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Lattices and Symmetry Scattering and Diffraction (Physics)
James A. KadukINEOS Technologies
Analytical Science Research Services
Naperville IL [email protected]
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Harry Potter and the Sorcerer’s (Philosopher’s) Stone
Ron:
Seeker? But first years never make the house team. You must be the youngest Quiddich
player in …
Harry:
… a century. According to McGonagall.Fred/George:
Well done, Harry. Wood’s just told us.
Ron:
Fred and George are on the team, too. Beaters.Fred/George:
Our job is to make sure you
don’t get
bloodied up too bad.
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Alastor
“Mad-Eye” Moody –
“Constant Vigilance”
Harry Potter and the Goblet of Fire
(2005)
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The crystallographer’s world view
Reality can be more complex!
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Twinning at the atomic level
International Tables for Crystallography, Volume D, p. 438
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PDB entry 1eqg = ovine COX-1 complexed
with Ibuprofen
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Atoms pack together in a regular pattern to form a crystal. There are two aspects to this pattern:
PeriodicitySymmetry
First consider the periodicity…
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To describe the periodicity, we superimpose (mentally) on the
crystal structure a lattice. A lattice is a regular array of
geometrical points, each of which has the same environment (they
are all equivalent).
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A Primitive Cubic Lattice (CsCl)
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A unit cell
of a lattice (or crystal) is a volume which can describe the
lattice using only translations. In 3 dimensions (for crystallographers),
this volume is a parallelepiped. Such a volume can be defined by six
numbers –
the lengths of the three sides, and the angles between them –
or three basis vectors.
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A Unit Cell
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a, b, c, α, β, γ a, b, c
x1
a + x2
b + x3
c, 0 ≤
xn
< 1 lattice points = ha + kb + lc,
hkl
integers domain of influence
–
Dirichlet
domain, Voronoi
domain, Wigner-Seitz cell, Brillouin
zone
Descriptions of the Unit Cell
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A Brillouin
Zone
C. Kittel, Introduction to Solid State Physics, 6th
Edition, p. 41 (1986)
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Bilbao
Crystallographic Server
http://www.cryst.ehu.es/
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The unit cell is not unique (c:\MyFiles\ACA\
ACA_big_07\index2.wrl)
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How do I pick the unit cell?
•
Axis system (basis set) is right-handed•
Symmetry defines natural directions and boundaries
•
Angles close to 90°•
Standard settings of space groups
•
To make structural similarities clearer
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The Reduced Cell
•
3 shortest non-coplanar translations•
Main Conditions (shortest vectors)
•
Special Conditions (unique)
•
May not exhibit the true symmetry
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The Reduced Form
a·aA
b·bB
c·cC
b·cD
a·cE
a·bF
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Positive Reduced Form, Type I Cell, all angles < 90°, T
= (a·b)(b·c)(c·a) > 0
Main conditions:a·a ≤
b·b ≤
c·c b·c ≤
½
b·b
a·c ≤
½
a·a a·b ≤
½
a·aSpecial conditions:
if a·a = b·b then b·c ≤
a·c
if b·b = c·c then a·c ≤
a·b
if b·c = ½
b·b then a·b ≤
2 a·c
if a·c = ½
a·a then a·b ≤
2 b·c
if a·b = ½
a·a then a·c ≤
2 b·c
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Negative reduced Form, Type II Cell all angles ≥
90°, T
= (a·b)(b·c)(c·a) ≤
0
Main Conditions:a·a ≤
b·b ≤
c·c |b·c| ≤
½
b·b|a·c| ≤
½
a·a |a·b| ≤
a·a( |b·c| + |a·c| + |a·b| ) ≤
½ ( a·a + b·b )Special Conditions:
if a·a = b·b then
|b·c| ≤
|a·c|if b·b = c·c then
|a·c| ≤
|a·b|if |b·c| = ½
b·b then
a·b = 0if |a·c| = ½
a·a then
a·b = 0if |a·b| = ½
a·a then
a·c = 0if ( |b·c| + |a·c| + |a·b| ) = ½
( a·a + b·b ) then a·a ≤
2 |a·c| + |a·b|
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There are 44 reduced forms. The relationships among the six terms determine the Bravais
lattice of
the crystal.
J. K. Stalick
and A. D. Mighell, NBS Technical Note 1229, 1986.
A. D. Mighell
and J. R. Rodgers, Acta
Cryst., A36, 321-326 (1980).
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“The Normalized Reduced Form and Cell: Mathematical Tools for Lattice
Analysis –
Symmetry and Similarity”, Alan D. Mighell, J. Res.
Nat. Inst. Stand. Tech., 108(6), 447-452 (2003).
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26International Tables for Crystallography, Volume F, Figure 2.1.3.3, p.52 (2001)
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“The mystery of the fifteenth Bravais
lattice”, A. Nussbaum,
Amer. J. Phys.,
68(10), 950-954 (2000).
http://ojps.aip.org/ajp/
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Symmetry Groups and Their Applications, W. Miller, Jr., Academic
Press, New York (1972), Chapter 2.
1 2 / 4 / 3
31 6 /
m mmm mmm m
m mmm
⊂ ⊂ ⊂ ⊂∩ ∩
⊂
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The 44 Reduced Forms
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A
= B
= C
Number Type D E F Bravais
1 I A/2 A/2 A/2 cF
2 I D D D hR
3 II 0 0 0 cP
4 II -A/3 -A/3 -A/3 cI
5 II D D D hR
6 II D* D F tI
7 II D* E E tI
8 II D* E F oI
* 2|D
+ E
+ F| = A
+
B
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A
= B, no conditions on CNumber Type D E F Bravais
9 I A/2 A/2 A/2 hR
10 I D D F mC
11 II 0 0 0 tP
12 II 0 0 -A/2 hP
13 II 0 0 F oC
14 II -A/2 -A/2 0 tI
15 II D* D F oF
16 II D D F mC
17 II D* E F mC
* 2|D
+ E
+ F| = A
+
B
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B
= C, no conditions on ANumber Type D E F Bravais
18 I A/4 A/2 A/2 tI
19 I D A/2 A/2 oI
20 I D E E mC
21 II 0 0 0 tP
22 II -B/2 0 0 hP
23 II D 0 0 oC
24 II D* -A/3 -A/3 hR
25 II D E E mC
* 2|D
+ E
+ F| = A
+
B
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No conditions on A, B, CNumber Type D E F Bravais
26 I A/4 A/2 A/2 oF
27 I D A/2 A/2 mC
28 I D A/2 2D mC
29 I D 2D A/2 mC
30 I B/2 E 2E mC
31 I D E F aP
32 II 0 0 0 oP
40 II -B/2 0 0 oC
35 II D 0 0 mP
36 II 0 -A/2 0 oC
33 II 0 E 0 mP
38 II 0 0 -A/2 oC
34 II 0 0 F mP
42 II -B/2 -A/2 0 oI
41 II -B/2 E 0 mC
37 II D -A/2 0 mC
39 II D 0 -A/2 mC
43 II D† E F mI
44 II D E F aP
† 2|D
+ E
+ F| = A
+
B, plus |2D
+ F| = B
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Indexing programs can get “caught” in a reduced cell, and miss the (higher) true symmetry. It’s always worth a
manual check of your cell.
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The metric symmetry
can be higher than the crystallographic symmetry!
(A monoclinic cell can have β
= 90°)
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Centered Cells conventional crystallographic basis/cell
•
Point group symmetry of the lattice (holohedry)-1, 2/m, mmm, 4/mmm, -3m, 6/mmm, m-3m
•
Basis vectors (and sides) contain symmetry elements
•
2, 3, or 4 lattice points / unit cell•
P, A, B, C, R, I, F
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39http://www.haverford.edu/physics-astro/songs/bravais.htm
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Definitions
[hkl]
indices of a lattice direction <hkl>
indices of a set of symmetry-
equivalent lattice directions (hkl)
indices of a single crystal face
{hkl}
indices of a set of all symmetry- equivalent crystal faces
hkl
indices of a Bragg reflection
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Now consider the symmetry…
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Point Symmetry Elements
•
A point symmetry operation does not alter at least one point upon which it operates–
Rotation axes
–
Mirror planes–
Rotation-inversion axes (rotation-reflection)
–
Center
Screw axes and glide planes are not point symmetry elements!
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Symmetry Operations•
A proper symmetry operation
does not invert the
handedness of a chiral
object–
Rotation
–
Screw axis–
Translation
•
An improper symmetry operation
inverts the handedness of a chiral
object
–
Reflection–
Inversion
–
Glide plane–
Rotation-inversion
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Not all combinations of symmetry elements are possible. In addition,
some point symmetry elements are not possible if there is to be translational symmetry as well. There are only 32
crystallographic point groups consistent with periodicity in three
dimensions.
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The 32 Point Groups (1)
International Tables for Crystallography, Volume A, Table 12.1.4.2, p.819 (2002)
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The 32 Point Groups (2)
International Tables for Crystallography, Volume A, Table 12.1.4.2, p.819 (2002)
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Symbols for Symmetry Elements (1)
International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002)
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48
Symbols for Symmetry Elements (2)
International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002)
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Symbols for Symmetry Elements (3)
International Tables for Crystallography, Volume A, Table 1.4.2, p. 7 (2002)
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2 Rotation Axis (ZINJAH)
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3 Rotation Axis (ZIRNAP)
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4 Rotation Axis (FOYTAO)
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6 Rotation Axis (GIKDOT)
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-1 Inversion Center (ABMQZD)
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55
-2 Rotary Inversion Axis?
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m Mirror Plane (CACVUY)
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57
-3 Rotary Inversion Axis (DOXBOH)
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-4 Rotary Inversion Axis (MEDBUS)
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-6 Rotary Inversion Axis (NOKDEW)
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21 Screw Axis (ABEBIS)
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31 Screw Axis (AMBZPH)
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62
32 Screw Axis (CEBYUD)
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63
41 Screw Axis (ATYRMA10)
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64
42 Screw Axis (HYDTML)
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43 Screw Axis (PIHCAK)
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61 Screw Axis (DOTREJ)
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62 Screw Axis (BHPETS10)
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68
63 Screw Axis (NAIACE)
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69
64 Screw Axis (TOXQUS)
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70
65 Screw Axis (BEHPEJ)
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71
c Glide (ABOPOW)
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n Glide (BOLZIL)
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73
d (diamond) Glide (FURHUV)
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74
What does all this mean?
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75
Symmetry information is tabulated in International Tables for
Crystallography, Volume A edited by Theo Hahn
Fifth Edition 2002
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76
Guaifenesin, P21
21
21
(#19)
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77
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78© Copyright 1997-1999. Birkbeck College, University of London.
http://img.chem.ucl.ac.uk/sgp/mainmenu.htm
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79
http://www.aps.anl.gov/Xray_Science_ Division/Powder_Diffraction_
Crystallography/SymmetryRietveld.html
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80
Hermann-Mauguin
Space Group Symbols the centering, and then a set of characters indicating the
symmetry elements along the symmetry directions
Lattice Primary Secondary TertiaryTriclinic None
Monoclinic unique (b
or c)
Orthorhombic [100] [010] [001]Tetragonal [001] {100} {110}Hexagonal [001] {100} {110}
Rhom. (hex) [001] {100}Rhom. (rho) [111] {1-10}
Cubic {100} {111} {110}
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81
Alternate Settings of Space Groups
•
Triclinic –
none•
Monoclinic –
(a) b
or c
unique, 3 cell choices
•
Orthorhombic –
6 possibilities•
Tetragonal –
C
or F
cells
•
Trigonal/hexagonal –
triple H
cell•
Cubic
•
Different Origins
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82
An Asymmetric Unit
A simply-connected smallest closed volume which, by application of all symmetry operations, fills all
space. It contains all the information necessary for a complete description of the crystal structure.
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83
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84
Sub-
and Super-Groups
•
Phase transitions (second-order)•
Overlooked symmetry
•
Relations between crystal structures•
Subgroups–
Translationengleiche
(keep translations, lose class)
–
Klassengleiche
(lose translations, keep class)–
General (lose translations and class)
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85
A Bärninghausen
Tree for
translationengleiche
subgroups
International Tables for Crystallography,
Volume 1A, p. 396 (2004)
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86
Mercury/ETGUAN (P41
21
2
#92)
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87
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88
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89
Not all space groups are possible for protein crystals.
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90
Space Group Frequencies in theProtein Data Bank, 17 June 2003
Space Group Number0 20 40 60 80 100 120 140 160 180 200 220
# En
trie
s
1
10
100
1000
10000
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91
Space Group Frequencies
Space Group Number0 20 40 60 80 100 120 140 160 180 200 220
Freq
uenc
y of
Occ
urre
nce,
%
0.01
0.1
1
10
100
PDB % CSD % ICSD %
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92
Some Classifications of Space Groups
•
Enantiomorphic, chiral, or dissymmetric
– absence of improper rotations
(including , = m, and )•
Polar
–
two directional senses are
geometrically or physically different
1̄ 2̄ 4̄
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93
Basic Diffraction Physics
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94
Bragg’s Law
2 sinn dλ θ=
1 sin2d
θλ
=
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95
Bragg’s Law
V. K. Pecharsky
and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 148 (2003)
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96
Optical Diffraction
PSSC Physics, Figure 18-A, p. 202-203 (1965)
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97
Optical Diffraction
PSSC Physics, Figure 18-B, p. 202-203 (1965)
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98
Optical Diffraction
D. Halliday
and R. Resnick, Physics, p. 1124 (1962)
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99
Oscillation directionof the electron
X-ray beam
Electric vector ofThe incident beam
Electron
ϕ
Scattering by One Electron
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100
Scattering by One Electron
•
Inelastic ( = Compton scattering = a component of the background) and elastic
•
Phase difference between incident and scattered beams is π
•
The scattered energy (intensity) is:22
20 2 2
1 sineleI I
r mcϕ
⎛ ⎞= ⎜ ⎟
⎝ ⎠
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101
Electromagnetic Waves
( 0; ) cos 2 zE t z A πλ
= =
E λ
AA
z
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102
During a time t
the wave travels over a distance tc
= tλν
Therefore at time t, the field strength at position z
is what it was at t
= 0 and
position z
–
tλν:1( , ) cos 2 ( )
cos 2 cos 2
E t z A z t
z zA t A tc
π λνλ
π ν πνλ
= −
⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
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103
( ,0) cos 2cos
E t A tA t
πνω
==
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104
Consider a new wave displaced by a distance Z
from the original wave:
Z
corresponds to a phase shift 2π(Z/λ) = α
E Z
z
new wave
original wave
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105
( ,0) cos
( ,0) cos( )orig
new
E t A t
E t A t
ω
ω α
=
= +
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106
cos( ) cos cos sin sincos cos sin cos( 90 )
A t A t A tA t A t
ω α α ω α ωα ω α ω
+ = −= + + °
cos( ) cos sini
A t A iAAe α
ω α α α+ = +
=
imaginary axis
real axisAα
Acosα
Asinα
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107
Scattering by Two Electrons
1
2
pq
r2
•1 s
s0 θ
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108
Scattering by Two Electrons
•
Let the magnitudes of s0
and s = 1/λ•
Diffracted beams 1 and 2 have the same magnitude, but differ in phase because of the path difference p
+ q
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109
0
0
sincos(90 )
1 cos(90 )
( )
p rr
r
qp q
θθ
λ θλ
λλ
λ
== −
= −
= ⋅= − ⋅
+ = ⋅ −
r sr s
r s s
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110
The phase difference of wave 2 with respect to wave 1 is:
0
0
2 ( ) 2πλ πλ⋅ −
− = ⋅
= −
r s s r S
S s ss0 s0
s S
θ
“reflecting plane”
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111
2 sin2sin
θθ
λ
=
=
S s
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112
Scattering by an Atomρ(r)
ρ(-r)
-r
+r
nucleus
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113
The Atomic Scattering Factor
[2 ]
[2 ] [ 2 ]
( )
( )
2 ( ) cos[2 ]
i
i i
f e d
e e d
d
π
π π
ρ
ρ
ρ π
⋅
⋅ − ⋅
=
⎡ ⎤= +⎣ ⎦
= ⋅
∫
∫
∫
r S
r
r S r S
r
r
r r
r r
r r S r
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114
Atomic Scattering Factors
V. K. Pecharsky
and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 213 (2003)
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115
Scattering from a Row of Atoms
M. J. Buerger, X-ray Crystallography, Fig. 14B, p. 32 (1942)
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116
Scattering from a Row of Atoms
cos
cos
cos cos(cos cos )
cos cos
OQ PR mOQa
PRa
a a ma m
ma
λ
ν
μ
ν μ λν μ λ
λν μ
− =
=
=
− =− =
= +
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117
Scattering from a Row of Atoms
M. J. Buerger, X-ray Crystallography, Fig. 15, p. 33 (1942)
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118
Scattering by a Plane of Atoms
M. J. Buerger, X-ray Crystallography, Fig. 16, p. 34 (1942)
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119
Scattering by a Unit Cell
J. Drenth, Principles of Protein X-ray Crystallography,
Fig. 4.12 p. 80 (1999)
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120
Scattering by a Unit Cell2
2
1( )
j
j
ij j
ni
jj
f e
f e
π
π
⋅
⋅
=
=
=∑
r S
r S
f
F S
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121
Scattering by a Crystal
The scattering of this unit cellwith O as the origin is F(S)
The scattering of this unit cellwith O as the origin is:
F(S)exp[2πita·S]exp[2πiub·S]exp[2πivc·S]
O a
c
ac
ta+ub+vc
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122
For a unit cell with its own origin at ta + ub + vc
the scattering is
and the total scattering by the crystal is
2 2 2( ) it iu ive e eπ π π⋅ ⋅ ⋅× × ×a S b S c SF S
31 22 2 2
0 0 0( ) ( )
nn nit iu iv
t u ve e eπ π π⋅ ⋅ ⋅
= = =
= × × ×∑ ∑ ∑a S b S c SK S F S
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123
Scattering by a Crystal
2πa.S t=0
t=1
t=2t=3t=4
t=5
t=6
t=7
t=8
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124
Ewald
–
oscillators Laue
–
spacing?
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125
The Fourier transform is an equation to calculate the frequency, amplitude and
phase of each sine wave needed to make up any given signal x(t):
( ) ( ) 2i ftF f x t e dtπ+∞ −
−∞= ∫
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126
Brightness image / Fourier transform
http://cns-alumni.bu.edu/~slehar/fourier/fourier.html
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127
Kevin Cowtan’s
Fourier Duck http://www.ysbl.york.ac.uk/~cowtan/fourier/magic.html
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128
Kevin Cowtan’s
Fourier Cat http://www.ysbl.york.ac.uk/~cowtan/fourier/magic.html
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129
Magnitudes from duck and phases from cat
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130
Magnitudes from cat and phases from duck
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131
A tail of two cats http://www.ysbl.york.ac.uk/~cowtan/fourier/coeff.html
Magnitudes only
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132
A similar structure – tail-less Manx cat
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133
Known magnitudes and Manx phases “an |Fo| map”
Despite the fact that the phases contain more structural information about the image than the magnitudes, the missing tail is restored at about half of its original weight. This occurs only when the phases are almost correct. The factor of one half arises because we are making the right correction parallel to the estimated phase, but no correction perpendicular to the phase (and <cos2>=1/2). There is also some noise in the image.
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134
“A 2|Fo|-|Fc| map”
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135
A crystal does not scatter X-rays unless
hkl
⋅ =⋅ =⋅ =
a Sb Sc S
These are the Laue
conditions
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136
Now remember
1
1
1
h
k
l
⋅ =
⋅ =
⋅ =
a S
b S
c S
s0 s0
s S
θ
“reflecting plane”
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137
The “reflecting planes” are lattice planes
reflecting planesdirection of Salong this line
a
b
a/h
b/kd
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138
Consider just one direction
1proj on ; 1, ,
2sin2 sin
or, since (or ) can be any integer2 sin
d but so dh h
d
dd h
n d
λθ
λ θ
λ θ
= ⋅ = =
=
=
=
a aS SS
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139
The Reciprocal Lattice
The idea of a reciprocal lattice predates crystallography. It was invented by J. W. Gibbs in the late 1880s, and its utility for
describing diffraction data was realized by P. P. Ewald
in 1921.
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140
The Reciprocal Lattice
For any lattice with basis vectors a, b, and c, construct another lattice with basis vectors a*, b*, and c* such
thata·a* = b·b* = c·c* = 1 and
a·b* = a·c* = b·a* = b·c* = c·a* = c·b* = 0Therefore,
a* = K(b×c) and K
= 1/[a·(b×c)]K
is 1/V
if a, b, and c form a right-handed system.
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141
Why do we care?
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142
Remember the Laue
conditions:
hkl
⋅ =⋅ =⋅ =
a Sb Sc S
S ⊥
a reflecting plane = a lattice plane. The
equation of such a plane through the
origin ishx
+ ky
+ lz
= 0
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143
The reflecting plane contains general vectors and lattice vectors:
r = xa + yb + zc
rL = ua + vb + wc u, v, and w
integers
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144
S is perpendicular to any vector in the plane, or
S·(r –
rL ) = 0 S·r = S·
rL
S·
rL = n (the planes don’t have to
pass through the origin)
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145
r = ua + vb + wc
S·a = h S·b = k S·c = l
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146
Consider the possibility of a different basis set for S:
S = UA + VB + WC
r = ua + vb + wc
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147
2
cos cos 1 cos 1 cos
1 cos cos cos cos 1
cos 1 cos 1 cos cos11 cos cos
cos 1 coscos cos 1
hkl
h h ha a a
h k lk k kb b ba b c
l l lc c c
d
γ β β γ
α γ α γ
α β β α
γ βγ αβ α
+ +
=
Triclinic
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148
(UA + VB + WC)·(ua + vb + wc) = n
( )
( )
( )u
U V W v nw
uU V W v n
w
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⋅ =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⋅ ⋅ ⋅⎛ ⎞⎛ ⎞
⎜ ⎟⎜ ⎟⋅ ⋅ ⋅ =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⋅ ⋅ ⋅⎝ ⎠⎝ ⎠
AB a b cC
A a A b A cB a B b B cC a C b C c
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149
Since U, V, and W
are general integers, the ends of S are points on a lattice
reciprocal to the direct lattice.
real unit cell
reciprocal unit cell
b
a
O
b*
a*
01
1011
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150
The Ewald
Construction
P
OMs0
S
2θ
s
reciprocal lattice
1/λ
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151
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152
Mosaicity
(Mosaic Spread)
B. D. Cullity
and S. R. Stock, Elements of X-ray Diffraction, p. 175 (2001).
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153
Intensity of a Diffraction Spot
23 22
int 02 2( ) ( )creI hkl V I LpT F hkl
V mcλω
⎛ ⎞= ⎜ ⎟
⎝ ⎠
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154
The Lorentz
Factor
•
With an angular speed of rotation ω
a r.l.p. at a distance 1/d
from the origin moves with a
linear speed v
= (1/d)ω•
For passage through the Ewald
sphere, we
need the component v⊥
= (1/d)ωcosθ
= (ωsin2θ)/λ
•
The time to pass through the surface is proportional to 1
sin 2λω θ⎛ ⎞⎜ ⎟⎝ ⎠
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155
The Polarization Factor
•
P
= sin2φ
where φ
is the angle between the polarization direction of the incident beam and the scattering direction
• φ
= 90 -
2θ
•
For an unpolarized
incident beam, P
= (1 + cos22θ)/2
•
Synchrotron radiation is polarized, so check with your beamline
staff!
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156
The Polarization Factorbeam
θ
90° - 2θp
psin2θ
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157
Transmission (Absorption)
( )0
1
d
iii
T AI eI
X
μ
μμ ρ ρ
−
= −
=
= ∑
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158
(Extinction)
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159
Calculation of Electron Density31 2
2 2 2
0 0 0( ) ( )
nn nit iu iv
t u vK F e e eπ π π⋅ ⋅ ⋅
= = =
= × × ×∑ ∑ ∑a S b S c SS S
A more accurate expression is2( ) ( ) i
cr realcrystal
K e dvπρ ⋅= ∫ r SS r
This operation is a Fourier transformation
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160
Calculation of Electron Density2
2
2 ( )
( ) ( )
1( ) ( )
( )
1( ) ( )
icr reciprocal
i
i hx ky lz
h k l
W e dv
eVx y z
x y zhx ky lz
xyz hkl eV
π
π
π
ρ
ρ
ρ
− ⋅
− ⋅
− + +
=
⎛ ⎞= ⎜ ⎟⎝ ⎠
⋅ = + + ⋅= ⋅ + ⋅ + ⋅= + +
⎛ ⎞= ⎜ ⎟⎝ ⎠
∫
∑
∑ ∑ ∑
r S
S
r h
h
r S
r F h
r S a b c Sa S b S c S
F
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161
Calculation of Electron Density1 1 1
2 ( )
0 0 0
( )
[ 2 ( ) ( )]
1/ 2
int
( ) ( )
( ) ( )1( ) ( )
( )( )
i hx ky lz
x y z
i hkl
i hx ky lz i hkl
h k l
hkl V xyz e dxdydz
hkl F hkl e
xyz F hkl eV
I hklF hklLpT
π
α
π α
ρ
ρ
+ +
= = =
− + + +
=
=
=
⎡ ⎤= ⎢ ⎥⎣ ⎦
∫ ∫ ∫
∑∑∑
F
F
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162
Anomalous (Resonant) Scattering
( )( )( )
normally ( )
( )
and ( )
I hkl I hkl
F hkl F hkl
hkl hklα α
=
=
= −
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163
Anomalous (Resonant) Scattering
http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html
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164
Anomalous (Resonant) ScatteringNear an absorption edge
fanom
= f
+ Δf
+ f”
f
fanomalous f”
fΔf
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165
Anomalous (Resonant) Scattering
FPH(+)
FP(+)
FH(+) without anomalous scattering
FH(+) with anomalous scattering
FP(-)
FPH(-)
FH(-) without anomalous scattering
FH(-) with anomalous scattering
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166
How do I determine f’
and f”?
•
Calculate them by programs such as FPRIME•
Measure the NEXAFS/XAFS, and calculate them using the (classical or quantum) Kramers-Kroning
transform
2 2
0
2 2
2
"2
2 ' "( ',0)' ''
ae dg
mc ddgfd
ff P dκ
κ
κ ω
πμε ωπ ω
ω
ω ω ωπ ω ω
∞
⎡ ⎤= ⎢ ⎥⎣ ⎦
⎡ ⎤= ⎢ ⎥⎣ ⎦
=−∑ ∫
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167
Symmetry in the Diffraction Pattern
( )
2
( * * *)( )or in matrix notation
in this notation, the structure factor is
( ) ( )T
T
ireal
cell
hx ky lz h k l x y z
xh k l y
z
F e dvπρ ⋅
+ + = + + + + = ⋅
⎛ ⎞⎜ ⎟ = ⋅⎜ ⎟⎜ ⎟⎝ ⎠
= ∫ h r
a b c a b c h r
h r
h r
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168
A symmetry operation can be represented by a combination of a rotation/inversion/reflection and a translation. The rotation… can be represented by a matrix R and the
translation by a vector t. By symmetry, ρ(R·r+t) = ρ(r).
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169
F(h) can also be written:2 ( )
2 2
2 2 ( )
( ) ( )
( )
( )
( ) ( )
T
T T
T T T
ireal
cell
i ireal
cell
T T T
i ireal
cell
F e dv
e e dv
F e e dv
π
π π
π π
ρ
ρ
ρ
⋅ ⋅ +
⋅ ⋅ ⋅
⋅ ⋅ ⋅
= ⋅ +
=
⋅ = ⋅
=
∫
∫
∫
h R r t
h t h R r
h t R h r
h R r t
r
h R R h
h r
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170
The integral is just F(RT·h), so
2( ) ( )( ) ( ) 2( ) ( )
Ti T
T T
T
F e F
I I
π
α α π
⋅= ⋅
= ⋅ + ⋅
= ⋅
h th R hh R h h th R h
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171
For a 21
axis along b
2 2 ( )
1 0 0 00 1 0 and 1/ 2
00 0 1therefore
( ) ( )[ ]T Ti i
realhalfthecell
F e e dvπ πρ ⋅ ⋅ +
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟
⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
= +∫ h r h R r t
R t
h r
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172
For the (0k0) reflections, h = kb*, so hT·r = hT·R·r = 0 + ky
+ 0
and hT·t = k/2 This simplifies:
2(0 0) [1 ] ( )ik ikyreal
halfthecell
F k e e dvπ πρ= + ∫ r
If k
is odd, F(0k0) = 0
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173
For C-centering
( ) 2
1/ 21/ 2 , so / 2 / 2
0and
( ) [1 ] ( )T
T
i h k ireal
halfthecell
h k
F e e dvπ πρ+ ⋅
⎛ ⎞⎜ ⎟= ⋅ = +⎜ ⎟⎜ ⎟⎝ ⎠
= + ∫ h r
t h t
h r
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174
The Patterson Function
2
2 2
Let ( ) ( )1( ) ( ) cos(2 )
or equivalently (no anomalous dispersion)1( ) ( ) i
P uvw P
P FV
P F eV
π
π
⋅
=
⎛ ⎞= ⋅⎜ ⎟⎝ ⎠
⎛ ⎞= ⎜ ⎟⎝ ⎠
∑
∑
h
h u
h
u
u h h u
u h
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175
What does the Patterson function mean?
Consider an alternate expression:
( ) ( ) ( ) realP dvρ ρ= +∫r
u r r u
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176
More about Structure Factors
2
1
1 1
( )
For noncentrosymmetric structures, it is useful:
( ) cos(2 ) sin(2 )
( ) ( )
jn
ij
j
n n
j j j jj j
f e
f i f
A iB
π
π π
⋅
=
= =
=
= ⋅ + ⋅
= +
∑
∑ ∑
r SF S
F S r S r S
S S
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177
|F(S)| decreases with increasing |S|
•
The falloff in f
(greater interference)•
Static/dynamic disorder (thermal motion)
2 20 (sin ) /
2 2where 8
Bi if f e
B u
θ λ
π
−=
=
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178
The falloff makes statistical analysis awkward. Intensities are normally measured on an arbitrary
scale. For large numbers of well-distributed S
2 2
2 2
2 sin /2 0 2
( ) ( , )
but
( ) i
ii
Bi i
F I abs f
f f e θ λ−
= =
=
∑S S
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179
The Wilson Plot2 22 sin / 0 2
0 2 2 2
( ) ( , ) ( )
ln[ ( ) / ( ) ] ln 2 sin /
Bi
i
ii
I K I abs Ke f
I f K B
θ λ
θ λ
−= =
= −
∑
∑
S S
S
http://www.ysbl.york.ac.uk/~mgwt/CCP4/EJD/bms/bms10.html
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180
Normalized Structure Factors
2 2
1/ 2
2
1/ 2
sin / 0 2
( ) ( ) /
( ) ( )
jj
Bj
j
E F f
F e fθ λ
⎛ ⎞= ⎜ ⎟
⎝ ⎠
⎡ ⎤= ⎢ ⎥
⎣ ⎦
∑
∑
S S
S
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181
Least Squares Refinement
•
Assume observations are uncorrelated•
IfError distributions are GaussianObservations are weighted by 1/σ2
LS gives maximum likelihood estimates of xj
•
Restraints count as observations (weights?)
E. Prince, Mathematical Techniques in Crystallography and Materials Science, Springer-Verlag
(1994)
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182
Accuracy vs.
Precision
•
The standard uncertainty (esd) is a measure of precision
•
LS yields minimum variances•
Correlations
•
Systematic errors•
Accuracy?
International Tables for Crystallography, Volume F, Section 18.5
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183
The Macromolecular CIF Dictionary (mmCIF)
International Tables for
Crystallography, Volume G Section 4.5
pages 295-443 (!)