April 1912: discovery of X-ray diffraction by crystals

Bragg reflections

The Braggs, father and son, develop methods for solving structures

Max von LaueNobel 1914

Walter Friedrich Paul Knipping

Munich 1912 :- Mineralogist (P. von Groth): crystals- Theoretician (A. Sommerfeld): Light-matter interaction- Experimental physics (W. Röntgen): X-rays

Ewald calculates the refraction index of lightin a 3D-periodic system and interviews Laue about it.

Cupric sulfatepentahydrate’’Blue vitriol’’

Diffraction by a periodic crystal

One atom/cell scattering factor 𝑓The complex amplitude is:

• Calculation of a geometric sum

Introduction: Crystal of 𝑁 × 𝑁 × 𝑁 cells

qx1 2 3 4 5

𝑁 = 8

Scattering function max at:

𝐴 𝒒 =

𝒖=𝟏

𝑵

𝒗=𝟏

𝑵

𝒘=𝟏

𝑵

𝑓𝑒−𝑖𝒒∙𝑹𝑢𝑣𝑤

= 𝑓

𝒖=𝟏

𝑵

𝑒−2𝑖𝜋𝑞𝑥𝑢

𝒗=𝟏

𝑵

𝑒−2𝑖𝜋𝑞𝑦𝑣

𝒘=𝟏

𝑵

𝑒−2𝑖𝜋𝑞𝑧𝑤

𝒒 = ℎ𝒂∗ + 𝑘𝒃∗ + 𝑙𝒄∗

𝒖=𝟏

𝑵

𝑒−2𝑖𝜋𝑞𝑥𝑢 =𝑒−𝑖𝜋(𝑁+1)𝑞𝑥sin(𝜋𝑁𝑞𝑥)

sin(𝜋𝑞𝑥)

𝐼 𝒒 = 𝑓2sin2(𝜋𝑁𝑞𝑥)

sin2(𝜋𝑞𝑥)

sin2(𝜋𝑁𝑞𝑦)

sin2(𝜋𝑞𝑦)

sin2(𝜋𝑁𝑞𝑧)

sin2(𝜋𝑞𝑧)

𝐼(𝑞𝑥)

𝑁3𝑓2

Laue equations- 1Crystal

• Total electron density 𝜌𝑡𝑜𝑡(𝑟)

• Electron density of a unit cell 𝜌 𝒓 (no disorder)

=

• Kinematic approximation• Perfect periodicity𝜌𝑡𝑜𝑡 𝒓 =

𝑢𝑣𝑤

𝜌𝑢𝑣𝑤(𝒓 − 𝑹𝑢𝑣𝑤)

𝜌𝑢𝑣𝑤 𝒓 = 𝜌(𝒓)

𝜌(𝒓) ∗

𝑢𝑣𝑤

𝛿 𝒓 − 𝑹𝑢𝑣𝑤 𝜎(𝒓) = 𝜌𝑡𝑜𝑡 𝒓

∗

Laue equation - 2 FT of 𝜌𝑡𝑜𝑡 𝒓

×

𝐴 𝒒 = 𝐹 𝒒 × 2𝜋 3 Σ(𝒒) ∗ 𝑣∗

ℎ𝑘𝑙

𝛿(𝒒 − 𝑸ℎ𝑘𝑙)

𝜌𝑡𝑜𝑡 𝒓 = 𝜌 𝒓 ∗ 𝜎(𝒓) ×

𝑢𝑣𝑤

𝛿(𝒓 − 𝒓𝑢𝑣𝑤)

∗

Laue equations - 3

• Each node of the RS replaced by S(q)

Intensity is max when𝒒 belongs to the RL

𝐴 𝒒 = 𝐹(𝒒) × Σ(𝒒) ∗1

𝑣

ℎ𝑘𝑙

𝛿(𝒒 − 𝑸ℎ𝑘𝑙) 𝐴 𝒒 = 𝐹(𝒒) ×1

𝑣

ℎ𝑘𝑙

Σ(𝒒 − 𝑸ℎ𝑘𝑙)

𝐴 𝑞𝑥 𝐹 𝑞𝑥

𝐼 𝒒 = 𝐹(𝒒) 2 ×1

𝑣2

ℎ𝑘𝑙

Σ(𝒒 − 𝑸ℎ𝑘𝑙)2

𝐹 𝑸ℎ𝑘𝑙 = 𝐹ℎ𝑘𝑙

𝐼 𝑞𝑥 𝐹 𝑞𝑥2

• Crystal size cell parameters, no cross terms:

Σ(𝒒) and coherence

Intensity around the 002 Bragg reflexionmesured on a synchrotron radiation facility (Diamond Light Source).From Maxime Dupraz Thesis, 2015

In order to measure Σ(𝒒), X-ray must interfereon the whole crystal:

Transverse coherence length > crystal size (1µm) 3rd generation x-ray source

Gold on sapphireSEM image

100 nm

S. Labat, N. Vaxelaire, IM2NP Marseille, CRISTAL beamline Synchrotron SOLEIL

Intensity around the 002 Bragg reflexionfrom a Au grain in a 200 nm filmmesured on a synchrotron radiation facility (SOLEIL).

Structure factor

Spherical approximation:binding electron are neglected

TF of electron density of the cell

• ℎ, 𝑘, 𝑙,Miller indices,

• 𝑢𝑗, 𝑣𝑗, 𝑤𝑗, atomic reduced coordinates (𝑟𝑗 = 𝑢𝑗 𝑎 + 𝑣𝑗 𝑏 + 𝑤𝑗 𝑐)

Ex: two identical atomes in +ua and -ua

𝐹 𝑸ℎ𝑘𝑙 = 𝐹ℎ𝑘𝑙

𝜌𝑢𝑣𝑤 𝒓 =

𝑗

𝜌𝑗(𝒓 − 𝒓𝑗)

𝐹 𝒒 = න𝜌𝑢𝑣𝑤 𝒓 𝑒−𝑖𝒒∙𝒓𝑑3𝒓 =

𝑗

𝑓𝑗 𝑒−𝑖𝒒∙𝒓𝑗

𝐹ℎ𝑘𝑙 =

𝑗

𝑓𝑗𝑒−2𝑖𝜋(ℎ𝑢𝑗+𝑘𝑣𝑗+𝑙𝑤𝑗)

𝐹ℎ𝑘𝑙 = 2𝑓𝑗 cos 2𝜋ℎ𝑢

Centrosymetry : 𝐹ℎ𝑘𝑙 real ; 𝐹ℎ𝑘𝑙 = 𝐹ഥℎത𝑘 ҧ𝑙

Diffracted intensity

• Position of the spots: lattice• Intensity of the spots: basis• Shape of the spots: crystal

Atom

Basis

Lattice

Crystal

Scattering factor

Structure factor

Reciprocal lattice

𝛴(𝒒)

Ewald construction

Geometricalinterpretation of diffraction

• Elastic scattering : 𝑘𝑖 = 𝑘𝑑 = 2𝜋/𝜆• Laue equations: scattering vector 𝒒 belongs to RL

𝒌𝑑

Crystal

𝒒

𝒌𝑖

O2𝝅/𝝀

Diffraction occurs for nodes that lie on the sphere

RS Origin

Ewald’s sphere of reflexion

Ewald’s sphere

© Gervais Chapuis

Laue Bragg

𝒒 = 𝑸ℎ𝑘𝑙q

dhkl

2p/l

O

If Qmh,mk,ml on Ewald’s sphere:

𝑞 = 2𝑘 sin 𝜃 =4𝜋

𝜆sin 𝜃

𝑞 = 𝑄ℎ𝑘𝑙 =2𝜋

𝑑ℎ𝑘𝑙2𝑑ℎ𝑘𝑙 sin 𝜃 = 𝜆

𝑞 = 𝑄𝑚ℎ,𝑚𝑘,𝑚𝑙 = 𝑚2𝜋

𝑑ℎ𝑘𝑙2𝑑ℎ𝑘𝑙 sin 𝜃 = 𝑚𝜆

Experimental techniques

In a 3D crystal, the number of nodes layingon the Ewald’s sphere is very small.

• The Laue method (many wave lengths l)• The rotating crystal method (many orientations)• The powder method (many crystals)

𝒌𝑑

Crystal

𝒒

𝒌𝑖

O2𝝅/𝝀

RS Origin

The Laue method

Crystal O

2p/lmin

2p/lmax

kd

1st diffraction pattern ever (CuSO4)Max von Laue, Walter Friedrich, Paul Knipping

Diffraction of a white beam

Laue pattern of MbCOOne pulse od 150 ps (ESRF ID13)2000 reflections ( E=7-38 keV )

Rotating crystal

All accessible nodespass through the Ewald’s sphere

𝒌𝑑

𝒒

𝒌𝑖

𝑂

The powder methodChaque nœud Qhkl décrit une sphère

𝒌𝑑

𝒌𝑖

O

Qhkl

Debye-Scherrer method2𝜃

One line: one distance 𝑑ℎ𝑘𝑙

Powder:Set of small crystallites (1-10 mm)

in any orientation.

2𝜃

Powder diffraction… on Mars… in 2012!

Rocks similar to Mauna Loa, Hawaii

15 kg24h exp. time

Exemple InSb under pressure• l = 0.447 Å

• Phase transition fcc orthorhombic

Ambiant pressure 4.9 GPa (49 kbar)

From M. McMahon

Cubic(111) Orthorhombic(220)

(311)

Diamond anvilcellDiamond anvil

cell

• 1-500 Gpa• 5000 K (Laser heating)

Laser source

X-rays

sample

gasket

ruby

XASmeasurements

Diffractionmeasurement

Principle of structure determination

Goal: to find electron density of the crystal

𝐹ℎ𝑘𝑙 are the Fourier coefficients of the Fourier series expansion of

the electron density 𝜌𝑡𝑜𝑡(𝑟)

Formally : with, for a periodical crystal:𝜌𝑡𝑜𝑡 𝒓 =1

(2𝜋)3න𝐴(𝒒)𝑒𝑖𝒒∙𝒓𝑑3𝒒

𝐴 𝒒 =(2𝜋)3

𝑣

ℎ𝑘𝑙

𝐹ℎ𝑘𝑙𝛿(𝒒 − 𝑸ℎ𝑘𝑙)

𝜌𝑡𝑜𝑡 𝑥, 𝑦, 𝑧 =1

𝑣

ℎ𝑘𝑙

𝐹ℎ𝑘𝑙 𝑒2𝑖𝜋(ℎ𝑥+𝑘𝑦+𝑙𝑧)

Phase problem

Only the intensity of Bragg reflexions, ∝ |𝐹ℎ𝑘𝑙|2

are measured

Generally, phases of 𝐹ℎ𝑘𝑙 cannot be obtainedexperimentally.

Measured intensity at 𝑸-vectors such that:

𝑸ℎ𝑘𝑙 < 𝑄𝑚𝑎𝑥.< 4𝜋/𝝀

rtot(r) is convoluted by function of width ~ 2𝜋/𝑄𝑚𝑎𝑥 :

La resolution is given by: 2𝜋/𝑄𝑚𝑎𝑥 ( mini = l/2 )

kd

q

ki

4p/l

Sphere of resolution

Resolution

𝜌𝑒𝑥𝑝 𝒓 =1

(2𝜋)3න𝐴 𝒒 𝑅(𝒒)𝑒𝑖𝒒∙𝒓𝑑3𝒒 = 𝜌𝑡𝑜𝑡 𝒓 ∗ 𝑇𝐹−1(𝑅 𝒒 )

Metal (Cu, Ni, Pt)

Determination of the phase by comparison…

Exemple : Structure of phthalocyaninJ. Monteath Roberston, J. Chem. Soc. 615, (1935); 1185 (1936)

Electron density mapCalculation of 1800 sums of 150 terms (270 000)

Phase +Phase -

‘‘Sines’’Fourierseries

Structure determination by Fourier series

The molecule is tilted

Extinctions

Screw axisExample: screw axis 21, 𝒄 direction

Structure factor contains terms such as:

(00l) l = 2nCondition:

q // axis (translation t)

q.t = 2n

In the general case

Reciprocal plane h=0

b*

c*

(x, y, z)

(-x, -y, z+1/2)

bc

c/2

a 𝑓 𝑒−2𝑖𝜋 ℎ𝑥+𝑘𝑦+𝑙𝑧 + 𝑒−2𝑖𝜋 −ℎ𝑥−𝑘𝑦+𝑙 𝑧+1/2 =

𝑓𝑒−2𝑖𝜋𝑙𝑧 𝑒−2𝑖𝜋 ℎ𝑥+𝑘𝑦 + (−1)𝑙𝑒2𝑖𝜋 ℎ𝑥+𝑘𝑦

1 s

10-3 s

10-6 s

10-9 s

10-12 s

10-15 sPrinciple of pump-probe

experiments

Stroboscopic measurements

t

Pump Probe

delay

Excited state

Repetition rate

Groundstate

• Study of metastable states (chemical reactions, e- desexcitations, Phase transitions)

• Very short livetime (fs to ms)• A first pulse (pump) initiates a modification of the system,

A second pulse (probe) probes it after certain delay.

Femtochemistry Ahmed H. ZewailNobel prize in chemistry (1999)

e- frequencies13.6 eV 3.2 as

Molecular vibrationsChemical reactions

Acoustic phonons

Photo-inducedtransformations

Tsonde ~ Tpompe << Tretard << Trép.

S

1 fs -> 0,3 µm

1.8 fs X-ray pulsesat LCLS in 2010

e-e int

e-ph int

Neutral (P21/n) Ionic/ferroelectric (Pn)

D+

A-

D+

A-

21

Long-range ferroelectric orderphoto-induiced in ~ 500 ps(Laser 800 nm)

ESRF ID9: E.Collet et al., Science 300, 612 (2003)

TTF

CA

Exciton

Mecanismof phase transitions in time

(not in temperature…)

Photo-induced phase transition: ~ 500 ps

n n n n

Electron density

Precise measurements of int. Electron density

• Chimical bonding• Electrostatic potential, charge transfer, dipolar moment, etc.

• Calculation of 𝐹ℎ𝑘𝑙 in the spherical approximation

Deformation map of electron density

𝜌𝑡𝑜𝑡 𝑥, 𝑦, 𝑧 =1

𝑣

ℎ𝑘𝑙

𝐹ℎ𝑘𝑙 𝑒2𝑖𝜋(ℎ𝑥+𝑘𝑦+𝑙𝑧)

𝐹ℎ𝑘𝑙𝑐𝑎𝑙𝑐 =

𝑗

𝑓𝑗 𝑒−2𝑖𝜋(ℎ𝑢𝑗+𝑘𝑣𝑗+𝑙𝑤𝑗)

𝜌𝑑𝑒𝑓 𝑥, 𝑦, 𝑧 =1

𝑣

ℎ𝑘𝑙

(𝐹ℎ𝑘𝑙−𝐹ℎ𝑘𝑙𝑐𝑎𝑙𝑐) 𝑒2𝑖𝜋(ℎ𝑥+𝑘𝑦+𝑙𝑧)

Examples of maps

Contour 0.05 eÅ-3 (Zobel et al. 1992)

Oxalic acid 15 K

C

OO

O O

C

H

H

Doublets libres

H2O in LiOH.H2O

Contour 0.005 eÅ-3

From Vainshtein

Static deformation mapHexabromobenzène C6Br6

rstat(r)= rmultipole(r)- rspherical(r)

d+

d-

+

++max

0 0

33 ),()'.(')()()(l

l

l

m

lmlmlvalvcoremul yPrRrPrr qrrr

The anisotropic distribution electron density around halogen atom originates the halogen-halogen interaction

Multipolar development of electron density(Hansen-Coppens model)

From S. Dahaoui et al., Angew. Chem. Int. Ed., 2009, 48, 3838

d-

d+