basics of diffraction

$: Basics of Diffraction$
JDN 22, September 21-24 2014 1

Basics on Diffraction

Pierre BeckerEcole Centrale Paris

SPMS CNRS [email protected]

New position : Augure of Jupiter Calchas

« Le cornichon (piclke) est confit dans du vinaigre, je suis confident du Roi »

JDN 2014 21-24 septembre Oleron

Scattering and diffraction of angstrom wavelength waves (X Ray, electrons, neutrons) is the key to study of atomic structure of condensed matter.

High resolution scattering allows for understanding vibrational, electronicand magnetic behaviour of materials: key importance of combined X, N, E studies

New pulsed sources allow for a time dependent study for out of equilibriumsystems

I shall focus on neutron scattering, but keep in mind the importance of combined studies


Outline

Nuclear Diffraction by a stationary target- Case of a crystal. Comparison with Xray and Electron diffraction- Taking thermal motion into account : total/Bragg scattering

Inelastic neutron scattering and study of phonons

Magnetic scattering of neutrons-Basics-Spin polarized diffraction: access to spin density of materials: complementarity with Xray high resolution scattering

Reality of crystals: extinction


Scattering cross section

X Rays, electrons, neutrons

2 θ

r u

R

r u 0

δ S d

( )0

02

F

uuFR

rr,=

Ω∂∂σ

( ) ( ) d000 SuuFdFuuN δσδ rrrr,, ==

F0: incident flux – F: scattered flux


( ) dd

2

00 EE

FuuN δδ∂∂σ∂δ Ω

Ω=rr

,

Case of energy sensitive detection

2

00

inc mF Φ= κh 2

ff

diff mF Φ=

κh

• Non relativistic particles

2

0

2

f

0

f2RΦ

Φ=

Ω∂∂

κκσ

• Photons: prefactor includes polarization factor of EM beam

JDN 22, September 21-24 2014

Scattering by a stationary target:

r r j

P

r R

r R j

O

( ) ( )trki0j0

0j0etrω−Φ=Φ

rrr ., jj rRR

rrr−= r j << R

Elementary scatterer:

Huyghens Fresnel principle: elastic scattering

( )( ) , 0

00

0j

jrRik

j

j

trki

j ebrR

etr

rrrr

rrr −

−

−Φ≅

ω

ϕ

particle oflength scattering jb


( )( ) , .

0

00jrQi

j

tRki

j ebR

etr

rrr

ω

ϕ−

Φ≅

f00f kkQukkrrrrr

−== r k 0

r Q=4πsinθ/λ

r k f

P

θθ

r r j

P

r R

r R j

O

Long distance scattering

jj ruRRrr

.−≅

( )( ) , 0

00

0j

jrRik

j

j

trki

j earR

etr

rrrr

rrr −

−

−Φ≅

ω

ϕ

r r j

P

r R

r R j

O

( )( ) , 0

00

0j

jrRik

j

j

trki

j earR

etr

rrrr

rrr −

−

−Φ≅

ω

ϕ


( )2

elastic

QAr

=

Ω∂∂σ

Complex target:

( )( ) ( )QA

R

etr

tRki

0f

00 rrω−

Φ=Φ ,( ) ∑

=

=N

j

rQij

jebQA1

.rrr

( ) ( )∑ −=j

jj rrbrnrrr δ

( ) ( )∫= rdernQA rQi rrr rr. ( ) ( ) ( )∫

−= QdeQA2

1rn rQi

3

rrr rr.

π

Scattering function:


Crystalline target (periodic scattering function)

One defines the structure factor :

Coherent scattering .

( ) ( )LrnrnL

rrr

r−=∑ 0

( ) ( )otherwise 0

cellunit if 0

=∈= rrnrn

rvr

( ) ( ) rdernQF rQi

cellunit

rrr rr.

0∫∫∫=

( ) ( ) ( ) ( ) ( )

−=

= ∑∑HL

LQi HQQFeQFQArr

rr rrrrrδπ

V

2.

3.

For a finite crystal, with N unit cells ( ) ( )HNFHArr

=

JDN 22, September 21-24 2014 10

Bragg diffraction conditions.

Qr

h is the momentumtransferred to the target by the wave.

Notice that the scattered wave has a phase shift ofπ/2 withrespect to the incident wave

λθπ sin4=H

For a finite crystal, there is an opening angle around Bragg condition, of order d/L and one must integrate the peak intensity

JDN 22, September 21-24 2014 11

Case of X Rays

Elementary scatterers are electrons, all undiscernable

Electron charge density is the effective « target » :

( ) rderrQA rQieeX

rrr rr.)()( ∫= ρ

( ) ( )( ) ( ) nRQi

nn

nnatn

e

eQfQA

Rrrrrr

rrr

.0

,,0

Σ=

−Σ≈ ρρ

Independent atomapproximation

For a 1A0 wavelength, the energy of the photon is about 12000ev, much higher than cohesive energies in a material

JDN 22, September 21-24 2014 12

r k 0

r k f

r E 0

r E //

r E ⊥

α2θ

r E //

r E ⊥

cmmc

erb 12

20

2

0 10 28,04

−==≡πε

C = cos2 2θ( )cos2 α( )+ sin2 α( )( )2

X2

0 QACrr

=Ω∂

∂σ

Formfactors at lowscattering angle are proportional to atomic number

They decay as the inverse of atomicradius


Neutrons result from fission of U235 and are produced with an energy > 1Mev Moderation via collisions with D2 O, liquid hydrogen, graphite

Also spallation sources, producing pulsed neutrons

Wave-particle duality 2

2222

222 λm

h

m

k

m

pE === h

kgm 2710 675.1 −=

( )

T

evE

evEA

8.30

082.0)(

)

286.0)(

2

0

=

=

=

λ

λ

λ

Neutron waves

For λ =1 Å

E =80 meV

T = 950K

v = 4000 ms-1

13Neutron is a fermion, with ½ spin

photonelectronprotonn γ++⇒Neutron life time is about 900 sec


One talks about « thermal Neutrons », behaving like a perfect gaz flow

It is necessary to monochromatise the beam, λ ≈ λ0 , which alsoproduces significant beam withλ/2 wavelength

The speed of neutrons is low enough to allow for time of flight detection, besides usual selective absorption

14


Basic interaction with matter is nuclear interaction, which for study of materials, can be considered as ponctual (range of fm)

( ) ( )Rrbm

rVn

rrhr −= δπ 22 b is called the scattering length by a nucleus, of the order of 10-12 cm

15

Very significant complementarity towards XRays


It can be re-written as [ ] ( )φδφ rbkr=+∆ 2

After collision, the neutron is scattered with the sameenergy, and can be describedas a spherical wave. One can show (1rst Born approximation, sinceb is very smallcompared to λ) that the proper solution (Green function) is

16

Let’s consider an incident neutron wave directed towards a nucleus located at origin.

rkierr.

00≈φ

( ) φφφδ 2222

222k

mErb

mm

hrhh ==

+∆−

When reaching the static nucleus, the neutron wave satisfies the Schrodinger equation

r

ebrG

ikr

≈)(

JDN 22, September 21-24 2014 17

( ) ( )∫ −−=

−

'''

'.'

0 rdernrR

eR rki

rRik

f

rrrr

r rr

rr

ψ

( ) ( ) rderVR

emQA

R

e rQiikR

eN

ikR

f

rr

h

r rr.

2, 2 ∫=−=π

ψ

[ ] ( )r4Gk2 rπδ=+∆r

ebrG

ikr

≈)(

[ ] ( )ψπψ rnkr

42 =+∆Complex target

JDN 22, September 21-24 2014 18

( )

( ) ( ) ( )

( ) ( )rnm

rV

rrrV

Rrbm

rV

ece

jjjN

rhr

rrr

rrhr

2

0

2

2

1

2)(

π

ρρε

δπ

=

−−=∆

−= ∑

Generalization to electron scatteringand beyond

( ) ( )

( ) ( ) ( )

( )[ ) ( )∫

∑

∫

=

=

−=

=

rderQA

eZQA

Q

QAQAmeQA

rdernQA

rQieX

RQi

jjc

Xce

rQiN

j

rrr

r

rr

h

r

rrr

rr

rr

rr

.

.

20

2

2

.

2

ρ

επ

It will also apply to magnetic interactions of neutrons with matterImportant complementarity of Xray / Electron scattering


( ) ( )22

2HFNHA

elastic

rr==

Ω∂∂σ

For a stationary crystal, forgetting about surface effects…

Phase problemremains to retrieve the scattering density

( ) ( ) rHi

H

eHFV

rnrr

r

rr .1 −∑=

Only the Patterson function can be obtained directly fromexperiment

( ) ( ) nneHFV

rP rHi

H

∗== −∑rr

r

rr .21

( ) ( )ijjunitcellji

i RrbbrPrrr −= ∑

∈

δ,


Also direct methods are available for getting the phases

Great thanks to David Sayre, Jerome Karle and Herbert Hauptman

( ) ( )rnrnrr

of thoseaslocation same have of Peaks 2

( ) ( ) ( )[ ]KHFKFHFrrrr

−, tripletssome of phases theamong relationsy Probabilit


ελ

θπ dd2sin

)2(d3

33 Ω=q

Due to finite size, scattering occurs out of exact Bragg condition

2 θ

r u

R

r u 0

δ S d

εqHQrrr

+=

For a given position of the crystal (ε), the cross section per unit volume isσ(ε), and after rotation of the crystal, one gets the total « kinematic » diffraction cross section per unit volume :

( ) )(A 2sin

1-32

θλεεσ

V

FdQkinematic == ∫

0

5000

10000

15000

20000

25000

30000

35000

40000

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

T = 300 KT = 10 K

h

Raies de surstrucutre (1/2 1/2 1/2) pour PbMg1/3Nb2/3O3 (PMN), à T = 300 K et 10 K,le long de la direction [hhh]spectromètre 6T2 LLB


0

500

1000

1500

20 40 60 80 100 120

Inte

nsité

(u.

a.)

2θ

10

0 11

0

41

0/3

22

40

0

41

1/3

30

11

1

20

0

21

0

21

1

22

0

22

1/3

00

31

03

11

22

23

20

32

1

33

1

42

0

43

0/5

00

42

13

32

42

2

43

1/5

10

33

3/5

11

43

2/5

20

52

1

44

0

Diagramme de diffraction des neutrons du BaTiO3 à 401 K : spectromètre 3T2 LLB


There are resolution limits due to the fact thatλπ4≤H

r

There can be a mixture of isotopes in a compound, and moreover there canbe a spin coupling between the neutron and scattering nucleus. As a consequence, the scattering length of a given atom can vary from one site to another one

( ) ( )LRQi

L jjL

jebQArrr

r

r +∑∑= .

The total cross-section is written as the sum of a coherent and an incoherent part

[ ]incoherent

if

2222jjjj

kjkj

bbbb

kjbbbb

−+=

≠=


IBbb N

rr.2 σ+=

1836/M2

e

JT10274.9m2

e

BN

124B

µµ

µ

==

== −−

h

h

SM

M

BS

NN

g:Electron

)2(:Neutron

µσµγ

−=−=

)1I(IBbbbb 22

N2N

22 ++−=−−+ +

++

+= bI

Ib

I

Ib

1212

1

One gets, besides a coherent Bragg diffraction, wherethe appropriatescattering length for a nucleus is<bj> an incoherent added intensity

[ ]∑ −=

Ω∂∂

jjj

inc

bbN22σ

Isotopic incoherence for NiSpin incoherence for H

σc

σi

H1,881,2

D5,62

O4,20

V0,025,0

Ni13,45,0


Diffraction experiments occur on systems at a given tamperature. This createsvibrational disorder of atomic positions. In crystals, those vibrations are phonons

( ) ( ) nLn uQiLRQi

L nn eebQA

rrrrr

r

r.. +∑∑=

222AAA ∆+=

Bragg diffraction

Thermal diffuse scattering

( ) ( )HFNHArr

=

( )n

n

uHin

nRHi

nn

eW

WebHF

rr

rrr

.

.

=

=∑Debye Waller factor

In the harmonic approximation, one can show exactly

( )2.2

1nuH

n eWrr

−=


For a givenfrequency 2

22 sin

2

2

ωλθπ

m

kT

n eW

−≈

Debye Waller factor lowers signal for high T, large scattering angle, light atoms, with major contribution from acoustic frequencies.

Anharmonic motion can also be considered

For systems made of molecules or for instance organicfragments linked to a protein, one can model vibrations in terms of intra-molecular (high frequency) and inter-molecular acoustic vibrations !

The other term <∆Α2> corresponds to thermal diffuse scattering. See laterit’sfrequency analysis. If no energy analysis is done, it isa diffuse contribution, which is considered as part of background

Static atomic disorder can also lead to a lowering factor, not dependent on T (see extinction !!!)

nuHierr

.


Parameters progressively introduced in the structure factor and the scatteringdensity function can be adjusted by a refinement of observed data towards the proposed model, via a minimisation with respect to adjustable key parameters(positions, thermal parameters…)

( ) ( )[ ]∑

−=H H

obs HIHIR

2

2

mod

σ

rr

Another approach is the maximum entropy method

Model refinement can be very critical

Besides crystals, many applications to aperiodic structures, disordered solids (glass) and liquids

JDN 22, September 21-24 2014 28

( ) ( ) ( )( )

∫ ∑∑

∑

−−−=

−−=Σ

kj f

rQijff

rQik

titi

fff

jkf eaeaedte

QAQ

,0

~.*

~.

0

0

2

0

0

2

1

,

ψψψψπ

ωεεδψψω

εεωrr

rr

h

h

hrr

Quantum approach

Fermi Golden Rule

( )ω∂∂σ∂

,0

2

Qk

k

Ef

f

rΣ=

Ω

( ) ( )∫∞

−=Σ0

ti dtetQI2

1Q ω

πω ,,

r

h

r ( ) ( ) ( )∑−=

kj

rQitrQikj

kj eeaatQI,

0..*,rrrrr

( ) ( ) ( )tQQtQI ,A0,A,rrr

∗=


Space and time correlation function

( ) ( )∫∞

−=Σ0

ti dtetQI2

1Q ω

πω ,,

r

h

r

( ) ( )∫ Γ= rdetrtQI rQi rrr rr.,,

( ) ( ) ( )∫ ′′+′=Γ rdtrrn0rntrrrrrr

,,,

JDN 22, September 21-24 2014 30

( ) ( ) ( )

( ) ( )∑ ∑

∑ ∑

−−=

−−=Σ

00

2

000

00

2

00

,~

,

~,

ffff

fff

krNkp

QApQ

ωεεδψψ

ωεεδψψω

hrrr

hrr

Most general situation

System and beam in a statistical initial state

JDN 22, September 21-24 2014 31

Ef + ε f = E0 + ε0 f0 kkQrrr

−=

ε0

εmax

ω f

ω0

ε f

r k f

r k 0

r Q

2θ

(a) ω > 0

ω f

ω0

ε0

ε f

r k f

r k 0

r Q

2θ

(b) ω < 0

dk

k0≈

d hω( )104eV

Photonsλ0 ≈ 1Å( )

r k 0

r k f min

r k f max

r Q min

r Q max

dk

k0≈

d hω( )1

40eVNeutronsλ0 ≈ 1Å( )

r k 0

r k f min

r k f max

r Q min

r Q max

JDN 22, September 21-24 2014 32

( )∫∫∞−

Σ=Ω

=

Ω

0

dQk

kdE

E 0

ff

f

2

total

ω

ωω∂∂σ∂

∂∂σ

,r

h

Total scattering

ω = ω0 −ω f ≤ ω0

Staticapproximation ( )∫∫

∞−

Σ=Ω

=

Ω

0

dQk

kdE

E 0

ff

f

2

total

ω

ωω∂∂σ∂

∂∂σ

,r

h

∂σ∂Ω

total

≈ h Σr Q ,ω( )dω

−∞

+∞

∫ ≈ Sr Q ( ) Mostly valid for X Rays

JDN 22, September 21-24 2014 33

Elastic scattering

( ) ( )( ) ( ) ( )∑

∑

=

=

00

*00

0

2

000

~~

~

ψψ

ψψ

QAQApQS

QApQS

tot

elastic

rrr

rr

Total scattering

JDN 22, September 21-24 2014 34

Phonon scattering

( ) ( ) ( )10

...

A A

.....

+=

+

+

≈= ∑∑∑ +

l

lQil

l

lQi

l

ulQi euQbebebQA l

r

rr

r

rr

r

rrr rrr

Consider inelastic scattering for a model crystal with one atom/cell, where the system will go from vibrational state ψn to ψm

Displacement can be decomposed in phonons. We simplify by considering only one phonon, of wave vector q

( )( )( ) ( ) 22

.

2

12/1

..1

ξωωε

εξ

qnq

cceNm

u lqil

rrh

rr rr

=+=

+=

JDN 22, September 21-24 2014 35

nmA mn ≠= for 00 ψψ

ωω

ωεεψψ

hmhh

hm

h

nn

ff

nm

mn

m

k

m

kEE

nmA

22

1for 0

20

222

0

1

==

±=±=≠

Creation or absorption of phonon

Energy change can be very pronounced with neutrons, verysmall with X Rays (need for very high resolution

( )

= ∑ ±±±

l

lqQinnnn e

Nm

QbA

r

rrrrr

.11

. ψξψεψψ

( ) ( ) ( )10

...

A A

.....

+=

+

+

≈= ∑∑∑ +

l

lQil

l

lQi

l

ulQi euQbebebQA l

r

rr

r

rr

r

rrr rrr

JDN 22, September 21-24 2014 36

( )

= ∑ ±±±

l

lqQinnnn e

Nm

QbA

r

rrrrr

.11

. ψξψεψψ

One gets diffraction condition (coherent effect)

HqQrrr

=±

In a given scattering direction thus, with energy analyser, one getsω, kf, and thereforeq, ω(q)


Key application of neutrons, in particular for phase transitions

JDN 22, September 21-24 2014 38

( ) ( )( ) ( ) ( )QQQ

Q.QF

LS

Mrrrrrr

rrrr

Φ+Φ=Φ

Φ= σ

Magnetic scattering occurs through dipolar interaction betweenneutron and electron magnetic moments

( )( ) ( ) ( )rmrmrm

rdrmV

1

ls

rrrrrr

rrrr

+=

= ∫µ12/ ±=⇒⇒ z of units in spin neutron σσ hr

Neutron Magnetic scattering

( ) BrV nm

rrr.µ−=

ABrrr

×∇= jj

j

jj

RA

AA

µπ

µ rrr

rr

×

∇=

=∑

1

40

JDN 22, September 21-24 2014 39

Spin component

( ) ( )( ) ( )

QQq

cmrrr

QMrqQMqr

QQF

ee

SSS

sMs

/ˆ

10 54.913.1

ˆˆ

.

120

00

r

rrrrr

rrrr

=

=×==

=××=Φ

Φ=

−

⊥

γ

σ

( ) ( )[ ]

( ) ( )∑ −=

=

jjjs

ss

rrsrm

rmFTQM

rrrrr

rrrr

δ Spin density

JDN 22, September 21-24 2014 40

Orbital component, related to current density

( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )[ ]rjFTQJ

QJqQ

irQ

QQF

prrrrpm

erj

l

lMl

jjjjj

rrrr

rrrr

rrrr

rrrrrrrr

=

×−=Φ

Φ=

−+−−= ∑

ˆ2

.

2

0

π

σ

δδ( ) ( )

orbital conduction

⇑⇑

×∇+∇= rmrj l

rrrrrrψ

( ) ( ) ( )QMrqQMqrQ L0L0L

rrrrrr

⊥=××=Φ

Orbital magnetisation density defined to any gradient, but magneticamplitude uniquely defined

( ) ( )( ) ( ) ( )QMQMQM

QM.rQF

LS

0Mrrrrrr

rrrr

+=

= ⊥σ

JDN 22, September 21-24 2014 41

Case of spin only magnetism, at zero temperature

Magnetization density in units of 2µB., with a quantization axis 0z (appliedfield or natural quantization axis)

( ) ( )jj

jS rrsrmrrrrr

−=∑ δ

Ground state eigenstate of S2 and Sz.

↓↑ −= nnM2 S

( ) ( )rsMmrm SSMSzSMSz SS

rr =ΨΨ=

Normalized spin density ( ) ( ) ( ) rrrs nn

rrr

↓↑− −=↓↑

ρρ1

JDN 22, September 21-24 2014 42

Orbital magnetization

( ) ( ) ( )

∑

∑

=

−+−=

jj

jjjjjL

lL

lrrrrl4

1rm

rr

rrrrrrrr δδIn units of 2µB

Often, orbital moment is not a constant of motion. Often, spin orbitcoupling remains localized, and L remains a reasonably good quantum number. (L2, S2, J2, Jz) are the good quantum numbers

J)g2(L

J)1g(S

)1J(J2

)1S(S)1L(L)1J(J1g

JgS2L

rr

rr

rrr

−=

−=

++++−++=

=+

JDN 22, September 21-24 2014 43

Neutron elastic scattering

220

2

0

2

0

.

.

⊥↓↑↑↓

⊥↓↓

⊥↑↑

×==

−=

+=

MrII

MrFI

MrFI

n

n

rr

rr

rr

σ

σ

σ

If no spin analysis for scattered beam, two cross sections

↓↑↓↓↓

↑↓↑↑↑

+=+=

III

III

JDN 22, September 21-24 2014 44

Unpolarized neutrons

( ) 22

2

1mn FFIII +=+= −↑

221 / xFIFFx nnm +=⇒=

Magnetic effects hard to observe with unpolarized neutrons

220

2

0n

2

0n

MrII

M.rFI

M.rFI

⊥↓↑↑↓

⊥↓↓

⊥↑↑

×==

−=

+=

rr

rr

rr

σ

σ

σ

↓↑↓↓↓

↑↓↑↑↑

+=+=

III

III

JDN 22, September 21-24 2014 45

Polarized neutron diffraction

Incident beam of polarized neutrons (spin )σr

polarisingmonochromator

flippersingle crystal

Qr

vertical magnetic field

downvertical

polarisation

( ) ( ) ( )2

N QMQFQIrrrrr

r ⊥⋅+∝ σσ

( )QIr

rσ

JDN 22, September 21-24 2014 46

advantage of polarized neutrons for for weak magnetism

MM FF =⊥simple case : in horizontal planeMagnetization along Oz

Qr

γ+≅ 41R Linear en FM

( )2M

2N FFI +∝

polarised

negligeablesi FM<< FN

non polarised

22

MN

MN

11

FFFF

II

R

γ−γ+=

−+==

−

+

N

M

FF=γwith

Flipping ratio

JDN 22, September 21-24 2014 47

component of ⊥MF

rMF

r

α

z

y

x

MFr

⊥MF

rz

MF⊥r

Qr

α=⊥ sinFF MM

( )2MMz

M sinFsinFF α=α= ⊥⊥

aligned along vertical axis zMFr

Qr

⊥

General case : out of horizontal planeQr


Strong complementarity with X Ray high resolution diffraction, leading to charge density: mainly cohesive contribution towards sum of independent atoms

P. Coppensand I, with T. Koristzansky, initiated such combined study

Presently a strong project with LLD, CRM2, Spring8 and our lab SPMScombined refinement of charge and spin, and also charge / spin and momentum.

( ) ( ) ( )( ) ( ) ( )rrr

rrr

atomsind

rrr

rrr

.ρρδρρρρ

−=+= ↓↑

( ) ( ) ( ) rrrs nn

rrr

↓↑− −=↓↑

ρρ1

∑

∑

↓↑↓↑ =

=

ii

ii

i

nn

n

,,

2

ααα ϕρ

JDN 22, September 21-24 2014 49

Real crystals: extinction

kinematick vQIP 0=

( ) )(A 2sin

1-32

θλεεσ

V

FdQkinematic == ∫

ελ

θπ dd2sin

)2(d3

33 Ω=q

ελ

rrrrr 1+=+= HqHQ

( ) ( )( ) λ

θαπεα

πεααεσ 2sin

sin2

21 ldvQv == ∫

−

JDN 22, September 21-24 2014 50

n planes : t=nd . Number of possible 3-fold scattering

Probablility of rescattering

t=nd

Reflected wave by a plane is out of phase by π/2 with incident wave

Critical extinction length Λ2/12/1

2sin

==

=Λ

Q

d

F

V

Q λθλ

JDN 22, September 21-24 2014 51

In case of perfect regionst > Λ, diffraction power

Generally, Λ around50µm

If t > Λ, dynamical regime

If t < Λ, mixed regime, for which an extinction theory is applicable. This isfavoured by using small wavelenths (gamma, synchrotron)

θλ

2sin

2

V

FQ =Λ

Natural width of perfect crystal scattering function : d /Λ

Width of σ(ε) : d / t

2/12/1

2sin

==

=Λ

Q

d

F

V

Q λθλ

JDN 22, September 21-24 2014 52

Mosaic model

( )( ) ( ) ( ) ( )

∆=+=∆≈

+== ∫/ :parameter extinctionsecondary ofwidth

dW becomes width of ,

width function, spread mosaic

QTx

d/tβ

W

sηβσ

τττεσεσεσ

ηε

( )

( )( ) : considered be toextinctionprimary case In this

/* with /*/ QW : extinction 1 type

// : extinction 2 type

2

2

pp

s

s

xQyQ

dtTtQTx

tTQTx

⇒

=Λ===>>

Λ===<<

ηηεσβη

βεσσβη

( ) ( )spspp xyyxyy =

Darwin’s equations

Coherence is forgotten about

( )

( ) 02

01

0

IIx

I

IIx

I

σµσ

σµσ

++−=∂∂

++−=∂∂

Energy conservation besidesabsorption

Many applications from BC approach

Strong improvements due to higher source flux and therefore use of much smaller crystals

Huge problems with Xray and neutron diffraction, in particular due to rather large samples, Unability to refine simple basic structures !

We could justify parameters studying the occurrence of domains at ferro-electrictransition in Rb and K dihydrogeno-phosphate, as a function of temperature

Our Doliprane seems still to be OK. But nobody talks anymore about it.Be aware of H1 N1…….

↓↑↓↓↓↓

↑↓↑↑↑↑

++

=IyI

IyIR If σ+− is small, one can simplify as

↓

↑=y

yRR k

JDN 22, September 21-24 2014 57

Extinction is like cold: everybody suffers from it, but very few die from it !

I do hope that studies about « extinction cold » in pathological labs will connectwith reliable prescription of home doctors to crystallographers and they will thenenjoy their better life in crystallography

Conversion from Darwinism to Katolicism

Norio Kato made key studies which were the basis of the workperformed with Mostafa Al Haddad

Λ= 1

hκ

Long range order parameter, static Debye Waller factor

BALHADDad in the world of PhysicsEquations de Takagi

Could be applied to eigen Bloch states of disordered solids


We must try to maximize the effect of our interaction with humans (each

being perfect if fully isolated). But weshould keep in mind the mosaicity of

human population

basics of diffraction

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