1 lattices and symmetry scattering and diffraction (physics) james a. kaduk innovene usa llc...
Post on 19-Dec-2015
226 Views
Preview:
TRANSCRIPT
1
Lattices and SymmetryScattering and Diffraction (Physics)
James A. KadukInnovene USA LLCNaperville IL 60566
James.Kaduk@innovene.com
2
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.
3
The crystallographer’s world view
Reality can be more complex!
4
Twinning at the atomic level
International Tables for Crystallography, Volume D, p. 438
5
PDB entry 1eqg = ovine COX-1 complexed with Ibuprofen
6
Atoms (molecules) pack together in a regular pattern to form a crystal.
There are two aspects to this pattern:
Periodicity
Symmetry
First, consider the periodicity…
7
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).
8
A Primitive Cubic Lattice (CsCl)
9
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.
10
A Unit Cell
11
a, b, c, , , a, b, cx1a + x2b + x3c, 0 xn < 1lattice points = ha + kb + lc,
hkl integersdomain of influence – Dirichlet domain, Voronoi domain, Wigner-Seitz cell, Brillouin zone
Descriptions of the Unit Cell
12
A Brillouin Zone
Kittel, Solid State Physics
13
The unit cell is not unique(c:\MyFiles\Clinic\index2.wrl)
14
15
16
17
How do I pick the unit cell?
• Axis system (basis set) is right-handed
• Symmetry defines natural directions
• Angles close to 90°
• Standard settings of space groups
• To make structural similarities clearer
18
The Reduced Cell
• 3 shortest non-coplanar translations
• Main Conditions (shortest vectors)
• Special Conditions (unique)
• May not exhibit the true symmetry
19
The Reduced Form
a·a
A
b·b
B
c·c
C
b·c
D
a·c
E
a·b
F
20
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·a
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 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
21
Negative reduced Form, Type II Cellall 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 = 0
if |a·c| = ½ a·a then a·b = 0
if |a·b| = ½ a·a then a·c = 0
if ( |b·c| + |a·c| + |a·b| ) = ½ ( a·a + b·b ) then a·a 2 |a·c| + |a·b|
22
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).
23
International Tables for Crystallography, Volume F, Figure 2.1.3.3, p.52 (2001)
24
25
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
26
A = B, no conditions on C
Number 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
27
B = C, no conditions on A
Number 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
28
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
29
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.
30
The metric symmetry can be higher than the crystallographic symmetry!
(A monoclinic cell can have β = 90°)
31http://www.haverford.edu/physics-astro/songs/bravais.htm
32
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 faceshkl indices of a Bragg reflection
33
Now consider the symmetry…
34
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!
35
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
36
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.
37
The 32 Point Groups (1)
International Tables for Crystallography, Volume A, Table 12.1.4.2, p.819 (2002)
38
The 32 Point Groups (2)
International Tables for Crystallography, Volume A, Table 12.1.4.2, p.819 (2002)
39
Symbols for Symmetry Elements (1)
International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002)
40
Symbols for Symmetry Elements (2)
International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002)
41
Symbols for Symmetry Elements (3)
International Tables for Crystallography, Volume A, Table 1.4.2, p. 7 (2002)
42
2 Rotation Axis (ZINJAH)
43
3 Rotation Axis (ZIRNAP)
44
4 Rotation Axis (FOYTAO)
45
6 Rotation Axis (GIKDOT)
46
-1 Inversion Center (ABMQZD)
47
-2 Rotary Inversion Axis?
48
m Mirror Plane (CACVUY)
49
-3 Rotary Inversion Axis (DOXBOH)
50
-4 Rotary Inversion Axis (MEDBUS)
51
-6 Rotary Inversion Axis (NOKDEW)
52
21 Screw Axis (ABEBIS)
53
31 Screw Axis (AMBZPH)
54
32 Screw Axis (CEBYUD)
55
41 Screw Axis (ATYRMA10)
56
42 Screw Axis (HYDTML)
57
43 Screw Axis (PIHCAK)
58
61 Screw Axis (DOTREJ)
59
62 Screw Axis (BHPETS10)
60
63 Screw Axis (NAIACE)
61
64 Screw Axis (TOXQUS)
62
65 Screw Axis (BEHPEJ)
63
c Glide (ABOPOW)
64
n Glide (BOLZIL)
65
d (diamond) Glide (FURHUV)
66
What does all this mean?
67
Symmetry information is tabulated in International Tables for
Crystallography, Volume A edited by Theo Hahn Fifth
Edition 2002
68
Guaifenesin, P212121 (#19)
69
70
© Copyright 1997-1999. Birkbeck College, University of London.
71
Hermann-Mauguin Space Group Symbolsthe centering, and then a set of characters indicating the
symmetry elements along the symmetry directions
Lattice Primary Secondary Tertiary
Triclinic 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}
72
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
73
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.
74
75
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)
76
A Bärninghausen Treefor translationengleiche subgroups
International Tables for Crystallography, Volume 1A, p. 396 (2004)
77
Mercury/ETGUAN (P41212 #92)
78
79
80
Not all space groups are possible for protein crystals.
81
Space Group Frequencies in theProtein Data Bank, 17 June 2003
Space Group Number
0 20 40 60 80 100 120 140 160 180 200 220
# Entries
1
10
100
1000
10000
82
Space Group Frequencies
Space Group Number
0 20 40 60 80 100 120 140 160 180 200 220
Frequency of Occurrence, %
0.01
0.1
1
10
100
PDB % CSD % ICSD %
83
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
84
Basic Diffraction Physics
85
Bragg’s Law2sinndλθ=
1sin2dθλ=
86
Bragg’s Law
V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 148 (2003)
87
Optical Diffraction
PSSC Physics, Figure 18-A, p. 202-203 (1965)
88
Optical Diffraction
PSSC Physics, Figure 18-B, p. 202-203 (1965)
89
Optical Diffraction
D. Halliday and R. Resnick, Physics, p. 1124 (1962)
90
Oscillation direction of the electron
X-ray beam
Electric vector ofThe incident beam
Electron
Scattering by One Electron
91
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:2220221sineleIIrmcϕ⎛⎞=⎜⎟⎝⎠
92
Electromagnetic Waves
(0;)cos2zEtzAπλ==
E λ
A
A
z
93
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(,)cos2()cos2cos2EtzAztzzAtAtcπλνλπνπνλ=−⎛⎞⎛⎞=−=−⎜⎟⎜⎟⎝⎠⎝⎠
94
(,0)cos2cosEtAtAtπνω==
95
Consider a new wave displaced by a distance Z from the original wave:
Z corresponds to a phase shift2π(Z/λ) = α
E Z’
z
new wave
original wave
96
(,0)cos(,0)cos()orignewEtAtEtAtωω==+
97
cos()coscossinsincoscossincos(90)AtAtAtAtAtωωωωω+=−=++°
cos()cossiniAtAiAAeω+=+=
imaginary axis
real axisA
Acos
Asin
98
Scattering by Two Electrons
1
2
pq
r 2
1 s
s0
99
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
100
00sincos(90)1cos(90)()prrrqpqθθλθλλλλ==−=−=⋅=−⋅+=⋅−rsrsrss
101
The phase difference of wave 2 with respect to wave 1 is:002()2πλπλ⋅−−=⋅=−rssrSSss
s0 s0
s S
θ
“reϕλectiν πλνe”
102
2sin2sinθθλ==Ss
103
Scattering by an Atom
r(r)
ρ(-r)
-r
+r
nucleus
104
The Atomic Scattering Factor[2][2][2]()()2()cos[2]iiifedeeddπππrrrπ⋅⋅−⋅=⎡⎤=+⎣⎦=⋅∫∫∫rSrrSrSrrrrrrrrSr
105
Atomic Scattering Factors
V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 213 (2003)
106
Scattering from a Row of Atoms
M. J. Buerger, X-ray Crystallography, Fig. 14B, p. 32 (1942)
107
Scattering from a Row of Atomscoscoscoscos(coscos)coscosOQPRmOQaPRaaamammaλνμνμλνμλλνμ−===−=−==+
108
Scattering from a Row of Atoms
M. J. Buerger, X-ray Crystallography, Fig. 15, p. 33 (1942)
109
Scattering by a Plane of Atoms
M. J. Buerger, X-ray Crystallography, Fig. 16, p. 34 (1942)
110
Scattering by a Unit Cell
J. Drenth, Principles of Protein X-ray Crystallography, Fig. 4.12 p. 80 (1999)
111
Scattering by a Unit Cell221()jjijjnijjfefeππ⋅⋅===∑rSrSfFS
112
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[2πitaS]exπ[2πiubS]exπ[2πivcS]
O a
c
a
c
ta+ub+vc
113
For a unit cell with its own origin at ta + ub + vc
the scattering is
and the total scattering by the crystal is
222()itiuiveeeπππ⋅⋅⋅×××aSbScSFS
312222000()()nnnitiuivtuveeeπππ⋅⋅⋅====×××∑∑∑aSbScSKSFS
114
Scattering by a Crystal
2π .a S =0t
=1t
=2t=3t=4t
=5t
=6t
=7t
=8t
115
A crystal does not scatter X-rays unlesshkl⋅=⋅=⋅=aSbScS
These are the Laue conditions
116
Now remember111hkl⋅=⋅=⋅=aSbScS
s0 s0
s S
θ
“ ”reflecting plane
117
The “reflecting planes” are lattice planes
reflecting planesdirection of Salong this line
a
b
a/h
b/kd
118
Consider just one direction
1proj on 12sin2sinor, since (or ) can be any integer2sindhhdddhndλθλθλθ==⋅=====aaSSS
119
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.
120
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.
121
Why do we care?
122
Remember the Laue conditions:hkl⋅=⋅=⋅=aSbScS
S a reflecting plane = a lattice plane. The
equation of such a plane through the
origin ishx + ky + lz = 0
123
The reflecting plane contains general vectors and lattice vectors:
r = xa + yb + zcrL = ua + vb + wc
u, v, and w integers
124
S is perpendicular to any vector in the plane, or
S·(r – rL) = 0S·r = S· rL
S· rL = n(the planes don’t have to pass through the origin)
125
r = ua + vb + wc
S·a = hS·b = kS·c = l
126
Consider the possibility of a different basis set for S:
S = UA + VB + WCr = ua + vb + wc
127
2coscos1cos1cos1coscoscoscos1cos1cos1coscos11coscoscos1coscoscos1hklhhhaaahklkkkbbbabclllcccd++=
Triclinic
128
(UA + VB + WC)·(ua + vb + wc) = n
()()()uUVWvnwuUVWvnw⎛⎞⎛⎞⎜⎟⎜⎟=⋅ ⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⋅⋅⋅⎛⎞⎛⎞⎜⎟⎜⎟ =⋅⋅⋅ ⎜⎟⎜⎟⎜⎟⎜⎟⋅⋅⋅⎝⎠⎝⎠ABabcCAaAbAcBaBbBcCaCbCc
129
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
130
The Ewald Construction
P
OMs0
S
2θ
s
reciprocal lattice
1/λ
131
Mosaicity (Mosaic Spread)
B. D. Cullity and S. R. Stock, Elements of X-ray Diffraction, p. 175 (2001).
132
Intensity of a Diffraction Spot
2322int022()()creIhklVILpTFhklVmcλω⎛⎞=⎜⎟⎝⎠
133
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
1sin2λωθ⎛⎞⎜⎟⎝⎠
134
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!
135
The Polarization Factor
beam
θ
90° - 2θp
psin2θ
136
Transmission (Absorption)()01diiiTAIeIXμμμρ ρ−=−==∑
137
(Extinction)
138
Calculation of Electron Density312222000()()nnnitiuivtuvKFeeeπππ⋅⋅⋅====×××∑∑∑aSbScSSS
A more accurate expression is2()()icrrealcrystalKedvπr⋅=∫rSSr
This operation is a Fourier transformation
139
Calculation of Electron Density222()()()1()()()1()()icrreciprocaliihxkylzhklWedveVxyzxyzhxkylzxyzhkleVπππrrr−⋅−⋅−++=⎛⎞=⎜⎟⎝⎠⋅=++⋅=⋅+⋅+⋅=++⎛⎞=⎜⎟⎝⎠∫∑∑∑∑rSSrhhrSrFhrSabcSaSbScSF
140
Calculation of Electron Density1112()000()[2()()]1/2int()()()()1()()()()ihxkylzxyzihklihxkylzihklhklhklVxyzedxdydzhklFhklexyzFhkleVIhklFhklLpTππrr++===−+++===⎡⎤=⎢⎥⎣⎦∫∫∫∑∑∑FF
141
Anomalous (Resonant) Scattering
()()()normally ()()and ()IhklIhklFhklFhklhklhklαα ===−
142
Anomalous (Resonant) Scattering
http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html
143
Anomalous (Resonant) Scattering
Near an absorption edge
fanom = f + Δf + f”
f
fanomalous f”
fΔf
144
Anomalous (Resonant) Scattering
FPH(+)
FP(+)
FH(+) without anomalous scattering
FH(+) with anomalous scattering
FP(-)
FPH(-)
FH(-) without anomalous scattering
FH(-) with anomalous scattering
145
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 transform220222"22'"(',0)'''aedgmcddgfdffPdκκκωπμeωπωωωωωπωω∞⎡⎤=⎢⎥⎣⎦⎡⎤=⎢⎥⎣⎦=−∑∫
146
Symmetry in the Diffraction Pattern()2(***)()or in matrix notationin this notation, the structure factor is()()TTirealcellhxkylzhklxyzxhklyzFedvπρ ⋅++=++++=⋅⎛⎞⎜⎟=⋅⎜⎟⎜⎟⎝⎠=∫hrabcabchrhrhr
147
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).
148
F(h) can also be written:2()2222()()()()()()()TTTTTTirealcelliirealcellTTTiirealcellFedveedvFeedvπππππrrr⋅⋅+⋅⋅⋅⋅⋅⋅=⋅+=⋅=⋅=∫∫∫hRrththRrhtRhrhRrtrhRRhhr
149
The integral is just F(RT·h), so
2()()()()2()()TiTTTTFeFIIππ⋅=⋅=⋅+⋅=⋅hthRhhRhhthRh
150
For a 21 axis along b22()1000010 and 1/20001therefore()()[]TTiirealhalfthecellFeedvππr⋅⋅+⎛⎞⎛⎞⎜⎟⎜⎟==⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠=+∫hrhRrtRthr
151
For the (0k0) reflections, h = kb*, sohT·r = hT·R·r = 0 + ky + 0
and hT·t = k/2This simplifies:
2(00)[1]()ikikyrealhalfthecellFkeedvππr=+∫r
If k is odd, F(0k0) = 0
152
For C-centering()21/21/2, so /2/20and()[1]()TTihkirealhalfthecellhkFeedvππr+⋅⎛⎞⎜⎟=⋅=+⎜⎟⎜⎟⎝⎠=+∫hrththr
153
The Patterson Function
222Let ()()1()()cos(2)or equivalently (no anomalous dispersion)1()()iPuvwPPFVPFeVππ⋅=⎛⎞=⋅⎜⎟⎝⎠⎛⎞=⎜⎟⎝⎠∑∑hhuhuuhhuuh
154
What does the Patterson function mean?
Consider an alternate expression:()()()realPdvrr=+∫rurru
155
More about Structure Factors2111()For noncentrosymmetric structures, it is useful:()cos(2)sin(2)()()jnijjnnjjjjjjfefifAiBπππ⋅=====⋅+⋅=+∑∑∑rSFSFSrSrSSS
156
|F(S)| decreases with increasing |S|
• The falloff in f (greater interference)
• Static/dynamic disorder (thermal motion)
220(sin)/22where 8BiiffeBuθλπ−==
157
The falloff makes statistical analysis awkward. Intensities are normally measured on an arbitrary
scale. For large numbers of well-distributed S22222sin/202()(,)but()iiiBiiFIabsfffeθλ−===∑SS
158
The Wilson Plot222sin/020222()(,)()ln[()/()]ln2sin/BiiiiIKIabsKefIfKBθλθλ−===−∑∑SSS
http://www.ysbl.york.ac.uk/~mgwt/CCP4/EJD/bms/bms10.html
159
Normalized Structure Factors
221/221/2sin/02()()/()()jjBjjEFfFefθλ⎛⎞=⎜⎟⎝⎠⎡⎤=⎢⎥⎣⎦∑∑SSS
top related