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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
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+