POST-CLASSICAL CRYSTALLOGRAPHY

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POST-CLASSICAL CRYSTALLOGRAPHY

A. Mackay

To cite this version:

A. Mackay. POST-CLASSICAL CRYSTALLOGRAPHY. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-153-C3-163. �10.1051/jphyscol:1986315�. �jpa-00225726�

JOURNAL DE PHYSIQUE

Colloque C 3 , supplément au no 7, Tome 47, juillet 1986

POST-CLASSICAL CRYSTALLOGRAPHY

A . L . MACKAY

Department of Crystallography, Birkbeck College, (University of London), Malet Street, GB-London, WClE 7 H X , Great -Bri tain

&sumg 0n donne une perspective *sur quelques d&veloppents dans la cristallographie au-delà des preoccupations classiques avec les cristaux re'lfuiers qui appartiennent aux 230 groupes spatiaux.

Abstract A perspective is given of some of the developments of crystallography beyond the consideration of the structures of regular crystals belonging to the 230 space groups.

0.0 Generalised crystallography

Ideas subversive of orthodox crystallography have a long history at Birkbeck College, so that the logical paradox of the title of this conference , quoted from Ehjin Schroedinger Il], is of no surprise. Bernal's program of "Generalised crystallography" was launched in 1967 [21 but had origins going back to the

"Theoretical Biology Club" of the 1930s and the recognition (presaged by Ernst Haeckel) of the key role of liquid crystals in living systems 133. Astbury's work on fibres had made people well aware of the existence of non-crystalline order. Later, Berna1 and Finney's statistical geometry of liquids extended crystallography in this direction. The program w a s re-articulated several times later [4,51 and is now being developed towards "Non-Euclidean Crystallography" [6], beginning with the examination of the packings of points in two- and three- dimensional curved manifolds 171. Pauling's development of the alpha- helix (for protein chains) gave the first clear atomic-scale structure (the diffraction effects of which were calculated by Crick, Vand and Klug) which fell outside the framework of classical crystallography by being a helix with a screw axis incompatible with any of the screw axes of the 230 space groups.

We can give here only a perspective of a general synthesis of what we have called "post-classical crystallography" and cm choose only a f e w topics for examination but we wish particularly to suggest that the question of textures is due for revision. We consider that the central paradigm of crystallography has nou changed from one of X-ray crystal structure determination (with the assimiption of a classical crystal with identical unit cells) to one of the electron microscopy of structures with non-periodic features.

0.1 The change of the experimental paradigm

Crptals have long been regardeà (correctly) as a metaphor for purity. M y

pure preparations crystallise. The basis of classical mineralogy w a s the recognition of the external forms of crystals. Indeed picking crystals for metallurgical purposes might be seen as the b i s of civilisation.

Classical crystallography, from 1912 considering the internal structure of crystals, was k e d on the information from X-ray diffraction. Such observations give FFY, the structure amplitudes, but c m be extended by direct methods of phase determination and by isomorphous substitution, essentially the methods of holography, to image the internal structure of crystals. However, in a "picture"

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986315

C3-154 JOURNAL DE PHYSIQUE

only 20% of the infomtion is earried by the amplitudes of the Fourier tenus and 80% resides in the phases. An X-ray microscope is based on the assumption of a crystal as the repetition of a large number of identical unit cells which act as a scattering amplifier.

In the classical view structures which were not perfect crystals were regarded as imperfect, or par or failed crystals. The change is that now we recognise that there are other structures which are intrinsically not crystals. Symmetry is not absolute but of limited range and the strict, unbounded equivalence of the structural units in the 230 space groups must be replaced by quasi-equivalence.

Almost al1 natural structures are hierarchic, where each level has a limited span.

The level above provides a quasi-static environment for the level in question and the level below is a statistical average, over space and time, of much finer detail.

We must also consider the information content of structures, even of inorganic structures, now that the dialectical relationship of DNA and protein structure has become evident. A crystal in the most general tenus has been defined as "a structure of a few types of identical components, the description of which is much smaller than the structure itself" [4], but we will not consider the question of information here, nor ask where and how the "Bauplan" is written, although we may indicate that in inorganic structures the rules are distributed and non-explicit in contrast with the D N A h t e i n system.

Classical crystallography emphasises the Fourier transform of the structure since the X-ray method gives the sinusoidal components directly, one diffraction spot corresponding to one sine m eof electron density. Thus, structures which do not have simple transforms are regarded, often erroneously, as disorganised.

Post-classical crystallography is concerned with less regular structures and the key instrument here is tkïe hi& resolution electron microscope (HRJPl) which gives a projection r(x,y) of a structure without loss of the phase relationships between the scattered waves (although there may be changes caused by the scattering transfer function). It raises the question of tamography, since one of the main technical problems is the combination of several projections to give a 3-dimensional picture r(x,y,z), a technique developed by Klug and others for the visualisation of biological macromolecular structures. HREM enables us to examine d lregions individually. This has already given new insights on the Berthollides - non-

stoichiometric c o m p o ~ ~ A ~ considered to be disordered crystals - as well as the Daltonides (stoichimetric, law-abiding crystals).

The other major tools are those of mode1 building by conputer graphics and of finding the physical properties of such systems by the methocls of molecular dynamics to check with macroscopically determined parameters, but in both cases working from the atomic interactions outwards to the macroscopic structure.

We must deal first with the crisis which has stimulated the present meeting - the appearance of icosahedral "crystals" [8]. The msition appears to be the f ollowing :

Every crystal belongs to one or other of the 230 space groups: each of the 230 space groups has diffraction effects (a reciprocal lattice weighted by the intensities of the structure factors) which belongs to one or other of the 32 crystallographic point groups. This material shows diffraction effects which belong to the icosahedral point group (53m): this is not one of the 32 point groups:

therefore the material cannot be a crystal. What is it? 1 assert that it must therefore be a kind of texture.

We do not have a very adequate vocahulary for dealing with textures and 1 suggest that this urgently needs examination. Any such study must begin with the paper on the synmetry of textures by Car1 Hermann (1931) 191 in which he

specifically points to the possibility of icosahedral textures. Texture does not necessarily imply statistical features. It is possible to define a completely deterrnined texture, such as that of a rope - a hierarchic arrangement of filaments.

1.1 Substantial regions (0.5-1 micron) of various alloys, in prticular Al/Mn (approx. 4:1), show electron diffraction patterns which have the full point group sywnetry of the icosahedron. That is, there are five-fold axes. High resolution electmn micrographs (point resolution about 2.5A) show highly ordered arrays of similar extent. ûptical diffraction from such micrographs confirms the five-fold symnetry. Thus, regarding the micrographs as superpositions of sinusoidal density waves (and recalling that in microscopy, in contradistinction to X-ray crystallo- graphy, the phases of these density waves are not lost), we must conclude that the density waves have amplitudes wfiich have icosahedral symmetry, but that the phases of these waves have a lower and perhaps variable symmetry. That is, there may be phase modulation of the density waves. Phase modulation, like the related fmquency modulation, produces sidebands, which here, if the modulation period is large (for example 100A) will be very close to the main spots and will be lost in their general bmadening. The micrographs nowhere show exact five-fold symoetry. There is everp indication that the diffraction patterns from different region of the specimens are the same. Different reseazch groups have published extremely similar diffraction ptterns (at least as reg- those along the five-fold axes). This observation revives the concept of "homometric structures" which was important earlier when there was less confidence in the uniqueness of sets of structure amplitudes of crystals, but has since languished. Homometric structures are those which have the same structure amplitudes but different sets of phases. This mode1 can easily be tested by performing the Fourier transfomation of the best high-resolution micrographs

.

We now suggest that the Penrose tiling is thus a device which can be used to generate homometric structures.

1.2 The icosahedral group can show altogether six special central sections frcw which to build up the whole of direct and of reciprccal space.

(a) perp. to a 2-fold axis and containing 2 five-fold axes, 2 three-fold and 2 two-fold axes

(b) perp. to a 3-fold axis and containing 3 diad axes.

(c) containing a 2-fold and a 5-fold at right angles.

(d) containing a 2-fold and two 3-folds.

(e) perp. to a 5-fold axis and containing 5 diad axes.

(f) containing a 2-fold and a 3-fold at right angles.

Views in these directions show that there are incoumensurate periodicities, mainly along the two-fold axes; the problem is to build a three-dimensional structure f m m these projections.

1.3 Topotactic relationship of crystal and icosahedral phases

The cubic crystal phase MEf32(M9Zn) 49 1101, as one of the most icosahedral structures known, is of clear relevance [Il] for the present controversy over quasi-crystals.

Electron microscopic examination (using a J W L 20ûCX microscope with side- entry double-tilt stage) of a sample of the splat-cooled alloy &32(fl,&)49,

kindly provided by Professor Ramachandrarao of BHU, Banaras, has shown the presence of the icosahedral phase, the cubic crystalline phase and a certain intermeàiate -

like the icosahedral phase, but probably commensurate. Electron diffraction ptterns identical with those published earlier by the l3anaras group [121, were obtained.

However, electron diffraction patterns from the crystalline phase, taken along the 11111 axis, showed that the reflections of the forms (8,-5,-3) and (5,-3,-2) were stronger than the remainder. They correspond exactly with the strong reflections in the three-fold axis patterns of the icosahedral phase and indicate a topotactic relationship between the phases, crystalline, icosahedral and T-phase (crystalline

in one dimension).

JOURNAL DE PHYSIQUE

The cubic crystal has the unit ce11 dimension a=14.16A and corresponds to the second of the series of cubic structures predicted by Elser [13] as rational projections through a six-dimensional primitive hyper-cubic lattice. (The first is exemplified by pyrite). The expected unit ce11 dimensions are 1.7013 x taun Xarh, where art, is the edge of the unit rhombohedra in the 3-D Penrose tiling,

in this case 5.1A. This gives 14.04A or, working backwards, 14.16 corresponds to 5.14A. (The spacing of the strong reflections from the planes perpendicular to the five-fold axis was measured as 2.45 A.[-> arh=5.25A] arh w a s taken to be tau3/2 times this)

.

With this edge of rhombohedron the cubic ce11 would contain (assuming mute to obtuse cells in the ratio of tau) 20.35 acute and 12.58 obtuse. The measured value of arh would only have to be 5.15A to give nwibers very close to 21 and 13, which are the expected Fibonacci convergents or to 20 and 12 which are what would be expected as the contents of the second cubic convergent. Two FEH each composed of 10 acute and 10 obtuse, but sharing 4 obtuse, would bring the total to exactly 20:12.

a=14.16 would correspond to ara= 5.144A. The 3-D Penrose tiling can be seen to be composed of rhombic triacontahedra (RTH), each either in contact, or sharing an obtuse rhombohedron or a rhombic dcdecahedron with its neighbours so tbat a crystalline b.c.c. array of FEH and a Penrose tiling are closely related. Since the unit ce11 contains 2 x 81 a t m we would expect that an acute rhombohedron should contain about 5.9 a t m and and an obtuse 3.65.

The row [tau, -1, -l/taul, on which the observed strong reflections nearly lie, &es an angle of 22.240 with the twofold axes (of the crystal phase). This angle is also evident in the lattice image micrographs. It is the angle through which the 111 layers of a cubic-close-packed structure must rotate about their normals [Ill] to move to a dense icosahedral packing in a kind of martensitic transformation (figure 1). (It might also be compared to the allosteric

transformation of oxy-haemoglobin to deoxy-haemoglobin). The two structures retain a conmion point subgroup m3. With respect to the icosahedron, the orientation of the cube is that of a cube inscribed in a dodecahedron, known since antiquity. The kinematics were developed [141 in considering the discovery of the packing of icosahedral particles in the polio virus 1151. The poliovirus crystal has the

"double diamond" structure, essentially a body centred cubic packing of isometric particles where each is deemed to make four tetrahedral bonds with its eight nearest neighbours to give two mutually inter- penetrating diamond networks. The structure of Mg~a(Al,Zn)rs is also almost b.c.c. and consists of rhombic triaconta-

hedral clusters, (containing 55 atoms in three successive spherical shells) and then extended by 60 more, still with icosahedral symmetry, to give a truncated icosahedron (12 pentagonal faces and 20 hexagonal) and then 12 more to make up a cuboctohedron (8 hexagonal faces and six squares, al1 with equal edges). These then pack in a body centred cubic lattice. ûnly 12 atoms out of 81 per cluster depart from exact icosahedral symnetry. This cuboctohedron must be reduced by slicing pualle1 to its hexagonal faces to give the cuboctohedron (with 6 squares and eight equilateral triangles) of figure 1, but it can also be seen as having the centres of its faces at the vertices of a rhombic triacontahedron which is its dual.

Bendersky and others 1161 have published electron diffraction photographs which show the same sublattice effect, but in this case comparing the crystalline A14Mn phase (hexagonal with a=28.4 and c=12.4 A) with the T" phase ([O0011 axis). Here the hkiO indices also have h,k pairs: (2,1), (5,3) and (8,5).

It is evident from figure 1 that icosahedra, in a body-centred cubic arrangement each sharing eight faces with its neighbours, can rotate in synchronism, so that the transformation cm take place simultaneously throughout an array in either direction. An array of icosahedra with their 222 axes in cornmon, may be seen in the icosahedral viruses and a similar geometry occurs in beta-tungsten. It is clear also that if clusters are to rotate into synchronism, bond-orientational- order, then the icosahdedral group, of order 60 (or 120) is the most favourable synnietry since the angle through which each cluster has to rotate is then a minimum.

Stephem and Goldman [17] have examined the scattering from a mode1 of a d o m

parallel packing of icosahedra but this mode1 is essentially a quasi-lattice packing, since al1 quasi-lattice points are given as integer multiples of the six icosahedral axes.

The occurrence of the Fibonami ratio

'Itio sine waves, starting in phase, take longest to corne to within a specified small distance of being exactly in phase again, if their wavelengths are in the ratio of 1 to tau. (Figure 2). That is, they are as far uncoupled as is possible. In this may be seen one of the physical principles occurring in incommensurate structures. An example may be found in the modulated structure of NbTes 1181 where the incomensurate vector related to the displacement modulation of the postions of the Nb a t m make the ratio tau with the crystal lattice repetition.

ûther compounds, such as CaaSi01 and SrzSiOlr show the same numerical value of the incAmensurate vector.

\

1.4 The anda au meory of' ~rystals

A numbe: of authors [ 19,201 have applied the Landau theory to the icosahedral phases. Thesk studies show, post facto (and Alexander and McTague, 1211 pre facto) that icosahec#ral combinations of density waves might be expected to give stable structures. We dish to remark here that the expansion for the free energy is in te- of tho~e triplets, quaduplets etc. of the reciprocal lattice vectors, which add to zero that these triplets (etc.) are just those te- which occur in the analysis of crystal structures by the direct methods of Karle and Hauptmann. Thus,

2 ) CRYST- ON 2-DIMENSIONAL NON-EUCLIDFAN MANIFOLDS

Since we live in a three-dimensional world, the consideration of packings of lunits on two-didensional manifolds affords us the opportunity of gaining experience with non-Euclideai) manifolds since we still have a third dimension into which the two-dimensional surface can be c w e d . More and more physical examples of these are appearing. They occur in silicates, lyotropic colloids, larger scale mineralised tissues and so on.

For example, given units which are equilateral triangles and given the local building rule which would require seven to meet at each vertex, a periodic minimal surface (the P-surface) results although in fact only 3/4 of the vertices have 7 tiles and 1/4 have 6. A planar tiling by triangles has always a mean coordination number of 6. Cunrature is a means by which other coordination nmbers can be a c c d t e d . Less than six lead to closed spaces (such as the surface of a sphere) which then have to be stacked by a new rule (to give a hierarchy of structures) and more than six gives saddle surfaces which can be ccnnbined to give an infinitely repeateing surface in 3 dimensions.

F'articularly interesting examples of actual two-dimensional manifolds are furnished by the periodic minimal surfaces. These are surfaces of zero mean curvature (as in a a soap film with the same pressure on each side) awl are obtained by hanging a soap film across the fundamental region in certain space groups, so that the coordinates of the film, and their first and second derivatives, are continuous across the boudaries. The whole of space is then divided by the film into two regions, usually congruent with each other. The film everywhere has a negative Gaussian curvature except at umbilic points where it is zero.

A minimal surface is one where at every point the mean curvature is zero. The simple but profound condition is div -0, where n is the unit normal at

every point (equivalent to the "director" in liquid crystal theory). At every point, except for the mbilic points where the curvature in al1 directions is zero, there are two principal cürvatures, Ki and K*, in planes at ri&t angles to each

JOURNAL DE PHYSIQUE C3-158

other (and containing g ) . The mean curvature (the "first curvature" of Weatherburn) is thus H = (1/2)(Ki+ Kz) and has dimensions L-1. The Gaussian

curvature ("the second curvature" of Weatherburn) is K = KiKz and has dimensions L-2. In terms of the unit n o m l 2K = div(g div + n x curl g).

A soap film, of surface tension T, requires a pressure difference proportional to T and to its mean curvature for its equilibrium and, if it has the the same pressure on each side, is a minimal surface and is thus a very convenient physical mode1 for visualisation. It is a theorem that every surface of constant mean curvature, such as that of a soapfilm with a pressure difference across it, has a surface of constant Gaussian curvature parallel to it, although if the mean curvature is zero then this surface is infinitely distant.

3) CRYST- IN 3-DIMENSIONAL NON-EUCLIDEAN MANIFOLDS

In two dimensions regular equilateral triangles tile the flat space in a lattice, but in three dimensions regular tetradedra do not fil1 space at al1 without interstices. There is great interest at present in the Fr&-Kasper phases (of metals) where the predominant feature is the filling of space by tetrahedra which cannot then be regular. A nwber of interesting mies have been obse~ed [22,23]

There are two approaches, one to give rules for filling in the interstices, and this is one of the properties of the Penrose tilings, and the other is to curve the space so as to close the interstices. The simplest example is the folding of regualar pentagons into a regular dodecahedron.

The question of crystallography in curved spaces (of positive, neetive and variable curvature) has become topical with the examination of the pcking of points in the 3-dim. space which is the surface of a hypersphere [7] but particularly with the developent of the idea that irregular packings in 34im space may appear as a result of local ordering which corresponds more to that appropriate in a higher dimensionality. Zn prticular, repeated local icosahedral syinmetry cannot appear in 3-dim. space groups but does m u r in higher dimensions, such as in the polytope [3,3,5]. It is clear that in quasi-crystals there is strong local order which, unusually dominates the long range order. These ideas have been workeci out since 1979 by, in particular, Mosseri, Sadoc and Kleman. We can mention here only limited aspects, but a study might begin with Mosseri et al. 1985 1241.

The "Boerdijk spiral"

When considering the packing of (nearly) regular tetrahedra, which comprises the principal characteristic of the l?rank-Kasper phases of metals, it is neces- to consider the Boerdijk spiral or helix [24,25]. This is an inconmensurate helix of regular tetrahedra, each sharing faces with two others, to give a colunm. The diad axis of each tetrahedron is rotated by an angle of 2 arctan (50.5) =131.8100 about the axis with respect to its predecessor. A complete turn of the three-start helix foxmed by the inishared edges of the tetrahedra is thus produced by about 30.5 tetrahedra. Considering the polytope [3,3,5] as the close pack% of 600 tetrahedra, the Boerdijk spiral can be found embedded in it, although regular tetrahedra in Euclidean space cannot form a ring. The helices can intersect each other by sharing tetrahedra. If we understand Coxeter 1251 correctly, the Euclidean Boerdijk spiral is to be found in the honeycomb [3,3,5.104

...

Multi-directional nematics

Any tessellation of 2-dim. s p e by parallelogrami can be converted to a weave or texture by where two threads cross on each tile, coming in at one edge and out at the opposite. Similarly, a 3-dim. pcking of parallepipeds can be converted to a texture where three threads cross in each tile. Thus, the Penrose tiling gives an algorithm for the construction of a 34im. nematic texture where 15 thread directions (parallel to the diad axes of the icosahedron) pass through each other as in a fabric. Even thought this may not be the case for quasi-crystals, multi-axis

nematics have to be considered as potential structures. Simpler structures have been observed experimentally [251.

Conclusion

Altogether we are entering a new phase of crystallography and the discovery of icosahedral "crystals" is just one indication that a new synthesis, of which the present paper is only an intimation, is urgently necessary. Table 1 suiranarises the viewpoint put in 1981 [26] which can now be seen to have experimental support in the materialisation of the concept of "quasi-lattice" there defined.

1 am much indebted to P.Ranachandrarao for the sample of his material and to Adria1 Walton for structure factor calculations.

TABLE 1. The transition from Classical Ckystailography to the Modern Science of Structure at the Atomic Levei.

[POST-CLASSICAL CRYSTALUXaZAPHYl

...

CLASSICAL CONCEPTS MODERN CONCEPTS

...

(1) Absolute identity of Substitution and components

.

...

(2) Operations of infinite Local elements of spmœtry

range. of f inite range.

...

(3) "Euclidean" space elements Curved space elements.

(plane sheets, straight lines) Membranes, micelles, helices.

Higher structures by curvature of lower structures.

...

(4) Unique dominant minimum in ûne of many quasi-equivalent free energy configuration states: metastability recording space

.

Progressive segregation and specialisation of information structure.

...

(5) Infinite number of unit~. Finite number of unit~.

Clusters. "Crystalloids".

...

(6) Assembly by incremental Assembly by intervention of other growth (one unit at a time). components ("crystalase" eiizyme).

Information-controlled assembly.

Hierarchic assembly.

...

( 7 ) Single level of organisation Hierarchy of levels of organisation.

(with large span of level). (SmalZ span of each level).

...

(8) Repetition according to Repetition according to program.

sywietry operatiom. Cellular automata.

...

(9 ) Crystallographic symmetry General syrmoetry operations operations. (=program statements).

C3-160 JOURNAL DE PHYSIQUE

(10) Assembly by a single pathway Assembly by branched lines in in configuration space. configuration space. Bifurcations

guided by "information", i-e. low energy events inthe hierarchy below

.

[reprinted from A.L.Mackay, Krist., E , 910-919, 1981).

TABLE2. T E X T U R E S

S-ested features of a GENERAL SYSTBWTISATION OF "TEXIZIRES"

Comprising non-crystal structures i n the fields of:

Metals Liquid crystals Fibres Polymers Helices, Coiled coils.

Sheets (lipid, silicate, macroscopic) Vesicles. Organic and inorganic.

1. Homogeneous textures. Nematics.

constant vector field constant tensor field (azimuthal as well as axial order) (with order parameters)

1.1 Multiple homogeneous textures. Polar/non-polar.

two or more axes of the same kind two or more axes of different kinds

c.f. 3-four-fol& and 4-three-fol& known in inorg. crystal chemistry (Andersson)

textures deriving from each of the point groups (including icosahedral where there may be 6 5-fold, 10 3-fold, 15 2-fold, 30 general direction. of equivalent non-polar axes)

the texture groups have been systematised by Shubnikov and his school (and include the infinite point groups. Int. Tab. A, p.777)

Tactosol Gels 1.2 Superimposed point groups.

Topotactic decomposition of higher symmetry crystals.

2. Periodic textures.

Cholesterics.

Blue phases (cholesteric in several directions, usually cubic) BoULigand textures Eutectics Spinodal decomposition Crystallisation of statistical objects, spherulites, vesicles Smectics

Molecular beam epitaxy (information-generated structures).

Periodic minimal surfaces

3. Quasi-periodic textures. (generated by eutactic stars).

4. Finite textures

spherulites cylindrical textures. tactoids micelles 5. Hierarchic structures.

Nematic textures of nematic textures Coiled coils etc.

FïGZnZB 1 (over) "Allosteric" transformation of a f.c.c. paclring to icmahedral [14] by rotation Oy 220 about four three-fold axes.

FJGiXE 2 (m) 'LtiD mm, of diffewt frequencies fi and fz. s t a r t b in

phase, take the greatest distance to came uithin a g i m small distance of being again in phase, when the rtio of their frequencies is tau. The closest approeches are after 2,3,5,8,13,21,34... cycles. If the waves are multiplied (as in amplitude modulation) then (fi+fi) and (fr-f~), the _{n o d} modes, are in the

ratioof t a u s to 1 and represent the CO- Bragg reflections C m , A.L.. Nature, 319, 102-104. (1986)J.

C3-162 JOURNAL DE PHYSIQUE

PI.- 3. Fivefold axis of imsahedrd phase.

FIGliRX 4. Pseudo-f ivefold axis partly "-teà" .

FI- 5. Threefold axis of icosahedral phase.

Pnxar& 6. 'lhreefold axis of cubic (crystalline) pbase.

Rkctron diffraction (200 IN, JB3L 200CX E.M.) from a specimen of splat-led

~ s x ( A l . Z n ) r ~ , .1-( Note the

-

FmxmmCES

111 Schroedinger, E . , (1944). "What is life?", C.U.P. p.3. [.."the chromosme fibre may suitably be called an aperiodic crystal". J

[2] Bernai, J.D., (1931). Cambr. üniv. Papers Box 22:A.4.7.

Bernal, J.D., "The ûrigin of Life", Weidenfeld and Nicolson, London, 1967.

Bernal, J.D. and Carlisle,C.H., "The Ftange of Generalised Crystallography", Soviet Physics - Crystallography, (1969). i3, No. 5, 811-831.

[3] Bernai, J.D. and Fankuchen, I., (1941). "X-ray and crystallographic studies of plant virus preparations", Jour. Gen. Fhysiol., 25, 111-146.

141 Mackay, A.L., (1975). "Generalised crystallography", Izv. Jugoslav. centra za kristalografiju, i0, 15-36. Mackay, A.L. (1969). "The structure of

structure", Chimia, 23, 433-437.

151 Mackay, A.L., and F i ~ e y , J.L., (1973). J. Appl. Cryçt., 6 , 284-289.

[6] Mackay, A.L., (1985). "Non-Euclidean Crystallography", paper at Janos Bolyai Symposium, Balatonszeplak, Hungary, May 1985. In press.

[7] Mackay, A.L., (1980). Jour. Phys. A, l3, 3373-3379.

C81 Shechtman, D., Blech, I., Gratias, D. and Cahn, J.W., (1984). "Metallic phase with long-range orientational order and no translational synaoetry", Phys.

Rev. Lett., 53, (20), 1951-1953.

191 Hermann, C., (1931). "Die Symnetriegmppen der amorphen und mesomorphen Phasen", Z. f. Krist., 7 9 , 186-221.

Cl01 Ber-, G., Waugh, J.L.T. and Pauling, L., (1957). "The crystal structure of the metallic phase Mg3z(Al,Zn) 4 9 " . Acta Cryst., l0, 254-259.

1111 Mackay, A.L., (1985). Nature, 315, 636.

Cl21 Mukhopadhyay, N.K., Subbanna, G.N., Ranganathan, S. and Chattopdhyay, K., (1986). "An electron microscopie study of quasicrystals in a quaternary alloy: Mg32(AlJZn,CU)49". Scripta Metal., (in press).

ibmwhandrarao, P. and Sastry, G.V.S., (1985). "A basis for the synthesis of quasi-crystals", Pramana, 25, L225-L230.

[131 Elser, V., (1986). "The Diffraction Pattern of Projected Structures". Acta Cryst. A X , 36-43.

1141 Mackay, A.L., (1962). "A dense non-crystallographic packing of equal spheres"

Acta Cryst., l5, 916-918.

[151 Finch,J. and Klug, A., (1960). Biochim. Biophys. A c t a , &l, 430-.

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