Diffraction in crystals and topology

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Diffraction in crystals and topology

V. Dmitrienko

To cite this version:

V. Dmitrienko. Diffraction in crystals and topology. Journal de Physique I, EDP Sciences, 1991, 1

(8), pp.1187-1193. �10.1051/jp1:1991199�. �jpa-00246404�

Classificafion

Physics

Abstracts

02.40 61.10D 61.70

Diffraction in crystals and topology

V. E. Dmitrienko

Insfitute of

Crystaflography, Leninsky

_pr. 59, Moscow, 117333 U-S-S-R-

(Received 20 March 1991, revihed 10

April

1991,

accepted

25

April

1991)

Abstract. It is shown that _some

simple

ideas of

topology

_can be

applied

to the diffraction

theory.

As an

example,

the

phase

of the wave transmitted

through

the

crystal

is considered for the

Laue diffraction geometry. ^{Vfhen the}

angle

of incidence varies across the diffraction

region,

the

phase

of the transmitted _{wave can} be

changed

on 2 wN, where N is _an

integer.

The number N is _a

topological

invariant

(topological charge)

of the diffraction

region

_: in the

general

_case, this number does not

change

its value when any parameters ^of

experiment

_are

slightly

varied, but it

changes by unity

^for some fixed values of those parameters. Therefore, from the

topological point

of view, the

regions

of diffraction _are

analogous

to the domain walls in ferroelectrics. Then, within this

approach,

the

singular points

are revealed inside the diffraction

region.

It is shown that the

topological

ideas _are

especially

fruitful for those _cases where the solution of diffraction

problem

is absent _or formidable

(for

instance, the

multiple-beam

diffraction _or diffraction in

imperfect crystals).

The

topological charge

is also

impo~ant

for the

dispersion

relations

connecting

the

phase

and the

amplitude

of the transmitted _wave in the diffraction

region.

1. Introducdon.

It _seems that there is _no need to convince

anybody

of the usefulness of

topological

ideas in the

solid state

physics.

Most _successes of

topology

_are connected with the classification of defects in ordered media _; the details _may be found in the review articles

[1-4].

The aim of the present paper is _to extend these ideas _on the _x-ray diffraction in

crystals

_;

certainly,

the _most results

are valid for electron and neutron diffraction _as well. The

general properties

of the defects in

electromagnetic

fields

(dislocations

and

disclinations)

have been discussed earlier

(see [5-9]

and references

therein).

The fields considered in the present paper are so

simple

that

they

have _no defects at all. In the first part ^{of the} _paper

(Sects. 2, 3),

_a traditional test-case of

dynamical

diffraction

theory

is

analyzed

_: the Laue-diffraction in

perfect crystal

without

absorption. Then,

in the section

4,

_we ^discuss

briefly

^the

multiple-beam

diffraction and the diffraction in

imperfect crystals.

In the latter _case the

general topological

consideration is

especially

useful because of the absence of the overall

theory

of diffraction in

imperfect crystals.

Note that in the

following

_we will not prove ^most of the mathematical statements the reader _can find the

proofs

and discussions in the review articles

[1-4].

Because the

topological approach

_{was never} used before in the

theory

of diffraction in

crystals,

_we

begin

with _some

introductory

illustrations.

l188 JOURNAL DE

PHYSIQUE

li° 8

Within the diffraction

theory,

a

topological

invariant

(it

is also called _a

topological charge)

is connected with the

phase

of the _wave transmitted

through

the

crystal

in the direction of incident _wave. This

phase (we

denote it 4l

(ho )) depends

_on the

physical _parameters

of

crystal

and _on

ho,

the deviation from the

Bragg angle.

When ho

- ± oo, the diffraction is absent and it _seems natural to

put WI

_{± oo}

)

=

0

(we neglect

the irrelevant

phase

shift

owing

to average dielectric

constant).

In

fact,

the

phase

is defined until the

integer

number of 2

wand, considering

the most

general

_{case, we} should write

4l(+oo)=2wN~~

4l

(-

_oo

)

=

2

wN_~ (l)

where

N~

~

and N_

~ are the

integer

numbers

(so

that _exp

(I

2

wN~

~

)

=

l).

It is

important

that these numbers _are not

arbitrary: indeed,

ho may be

coflinuously changed

from

oo to _{+ oo}

through

the diffraction

region

and _{we can} fix the difference of the

phases

:

4l(+oo)-4l(-oo)m2wN (2)

where N

=

N~~ N_~. Only

this difference has

physical

sense and _may be measured _{; at} least in

principle,

it may be measured either

by

the _x-ray interferometers _or

using

the

polarization phenomena (the

review of _x-ray

polarization phenomena

is

given

in

[10]).

The

integer

number N is _a

topological

invariant _{: if any} conditions of diffraction _are

slightly changed,

the invariant remains

unchanged (in general case).

^This situation is illustrated

by figure

I where N is the number of _tums of _a

string

on a

pivot.

Because of this

analogy,

such

type

^of

topological

invariants is called the

winding

number _; the

sign

of N

depends

_on the direction of the _turns. If the ends of the

string

_are

fixed,

N remains

unchanged during

_any deformations of the

string (we

_{can say} that this

geometry

^is

topologically stable).

The

only

_way

to vary N is to cut the

string,

to

change

the number of tums and then to

glue

the

string again.

Other

physical systems

^with

analogous topological properties

can be found _among domain walls in ferroelectrics _or

ferromagnetics.

These assertions will _now be

justified

for _some

simple examples.

m+r

aw

m#

WV

Fig.

1. Illustration to a

topological

invariant N _: a string

turning

around a

pivot.

If the ends of the

string

_are

fixed,

a nuJnber of tums N cannot be

changed

without

cutting

the

string.

2. Laue-case diflracfion without

ahsorpfion.

In the _case of

perfect crystals,

the

expressions

for the

amplitudes

of diffracted and transmitted

waves can be found in many books and review _papers

[10-12].

To make the

problem

_more

transparent,

^let us consider the

symmetrical

Laue-case diffraction of _an incident

plane

wave in

a

crystal plate (see

insertion at

Fig. 2).

In the two-wave

approximation,

when

only

direct and diffracted _{waves are} taken into account, the solution of Maxwell

equations together

with

~----~---~---tlf?

i _L

#

~

5 I

t

N=0

A8/A80

Fig.

2.- The

phase

~P of the transmitted _{wave versus} the deviation from the

Bragg angle

ho for different thicknesses L of the

crystal: I)

A =1.0, 2) A =1.55, 3) ^A =1.6,

4)

A=4.7,

5)

A

= 4.75, 6) A

= 7.85

(A

= arL/L~X~). Diffraction geometry

(the

symmetrical Laue

case)

is shown at the insertion 2 arN levels are marked

by

the dashed fines.

relevant

boundary

conditions

gives

_us the

following expressions

for the reflection and transmission coefficients _r and _t

(for

the

amplitudes

of

waves)

_:

r ₌

xA( Yfi)~

_{sin YA}

₍₃₎

t ₌

(cos

YA

iyY~

sin YA

) exp(iyA) (4)

where Y~

=

y~

₊

_I,

_y is

proportional

to ho

= ~ and A is the normalized thickness L of

the

crystal plate

y = ho sin 2

~/(C fi)

W

A@/A@o (5)

A

= Lw

(ko(

^C

filcos

@~ m

wL/L~~~ (6)

In the

equations (3)-(6),

_xA and x_A are the Fourier-harmonics of the x-ray

susceptibility corresponding

to reflections h and -h ; C

= I for

«-polarization

of the x-ray wave and C

= cos 2

~ for

«-polarization

_;

ko

is the _wave vector of the incident beam _; ho

o is the

typical

width of the diffraction band

(usually

about few seconds of

arc)

and L~~~ is the extinction

length (about

10~^~

mm). Note,

that in

equation (4)

_we omit the irrelevant

phase

shift

owing

to the average dielectric constant, so that t

=

I if the diffraction is absent

(that

is if xh =

°).

Rewriting

t in the

exponential

form _t

=

exp(14l

_K

),

_{we can}

easily

obtain the

phase

4l of t

from

equation (4)

_:

4l

=

yA

arctg

(y

Y~ ^' tg ^YA

) (7)

(we

take into account that in

non~absorbing crystals

_x-

h =

xi

and therefore both _y and A _are

real).

The

dependence

of 4l _on ho

(or y)

is shown at

figure

2 for different A. It is

evident,

that both the difference

(2)

and the

corresponding topological

invariant N _are constant inside

some intervals of A : if

w

(n

< A

< w

(n

^{+ then N = n and N is}

^changed

^{from n}

2 2

1190 JOURNAL DE

PHYSIQUE1

li° 8

to n + I for A

=

A

~ ⁼

w

(n

⁺

),

^where n =

0, 1, 2, 3,

^Even a very small variation of A

near

A~

can

change

the

dependence

4l

(ho ) drastically (compare

the _curves 2 and

3,

4 and 5 at

Fig. 2).

If A

= A

~,

the

topological

invariant _cannot be defined in _a

unique

_way because in this

case t

= 0 for y =

0 and the

phase

oft is indefinite _; this is _an

example

of

singularity

which will be discussed later _on. To avoid

confusion,

it should be noted that the

arctg-function

has all its

values between

-w/2

and

w/2. Therefore, calculating

4l from

(7)

_{as a} function of

ho

(or y),

_we should add

integer

number of _w to restore _a continuous

dependence

of Won ho

(or y);

such

procedure

is

impossible only

in the

singular points

where A

=A~

^and

y = 0. Note also that it is convenient to put ^4l _~ 0 when _y

- oo because this

region

_can be

continuously

transformed to

large-wavelength region

A _- oo where _any diffraction is absent.

If A - oo, the ratio

4l/A

tends to the universal function _: '°

= + y +

sign (y) (i fi) _(~)

where function

sign (y)

denotes the

sign

of _y.

3.

Singular points.

Let _us consider _more

carefully

the

dependence

of 4l _on the variable diffraction

parameters.

In

principle,

we can vary both o and A _; the latter _can be varied

by

the rotation around the diffraction vector h _so that

A(q7)

=

A(0)/cos

_q~ where _q7 is the azimuthal

angle

of this rotation.

Figure

3 shows the

part

of the diffraction band in coordinates o and _q7; the

region

of strong diffraction is shaded. Let _us suppose that when _{we cross} the diffraction band

along

the line

ab,

the

topological

invariant N is

equal

to _n

(we

shall write

N(ab)

=

n)

and when _we go

along

cd N

(cd )

= n + I. For the

beginning,

_{we suppose} also that all the

points

_a,

b,

_c, and d

are very far from the diffraction

band,

_so that at these

points

the diffraction corrections to the

phase

4l _are

negligible

_{; as a}

result, N(ac)

=

N(bd)

= 0. If _we

path

in the counterclockwise direction

along

the closed circuit

(or

the

loop) acdba,

then the

integral phase

difference

a

c

p

or (arb.

units)

Fig. 3.

-

The _singular

point

inside

the

ha is

equal

to 2 _w and _we find _a number

lit

_which

corresponds

to this

loop iii

=

N(acdba)

=

N(ac)

₊ N

(cd)

₊ N

(db)

₊

N(ba)

=

(9)

Now _{we can} deform the

loop however, iii

_{remains the}

same until all the deformations _are inside the

region

where 4l is continuous. For

arbitrary loop

_{we can} write

lit

as the

following integral

_over the

loop

_:

lit

=

~'~

=

iV4l ^{di. (10)}

2 _{" "}

If _we shrink the

loop,

_we find inside the

loop

at least _one

point

where 4l is discontinous and

lit

=

I for _any

loop containing

this

point ~but

_no other

point

of

discontinuity).

Thus _{we can}

assign

^the

topological charge lit

to this

point.

It is _easy to show from

equations (4-6)

that for the Laue-case diffraction in

perfect crystals

the

singular points

_are at the

positions

y =

0,

A

=

A~. Expending equafiion(4),

_we obtain for any small circle

surrounding

these

points

t =

(- 1)~+ (AA

₊

iy

m p

e'* (I I)

where AA

=

A A

~,

AA «

I,

_y «

I,

_p

=

~~,

p is the radius of the circle. It is evident from

(I I)

that 4l is

equal

to the azimuthal

angle

of

points

at the circle relative _to the

singular point

and therefore ha

=

2 _w when _{we pass} the whole circle in the counterclockwise direction.

4.

Beyond

the solvable models.

In the

previous

sections _we calculated the

topological charges

N and

lit

_{for some}

simple

model

exhibiting

the

simple solution,

but the main

advantage

of the

topological

treatment is that it

can be extended _on the _cases where the

explicit

solution is absent. For

example,

it is clear that

even the exact solution of the Maxwell

equations

in

crystals

should contain the

singular

points. Indeed,

the difference between exact and

approximate

solutions is

expected

to be _very

small in

regular region; therefore,

the difference in

lit

_{should be also}

_small;

_but

lit

_{has got}

_integer

_values

_only

;

hence, lit

_{does not}

change

at all

and, shrinking

the

loop,

_we

can find the

singular point (but

at

slightly

different

position).

Now,

let _us consider the intersection of _two diffraction bands and the

multiple-beam

diffraction in the intersection

region (Fig. 4).

For this

geometry,

the numerical

computations

of reflection and transmission coeffcients _are

usually

needed and _we should be careful

dealing

with

singular points

in

computer

calculations.

Suppose,

^{that both bands contain} no

singular points

if the

multiple

beam diffraction is absent _;

therefore, N(ab)

= N

(dc)

for _one band

and

N(ac)

=

N

(db)

for another.

Hence,

for the

loop acdba, N(acdba)

=

0 and _{we can}

conclude,

that

only

_even number of

singular point

can arise _{as a} result of interaction between the bands for half of these

points lit

= I and for another half

I

= I. At

figure

4 _we show two of such

points

_; the

loop

acdba _can be

continuously

transformed to the small circles

around both

points

with different

signs

^of

iii.

_If

we

change

the

parameters

^of

multiple-wave diffraction,

_some

points

_can arise and

annihilate,

but

only

in

pairs

with

iii

=

I and

iii

=

I.

Occasionally,

two

points

with

iii

=

I _can

produce

one

point

with

iii

= 2 but the

latter is

topologically

unstable

(it splits

into two

points

^after an

arbitrary

small variation of diffraction

parameters).

1192 JOURNAL DE

PHYSIQUE

1 li° 8

b d

a c

p (arb. units)

Fig.

4. The

singular points

at the intersection of two diffraction bands where the

multiple-beam

diffraction _occurs.

Contrary

to

figure

3, it is

supposed

that each band had _no

singular points

before the interaction between them _was taken into _account.

If _we introduce _some

sufficiently

small

degree

of

imperfectness

into the

crystal

structure, the

topological charge

N holds its value

(in

the

general case),

but the

increasing imperfectness

results to

decreasing

of N

by unity

_step

by _step. Therefore,

even in the

imperfect crystals

the transmission coefficient _t

(for

the coherent

part

^{of the} transmitted

wave)

_can be

equal

to _zero.

If the

imperfectness

is very

large,

_so that the

typical

size of

perfect regions

is much smaller than L~~~, the

phase

4l

(ho )

looks like _curves I and 2 at

figure

2 for _any thickness of the

crystal [10]

and N

= 0 for any thickness.

Hence,

from the

topological point

of

view,

_{we can} divide all

imperfect crystals

into two classes _: class _« 0 _» if N

=

0 for any thickness of the

crystal

and

class« I

(which

includes the

perfect crystals

_as

well)

if N

=1,2,..

with

increasing

thickness. It is

unclear,

what is the transition between these

classes,

but it _seems that such

topological

classification _may be instructive.

The last

item,

_we want to

discuss,

is the

dispersion

relations for the

phase

and the absolute value of the transmission coefficient _t. It is well known

[13]

that these relations _are the result of such fundamental

principle

_as

causality

_; therefore

they

should be valid in all _cases

(for perfect

and

imperfect crystal, etc.).

In the conventional

dispersion relations,

t is the function of the

frequency

_w, but in the _case of diffraction _{we can} rewrite _{t as} the function of ho _or y because _near the diffraction band Am is

proportional

to ho and y.

Considering

_{t as} the

function of y ^on the

complex plane

_{we can} write the

following dispersion

relation

j ~

~°

in

jt(z)

dz

~

(

~~~

Y ~Y"

(12)

4~

(y)

= «

~

z y

,

y

y;*

where

symbol

^P denotes the

principle

value of the

integral

and the summation is made _over all those

points

_{y; in} the upper half of

complex plane

where

t(y;)

= 0. If y - ± oo, the

integral

in

(12)

vanishes and after the summation _we have

4l(+

_oo

4l(-

_oo

)

=

2

«No

_;

comparing

this

expression

with

(2),

_we find that the

topological charge

N of the diffraction band is

equal

to the number of _zeros

No

^of t in the _upper

half-plane

of

complex

_y.

5. Discussion.

We have considered the

topological properties

of the diffraction bands for

simplest

geometries.

The natural

question

arises _: what is the

relationship

between _our consideration and the

general theory [5-9]

of defects in

electromagnetic

wavefields ? Those defects _occur in

suffciently complicated wavefields;

if _we illuminate

by

the

divergent

_wave the part ^of diffraction band with the

singular point,

the transmitted

divergent

wavefield contains _one dislocation in the _sense of works

[5-9]

and the _core of the dislocation

just corresponds

to the

singular point.

The

topological

ideas may have

interesting applications

^for other

types

of _waves

diffracting

in

crystals

_:

changes

in the electron _wave

phases

at the exit surface of

crystal

is

important

for electron

microscopy

in the Mossbauer y-ray diffraction _new

singular points

_can be

produced by

small variation of the y-ray energy. The

polarization

of _waves also adds _new dimensions _:

the disclination-like

singular points

_{can occur} at the

polarization

pattems

[7, 8]

; we can

conclude that the

topological approach

_may be fruitful for the different diffraction

problems.

References

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Lignes,

Parois dansles Fluides

Anisotropes

et1es Solides Cristallins. Vols. I- II

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Les

llditions

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Physique, 1977).

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St

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591-648.

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