital letter giving the centering of the geometrical lattice; then a slash; then a capital letter giving the centering of the one-colored subgroup lattice. (P = primitive, C = centered in the ab mesh, B = centered in the ca mesh, A = centered in the bc mesh, F = centered in all faces of the unit cell, I = centered in the unit cell's body, R = rhombohedral centering.) Then follows the matrix, (tjk), of the transformation that produces the one-colored subgroup lattice, L', from the geometrical lattice, L. The magnitude of the determinant A of this matrix gives the number of colors; this number may follow the ma- trix.

Examples may clarify the meaning of this notation. OP/P (pOO/OqO/OOr) means that the L and L' lattices are

both primitive orthorhombic, with primitive unit cells in the form of rectangular parallelepipeds. The relationship of these two lattices is determined by the transformation in parentheses. The number of colors, n, is the determinant of the transfor- mation; it is pqr = n, in this case. The abstract structure of the permutation group can be derived from Cp X Cq X Cr. It is Cn, unless n is divisible by a perfect square, or a perfect cube, in which case other structures may be present, and should be considered, as described in ref. 1.

MC/P(ppq/rrO/sst) means that the color-blind lattice is C centered monoclinic and the one-colored subgroup lattice is

51

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5752 Chemistry: Harker

Table 1. Symbolism for the Bravais lattices

Minimum Old Crystal axial Old centering New system symmetry* symbol symbol symbol

Triclinic 1 Ftr P GP Monoclinic 2 I'm P MP

TF' A, C, I MA, MC, MI Orthorhombic 222 Fr P OP

OF A, B, C OA, OB, OC rO F OF r'o I 01o

Tetragonal 4 Ft P, C SP, SC r', I, F SI, SF

Hexagonal 3 rh P, H, C HP Rhombohedral 3 Frh R HR Cubic 23 Fc P KP

Fc F KF rIc I KI

The new lattice symbols always consist of two capital letters. The first denotes the crystal system, the second the centering. The first letters were chosen thus: G for "General," M for "Monoclinic," 0 for "Orthorhombic," S for "Square" (Tetragonal), H for "Hexagonal," and K for "Cubic." * International Hermann-Mauguin notation.

primitive monoclinic. The transformation, in parenthesis, uses axes as defined in Table 2. Its determinant is 2r(pt - qs) = n. This colored lattice symmetry thus requires an even number of colors.

SI/P(011/101/110)2 means a color-blind body-centered tetragonal lattice given two colors (because A = n = 2) in an array that produces a primitive tetragonal one-colored lattice. (This is the ideal structure of the ordered alloy AuCu if the two "colors" are taken to mean the chemical natures of gold and copper atoms, respectively, and the axial ratio c/a of the P lattice is near /-2 .)

Table 3 contains the above-described general symbols for the three-dimensional colored lattices with the various centering combinations possible in each crystallographic system, as well as formulas from which the value of A can be found in each case. It is noteworthy that an arbitrary number of colors is possible only for a few combinations of L and L'; Table 4 lists some possible, useful numbers of colors for each of the 39 types of colored lattice symmetry.

Application to colored space group theory

In an object having a colored space group, each asymmetric unit has a single color, and the distribution of subunits and their colors is changed by a group of combined geometrical and color permutation operations that leave the geometrical and color aspects of the object unchanged. There are colored space-groups that involve colored lattice translations, and there are others that do not (2). Table 4 shows the numbers of colors possible for each of the 39 colored lattice symmetries. In the column headed 1, the 14 Bravais lattices (lattices with only one color) are marked with a +. The other columns contain +s where the lattice symbolized at the extreme left can have that number of colors. When a certain combination of L and L' is not compatible with a certain number of colors, then a colored space group with a multiple of that number of colors cannot have that colored lattice as a subgroup. When the number of colors of a colored space-group is factorable-e.g., 6 = 1-2-3-then each factor, as well as the number itself, may correspond to a number of colors on a colored subgroup lattice.

In the book Shubnikovskie Gruppy ("Shubnikov groups") by Koptsik (3), the two-colored space groups are listed. Their

Proc. Natl. Acad. Sci. USA 75 (1978)

Table 2. Axes to be used in describing Bravais lattices

MPG* Lattice Axes

1 GP Any a, b, c not in one plane

2 MP a l b, c I b, b along 2

2 MC c 1 2,a = b,a+balong2

2 MA a 1 2,b = c,b + calong2

222 OP a I b, a I c, b I c, each along a 2

222 OC a I c, b I c, a = b, a +b along a 2, a - b along a 2, c along a 2

222 OA, OB Appropriate permutations of above

222 OF a, b, c each I to a 2, but not along a 2; each of -a + b + c, a - b + c, a + b - c is along a different 2

222 01 a is a shortest general lattice vector not along a 2, b, and c are related to a by rotations about two of the 2-fold axis

4 SP a I c, b I c, a = b, c along 4, b from a by 4

4 SI Origin on 4, a to lattice point nearest 4, -c from a by 4, b from -c by 4

3 HP c along 3, a I c, b from a by 3

3 HR a = b = c, b from a by 3, c from b by 3; a+ b + c along 3

23 KP a = b = c, a, b, c along different 2s

23 KF a bisects angle between two 2s; using nearest 3, b from a by 3, c from b by same 3

23 KI a along one 3, b from a by one 2, c from b by another 2

Note: All these axial triples a, b, c are to be set up in the right- handed sequence. * Minimum point group.

lattices are all either one-colored or two-colored. Those with two-colored lattices are in agreement with the entries in the column headed 2 of Table 4.

Application to "ordered" alloy structures

Many metallic alloy systems exist as "random" solid solutions at high temperatures, but change to "ordered" structures based on the same (or almost the same) atomic positions at lower temperatures. There are several cases for which the high- temperature state is a face-centered cubic structure with one atom per lattice point, in which each atomic position is occupied at random by all of the atomic species present. At lower tem- peratures the structure often consists of an ordered arrangement of the different atoms, using the same positions. The face- centered cubic lattice, or a slight distortion of it, then corre- sponds to the "color-blind" lattice L, while the "ordered" structures correspond to various colored lattices Lc. If we take the position that L and L' must belong to the same crystal sys- tem (as we have in the preceding sections), we must allow L to be any of KF, HR, or SI, in order to describe the known ordered

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Chemistry: Harker

Table 3. Three-dimensional colored lattices Lc for which thi subgroup lattice, L', belong

System Symmetry S3

Triclinic 1 GP/P (pqr/jkl/stu)

Monoclinic 2 MP/P(pOq/OrO/sOt) MP/C(prq/prqI/sOt) MC/P(ppq/rrOl/sst ) MC/C(pqr/qpr/sst)

Orthorhombic 222 OP/P(pOO/OqO/OOr) OP/C(pqO/pqO/OOr) OP/F(Opq/rOq/rpO) OP/I (pqr/pqr/pqr) OC/P(pjO/qqO/OOr) OC/C(pqO/qpO/OOr) OC/A(ppO/qqr/qqr) OC/F(ppq/rrq/p + r, I OC/I (pqr/pqr/qp?) OF/P(jppp/qqq/rrr) OF/C(ppq/qqp/rrr) OF/F(pqq/F,p-q + r, OF/I(pqr/q,p, - (p +

OI/P(Opp/qOq/rrO) OI/C(p,q,p + q/p,q,p OI/F(p + q,q,p/q,q + OI/I(pqr/q - r,p - r,r

Tetragonal 4 SP/P(pqO/qpO/OOr) SP/I(pqr/pqr/qpr) SI/P(p,q,p + q/q,j,q -

SI/I(pqr/q - r,p - r,r,

Hexagonal and 3 HP/P(pqO/q,p - q,O/( rhombohedral HP/R(pqr/q,p - q, r/q

HR/P(p,q - p,q/q,p,q HR/R(pqr/rpq/qrp)

Cubic 23 KP/P(pOO/OpO/OOp) KP/F(Opp/pOp/ppO) KP/I(jpp/pjpp/ppp ) KF/P(ppp/ppp/ppp ) KF/F(pOO/OpO/OOp) KF/I(p 53pp/p3pp/p3 KI/P(Opp/pOp/ppO) KI/F(2ppp/p2pp/pp KI/I(pOO/OpO/OOp)

All lattices are referred to primitive unit cells defined by the axes desci axial symmetry required to establish the crystal class of the L and L' lat be raised to 422 by making either p = 0, q = 0, or p = q. In the hexagonal s, if L is HP by setting either p = 0, q = 0, q = 2p, or p = 2q; if L and L' are of merely to 32; if L is HR and L' is HP by setting p = 2q, q = 2p, p = 0

structures derived from cubic closest packing. It also will be convenient, in some cases, to agree that the colors of chemically similar atoms are "crystallographically different, but indis- tinguishable chemically." Using these concepts we obtain the following descriptions of ordered structures:

Gold-copper: AuCu, SI/P(101/011i/110)2.

AuCu3, KF/P(111/111/111)4.

Nickel-Molybdenum: MoNi4, SI/I(201/111/012)5.

Copper-Platinum: CuPt, HR/R(110/011/101)2.

Tetradymite: Bi2Te2S, HR/R(221/122/212)5.

In a similar way, for cases in which the random solid solution

Proc. Natl. Acad. Sci. USA 75 (1978) 5753

i geometrical (or color-blind) lattice, L, and the one-colored Lo the same crystal system

mbol No. of colors = A

p(ku - It) + q(ls -ju) + r(jt - ks)

r(pt - qs) 2r(pt - qs) 2r(pt - qs) (p + q)[(p - q)t + 2rs]

pqr 2pqr 2pqr 4pqr 2pqr r(p2 - q2) 4pqr

-r, 0) 4pqr 2r(q2 - p2) 4pqr 2r(q2 - p2)

r/q - r,r - q,p + r) (q - p - 2r)(q2 p2) q + r)/- (p + q + r), r,q) 2(p + q)(q + r)(r + p)

2pqr - q/rrO) 4pqr ',r/p,r,p + r) 4pqr Ir - q,q,p -q) (p + q - r)(p - q+ r)(p -q-r)

r(p2 + q2) 2r(p2 + q2)

-p/rrO) 2r(p2+ q2) q,r - q,p - q) (p + q - r)[(p - q)2 +'r2]

)Or) r(p2_ pq + q2) - p,,r) 3r(p2 -_ pq + q2)

-p/rrr) 3r(p2 + pq + q2) p3 + q3 + r3- 3pqr

p3 2p3 4p3 4p3 p3

p) 16p3 2p3

p) 4p3 p3

ibed in Table 2. The column headed "Symmetry" shows the minimum tices. In the tetragonal system, the common symmetry of L and L' can ystem, the common symmetry of L and L' can be raised to 32, as follows: both HP, their common symmetry is raised to 622 in this way, instead , q = 0, or p = q; if L and L' are both HR by setting p = q.

is body-centered cubic, the ordered structure can be described thus:

Fe3Al, KI/F(211/121/112)4.

CuZn, KI/P(011/101/110)2.

Na2Tl2, KI/F(211/121/112)4.

Other simple crystal structures can also be described by using similar principles. For instance:

NaCl, KP/F(011/101/110)2. CsCI, KI/P(011/101/110)2.

If we denote absent atoms by Ze (for "Zeronium"), then still other simple structures can be treated in the same way:

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5754 Chemistry: Harker

Table 4. Number of colors possible for (

Colored lattice Number symbol 1 2 3 4 5 6 7

GP/P + + + + + + +

MP/P + + + + + + + MP/C + + + MC/P + + + MC/C + + + + + + +

OP/P + + + + + + + OP/C + + + OP/F + + + OP/I + OC/P + + + OC/C + + + + + + + OC/A + OC/F + OC/I + -i + OF/P + OF/C + + + OF/F + + + + OF/I + OI/P + + + OI/C + OI/F + 01/I + + + +

SP/P + + + + + + + SP/I + + + SI/P + + + SI/I + + + + +

HP/P + + + + + + + HP/R + + HR/P + + HR/R + + + + +

KP/P + KP/F + KP/I + KF/P + KF/F + KF/I KI/P + KI/F + KI/I +

Totals: 14 20 12 29 11 19 11

Fluorite CaF2Ze, KI/F(211/121/112)4. Perovskite CaTiO3Ze3, KP/P(200/020/002)8.

Diamond C2Ze2, KI/F(211/121/112)4. Graphite C4Ze2, HP/P(110/120/002)6.

Application to magnetic structures

If certain atoms in a crystalline substance have magnetic mo- ments, or "spins," these atoms will generate local magnetic fields. The neighboring magnetic atoms will be oriented in this field according to the laws of quantum theory. In consequence, a "magnetic structure" may arise, in which the spins of the magnetic atoms form a repeating pattern of orientations. Symbolizing each orientation by a color, these structures could be described by their colored symmetries. If the magnetic atoms were of only one chemical element, and if they can be thought of as occupying points on a lattice, then the theory of colored lattices described above should be applicable. However, more

Proc. Natl. Acad. Sci. USA 75 (1978)

ach of the 39 colored lattice symmetries

of colors possible 8 9 10 11 12 16 24 48

+ + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + +

+ + + + +

+

+ +

+~~~~~~~~

+~~~~~~~~

31 13 18 11 27 31 29 29

general magnetic structures would require the use of colored space groups for their description.

The two-colored space groups have been thoroughly treated, and listed, by Koptsik (3), but other colored space groups have not been completely investigated, although a considerable number of articles concerning them have been published (2).

The author is grateful for many improvements and clarifications of this paper which resulted from conversations with Professors W. J. Buerger, J. D. H. Donnay, and Marjorie Senechal. He is also thankful for the financial support made available through National Science Foundation Grant DMR76-21344(01).

1. Harker, D. (1978) Proc. Natl. Acad. Sci. USA 75, 5264-5267. 2. Shubnikov, A. V. & Koptsik, V. A. (1974) Symmetry in Science

and Art (Plenum, New York), pp. 389-408. 3. Koptsik, V. A. (1966) Shubnikovskie Gruppy (in Russian) (Izda-

telstvo Moskovskogo Universiteta, Moscow).

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