A quasi-crystalline sphere-packing with unexpected high density

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A quasi-crystalline sphere-packing with unexpected high

density

Jörg M. Wills

To cite this version:

Short Communication

A

_{quasi-crystalline}

_{sphere-packing}

with

_unexpected

_{high density}

Jörg

M. Wills

Math. Inst. Univ.

_Siegen,

Hoelderlinstr.

3,

D-5900

_Siegen,

F.R.G.

(Reçu

le

_{23 janvier}

_1990,

_{accepté sousfonne définitive}

le 5 mars

₁₉₉₀₎

Résumé. 2014

_{Henley, puis Olamy}

et Kléman ont construit des _pavages de Penrose à 2D _par

em-pilements

de

_disques

de

_grande

densité. A l’aide de ces

_empilements,

on a construit un _pavage de Penrose à 3D par

empilement

de

_sphères

d’une densité de

_{0,67798, plus}

_{élevée que la meilleure} den-sité connue _pour les

_{quasicristaux}

usuels. Cet

_{empilement pourrait}

fournir un modèle de

_{quasicristaux}

cylindriques,

récemment découverts _par

_{Bendersky et}

_al.,

Schaefer et al. et

_{Fung et}

al. Abstract. 2014

Hendley

and later

_Olamy

and Kléman constructed circle

_packings

in 2D-Penrose

_tilings

of

_{high packing}

densities. From these circle

_packings

one obtains with an additional idea a

_sphere

packing

in a

_cylindrical

3D-Penrose

_tiling

of

_{packing density}

_0.67798..,

which is

_higher

than the best known

_packing

densities of 0.62... for usual

_{quasicrystals.}

This

_{packing might}

be a model for the

cylindrical quasicrystals

discovered

_by

_{Bendersky et}

al.,

Schaefer et al. and

_{Fung et al.}

Classification

Physics

Abstracts 61.50E

1. Introduction.

Atomic dense

_packing

is a fundamental

_property

of condensed matter. So

_recently

several authors have

_investigated

dense

_{sphere packings}

in

_{quasicrystalline}

_{structures, e.g.}

_Henley

_[1],

Olamy

and Alexander

_[2],

_Olamy

and Kléman

_[3],

and

_Oguey

and Duneau

_[4].

_Moreover,

_Henley

and

_Olamy

and Kléman

_investigated

dense circle

_packings

in 2D-Penrose

_tilings.

In this

_paper

we. construct

_{cylindrical sphere packings}

based on these 2D-Penrose

_tilings;

i.e. our two basic

rhombohedra are not the usual

_golden

or

_{Ammann-rhombohedra,}

but

_they

are

_orthogonal

_prisms

over the two Penrose rhombs. Because of their

_{high density}

these

_packings

_might

be

_interesting

from the structural

_{viewpoint. Quasicrystals of cylindrical}

_type,

i.e. with one translation axis were discovered in 1985

_by

_{Bendersky et}

al.

_[5],

Schaefer et aL

_[6]

and

_{Fung et}

aL

_[7];

so our

_sphere

packing might

be a model for

_{quasicrystals}

of this

_type.

2. Construction.

For our 3D-

_tiling

we need the

_2D-tiling

which is described in detail in

_Olamy

and Kléman

_(pp.

28,

29).

So to save

_space

we use their

_terminology

and results. As

_Olamy

and Kléman

_point

out

1062

(p. 29)

their

_"matching

rules" are no

_"forcing

_rules",

i.e.

_they

do not enforce

_{quasiperiodicity.}

So

"packing

rules"

_might

be more

_appropriate.

But we use

_"matching

rules" further on in accordance with the

_{Olamy-Kléman-terminology.}

We mention further that

_{Henley’s matching}

rules

_[1]

work as

_well,

but

_they

lead to

_slightly

smaller densities.

The construction: to fix a scale we consider

_only

unit

_spheres,

i.e. of radius 1. Then the two basic rhombohedra have

_edge-length

V3

and

_height

2.

The fat rhombohedron has basic

_angles

of 72° and

_{108° ;}

the thin rhombohedron has basic

angles

of 36° and 144°. Five of the fat rhombohedra

_generate

a

_{5-star-prism E of height}

2

_(see

_Fig.

1);

three fat rhombohedra and one thin rhombohedron

_generate

a

_{diamond-prism A of height}

2 and one fat rhombohedron

_together

with two thin rhombohedra

_generate

a

_{hexagonal prism}

H of

height

2

_{(see Fig. 2). Elementary}

calculation shows for the volumes of the thin rhombohedron:

V = 6 sin 36 and for the fat rhombohedron: V = 6T sin

36,

where r

= 2

(Bf5 +-l) .

So one

’

2

gets

for the volume of the

_prisms

E, 0

and H :

VE =

30T sin

_{36, Vo -}

_6(3T

+

₁₎

sin

_{36, VH}

=

6( T

+

₂₎

sin 36.

Fig.

1. -

The 5-star E built up of 5 rhombic

prisms,

its

_sphère

_packing

and the

_matching

rules.

Now we

_arrange

the centres of the

_spheres

such that

_they

coincide with the

_matching

rules for

E, A

and H

_{given by Olamy}

and

Kléman,

_(cf.

pp.

28, 29) :

For each of the three

_prisms

E, A,

H the centres are

_arranged

at the

_vertices,

denoted

_by

1 and the centres in the

_midpoints

of the vertical

_edges,

denoted

_by

0.

Because of the basic

_edge-lengths

V3

of E , à

and H the mutual distances of

_neighboured

centres are either 2 or

2V3

sin 36 =

2.036...,

so no two balls

_overlap.

Now 17

_spheres

contribute to the

_packing

in

_E,

which all

_together

have the volume of

_exactly

5

_spheres.

So the number of

_packing

_sphères

for E is nr, = 5. Similar considérations show _nA = 3

and nH =

2,

which are the same numbers as for the circle

_{packing by Olamy}

and Kléman. The

Fig.

2. - The diamond

prism à

and the

_hexagonal

_prism

H,

their

_{sphere packing}

and the

_matching

rules.

with

and

So the _average volume of the

_sphere

_packing

is

The _average volume of the

_prisms

_E,

A and H is

So one obtains for the

_density

Until now we have an infinite slice of

_height

2 endowed with a 2D-Penrose

_tiling.

If we take

in-finitely

many

congruent

copies

of this slice and fit them

_together

such that in the vertical direction the whole

_packing

is invariant under translations of

_length

2,

we obtain the desired

_{cylindrical 3D}

Penrose

_packing.

Of course, this

_packing

has the same

_density

which is much

_higher

than the best known densities in

_{quasicrystals}

_{generated by}

the Ammann-rhombohedra

_(with

8 =

0.62...).

Moreover this

packing

is

stable,

i.e. no

_sphere

can be moved

essentially

from its

_position

without

_touching

another

_sphere.

References

[1]

HENLEY

_C.L.,

_Phys.

Rev. 34

₍₁₉₈₆₎

797-816.

1064

[3]

OLAMY

Z.,

KLÉMAN

M., J.

Phys.

France 50

₍₁₉₈₉₎

19-37

[4]

OGUEY C. and DUNEAU

M.,

Europhys.

Lett. 7

₍₁₉₈₈₎

49.

[5]

BENDERSKY

_L.,

_Phys.

Rev. Lett. 55

₍₁₉₈₅₎

1461-1463.

[6]

SCHAEFER

R.J.,

BENDERSKY L.A. and BIANCANIELLO

F.S., J.

Phys.

France 47

₍₁₉₈₆₎

311.