Electron microscopy diffraction contrast in quasicrystals: some rules

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Electron microscopy diffraction contrast in quasicrystals:

some rules

Maurice Kléman, Wolfgang Staiger, Dapeng Yu

To cite this version:

Maurice Kléman, Wolfgang Staiger, Dapeng Yu. Electron microscopy diffraction contrast in quasicrys- tals: some rules. Journal de Physique I, EDP Sciences, 1992, 2 (5), pp.537-543. �10.1051/jp1:1992165�.

�jpa-00246517�

Classification

Physics

Abstracts

61.42 61.70J

Short Communication

Electron ndcroscopy diffraction contrast in quasicrystals:

_some

rules

Maurice

K14man, Wolfgang Staiger

and

Dapeng

Yu

Universit4 de

Paris-Sud,

Centre

d'orsay,

Laboratoire de

Physique

des

Solides,

B£timent s10, 9140s

Orsay

Cedex, France

(Received

10 October 1991, revised 27

January1992, accepted

19

February 1992)

Abstract A number of authors have observed dislocations in icosahedral

quasicrystals

and advanced

some rules to the effect of

measuring

their

Burgers

_{vectors, on} the basis of the two-beam method in the kinematical

approximation.

We first

shortly

review their

results,

which _state that the determination of the direction of the

Burgers

vector b

=

bjj+b

_i,

_(bjj

in ll~jj, ^bi in lI~

i),

which has six components, can be

suitably simplified

if the five necessary extinctions fall into two classes, ^viz. four so-called

'strong

extinction conditions'

(SEC) G( _bjj

₌ G

[

b i=o, and one so-called'weak extinction condition'

(WEC) _G(j bjj+G[ bi

₌ 0. We then state the

relationships

between the

bjj

^{and the b}i components which result from these conditions.

We show that the modulus and

sign

_can be determined without

ambiguity by examining

two incommensurate

non-extinguishing

diffractions

(G"

b

= n

#

o, _n

integer).

Any

'perfect'

dislocation

obeys

at least

approximately

the rules which follow from this

an4lysis.

Dislocations which fail _to

obey

them _can be

'imperfect'

dislocations

(we

define

them),

whose existence should not be discarded.

1 Introduction.

An

important question concerning quasi-periodic crystals

with icosahedral

symmetry,

^since their

discovery

in Al-Mn

alloy [I]

is how far and

why

do

they

differ from the

perfect quasicrys-

talline order. The nature of the

imperfections,

their

origin

and their role in

plastic

deformation

are far from

being

understood. Dislocations and

phason

defects

(a special type

of defects in

aperiodic crystals)

have been

recognized

_as

specific

defects

by

Socolar et al. [2] on the basis of

a

density

wave

description

or a unit cell

picture.

Dislocations have been first classified

by

the

topological

method

[3, 4].

^A

generalized

Volterra _process has also been

employed _{[5, 6],} starting

from _a dislocation built in _a

hypercubic crystal ^lZ~ belonging

to _a 6-dimensional _space

IE~,

and

projecting

it in the

physical

_space

IPjj.

^{It is this method} we use in the

sequel:

the

Burgers

538 JOURNAL DE

PHYSIQUE

I N°5

vector in

IE~,

viz. b

=

bjj

+

bi,

has

components

in both

physical _(bjj

e

IPjj)

^and

complemen-

tary

(bi

e

IPi)

_spaces, ^hence 6

parameters

are needed to index it. As in usual

crystalline

materials,

the

bjj ^components

are a measure of

plastic deformation;

the

bi components

are

a measure of the

matching

rule violations

(the phason defects)

which

necessarily

_accompany

the dislocation line.

Experimentally,

dislocations have been observed in HREM

images,

either

directly

_as in icosahedral

quasicrystals

of Al-Mn-Si

by Hiraga

et al.

[7],

or

by

_means of

image processing

_as in icosahedral Al-Mn

quasicrystal (Wang

et al.

[8]).

Dislocations in icosahedral

Al-Cu-Fe have been observed in conventional electron

microscopy

and

analyzed by

Devaud-

Rzepski

et al. [9] and

Zhang

et al.

[lo]

the component ^{of the}

Burgers

vector in

physical

_space

was found _to be

parallel

to a 2-fold direction. The contrast

theory

of _a

quasicrystal

dislocation with electron

microscopy

has

already

been

approached [9, 11, 12].

Dislocationlike

images

and

grain

boundaries have also been observed

by

Chen et al.

[13]

^{and Yu et al.}

[14]

^{in Al-Li-Cu}

alloys.

In _a usual

crystal,

in the kinematical twc-beam mode of

observation,

_a dislocation of

Burgers

vector b

gets practically

out of contrast for diffraction vectors G" which

obey

the

relationship

G° b

= 0

[15]. Geometrically,

this

relationship

_means that the relevant

diffracting

vectors all

belong

to the _same

plane and,

_{as a} matter of

fact,

all the

diffracting

vectors

belonging

to this

plane

are relevant. In

fact,

extinction is

strictly

observed when the dislocation is

parallel

to the film

surface,

is of _screw character and the

elasticity

is

isotropic [16].

^Small

departures

to these conditions

yield

_a residual contrast which in _{many cases} is _not

perceived (in particular

mixed dislocations

parallel

to the

film).

As _a consequence the

Burgers

vector _can be

unambiguously

determined in direction

(but

not in

length),

when two extinctions with

linearly independent

diffraction vectors G° and

GP

are known. The

length

and

sign

themselves _can be determined

by considering

^other

reflections,

of the

type

^{G° b} _{= n,} ^where n

~

0 is _an

integer.

The condition of

isotropic elasticity

is in

principle

satisfied in

quasicrystals,

whose icosahe- dral

symmetry

is

larger

than the

symmetry

of cubic

crystals,

and it has been shown in other

respects [11, 12]

^{that the}

generalization

of the condition of extinction G.b

= 0 to _a

quasicrystal

is

simply:

Gjj bjj

⁺

Gi bi

₌ 0.

(1)

Therefore _one needs five

independent

extinctions in order _to

get

^the direction in

E~

_{of the}

Burgers

vector. Here

Gjj

^and

^Gi

are

projections

in

IPjj

^and

IPi

of the total diffraction vector

G =

Gjj

+ G

i, with

Gjj ^Gi

= 0.

Only Gjj

^is

experimentally known,

but because of the

incommensurability, Gi

_can be deduced

unambiguously

from the

knowledge

of

Gjj.

^{Since six}

vectors cannot be

independent,

there is _{no way} to obtain the modulus and the

sign

of the

Burgers

vector

by

the

only

_use of

extinguishing

diffractions. We revisit the

geometry

^{of the}

extinguishing

diffraction vectors in the line of reference

[12],

^{and then turn} our attention to the diffraction _vectors

Gjj bjj

⁺

Gi bi

_{= n}

~ 0, showing

that

they

_can be used to _measure the modulus and the

sign

of the

Burgers vector,

_as in the usual 3D _case.

Finally

_we

analyse

what

happens

when the extinctions do not

obey

the rules which _are established.

2,

strong

and weak extinction conditions.

This section

presents

^{the results of reference}

[12]

in _a

slightly

different

geometrical

_manner, and

brings

_some new results to the

question

of the determination of the direction of the

Burgers

vector of _a

dislocation,

in

particular equations (5, 6, 11, 12)

and

(13)

and the final comments.

An extinction G° is of the

strong type (SEC)

if it

obeys

the two conditions:

G( .bjj

₌ 0

(2)

G

[ bi

= 0.

(3)

If it h _so, it _can be

proved

that all the diffractions

lG(

^{in the} same row

yield

extinction. I is either _an

integer (in

this _case the extinction

AG(

^{is of}

^evidence)

^{or it is an irrational of the}

type (p

+

Tq)

^where _p ^and _q are

integers

^and T =

(1+ v$)/2

is the

golden

ratio

(~). Assunfing

now that these SEC _are true for _any diffraction vector

belonging

to two different _rows, then

bjj

^{is known in direction}

bll

" fill

(~i

_~

_~~

_~~~~

Furthermore,

since

G( ^(resp. G()

^{is the}

^projection

_{in IPjj of} a lattice _vector

G( (resp.

G

(),

its lift G

[ (resp. G()

_in

_IPi

_is determined. Hence

bi

is also known in direction:

bi

₌

Pi (G ^[

A

G( _{)1 (4b)}

fljj and

pi

_are two constants which _are

related,

since

bjj

^and

bi

_are

components

of the _same

vector b

(see below).

Since the direction which carries

bjj (resp. bi)

is the

projection

in

IPjj

(resp. IPi)

of _a 4-dimensional

plane

in

IE~,

b

belongs

to the intersection of two 4-dimensional

planes

in

IE~,

I-e- to _a 2-dimensional

plane.

Any exiting

diffraction vector of the _set

Gjj

e

(woG(

+

wpG(

w~ = pa + Tqo, wp =

pp + Tqp; p~, q~, pp, qp,

integers)

would also

yield

_an extinction of the _same dislocation with

Burgers

vector

b;

_we call

Ifl(P (resp.

Ill

[P)

the

2-plane

in IPjj

(resp.

in

IPi)

which contains

Gjj (resp. Gi). _bjj (resp. bi)

is

perpendicular

to the

plane D(P (resp.

^D

[P).

The

explicit

relation between fljj and

pi

is obtained _as follows. Let _us

introduce,

after Cahn et al.

[17],

the notation b

=

q(h/h', k/k', I/I')

for _a 6-dimensional vector b in

lZ~, meaning

that the

components

of b in IPjj ^are

bjj

⁼ ~ ^< h +

h'T,

k +

k'T,

I + l'T _> and its

components

ⁱⁿ

IPi

are

bi

₌

11 < h'-

hT,k'- kT,

I'- lT >, ^these

components being respectively along

a set of three

orthogonal

^2-fold axes e(j

(I

₌

1, 2, 3)

^{of the} icosahedral

phase

in IPjj ^and

along

the lifts

e[

in

IPi

Here ~~ =

l/2(2

₊

_T)

and the

hyperlattice parameter

^is taken

equal

to

unity (a

vector G in

reciprocal

_space writes

similarly

G

=

~~~ (t/t', u/u', v/v');

Gjj

=

~~~

< t +

t'T,

_u +

u'T,

_v + v'T >;

etc.). Letting

fljj ⁼ p + Tq,

pi

= T + _Ts,

(5)

where p, q, r, and _{s are}

integers (we drop

here for convenience the

prefactors

_{~ or}

~~~),

and

writing

that

bjj

^and

bi

_are

components

^{of the} same vector in

lZ~,

_we

get:

r " P +

2q,

s =

-(p

+

q). (6)

(~ ^A diffraction vector G

= Zn;ei in 6-dimensional

reciprocal

_space ^IE~ defines _{a row} of diffraction

vectors

AG°,

where is _a rational. If G° is the diffraction _vector of this set such that the

n;'s

_are

relative

primes,

all the set obtains with

integer

A's. A _row of

diffracting

vectors in

ll~jj obtains _as the

Projection

of _at least two _rows in

IE~;

G° and

G°'

_differ

_by

a factor of _r.

540 JOURNAL DE

PHYSIQUE

I N°5

For

example,

with

fljj ⁼

I,

_we have

pi

_"

^-T~~

More

generally,

with fljj ^{= T" =}

In-

i +

Tin,

(the

Fibonacci numbers _are:

lo

₌

0, Ii

_"

1, f2

₌

1, f3

₌

2, etc..., f-n

₌

(-1)"~~ In, In

₌

In-i

+

fn-2),

_we

get Pi

_"

(-I)"~~T~"~l,

for _n

integer positive

_or

negative,

viz.

fli/fly(

_"

(_i)n-l~-2n+1

can be deduced

from

a

_complete

set

of

SEC.

p

_and q

are

arbitrary ntegers. The

one "weak extinction contrast"

diffraction

_vector G~ =

G(j

+

G[ obeying

the relation:

'~~ _~ ^~

~(l '~ll

"

-T~~ Gi bi (G~

₌

~~~ (ii /t(,

_vi

/u[,

_vi

/v();

b

=

~(h/h', k/k', I/I) ^m"

₌

t(h

+

h'ti

₊

t[h'+ u(k +k'u;

₊

u(k'

+

vii

+

l'vi

+

v(I')

lifts

partially

this arbitrariness.

Equation (7) _expresser

the fact that all the

G(j's (resp.

^{all the}

Gl's)

which

obey

WEC rules

can be

partitioned

into sets which have the _same

projection ^m'

_on

_bjj

and _on

bi Therefore,

if

one knows several _rows in

IPjj (and consequently

_en

IPi) along

which WEC rules _are

satisfied,

there is in IPjj

(resp.

in

IPi)

_a

family

of

parallel 2-planes

_lTjj

(resp. Ifli)

which cut those _rows at the extremities of

diffracting

vectors

G(j's (resp. Gl's). _bjj (resp. bi)

is

perpendicular

to this

family

of

planes

Ifljj

(resp. Ifli).

See

figure

2 in reference

[12].

This result is _no

surprise,

because _we

already

know that the

2-plane Ifl(P

ⁱⁿ IPjj

(resp.

the

2-plane Ifl[P

ⁱⁿ

IPi)

is

perpendicular

to

bjj (resp. bi).

Therefore the

family

of

planes

_Ifljj

(resp.

Ifli)

is

parallel

to

Ifl(P (resp.

Ill

[P).

We have anyway

got something

_new

by finding

one WEC diffraction vector

G(j (it

is

enough

to know

only one):

the

integers

_p and _q introduced above

can be

expressed

but to _some

integer

constant.

Let _us introduce the notation G° ₌

~~~(t~/t[, uo/u[, va/v[)

and

GP

=

~~~(tp/tj,

up

/uj, vp/vj),

we

have, according

to

equations (4)

and

(6):

~~~bjj

=

pA

+

qC

+

T(qA

₊

(p

+

q)C) (8)

~~~bi

=

qA

+

(p

+

q)C T(pA

₊

qC) (9)

where A and C _are two 3-vectors whose

components

are written:

AI

_{" uovp upva} +

u[vj ujv[ A2

₌

A3

₌

Ci

₌

u[vp viva

+

u«vj upv[

+

u[vj ujv[ C2

₌

C3

=

Introducing similarly

the vector notation

t,

₌

(t;,

_u;,

v;)

,

t(

₌

(t(, u(, _vi

,

we write the scalar

product:

G(j bjj

⁺

G[ _bi

=

(2

+

T) (t, (pA

+

qC)

+

t( (qA

+

(p

+

q)C))

= 0

(10)

We have therefore

p =

ii (-(t,

+

t()

C

t( A) (11)

q =

ii (t(

_{C +} t;

A) (12)

where

ii

is _some

integer.

But note that

i;G~

is also _a vector

satisfying

_a WEC rule.

Renaming it,

_we

get,

after _{some easy}

algebra:

~~~~~' '~"

~

~~~~~~ '~~

₌ ~~ P~ Pq

~i~~

If q~

p~

_pq takes the smallest

possible

_non-2ero

values,

^I-e- q~

p~

_{pq =}

(-1)"+~ (n arbitrary integer),

_we have _p

=

In-ii

_q

=

in.

Therefore _p and _{q are}

proportional

to two successive Fibonacci numbers.

Note that all the vectors

bjj (resp. bi)

which

satisfy

the _same SEC and WEC _are

necessarily

commensurate.

If,

for

example, _bjj

is _a

possible choice,

so that

according

to

equation (7)

_we

have

T~~G(j bjj

=

m',

_{it cannot} be _so for

b(j

⁼

(p'+ _Tq')bji,

because

T~~G(j

_b~j #

lll~(/

₊

Tg')

_#

lll~p'

+

Tlll~q' (14)

does _not

obey equation (7).

In

fact, _G(j

would not

extinguish

a dislocation of

Burgers

vector

~~l'

3. Other dilEraction vectors: modulus and

sign

of the dislocations.

Let G ₌

~~~(t It', u/u',

_v

Iv')

be _a diffraction _vector which does not

extinguish

the dislocation b =

~(h/h', k/k', I/I').

We have:

Gll '~ll "In

+

(n ill)Tl

Gi bi

=

In

+

mT)

G b

= n.

(15)

where _n and _{m are}

integers

which

read,

in the notation _we have

already

used:

n=t.h+t"h', m=t.h-(t.h'+t"h) (16)

In

principle,

_{as we} shall

show,

the value and

sign

of _{n can} be determined

experimentally by

the _same method _as in _a usual

crystal.

The

important

issue is to determine

Gjj bjj

^and

Gi bi separately.

Let _us show that the consideration of another diffraction vector

G',

whose

component

in IPjj ^is

parallel

to G and whose~modulus

G(j)

is incommensurate with

(Gjj(

^is

sufficient to measure m, from which the two scalar

products Gjj bjj

^and

^{Gi bi}

^of

equation (15)

can be deduced. Let indeed

G(j

^{= (tY +}

flT) Gjj

^{be another} diffraction _vector.

G(j

^suffices to determine

G[

and _we have indeed:

G[

=

(a'

₊

fl'T)Gi (17)

where tY' _{= tY} +

fl

and

fl'

₌

-fl

_are

integers.

It is _{now easy} to show that the values of n' and m' relative to G'=

G(j

⁺

G[,

v12.:

read _now:

m'

= am

fin, n'

= an +

fl(n m). (19)

Since _tY and

fl

_are

known,

the value of

n',

which is measured _as the rank of _an extinction

contour

[16], yields

_m. Observe that _{m can}

only

be reached in _an observation in which

fl ~

0.

542 JOURNAL DE

PHYSIQUE

I N°5

When

fl

=

0,

^G'is

parallel

to

G,

^while it is not when

fl ~ 0, although _G(j

and

G[

_are

respectively parallel

to

Gjj

^and

Gi (incommensurability

of G and

G').

The

integral

^values _n,

n', etc.,

^relative to the vectors

G, G', etc.,

are determined _exper-

imentally by

the _same method _as in usual

crystals,

which consists in the observation of the

extinction contours at their intersection with

dislocations,

arid the _measure of the

displacement

of the dislocation

image

when the diffraction _vector varies. This

displacement

is

directly

_re- lated to n. In

effect,

_as shown in _many instances

(see

for

example [12]),

the

phase

which _enters the

amplitude

^diffracted in _a dark field

experiment

with _a diffraction vector

Gjj

^{does include}

the total

phase

shift 2xG b

= 2xn due to the 6D

dislocation,

and not the

partial phase

shift

Gjj bjj

^{due to the}

diffracting

vector in

physical

_space. This remarkable

property

enables _us to extend to

quasicrystals,

mutatis

mutandis,

the whole

theory

of contrast of usual

crystals,

with this

unique (but fundamental)

difference that _now each relevant diffraction G has to be

supplemented by

_a diffraction G'

parallel

_to G and incommensurate with it.

4. Other extinction conditions?

Coming

back to the _use of

extinctions,

^let _{us now}

investigate

the

possiblity

of

having only

one row of diffraction vectors G"

=

G(+G ^[

^which

obey

a SEC rule. What about the other

extinguishing

diffractions

G~?

_{We have}

bjj

=

G(

A

ujj bi

_" G

[

A _vi where

ujj (resp. vi)

is some unknown 3-dimensional vector in

Vii (resp. IPi)

which has to be determined

by

the other relevant diffractions

G~

which

yield

relations of the

type G(j ^bjj

⁺

G[ b1

₌ 0. Two

cases arise:

a)

in the most usual case, one

expects

b to be _a lattice vector of the

hypercubic

lattice

(a

sc-called

perfect dislocation);

then

necessarily _ujj (resp. vi)

is _a lattice vector

too,

and _ui

(resp.

_vii is

unambiguously

defined.

Consequently

_one

has, by

_a calculation of the _same

type

as

above,

~" ^~

~~~~i

~

~"' ~i

₌

-~~~Gl

A Ui

(20)

These

expressions yield _ujj

₌ T~

vii

^and vi =

-T~~

ui

(we neglect

all

components parallel

to

G(

^and G

[).

The _{vectors u} and _v

(which incidentally

_are not

parallel

in

IE~) belong

to the

same row of

possible

lattice

reciprocal vectors,

but to a different _row than G°. Therefore _we

are in the _same situation _as above and the SEC rule is

obeyed by

_a set of diffraction vectors

Gjj

⁺

(w«G(

⁺

wujj

w~ = p~ + Tqa, w ₌ p + Tq; p~, qa, p, q

rationals) belonging

to the

same

plane.

b)

if b is not a lattice _vector in

IE~,

^then the SEC rule does not

apply

to all the diffraction vectors.

Reciprocally,

if the SEC rule does

apply

to

only

_one diffraction _row

(and

not

two),

the

corresponding Burgers

vector is not _a lattice vector in

lE~.

This is worth _{some more}

comments;

we restrict _our discussion to a few. A first

possibility

in

E~

_{is of the usual}

_type,

the

corresponding

dislocation

being

what is

commonly

called _a

partial dislocation,

for

example

if the

hypercubic

lattice is not

primitive (it

has been shown for

example

that the I-Al-Fe-Cu

phase

is face-centered

[9]);

another

possibility,

which would be characteristic of _a

quasicrystal

and cannot be

excluded, especially

if the

"phason" part bi

and the

"phonon" part bjj

^of the dislocation have

quite

different

energies,

_or if the _energy of interaction between these two

parts

is

excessively large,

^{would be} to have _a total

Burgers

vector where the

"phonon" part bjj

^{and the}

"phason" part bi

are not

topologically

related

[18].

^{We shall call such} an

object

an

imperfect dislocation, reserving

the term of

partial

dislocation in IE~ _to the usual

type

of

partial.

But _even in such _{a case can} the

SEC

conditions be

obeyed by

diffraction _vectors

lying

in _a

plane,

for

example

if

bi

₊ 0.

Pure

WEC,

I-e- without _any SEC

diffracting vector,

^is the

signature

of _a dislocation with _no

topological relationship

between

bjj

^and

^bi

^{This is the}

general

_case of _an

imperfect dislocation,

for _any

perfect dislocation,

whose

Burgers

vector in IPjj is of ^the

type bjj

=

AG(

^A

G(,

^should

produce

SEC. But note that the

analysis leading

to

equations (11)

and

(12)

is _no

longer valid, and, consequently,

the ends of the WEC diffraction vectors in IPjj ^{cannot be}

partitioned

in _a set of

2-planes perpendicular

to

bjj.

Acknowledgements.

We wish _to thank Proh

Gratias,

Urban and

Wollgarten,

and Dr.

Zhang,

for

letting

_us have

a

preprint

of their work

(Ref. [12]) prior

to

publication,

and _one of the referees for

inviting

_us

to

investigate

the non-extinction _case.

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