A note on quasielastic neutron scattering

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A note on quasielastic neutron scattering

G. Coddens

To cite this version:

G. Coddens. A note on quasielastic neutron scattering. Journal de Physique I, EDP Sciences, 1994,

4 (6), pp.921-938. �10.1051/jp1:1994236�. �jpa-00246955�

Classification Physics Abstracts

07.90 fil.12E 66.30

A note

_on

quasielastic neutron scattering

G. Coddens

Laboratoire Léon Brillouin, C.E.A.

/C.N.R.S.,

F-91191 Gif-sur-Yvette Cedex, France Neutron Scattering Project, I.I.K.W., University of Antwerp, B-2610-Wilrijk, Belgium (Received 7 October 1993, revised 11 February1994, accepted 24 February

1994)

Abstract, We present _a theorem thon in coherent quasielastic neutron scattering _{on a}

confined and limited system of identical partiales only the configuration-configuration correla- tions have to be examined without _any hindsight _on the distribution of the partiales inside the

configuration. This constitutes _a strong simplification with respect to the problem where ail partiale-partiale correlations would have to be accounted for. The method is illustrated _{on an} example of correlated hopping _{on a} piece ^of _an octagonal quasipenodic tiling. With _some mod-

ifications the method

con aise be used for incoherent scattenng. A further simplification of the calculations bath for coherent and incoherent scattenng functions is obtained by introducing _a

new matrix notation thon allows fully for the symmetry properties of the problem.

1 Introduction.

Quasielastic

neutron

scattering

is _a

powerful technique

to

study

diffusive motions _m condensed

matter

il,

2]. ^In

general

the

experimental

data _are

analysed by comparing

them with the

scattering

^functions

Sin~(Q,w)

and

S(Q,w)

calculated from _a model. In _a

large

number of situations _{one can use} so-called

jump

models.

Although

this

approach definitely

contains

approximations

[3,

4],

in practise it very often works rather well. Correct quantum ^mechanical calculations _are diilicult [4]. A

large

amount of such

jump

model calculations have been carried out for trie incoherent

scattering

function

Sin~(Q,w);

for bounded motion trie concept ^of trie EISF

(elastic

incoherent structure

factor)

_was introduced [5], ^and

developed

in

depth

for _many

physical applications

_[6]. For coherent scattenng, calculations of

S(Q, w)

_are however _{scarce, as} it _turns out to be diflicult _to take into account the correlations between the different

partiales.

In the present note we propose a method that _may be instrumental in

tackling

this

problem.

2. A theorem for coherent

scattering.

According

to Van Hove I?i ^{the incoherent and total} scattering ^functions are

given by

Sinc(Q'~°) ÎÎ ^JÎh /cÎ

~ ^~~~

~

_~~~~~~ ^~~~~~~ ~~~~~~~~~~~ ~~~

S(Q, w)

=

~~

/~

_e~~~~

~j _bjbk

(e~Q'(~~~~~~~?1°~)

)th

^dt

(2)

ki 2Jr&

_~ ~

Here

kj,

kf are the _wave vectors of the

incoming

and

outgoing

neutron,

hQ

and hw _are the

neutron momentum and _energy

transfers, bj

are the

scattering lengths

and

rk(t)

is the position of

partiale

k at time t;

)th

stands for thermal _average. Whereas the incoherent

scattering

function involves

only

correlations between _a

partiale

and

itself,

the total

scattering

func- tion contains also correlations between different

partiales.

Incoherent and coherent neutron

scattering

provide _a beautiful illustration of the

particle-wave duality

in quantum ^mechanics.

Coherent

scattering corresponds

to the _neutron

acting

as a wave, while incoherent

scattering

describes the

partide

behaviour of the neutron. This has been

expounded

in _a

superb

_way

by Feynman

_[8]: it becomes

possible

to reconstruct the

path

of the neutron in the

scattering

_pro-

cess if its interaction with the

scattering

^nudeus tags ^{this nudeus}

by changing

its state.

E-g-

if the

spin

of _a

spin 1/2

nudeus is

flipped

_we know afterwards that it is this

particular

nudeus that has scattered the neutron. Hence the neutron has behaved _{as a} partide with _a well defined

path

and the scattering process is incoherent.

If,

_on the contrary, ^there is _no way ^to tell which

path

the neutron has

taken,

since _no nudeus has been

tagged,

then the neutron has behaved

as a wave and been scattered

coherently.

The fact that _no atoms have been

tagged

_means that

they

stay

indistinguishable

in the

following

_sense ^if _a

configuration

_or cluster of _n identical atoms Ci

(0)

at time 0 evolves to _a

configuration C2(t)

at time t, ^then _no information will be available about the atoms in

C2(t)

apart ^{from the}

configuration;

there _con be _no information about

possible permutations

between the atoms within C2 since this would

imply

the _presence of _a kind of

tagging allowing

us to find out about these permutations. From this argument

we conclude that the

appropriate language

to

speak

about coherent

scattering

should not be

m terms of partiale correlations but rather in terms of

configuration

correlations. This _can be

proved ngorously

if _{we assume} ail

partiales

to be identical. Let _us introduce the notation _:

n

(vi,

_v2,

,

vn)

_llQ

(ui,

_u2,

.,

un)

=

£

e~Q'~?

e~~Q'~J (3)

j=i

Here

(vj)j=i...n

and

(uj )j=i...n typically

descnbe _a

configuration

of _n

partides

with position

vectors vj or uj. ^Let

Ç(n)

be the group of n! permutations of

(1,

2,...,

j,...n).

We further define the

cydic permutations Pk (1, 2,. ,j...n)

_-

(1fIl k,

2 fil

k,.

,

n e

k),

where the

signs

fil indicate _{sums to} be taken moduto _n. We also write

P(ri,

_r2,

...,

rn)

=

(rp~i~,..

,

rp~n~),

^VP E

Ç

(n).

Now

by definition,

the intermediate

scattering

function

I(Q, t)

is

given by

2Jrh ki

~~q

_{~) =}

f f

(e~Q'~?~~~e ~Q _~~~°~)th

b2

kf

^'

~=ik=i

=

(É(Pj-i(ri(t),r2(t); >rn(t)))

_llQ

(ri(°),r2(°)>. >rn(°))lth (4)

In

fact,

_suppose that _we write the terms

exp(iQ (r~(t) rk(0)))

in _a matrix at positions

(k, j).

The first fine of

equation (4)

_says then that _we have to make the _{sum over} ail the elements of this matrix. In the second fine the _{same sum is} obtained

by

^first

adding

the _n terms _on the

diagonal,

1-e- trie elements

(1, 1), (2, 2), (n, n).

This is the contribution from

the _terms

containing

Po. ^The terms

containing

Pi

correspond

to the elements _on the

parallel

diagonal (1, 2), (2, 3), (n

1,

n), (n, 1),

1-e- to the elements

(k,

Pi

(k)).

This way the n2 combinations

(k, j)

_are counted

by introducing

the _n

cyclic permutations

and the

symbol

_cQ,

which also contains _n terms. We abbreviate further C~

=

(ri,r2,

...,

rn)

and

f

₌ 2Jrhki

/kfb2.

We _cari put

C(t)

₌

P°T(t)C(0)

with _some P _E

Ç(n);

here

T(t)

_con be _a matrix but could also be _{a more}

general

matrix

function,

and it describes the

configurational changes,

^while P

describes _a

possible permutation

of the

particles

that has taken

place

inside the duster.

In this _way

fIlQ, t)

₌

(É(Pk-iC(t))

_aQ

C10)lth

= ~~ ~~,

~ _(JrC(t))

_aQ

_C10)lth

«eg(n)

=

j~

(

~~,

~ l(Jr°P°T(t)C(0))

_QQ

C10))lth

«eg(n)

=

~ jJr°T(t)C(0))aQC(0)jth i~

_~)~

«EG(n)

=

(ÉlPk-i _°T(t)C(0))

_aQ

_{C10)lth (5)}

The idea behind the

preceding

fines is _as follows.

Suppose

_we write

again

the elements

exp(iQ (r~(t) rk(0)))

in _a matrix.

However,

this time _on the

diagonal

_we write the el-

ements

ii, II(j)),

with II E

Ç(n),

and _on the

parallel diagonals

_we write

ii, P(II(j)).

The

sum of ail the elements of the matrix will still be the _same. The relation Hi _{+ 112 m}

(3k

E

[0,n

_1] n

iQ)(P(IIi

_{= 112)} is _an

equivalence

relation. The

corresponding quotient

struc- ture defines _a

partition

of

Ç(n)

in

(n -1)1

classes

Cl(II).

We _can thus rewrite the _sum

£)~~ £(~~ term(j, Pin( j))

_as

£)~~ _£n~~ojn~ term(j, IIk(1)).

Each of these classes _corre-

sponds

to _a matrix

yielding

_an identical value for the _sum. This

explains why

_{we con}

replace

these _sums

by

~~~ _,

£)~~ £~~g~~~ term(j, Jr(j)).

The final result _expresses that _{we con} cal- culate the coherent scattering ^function

by using

tue time evolution of the

configurations,

since any ^trace of P has been lost in the final fine of equation

(5),

due _to the _group

properties

of

Ç(n) (which

intervene when _we

replace

the _{sum over} Jr°P

by

the _{sum over}

Jr).

^{Note that in} trie

preceding

fines P

corresponds

to a

physical reality

while trie Pk-i _are

just

mathematical

expedients

introduced _to count the combinations

(k, j).

When _a

partiale

jumps ^between two

neighbouring

sites with _a characteristic time _T, in the

language

of clusters _{one con} say that the system

jumps

between two

("neighbouring")

configurations

with the _saine characteristic time. This _{cari even} be

generalised

to cases where several

partiales jump

in _a correlated _way,

provided

that each _move still corresponds to _a

change

^of

configuration.

Each

jump

^of a

partiale (or

_set of

partides) corresponds

then to _a

jump

^{of the} system. ^The system can thus be considered _{as an} abstract

partiale

and the difTerent

configurations

_as abstract sites. In the _case when the number of

configurations

is limited this abstract

isomorphism

will make it

possible

to write the rate equations

linking

various dusters

in the _saine matrix form _as for the

jumps

of _a

single partiale

between different sites _:

$

+

S~~A(t)SP

= 0

(6)

The sc-called

jump

matrix M

=

S~IA(t)S

,

where A is

diagonal

_{or more}

generally

of the Jor- dan form, represents the cluster-cluster transition

probabilities;

often A wîll be

independent

of t.

The solution of the matrix

equation (6)

is

given by

_:

SPjt)

=

expj- f~ Aju)du)SPj0) ii)

In

equations (6)

and

(7)

P is _a 1 _{x m} matrix and Fg

(t) gives

^the

probability

to find the system

m

configuration

_Cg at time t; m is the number of different

(indistinguishable) configurations.

In the _more conventional methods _to calculate the incoherent

scattering

^function as described

m reference _{[2] a set} of linear

equations analogous

to

(6)

is

obtained; however,

_m those _cases the elements Pi of trie column matrix P

give

tue

probabilities

to find _a

partiale

at site _at time t. The meaning of these

quantities

is thus quite

different, despite

the formol

analogy (which

is

the result of the abstract

isomorphism); they

do Dot refer to

partiales

_m the present case. The formalism _can

only

be carried out

easily

if each

jump

of _a

partiale (or

each set of simultaneous

jumps

^of

partiales) corresponds

to _a

change

^of

configuration.

In the calculation of the thermal

average one has _to consider _m different

problems

of the type

(6), depending

_on the system

being

in the

configuration Ci,

C2,

...,

Cm at the moment t ₌ 0. We will further note these _m

different matrices _as

P(~), P(2),

,

P(~)

_{Ii is dear that}

P~~)(t)

=

S~~ exp(- / ~ A(u)du)S(pi, 0,

0,

0,. ,0)~ ₍₈₎

where _pi is the

probability

that the system is in

configuration

Ci at t

= 0.

P)~~(t)

=

pji(t)

,

is the

probability

that the system ^{is in}

configuration

_Cg at time t, if it _was in

configuration

Ci at time 0.

By

juxtaposition ^of the _m equations of the type

(8)

_{we see} that _:

ll

^{P12 Plm}

P21 P22 P2m

=

S~~ exp(- / _{A(u)du)SAO (9)}

0

Pml Pm2 Pmm

where

Po

is a

diagonal

matrix

containing

the pj. ^For

given configurations

_Cg

(t)

and

Ck(0)

~(l~s-l~j

n

_(~))

_{~Q ~k(Ù)}

"

~ ~

_~~~~"~~~~ ~~ ~"~~~

s=1 _ru ECj _rw ECk

=

~ eio.r»~t))( ~ e-iQ.r"i°))

=

>i~~i~(Q)>c~~o~(Q)

r»ec~ r~ec~

where

Jlc(Q)

"

~

e~~Q'~"

(10)

ru EC

The thermal _{averages occurring} in

equations (4)

or

(5)

can thus be

expressed

_{as a sum} of terms

Jl] j~~(Q)Jlc~jo~(Q), weighted ^by

^their

probabilities pjk(t),

1-e-

£)~~~ _Jl], (Q)pjk(t)Jlc~ (Q) wllere _{(j, k)}

,

run over ail possible

configurations.

Hence it is convenient to introduce the _row matrix _or vector

iF(Q)i

=

i>i~(Q),>i~(Q),

,

>i~(Q)i (11)

to describe the coherent

scattering

function _as

S(Q,

_ùJ)

=

f~ e~~~'~jF(Q)iS~~ exp(- f~ A(u)du)SPOIF(Q)i~dt

=

lIF(Q)IS~~A*(ùJ)SPOIIF(Q)l~ (12)

A* is the Fourier transform of

exp(- fj _{A(u)du); t}

_{stands for Hermitian}

conjugation.

It may

transpire

that

equation (12)

is rather remote from _a

Vineyard

type of

approximation

_[9].

It must be

clearly

understood that in _our calculations each

jump

_or set of simultaneous

jumps

must

correspond

to _a

change

of

configuration.

The formalism has _to be

applied

with caution if several partiales make _{up a}

rigid body,

as e-g- the D atoms in _a CD4 rotor _: such

totally

correlated simultaneous jumps ^have not been accounted for in the formalism if

they

lead to the situation that after _a

"jump"

the final

configuration

is identical to the initial _one. This effect could be

possibly

allowed for in the

formalism,

but in the _case of

2Jr/3 jumps

around the symmetry axes of the CD4 molecule _we will end _up with

Orly

_one

possible configuration

and Dot very much will be

gained by using

the present ^formalism. Let _us

finally

mention that the factorisation of

equation (2)

obtained in

equation (10)

_cari be extended _{so as to} include also the

scattering lengths

bi, which suggests ^{that the formalism} m terms of

configurations

also

applies

to _cases with non-identical

partides.

Several subsets wherem the

partiales

remain

indistinguishable

_may then exist within _a

configuration.

3. An

example

_{: an}

octagonal tiling

mortel.

Before

formulating

the second remark of _{our note we} will first illustrate _our

technique by applying

it to _a

problem

of atomic

("phason") hopping

in _a duster taken from _an

octagonal

quasiperiodic tiling

_as described

by Kalugin

and Katz [10]. ^{Three atoms} cari move as shown in

figure

1a. Inside the

tiling

_we

perceive

assemblies of three tiles that look _as the

drawing

of

a cube in perspective

(e.g.

inside the

hexagon LOGHAB).

This is _a consequence of the fact that this

quasiperiodic tiling

model is obtained

by projection

from _a 4D

hypercubic tiling.

The

projection

of the 4D

hypercube

onto IR~ is shown in

figure

1b. An atomic jump

corresponds

to _a

change

m

perspective

in _a cube and is called _a

phason).

After

eight

_{moves we} get bock to the initial

configuration

but the three atoms in

I,

L and O have

undergone

_a

cyclic permutation.

The three atoms diffuse _on the small octagon ^{IJKLMNOP of}

figure 1b,

but in

a

strongly

correlated _way. Such toy ^models are

thought

to

give

the _essence of the

problem

of atomic jumps ⁱⁿ

quasicrystals.

On _a

larger quasiperiodic tiling they

_can lead to

long-range

diffusion and comphcated

cooperative dynamics.

The point is that with the present

technique

we

might

get a first _{grasp on} such

complicated

correlated systems. ^In icosahedral Alfecu there exist

probably "floating

clusters" of ₇ Cu _{atoms on a} dodecahedron [11]. ^{The rule of the} game is here that pairs ^of atoms should _{never occupy} first

neighbour

sites with respect to each other.

Moreover, they

should _{never occupy}

symmetrically opposite

sites on the dodecahedron.

Quasielastic

coherent

scattering

from Cu atoms has been observed in this system [12], ^which

justifies

a

study

of the

problem

evoked. There _are 320 different

configurations meeting

the first condition. This number is further reduced to 100

by

the second _one. Without the theorem the rank of the

jump

matrix to be considered would be much

higher,

since _we would have to allow for ail

possible permutations

of the

particles

that _{can occur} within _a

configuration,

and the rank could be 100 _x 7! _m the worst _case that ail of them _are

actually

allowed. Just

finding

out which

permutations

have to be accounted for constitutes

already

_a

major

effort in its _own

right.

We

just

mention this

example

to indicate how the new method presents a definite

advantage

over the older _ores. From _{Dow on we} will deal

only

with the much

simpler

calculation of

figure

1. As _cari be _seen from

inspection

of

figure

there _are

only

8

indistinguishable configurations.

If _{we assume} that the transition

probability

^between two clusters is

always

the _same

(1/T)

the rate

equations

^become

formally

identical _to those for the motion of _a

single partiale

_on

a

regular

octagon, as described

by

Bée [2], ^who

gives

the _more

general

solution for _a

single partiale

_{on a}

regular

_n-gon

~~

=

~Pj_i ~Pj

₊

_{~Pj+1 (13)}

~M

^{_ D}

a~

~j

^O

i~

~

~~

~ M

~ M

~ ~

A o~ J

I P

~ ^~

B K

F

~ M

N K

j

~

P

~ ~

~

K

~~

N K N G

~

Fig. 1. Left: model of three correlated partiales diffusing _{on an} octahedron. Right _: 2D projection

of _a 4D hypercube which helps to visualise the octagon of sites visited by the partiales

(IJKLMNOP)

Here Fg

(t)

is the

probability

to find the

partiale

in site

j

at time t.

Keeping

in mind the caveat about the different

meanings

^{of the} quantities involved

(as explained above),

_{we con}

exploit

this formai

identity

and

apply immediately

the results of reference _{[2] to our} duster transition

probabilities.

The matrix to be

diagonalised

is thus

(~ôj;k-1

₊

_{26j;k ôj;k+1)} (14)

It has the

topology

of the

problem

of the

phonons

of _a linear chain of _{masses p} with uniform distances _a,

spring

constants _~ and

cydic boundary

conditions. This makes it

possible

to map the

problem

of the transition

probabilities

between the dusters _{on a}

problem

of

phonons

_on

a

periodic

lattice. The symmetry _con ^be

expressed

_{as a} translational invariance where the coefficients of trie basis vectors of the lattice _are mtegers ^modulo J E lZ e-g- the diffusion of _a

partide

_{on a} ^cube is described

by

the _same

equations

_as for _a cubic lattice if trie coefficients _are taken moduto

2).

In this _{way one can}

immediately postulate

the

eigenvalues

and

eigenvectors

based _on trie Bloch

theorem,

and trie

diagonalisation

of the matrix becomes

simple.

This _comes down to

postulating

_:

Pr ₌ Uqe~~'~

(15)

A

M o=M+5

(O) fa')

= 2 +

Q~ _{2 m}

+ tx

(b)

Fig. 2.

(a)

Schlegel diagram of _an icosahedron

(a

topological connectivity scheme that shows which vertices _are first

neighbours).

The point A

(represented

_as being ai

infinity)

is opposite to A

on the icosahedron. Matrix M.

(a')

Matrix Q

= M + 5.

(b)

Q~ calculated with the diagrammatic method and the relation Q~

" 2 m+a. In

Q(a')

P has B and F _{as non-zero} neighbours, thus yielding 2 _on P

m

Q~(b).

The part in Uq ^{leads to rate}

equations

^and

eigenvalues.

^The

exponential

part ^leads to the

eigenvectors.

For tue linear chain this

yields

_:

~kj

_"

) _{~XP(~~)(~ ~)() ~))}

Ajk

_" ^~

sin~(

^~

(j 1))ôkj

T m

P0 _" Émxm

(16)

Identification with the

phonon problem

_is obtained

by

~~~~ _~~

- qa; -

~;Ajk

_- w~

(17)

m T /L

We number the clusters _as in

figure

1, _1,e. ILO

-

Ci,

LOJ _- C2, ^OHJ _- C3, We

assign

vectors ei =

(cos(Jr(1-

1)

/4), sin(Jr(1-1) /4))

to the

positions

of the sites IJKLMNOP. The

8 different dusters

Cj

are then

given by triplets(ei~~~-1),

_eie3ji

ei~~~+i~),

where here agoni the _sums in the indices have to be taken moduto 8.

E-g-

^for Ci we find _a

triplet (ei,

_e4,

e7).

The

probability

that at time t the system has taken _up the

configuration

_Cg is

given by

the quantities Fg

obeying

the set of differential

equations (13).

As clusters _Cg and

Cj+i

have the sites eie3j and eie3u+i) ^{in common the transition from the first to the second} one is achieved

by

_a

jump

^{from the} site eie3u-1) ^{to the site} eie3u+2). ^These conventions lead to:

~C, (Q)

"

~XP(~~Q '~le3(j-1))

₊

~XP(~~Q

_~lfll3j +

~XP(~~Q

_~le3(j+1)

~~~~~~~~~

uJ21rj ^~~~'~~

^{~~~~~ ~~ ~~~ ~~~~}

Calculation from equation

(12)

of

S(Q,

_w) is _now

straightforward,

be it tedious. The details

are outlined in the

Appendix.

The final results _are summarised in table I and

figures

3 and 4.

~ ~

sellù) 5~(ù)

₈

w ~

c c

i

Siio)

S,lQ)

_i

S~IQ)

Sslo)

2

0 2 ( S 0 2 ( 6 8

QR

Fig. 3. Coherent structure factors _as bsted in table I for the various contributions to the energy

fine-shape. Sel is the _sum of the structure factors _as described in the text.

4. Incoherent

scattering

In the

previous example

the

problem

of incoherent scattenng ^would

require

the _use of 24 _x 24 matrices, which underlines the _power of the method. In fact for incoherent

scattering

the rank of the matrix is _nm due to the fact that

only

_one

partide

has been

tagged

and

(n

₁₎

partides

^remain

indistinguishable.

This _can be

easily

understood _as follows. It suilices _[3]

to

replace

each cluster in

paragraph

2

by

_n clusters which

only

differ from _one another

by

the position of the

tagged partide

in tue cluster. One then has to write the

jump

matrix for the transition

probabilities

between _nm

tagged

dusters. Even this _cari be _a considerable

simplification

in comparison to the dassical

approach,

where it is customary to try to write down differential

equations describing

the

single-particle jumps.

The

previous example clearly

shows the

complications

this

entails,

smce tue

jump probability

for _one

partide depends

_on

oe~

2j~

~ à

?

p~

²

j3

~

_~ iÉ

-'

0 2 1 6 8

OR

Fig. 4. Total compounded coherent quasi-elastic signal

(S(Q,

w) without the elastic peak) _{as a}

function of Q and _w. A 3D-plot

(left)

and _a contour map

(right)

_are shown. The dashed fine

on the contour map shows the

(Q,

w) values where the intensity _is ai half the value of the maximum ai

(Q,0).

Ii is clear thon the signal is largely dominated by the first Lorentzian in table I, such that the FWHM

is almost Q-independent.

Table I.

Components

of tlle total _structure factor

S(Q,w)

for atomic

llopping

_on tlle _oc-

tagonal tiling

in tlle model of

figure

1

(average

_{over a}

powder sample

witllout

texture).

Tlle Lorentzians A _are

specified by

tlleir HWHM r.

Line shape Structure factor

9

6

p

^{1 +}

^2j0

^{(QDI + 2}

^j0

^{(QDz + 2}

^j0

^{(QD~ +}

j~

_QD,

~(^~

[ _j

3 +

25)

₍₁

_{S Jo}

(QDI +

S Jo

_(QD~

Jo

(QD, j

~(p)

2 _~- ^{(1 '2}

^j0

(QDzj +

j0

(QD,

~(~~[ j

₍₃

_25>

(1 +

S Jo

(QD,

S j~

_jQD~

j~

_(QD,

~(pl

4

_p

(1

2jo

_(QD, +

2jo

_(QD~)

2j~

(QD~j +

j~

(QD, ))

the issue if another

partide (the preceding one)

has

already jumped.

It is in fact hard to

imagine

how _one could solve ibis

problem

without

eventually falling

back _{onto a}

description

in terms of

configurations, possibly allowing (unnecessarily)

for ail allowed permutations ^of

trie

partiales.

Hence _even in this less favourable _case the _new

approach

presents a

significant

simplification.

Trie mcoherent _case will of _course require ^the use of

single-partiale

structure

3

m Sllo) S~IQ)

.~ 2 C OE

~ i

S~IQ) S~IQ)

i

o

S31Q) _S~~IQ)

o

i

S,

iQ)

SIIQ)

o

S~IQ)

^Q)

o

S61°)

S131°)

0

0 2 ( 6 2 ( 6 8

QR

Fig. 5. Incoherent _structure factors _as listed in table II for the various contributions to the _energy fine-shape. The structure factor Sg is onùtted _as ii is _zero.

UT

É Q~ 2 0

~[

^/,

6

-' 2

0

~

ÙJ~ 2

,

Fig. 6. Total compounded incoherent quasi-elastic signal

(Sinc(Q,

w) without the elastic

peak)

as a

function of Q and _w. Three dimensional plot and contour map. The dashed fine _on the contour map shows agam the location of the points ^where the intensity _is ai half the maximum value ai _{w =} 0.

factors

only,

instead of the _more

complicated quantities

Jl. In the

Appendix

_we derive the incoherent scattenng ^function

corresponding

to the

problem

of the

octagonal tiling

described

m

paragraph

3. The final results _are summarised in table II and

figures

5 and 6. The

jump equations

_are obtained

by using

a

description

in terms of 24

configurations.

Since the

partiales

cannot

leap-frog

_on trie

tiling,

this is also the number of ail allowed

configurations

,

1-e- there is _no

simplification

here

by

_{a presence} of

indistinguishable permutations.

Table II. Com ponents ^of tlle incollerent structure factor Sin~

(Q, w)

for tlle

problem

of atomic

hop pin

_{g on an}

octagonal tiling

illustrated in

figure (case

ofa

powder

_sam

pie

witllou t texture

).

Tlle Lorentzians A _are cllaracterised

by

tlleir HWHM r. Tlle structure factor

corresponding

to

à(w)

is also called elastic incollerent structure factor

(EISF).

Tlle _structure factor Sg is _Dot listed _as it is _zero.

Line shape Structure Factor

6 _{1 +} 2

jo

(QDI + 2

jo

_(QDz + 2

jo

_lQD3 + jo 1QD,j /8

~(^~

~)

3 +

2$É

+

Ù

_il

+

Ù j~

_(QD~

Ù _j~

(QD3 Jo QD, l'2

~(~fi)

_{(2 +}

^S

₎₁₁

_2jo

_{(QDzj +}

_j~

_{jQD, )j/à}

~~~

-j~

_~

+

~«~

_ii

ùn Jo

QD, ⁺

~

_~°

~~~~ ~° ~~~' ^'~~ ~

~(p)

' Il 2j0 (QDI +

2j~

_(QD~

2j~

_(QD~ +

j~

_(QD, )>/3

~(~~~)

(3 +

2~ _{Ù)[1 Ù j~}

(QD, +

Ù j~

_(QD~

j0

(QD, )>/12

~(p)

¹

^2jo

(QDz ⁺

j~

(QD, ^)j/iz

~(~~~)

3

2~É _Ù)

il +

Ù j~

_(QD~

Ù j~

_{(QD~ )}

j~

(QD, ) Il 2

~~~

j~

(QD, ^)]/l~

~z+~Ù~

2Ù)

_Il

+

Ù

Jo ~~~'

~

~~ _~

~jÎ) S)11 _2j~

_(QDz)

+

jo

(QD, )1/6

z+&4

~-~-~- _~p ~pj~<~D,~

₊

«j~jQD~) JOIQD,)1/<2

~~

~

2 z 3 + 3>11

~(pl

4

2j0

_(QDI +

2j0

_(QD2

2j0

_(QD3 +

j0

(QD, ^jj/24

5.

Diagrams

_as matrix

representations.

Calculations of

scattering

functions often lead _to

boring lengthy

calculations to cast the

problem

into the Jordan form

(6)

and

powder averaging

of

(12) generally

introduces Bessel

functions,

which make the final result look very

impressive.

From trie _use of the Bloch theorem _m the previous

example

it may have become clear that _{one cari} often

simplify

the calculations

considerably by exploiting

ail the symmetry properties of the

problem.

The 8 Bloch _waves

are in _{a sense} symmetry

adapted

functions.

Very

often the _mere notation of the matrix M in

the form of _{a square}

already

will conceal _{us some} of the properties. ^{We therefore} _propose to write down the matrix m the form of _a so-called

Schlegel diagram

_as shown in

figure

2a for the matrix that

corresponds

to the differential

equations

for the diffusion of _a

partiale

_{on an} icosahedron I.

(A Schlegel diagram

of _an

object

is _a

drawing

that

gives

its

topology,

1-e- the connections between its

vertices).

The

diagram corresponds

to _one fine of this matrix M. In

each vertex

j

of the

diagram

_we put the term

M~,

1-e-

by writing

-5 in A and +1 in

B, C, D,

E anf F in

figure

2a _we express

~) =-SPA+PB+Pc+PD+PE+PF (19)

However, apart ^{from the}

labelling

with A, B,

C,...

ail 12

diagrams

will be the _same due to

symmetry. Thus,

figure

2a represents, _{in a sense,} the whole matrix M. We _now consider the matrix

Q

= sfl12x12 + M, which is

presented by putting

a zero in A and _a in

B, C,

D, E, and F. To calculate

Q",

_we just have to

give

a rule to calculate

TQ

for _a matrix T with the

same symmetry so that it _can be

represented

_on the _same type ^of

Schlegel diagram.

Let _us consider _a

point

P _on fine A of matrix

T,

1e. _a point ^P on

drawing

A of matrix T. This

point corresponds

to the element TAP We _now want the value for P _on

drawing

A of

TQ,

1-e-

(TQ)AP

"

LjTAJQJP.

Trie values of TAJ can be found in trie points ^{J of trie}

corresponding

drawing

A of matrix T. Of trie elements

QJP only

those values of J that

correspond

to the

neighbours

of P will have _{a non-zero} contribution

(equal

to

1),

such that

:

~TAJQJP

_"

~

_{2~AJ (~Ù)}

JEz JEfif(P)

where

tif(P)

is the set of the first

neighbours

of P; 1-e- to find the number _to put _on vertex P of

diagram

A of

TQ,

_we

just

have _{to sum} ail numbers _on the

neighbours

of P _on

diagram

A of T. This is child's

play

_as illustrated in

figure

2b for

Q~.

It is much less laborious thon tue standard _way of

performing

matrix calculations _{since we} bave _to calculate

only

_one fine and _one does not have to strain one's eyes ^to ensure that the

right

elements _are

multiplied.

Moreover the notation _is much

quicker

than

writing

down _a square of n2 _elements.

Figure

2b shows how

Q2

_cari be rewritten _{as a sum.} The first

diagram corresponds

to 2m12x12

, a 12 _x 12

matrix

containing

numbers 2

everywhere.

Tue second

diagram

shows that if _we index the point

corresponding

to -rA

by index(A)

₊ 6 then the

diagram

_can be written under block form _as

l~Î~~6 ^~~~~

~~~~

which _we will further note down _{as a.} As _ma

= m and _o is _{very easy} to

diagonalise

it just takes

a few fines to solve

Q2"

and

Q2"+~,

and thus also

exp(M)

_as

exp(-5) exp(Q).

This shows how this

diagrammatic

method allows _us to calculate

exp(M)

in a few fines without

reducing

M to the Jordan form. To solve the

problem

of _a

partiale diffusing

_{on a} dodecahedron _one has to calculate _up to

Q~,

to find the

general

formula for

Q".

Of _course each _case will _{require some}

ad hoc ski]] in

summing

the

Taylor

series, such that the method at this stage of

development

is not _as

systematical

_{as an}

approach

based _{on group}

theory,

but it may constitute in several

cases a

genuine

short-cut for the calculations. In fact the matrices of the type _m are bound to

occur sooner or later _as

they

represent the infinite time solution where ail the _memory of the

initial conditions bas been lost. This is

intimately

related to trie fact that in _a finite system

we will find

trajectories

^that are

loops.

Appendix

1.

Further details _on trie calculation of trie

scattering

functions for trie

octagonal

tiling.

in tuis

Appendix

_we will

give

_{some more} details _on tue calculations for tue total

scattering

function

corresponding

to the

example

^of

paragraph

3. First we show that S defined

by equation (16) obeys

SS*

= fl, such that S~~

= S*. In fact

:

~j _Skjsji

=

j ~jexP(-~)(k i)(1-1)) exP(~)(i ^{i)(t i))}

li li

=

f exp(-

^~~~

(k

₁₎

ii 1))

_m ôki

(A1)

m m

J=i

Further SAS*

yields

trie matrix M defined in

equation (13)

_:

m m

~j ~j _{Skj Ajis(~}

=

j=i i=1

=

~

~j exp(- ~~~(k -1)(j -1))

^~

sm~(

^~

ii -1))ôji exp(~~~(t 1)(h 1))

m m T m m

J>1

=

f _exp(-~~~(k

1)(t 1))

^~

sin~(

^~

(t 1)) exp(~~~(t 1)(h 1))

m

~~~

m T m m

=

É(( _exP(-j(k

h)(t i)) j _exP(-j(k

h ₊

i)(t -1))

expj-

^~~~

(k

h

1)jt 1)))

₌

26kh ôk,h-i _ôk,h+1 (A2)

m m

where _we have

expressed

the sine function in _terms of

exponentials.

Let _{us now} calculate the elements _pkh of the matrix

exp(-Mt).

We

assign

trie

probability 1/m

to each initial condition:

pkh(t)

=

~j Skj(exp(-At))pis(~

m J,

=

~

~j exp(-~~~ _{(k 1)(j 1))} exp(-~~ _sin~(

^~

(j 1)))ôji

m m T Yll

J,

x

exp(~~~ (t 1)(h 1))

m

m

"

à _{~ ~~P(~ ~Î (k h)(1- 1)) exP(-} ) _sin2(

^~

_{(i i)))}

(A~~

i=i m

After Fourier transform trie terms

exp(-)~ sin~(((t _-1)))

will

give

rise to trie Lorentzians

A((w)

defined in

equations (16)

^and

(18).

It is convenient to calculate trie _structure factor of each Lorentzian

separately.

Trie

prefactor

of the t-th

exponential

in the total

scattering

function

£~ ~Jlc~(Q)Pkh(t)Jl]~(Q)

is:

2 É É

FCk

(Q) ~~P(~ ~) (k h)(t i))Fih (Q)

"

k=1h=1

"

(É )>CL ^{lQ) ~~Pl~} ~)k(1~ i)))(É _{)>Ch (Q) ~~P(~ ~)h(1~ i)))~}

k=1 h=1

=

j É _{Fc~(Q) exP(-)k(t i))}

l~ =

1Ui(Q)

_l~

(A4)

By introducing

_xi

"

exp(-£(t 1))

the ternis

Ui(Q)

_cari be rewritten _as

Ui(Q)

"

à _L~Î=i

_Jlc~

(Q)(xi)~. By grouping

ail the ternis which contain

exp(-iQ

_ej inside this expression we obtain _:

~ _~~ ~ ~~

x)~~~~~

exp(-iQ

_ej

(A5)

ÎÎ

xi

~

In _{our case n}

= m = 8 such that

j

ruas from _to 8. From formula

(A5)

it follows that

Ui(Q)

is the Fourier transform of the distribution:

~ ~~ ~ ~~

~j _(xi

_{)~~~~~~ô(r ej}

_(A6)

xi

~

J=i

For t

= 1, this _is

just

the Fourier transform of the octagon IJKLMNOP normalised

by

_a factor

3/8,

which is _a very intuitive result. For

higher

values of t the ternis

(A6)

_can be considered _as Fourier transforms of

higher

order _moments. The t

= 1 terni _m

S(Q, w)

is thus

)S(Q)o~t. ô(w).

We have _xi

= 1 and x)~~~~~~ =

exp(1((t 1)3(j 1))

_{= x)~~}

~~, where

Lit, j)

is defined

by L(t, j)

1 _e

(1-1)3(j 1)(mod8).

The values of

L(t, j)

_are

listld

_{in table III for} convenience.

Furthermore, x)xK

_"

x)_~

~i. The

product

_U~*Ui is

given by

~ ~~ ~ ~~

(~

~j

x)~~~~~~x)~~~~~

exp(iQ (ej ek)) (Ai)

64 _xi

J,~

There _are 64 combinations

(j, k).

The 8 combinations with

j

₌ ^k

give

rise to terms under trie

sum that _are

equal

to 1. Trie 16 combinations with

j

k =1

correspond

to vectors ej ek that _are

edges

of the octagon and have _a

length

Dl

"

fiR,

_{where 2R is the diameter}

of the octagon ^{IJKLMNOP. Trie 16} combinations with

j

k

= 2

correspond

to _a

diagonal

with _a

length D2

"

ViR.

The 16 combinations with

j

^k ₌ 3

correspond

to _a

diagonal

with _a

length D3

"

~/R

_and

_finally

_{the 8} combinations with

j

k

= 4

correspond

to

a diameter of the octagon

(D4

_"

2R).

In ail _cases with

j #

k the _term

(k, j)

_is the

complex

conjugated

^of trie term

(j, k)

and

gives

_rise to the _same type ^of distance. In total _we bave