Pecharsky V.K., Zavalij P.Y. Fundamentals of Powder Diffraction and Structural Characterization of Materials

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Index

initio

indexing 344,402,404,420

422,425,433,436,443,444,45

478,485,488,490,562,575

initio

structure solution 244,494, 5 15,

5 16,603

absolute intensity 188

absorption factor 187, 193, 194

accuracy of intensity measurements 328,

329

accuracy of lattice parameters 305

accuracy of peak positions 344, 346,402,

450

acute exposure 282

ALARA

283

angstrom 7, 100

anisotropic broadening parameters 179

anisotropic peak broadening 180, 554,

555,559,665

anomalous scattering 2 15,217, 622,648

aperiodic crystal 2, 88, 93

assignment of indices

See

indexing

asymmetric unit 8, 9, 61,64, 502,668

atomic displacement parameters 207,606,

643

anharmonic 21 1

anisotropic 208

isotropic 208

atomic radius 100

atomic scattering factor 144,204,212,

213,215,238,249,250

atomic scattering function 114, 143

145

augmented matrix 8 1,2 19

augmented vector 8 1,86

automatic indexing 436,439,443,444,

450,478

axial divergence 1 18, 168, 170, 180, 182,

184,258,268,312,313,366,531,532

background 604,606,609,643,650

background function

Chebyshev 35 1,643

diffuse 352

Fourier polynomial 35 1

polynomial 35

background subtraction 346,348

automatic 349

box car curve fit method 350

manual 349

balanced filters 121

base-centered unit cell 35

basis set 445

basis vectors 7

right-handed set of 58

Bayesian estimation theory 497

BGMN 181

body-centered unit cell 36

bond valence sum 648

Bragg angle 147

Bragg residual 512, 526,608

Bragg, William Henry

Bragg, William Lawrence 3 1

Bragg-Brentano focusing 1 16,267,270

272,309

Braggs' law 148

Bravais lattice 32, 37,38,55,56,64, 96,

224,227,416,436,443,503,504

Bravais, Auguste 37

brightness 104

brilliance 104

broadening parameter 179

calculated density 564

Cartesian basis 72

CASTEP 577

center of inversion 16

CeRhGe, 240,241,247,255,258,483

488,530

548,628

635,671

characteristic radiation 109

characteristic wavelengths (Table) 109

characterization goals 340

crystallinity 342

lattice parameters refinement 342

phase identification 342

structure determination 342

charge coupled device detector 137

ChekCell443

chemical analysis 6 16

chemical composition 693

chi-squared 5 13

chronic exposure 282

coherent scattering 102, 140

coherent scattering length 212,543,634

Collaborative Computational Project No.

14 415,420,444,475,507,601

collimated beam

1 16

collimation 1 15

color symmetry groups 53

Index

combined Rietveld refinement 603, 623

628,650

653

comparing patterns 379

constant background 348

constrained variable 6 15

constraints 609,615

617,673

definition of 654

constructive interference 139

continuous scan data collection 322

conventional neutron sources 114

conventional x-ray sources 104

convolution 17

corundum number 388

counting statistics 329

counting time 320, 321,323

cross-contamination 289

CrysFire 444

CRYSMET

Metals and Alloys Database

375

crystal families 32

crystal monochromator 123, 124,268,

269

geometry 125

crystal structure model 494,496

crystal systems 26,27,503

characteristic symmetry element 27

crystalline state 2

crystallites 103

crystallographic

axes 7

basis 7

coordinate system 7

directions 49

indices 46

planes

45,46

point groups 29

space groups 53

symmetry

crystallographic databases (Table) 375

crystallographic space groups 54

CSD

Cambridge Structural Database

375

cylindrical sample 27

damping 641,642

damping parameter 641

data processing steps 341

de Broglie equation 10

Debye rings 153,154, 158,262,302

curvature of 168

spottiness 302

Debye-Scherrer camera 262,263,264

Delaunay-Ito

method 441,443

reduction 44 1

technique 448

transformation 44 1

destructive interference

139

detectors

area 131

characteristics of (Table) 129

dead time of 129

efficiency of 128

line 13 1

linearity of 128

point 13 1

proportionality of 129

ratemeters 130

resolution of

130

true counters 130

DICVOL 444,447

449

DICVOL (example) 454

difference Fourier map 566, 567, 65

655,665,673

diffracted beam aperture

See

receiving

slit

diffracted beam monochromator 126

diffraction groups 228

diffraction peak 142

diffuse scattering 650

digitized pattern 371, 374

direct methods of structure solution 252

direct phase determination methods 249,

495,550,564

direct space modeling 244,496

direct space techniques 495

displacement ellipsoids

0,618, 623,

634,637,638

divergence

axial 1 17, 1 1 8

in-plane

1 17

divergence slit 1 16, 117, 268

effect on intensity 3 10

effect on resolution

selection of 309,3 10

variable 275,3 16

DLS-76 579

DM013 577,579,580

Index

DMSNT 347,349,350,362,368

373,

378

dosimeter 286

d-spacing 46

Durbin-Watson statistic 514, 608

dynamical diffraction 103

effective linear absorption coefficient 194

elastic scattering 138

electron density 238,495,499,528

deformation 243

difference 242

distribution 529, 537, 538, 539, 540

E-map 252,254,495,515,550,551,557

559,565

enantiomorph 1 1

energy minimization 577

energy minimization methods 498

entropy function 498

estimated density 556

Euler's formula 21 6

Ewald, Peter Paul 50

Ewald's sphere 149

154, 164, 190, 199,

33 1

excluded range 659,665

expected profile residual 5 13,608

EXPGUI 640

EXPO 507,s 15

extinction correction 628

extinction factor 187

face-centered unit cell 36

false peaks 529

Fe7(P04)6 460,462,463

Fedorov, Evgraf Stepanovich 53

FeP04 152,343,575

581,677

684,

697

Fibonacci numbers 9 1

Fibonacci, Leonardo Pisano 91

figure of merit (indexing) 450,452

41 8,419,427,452,572

MZ0 419

MN 420,446,447,455

finite symmetry elements 12

five-fold symmetry 88

flat sample 270,271

flotation approach 501

focal point of the source 268,269

focusing optics 268

four-fold inversion axis 18

four-fold rotation axis 18

Fourier integrals 237

Fourier map 239,526

528,536,55 1,

586,630,651,688

difference 242,558

Fourier transforms

forward 23 8

reverse 238

fractional coordinates 7

Friedel pairs 190, 219

Friedel's law 220,221

full pattern decomposition 347, 504, 5 17,

518, 525, 533, 541, 548, 549, 555,

584,585,589

figures of merit 5 12

full profile refinement 599, 601, 697

full width at half maximum 174, 176,

544,555,563

Fullprof 507,601,640

fundamental parameters approach 18 1

gas proportional counter 132

Gd5(Si,Gel.,)4 580,581, 587, 590,691,

696

Gd5Ge4 580

587,684

696

Gd5Si2Ge2 581,582,587

590,684,692

697

Gd,Si,

580

582,585

587, 590,684,

687

696

genetic algorithm 497

geometrical method 580

geometry optimization 575,577,678

glide planes

40,41

symbols of 40

translation vectors 40

global minimum 5 1 1

goniometer 267

axis 270,273

circle 268,269

horizontal 269,274

horizontal axis 274,275

radius 269, 276

vertical 277

vertical axis 277,278

goodness of fit 5 13,608

gravimetric density

500,5 16,564

group

examples 25

Index

properties of 24

theory 24

GSAS 58,507,515,563,600,606,610,

639

645,650,660,661,667,669,

673,676,683,693

half-value layer 28 1

Hanawalt search 372

Hauptman, Herbert

25 1

heavy atom method 248,495

Hermann-Mauguin symbols 53

hexamethylenetetramine 159,210,

594,

595

Hf2Ni3Si4 596,597,700

histogram See powder diffraction pattern

hkl

triplets See Miller indices

ICDD See International Centre for

Diffraction Data

ICSD

Inorganic Crystal Structure Data

375,569,573

image plate detector 137,271,277,278

incident beam aperture See divergence

slit

incoherent scattering 102

indexing 553

initio

403,415,421,422,424,432,

436,450,457,587

initio

(example) 427,429

automatic 576

automatic (algorithms) 436

cubic crystal system 422

difficulty of 401,403

figure of merit 4 1 8

forbidden integers 423,424,428

four cornerstones 421

from first principles See

initio

hexagonal crystal system 429

hexagonal crystal system (example)

43 3

in direct space 436

in reciprocal space 436

known unit cell 402,405

known unit cell (example) 407

of powder diffraction data 401,413

permutation of indices 437,444

reliability of 401

semi-exhaustive 444

solution 459,462

tetragonal crystal system 429

trial-and-error 438, 444

unit cell selection 415

zone search method 439,448

infinite symmetry elements 39

in-plane divergence 169,268

instrumental function 17 1

instrumental parameters file 640

integrated intensity 186,504

interaction of symmetry elements 19,22,

23,43, 86

interatomic distances 527,536

interatomic vectors 245, 495, 535

distribution of 545

interference function 141

International Centre for Diffraction Data

330,343,372

International Tables for Crystallography

14,29, 50,54, 57

65,71, 82,93,

94,96,97, 101, 109, 123,212,213,

215,230,249,256,280,333,334,

414,481,525,535

International Union of Crystallography

10,

14,29,93,94,207,256,374,415,

420,444,475,507,525,601,699

interplanar distance 46, 148

absolute error in 465

inverse matrix 468

ionizing radiation 281

non-stochastic effects 28 1

stochastic effects 28 1

isostructura1495,503, 570

IT0 444,448,449,562,575,576

IT0 (example) 453

IZA- Zeolite database 375

K absorption edge 120, 121

Ka and

radiation 109

Kaz stripping 346,355

Karle, Jerome 25 1

kinematical diffraction 103,493

Klug's equation 385

KoalarietIXFIT 18 1

Index

451

456,516

530,537,601

628,

697

lattice 4, 5

arbitrary origin of 5

centering 37

origin of 5

point 7

translations 7

translations due to centering 37

lattice parameters 7

Laue classes 3

1,32

Laue classes, "powder" 3 1

Laue equations 147

Laue, Max von 3 1,100

Le Bail method 505, 506,515

least squares

correlation coefficients 5 1 1

iterative 5 10

linear 466

method

468,5 10

non-linear 507,5 10

solution 467,470,473, 508

standard deviations 474,509

standard uncertainties 474,509

technique 466,467,508

utilities 475

weighted 474

weights in 474, 509

LEPAGE 443

Levenberg-Marquardt damping 64 1

LHPM-Rietica 507,5 15, 5 17,5 18, 53 1,

541,600,610- 612,614,615,617,

624,631,637,640,641,643,684

linear absorption coefficient 1 19, 120,

122

effective 297

LMGP 443,444

location of hydrogen atoms 644

Lorentz factor 187, 190

Lorentzian broadening function 178

Lorentz-polarization factor 191, 192

low resolution 677

rna2M070~~ 457,568

571,662

668,

697

mass absorption coefficient 122

mass absorption coefficients (Table) 123

Materials Studio 577

maximum entropy method 497,498

maximum likelihood method 498

microstrain broadening 180

Miller indices 46,402,469

generation of allowed combinations

406,413

symmetrically independent

combinations of 414

Mineralogy Database 374, 375

mirror plane 16

mixing parameter 174,544

Mn7(OH)3(V04)4 571

575,669

676

modulation

commensurate 88

incommensurate 90

saw-tooth-like 93

sinusoidal 90

vector 89,401

monochromatization 1 19,274

monoclinic distortion 587,588

multiplicity factor 187, 189,221

multi-term Simpson's integration 178

Nd5Si4 548

553,560,635

639,689

Niggli reduction 442

NiMnO,(OH) 348,363

369,379

381,

553

555,560,643

653

NIST

Crystal Data 375

non-linear least squares method 600

non-linear least squares minimization 603

normal equations 468

normal equations matrix 468

properties of 468

normalized structure factor 250

nuclear density

239,499,546,547,65 1

numerical conditioning

5 10

occupation factor

See

population factor

occupational disorder 206

one-dimensional modulation 401

one-fold inversion axis 16

one-fold rotation axis 16

optical axis 270

order of an axis

80,s 1

origin of coordinate system 58

partially vacant sites 669

particle agglomeration 288

particle size 287

particles

Index

needle-like 290

packing density 298

platelet-like 290

ribbon-like 29 1

pattern recognition 371

pattern scale factor 603

Patterson function 245,246,535,544

Patterson technique 495

Pauling File 374, 375

Pawley method 505,506

PDB

Protein Data Bank 375

PDF See Powder Diffraction File

PDF quality mark 382

PDF-2 375

PDF-4 374

PDF-4 Full File 375

PDF-4 Organics 375

peak asymmetry 168, 182,642

peak broadening 575,642

peak positions 164

factors affecting 167

peak range 186,521,522

peak search 346,356

automatic 356

first derivative method 357,358

manual (visual) 356

profile scaling method 356, 359

second derivative method 356,357,

358,359

peak shape function 17 1,642

Gauss 173, 175

Lorentz 173, 175

Pearson-VII 174,585

pseudo-Voigt 173,544,584,589

split Pearson-VII 183

split pseudo-Voigt 183

Thompson modified pseudo-Voigt

178

peak shape function parameters 606

Pearson's classification 503

Pearson's Handbook 504,524

Pearson's symbol 503, 504, 524,540

phase identification 377

in a multiple phase sample 382

phase problem

18,243

photoelectric effect 12 1

photon counting errors 33

pks2ind 45 1

point group symmetry 26

Poisson's probability distribution 328

polarization factor 140, 187, 191

poor crystallinity 575, 677

population factor 204, 205, 206, 643

population parameters 606,676

porosity 649

effect 195,642

position sensitive detector 136, 271

curved 276

positional parameters 606

powder diffraction cone 153,

155

powder diffraction data

resolution of 265

powder diffraction experiment

fast 3

18,34 1

overnight 3 19, 341

weekend 319,341

Powder Diffraction File

record (Figure) 373

Powder Diffraction File 343,372

record 372

Powder Diffraction File 375

powder diffraction method 262

powder diffraction pattern 155

digitized 345

reduced 345

powder diffractometer 264,265,267

preferred orientation 196, 561, 567, 606,

629,636,642,653

complex 553

in-plane

29 1

uniaxial 290

preferred orientation axis 197,646

preferred orientation factor 187

preferred orientation function

ellipsoidal 198

March-Dollase 199,645

spherical harmonics expansion

200

preferred orientation magnitude 201,646,

653,668,671

primary beam monochromator 126

primary extinction 202

primitive unit cell 35

probability of phase relationship 250

profile fitting 346,553

conducting of 36 1

difference plot 363

examples of 362

semi-manual 562

Index

weights in 364

profile fitting parameters

approximate 361

integrated intensity 360

peak positions 360

peak shape 360

profile intensity 186

profile residual 5 12,608

propagation vector 102

pseudo-Voigt function 544

pycnometric technique 501

qualitative analysis 371

quantitative analysis 37 1, 384

absorption 385

absorption-diffraction method 385

accuracy 3 88

by full pattern deomposition 388

by Rietveld refinement 388

internal standard method 387

limits of detection 388

method of standard additions 386

preferred orientation 385

reference intensity ratio method 388

spiking method 386

quasicrystals

15,9 1

radiation

absorbed dose 280

dose equivalent 280

exposure 280

shielding 28 1

608

61 1,621,624,628,629,633,

636, 645, 657,663, 665,671, 679,

685, 688, 693,

See

Bragg residual

receiving slit 268

effect on intensity 3 14

effect on resolution 3 14

selection of 3 13

reciprocal lattice 50, 149, 150, 15 1,400

one-dimensional projection of 400

reciprocal lattice vectors

loss of directions 400

measurable lengths 400

reciprocal space methods 244

reciprocal space techniques 495

refinement of the chemical composition

669

reflection conditions 64

cubic crystal system (Table) 237

hexagonal crystal system (Table) 235

monoclinic crystal system (Table) 23

orthorhombic crystal system (Table)

23 1

tetragonal crystal system (Table) 236

trigonal crystal system (Table) 235

reflection geometry 270

residual

expected 330

observed 330

restraint weight factor 680

restraints

definition of 654

Re,, 608,

See

expected profile residual

526,608

rhombohedra1 unit cell 36

Rietan 601,640

Rietveld method 506,584,599

608, 699

minimized function in 604,624

Rietveld refinement

figures of merit in 608

multiple phase 605

sequence of parameters 607

termination of 609

Rietveld, Hugo

599

right-hand rule

rigid-body 660

Roentgen, Wilhelm Conrad 100

rotation angle 14

rotation matrix 77, 79

Rp 608

61 1,62 1,624,629,633,636,

645, 657, 663, 671, 679, 685, 688,

693,

See

profile residual

Rwp 608

61 1,621,624,629,633,636,

645, 656, 657, 658, 663, 671, 679,

685, 688, 693,

See

weighted profile

residual

sample displacement 427,606,643,

See

specimen displacement

sample displacement error 407,45 1,460,

471,473,476,477,562

sample holders

flat 292

sample preparation

back filling 293

capillaries 294

dusting 293,294

Index

filling 293

irradiated length 296

sample length (Equation) 295

sample transparency 606

sample transparency error 407

sampling interval 322,323

Sayre equation 250

scale factor 146, 187, 188

scale factors 606

scan range

end angle 324

start angle 324

scan rate 322,323

scatter slit 268, 3 15

selection of 315,316

variable 275,3 16

Scherrer constant 180

Schonflies symbol 60

Schonflies, Arthur Moritz 53

scintillation detector 133, 275, 276

screw axes 42

enantiomorphous 42

symbols of

translations along 42

searchable index 377

search-and-match 372

automatic 377

search-and-match options 378

search-and-match parameters

number of Bragg reflections 377

number of strongest reflections

377

window 378

secondary extinction 202

serial correlation 5 14,608

shape of the crystallites 553

SHELXL-97 5 15,557

559,565

SHELXS-90 515,557,565

simulated annealing 498

site occupancy 610,693

site position

general 66

multiplicity of 64, 65

special 66

Wyckoff letter 65

Wyckoff notation 65

site positions 64, 65

six-fold inversion axis

six-fold rotation axis 18

smoothing 346,352

box car 352

coefficients 353

fast Fourier transformation method

353

loss of resolution due to 352, 355

peak broadening due to 352

soft restraints 657,660,678,680

solid-state detector 133,274

Soller slits 117,268

effect on asymmetric broadening 3 12

effect on resolution 3 12

space groups

determination of 229

diagrams 63

full international symbols of 56

in nature 59

symbols of

54,60

symmetry 53,494

visualization of 58

spallation neutron sources 114

special sites

on centers of inversion 68

on inversion axes 67

on mirror planes 66

on rotation axes 67

specimen displacement 169, 170,298,

299,300

specimen function 172

spectrum of the electromagnetic waves

103

spread 328

SrSi, 595, 596, 700

standard deviation 6 14

statistical error 329

step scan data collection 3 19

step size 320, 321

stereographic projection 27

structural characterization steps 341

structure amplitude 145,203,204, 216

complex 2

16,2 17

magnitude 2 17

normalized 557

phase angle 2 17

structure factor 145, 187, 504

observed 495,499,525,542

structure solution from first principles

340,553

structure solution process 499

superspace 90,401

Index

surface roughness 195

symmetry 1

non-conventional 88

rotational 14

symmetry element 9, 10

complex 9, 1

finite 12

infinite 12

multiplicity of 12

order of 12

proper 12

simple

symbols of 13

symmetry groups 21

symmetry operation 8,9, 10,64

algebraic description of 72

improper

simple 10

symbolic description of 69,71

synchrotron radiation sources 104, 112

synchrotron spectrum 1 13

systematic absences 222,5 16

analysis of 494,531, 548,554,562,

568,572,576

and space groups 227

due to glide planes (Table) 225

due to lattice centering (Table) 224

due to screw axes (Table) 227

tangent formula 25 1

Taylor's expansion 508

temperature factor 204,207

texture

See

preferred orientation

Thomson equation 140

three-dimensional modulation 401

three-fold inversion axis 17

three-fold rotation axis 17

time-of-flight 1 15, 159

tmaV307 561

566,654- 661,668

translation vector 79

transmission geometry 270, 271, 276

transparency shift 169

transpose matrix 468

transverse electromagnetic wave 103

TREOR 444,445,447,449,553,572,

575

TREOR (example) 45 1

trial-and-error methods 496

true absorption 120

two-fold inversion axis 16

two-fold rotation axis 16

U3Ni6SiZ 427,480

unimodularity 75

unit cell 4,5,6

centered 35

content of 8,494,500,5 16

mass of 500

Niggli-reduced 44 1,443

origin of 5, 63

primitive 35

reduction 440

selection of 32

shape of 33

standard choice of 34

unit cell dimensions 7,606

precise 469

precision of

465

vacancies 672, 674,676

VO(CH3C00)2 597,598,700

VRML 58

wavelength dispersion function 172

wavevector 149

weigh in Rietveld refinement 605

weighted profile residual 5 12,608

white radiation 107

WinCSD 367

371,515,550,562

WLepage 443

x-ray

capillary 279

mirror 279

x-ray density 50 1

x-ray emission spectrum 108

x-ray fluorescence 102

x-ray holography 100, 101

x-ray tube 104

efficiency of 105

line focus 106

point focus 106

rotating anode 1 10,275

sealed 106

spectrum of

107

x-rays 100, 102

absorption of 102,119,120

fluorescent 121