Cubic approximants in quasicrystal structures

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Submitted on 1 Jan 1990

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Cubic approximants in quasicrystal structures

V.E. Dmitrienko

To cite this version:

Cubic

_approximants

in

_quasicrystal

structures

V. E. Dmitrienko

Institute of

_{Crystallography,}

117333, Moscow, U.S.S.R.

(Received

7 mai 1990,

accepted

in

_{final form}

7 _August

₁₉₉₀₎

Abstract. 2014 The

regular

deviations from the exact icosahedral _symmetry,

_usually

observed at the diffraction _patterns of

quasicrystal alloys,

are

_analyzed.

It is shown that

_{shifting, splitting}

and

asymmetric broadening

of reflections can be attributed to

_{crystalline phases}

with the cubic

symmetry very close to the icosahedral one

_{(such pseudo-icosahedral}

cubic

approximants

may be called the Fibonacci

_crystals).

The Fibonacci

_crystal

is labelled

_{as Fn+1/Fn>,}

if in this

crystal

the most intense vertex reflections have the Miller indices

_{0,

_Fn,

_{Fn + 1}}

where

_Fi

are the Fibonacci numbers

_(Fi

= 1, 1, 2, 3, 5, 8, 13, 21,

34...).

The deviations of _x-ray and electron reflections from their icosahedral

positions

are calculated. The

comparison

with available

_experimental

data shows that at least four different Fibonacci

crystals

have been observed in Al-Mn and Al-Mn-Si

alloys :

2/1>

(MnSi

structure), 5/3> (03B1-Al-Mn-Si), 13/8>, and 34/21>

with the lattice constants

4.6

_Å,

12.6

Å,

33.1

_Å,

_{86.6 Å respectively.}

It is

interesting

to note that there are no

_experimental

evidences for the intermediate

_{approximants 3/2>, 8/5> and 21/13>.}

The

possible

space groups of the Fibonacci

crystals

and their

_{relationships}

with

_{quasicrystallographic}

space groups are discussed.

Classification

Physics

Abstracts

61.16 - 61.50e - 61.55H

1. Introduction.

Among

many

problems

concerning quasicrystals

the most

_important

ones are

₁₎

the

characterization of

_{quasicrystalline}

atomic structure and

₂₎

the nature of the disorder in

quasicrystalline alloys

[1,

2]. Imperfectness

of

_quasicrystal

structures is observed as

_peak

splitting, shifting

and

_{asymmetric broadening}

in x-ray and electron diffraction

pattern

[1-9J.

In

the icosahedral

_glass

model,

the disorder lies in the basis of the

_{quasicrystalline}

structure. In

other theoretical

models,

the

_{perfect quasicrystalline}

structures are

_possible

and the

imperfectness

of real

_samples

is

_supposed

to be a result of frozen

_phason

strains

[1,

2, 5,

6].

In the

present

paper another

approach

is

developed.

It is shown that in many cases the

asymmetry

and deviations of the

Bragg

reflections from the icosahedral

_positions

can be connected with the presence of

crystalline ,phases

which

_approximate

the

_{quasicrystalline}

phase

very

closely.

Below, only

cubic

_approximants

are considered in detail

_(they

are called

the Fibonacci

crystals).

The sequence of Fibonacci

crystals

with ever

larger

unit cells forms a

chain between conventional

_{crystals (with}

small unit

_cells)

and

quasicrystals.

Thus,

above

_all,

the Fibonacci

crystals

give

us a convenient

_{starting point}

for the

understanding

of the

quasicrystalline

phase.

Although

the

_{possibility of crystal approximants}

_{has been discussed in many papers}

[1,10-17],

the solid

_arguments

for their real existence are not numerous

_{(excepting a-Al-Mn-Si).}

It

can be shown from the

_comparison

between theoretical

_predictions

and available

_experimental

data that at least four different Fibonacci

_crystals

can exist in Al-Mn and Al-Mn-Si

_{alloys ;}

their lattice constants are about 4.6

À (MnSi

_type

_structures),

12.6

Á

_{(a-Al-Mn-Si),}

33.1 À

[15],

and 86.6

_{Á (see below). Although}

the latter

_approximant

looks rather fantastic

(the

unit cell contains more than 4 x

104 atoms !),

its existence seems to be confirmed

_by

_x-ray and electron diffraction. In other

_{alloys (Al-Li-Cu, etc.)}

the deviations from the icosahedral

symmetry

are also observed

_[6-9]

and their

_analysis

will be

given

elsewhere. For

two-dimensional

_{quasicrystals,}

the

_{periodic approximants}

have been considered in details earlier in reference

_[18],

where the

_importance

of such

_approximants

was shown.

The _{remainder of this paper will}

_proceed

as follows : in section 2 we discuss the

_symmetry

of the Fibonacci

_crystals

and relations between the

_Bragg

reflections in cubic and icosahedral

structures. In sections 3-4 the deviations of cubic reflections from their icosahedral

_positions

are calculated.

Then,

in section

_5,

the

possible

structure of the Fibonacci

_crystals

is illustrated

by

the

_simple

low-order

_approximant

with 8 atoms _per unit cell

_(MnSi

structure,

P213

space

group). Finally,

in section 6 the

_comparison

of our results with the

_experimental

data are

given.

2.

_{Crystallography}

of Fibonacci

_crystals.

Both the icosahedral

_{quasicrystals}

and their Fibonacci

_approximants

can be obtained from the suitable

_projection

of six-dimensional cubic lattice on our three-dimensional space

_[10-12,

_16,

17,

19].

In _{this paper} we use more

_{simple approach, considering quasicrystal (or crystal)}

as a

superposition

of

_density

waves with some set of the

_reciprocal

lattice vectors

_{Q :}

where p

(r )

is a

_density.

This

_approach

is convenient if one focuses on the diffraction

_patterns

rather than on the

_real-space

structure.

_{Any reciprocal}

lattice vector

_Q

of icosahedral

quasilattice

may be written as a linear combination of six basic vectors

_{Qm :}

where the

_{Nm are integers}

_{and may be}

_thought

of as the six-dimension Miller indices of the

Bragg

reflections. At first we consider the

_primitive

icosahedral lattices. As the basic vectors

we will use six vertex vectors

_Qvm

directed toward the icosahedron vertices

_(the

fivefold

_axes).

We have chosen as

_{(100 000)}

the most intense reflection of this

_series,

which for Al-Mn occurs

at

2.9 Â -

and corresponds roughly

to interatomic distances :

_d

= 2 w

/

1 Qv.

1

= 2.17

Á.

The

advantage

of this choice is that the most of the intense reflections can be indexed with a small

number of nonzero

_{Nm [19]. Using}

three-dimensional Cartesian

coordinates,

those six basic

vectors

_Qvm

_may be written as

basic vectors

_{Qvm by}

six vectors

_Qnm

of a cubic lattice with the Miller indices

equal

to the

Fibonacci numbers :

an is the lattice constant. The n-th order

approximant

is convenient to label as

(F n + 1/ F n> ;

its lattice

constant an

increases with n :

The

_relationship

_{between an}

and _aR was first obtained in reference

[10].

For

example,

in our

notation the well-known a-Al-Mn-Si

_crystal

should be labelled

_(5/3)

_(because

of another choice of basic vectors, in reference

_[10]

this

approximant

was

_{labelled (1/1),}

but this

produces

some difficulties in

labelling

of low-order

approximants

with n =

1, 2).

To avoid confusion it should be noted that in the

_{crystallographic}

literature some definite

choice of cubic axes is used

_{[20] ;}

if the space group is Pa3,

then,

as a result of this

choice;

the

first index of hk0 reflections must be even ; to meet the case,

Fn

and

_{F n +}

1 in

equation (4)

must be

_permuted

if

_{Fn +}

1 is even

(see

an

example

in Sect.

5).

Using equations (2)

and

_(4),

the

h,

k

and f

Miller indices can be

_expressed

via

Nm:

For another

_indexing

scheme,

that of reference

_[21],

the

_quasicrystal

reflection

_{(h /h’,}

k/k’,

f /Î’)

corresponds

to the cubic reflection with the Miller indices

_hFn-3

+ h’

Fn-2’

kF,,-3 + k’F,,-2

and iF,, -3 + iF,,

-2-The cubic vectors

_Qnm

can

_approximate

the icosahedral vectors

_Qvm

_very

_closely,

even if the

order n is not _very

_large.

For

_example,

the

_angular

deviation from the fivefold axis is about 2° for

_(230),

0.75° for

_(350),

0.29° for

_(580),

and 0.11 ° for

_(8,

_13,

_0).

Using Qnm

as the basic vectors in

equation (2),

we obtain a set of

_{reciprocal lattice}

vectors

with cubic

_symmetry.

In

_fact,

for cubic lattice any three basic vectors _may be

used ; however,

it is convenient to use those six-vector

_basis,

because in this case the vectors of cubic lattice

are close to the vectors of

_quasicrystal

with the same

_Nm

_(examples

are

_{given below).}

The

point

symmetry

of cubic lattice is 23 or

m3 (that is,

the

_symmetry

is lower than icosahedral

point

symmetry

532 or

532/m).

The

_{Landau-theory}

calculations of the

_spatial

structure of Fibonacci

_{crystals [15]}

show that the most

_possible

_{space groups} are

Pm3

or

Pa3

_(if

both

Fn

and

_{Fn +}

_{1 are}

_odd,

the

_possible

space groups are

Im3

or

_I213).

It is

_interesting

to note that the Landau

_theory

_predicts

the same absolute values of the

strongest

harmonics in

Pm3

and in

_Pa3,

whereas the

signs

_{of p Q}

are

_predicted

to be different.

_Although

the

Landau-theory

approach

is

_hardly

suitable for a detail

_description

of

_crystal

structure, there is a

_hope

that the

qualitative

results,

such as

_spatial

_symmetry

of a cubic

_phase

and the

_signs

of the

Fourier harmonics

_{p Q,}

are correct.

_{If n = 1,}

the basic vectors

_{become 111 0}}

and

_symmetry

of the Fibonacci

_crystal

is

Im3m

_[17].

When the

point

symmetry

reduces from icosahedral to

_cubic,

the icosahedral

_symmetry

axes

(planes)

become

_unequal.

As a

result,

the icosahedral reflections _may

_split

_up into several

types

of cubic reflections.

_Inequality

of the

_symmetry

axes

_(planes)

is

_especially

evident if we

put

icosahedron into cube

_(see

_{Fig. 1).}

Fig.

1. - The

relationships

between symmetry axes in cubic

crystals

and icosahedral

_{quasicrystals.}

The

symbols

of axes and

_{pseudo-axes (with asterisk)}

are evident from the

_figure.

It is clear that instead of

_thirty

twofold icosahedral axes,

only

six

twofold (100)

axes are

held in cubic

_crystal.

The other

_{twenty-four (,r 2 r 1 >}

directions _may be

_regarded

as twofold

pseudo-axes,

because for these directions the deviations from twofold

symmetry

are not very

large (in

the Fibonacci

_crystals

but not in an

_arbitrary

cubic

_{crystal !).}

Then,

instead

of twenty

threefold axes, in a Fibonacci

_crystal

we have

eight

_{threefold (111)}

axes and twelve threefold

(7» 2 Io>

pseudo-axes.

The twelve

_{fivefold (lrO>}

axes are turned into twelve

pseudo-axes (all

of them are

_{equivalent). Finally,}

instead of fifteen mirror

planes,

we have three mirror

planes

and twelve

_{pseudo-planes}

which are normal to twofold axes and

pseudo-axes

of cubic

crystal

respectively.

The

_inequality

between axes and

pseudo-axes

leads to

_{inequivalence}

and

_splitting

of reflections. For

_example,

one can

_easily

obtain from

_(2)-(4)

that

thirty

twofold

_(110000)

reflections

_{(directed along}

icosahedron

_{edges) split}

up into two

_types

of cubic reflections : six

(2

Fn,

0,

0 ?

reflections and

_twenty-four

_{{F n +}

1,

Fn,

F n - 1}

reflections which are directed

along

twofold axes and

_{pseudo-axes (approximately).}

The vectors of these reflections form

deformed « icosahedron » in the

_reciprocal

space of the Fibonacci

crystal.

The threefold icosahedral reflections

_split

_up also into the cubic reflections of two

_{types ;}

for

_example,

the

( 110 001 )

reflections are turned into

_eight

_{{Fn - , F n -1’ F n - 1}}

reflections and twelve

{F n’

Fn - 2,

0}

reflections. Those icosahedral

_reflections,

which are

_parallel

to mirror

_planes,

split

up into cubic reflections of three

types :

twelve of them are

_parallel

to mirror

_planes

of cubic lattice and the rest 48 reflections are close to

_{pseudo-planes.}

As a

_result,

the

_{(111 000)}

reflections

_split

_up into

_{{F n + 1 + F n - , F n + , O} ,}

_{{F n + 2, F n - 1, F n - 1 } ,}

, and

{2

Fn, Fn,

F n +

t}.

Finally,

an

arbitrary

icosahedral reflection

splits

up into five

types

of cubic reflections

₍₅

x 24 = 120 reflections in

all).

The

quantitative description

of reflection

splitting

and

_shifting

is

_given

in the next sections.

At the same

_time,

the Fibonacci

crystals

have many common features with

quasicrystals.

The

_general

feature of

_{quasicrystalline}

lattices is the inflation of the reflection

_wavelengths

with factors T or

r 3[

_{1, 2].}

In the Fibonacci

_crystals

the

_approximate

inflation takes

_{place [ 151.}

For

_example,

in addition to the basic « icosahedron », formed

by thirty

vectors

{F n +

1,

F n,

F n - 1}

and ( 2

Fn,

0,

0 ) ,

we see

(for

rather small

Nm

in

(2))

the sequence of

larger

(2

Fn +

k,

0,

0 },

where k = ±

1,

±

2,

... ; each « icosahedron » differs from the basic one

_by

factors about

_{r k (the}

_{T inflation).}

The Fibonacci

_crystal,

of course, contains the smallest

« icosahedron », formed

by

the

_{vectors (21 1 ) and {200},}

_(or

_very

_degenerate

«

icosahed-ron _», formed

_by

_{vectors ( 1 1 0) and ( 200 )}

of the same

_series)

whereas the

_reciprocal

_{space of}

quasicrystals

contains

_(in

the

_{general case)}

the infinite series of ever smaller icosahedra. On

the other

_hand,

the

_approximate

T3-inflation

takes

_place

for the

_{pseudo-fivefold}

{F n’ F n + l’ O}

reflections: if all

_{Nm are}

not _very

_large,

one obtain from the basic

{F n’ F n + l’ O}

vectors the

_{{Fn - 3, F n - 2, O}}

reflections,

but neither

_{{Fn - l’ F n’ O}}

nor

{F n -

2,

F n -1,

O}.

However,

in the Fibonacci

crystals

the inflation rules are not

rigorous

and,

for

_example,

the

_{{F n - 3’ F n - 2’ O}}

reflections can be obtained for

_sufficiently

_large

Nm

(but intensity

of reflections is small if

_Nm

are

_large).

It is

_interesting

to note that the

_approximate

T

3-inflation

becomes more

_pronounced

in the

Fibonacci

_crystals

with

_{Pa3, Im3,}

and

_I213

_{space groups} because of the extinctions rules and because of the

_parity

of the Fibonacci numbers whose

_cycle

is three

_(odd,

_odd,

_even,

_...).

For

example,

let us consider

Pa3

group ; if n = 2 and the basic reflections _are

(210) ,

then the

T3-inflated {850}

reflections are

possible,

whereas the intermediate

_{reflections (320)}

and

{530}’

are forbidden

_by

the extinction rules of

Pa3

group

(in

{hkO}

reflections h must be

even).

In the end of this section we discuss the

_symmetry

of those Fibonacci

_crystals

which

approximate body-centered

and face-centered icosahedral

_{quasilattices.}

In the

_{body-centered}

quasicrystals E

Nm

is even ; in this case, h + k

+ f

is even for all reflections and the Fibonacci

m

crystals

are bcc too. More

_intriguing

situation is in the case of the face-centered

_{quasicrystals}

(like

A165Cu2oFe15),

where all

_Nm

are of the same

_{parity :}

it follows from

_{equations (6)}

that all Miller indices are even for all reflections of the

_{corresponding}

Fibonacci

_crystals.

_Thus,

choosing

another cubic unit cell

_(twice

smaller in each

_direction),

we return to the same

indexing

of the Fibonacci

_crystal

as in the case of

_primitive

icosahedral lattice

_(with

one

important exception :

the inflation factor is now about T for all

_types

of

_{reflections).}

3. Déviations of cubic reflections from icosahedral

_positions.

In the Fibonacci

_crystal,

the deviation

_AQ

of an

_arbitrary

cubic reflection from its icosahedral

position

can be written as a sum of the deviations

_âQnm

of the basic reflections :

where

_{àQ_ = Qnm - Qvm.}

To obtain the

_explicit

form of

_AQ,,.,

we use the

_relationship

between an

and _aR

_(see

_Eq.

₍₅₎₎

and the

_{following expression}

for

_{Fn :}

After

_simple

calculations we obtain for

_{âQnm :}

where m =

1,

For any Fibonacci

crystal,

dQnm

are normal to the

_{corresponding}

basic vectors

Qvm;

thus,

the deviations of the absolute values of

_Qnm

from the absolute value of

are

very small

(of

the order of

A n 2

Qv,,,

This is also valid for any fivefold reflection.

For other

_strong

reflections the deviations of

Q

are of the order

_of

_{1 An 1}

for weak reflections

(large Nm)

the deviations _{may be} not _{very small. The}

_simple

calculations show

_{(see figures}

below)

that both the vertex vectors and the

_edge

vectors of the Fibonacci

_crystals

fit the

corresponding

icosahedral vectors _very

_closely.

_Therefore,

careful

_analysis

of _x-ray and electron diffraction

_patterns

is needed to

_{distinguish quasicrystals}

and their Fibonacci

approximants.

,

4. Diffraction

_patterns

of the Fibonacci

_crystals.

To show the main features of the cubic distortions of icosahedral

_{quasicrystals,}

we illustrate

the above

_equations

with

_{computer-simulated pictures.}

At

first,

let us consider the x-ray

diffraction on

powder

samples.

In this case,

only

the absolute values of the

reciprocal-lattice

vectors are relevant

_(but

not their

_{directions). Figure}

2 shows a

_computer

simulation of _x-ray

powder

diffraction

_patterns

for three Fibonacci

_crystals.

The half-width at half-maximum of individual reflection is chosen 0.009

Â-

1 and

_quasicrystal

_parameter

_{aR = 4.60 À}

(about

those ones observed in real

_{samples) ; positions}

of icosahedral and cubic reflections are

calculated from

equations (2)-(4).

For

_simplicity,

the total intensities of all reflections

(before

splitting)

are chosen of the same value and the intensities of the

_split

_components

of each

reflection are

_supposed

to be

_proportional

to their

_{general multiplicity}

factors

₍₂₄

for

hkf,

12 for

_hk0,

8 for hhh and 6 for

_{h00). Figure}

2 shows that in

_{the (5/3)}

_approximant

the

splitting

of

_strong

reflections is

_complete.

In

_{thé (13/8)}

Fibonacci

_crystal,

the values of

splittings

are

_comparable

with the reflection

widths ;

as a

_result,

the reflections are broadened

asymmetrically

and their

_apparent

widths are different. For

_{the (34/21)}

Fibonacci

crystal,

the

Fig.

2. -

Computer

simulation of x-ray

powder-diffraction picture

for (5/3),

(13/8),

and

(34/21 )

Fibonacci

_crystals.

The

peak positions (when

not

_evident)

are marked

_by

vertical lines ; for

splittings

are so small that the reflections are almost

_symmetrical

and their shifts from the

icosahedral

_positions

are

_hardly

observable

_(see

below

_{Fig. 7).}

For non-shown intermediate

approximants (8/5) and (21/13),

the

_signs

of the

_splittings

and, hence,

the

_signs

of

asymmetry

are

opposite.

The electron-diffraction

_pattern

is shown at

_figure

3 for the case of

_{pseudo-fivefold}

symmetry

(here

and below the incident beam is assumed to fall

_{along -}

z direction so that the

x, y and z axes form the

_{right-hand triad).}

This

_picture

is universal for the Fibonacci

crystals

of

any

order ;

the

only

difference is on a scale. The scale can be determined from the

positions

of the

_strongest

_edge

reflections

₍₂

_Fn,

0,

0 }, {Fn + l’ F n’ F n - 1}

which are at about 3

Â-’.

The

positions

of all

_spots

in

_figure

3 are determined from the

projections

of the

reciprocal-lattice

vectors

hkf

on the

_properly

chosen x

_{and y}

axes : _{xhk p} = h and yhk p =

(Tk + f ) / It

is

_important

that the most of relevant vectors deviate from their icosahedral

positions

not

_only

in the

_plane

of the

_pattern

but in

_off-plane

direction too : _{Zhkf =}

(rf2013A;)/B/1+T;

such

off-plane

deviations can cause a

_{disappearance}

of some weak

_spots.

Fig.

3. -

Universal

_{pseudo-fivefold}

electron-diffraction _pattern of the Fibonacci

_{crystals (incident}

beam falls

along -

z

_direction).

All x-coordinates are

_{integer :}

_xhkl = h.

Only

about _one fourth of the _pattern is

shown ; the

_{eomplete picture}

can be obtained

by

the mirror reflections : x --+ - x and

(or)

y - - y. The scale of the pattern are determined

_by

the

_positions

of the most intense

_edge

reflections

(Q =

3 A - 1 ) ;

for the different Fibonacci

_crystals

those

positions

are marked

by

arrows. The Miller indices are shown for outer reflections

_{only ;}

the indices of :inner reflections can be obtained from the

outer ones after

_{multiplication by}

factor T -

_{i (i}

= 1, 2,

3, ...)

and

round-up

_to the closest

integer.

The

innermost ten reflections are of the

_{type (200)}

_{and ( 21 1 ) .}

The

approximate

T inflation is

clearly

seen

in the

_{figure (pentagons}

with _strongest distortions are marked with solid

_lines).

Pseudo-twofold axes are

shown

by

dashed lines.

Several features of the

pattern

should be noted. The reflections are deviated from their

icosahedral

_positions

in x-direction

_{mainly ;}

_therefore,

the twofold h00 reflections are

deviated at radial

direction,

whereas the

_{pseudo-twofold}

reflections

_(shown

with dashed

_lines)

pseudo-twofold axis directed at 72° to the

_x-axis).

For

_{(pseudo-)twofold}

reflections the deviations are

usually

smaller than for other reflections with

_comparable

_{1 Q 1.}

. Note also the

« zigzag »

character of the deviations

_(this

is

_clearly

seen at

_{glancing angle).}

The deviations are

_larger

for reflections with

_small

_{[ Q}

₁

_and(or)

with

_{large Nm ; they}

are

_especially

visible when at least one

of the Miller indices becomes small. The inflation _process

_{(multiplication}

of

hkf

_by

the factor

T and

round-up)

decreases the distortions of the

_spots

_from their icosahedral

_{positions (see,}

for

_{comparison, proposition}

4 from Ref.

_[16]).

The

_{(pseudo-)threefold}

_patterns

are obtained as a

_projection

of the

_{reciprocal-lattice}

vectors on _any axes which are normal to a

_{(pseudo-)threefold}

axis. These

_patterns

can be of three different

_{types ;}

two of them are shown at

_figure

4 and the third one is the mirror

reflection of

_figure

4a

_(the

choice of axes is

_displayed

at the

_figure).

At the threefold

_patterns

the spots

are deviated in

_tangential

directions in

_zigzag

manner, whereas at the

pseudo-threefold

_pattern

the deviations are directed

_along

x-axis and all x-coordinates are

_integer.

Fig.

4. - Threefold

(a)

and

_{pseudo-threefold (b)}

electron-diffraction _patterns. Other outer reflections

can be indexed for

_{(a) using}

60°-rotation

_{(cyclic permutation}

of the indices and

_change

of their

_signs)

and for

(b) using

the mirror _symmetry

_{(x - -}

x and

(or) y - - y).

The indices of inner reflections can be obtained as for the

pseudo-fivefold

pattern. The twofold and

_{pseudo-twofold}

axes are shown with the dashed lines. The third _type of the _pattern

_(the

threefold one with z

Il 111 ])

is the mirror reflection of

(a).

There are also three different

_types

of

(pseudo-)twofold

patterns

(see

Fig. 5).

To avoid

confusion,

it should be

_emphasized

that the

pseudo-twofold

patterns

(which

correspond

the

incident beam direction

_along

the

_{pseudo-twofold axes)}

have the exact twofold

symmetry

because of the inversion

_symmetry

of the electron-diffraction

_patterns.

The different

_types

of

spot

displacements

are observed for the reflections

lying along

symmetry

axes

_(radial

displacement)

and

_{along pseudo-axes (tangential displacement mainly).}

The

_off-plane

deviations are also

_present

both at the

_{pseudo-twofold}

and at the

_{pseudo-threefold}

_patterns.

5. A

_{simple example :}

MnSi cubic

_crystal.

It was shown earlier that two

_approximants

were observed in Al-Mn and Al-Mn-Si

_{alloys :}

Fig.

5. -

Twofold

(a)

and

pseudo-twofold (b)

electron-diffraction patterns ; the third type of the _pattern

(pseudo-twofold)

is the mirror reflection of

_(b).

The _spots show the

_positions

of reflections but not their intensities.

approximant

of this series

should

be

_(2/1)

with the space group either

Pm3

or

Pa3

[15] (or

close to

_them)

and with the

following

strong

reflections:

_{{ 210 } , {211}}

and

{200}, {231}

and

_{{400} ;}

the lattice constant should be about

T 2

smaller than that of a-Al-Mn-Si.

_Turning

to the

_{crystallographic}

handbooks,

we find a

_possible

candidate

_{[22] :}

MnSi

cubic

crystal

(a

= 4.6

Â)

with

P213

space group

(close

to

_Pa3).

This structure contains

_only

eight

atoms _per unit cell : Mn in 4a

_positions

x, x, x with xMn = 0.138 and Si in 4a

positions

with xs; =

0.846 ;

each Mn atom is surrounded

by

seven Si atoms and vice versa. There are a

lot of

_compounds

with this structure :

_{FeSi, CoGeSi,}

_{AlNi2Si, NaBr03,}

_AIPt,

_Al4GePd5

and

even

_{AI,Mn,,,Si2; they}

_may be used for the search of new

_{quasicrystals.}

To demonstrate the hidden icosahedral

_symmetry

of MnSi

crystal (both

in the real _space and in the diffraction

_patterns),

let us consider the idealized structure AB with _xA =

1)/4=t

and _xB = 1 - xA = 0.845. This _structure looks like a

_periodic

_system

of

interpenetrating

dodecahedra

_{(see Fig. 6) :}

each A atom is at the center of an ideal

dodecahedron and seven vertices of the dodecahedron are

_{occupied by}

B atoms

_(and

vice

versa) ;

the closest A-B distance is

/3/(2 )

and the closest A-A

(or

B-B)

distance is

T - 1.

If we do not

_distinguish

A and B atoms

_{(A = B ),}

the space group becomes

Pa3

instead

_ofP2t3

for

_{A =1= B)}

as

_expected

from the Landau

_{theory [15].}

The structure factor

F(hkf)

for such

Pa3

crystal

is

_{given by}

the

_{following equation}

where

_{f = f (h 2+ k 2+ J2)}

is the atomic factor. From this

equation

the

approximate

icosahedral

_symmetry

of diffraction

_patterns

can be seen

_(at

_least,

in

high-Q region).

For

example,

the structure factors of the twofold and

_{pseudo-twofold}

reflections are almost

_equal

Fig.

6. -

Idealized MnSi structure. Cube with the inscribed dodecahedron shows the unit-cell dimensions

_{(conventional}

unit cell has another

origin [20]).

Full circles : A-atoms and empty circles :

B-atoms. Atom 1 is at the center of the cube and atom l’ is at the cube

_{diagonal ;}

the rest atoms are at the cube faces.

whereas 1 F (Fi ,

1

Fi Fi - 1 ) 1

8

If 1.

. This is also valid for the threefold and

pseudo-threefold

reflections : if i = 3 k then

and if i =A 3 k then

All these

_{approximate equalities}

become more and more accurate with

_{growing i}

(to

prove

this,

one should remember that

_{Fi +}

₁

_7-F,

and that the

_cycle

of the

_parity

of the Fibonacci numbers is three :

_{odd, odd,}

even,

...).

It is

_interesting

to note that there is a connection between MnSi structure and more

_simple

1 / 1 >

approximant.

The latter

_approximant

should have

Im3m

symmetry

if A = B

_[17]

or

Pm3m

symmetry

(like FeAl)

if

_{A =F}

B. We can see at

_figure

6 that there is a cluster of

_eight

atoms which are

_arranged

as in

Im3m

cubic structure

_(A-atoms

marked with

_{1, 5, 6,}

7 and B-atoms marked with

_{l’, 5’, 6’, 7’).}

The threefold axis of the cluster is directed

_along

the

threefold axis of

_MnSi,

whereas the fourfold axes of

Im3m

structure are directed

along

pseudo-twofold

axes of MnSi structure. It seems that MnSi structure looks like

_{microtwinning}

of

_FeAl-type

_{clusters ;}

it _{would be very}

_interesting

to

_generalize

this connection between

neighbouring approximants

for the

_high-order

case.

Note also that the hidden icosahedral

_symmetry

of MnSi

_crystal

results in some

physical

properties.

For

_example,

the elastic

_properties

of MnSi are almost

isotropic (the

ratio of the

shear moduli is very close to unit : 2

_{c441 (CI I -}

_{C12) = 1.05) ;}

then,

the tensor form of the

tensor structure

_{amplitudes [23]}

of MnSi is close to that of

_{quasicrystals [24].}

The direct

_proof

shows that there is no

_tendency

to the

_weakening

of those MnSi reflections

which

_correspond

to the forbidden reflections of

_{nonsymmorphic}

icosahedral space groups

[25].

That’s

_why

we should conclude that the

nonsymmorphic

MnSi

crystal

is the

approximant

6. Discussion.

Now we turn to the

_high-order

Fibonacci

_crystals

_((13/8)

and

_{(34/21 )}

and

_begin

with discussion of

experimental

evidences in

_support

of their existence. The direct evidences of this

sort can be seen at diffraction

_patterns.

Most of the _x-ray diffraction

_patterns

were obtained

from

_powder

_{samples ;}

in this case

_(see

_{Fig. 2),}

one can

_try

to

_{identify (13/8)}

_approximant

using

careful

_analysis

of the

_shape

of reflections. The

_only

available

_example

of such

_analysis

provides

reference

[4],

where reflection with

_{Q -}

3.04

 -

was

fitted to two Gaussian

_peaks.

It was shown that the widths of both

_peaks

are

_equal

to the width of fivefold

_peak

(Q

2.894

 - 1) ;

their fitted

_positions

are 3.031

Â-

1 and 3.046

A-

I

(those

values are

marked

by

squares at

_{Fig. 7),}

and the ratio

_{of peak’s}

intensities is about

_1/3.

Those values are

very close to the theoretical

_predictions

for

_{16,0,0}

and

_{13,8,5}

reflections in

(13/8)

approximant :

3.033

 -

and

3.045

 -

_{l respectively}

(the

ratio

_{of peak’s}

intensities is

expected

to be about

_{1/4) ;}

icosahedral

_peak

should be

_positioned

at 3.043

A-

1.

In the

pioneering

paper

[3]

the

splitting

of this reflection was also

_observed,

but the

_intensity

ratio was estimated as

_1/1.

_{In any case,} the proper

_fitting

of

_{asymmetrically}

broadened reflections

(like

in Ref.

_[4])

may

give

decisive

_arguments

for the choice’ of

_{approximant (or}

another

phason-strain model).

Fig.

7. - The deviations of cubic reflections from their icosahedral

positions

in different Fibonacci

crystals : (A)

for the

_{twofold { 2}

_Fi,

0,

_{0 }}

reflections and

_(B)

for the

_{pseudo-twofold {Fi + h}

_{F i, F i -}

_i}

reflections. Theoretical values are at the vertices of broken lines :

_{(34/21) -}

dash lines and

13 /8 > -

solid lines.

_Experimental

data : circles

[5] -

Mn

_implantation

into

_{single-crystal}

_{Al ;} squares

[4]

-

AlnMn22Si6;

rhombi - from

figure

5 of reference

_{[7] - A186Mn14’}

Indices of reflections see at

_figure

2. For

_(5/3)

_{approximant (a -Al-Mn-Si)}

all deviations

_(not-shown)

are about

T 4 times

greater than for

(13/8)

approximant.

For

_(34/21)

_approximant,

the

_powder

diffraction

_pattern

is

_{practically indistinguishable}

from

_quasicrystal

one

_(see

_{Fig. 2). Fortunately,}

in reference

_[5]

the _very accurate

measure-ments of reflection

_positions

were made for an orientated

_sample

obtained

_by

Mn

implantation

into Al

_monocrystal.

The measured deviations of reflections from their

icosahedral

_{positions (shown}

_by

circles at

_{Fig. 7)}

are in well accordance with the theoretical

More obvious

_(but

_may be less

_accurate)

evidences for the Fibonacci

_crystals

can be found

at electron diffraction

_patterns.

_Although

some deviations from icosahedral

_symmetry

were

observed in many

works,

for the decisive conclusion about their nature

_convergent

beam electron diffraction

_[7,

_8]

is

_preferable.

The size of the area should be less than

_(or,

at

_least,

of the order

of)

the coherent

_length

_{f coh}

of the

_positional

_{order ;}

in AIMnSi

_{quasicrystal alloys}

that

_length

is about 300-500

 (it

can be estimated from the intrinsic width of _x-ray

_peaks).

The shifts of reflections at the fivefold

pattern

from reference

_[7]

are close to those

_expected

for the

_(13/8)

_{approximant (see}

discussion in Ref.

[15]);

as an

_example,

the shifts of two

reflections are shown

_by

rhombi at

_figure

7. The deviations observed in electron diffraction

patterns

from

_Al78Cr17Rus

_{alloy [19]}

can be also attributed

_{to (13/8)}

_{approximant : figure}

3

Fig.

8. - The

superposition

of the

_{pseudo-fivefold}

_patterns from different domains with coherent

orientations ; (a), (b)

and

(c) -

for the same

_approximant

with different orientations of twofold axes :

_(a)

- 360 and

(b) -

72° between twofold axes in different

domains ; (c) -

five different orientations, i.e. the

polydomain

diffraction _{area ;}

_{(d) -}

for the

_{F n + 1/ F n >}

_approximant

in one domain and the

F n + 3/ F n + 2>

approximant

in another with the same direction of the twofold axes. Coordinate axes are

labelled for the

_{(Fn + 1/ F n >}

_approximant

with the twofold axis

_along

x-direction. Note that the spots are

from reference

_[19]

is very close to the innermost

_part

of our

_figure

3

_{for (13/8)}

approximant.

Those

_pentagons

with the

_{right angles,}

marked at

_figure

_3,

can be seen at

figure

11.2 of reference

_[26].

On the other

_hand,

the

_convergent

beam electron diffraction

pattern

of an

_Al74Mn2oSi6

_{alloy (Fig.}

2 of Ref.

_[8])

seems to be close to the

_pattern

_expected

for the

_{(34 /21 )}

_approximant.

Note

_also,

that the absence of some weak

_spots,

_violating

the

symmetry

of

_patterns

_[8],

_{may be caused}

_by

the

_off-plane

deviations of

_reciprocal

lattice

vectors in cubic

_{approximants (those}

deviations are _stronger for weak

_spots).

As a

_result,

some weak reflections can _appear or

_disappear

when the direction of incident electron beam is

slightly changed.

The

_qualitative

_analysis

of electron diffraction

_patterns

is rather difficult

because,

even on

the best

_patterns,

the

_shape

of diffraction

_spots

is distorted too : the

_spots

are

_elongated

in one or in different directions and the faintest

_spots

show the

_greatest

distortion. Such

_picture

can

be

_explained

if we

_adopt

different orientation of the

_symmetry

axes of the Fibonacci

_crystals

in different domains

_(the

size of the domains is about

_icoh)-

It is

_interesting

that the small value of

_ecoh

in

_{quasicrystalline alloys}

can be also

_explained

within this model.

_Indeed,

let us

suppose that there is the

phase

transition from

_quasicrystal

to its

_approximant

_(the

transition of this sort seems to be observed in Ref.

_[27]).

If the transition is

_{rapid enough,}

then the domains with lower

_(cubic)

_symmetry

can _grow

independently

and their

_symmetry

axes _may

be orientated

_along

different axes of

quasicrystal

matrix. For

example,

some twofold axis of

quasicrystal

can transform either into twofold or into

_{pseudo-twofold}

axis of the Fibonacci

crystal ;

the same is true _{for any threefold} axis. Such coherent orientation of axes and

pseudo-axes is well known for Al-Mn

_alloys.

There is no need of visible

_{sharp boundary}

between different domains of

_{high-order approximants,}

because their local structures are similar and

the differences between them are accumulated at the distance of the order of

_fcoh-

In this

model,

the natural estimation for

_ecoh

is

_{fcoh -}

2

_{’TT /}

_{1 dQnm 1,}

where

_àQnm

is the difference between the basic vectors in cubic and in icosahedral lattices

_(see

_Eq.

_{(7)) ;}

_at

that distance the mismatch between

_quasicrystal

matrix and its cubic

_approximant

riches about one

interatomic

_spacing.

The

_{superposition}

of diffraction

_patterns

of

_differently

orientated domains are shown at

figure

8a, b ;

the close

_pairs

of reflections look like

_elongated

_spots.

If the electron beam size is much more than

fcoh,

then all

_possible

orientations of cubic axes with

_respect

to those of

Fig.

9. - Threefold

quasicrystal

are

_presented

in the diffraction area and the calculated diffraction _patterns for such

_polydomain

structure are shown at

_figures

8c and 9. Note _strange «

_{triangular »}

form of the

_spots

which is

_frequently

observed in Al-Mn and Al-Li-Cu

_alloys

_[1,

_19,

_{26] ;}

the centers

of the

_spots

are almost

_exactly

at

_{correspondent}

icosahedral

_positions.

Another

_type

of the

pattern

(with

spots

elongated

in the same

_direction)

may be a result of the same orientation of

axes in domains of the Fibonacci

_crystals

of different order

_(Fig.

_8d).

7. Conclusion.

We have

_presented

above the

arguments

for existence of

_high

order Fibonacci

_approximants

in Al-Mn-Si

_{alloys (for}

other

_alloys

and for

_{low-symmetry approximants}

see Ref.

_[28]).

This

problem

has

_given

rise to _{many controversies} and in conclusion we shall

_try

to relieve them. The most

_popular

_argument

_{against crystal approximants}

with a =

30 Â

is that these

approximants

cannot

_explain

the

_low-Q

reflections with the

_d-spacings

of

_8.85(7)

and

5.42(3) Â

[29].

However,

this

_argument

seems to be

_unacceptable

because in the Fibonaci

crystals

the ratio of these

_d-spacings

should be close to

[(Ff+l +Ff+Ff-I)/(Ff+F’f-1 + Fl-2)]o.s

rather than to

_{Fi /Fi _}

₁

_(see

_{Fig. 2).}

_Thus,

within the

_experimental

errors, the

d-spacings

of

8.85(7)

and

5.42(3) Â

can be attributed to

pseudo-twofold almost-parallel

reflections (321 ) and {532}

in

_{the (13/8)}

Fibonacci

_crystal

with a =

_{33 À ;}

_therefore,

this

_approximant

cannot be ruled out.

_Moreover,

as it was

emphasized

above,

the average

position

of the

_{partially splitted}

cubic reflections is very close

to

_{corresponding}

icosahedral

_position

even for

_low-Q

reflections

(see Figs.

_2,

8c and

_9).

In

any case, more careful

_fitting

of reflections is desirable.

Now we must

_emphasize

that there is no contradiction between the

_{phason-strain}

description

of the distortions of diffraction

_patterns

_{[19, 27]}

and our

_description.

_Indeed,

the

Fibonacci

_{crystals correspond}

to some fixed values of

_phason

deformation

_{[14, 16-18,}

_{30] ;}

those values are fixed

_owing

to the commensurate character of

_crystal

structures.

_Speaking

in the

_{phason-strain language,}

we should _say

_accurately

that the observed shifts of reflections can be caused

by

those

anisotropic phason

strains,

which

correspond (at

least,

approximately)

to (13/8)

Fibonacci

_{crystal (in}

many cases

_{[4, 7, 19])}

or

_{to ( 34 /21 )}

Fibonacci

_{crystal (in}

the

best case

[5]).

The next

_controversy

is connected with the

_Pauling

model

[31]

(twinned

cubic

crystal

with

a - 23.36

Â).

It should be noted that all

reflections,

which were indexed

_{by Pauling}

at

_low-Q

region

(where

the differences between

_crystal

and

_quasicrystal

reflections are more

pronounced),

can be

_easily

indexed

_{in (13/8>}

Fibonacci

_crystal

because the ratio of lattice

constants

_{(33.1 /23.36)}

is _very close to

J2

_(hence,

the ratios of

h2 +

k2

+ £2

are very close to

2).

Moreover,

those

_reflections,

which can not be indexed in the

_Pauling

model

_{[31 ],}

may be

indexed in

_(13/8)

_approximant;

for

_example,

reflection in

_A174Mn2lSi5

with

_d-spacing

of

8.98 Â

_(or

_{8.85(7) Â}

_[29])

can be indexed as

_{321}.

Some theoretical

_arguments

_{against crystal approximants}

are

_given

in reference

_{[32] ;}

it is

shown that the

_{gain owing}

to lock-in terms _{in free energy is smaller} than the loss

_arising

from the

_gradient

term

_(at

_least,

for

_{high-order approximants).}

_However,

within the model used in

reference

_[32],

the Fibonacci

_{crystals give}

no loss in

_gradient

term, because the

_lengths

of

wave-vectors are

_equal

for all basic harmonics

_{(compare Eq. (3)}

and

Eq.

(4)).

Moreover,

even more

_complex

model is unstable with

_respect

to variations in the

lengths

of icosahedral

wave vectors

_[33]

and the

_phason

strains may be favorable.

Indeed,

most _{of harmonics may be}

shifted toward lower

_gradient

_{energy when}

_symmetry

is broken to cubic. The mode

_locking

can fix those

_phason

strains on the values

corresponding

to the Fibonacci

_crystal

of some

order

_(more

detail discussion see in Ref.

_[34]).

The fixed

sign

of the

phason

strains can

intermediate

_approximants

seem to be absent. It is

_{unclear, however,}

whether

_All2Mn

structure

_{(Im 3}

_{space group, a} = 7.47

Á)

can be

_regarded

as n = 3 member of this series of

approximants, becausein (3/2)

approximant

the

_{reflections {320} or (230)}

_{are expected to}

be

strong

but those reflections are forbidden in

Im3

structure.

In the

_end,

it should be noted that the Fibonacci

_crystals

are not

_unique

and other

_types

of cubic

_approximants

can exist. As a curious

_(but

not

_impossible)

_example

we can consider the

(Ln + 1/ Ln)

Lucas

approximants,

where the Lucas numbers

_Li

are connected with the

Fibonacci numbers :

_{Li = Fi +}

_{1 +}

_{Fi -}

1

(jL,

=

1, 3, 4, 7, 11, ...,

i =

1, 2, 3,

4, ...).

The dif-fraction

patterns

of the Lucas

_approximants

are similar to the

_patterns

of the Fibonacci

approximants (zigzag

shifts of

reflections,

asymmetric broadening, etc.),

whereas the values of all those effects are

_{different ;}

the consideration of the Lucas

_approximants

will be

given

elsewhere.

Note added

during

revision : .’

In the recent work

_[35],

Prof. L.

_Pauling

also considers the Fibonacci

approximants

instead of his twinned model. He

_interprets

the

_experimental

data of reference

[5]

as (13/8)

approximant,

whereas it seems evident from

_figure

7

_{that (34/21)}

_{approximant gives}

better

fitting

of the data. In reference

_[36],

the series of cubic and non-cubic

_approximants

are

constructed

_using

the canonical-cell

_{tiling (the Fn +}

_{1 IF,}

approximant

of Ref.

_[36]

_corresponds

to

_{the (Fn + 4/ Fn + 3)}

_approximant

of the

present

paper).

Note also reference

_[37],

where the

computer

simulations of x-ray diffraction

patterns

have been

_presented

for the Fibonacci cubic

_approximants

of different order.

_{Many approximants}

other

than just

cubic

_approximants

may occur both in Al-Mn-Si

[1, 28, 38]

and in other

alloys (see

Refs.

[28,

39,

40]

for Al-Li-Cu and Ref.

_[41]

for

_Al-Cu-Fe).

Such

_{crystal-quasicrystal polymorphism}

seems to be a common

feature of different

_alloys

and further studies are needed to

_clarify

its

_origins.

Acknowledgments.

The author is

_grateful

to Drs. V. M.

_Kaganer,

E. B.

_Loginov,

E. M.

_Terentjev

and S. B.

Tochilin for the aid in

_computer

calculations and

_graphics.

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