Blake A.J.(ed.) Crystal Structure Analysis

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382 Questions and answers

Mean |E*E-1| = 1.327 [expected .968 centrosym and .736 non-centrosym]

Systematic absence exceptions:

b-- c-- n-- 21-- -c- -a- -n- -21- --a --b --n --21

N 347 317 316 7 240 235 239 12 85 97 94 26

NI>3s 4 70 68 0 79 72 73 0 28 26 22 8

<I> 0.4 115.8 116.2 0.2 204.6 335.6 275.4 0.2 59.9 40.3 78.9 347.8

<I/s> 0.5 1.9 1.8 0.2 2.6 2.6 2.5 0.4 2.3 2.1 2.0 2.8

Identical indices and Friedel opposites combined before calculating R(sym)

No acceptable space group - change tolerances or unset chiral flag

or possibly change input lattice type, then recheck cell using H-option

XPREP output for oxalyl chloride

c = 16.676(11) Å. The following (table above) was

found assuming orthorhombic symmetry using

XPREP. Show that these data are consistent with

the correct space group being P2

/c with a = 16.67,

b = 5.34, c = 7.26 Å, β = 90

◦

There are absences in the 0kl (k odd, b–) and h00 (2

–)

index classes. The absences in the –2

– class are just

a subset of the former. These absences do not cor-

respond to any orthorhombic space group, but do

correspond to the monoclinic space group P2

/b11.

This is just a non-standard setting of P 2

/c, with the

new b = old a and new c = old a.

(d) Calculate Z



and comment on the mean value |E

−

1| = 1.327.

The volume of the cell is 646.3 Å

. Applying the

18 Å

rule gives six molecules per cell. This is rel-

atively unusual, though not impossible here, as the

oxalyl chloride molecule is centrosymmetric, and so



= 1.5 is possible. The value of |E

−1| is unusu-

ally high, and implies a distribution comprising both

strong data and weak (or absent) data.

(e) A Patterson map calculated using the second cell

given in part (c) showed a very strong non-origin

peak at [1/3 0 2/3]. Suggest a transformation to a

smaller unit cell.

⎛

⎝

3 0

010

001

⎞

⎠

is possible, though this would give a very acute β

angle (cell OABC in the diagram), and it is better to

use

⎛

⎝

3 0 −

01 0

00 1

⎞

⎠

cell 0ACD.

(f) What are the dimensions of this smaller cell?

From the ﬁgure above



= length of the vector (

3 0 −

3) =[(16.67/3)

(7.26/3)

]

1/2

= 6.06 Å



= b = 5.34 Å



= c = 7.26 Å.

β can be obtained from the dot product

(a/3 − c/3).c = a.c/3 − c.c/3

= (16.67 × 7.26 × cos 90)/3 − (7.26

/3)

= 0 − (7.26

/3)

= 6.06 × 7.26 × cos β

so β = 113.5

◦

(g) The structure of oxalyl chloride was successfully

modelled as a twin. What is the likely twin law?

The likely twin law is a two-fold rotation about the

a-orc-axis of the larger pseudo-orthorhombic cell.

This can be expressed on the axes of the small

Questions and answers 383

monoclinic cell by forming a triple matrix product:

⎛

⎝

3 0

−1

01 0

00 1

⎞

⎠

⎛

⎝

10 0

0 −10

00−1

⎞

⎠

⎛

⎝

301

010

001

⎞

⎠

⎛

⎝

0 −10

00−1

⎞

⎠

Chapter 19

No exercises.

Chapter 20

No exercises.

Chapter 21

No exercises.

Chapter 22

No exercises.

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Index

α−doublet 254–5

absences, systematic 44–7, 49, 51, 61, 94,

176, 184, 264, 280, 282, 284, 286,

287–8

absorption 43, 75, 174, 189, 197, 233–4,

243, 247–8, 256, 341

absorption correction 41, 65, 70, 75, 90,

94–5, 177, 198–9, 282

absorption edge 1, 191, 255, 257

accuracy, see precision and accuracy

amplitudes 4, 5, 54–5, 170

in direct methods 133–4, 140

in Fourier syntheses 104–9, 110–14,

117–18, 172–3

normalized, see structure factors,

normalized

analogue of diffraction, optical,

see microscope, analogy for X-ray

diffraction

analysis of variance 234, 236, 245

angle, dihedral 211–13

area detector 41, 53, 54, 62, 65, 67, 70,

73–75, 77–82, 90, 98–9

body, rigid 217–8

bond valence 195–6, 201–2

Bragg angle 6, 71, 80, 95, 175

Bragg–Brentano geometry 254

Bragg’s law (Bragg equation) 6, 7, 55–6,

170, 257

Bravais lattice, see unit cell centring

capillary tubes 34, 37–8, 256–7

cell, unit 2, 3, 11–12, 51, 104–5, 118–19,

209, 271–2

centring 14–15, 19, 22, 45–6, 61, 120,

197, 279

contents 21, 43–4, 88, 247, 265

determination 58–64, 84–8, 94, 264, 283

origin 18, 51, 122, 129–30, 140–1, 144

parameters 2–3, 14, 49, 56, 57, 62–3,

211, 243

Central Limit Theorem 227–9, 233

chirality 13–14, 22–3, 42, 48–9, 184,

207, 272

CIF (crystallographic information

ﬁle) 299, 310, 315, 319–26

conﬁguration, absolute, see structure,

absolute

conformation 207, 213, 246

constraints

in direct methods 134–41

in reﬁnement 21, 160–2, 180, 184, 191,

192, 210, 214, 216, 245, 248, 311

convolution 3, 119, 133–4, 135, 258, 351–2

correlation, see covariance and correlation

coset decomposition 279–80, 290, 291,

291–2

covariance and correlation 158, 160, 178,

191, 209–11, 216, 240–2, 263

crystal growth 28–34, 189, 271

crystal morphology (shape) 19, 42, 90

crystal mounting, sample mounting 36–9,

62, 254–6

crystal packing 18, 232, 304–5

crystal screening, crystal

evaluation 35–36, 42–3, 82–4

crystal, single 9, 28, 33, 34, 35, 42

crystal system 14–15, 17–20, 22, 35, 43–4,

51, 54

damping, see restraints, shift-limiting

databases 34, 87–8, 195, 230, 258–9, 294,

321, 327–31

data collection 41–2, 56, 61–2, 64–7, 73–90

data completeness 64–5, 89–90, 95, 176

data reduction 67–71, 93–8

data, unique, see set of data, unique

Debye–Scherrer rings 254

degrees of freedom 158, 224, 231, 239

density of crystals 43–44, 192

design matrix 157, 159, 180

deviation, estimated standard,

see uncertainty, standard

deviation, standard 155–6, 222, 224–5,

229–30, 232, 236

difference electron density 110–13, 114,

172–3, 174, 245, 248

diffraction, multiple, see Renninger

reﬂection

diffraction pattern 2, 3, 4–5, 9–10, 18–20,

54, 64–5, 74, 99, 104–5, 110, 170, 192,

193, 194–5, 252–3, 257, 258–65,

274–6, 280–1, 282

diffractometer 36, 39, 62–4, 65–6, 73–5, 80,

243, 254–8

diffusion, liquid 30–1

diffusion, reactant 31–2

diffusion, vapour 31

disor

der

28, 50, 76, 104, 108, 174, 176,

182–3, 189, 190–4, 197–8, 214, 215,

218, 246

dispersion, anomalous, see scattering,

anomalous

displacement ellipsoid 215, 303–4

displacement parameters 103–4, 172,

181–2, 214–8, 243

anisotropic 103, 215–6

displacements, atomic, see displacement

parameters

distribution, normal (or Gaussian) 155,

211, 225–7, 230–1

distributions, statistical 47–8, 222–9, 233

electron density 5, 6, 7–8, 103, 134–5,

135–40, 170–2, 182–3, 184

from direct methods 141, 145–6

from Fourier synthesis 105–7, 108,

109–10, 113–14, 170, 172–3

and maximum entropy 150, 151, 153–4

ellipsoid, thermal, see displacement

ellipsoid

entropy, maximum 140, 149–54

equations, normal 157, 161, 162, 166, 183

equations, observational 156–7, 158, 159,

162–4, 165–6

errors, random 177, 178, 221–2, 221–32

errors, systematic 76, 145, 174, 176, 178,

214, 221–2, 242–7

E-map 109, 145

E values, see structure factor, normalised

Ewald sphere 56–8, 69

extinction (optical) 35–6, 42–3

extinction (primary and secondary) 70,

94, 181, 182, 243

385

386 Index

ﬁgures of merit 141, 144–5

Flack parameter, see structure, absolute

Fourier transform, Fourier synthesis 6, 9,

54–5, 64, 103–14, 133–4, 135, 146

in structure reﬁnement

in structure solution 117–19, 126–7,

133–4

Friedel pairs 183

Friedel’s law 4, 19–20, 64–5

Gaussian distribution, see distribution,

normal

gel crystallization 32–3

geometry of diffraction 9–10, 55–8, 254

geometry of molecular structure 128, 181,

205–7, 209, 210–11, 217, 232, 242,

245, 310–12, 328

goniometer head 36–7, 39, 62

goodness of ﬁt 173, 183, 235–6, 261

graphics, molecular 300–9

group, rigid 128, 210–11

Harker lines and planes (sections) 123–6

Hermann–Maugin notation 13–14

high-pressure data collection 77, 266

hydrogen atoms 47, 111–13, 182, 210,

213–14, 245, 308, 340

hydrogen bonding 213–4, 303, 305, 311–12

ill-conditioning and singular

matrix 164–5, 180, 246

indices, indexing 6, 55–62, 85–7, 104, 106,

264, 282–3, 285

inequality relationships 136, 138, 140

integration, see data reduction

intensity 3, 4, 5, 9–10, 44, 54–5, 64, 65–7,

67–71, 81, 88–90, 93–4, 142, 170, 175

252–4, 255–6, 257, 259, 261, 262, 265,

274, 337

intensity statistics 47–8, 141, 228–9, 264,

275–6, 284

interactions, intermolecular 213, 273

International Tables for

Crystallography 17, 50–1, 122, 278

Karle-Hauptmann determinants 136–7

Lagrange multipliers 160–1

lattice 2–3, 11–12, 14–15, 45, 58–9, 69

lattice centring, see cell, unit

lattice planes 6, 55–8, 69, 70, 71, 136–7

lattice, reciprocal 2–3, 36, 45, 57, 58–62, 87,

170, 208

Laue class, Laue symmetry 19–20, 21–22,

41, 43–4, 48–9, 61–2, 64–5, 89–90,

120, 276, 283

least-squares planes 211–13, 310

least-squares reﬁnement 156–66, 169–85,

200–1, 232–4, 240, 261

leverage 176

libration 182, 217–18

Lorentz-polarization corrections 69–70,

94, 261

low-temperature (and high-temperature)

data collection 38–9, 76–7, 176,

197–8, 200, 218, 243, 246, 265–6

matrix, singular, see ill-conditioning

mean 155–6, 224–5, 229–30, 231–3

methods, direct 133–46

microscope 35–36, 42

analogy for X-ray diffraction 7, 9,

53–55, 105–6

Miller indices, see indices

minimum, false 180, 263

model, riding 210

monochromator 69–70, 254–5, 257,

334, 337

mosaic spread, mosaicity 35, 89

mother liquor, see solvent

multiplicity of atom site, see site

occupancy factor

neutron diffraction 191, 214, 257–8, 266,

339–41

orientation matrix 62–4, 84–7, 93

origin, ﬂoating, see polarity

outliers 178, 179, 225, 233–5

parameters, in reﬁnement 104, 155–8,

161–6, 172–3, 175, 178, 182–3, 205,

231, 240–1, 261–3

parameters, thermal, see displacement

parameters

Patterson search 128–30

Patterson synthesis, Patterson map 109,

117–30

phase change, phase transition 38, 77, 99,

176, 189, 190–1, 194–5, 265–6,

273, 284

phase identiﬁcation and analysis 258–9,

262–3

phases (of reﬂections) 1–2, 4, 5, 6, 8,

54–5,

104–14,

133–5, 142–4, 170,

172–3

determination 134–46

point group 10, 12, 14, 19–20, 21, 22–3, 42,

51, 213

polarity, ﬂoating origin 22–3, 51, 125,

126–7, 129–30, 165, 210, 244

polarization of light, see microscope

positions, general and special 21, 23, 44,

51, 122–7, 210, 216

powder diffraction 99, 194, 251–67

precipitant (antisolvent) 28, 30–1

precision and accuracy 183, 205, 210, 211,

213–14, 218, 222, 240, 242, 264

probability distribution function

(pdf) 138, 225–6

probability plot, normal 236–8

pseudo-symmetry 49, 50, 126–7, 183–4,

190, 194–5, 196–9, 199–202, 271–2,

280–2, 283, 284, 293–4

publication 299, 304, 307, 309, 310, 311,

315, 321

quartets, negative 137, 144

recrystallization, see crystal growth

reﬁnement

of crystal structure, see least-squares

reﬁnement

of unit cell 87, 94

reﬂection proﬁles 65–7, 68–9, 93, 260–1,

263

reﬂections, equivalent (by symmetry) 43,

61–2, 64–5, 90, 95, 105, 107, 194,

222–3

Renninger reﬂection (multiple

diffraction) 49, 94, 176

replacement, molecular 140

residuals, see R indices

resolution, series termination 7–8, 108,

114, 135, 173, 175–6, 253

restraints 158–60, 180, 181, 182, 183, 184,

191, 192, 201, 210, 214, 218, 245, 248

shift-limiting 180, 181

Rietveld reﬁnement 261–4

R indices, R factors 174, 177, 181, 191,

198–9, 238–9, 261–2

rule, 18Å

samples, air-sensitive 38–9

Sayre’s equation 139–40, 146

scattering, anomalous 1, 2, 4, 19, 48–9,

183, 191, 243–4, 257

use in absolute structure determination,

see structure, absolute

scattering factor, atomic 1–2, 4, 103–4, 112,

135, 139, 172, 189, 215, 243, 245,

339–41; see also scattering,

anomalous

scattering, (thermal) diffuse 71, 94, 100,

176, 193–4, 243

series termination, see resolution

set of data, unique 41, 56, 61–2, 64–5,

90, 107

signiﬁcance level in statistical tests 239

site occupancy factor 104, 182, 183, 190,

191–2

Index 387

solvent 28–32, 33, 39

of crystallization 27–8, 29, 35, 38, 44,

88, 182

Soxhlet apparatus 30

space group 17–18, 18–20, 21, 22–23,

41–51, 121–8, 190, 192, 197, 200,

246, 264

of Patterson function 119–20

stereoscopy 306–7, 309

strain 196, 260–1

structure, absolute 75, 182, 183, 248, 276–7

structure factor 4, 5, 6, 54–5, 64, 68–9,

104–6, 109, 117, 133–4, 170, 172, 178,

345–6

calculated 105, 110

normalized (E-value) 47–8, 121, 135–6,

137–8, 139, 140–1, 142

observed 105

structure, incommensurate 74, 99, 190,

194–5

structure invariants 140–1, 143

structure, model 103, 110–14, 172–3, 175,

177, 178–9, 180–3, 190, 191–2, 195,

196, 200–1, 215, 221, 236, 238, 239,

244–7, 259, 261, 262, 264–5

structure validation 173–4, 195–6, 201–2,

247–8, 325–6

subcell, supercell, see pseudo-symmetry

sublimation 33

substructure, superstructure, see

pseudo-symmetry

symmetry element, symmetry

operation 12–14, 15, 16–18, 19, 21,

42, 47, 51, 123, 210, 216, 242, 246,

271–3, 280

symmetry, metric 21–2, 43, 49, 61–2, 253,

279, 283–4

symmetry, molecular 12–15, 20, 311

synchrotron radiation 71, 76, 257, 266,

335–9

tangent formula 139–40, 141, 142–3, 144

tensor, metric 208

temperature factors, see displacement

parameters

Thomson scattering 1

torsion angle 207, 212, 213

translation symmetry 10–12, 16–17

twin law 272, 274–6, 279–80, 282–3

twinning 28, 29, 42–3, 49, 74, 85, 98–9, 170,

179, 189, 192, 194, 271–95

by inversion 276–7

merohedral 277–8, 279, 289–91

non-merohedral 280–3, 292–4

pseudo-merohedral 278, 285–9, 291–2

twin scale factor 272, 276

uncertainty, standard 205, 222

in data 177, 261

in molecular structure 209–10, 211, 242

in reﬁned parameters 178, 210–11, 240

unit, asymmetric 20, 21, 51, 103–4, 107,

121–6, 140, 195, 199–200

variance 143–4, 151, 155–6, 158, 165, 212,

224, 229

weights

in direct methods 144

in Fourier syntheses 112–13

in least-squares reﬁnement 157–8,

159–60, 161–3, 178–9, 183, 197, 212,

214, 232–8, 240, 244, 261–2

in mean values 155–6, 224–5, 229–31

Wilson plot 142

X-ray photographs 36

X-ray sources 75–6, 254, 333–9

X-ray wavelength 6, 12, 55–6, 70, 75–6,

257, 334, 337

Z and Z



(unit cell contents) 21, 43–4, 51,

183–4, 247