On the origin of pseudosymmetry

Journal

ofCrystal Growth 6 (1970) 323-326 8: North-Holland

Publishing

Co., Amsterdam

ON THE ORIGIN OF PSEUDOSYMMETRY*

B. G. BAGLEY** Division

of Engineering

and Applied

Physics,

Received

Harvard

t?

University,

7 October

-f

Cambridge,

Massachusetts,

U.S.A

1969

It is suggested that the growth of a low energy atomic configuration is an alternative to statistical faulting for the origin ofpseudosymmetry, and that structures observed to have a single five-

fold pseudosymmetric axis may have resulted from the continued growth of a low energy pentagonal dipyramid nucleus.

A previous note’) described the properties of a noncrystallographic five-fold pseudosymmetric structure with a density of 0.7236 (compared to 0.7405 for close packing). Contrary to the criticism by Clarke2), the structure is generated by a simple process of continued packing on the close packed faces of a pentagonal pyramid (or dipyramid) nucleus, just as the continued packing on the close packed faces of a tetragonal pyramid (or dipyramid) nucleus results in cubic close packing3). It was suggested’) that the growth of a pentagonal dipyramid nucleus was a simple mechanism which would account for observations of five-fold pseudosymmetry. However, the origin of such a noncrystallographic nucleus, whether accidental or otherwise, was not considered further. Recently, there have been an increasing number of observations of five-fold pseudosymmetry in many diverse materials (see table 1). With few exceptions, these observations were characterized as quintuple twins ((111) twinning plane) consisting of five face centered cubic (fee) individual crystals about a common [l lo] axis, with the 7’ 20’ difference between 5 x 70” 32’ and 360” accounted for by lattice strain or imperfections (70” 32’ is the smallest angle between (11 l} faces).

In classical crystallography, two (or more) crystals constitute a twin if the orientation of one can be brought into congruence with the other by rotation of 180” about the twin axis or reflection across the twin plane26). According to this definition, a five-fold pseudosymmetric structure does indeed consist of five twinned crystals. However, a geometrical definition such as this neglects completely the origin of twins and the mechanism whereby twins occur, and is so general that it includes, for example, all pure tilt grain boundaries. Buerger (ref. 27, p. 176) has stated that an explanation of twinning, “lies not in geometry but in the physics of crystal growth”. To provide a mechanism for the origin of growth twins, Buerger 28,2g) introduced th e concept of statistical faulting (stacking fault twins). The arguments presented by Buerger, and experimental evidence, demonstrate clearly that statistical faulting is indeed a

* This work

was supported in part by the Office of Naval Research under Contract NONR 1866(50) and by the Division of Engineering and Applied Physics, Harvard University. t This work constitutes part of a Ph.D. Thesis submitted to Harvard University by B. G. Bagley. ** Xerox Corporation Predoctoral Fellow; American Society for Testing and Materials Predoctoral Fellow. tt Present address: Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey 07974, U.S.A.

323

mechanism whereby growth twins can occur. Can this mechanism, however, account for the observations of five-fold pseudosymmetry? If multiple statistical faulting were the mechanism whereby five-fold structures developed, perfect fivefold symmetry should be highly improbable and the other multiple twins should also be observed. Thus, it seems unlikely that multiple faulting could generate the five-fold structures having the small size and perfection observed by Melmed and Hayward’), Ino”), Schwoebelr6), Milman etal.’ 7),Allpress and Sanders’ 9), Kimoto and Nishida2’), and Komoda22). In addition, the frequency of occurrence of five-fold pseudosymmetry has been demonstrated by Melmed and Hay-

324

Material

How Observed

Cu

Ni,

Fe, Pt

Vapor

Optical

oxygen free ity copper sheet

Field emission Electron by

Optical

crystallization at high pressure in the prcsencc of a liquid mctnl c‘u AgBr

X-ray

Hydrogen reduced CoBr, Grown from metal solutions

Optical Optical

Hydrogen

X-ray

si, Ge Ni

Vapor

Au

reduced

deposited

NiBr2

(external

I IO

dilfraction

IO

microscopy microscopy diffraction

II

and gro\Lth and

onto a gold

electron ditfraction Electron microscopy

(surface

Vapor

Bacteriophage

‘)

dendrite\

microscop!

morphology Eleclron microwopy

substrate Isolated by dill‘erential

R I7

microscopy

onto cleaved

Au

(or a contaminant of the same virion

microscop)

microscop)

Flectron

NaCI deposited

of thermal

morphology)

catalyst Electrodeposited Precipitated from solution

Co

microscopy

etch tigure

deposited

Elcctrodcposited Synthetically prepared

Diamond

Ho\v observctl

in one crystal of ;I

polycrystallinc high conducti\ Cu. Ni

prepared

Whiskers

I? 13. II

J

(ol’tcn hello\\

morphology) Electron microccopy

centrifugation

six) Ni

Thermally

Au

carbonyl Vacuum

decomposed deposited

nickel

Electron

onto cleaved

mica Au, Ag, Pd. Ni, Co Herpes virus

Evaporated argon Observed

simplex

and condensed

diffraction

Electron

microwopy

dilTraction Electron microscopy

in

in the nucleus of a likcr

hepatitis Vapor deposited

and electron and electron

Electron

microscopy

with

onto clca\al

Electron microxopy electron difl‘rnction

NaCI Electrodeposited

X-ray

Ni, Cu, Au

Evaporated

microscopy Electron microw~pq

Au

xenon Vapor

Ag

and

diffraction

cell of a mouse infected

Au

microscopy

electron

and condensed

deposited

in

onto cleaved

diffraction

Electron

and

and optical

microscop)

mica

ward”),

Schlijtterer’.‘),

hell’), and

Downs

and

Braun18).

Ino”),

Kimoto

and

Schwoe-

the

symmetry

Nishida2”).

could

Clearly.

implications

(ref.

states,

and

concerning they

may

perfect

fortuitous.

five-fold

or highly

pseudo-

statistical

early

p. 476). of

not

his

growth

of

discussion faults

the

during

crystal

“conditions

be less unfavorable,

on

have

energy

the

twins

nucleus.

are thus if not

difposi-

energy

further

than

are

once

kinetically

for

material

accidents

which

for

because

the

faults

are

slow.

The

the fault

all

volumes.

energy

probability

wa\

it. ho%-

is an alternative

Th~ls

growth

formed.

one

ofpseudosymmetry.

material. of

idea

structure

If we pursue

the origin

has.

unfaulted

fhc

a faulted

a nonfaulted

by Buerger.

faulting

than

ol’ twins”.

stage

we are led to a mechanism

observed fore.

formation

nucleation

a lower

to statistical

the

to the

the

Faulted in

statistical

the energy:

favorable. during

not discussed ever,

improbable

without

be operating. 28,

occurrence

formation

of

a mechanism

must

Buerger probable

observation

is not a rare,

occurrence.

tively that

Komoda”).

Thus

ferent;

DeBlois’“*“).

;I

higher

stacking

fault

process. is small

their

removal of

their

They and

arc

therecan

formation

hc

ON

increases

with an increasing

growth

THE

ORIGIN

OF

rate (increasing

supersaturation). The faulting process can also lead to faster growth kinetics by providing re-entrant corners for the nucleation of new layers without the usual energy barrier. Thus faulted crystals are observable also because they can grow faster than their unfaulted counterparts. This is an important consideration at low supersaturation where, because of surface energy, the slower growing small unfaulted crystals are unstable with respect to, and subject to dissolution and reprecipitation on, the large faster growing faulted crystals. Now let us suppose that during crystallization,conditions can prevail such that in the nucleation stage a particular molecular configuration (crystallographic or noncrystallographic) has a lower free energy than an equal volume (any shape) of the equilibrium infinite crystal. In general, the energy of this initially lower energy nucleus, as the nucleus grows, will equal and then exceed the energy of an equal volume of the regular crystal. This low energy nucleus is the stable configuration for small volumes, however. Therefore, in growth from a dilute phase at low supersaturations where the nucleation frequency is a strong function of nucleation energetics, the occurrence of a normal crystal becomes improbable. Contrary to statistical faulting, increasing the supersaturation and lowering the nucleation temperature, makes large nuclei, and therefore normal crystals, more probable. As a low energy nucleus grows larger, its energy finally exceeds that of an equal volume of the regular crystal. In the case of five-fold packing of a normally cubic material, the structure can relieve the strain energy associated with the lower density by cleaving (the 7” 20’ angular deficit subsequently filling up with single crystal or polycrystalline material) or by introducing dislocations. The low energy of the faults can be released by diffusion, or dissolution and reprecipitation. The regular packing of the pentagonal dipyramid can itself be terminated by stacking faults (or twins) in the close packed faces, or by the nucleation of the regular crystal on one of the close packed planes. An increasing supersaturation increases the probability of this termination occurring and thus decreases the possibility of observing the five-fold symmetry. The role of impurity gases is not clear. Melmed and

PSEUDOSYMMETRY

325

Hayward’) observed that five-fold whiskers predominate at a residual gas (strongly adsorbing) pressure of 1O-6 Torr. They suggest that gas adsorption may make the (100) plane that of lowest energy and, since the sides of five-fold whiskers are (100) planes, this would favor the structure. They also point out that the high surface-to-volume ratio makes surface energy effects important. Schwoebeli6), on the other hand, observed five-fold surface growths, but no whiskers, in gold deposited in a vacuum of lo-” Torr. He explains their origin as arising from impurity atoms of a particular diameter (Z 0.7 gold atom diameter) serving as nuclei around which five gold atoms cluster with subsequent lateral growth of (110) planes parallel to the substrate and vertical growth developing close packed faces. An alternative explanation, however, is that impurities are not necessary for the origin of five-fold symmetry, but are important in determining whether the final morphology will have an acicular or polyhedral habit. Changes in surface energies, as discussed by Melmed and Hayward, are relevant in this context. The previous discussion has been focused on fivefold pseudosymmetry because of its non-crystallographic nature and the experimental evidence available. The same arguments apply to the observation of pseudosymmetries other than five-fold, such as the sixfold pseudosymmetry commonly observed in chrysoberyl (BeAl,O,), aragonite (CaCO,), witherite (BaCO,), strontianite (SrCO,), cerussite (PbC03), and muscovite (KAl,Si,O, o(OH),)30). In polyatomic materials such as these, a low energy pseudosymmetric nucleus may be the result of atoms attempting to satisfy, locally, a particular coordination. The author is indebted to H. S. Peiser and D. Turnbull for many helpful discussions. Appendix The idea that for a small number of atoms a noncrystallographic array has a lower energy than a crystallographic one forms the basis for a structural model of the liquid state as proposed by Frank3’). He suggests that for coordination twelve, the noncrystallographic icosahedron is a lower energy configuration than the hexagonal or cubic close packed arrangements, and therefore that the icosahedron is an important element of liquid structure.

IL G.

326

BAGLEY

Berna132-34 ) also has a structural model of the liquid state according to which atoms are packed in a dense random array subject to the constraint of no overlap. One of the important structural units of this model is the super dense 7 atom pentagonal dipyramid33.34). If the pentagonal dipyramid is in fact a low energy configuration. as the discussion in this note indicates, it will occur as a structural element in liquids with a frequency higher than that suggested by the Bernal random model.

12) J. W.

Faust,

(1964)

1407.

Ii)

Jr. and H.

R. W. DeBlois,

F. John.

.I. Appl.

.I. Phy\.

Phys. 36 (19651

Chem.

Solids 25

1647.

14) R. W. DeBlois, J. Vacuum Sci. Technol. 3 (1966) 15) S. Ino, J. Phys. Sot. Japan 21 (lY66) 346. 16) R. L. Schvocbcl,

J. Appl.

B. G. Urman, 17) G. Milman, Science 152 (1966) 1381.

Phys. 37 (1966) A. Mitchell

146.

‘515.

and

R.

Langridgc.

IX) G. L. Downs and J. D. Braun, Science IS4 (1966) 1443. 19) J. G. Allpress and J. V. Sandcra, SurT~cc Sci. 7 (1967) I. 20)

K.

Kimoto

and

940. 21) J. L. Melnick,

I. Nishida.

E. R. Rabin

J. Phb\.

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and A. B. Jcnson,

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