Synthesis, Structure and Physical Properties of Selected Transition Metal Oxides

Physical Properties of Selected

Transition Metal Oxides

J. E. L. Waldron

Royal Institution of Great Britain

London

thesis submitted for the degree o f Doctor o f Philosophy

from the University of London

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M onoclinic N b j20 29 has been synthesised by topotactic reduction o f H -N b2 0^. It

has a com plex crystallographic shear structure m ade up o f 3x4 blocks o f com er

shared N bO ^ octahedra. N b 1 2 ^ 2 9 is a m ixed d^/dj com pound w hich, can be w ritten

as Nb^^2^t)^^io029" It has been found, using Rietveld refinem ent o f pow der neutron

diffraction data, to undergo a charge ordering transition at around 12K resulting in a

low ering o f sym m etry from A 2/m to Am. B elow this transition, one Nb^+ ion is

localised on a crystallographically distinct site in the centre o f the 3x4 blocks. This

charge ordering transition has an associated transition to an incom m ensurate one

dim ensional antiferrom agnetic state. The nature o f the m agnetic order has been

in v e stig a te d u sin g s u sc e p tib ility and |xSR m easu re m en ts. S u scep tib ility

m easurem ents reveal that only 48% o f the Nb"^+ in the system is localised, and its

tem perature d ependence is w ell fitted to a B onner-F isher m odel for S = l/2 one

dim ensional antiferrom agnets. It is found that the rem aining valence electrons in

^ ^1 2 ^ 2 9 itinerant and contribute to its m etallic conductivity.

Further evidence for the unique properties o f m onoclinic N b^2029 been

obtained by doping the structure w ith sm all am ounts o f diam agnetic Ti^+, w hich

rapidly causes a destruction o f the long range m agnetic order at only 3% doping, but

leaves the m etallic conductivity unaffected, as is expected for a one dim ensional

F irstly I w ould like to thank M um m y and D addy for all th eir help and

support in so m any w ays over the last 4 years.

I w ould also like to say a huge thank you to M ark G reen for his endless

guidance and backing th ro ug ho ut and particularly during the last few m onths o f

w riting up. W ithout h im none o f this w ould have been possible. T hank you to

Professor Day for allow ing m e to do a PhD at the RI and for all the opportunities

this has afforded me. T hank you too to Steve Bram w ell for his sharp insight into all

aspects o f m agnetism .

I am m uch indebted to Professor Iw asa for all his help during my trip to

Japan and especially to Chris N uttall who spent so long trying to m ake the pressure

rig w ork w ith me. T hank you also to A lex Lappas for all his hard w ork fitting the

pS R data w hich has m ade such a difference to the analysis o f my SQUID data. I am

naturally grateful to D an N eum an at N IST and A lan H ew at at ILL for all their help

during neutron diffraction experiments.

Thank you to Justin, M att, Gavin, Steve and Jo for m aking all those hours

spent in the XRD room doing GSAS alm ost fun! Thanks also to Sarah and G ayna for

all those swift halves, to Fiona for being a com plete star and alw ays there to pick up

the pieces w hen things w ent w rong and to Al, G reg, Justin and Jo w ho practically

forced me to sit at m y com puter and type.

Finally, thanks to Rod, D oug and everyone else at K illin for giving m e the

perfect place to escape from everything. W ithout all o f y ou I never w ould have

finished.

C hapter 1 Introduction

1.1 Introduction

1.2 Low D im ensionality in H igh Superconductors

1.3 One D im ensional M agnetism

1.4 Spin Peierls System s

1.5 A lternating C hain C om pounds

1. 6 Spin Ladder System s

1.7 V anadium Oxides

1. 8 N iobium Oxides

18 18

20

C hapter 2 T heory

2.1 D iffraction

2.1.1 Rietveld Refinem ent

2.2 M agnetism

2.2.1 M agnetic U nits

2.2.2 Types o f M agnetic O rdering

2.2.2.1 Long Range Order

2 2 2 2

2.2.3 Types o f M agnetic Exchange

2.2.3.1 Direct Exchange

2.2.3.2 Superexchange

2.2.3.3 RK K Y

2.2.4 Curie W eiss

2.2.5 Low D im ensional M agnetism

2.2.5.1 D im er M odel

2.2.5.2 One D im ensional M agnetic Systems

2.2.5.2.1 Spin Peierls

2.2.5.2.2 A lternating Chain M odel

2.2.5.2.3 Spin Ladders

2.2.5.3 Tw o D im ensional M agnets

2.3 C onductivity

2.4 iliSR

66

66

67

C hapter 3 E xperim ental M ethods 71

3.1 D iffraction 71

3.1.1 N eutro ns vs X -rays 71

3.1.2 X -ray D iffraction 71

3.1.3 N eutron D iffraction 72

3.1.3.1 Institut Laue-Langevin, G renoble, France 72

3 .1 .3 .2 D 2 B 73

3.1.3.3 D I A 74

3.1.3.4 N ational Institute o f Standards and Technology, N IST,

G aithersburg, U SA 75

3.1 .3 .5 B T -1 75

3.2 M agnetism 76

3.2.1 S QUID m agnetom eters 76

3.3 C onductivity 77

3.3.1 H igh Pressure Conductivity 78

3.4 M uon Spin Relaxation, p S R 80

C hapter 4 N bj2029 : B ackground and Synthesis

4.1 Introduction

4.2 Crystallographic Shear Structures

4.3 Previous Studies o f N bj2029 and R elated Structures

4.4 Synthesis

4.4.1 Synthesis M ethod

4.4.2 Low Tem perature Reactions

4.4.3 R eactions at 1200°C

4.4.4 H igh Tem perature Reaetions

4.5 Summary

99

102

C h a p te r s • Structure 105

5.1 Introduction 105

5.2 Previous Structural W ork on N bj2029 107

5.3 X -ray Structure 109

5.4 N eutron Structure 110

5.4.1 N IST data 110

5.4.1.1 Refinem ent R esults, N IST data 110

5.4.1.2 A tom Positions and R efinem ent Profiles, N IST

data 1 1 2

5.4.2 A m Space G roup 118

5.4.3 N eutron D iffraction at the Institut Laue Langevin 119

5.4.3.1 Refinem ent Results, ILL data, A 2/m 119

5.4.3.2 A tom Positions and Refinem ent Profiles, ILL

data, A2/m 121

5.4.3.3 Refinem ent Results, ILL data. Am 126

5.4.3.4 A tom Positions and Refinem ent Profiles, ILL

data. Am 127

5.4.4 C om parison o f A 2/m and A m 132

5.4.5 N b - 0 Bond D istances 134

5.4.6 B ond D istance A nalysis 139

C hapter 6 N bj2029 : P hysical Properties

6.1 Introduction

6.2 T ran sport Properties

6.3 M agnetism

6.3.1 Susceptibility o f M onoclinie N bj2029

145

145

146

148

6.3.2 Curie W eiss Fitting 151

6.4 M uon Spin Relaxation (|iSR ) 152

6.5 Fitting to Low D im ensional M odels 157

6.5.1 D im er M odel 157

6.5.2 B onner-Fisher M odel 158

6.5.3 A lternating Chain M odel 161

6.5.4 Spin Ladder M odel 162

6 . 6 Summary 162

C hapter 7 Tij^Nbj2_x029 167

7.1 Introduction 167

7.2 Synthesis 172

7.3 Structure 178

7.4 M agnetism 182

7.5 C onductivity 190

7.6 Conclusions 193

Figure 1.1 Phase D iagram o f L a2_xSr] _xCu0 4 19

Figure 1.2 Structures o f LaNiOg and PrNiOg. Ln in green, NiO^ octahedra

in grey. 19

Figure 1.3 Structures o f the R uddlesden-Popper series general form ula

^ n + i ^ n ^3n+i- ^ ^ 6 octahcdra are represented in blue, A

is shown in green. 2 0

Figure 1.4 Structure o f L a2_^Sr,^C u0 4 21

Figure 1.5 V ariation o f w ith the num ber o f connected planes in various high

tem perature superconductors. 2 2

Figure 1.6 Structure o f the n = l, 2, and 3 m em bers o f the TlBa2Ca^_ ^ ^ ^ n ^3n+ 2

series 2 2

Figure 1.7 D ifferent possible exchange pathw ays in low dim ensional m agnets

24

Figure 1.8 Structure o f D iC hlorobi (thiazole) copper (II) (Cu is red) 25

Figure 1.9 Structure o f SrCu0 2 (Cu is red, Sr is green, O is blue) 25

Figure 1.10 Structure o f Sr^CuOg (Cu is red, Sr is green, O is blue) 26

Figure 1.11 Structure o f CsNiClg (N i is red, Cs is green. Cl is purple) 28

Figure 1.12 CuGeOg (Cu is red, Ge is green, O is blue) 29

Figure 1.13 Structure o f (V O)2P2 0 2 (V in red, P in green, O in blue) w ith spin

ladder exchange pathw ay superim posed 30

Figure 1.14 Structure o f (V O)2P2 0 2 w ith the alternating chain exchange

pathw ay superim posed 30

Figure 1.15 Structure o f LiCuClg H2 0 (Cu in red. Cl in green, Li in blue, H2O in

grey) 31

Figure 1.16 Structure o f Cu2(l ,4-diazacycloheptane)2Cl4 (Cu in red. Cl in green)

32

Figure 1.17 Possible exchange pathw ays in alternating chain com pounds 33

Figure 1.19 Structure o f Sr2CugOg (Cu in red, Sr in green, O in blue) 3 5

Figure 1.20 Structure o f L aC uO j 5 (Cu in red. L a in green, O in blue) 36

Figure 1.21 Structure o f N aV2 0 ^ (V^^O^ in green, in red, N a in blue) 38

Figure 1.22 The two distinct vanadium environm ents in N aV2 0g

corresponding to V 4+and 39

Figure 1.23 Structure o f M gV2 0^ (VO^ in grey, M g in blue) 40

Figure 1.24 Structure o f LiV2 0 ^ showing V2O5 layers (V^"*" in blue,

in grey, Li in pale blue) 41

Figure 1.25 Structure o f LiV2 0 g showing zig-zag chains o f (blue) and

(grey) 41

Figure 1.26 Structure o f CSV2O5 (square pyram idal in blue, tetrahedral

in grey, Cs in pink) 42

Figure 2.1 Pictorial D epiction o f B ragg’s Law 49

Figure 2.2 E w ald’s Sphere 50

Figure 2.3 Cation-anion-cation m agnetic super-exchange in a (top) and

K

(bottom ) bonds w ith an interaction at 180° 57

Figure 2.4 Exchange as a function o f distance for a m agnetic impurity

em bedded in a sea o f conduction electrons as described by

the RK K Y interaction. 58

Figure 2.5 Exchange param eters in an alternating chain 63

Figure 2.6 Exchange param eters for next nearest neighbour interactions 65

Figure 2.7 Exchange param eters for zig-zag chains 65

Figure 3.1 Schematic diagram o f diffractometer D2B 73

F igure 3.2 Schematic diagram o f diffractometer D IA 74

Figure 3.3 Schematic diagram o f diffractometer B T l 75

Figure 3.4 M ain com ponent o f a SQUID m agnetom eter 76

Figure 3.5 Set up o f pellets for conductivity m easurem ents 77

Figure 3.7 Longitudinal setup o f a jiSR experim ent 80

Figure 4.1 Form ation o f crystallographic shear planes 83

Figure 4.2 The structure o f H -N b2 0g 84

Figure 4.3 The structure o f N b2 5 0 ^ 2 85

Figure 4.4 The structure o f m onoclinic N bj2029 8 6

Figure 4.5 The structure o f orthorhom bic N bj2029 87

Figure 4.6 N b2 0^ im purities in Nb^2029 90

Figure 4.7 X -ray diffraction pattern o f N bj2029 after extended heating

showing an increase o f the [hOO] reflection, w hich occurs w hen

the m aterial is over reacted. 91

Figure 4.8 X -ray diffraction pattern o f sam ple m ade from N b and N b2 0^ and

heated at 700°C for 12 hours 93

Figure 4.9 X -ray diffraction pattern o f sam ple m ade from N b0 2 and N b2 0^ and

heated at 700°C for 12 hours 93

Figure 4.10 X -ray diffraction pattern o f sam ple m ade from N b and N b2 0^ and

heated at 1100°C for 12 hours 94

Figure 4.11 X -ray diffraction pattern o f sam ple m ade from N b0 2 and N b2 0g and

heated at 1100°C for 12 hours 95

Figure 4.12 X -ray diffraction pattern o f sam ple m ade from N b and N b2 0^ and

heated at 1200°C for 48 hours 96

Figure 4.13 X -ray diffraction pattern o f sam ple m ade from N b and N b2 0g and

heated at 1200°C for 3 hours 96

Figure 4.14 X -ray diffraction pattern o f sam ple m ade from N b and N b2 0 ^ and

heated at 1200°C for 90 m inutes 97

Figure 4.15 X-ray diffraction pattern o f sam ple m ade from N b and N b2 0^ and

heated at 1200°C for 1 hour 98

heated at 1200°C for 30 m inutes 98

Figure 4.17 X -ray diffraction pattern o f sam ple m ade from N b0 2 and N b2 0g

and heated at 1200°C for 30 m inutes 99

Figure 4.18 X -ray diffraction pattern o f sam ple m ade from N b and N b2 0g and

heated at 1300°C for 5 m inutes 100

Figure 4.19 X-ray diffraction pattern o f sam ple m ade from Nb and N b2 0^ and

heated at 1350°C for 8 m inutes 100

Figure 4.20 X -ray diffraction pattern o f sam ple m ade from N b0 2 and N b2 0g

and heated at 1300°C for 5 m inutes 101

Figure 4.21 X -ray diffraction pattern o f sam ple m ade from N b0 2 and N b2 0^

and heated at 13 50°C for 10 m inutes 101

Figure 4.22 X -ray diffraction pattern taken over 12 hours o f a 1 gram sam ple

o f m onoclinic N bj2029 m ade from N b and N b2 0g and heated at

1200°C for 1 hour 102

Figure 5.1 M onoclinic N bj2029 w ith the [100] lattice plane 105

Figure5.2 Key for N b positions in all structures 108

Figure 5.3 M onoclinic N bj2029 in the A 2/a space group as solved by N orin 108

Figure 5.4 M onoclinic N bj2029 the A 2/m space group 109

Figure 5.5 Refinem ent profile o f room tem perature N IST data refined

in A 2/m 113

Figure 5.6 Refinem ent profile for 25K N IST data refined in A 2/m 115

Figure 5.7 Refinem ent profile o f the 5K N IS T data refined in A 2/m 117

Figure 5.8 M onoclinic Nbj2029 in the A m space group showing the 12

crystallographically distinct sites in each 3x4 block 118

Figure 5.9 Refinem ent profile o f 1.594Â 2K ILL data refined in A 2/m 122

Figure 5.10 Refinem ent profile o f 2.398Â 2K ILL data refined in A 2/m 123

Figure 5.11 Refinem ent profile o f 2.398Â 3OK data refined in A 2/m 125

Figure 5.13 R efinem ent profile o f 2K 2.398Â ILL data refined in A m 129

Figure 5.14 R efinem ent profile for 3 OK ILL data refined in Am 131

Figure 5.15 Peak at 20 = 60° (2K, 1.594A, ILL data) refined in A 2/m 133

Figure 5.16 Peak at 20 = 60° (2K, 1.594À, ILL data) refined in Am 133

Figure 6.1 R esistivity as a function o f tem perature for monoclinic

^ ^ 1 2 ^ 2 9 147

Figure 6.2 C onductivity m easurem ents at am bient pressure and lOkBar 148

Figure 6.3 Field cooled and zero field cooled Susceptibility plot for

m onoclinic N bj2029 at lOOG 149

Figure 6.4 Field cooled and zero field cooled Susceptibility plot for

m onoclinic N b j2Û2 9 at lOOOG 150

Figure 6.5 Field cooled and zero field cooled Susceptibility plot for

m onoclinic N b|2029 at lOG 150

Figure 6 . 6 1/% as a function o f tem perature for N b j2Û2 9 152

Figure 6.7 M uon spin relaxation function at 0.06K 154

Figure 6 . 8 M uon spin relaxation function at 8K 155

Figure 6.9 M uon spin relaxation function at 14K 155

Figure 6.10 Tem perature dependence o f the oscillation frequency o f

^ ^

Figure 6.11 D im er m odel fit to the susceptibility o f N bj2029 at lOOG 158

Figure 6.12 B onner-F isher fit to susceptibility o f Nb^2029^^ lOOG 159

Figure 6.13 B onner-F isher fit to the susceptibility o f N bj2029 at lOOG

w ith g fixed at a value o f 2 160

Figure 6.14 B onner-F isher fit to the susceptibility o f N bj2029 at lOOG using

values o f g and J/k that fitted high tem perature data well 160

Figure 7.1 A one dim ensional chain o f N b atom s disrupted by the inclusion

o f one non m agnetic Ti atom. 168

Figure 7.3 K ey show ing colours for different m etal positions in

^^2^ ^ 10^29 170

Figure 7.4 Structure o f m onoclinic T iN b |,0 2 9 (A 2/m ) 170

Figure 7.5 Structure o f orthorhom bic T iN b ^0 2g 171

Figure 7.6 X -ray diffraction pattern o f Tig o^N b^ 9 5O2 9 174

Figure 7.7 X -ray diffraction pattern o f Tig ]N b jj 9O2 9 174

Figure 7.8 X -ray diffraction pattern o f Tig 2 ^ 6 ^ j g029 175

Figure 7.9 X -ray diffraction pattern o f TiQ2g N b ^ 7 5O2 9 175

Figure 7.10 X -ray diffraction pattern o f T iN b ^ 0 2 9 176

Figure 7.11 V olum e as a function o f Ti doping in Tij^Nbj2_x029 177

Figure 7.12 X -ray diffraction patterns o f N bj2029 and TiNb^ , 0 2 9 178

Figure 7.13 R efinem ent profile o f TiN bj JO2 9 refined in A2/m space group 181

Figure 7.14 O rdering tem perature as a function o f Ti doping in Ti^Nb22_x^29 183

Figure 7.15 Susceptibility as a function o f tem perature for Tig osN bjj 9 5O2 9 183

Figure 7.16 Susceptibility as a function o f tem perature for Tig |N b , j 9O2 9 184

Figure 7.17 Susceptibility as a function o f tem perature for TiQ 2X6 , ^ g029 184

Figure 7.18 Susceptibility as a function o f tem perature for Tig 2 5 ^ 6 1 1 7 5O2 9 185

Figure 7.19 Susceptibility as a function o f tem perature for T iN b ^ 0 2 9 186

Figure 7.20 1/% as a function o f tem perature for Tig g^N b^ 9 5O2 9 186

Figure 7.21 1/% as a function o f tem perature for Tig ^N b^ 9O2 9 187

Figure 7.22 1/% as a function o f tem perature for Tig 2X6 ^ g029 187

Figure 7.23

\l%

Figure 7.24 1/% as a function o f tem perature for T iN b jj0 2 9 188

Figure 7.25 V ariation o f C w ith x in Ti^^Nbj 2 . ^ 0 2 9 190

Figure 7.26 R esistivity as a function o f tem perature for Tig g^N b^ 95O29 191

Figure 7.28 R esistivity as a function o f tem perature for Ti^ 2N bj j g0 2g 192

Figure 7.29 R esistivity as a function o f tem perature for TiQ 2gNb] ^ 7 5O2 9 192

Figure SI Structure o f m onoclinic N b j2Û2 9 solved in the A m space group

from 2k ILL data w ith N b position 12 highlighted as Nb^+

p o sitio n s 199

Figure S2 M onoclinic N b j2Û2 9 w ith distances betw een N b position 12

Table 5.1 G oodness-of-fit factors for refinem ents o f N IST data in A 2/m 111

Table 5.2 Therm al factors for refinem ents o f N IST data in A 2/m 111

Table 5.3 Lattice param eters for refinem ents o f N IST data in A 2/m 111

Table 5.4 A tom positions for refinem ent in A 2/m o f the room tem perature

N IS T data 112

Table 5.5 A tom positions for refinem ent in A 2/m o f the 25K N IST data 114

Table 5.6 A tom positions for refinem ent in A 2/m o f the 5K N IST data 116

Table 5.7 G oodness-of-fit factors for 2K ILL data refined in A 2/m 120

Table 5.8 Therm al factors for 2K ILL data refined in A 2/m 120

Table 5.9 G oodness-of-fit and therm al factors for 3OK ILL data refined

in A2/m 121

Table 5.10 A tom positions for 2K ILL data refined in A 2/m 121

Table 5.11 A tom positions for 3OK ILL data refined in A 2/m 124

Table 5.12 G oodness-of-fit factors for ILL data refined in Am 126

Table 5.13 Therm al factors for ILL data refined in Am 126

Table 5.14 A tom positions for 2K ILL data refined in A m 127

Table 5.15 A tom positions for 3OK ILL data refined in A m 130

Table 5.16 Lattice param eters for refinem ents o f all ILL data 132

Table 5.17 N b -0 bond lengths for N IS T room tem perature A 2/m

refinement. All distances in

A

Table 5.18 N b -0 bond lengths for N IS T 25K A 2/m refinement.

All distances in

A

Table 5.19 N b -0 bond lengths for ILL 3OK A 2/m refinement.

All distances in  135

Table 5.20 N b -0 bond lengths for ILL 3OK A m refinement.

A ll distances in  136

Table 5.21 N b -0 bond lengths for N IS T 5K A 2/m refinement.

All distances in

A

Table 5.22 N b -0 bond lengths for ILL 2K A 2/m refinement.

Table 5.23 N b -0 bond lengths for ILL 2K A m refinement.

All distances in  138

Table 5.24 Average bond lengths, bond valence sum and sample

variance calculations 139

Table 6.1 J values for various low dim ensional S = l/2 antiferrom agnets 164

Table 7.1 Synthesis conditions for T i^N b j2_xCl2 9 172

Table 7.2 Synthesis conditions for Ti^Nb^2_x^ 2 9 173

Table 7.3 Lattice param eters for Ti^Nbj2_x029 177

Table 7.4 Lattice param eters, Therm al factors and goodness o f fit

factors for T iN b ^0 2g neutron diffraction refinem ent. 179

Table 7.5 A tom positions for the refinem ent in A 2/m o f TiNbj j029 180

Table 7.6 Summary o f results from C urie-W eiss fits to Ti^Nb^2_x029 189

Table SI Bond lengths, average bond length and sample variance for the

1.1 Introduction

Transition metal oxides are w idely studied system s due to their huge variation o f

properties and structure types. Their physical properties can vary greatly, often

only with subtle changes to structure or composition. Conductivity properties are

found to vary hugely, from insulating behaviour in T iÛ2 through semiconductors

such as FcggO and metals such as ReO^, to high temperature superconductivity in

compounds such as Y B a2CugOy_^ (T^. = 92K)*’ Many compounds show metal

-non-m etal transitions as functions o f temperature (V O2), pressure (V2O3) or

com position (Na^WOg)^» Transition metal oxides also show interesting magnetic

properties which are, in turn, often related to the structure. The structure types

shown by transition metal oxides are vast, varying from three dimensional structures

such as perovskite, rutile and spinel, to layered and highly defective structures^»

One o f the most advantageous aspects o f transition metal oxides is their versatility in

oxidation states, enabling m ixed valence or non-stoichiom etric oxid es to be

synthesised.

One o f the most important phenomena observed in transition metal oxides in

recent years is high temperature superconductivity. The onset o f superconductivity

can be controlled in many cases by doping, which changes the oxidation state o f the

transition metal. The La2_xSr^CuO^ phase diagram is an intriguing example; its

properties change from antiferromagnetism to superconductivity via spin glass

behaviour with increasing proportions o f Sr dopant or with application o f pressure.

A diagrammatic representation o f the La2_^Sr^CuO^ phase diagram is shown in fig

1.1. The high transition temperatures found in cuprates are still not understood. It is

therefore important to study other system s which have unusual low dimensional

superconductivity

AF insulator metal

Spin glass

No long range magnetic order Or superconductivity

Figure 1.1 Phase Diagram o f La2-xSrxCu0 4

As m entioned previously, it is not only changing the oxidation state o f a

transition metal oxide that can affect its properties; small changes in structure can

also have huge effects. For exam ple, a m etal-insulator transition is seen in the

LnNiOg (Ln = lanthanide) series as a function o f lanthanide size. LaNiO^ has an

undistorted perovskite structure and shows m etallic conductivity, but w hen La is

replaced by Pr, the structure becomes distorted and the material becomes insulating

below 150K; the structures are shown in figure 1.2. The change in Ln ion creates

chem ical pressure and distorts the Ni-O -Ni bond angle away from 180° thereby

reducing the overlap between orbitals and correspondingly reducing the bandwidth o f

the conduction band. Narrow band width systems become insulating despite being

p artially o ccup ied u nder certain conditions. These system s are know n as

M ott-H ubbard insulators.

LaNiO

PrNiOg

Figure 1.2 Structures o f LaNiO] and PrNiOg. Ln in green, NiO^

Dimensionality can also markedly affect physical properties. An exam ple is

given in the Ruddlesden Popper series with the general formula A

change in property can be seen to occur with the change in the dim ensionality. The

structure can be thought o f as planes o f MO 2 octahedra separated by AO rock salt

layers (shown in Figure 1.3) such that as n increases, the number o f octahedra with

shared oxygen ions, and consequently, the dim ensionality increases. In the

^ V i ^ n ^3n+i series for example, the n=l m ember is a M ott Hubbard insulator due

to the narrow bandwidth. However, the conductivity increases as the dim ensionality

increases so the

n-2

metallic. This is due to an increased bandwidth as the dim ensionality o f the system

is increased.

n = o o

n = l n = 2

Figure 1.3 Structures o f the Ruddlesden-Popper series o f general form ula

An+iM n0 3n+i. MOe octahedra are represented in blue, A is show n in

green.

1.2 Low Dimensionality in High

Superconductors

One o f the m ost interesting areas o f solid state chem istry in recent years is low

dim ensionality and its effect on physical p roperties. The high tem perature

structure o f w hich is shown in figure 1.4 below. The CUO2 layers provide

electronically active planes which are separated from one another by L a/S r-0 rock

salt layers. The superconductivity occurs between specific doping levels o f Sr, but

the parent compound La2CuO^ is an example o f a two dimensional antiferromagnet.

A

CUO

(La,Sr)02

(La,Sr) 0 2

CUO2

(La,Sr) 0 2

(La,Sr)02

CUO2

Figure 1.4 Structure o f La2-xSrxCu0 4

M any o f the cuprate superconductors have low dim ensional structures and

T

variation o f T^. with the num ber o f connected planes (related to the dim ensionality)

for three different series o f high tem perature superconductors. For exam ple,

members o f the TlBa^Ca^ |C u^0 2 ^ + 2 series have increasing T^, as the separation

between the CUO2 layers is increased, with T^. = 80K for n = l, 108K for n= 2 and

125K for n=3 as seen in figure 1.5. The structures o f this series are shown in figure

140

120

100 80

T

60

. — Bi S r C a C u , O, 40

20

T l^ B a^ C aC u ,^ O,

2.5 3 3.5 4

1.5

Figure 1.5 Variation o f Tc with the number o f connected planes in various high temperature superconductors.

1-Tl n=l

TlBa.CuOs l-Tl n=2

TlBa^CaCup^

2-Tl n=3

TlBa^CaCu^Og Figure 1.6 Structure o f the n = l, 2, and 3 members o f the

1.3 One Dimensional Magnetism

A lthough many superconductors are m ainly concerned with tw o dim ensional

structures, one dimensional magnetism is an area o f considerable recent interest^.

This is because it is possible to calculate the ground state energy exactly in som e

cases and they exhibit many exotic properties such as spin-Peierls transitions and

spontaneously dimerised states, which will be described in more detail in Chapter 2.

One dimensional magnetism is usually inferred from the crystal structure. Although

it may appear that chains o f magnetic atoms are present there are several different

possible types o f exchange pathway as shown in figure 1.7. The sim plest one

dimensional magnet is illustrated in figure 1.7a. In this case, the metal atoms are

equally spaced with equal exchange between them. It is possible for next nearest

neighbour exchange to occur as illustrated in figure 1.7b. Here although the atoms are

equally spaced, there is som e exchange between the first and third atom. This could

be due to a superexchange interaction through metal 2 but it could also be found if the

atoms were arranged in a zig-zag structure as illustrated in figure 2.7. In these

systems the ground state is dimerised and therefore, for an S = l/2 antiferromagnet, a

singlet. Figure 1.7c shows a one dimensional magnet with a lattice distortion known

as a spin-Peierls distortion. In a spin-Peierls system the ground state is again a

singlet but is distinguishable from that in a one dimensional magnet with next nearest

neighbour exchange as it is associated with a lattice distortion. In figure 1.7d, a spin

ladder is illustrated where two one dimensional chains interact with one another to

form a ladder type structure. The ground state o f this type o f system has an energy

gap i f J J is antiferromagnetic. However, perhaps the most interesting aspect o f spin

ladders is the prediction that upon doping with holes superconductivity is expected.

O C M J O

Figure 1.7 Different possible exchange pathways in low dimensional magnets

Synthesis o f one dimensional magnets can be difficult because the magnetic

chains must be a large distance from one another with non magnetic atoms between

them. There are no examples o f true one dimensional Heisenberg antiferromagnets

because there is alw ays a sm all am ount o f interchain interaction at finite

temperatures. However, there are many examples o f quasi one dim ensional systems,

usually organom etallic systems, which have much stronger intrachain interactions

than interchain interactions. Tétram éthylam m onium m anganese trichlorate (II)

(TM M C) has intrachain interactions that are four orders o f magnitude larger than the

interchain interactions^. Dibromo- and dichlorobi (thiazole) copper (11)^ (shown in

figure 1.8), copper benzoate^ (B E D T -T T F )2[A uIII(I-m nt)2] (I-mnt = 1,1-dithio-

2 ,2 d ic y a n o -e th y le n e ), (B E D T -T T F )2[ B i B r J' 0 and [ 3 ,3 ’- D im e th y l- 2 ,2 ’

Thiazolinocyanine]-T C N Q i • are also good examples o f one dim ensional systems

Figure 1.8 Structure o f DiChlorobi (thiazole) copper (II) (Cu is red)

There are few examples o f inorganic one dimensional magnets. S r2C u0 3 and SrCu0 2

are alm ost ideal ID Heisenberg a n tife rro m a g n e ts ^ T h e y have J values o f 2200 and

2100K but 3D long-range anti ferromagnetic order does not occur until 5K. They are

low dim ensional because structurally the m agnetic copper is arranged in chains.

S rC u0 2 has a zigzag chain structure (figure 1.9) with anti ferrom agnetic exchange

along the legs which creates frustration. The frustrated zigzag exchange is weak, so

the system can be regarded as tw o S = l/2 one dim ensional chains at higher

temperatures. Although static magnetic order is observed, no Bragg peaks are seen in

the neutron diffraction patterns, due to the f r u s t r a t i o n T h i s frustration is not seen

in Sr2Cu0 3, which has a simple linear chain structure as illustrated in figure 1.1 0.

O f one dimensional compounds, S = l/2 complexes have been the most w idely

studied. The Bonner Fisher expression, see section 2.2.5.2, g ives a theoretical

expression for the susceptibility. Other com plexes with integer spins have also been

w ell studied due to Haldane’s conjecture (section 2.2.5). A w ell studied series is the

A B X3 series where A is a group 1 metal. B i s a divalent transition metal cation

and X is F, Cl or Br. Many o f these compounds have a hexagonal perovskite

structure and are pseudo one dimensional but with significant interchain interactions.

The structure o f these materials means that the magnetic metal atoms are often

arranged in a triangular structure which leads to frustration o f the moments. One

exam ple from this series is CsCrClg, w hich is a S=2 quasi one dim ensional

Heisenberg antiferromagnet (shown in Figure 1.11). The magnetic properties are

dominated by the chains o f Cr2+ ions.

The nickel analogue, CsNiClg provided the first m odel system to test

Haldane’s conjecture. It is not only pure one dimensional materials that have

attracted attention in recent years. Impurities in one dimensional system s are much

more pronounced than in two and three dimensional exam ples because even one

impurity breaks the interaction in a one dimensional chain. When CsV^^Mg^Clg is

doped with Mg on the vanadium sites (x > 0.052), the one dimensional magnetism is

destroyed and the system behaves like a collection o f paramagnetic spins obeying the

Curie W eiss law^^. Doping o f one dimensional systems with non magnetic atoms

such as this w ill become important in Chapter 7.

1.4 Spin Peierls Systems

The majority o f exam ples o f spin Peierls compounds, as with one dim ensional

m agnets, are organic or organom etallic com pounds such as M E M -(T C N Q) 2

(methylethylmorpholinium tetracyanoquinodimethanide2) (Tgp=19) a

’-(B E D T -T T F)2A g(C N )2(Tgp=7K)^^'^ ^, aqua [N-(salicylaldiminato)glycinato] copper

1,2-perfluoro-m ethylethylene-l-2-dithiolato)-gold [TTF-BDT(Au)] (Tgp=2K) or w ith Cu (Tgp=

1 However , in 1993^^’ CuGeOg was discovered to be the first inorganic

example o f a spin Peierls compound (Tgp=14K). The structure is shown in figure

1.12. This discovery made the study o f the spin Peierls phenom enon much more

accessible as CuGeOg is easy to synthesise com pared with the com plex organic

systems previously studied. It also has a sim pler structure that consists o f Cu

chains separated by GeO^ units^^. When doped with small am ounts o f either Zn or

Si 28,29 the spin Peierls state is destroyed and a long range ordered antiferromagnetic

state is created.

Figure 1.12 CuGeO] (Cu is red, Ge is green, O is blue)

1.5 Alternating Chain Compounds

For alternating chains, the next nearest neighbour interaction becomes im portant and

influences the susceptibility. Cu(N0 2 ) 2 2.5H2O is a widely studied alternating chain

com pound w hich undergoes a structural transition at low tem peratures to a spin

ladder. The susceptibility o f a spin ladder is very sim ilar to that o f an alternating

chain. (V0 )2P2 0 7 was widely thought to be a spin ladder^^'^^ but was found to be

an alternating chain after inelastic neutron scattering e x p e r i m e n t s34 y ^ e nature o f

the exchange pathw ays that m ake up the spin ladder and alternating chain

•

•

•

s%fi

•

•

•

e e #

•

• •

Figure 1.13 Structure o f (VO)2P2 0 7 (V in red, P in green, O in

blue) with spin ladder exchange pathway superimposed

• • • • •

% *

4 P

%

#

T

d # *

d # #

• C P

P C • C P

f i B • C B S B B

B #

% B #

B C P

B

% •

Figure 1.14 Structure o f (V0 )2P2 0? with the alternating

An alternating chain o f C u2Cl^^‘ dimers with CUCI2 bridging units is the

backbone o f two alternating chain com pounds; Isopropylam m onium CuCl^ and

LiCuCl3'2H2 0^^ (figure 1.15). Complex organic ligands are also capable o f creating

alternating chains o f copper ions such as catena-D i-m -chloro-bis(4-m ethylpyridine)

copper(II), catena-di-m -brom o-bis(N -m ethylim idazole)copper(II), catena [Hexane-

d io n e b is ( th io s e m ic a r b a z o n a to ) ] c o p p e r ( I I ) a n d c a te n a - [ O c ta n e d io n e

bis(thiosemicarbazonato)]copper(II)36.

Figure 1.15 Structure o f LiCuCl3'H2 0 (Cu in red, Cl in green, Li in blue, H2O in

1.6 Spin Ladder Systems

There are several examples o f compounds exhibiting two-leg spin ladder magnetism,

how ever, m any o f them are confused w ith altern atin g chains. C u2( l ,4-

diazacycloheptane)2Cl^ is an example o f a com pound where there has been dispute

about the exact nature o f the magnetic properties. The structure is shown in figure

1.16.

Figure 1.16 Structure o f Cu2( l ,4-diazacycloheptane)2Cl4 (Cu in red. Cl in

green)

Cu2( 1,4-diazacycloheptane)2Cl^ was first synthesised in 1990 by Chiari et al who

stated that there were five different possible exchange pathways in the system. This

could be sim plified to three if the assum ption that m etal atom s w ith sim ilar

o f the exchange constants within the structure is then approximated to that shown in

figure 1.17.

J3

Figure 1.17 Possible exchange pathways in alternating chain compounds

M aking this assumption, it was possible to fit the susceptibility to an

alternating chain model. Although this was successful, there was som e doubt as to

the validity o f the assumptions. In the follow ing y e a r s i t was assumed that the

exchange constant J3 was much smaller than the other two and could be discounted

making the alternating chain mechanism correct. However, although susceptibility

data was fitted satisfactorily, there was still evidence that the third exchange constant

needed to be included. The spin ladder m odel with all three exchange constants

included has been the accepted solution to the p r o b le m a n d has been confirmed

with ESR,^® theory*^, and inelastic neutron s c a t t e r i n g ^ ^

Another example o f a spin ladder is [Cu2(C2 0^)(CjQHgN2)2](N0 3 ) 2 which

has a broad peak at around 300K and a Curie tail at low temperatures. It was fitted

to the spin ladder Hamiltonian using Quantum Monte Carlo calculations and it was

found that was approximately ten times the size o f

Strontium cuprates are com m on exam ples o f spin ladders. A series o f

com pounds, Sr^^Cu^^j0 2 ^ produces an exam ple o f a tw o leg ladder for n=3

(Sr2Cu^0 ^ or SrCu2 0g) and an example o f a three leg ladder for n=5 (Sr^Cu^O^Q or

Sr2Cu3 0g). SrCu2 0 3 is a two leg ladder with a spin gap^^ the structure o f which is

Figure 1.18 Structure o f S1CU2O3 (Cu in red, Sr in green, O in blue)

The tem perature dependence o f % is as expected for a two leg ladder with

observation o f a spin gap and zero susceptibility at T=OK. No static magnetic order

was observed by pSR down to 20mK'^^, but short range spin order was observed

using neutron scattering and pSR"^^. S r2CugOg by contrast is a good exam ple o f a

Figure 1.19 Structure o f Sr2Cu3 0$ (Cu in red, Sr in green, O in blue)

There is no spin gap observed in this sy stem ^ and long range magnetic order

is observed from T=52K using pSR^^. At T=OK the susceptibility has a finite value

so the ground state o f the system is seen to respond to an external m agnetic field.

Three leg ladders tend to behave more like one dim ensional chains than two leg

ladders with no spin gap and a non zero ground state. Another intriguing example o f

the strontium cuprate family is (Sr,Ca) i^Cu2^0 ^ |^ . This is a com pound com posed

o f three different layers; a one dimensional CUO2 chain, a (Sr,Ca) layer and a CU2O3

two leg ladder as seen in SrC u2 0 3. W hen there is no Ca in the system , the

m agnetisation is dominated by the one dim ensional chains which are in a dim erised

state47. The presence o f the Ca creates holes, w hich have been shown using bond

this com pound also shows a spin gap when doped with Ba as well as w ith Ca

W hen undoped the Cu^^ and Cu^^ ions order along the chains and a spin gap is

observed H owever doping with on the Sr sites suppresses the spin gap

W hen Sr is rep laced w ith La to form L a^C agC u2^0 ^ |, one dim ensional

anti ferromagnetism is p r o d u c e d ^ ^

One o f the most intriguing aspects o f spin ladders is the prediction that hole

doped, even leg, ladders would exhibit high superconductivity. This was found

by calculation by D agotto et a P \ who found that there w ere strong pairing

correlations between the holes which would pair up resulting in superconductivity.

It has how ever proved difficult to dope two leg ladders with holes. It has been tried

with S rC u2 0 g but no evidence o f superconductivity was found. A new ladder

com pound L aj^S r^C u0 2 5 was found and doped with holes but although an insulator

to metal transition was observed there was no superconductivity down to 5K.

ê

The parent compound LaCu0 2 5 (the structure o f which is shown in figure 1.20) is

almost insulating and hole doping system atically decreases the resistivity. A spin

gap o f 473K was observed showing that the interladder interactions are very weak^^.

It is possible that superconductivity does not arise, because there is insufficient

coupling between ladders preventing the three dimensional superconducting phase

transition. Akimitsu et aM^ doped (Sr,Ca)^^Cu2^0 ^ 2 with Co. They found that the

cobalt was doped onto the chain sites which releases the excess holes onto the ladder.

This did result in a metal-insulator transition but no superconductivity. It was not

certain whether the spin gap survived the doping, because the paramagnetic Co ions

dominated the susceptibility measurements. grQ^Ca^g 6^ ^2 4^4i 84 s t u d i e d t o

examine the nature o f the hole doping. They found that at pressures o f 3.5-4 GPa

there was a 5% superconducting fraction with T^=9-12K. It was postulated that the

holes had been released from the chain sites onto the ladder sites because o f the

pressure. It was also found that doping with Ba and Nd produced a compound that

showed superconducting behaviour at elevated pressures

1.7 Vanadium Oxides

Vanadium oxides show a variety o f structural and electrom agnetic properties.

Vanadium ions form a variety o f different kinds o f coordination in an oxide

environment, for example the oxygen coordination can be octahedral, tetrahedral,

trigonal and square pyramidal. A typical vanadium oxide with a square pyramidal

coordination is V2O5. It has a layer structure with the layers formed by sharing the

edges and comers o f square pyramids o f V2O5. Various different types o f ions can

be intercalated between the V2O5 layers^^. The most w ell known are the alkali or

alkaline-earth metals which form A^V2 0g. D oping with these metals reduces the

vanadium and introduces som e V^'*’ d* S= 1/2 into the system . This includes

N aV2 0g which undergoes a spin Peierls transition at T(.=35K, shown in figure

•

V

•

r • " V

Figure 1.21 Structure o f NaV2 0 5 (V5+0$ in green, V4+O5 in red, N a in blue)

The V O5 square pyramids are joined to one another by a com bination

o f edge and com er sharing. It was originally thought that this system, which has an

average vanadium oxidation state o f 4.5, had chains o f ions separated by chains

o f ion^^. It has since been found using X-ray and neutron diffraction^^ and

NMR^o that at room tem perature, the structure does not consist o f charge ordered

chains but o f zig-zag chains o f vanadium ions which are all equivalent and have the

oxidation state 4.5+. At 34K, which was thought to be the spin Peierls transition

temperature, a phase transition occurs and the electrons move to becom e localised on

vanadium is in the +5 oxidation state (red in figure 1.21) and 50% is in the +4

oxidation state (green in figure 1.2 1).

V2O3 and V O2, the d^ and d^ binary oxides, have very

different vanadium environments. V O2 has VO^ octahedra m ade up o f six V -0

bonds o f around 1.92Â. The vanadium in V2O5 is only surrounded by five oxygen

atoms in a highly distorted arrangement, the bond lengths are 1.75, 1.77, 1.84, 1.57

and 2.04Â. In N aV2 0 5, there is a significant structural difference between the

and ions. The blue atom on the left in figure 1.22 below has four bond lengths

which are very similar and one shorter one going into the Na layer whereas the grey

atom on the right has very irregular bond lengths, 1.76, 1.87, 1.98, 1.87 and 1.53Â,

similar to those found in V2O3.

Figure 1.22 The two distinct vanadium environments in N aV2 0$ corresponding

to V4+ and V5+

Both CaV2 0 3 and MgVO^^^ (shown in figure 1.23) have sim ilar structure types to

N aV2 0g^^ but as they are divalent there is no in the structure and therefore no

charge ordering. The only difference between the three compounds is the m anner in

which the layers are stacked. As there is no all the vanadium is magnetic

and is arranged in a ladder structure and both compounds show the susceptibility o f a

Figure 1.23 Structure o f MgV2 0$ (VO5 in grey, Mg in blue)

The lithium doped compound has an average valency o f 4.5+ and has a layer

structure similar to N aV2 0^. However, this structure differs from the other members

o f the series through its charge o r d e r i n g F r o m structural studies, the vanadium

was found to be charge ordered but in zig-zag chains rather than one dim ensional

columns. Within the layers zig-zag chains are linked to zig-zag chains

by com er sharing. In this structure there are also two types o f pyram ids with

apex oxygens above and below the sheet. The structure is shown in figure 1.24 and

Figure 1.24 Structure o f LiV2 0$ showing V2O5 layers (V4+ in blue, V5+ in grey, Li

in pale blue)

Figure 1.25 Structure o f LiV2 0$ showing zig-zag chains o f

The crystal structure o f CSV2O5 is different from both LiV^O^ and

N aV2 0g^^. CSV2O5 is a layer structure, shown in figure 1.26, but there are square

pyramidal and tetrahedrally coordinated vanadium ions. The ions occupy the

square pyramidal sites and the ions occupy the tetrahedral sites. R ather than

being aligned in chains the square pyram ids are joined as dim ers. This is

reinforced by the susceptibility w hich shows the tem perature dependence o f a

dimer^^.

Figure 1.26 Structure o f CsViOg (square pyramidal V4+ in blue, tetrahedral V5+ in grey, Cs in pink)

N aV2 0g includes one dim ensional chains which are formed by only com er

sharing square pyramids, whereas CSV2O5 includes dim ers form ed by only

is much larger than that o f CsVjO^. This means that the exchange interaction is

much stronger in the com er sharing pyramids than in the edge sharing pyramids.

^^^2 ^ 5 which has a combination o f edge sharing and comer sharing pyramids has an

exchange integral which is close to the difference between those in N a V2 0 g and

CsV^Og.

1.8 Niobium Oxides

Niobium oxides mainly show itinerant properties; Niobium metal is a superconductor

(T(.= 9K) showing the highest transition temperature o f any element. The monoxide

NbO which has a defect rock salt structure, is also a superconductor but with a much

lower

Tç.

However, when it is reduced by doping with Li, the stmcture changes to a layered

stmcture made up o f NbO^ trigonal prisms separated by Li atoms and Lig gNbÛ2^^

is a superconductor with a T^. o f 4.5K. Another niobium based superconductor is Li

intercalated KCa2N b3 0 jQ which has a T^ o f IK^^. Although there are many

examples o f niobium based superconductors there are very few known magnetic

compounds as metal-metal bonding is prevalent among early transition metals in low

oxidation states such as Nb^+ dL It is therefore interesting to attempt to create a

m agnetically ordered state in a niobium oxide and to study effects o f changing the

oxidation state using doping to see if there is an equivalent system to La2_^Sr^CuO^

with a boundary between antiferromagnetism and superconductivity.

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2.1 Diffraction

A crystal structure can be envisaged as an infinite array o f lattice points to w hich a

basis is connected. A basis is com posed o f a set o f atom s w hich m ake up a unit cell.

The structures o f the crystal system s presented in this thesis w ere determ ined by

pow der neutron or X -ray diffraction. The reciprocal lattice is a useful concept in

diffraction theory. The direct lattice is used to describe the atom ic positions in the

crystal. The recipro cal lattice is used to u nd erstand the p o sitio n o f diffracted

radiation. The reciprocal lattice (a * ,

b*, c* )

following equations

b

---a b X c a b X c

* a X

b

b

= h â + kb + Ic . The m agnitude o f this vector is the reciprocal o f the

shortest distance betw een the M iller (h,k,l) planes,

| a * l = ^ d

hkl

The condition for elastic scattering is expressed qualitatively by B ragg ’s law (figure

2.1) and geom etrically, in reciprocal space, by Ew ald’s sphere (figure 2.2).

For diffraction to occur then the d iffracted beam s from lattice p oints m ust

interfere constructively, i.e. the difference in p ath length betw een tw o d iffracted

beam s m u st be an integral num ber o f w avelengths. In the diag ram b elow , the

difference betw een the tw o path lengths is A B + BC. AB = BC = d sin0 so the

difference in path length is 2d sin 8 w hich m ust be an integral num ber o f w avelengths

for diffraction to occur, giving B ragg’s law

n ^ = 2d sin0

In E w a ld ’s constru ctio n, O is the o rig in w here the in cid en t w ave, kg,

term inates and C is the centre o f a sphere o f radius

MX.

reflection is achieved only if one or m ore points such as P, lie anyw here on the

sphere and happen to coincide w ith a reciprocal lattice point, m aking an angle called

the Bragg angle 0 ^ such that Z OCP = 20

Diffracted Beam

Incident Beam

Figure 2.2 E w ald’s sphere

W aves scattered by different atom s interfere w ith one another and the overall

scattering pow er or structure factor, for the reflection h,k,l, is given by:

E = X! f ; exp( - 27ii(hx + k y + lz )exp( - W ) (X -rays)

1 1 1

where Xj, y. and Zj are the fractional coordinates o f the ith atom , f- is the scattering

am p litude for X -ray s and

b-

D ebye-W aller isotropic tem peratu re facto r w hich is; W ,= s in ^ 9 ( u )

( u ^ ) i s the m ean square displacem ent o f the atom. The intensity o f the reflections is

proportional to the square o f the am plitude o f the structure factor.

2.1.1 Rietveld Refinement

Structures w ere solved using R ietveld refinem ent f ^ o f both X -ray and neutron

diffraction profiles. T his is a least squares m ethod, w here structure inform ation

including atom positions is refined to b est m atch the observed data. In the R ietveld

m ethod, the profile is assum ed to be the sum o f a num ber o f Bragg peaks and the

calcu lated in ten sity at each p o in t o n the p ro file is o btain ed by su m m ing the

contributions from the background and each reflection at that point. This calculated

profile is com pared w ith the observed p ro file and the param eters are refined to

m inim ise the function:

S^ = EwXy-y

w here y^ is the observed intensity for i * step and y^j is the calculated intensity for

the i^^ step. The w eighting o f the observables is calculated as w = — . F or X -ray

i y O b s

diffraction, the calculated intensity for each

hkl

y = s V L

IF P A ( | ) ( 2 e - 2 0

T+ y

ci ^ hkl hkl i i hkK hkl hi

hkl

where: s is the scale factor,

L^j^j is a contribution from the Lorentz, polarisation and m ultiplicity factors,

is the structure factor for the particular hkl reflection,