An Introduction to the Rietveld Method James A. Kaduk Senior Research Associate Analytical Sciences Research Services INEOS Technologies [email protected]
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An Introduction to the Rietveld Method
James A. KadukSenior Research Associate
Analytical Sciences Research ServicesINEOS Technologies
Use all the crystallography and diffraction physics we know to model the powder pattern
( )2hkl o c
hklw F k F−∑ ( )2
i oi cii
w y y−∑
Quantity Minimized in the Least Squares
Single Crystal vs. Rietveld
What we’ll try to do
Introduction to practiceUnderstanding the instrumentCommon global parametersQuality measuresMinimize the fear
What we’ll not try to cover
Basic crystallographyLeast-squares theoryDiffraction theoryProgram tutorial (GSAS, Fullprof, Topas,
Riquas, HighScore Plus…)
The Rietveld Method is a refinement technique, not a
structure solution method. A good starting model is required!
How good is good?
• All atoms within 0.5-0.6 Å of their true positions, but…
• x% of the total scattering power
How do you get the model?
• Powder Diffraction File (PDF/LPF)• Inorganic Crystal Structure Database (ICSD)• Metals Data File (CRYSTMET)• Cambridge Structural Database (CSD)• (Protein Data Bank) (PDB)• Crystal Data Identification File (CDIF)• Primary literature• ab initio structure determination
What determines the intensities?(1) Structure Factors
Atomic positionsOccupanciesAtomic scattering factorsDisplacement coefficientsLattice parametersSymmetry
What determines the intensities?(2) Global Parameters
ConcentrationIncident intensityBackgroundDiffuse scatteringExtinctionAbsorption
Preferred orientationMultiplicityLp factorProfile functionDiffractometer parameters…
To get accurate results, we must model all these quantities correctly!
The advantage of the Rietveld method is that it uses all the
information in the powder pattern, and yields the most information.
We need to remember that we are fitting a model to data, and that
our answers will only be as good as the model is appropriate.
Two StepsPreparation and Refinement
In some programs (like GSAS) these two steps are separate,and in others, they are combined into a single operation.
The Sample
• Need a powder, but…• Random is best, but…• Resolution – more is better, but can generate
size/strain by trying to get powder• Phase purity• An advantage of the Rietveld method is that
ideal samples are rare, and the method provides a way of dealing with real samples
The Instrument
• Alignment/systematic errors • (zero, shift, trns)• Wavelength(s)• Profile function
Data Collection• Compromises!• Fixed step sizes (but “new” GSAS format)• Wide 2θ range• ≥ 5 steps across FWHM of sharpest peaks• Constant or variable counting time• Step or continuous scan• ~10,000 counts for strongest peaks• Programs assume fixed slits!
Background
• Crucial to get right – affects integrated intensities (and thus the structure) – especially the displacement coefficients
• Interacts with the profile function• Peak tails• Use a few parameters as possible• Crystalline sample – slowly varying• Background parameters are highly-correlated
Background Functions
Shifted Chebyshev Polynomials of the First Kind
11
( )N
b j jj
I B T x−=
=∑T0(x) = 1, T1(x) = x, T2(x) = 2x2-1, Tn+1(x) = 2xTn(x) - Tn-1(x)
min
max min
2(2 2 ) 12 2
x θ θθ θ
−= −
−
Cosine Fourier Series
12
cos[ ( 1)]N
b jj
I B B x j=
= + −∑
Polynomial
11
0
2 1m
b mm
I BBKPOS
θ=
⎛ ⎞= −⎜ ⎟⎝ ⎠
∑
Diffuse ScatteringThe Debye Equation
2
4 sinsin
( ) ( ) 2 ( ) ( ) 4 sin
ij
n i jijn i j
r
I f f f r
π θλ
θ θ θ θ π θλ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥= +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
∑ ∑∑
P. Debye, Annalen der Physik, 351, 809 (1915)
Diffuse Scattering in GSAS (#1)
2sin( ) 1exp2DS
RQI A UQRQ
⎛ ⎞= −⎜ ⎟⎝ ⎠
Q = 2π/d
GSAS Diffuse Scattering Function #1 Terms
2θ, deg
0 20 40 60 80 100 120 140 160
Back
grou
nd In
tens
ity
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
R = 1.6, U = 0.05R = 2.4, U = 0.05R = 1.6, U = 0.2
Differences in Diffuse Scattering Terms
2θ, deg
0 20 40 60 80 100 120 140 160
Back
grou
nd In
tens
ity
0
1e-5
2e-5
3e-5
4e-5
5e-5
1.6/0.05-2.4/0.051.6/0.2-2.4/0.2
χ2 = 1.112
Profile Coefficients
• Crucial to getting the right answers• Structure/intensities/overlap/tails• Valuable information in profile coefficients
X-ray Profile Functions
• Gaussian• Lorentzian (Cauchy)• Modified Lorentzians• Split Pearson VII• (pseudo)Voigt• Empirical “learned”• Stacking fault model – DIFFax, …• Fundamental parameters
Instrument Profile Function
• Some programs (GSAS) require one• Helps interpret refined values• Use a sample free of size and strain broadening
– SRM 660a/b (LaB6)– SRM1976/a (corundum plate)– SRM 640c/d (Si)
Peak Position Errors Bruker D8 AdvanceSRM 660a LaB6
New Tube 13 August 2006
X Data
0 20 40 60 80 100 120 140 160
Y D
ata
0.00
0.01
0.02
0.03
0.04
KADU1044Δ2θ = 0.009(2) + 0.012(2)cosθ + 0.016(2)sin2θ
Bruker D8 Advance Resolution FunctionSRM 660a LaB6
New Tube 13 August 2006
X Data
0 20 40 60 80 100 120 140
Y D
ata
0.00
0.05
0.10
0.15
0.20
0.25
KADU1044FWHM = 0.0006(295)tan2
θ + 0.021(4)tanθ + 0.0441(295) + 0.005(29)/cos2θ
Bruker D8 Advance Peak ShapesPearson VII Exponent
SRM 660a LaB6 New Tube 13 August 2006
2θ, deg
0 20 40 60 80 100 120 140 160
Pear
son
VII E
xpon
ent
0.9
1.0
1.1
1.2
1.3
1.4
1.5
123456789012345678901234567890123456789012345678901234567890 INS BANK 1 INS HTYPE PXCR INS 1 IRAD 3 INS 1 ICONS 1.540629 1.544451 -0.990 0 0.5 0 0.5 INS 1I HEAD NIST SRM 660a LaB6 VANTEC-1 0.3 mm div slit 29 Apr 2004 INS 1I ITYP 0 5.0000 150.0000 1 INS 1PRCF1 2 18 0.01 INS 1PRCF11 0.287900E+00 0.000000E+00 1.124000E+00 2.477000E+00 INS 1PRCF12 2.103000E+00 0.442000E+00 2.052000E+00 -4.818000E+00 INS 1PRCF13 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF14 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF15 0.000000E+00 0.000000E+00INS 1PRCF2 3 13 0.01 INS 1PRCF21 0.336500E+00 0.000000E+00 1.032000E+00 0.000000E+00 INS 1PRCF22 2.526000E+00 2.051000E+00 0.269500E-01 0.005000E-01 INS 1PRCF23 0.444100E+00 -5.024000E+00 0.000000E+00 0.000000E+00INS 1PRCF24 0.000000E+00 INS 1PRCF3 4 12 0.01 INS 1PRCF31 2.000000E+00 -2.000000E+00 5.000000E+00 0.000000E+00 INS 1PRCF32 0.100000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF33 0.000000E+00 0.150000E-01 0.150000E-01 0.750000E+00
Peak Position Errors, PANalytical X'Pert Pro MPDSRM 660a, LaB6, 25 January 2008
1/2 deg divergence, 0.02 rad Sollers
2θ, deg
0 20 40 60 80 100 120 140 160
Δ2θ
, obs
-cal
c, d
eg
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
LAB6_05x column 2 vs y column 2
PANalytical X'Pert Pro MPD Resolution FunctionSRm 660a LaB6 25 January 2008
1/2 deg divergence, 0.02 rad Sollers
2θ, deg
0 20 40 60 80 100 120 140 160
FWH
M, d
eg
0.00
0.05
0.10
0.15
0.20
LAB6_05FWHM = -0.016(57) + 0.0537(32)/cosθ, r2 = 0.93
PANalytical X'Pert Pro MPD Peak ShapesSRM 660a LaB6 25 January 2008
1/2 deg divergence, 0.02 rad Sollers
2θ, deg
0 20 40 60 80 100 120 140 160
Pear
son
VII E
xpon
ent
0.5
1.0
1.5
2.0
2.5
3.0
123456789012345678901234567890123456789012345678901234567890 INS BANK 1 INS HTYPE PXCR INS 1 IRAD 3 INS 1 ICONS 1.540629 1.544451 -0.990 0 0.5 0 0.5 INS 1I HEAD NIST SRM 660a LaB6 PIXCEL 1/2 deg div 0.02 Soller 25 Jan 2008 INS 1I ITYP 0 5.0000 150.0000 1 INS 1PRCF1 2 18 0.01 INS 1PRCF11 0.756500E+00 0.000000E+00 3.646000E+00 2.428000E+00 INS 1PRCF12 1.906000E+00 1.308000E+00 1.063000E+00 0.000000E+00 INS 1PRCF13 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF14 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF15 0.000000E+00 0.000000E+00INS 1PRCF2 3 13 0.01 INS 1PRCF21 1.153000E+00 -0.928000E+00 4.161000E+00 0.000000E+00 INS 1PRCF22 2.472000E+00 1.814000E+00 0.157700E-01 0.005000E-01 INS 1PRCF23 1.232000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF24 0.000000E+00 INS 1PRCF3 4 12 0.01 INS 1PRCF31 2.000000E+00 -2.000000E+00 5.000000E+00 0.000000E+00 INS 1PRCF32 0.100000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF33 0.000000E+00 0.150000E-01 0.150000E-01 0.750000E+00
GSAS Profile Function #2 (3-5)pseudo-Voigt
2 22tan tan
cosPU V Wσ θ θθ
= + + +
( )cos cos tancos
X ptec Y stecϕγ ϕ θθ
+= + +
2 cos sin 2tan 2
if asymzero shift trnsθ θ θθ
⎛ ⎞Δ = + + +⎜ ⎟⎝ ⎠
Size Broadening
18000( )iso
inst
KLX X
λπ
=−
18000( )inst
KLX ptec X
λπ
=+ −
18000( )inst
KLX X
λπ⊥ = −
Strain Broadening - Isotropic
( )100%18000 instS Y Yπ
= −
Strain Broadening - Anisotropic
( )100%18000 instS Y stec Yπ
= + −
( )100%18000 instS Y Yπ
⊥ = −
2
( ) 100%18000
H K LS
HKL
dS hkl h k lπ= ∑
Constant Microstrain Surface
How do you know when you’re finished?
“A Rietveld refinement is never finished, only abandoned”
P. W. Stephens
(or when you’ve answered the question you set out to answer)
J. A. Kaduk
There is no one measure of the quality of a Rietveld refinement!
Statistical Measures
( )22 2/wp i oi ci i oii i
R w y y w y= −∑ ∑
/p oi ci oii i
R y y y= −∑ ∑
2exp 2
i oii
N PRw y−
=∑
22
2
( )1( )
oi ci
i oi
y yN P y
χσ−
=− ∑
2
2
exp
wpRR
χ⎛ ⎞
= ⎜ ⎟⎜ ⎟⎝ ⎠
At the end of refinement
• χ2 < 1: the σ(yoi) are wrong• χ2 = 1: the errors are dominated by statistics• χ2 > 1:
– The model is reasonable, but the σs are underestimated
– The model contains systematic errors– The model is wrong
Graphical Measures
The Rietveld Plot
The Weighted Plot
The Error Plot
The Normal Probability Plot
Low Angle Plot
High Angle Plot
Background Plot
Cumulative χ2 Plot
A Case Study
Effluorescenceat the base of a door
601 Door Effluorescence
x103
5.0
10.0
15.0
20.0
Inte
nsity
(Cou
nts)
00-041-1476> Sylvite - KCl01-070-2509> Halite - NaCl
01-078-1064> Trona - Na3H(CO3)2(H2O)2
10 20 30 40 50 60 70 80 90Two-Theta (deg)
[kadu1220.raw] 601 door effluorescence (40,40,0.3) JAK
Use 17-100°. Refine 3 phase fractions and 3 background terms.
Get in the right neighborhood, then deal with the largest errors.
χ2 = 71.61
Add shift and 3 cells.
χ2 = 29.75
Add profile X and Yfor all three phases.
X < 0 for phases 1 and 2.Phase 3 Y = 7.1(23).
Fix all three at instrumental values.
χ2 = 10.58
Add a common Uiso for KCl.
χ2 = 8.619Uiso = 0.075!
Add five C-C diffuse scattering terms.
χ2 = 5.94
Has the feel of granularity, so micronize the sample.
x103
5.0
10.0
15.0
20.0
Inte
nsity
(Cou
nts)
00-041-1476> Sylvite - KCl01-070-2509> Halite - NaCl
01-078-1064> Trona - Na3H(CO3)2(H2O)2
10 20 30 40 50 60 70 80 90Two-Theta (deg)
[kadu1220.raw] 601 door effluorescence (40,40,0.3) JAK[kadu1221.raw] 601 door effluorescence, micronized (40,40,0.3) JAK
Copy KADU1220.EXP to KADU1221.EXP, change the data file, and add profile Y for phase 3.
χ2 = 3.789Uiso = 0.0290(4) Å2
Extra phase(s)?
0
250
500
750
1000
Inte
nsity
(Cou
nts)
01-077-0114> Aluminum - Al(OH)3
00-001-0909> Nahcolite - NaHCO3
00-046-1045> Quartz - SiO2
10 20 30 40 50 60Two-Theta (deg)
[kadu1221.raw] 601 door effluorescence, micronized (40,40,0.3) JAK
Add trona 2nd-order preferred orientation.
χ2 = 3.040
Corundum?
Quantitative Phase Analysis
43.0(1)31.5(1)25.4(1)wt%
TronaHaliteSylvitePhase
Profile Y before and after micronizing
0.21(1)0.62(1)0.26(1)Strain, %
14.0(10)37.5(2)17.1(1)Micronized
2.10326.1(3)13.8(2)Hand ground
TronaHaliteSylvitePhase
A Case Study
IUCr CPD QPA RRSample 1H
Sample 1H
29.4(2)34.5(1)36.1(1)INEOS
30.0334.2635.35XRF
30.1934.6935.12Weight, %
ZinciteZnO
FluoriteCaF2
Corundumα-Al2O3
Phase
QPA by the Rietveld Method
S M ZXSMZ
α α αα =
“Star” Quality Structures from the PDF-4+ 2007
04-008-819704-005-4266 (I)04-004-2852PDF Entry
⅓,⅔,0.38190.006
¼,¼,¼0.004
0.30624,0,¼0.003
Atom 2, xyzUiso
⅓,⅔,00.007
0,0,00.005
0,0,0.352160.003
Atom 1, xyzUiso
3.2505.207
5.4634.760
12.993Cell a
c
P63mcFm mR cSpace Group
ZinciteFluoriteCorundumPhase
Use 20-148°. Three phase fractions and 3 background terms.
χ2 = 19.08
Change to 25-148°. Add 3 cells and shift.
χ2 = 17.54
Add profile X and Yfor each phase.
X = 2.428, Y = 1.906
χ2 = 2.835
Change peak cutoffs from 1% to 0.2%.
Add the structural parameters.
Quantitative Phase Analysis
28.43(7)33.67(8)37.90(16)Observed
30.1934.6935.12Expected, wt%
ZinciteFluoriteCorundumPhase
AKLD = 0.055
Absolute Concentration ErrorsIUCr CPD QPARR Samples 1n
2008 Results, Triplicate Analyses
Expected Concentration, wt%0 20 40 60 80 100
Con
cent
ratio
n Er
ror,
wt%
-1.0
-0.5
0.0
0.5
1.0
Al2O3CaF2ZnO
Another Case Study: Alka-Seltzer
x103
5.0
10.0
15.0
20.0
25.0
Inte
nsity
(Cou
nts)
02-061-2110> C6H8O7 - Citric acid
02-060-0435> C9H8O4 - 2-(Acetyloxy)-benzoic acid
00-015-0700> Nahcolite - NaHCO3
10 20 30 40 50 60 70Two-Theta (deg)
[alka-seltzerm.raw] Alka-Seltzer, micronized (40,40,0.3) JAK
The package says Alka-Seltzereach tablet contains:
30.861000C6H8O7
Citric acid
10.03325C9H8O4
Acetylsalicylic acid
59.121916NaHCO3
Sodium bicarbonate
Concentration, wt%Amount, mgCompound
Sum = 3241 mg. The tablet used weighed 3223.3 mg (99.45%).
Refine 3 background terms and 3 phase fractions
χ2 = 139.9
To get peaks in the right places, refine cells and a common
specimen displacement term
χ2 = 96.48
Add profile Y (strain) for each phase
χ2 = 22.44
Change to anisotropic strain broadening (Profile #4)
for NaHCO3 and citric acid
χ2 = 16.10
Add Uiso for NaHCO3(guessed initially)
Add diffuse scattering terms for Kapton background
χ2 = 9.147
Get the aspirin CIF from CCDC to use the experimental Uiso, and refine
(grouped) the Uiso for citric acid(used default values initially)
Some citric acid Uiso go < 0. Reset to “reasonable” values.
χ2 = 8.741
Add 2nd-order spherical harmonic preferred orientation coefficients
for each phase
χ2 = 5.355
Alka-Seltzer Analysis
28.3(1)30.86Citric Acid
8.6(1)10.03Acetylsalicylic
Acid
63.1(1)59.12Sodium
Bicarbonate
RefinedExpectedwt%
AKLD = 0.080
Take to extreme
XPS anode deposit on rag
A Challenging Case StudyMullite
Kyanite Mining CompanySupplied by Dilip Jang
Data collection by Fangling Needham, ICDD
x103
100
200
300
I(Cou
nts)
01-070-3755> Quartz - SiO2
01-074-4143> Mullite - Al4.44Si1.56O9.78
01-079-1456> Mullite - Al4.54Si1.46O9.73
04-008-9528> Al2.13Si0.87O4.93 - Aluminum Silicon Oxide
00-015-0776> Mullite - Al6Si2O13
10 20 30 40 50 60 70 80 90 100 110 120 130Two-Theta (deg)
[Fangling_1_mullite_vslit_9hrs_Mul_A.RAW] Mullite
Mullite Crystal Structure
Mullite and quartz phase fractions, and 3-term shifted
Chebyshev background
Add lattice parameters for both phases, shift, and profile X and Y for each phase.
Quartz profile X refines to 1.71(32) < 2.477; fix.
χ2 = 239.1
There are additional peaks (phases)
2021628810033I
1.7651.9633.1893.2494.0486.652d
51.7646.2127.9627.4321.9413.332θ
The strongest two peaks correspond to the strongest peaks of cristobalite and rutile. Add as phases 3 and 4 (phase fractions).
Change excluded region to 0-13°, and add 3 more background terms.
χ2 = 106.6
Some amorphous material is present
POWPLOT/R
Add 3 Type 1 diffuse scattering terms – amplitudes only
χ2 = 90.37
Refine the mullite structure with constraints and restraints
Al1-O5 = Al1-O6 = 1.91(2) ÅAl2/Si3-O5, O6, O7 = 1.67(2) Å
O-O = 2.73 (3) ÅAl4-O = 1.80(2) Å
χ2 = 73.37
Composition of Al2/Si3
M-O = 1.693. Interpolate between 1.61 and 1.74 to get Al/Si = 64/36.
Set fracs to 0.66/0.33 and constraint 2/1. Add Al4, O7, O8 fracs, as well as M and O Uiso.
wRp = 0.0733, χ2 = 63.33
Check the chemistrya = 7.5460(2), b = 7.7026(2), c = 2.88511(5) Å
ICSD 2008/1 has 24 mullites ±0.03 Å
Al5.14Si1.20O10.050.186(3)0.300(1)0.599(2)Current
Al4.59(6)Si1.41(6)O9.71(3)0.15(2)0.35(2)0.50(1)Average
FormulaAl4 fracSi3 fracAl2 frac
+20.22/-20.10, unrestrained
Look at the difference plot.Mineral-related strong
and/or 13.33° among long.Kyanite (!), Al2Si2O5
01-071-6298, ICSD 77538.
Add kyanite as phase 5.
Change mullite profile to #4, anisotropic strain broadening
Add quartz structure with restraints, and 2nd-order spherical harmonic preferred
orientation for mulllite
Add two more distances to the diffuse scattering function
Add profile U for mullite and quartz
Quartz not significant, so fix.
Add cells and profile Yfor phases 3, 4, and 5.
Add 4th-order spherical harmonics for mullite and quartz preferred orientation
wRp = 0.0340, χ2 = 14.27
Largest errors at non-mullite peaksΔF = 0.41/-0.30 eÅ-3
Lattice constants are a = 7.54693(10) b = 7.70404(9) c = 2.885182(23) Alpha = 90 Beta = 90 Gamma = 90 Cell volume = 167.7497(33)
Name X Y Z Ui/Ue*100 Site sym Mult Type Seq Fractn Al1 0.000000 0.000000 0.000000 1.187(17) 2/M(001) 2 AL 1 1.0000 Al2 0.14783(12) 0.33997(11) 0.500000 1.187(17) M(001) 4 AL 2 0.599(1) Si3 0.14783(12) 0.33997(11) 0.500000 1.187(17) M(001) 4 SI 3 0.300(1) Al4 0.26732(80) 0.20424(64) 0.500000 1.187(17) M(001) 4 AL 4 0.142(1) O5 0.35892(23) 0.42105(16) 0.500000 1.020(32) M(001) 4 O 5 1.0000 O6 0.12749(23) 0.22258(20) 0.000000 1.020(32) M(001) 4 O 6 1.0000 O7 0.500000 0.000000 0.500000 1.020(32) 2/M(001) 2 O 7 0.60(1) O8 0.4426(22) 0.0493(21) 0.500000 1.020(32) M(001) 4 O 8 0.126(7)
Al1-O5 1.893 ×4Al1-O6 1.966(2) ×2
Al4-O5 1.808(5)Al4-O6 1.793(4) ×2Al4-O8 1.782(19)Al4-O5 2.381(5)Al4-O7 2.358(5)
Al2/Si3-O5 1.711(2)Al2/Si3-O6 1.709(1) ×2
Al2-Si3-O7 1.663(1)average = 1.697
Al2-Si3-O8 1.751(14), 1.768(13)
Chemical Reasonableness
Al2/Si3-O and fracs: 2/3 AlAl4-O: Al
Al4.96Si1.20O9.70+19.68/-19.40
77.81 wt % Al2O3, 22.19 wt% SiO2a: 75 wt% Al2O3, V: 25% SiO2
Quantitative Phase Analysis
2.15(6)Kyanite0.60(2)Rutile0.61(3)Cristobalite7.15(6)Quartz
89.38(3)MulliteConcentration, wt%Phase
Typically 10-12% glassQuartz granularity?Exceptionally high-quality data
Chemical Reasonableness
Chemical Reasonableness“Chemical reasonableness in Rietveld analysis; organics”,
J. A. Kaduk, Powder Diffraction, 22(1), 1-9 (2007).
“Chemical reasonableness in Rietveld analysis; inorganics”, J. A. Kaduk, Powder Diffraction, 22(3), 268-278 (2007).
“Structure Refinement”, J. A. Kaduk, Chapter 8 in Practice and Applications of Powder Diffraction, A. Clearfield, J. H.
Reibenspies, and N. Bhuvanesh, editors. Blackwell (2008).
Chemical Reasonableness• Convergence (symmetry)• Molecular Geometry
– Bonded and Non-Bonded Distances– Angles– Torsions– Planarity– Hydrogen Bonds– Displacement Coefficients– Atomic Valences
• Magnitude of σs• Difference Fourier• Bulk and Individual Phase Compositions
Organic Structures
• Cambridge Structural Database• ConQuest/Vista• Mogul (intramolecular)• Gold (intermolecular)
Inorganic Structures
• PDF-4+• ICSD• CRYSTMET• Pearson’s Crystal Data• Bond Valence• Ionic Radii (R. D. Shannon and C. T. Prewitt),
“Effective ionic radii in oxides and fluorides”, Acta Cryst., B25(5), 925-946 (1969).
Bond ValenceThe Chemical Bond in Inorganic Chemistry, I. David Brown.
IUCr Monographs on Crystallography 12. Oxford University Press (2002).
“Bond valence parameters for solids”, N. E. Brese and M. O’Keefe, Acta Cryst., B47, 192-197 (1991).
http://www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown/bond_valence_parm
0exp ijij
R RS
B−⎛ ⎞
= ⎜ ⎟⎝ ⎠ i ij
jV S=∑
Implement chemistry knowledge inRestraints (soft constraints)
distances Dangles Atorsions Tplanar groups Pchemical composition Cchiral volume Kϕ/ψ Rmagnetic moments M
Rigid Bodies
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