———* iTiiiixiw\\\\\\\\mmm\\\\\\\\\\\\u\\\ 131293 LIBRARY Michigan Std“ University This is to certify that the dissertation entitled Solid State Alkali Metal NMR Spectra and Magnetic Susceptibilities of Alkalides, Electrides and Their Related Compounds ' presented by J ineun Kim has been accepted towards fulfillment of the requirements for Ph 0 D 0 degree in ChemiStry V Date 6/29 /‘?7 MS U is an Affirmative Action/Equal Opportunity Institution fldoflflsl 0-12771 imég PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE MSU Is An Affirmative Action/Equal Opportunity Institution SOLID STATE ALKALI METAL NMR SPECTRA AND MAGNETIC SUSCEPTIBILITIES OF ALKALIDES, ELECTRIDES AND THEIR RELATED COMPOUNDS By Jineun Kim A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1989 Co<3<¥<3225755 ABSTRACT SOLID STATE ALKALI METAL NMR SPECTRA AND MAGNETIC SUSCEPTIBILITIES OF ALKALIDES, ELECTRIDES AND THEIR RELATED COMPOUNDS By Jineun Kim The position of NMR lines in single crystals as a function of orientation was described as a function of the chemical shift parameters (SXX, 5yy. 522), a quadrupolar coupling constant (x), an asymmetry parameter (no) for quadrupolar interactions, and Eulerian angles (a, 3. 1). Solid state NMR powder patterns were explained by the superposition of single crystal NMR lines at different orientations of the crystal with respect to the external field, taking into account the probabilities of various orientations. A nuclear quadrupolar coupling constant of 0.1763 MHz 0 ('7 = 0) for Na- in Na+C222-Na- was obtained by a single crystal NMR method, in addition to a quadrupolar coupling constant of 1.268 MHz (no = 0) for Na+. A quadrupolar coupling constant of 0.605 MHz (no = 0) for Na+ in Na+C222-Br- was determined by a single crystal NMR StUdY- ‘Nonaxially symmetric quadrupolar coupling and chemical shift tensors (X = 0.6810 MHz, no = 0.4755, 5xxi= 62.0 ppm, 5YY= 46.1 ppm, and 522: ~28.4 ppm) for Cs+ in tetragonal Cs+(15C5)2-I- were obtained by a single crystal NMR study. Simulations of the powder patterns of numerous Na+ and Cs+ salts by the program VMASS yielded quadrupolar coupling constants, asymmetry parameters, and chemical shift parameters. The powder patterns are inhomogeneously broadened in spite of the presence of strong proton-proton and proton-alkali cation interactions. Alkali metal NMR spectra of M+BH4— (M = Li, Na, K, Rb, Cs) and M+TPB— (TPB- = B(C6H -) were investigated. The most s)4 diamagnetically shifted alkali metal cations ever found were present in the M+TPB- salts. A spin echo technique with phase cycling allowed the observation of iii and Rb+ ix; crystalline alkalides, electrides and their related compounds. The magnetic behaviors of three electrides, Cs+(1806)2-e-, Cs+(15C5)2'e-, and K+C222 e- were discussed in relation to their structures. The magnetic susceptibility data of Li+C211'e- fit a modified Oguchi theory of magnetism, Yielding a coupling constant, J/k = -13.9 K, a Weiss temperature, -20.6 K, and a Néel temperature, TN = 17.0 K. 70 mg. mfg, Meow} ii ACKNOWLEDGEMENTS I wish to express my sincere appreciation to Dr. James L. Dye for his guidance, encouragement, and support through the entire course of my study. I would like to thank Dr. J. P. Yesinowski, Dr. D. Nocera, Dr. R. H. Schwendemann, Dr W. P. Pratt, Dr. S. A. Solin and Dr. K. C. Hunt for their guidance and helpful discussions. I would like to thank Dr. L. D. Le, J. Kermit, and D. Jablonski for continuous support and help with instrumentation in the NMR facility and Dr. D. L. Ward for the crystal structure determination. I woubd also like to thank Prof. E. Oldfield and his group in the Department of Chemistry in the University of Illinois and the Department of Radiology in M.S.U. Clinical Center for their help and the use of their instrument. My special thanks go to my collaborators, A. S. Ellaboudy. R. s. Bannwart, L. E. H. McMill, and J. L. Eglin. I acknowledge the encouragement and help of the Dye group, R. H. Huang, M. E. Kuchenmeister, J. Skowyra, M. DeBacker, S. B. Dawes, M. L. Tinkham, o. Fussa, M. Faber, J. Papaioannou, F. Tientega, K. Moeggenborg, G. Xu, D. H. Shin, K. L. Tsai, S. DOeuff, E. Jackson, and I. Behbahani. iii Special recognition goes to the glassblowers K. Mistry, M. Langer and EL Bankroff for their excellent service and encouragement. Thanks to the machinists R. Geyer, D. Watters, and D. Menke. Thanks to all Korean friends in the Department, in particular H. ii. Nam, H. (3. Cho, ii. D. Kim» Ii. R. Kim, LL Shin, T. Kwon, and many others for encouragement and help. They also helped for me to solve many problems in writing three programs and setting up a personal computer. I am. grateful to receive financial support from. the Department of Chemistry, Michigan State University and the National Science Foundation Grants DMR 84-14154 and DMR 87-14751. Finally, I would like to thank my family for their encouragement, support and understanding. Above all, I dedicate this thesis to my wife, BoYoung, whose support and extremely long patience were vital to the completion of this degree. iv TABLE OF CONTENTS Page LIST OF TABLES ................................................... Vii LIST OF FIGURES .................................................. ix CHAPTER I, INTRODUCTION ..................................... 1 CHAPTER II. THEORY OF SOLID STATE NMR """""""""""""""" 10 II. A. Nuclear Spin Interactions in Solids -------- 11 II. B. Transformation Properties of Spin Interactions in Real Space and a Single Crystal NMR Line .............................. 16 II. C. Powder Patterns ............................... 37 II. C. 1. Static Powder Patterns ------------------ 37 II. C. 2. Variable Angle Sample Spinning ---------- - 43 II. D. Magnetic Dipolar Broadening ---------------- - 48 II. E. Homogeneous, Inhomogeneous, and Heterogeneous Interactions ------------------ 53 CHAPTER III. MAGNETIC SUSCEPTIBILITY ------------------- ° 57 III. A. Definitions of Terms and Units ------------- 58 III, B. Diamagnetism .................................. . 59 III. C. Paramagnetism, Ferromagnetism and Antiferromagnetism ........................... . 60 CHAPTER IV. EXPERIMENTAL METHODS --------------------- 80 IV. A. Synthesis and Recrystallization of Model Salts ..................................... 80 IV. A. 1. Method 1 ...................................... 80 IV. A. 2. Method 2 ...................................... 80 IV. A. 3. Recrystallization ........................... 81 IV. B. Synthesis and Recrystallization of Alkalides and Electrides -------------------- - 81 IV. B. 1. Synthesis .................................... 81 Iv. B. 2. Recrystallization ........................... 85 IV, C. NMR Experiments .................................. 85 CHAPTER V. RESULTS AND DISCUSSIONS ......................... 93 V. A. Alkali Metal NMR ............................... 93 V. A. l. A ingle Crystal NMR Study of Na CZZZ‘BP ................................... 93 V. A. 2. A ingle Crystal NMR Study of Na €222.Na ................................... 97 Page V. A. 3. A Single Crystai NMR Study and a Powder Lineshape of Cs (15C5) 'I ---------------- V. A. 4. Lineshape Analysis of 3Na NMR+Spectra of Model Salts that contain Na ---------- V. A. 5. Lineshape Analysis of 13303 NMR Spectra of Sgsium Madel Salts ----------- - V. A. 6. A Study of K and Rb NMR Spectra by the Spin Eého Method .................. . V. A. 7. Chemical Shifts of Alkali Metal ‘ Tetraphenylborates ........................ . V. B. Magnetic Properties of Some Electrides"“ V. B. 1. Magnetic Properties and Structures ------ V B' 2. Application of the Modified Oguchi Theory of Magnetism ........................ CHAPTER VI. CONCLUSIONS AND SUGGESTIONS FOR FUTURE \ WORK ............................................... VI, A. Conclusions ............................... VI. B. Suggestions for Future Work ----------------- APPENDICES ....................................................... Appendix A, program XTAL ................................. Appendix B, Program VMASS ................................ Appendix C, Program ANTIMAG ............................. LIST OF REFERENCES .............................................. vi 110 139 164 185 196 196 198 202 202 205 207 207 214 225 230 Table Table Table . Table Table Table Table Table Table Table Table Table Table Table Table Table w 10 11 12 13 14 15 16 LIST OF TABLES Proton decoupled and frequency dependence of the chemical shift and linewidth of Na NMR [18] ..................................................... . 39 K MAS NMR results [19] .............................. 87Rb MAS NMR results [19] ............................. 133CS MAS NMR results [18] ........................... . Connection between Cartesian tensors and spherical tensors for spin interactions [27] ----- ' Spherical tensor representation of spin operators [27] .......................................... Synthesis and recrystallization of model salts'-~ T e results obtained+from a single crystal Na NMR study of Na C222'Na ........................ 133Cs chemical shift, quadrupolar coupling tensors and direction cosines ------------------------ NMR parameters obtaiaed by simulations of the static and MAS Na NMR spectra at two different fields ..................................... u 3 A summary of the 13 Cs NMR results ----------------- 4.. Mean interatomic distances between Cs and O ----- Nuclear quadrupole moments and Sternheimer antishielding factors for K and Rb ----------------- 39 K and B7Rb NMR parameters ......................... MAS NMR chemical shifts of alkali metal nuclei in some alkali metal salts """"""""""""""""" + - Alkali metal NMR parameters for M TPB Obtained by‘ simulations ............................ vii Page 29 30 82 '108 114 126 158 161 182 184 186 195 Table Table . Page Table 17 Parameters obtained by the KINFIT analysis of ‘ the Li C211'e magnetic susceptibility data ----- 201 Table 18 External magnetic field effect on NMR 204 lineshapes ............................................ viii Figure I Figure 1 Figure Figure Figure FiSUre Figure FiSUre FiKUre Fi§Ure FiSUre Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure FiSure LIST OF FIGURES 1 Representative complexants. ------------------------ 2 Relation between the goniometer axes, X9, Y , Z9 and the laboratory axes, X“, Y", Z“. 3 Rotational transformations from the axis system, x, y, z to a new axis system, x”', y”', z"' through Eulerian angles, a, B. 1. ------ - 4 Quadrupolar splitting of the magnetic resonance of spin 3/2 in a single crystal at an arbitrary orientation. a) Energy level diagrams; b) Theoretical NMR spectra without linebroadening. The numerals indicate relative intensities. ----- 5 Computer simulated static CSA lineshapes for a transition (1/2’-1/2). .............................. 6 Computer simulated static second order quadrupolar lineshapes for the central transition of a quadrupolar nucleus. ---------- ' 7 Computer simulated VAS CSA lineshapes for a transition (1/2’_1/2). .............................. 8 Computer simulated VAS second order quadrupolar lineshapes for the central transition of a quadrupolar nucleus. ------------ - 9 Specific heat a) and Susceptibility b) of one-dimensional Ising ferromagnet [42]. ---------- 10 Exact specific heat C/k versus kT/J for isotropic nearest~neighbor, square-lattice Ising model [45]. ................................... 11 Specific heat C/k versus kT/J for isotropic, nearest-neighbor, fcc Ising model. Results of aeries methods exact everywhere except near r [45]. ............................................. . C ix Page 2 18 20 36 41 42 46 47 67 67 Figure Figure Figure Figure Figure Figure FiSure FigUre FiSUre FiKUre FiEUre Page Figure 12 Calculated reduced magnetic susceptibility versus reduced temperature for K = 0.0 to 3.0. The Néel temperatures are indicated on the curve in the inset. -------------------------- 79 Figure 13 Apparatus for the synthesis of alkalides and electrides [K _ cell]. .............................. 83 Figure 14 Single crystal holder and cap used in the single crystal NMR studies. --------------------- 87 Figure 15 Baseline distortion removal by spin echo technique. a) Distorted NMR line with single pulse experiment; b) Partial removal of baseline distortion by spin echo experiment with insufficient delays; c) Distortion free NMR spectrum by spin echo experiment with sufficient delays. ................................... 91 Figure 16 Single crystal 23Na NMR spectra of Na CZZZ-Br- at UL = 47.61 MHz. a) Proton coupled; b) Proton decoupled. --------------------- 94 Figure 17 Angular dependence of the central transition of 23Na in a single crystal Na+0222‘Br-. --------- 96 Figure 18 23Na NMR spectra of a single crystal Na+C222'Na at VL = 47.61 MHz. a) Proton coupled; b) Proton decoupled. """"""""""""""" 99 Figure 19 Si gle crystal 23Na NMR spectra of Na C222-Na at three different orientations, at at ”L = 47.51 MHz. ............................... 100 Figure 20 MAS 23Na NMR spectrum of Na+C222‘Na_ at “r = 52.94 MHZ [17]. ...................................... 102 Figure 21 Three orientation dependent 23Na NMR spectra at UL = 47.61 MHz for a black single crystal of Na+C222 Na- at room temperature. ------------- 104 . 2 . F18ure 22 3§a NMR spectra of two Single crystals of Na 0222-Na at different temperatures, at UL = 47.61 MHz. a) Crystal 1 at ~-50°C.; b) 3 Crystal 2 at ”-30 C.; c) Crystal 2 at V-20 C." 105 Figure 1 Figure I Figure Figure Figure FigUre Figure Figllre Figure Figure Figure Figure Figure Figure Figure Figure 23 24 25 26 27 28 30 23 Page Angular dependence of the Na NMR transitions in a single crystal of Na C222°Na . a) Chemical shift variation of the central transition of Na ; b) Half the separation between the satellites of Na . ------- 107 Angular dependence of the central transitions of Cs in a single crystal Cs (15C5)2 I . a) Site A; b) site B. ............................... 112 Angular dependence of half the distance between two transitions, (3/2,1/2) and (- /2,-3/2) pf Cs in a single crystal Cs (15C5)2-I . a) Site A; b) Site B. ---------- ' 113 133Cs NMR spectra of a single crystal Cs+(15C5)2 I‘ at uL = 23.61 MHz. a) B0 is along the crystallographic a axis; b) B0 is along the crystallographic c axis; c) At an arbitrary orientation. """""""""""""" 116 133 Cs NMR spectra of a powdered sample of Cs+(15C5)2 I" at uL = 52.482 MHz. a) MAS spectrum with 0r s 4 kHz; b) Observed static powder pattern; 0) Computer simulated powder pattern. ---------- 119 133 Cs MAS NMR spectra of a powdered sample of Cs+(15C5)2'I- with decoupling at UL = 23.61 MHz. a) are 470 Hz; b) wr~ 650 Hz. -------------- 122 23Na NMR spectra of Na+C222~I- a), b), and c), and Na c222-SCN‘ d) at ”r = 47.61 MHz. a) Proton coupled static spectrum; b) Proton decoupled static spectrum; c)Proton decoupled MAS spectrum; d) Proton decoupled static spectrum. Observed (——-)and simulated ("'). ----------- 127 Pr ton decoupled static+23Na NMR spectra of Na (12C4)2°I a) and Na (12C4)2-TPB b) at uL : 47.61 MHz. Observed (-—) and simulated xi Figure Figure Figure Figure Figure FiSure Fiiure Fiiur: Page Figure 31 Prgton depoupled static 23Na NMR spectra of Figure 32 Figure Figure Figure Figure Figure Figure 33 34 35 36 37 38 Na ISCS-I a) and Na 15C5-SCN- b) at uL = 47.61 MHz. Observed (--) and simulated (-~).- 130 23§a NMR spectra of N +18C6-Br: a), Na 1806-1 b), 0), Na 18C6-SCN d) and e). a) Proton decoupled static, UL: 47.61 MHz; b); d) Proton coupled static, vL= 105.482 MHz; c) Proton coupled MAS, 0L: 105.482 MHz; e) Proton decoupled MAS, DL = 47.61 MHz. Observed (-—-), Simulated ( ~-). -------------- 131 Square-antiprism;arrangement of oxygen atoms in Na (12C4)2-Cl 'SHZO. The inter-oxygen distances and the crystallographic two fold axis are shown. ---------------------------------- 134 Local structure of Na+18C6'SCN-. a) View along a direction in the mean plane; b) View in a direction normal to the mean plane. -------- 136 133Cs NMR spectra of Cs+C222-I-. a) Proton coupled static, 9L: 52.482 MHz; b) Proton decoupled static, UL: 23.61 MHz; c) Proton decoupled MAS, UL: 23.61 MHz. Observed (-—-), Simulated ( -~). --------------- 140 133Cs NMR spectra of Cs+C222-SCN-. a) Proton coupled static, UL: 52.482 MHz; b) Proton coupled MAS, UL: 52.482 MHz; c) Proton decoupled static, UL: 23.61 MHz; d) Proton decoupled MAS, UL: 23.61 MHz. ),Simulated ("H, ----- ). ---------- 142 Observed ( 133Cs NMR spectra of Cs+C222-TPB-. a) Proton coupled MAS, UL: 52.482 MHz; b) Proton decoupled static, UL: 23.61 MHz; c) Proton decoupled MAS, VL= 23.61 MHz. Observed (-——), Simulated ("”). -------------- - 144 Prgton coupled_static 133Cs NMR spectra of Cs (18C6)2-TPB a), Cs (18C6)2-SCN b), and Cs+(18cs)2-I' c) at u = 52.482MHz. L Observed (___._), Simulated (W). ................. 147 xii Figure 39 Figure 40 Figure 41 Figure 42 Figure 43 Figure 44 Figure 45 Fi‘a’ure 46 FigUre 47 FiSUre ii Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 39 40 41 42 43 44 45 46 47 48 Prgton coupled st tic ‘33c5 NMR spectra of Cs+1806-I a), Cs 18C6°TPB b), and Cs 18C6-SCN- c) at ”r: 52.482 MHz. ---------------- 149 Proton coppled static 133Cs NMR spectra of Cs l8C6-I -Toluene at UL: 52.482 MHz. a) Observed; b), c), d), e), f) and g) Simulated. ------ p ----- 151 Stereoscopic_view of the crysta structures of Cs 18C6 I 'Toluene a) and Cs 18C6-SCN b) [54]. .................................................. 154 Proton coupled_133Cs NMR spectra of Cs (15C5)2 TPB at DL= 52.482 MHz. a) Static; b) MAS. ................................... 155 133Cs MAS NMR spectra at UL = 52.482 MHz under slow spinning conditions. b) Cs+C222-TPB-, wr~1 kHz; 0) Cs+C222'TPB-, “r“ 2 kHz; a) CS+18C6‘TPB-y car». 1 kHz. ......................... 156 39K spin echo NMR spectra of K+SCN- at 0L = 18.67 MHz. Observed (-——), Simulated ("~).'-‘“ 165 39K spin echo NMR spectra of K+(12C4)2 I- a), K+(15C5)2 I" b), and K+18C6-I_ c) at ”r = 18.67 MHz. Observed (-—-), Simulated ( -~).---' 166 + _. 39K spin echo NMR spectra of K (12C4)2-SCN a), K+(15C5)z-SCN_ b), K+18ce-SCN' c), at ”L = 23.32 MHz and K+C222‘SCN-d) at ”L = 18.67 MHz. Observed (———), Simulated (-°”)- """""" 167 87Rb spin echo HMR spectra of Rb+1806-Cl- a) and Rh 18C6-SCN b) at U = 130.93 MHz. -------- ' 170 L + .— 87Rb spin echo NMR spectra of Rh C222 Br at UL = 98.16 MHZ. ...................................... 172 xiii Figure 49 Figure 50 Figure 51 Figure 52 FiSure 53 FiSure 54 Figure 5 I“ FiSUre 5' . 39 , Page Figure 49 K spin echo NMR spectra of alkalides and elec rides containing crown ethers. a) K (15C5)2 e , U = 23.32 MHz; L + P - b) K (15C5)2 K , VL = 18.67 MHz; c) K+(1505)2 Na’, ”1 = 18.67 MHz; d) 2*1806-K', ”1 = 23.32 MHz. Observed (-—-), Simulated (-~'). --------- r ------ 173 . 39 . Figure 50 K spin echo NMR spectra of alkalides and electrides containing C222 at UL = 18.67 MHz. a) K+C222 Na‘; b) K+C222 K“; c) K+0222 e‘. Observed (-——), Simulated (..H). ----------------- 176 . .. 8 7 , - Figure 51 Rb spin echo spectra of Rb+(15C5)2'Rb a) and Rh (15C5)2 e b) at UL = 130.93 MHz. ------- - 178 . a7 _ Figure 52 Rb spin echo spectra of Rb+1806 Rb a) at vL = 163.7 MHz and Rb+18C6-Na- b) at ”1 = 130.93 MHZ. ........................................... 179 . 87 _ Figure 53 Rb spin echo spectrum of Rb+C222-Rb at UL : 130.93 MHZ. ..................................... 181 Figure 54 Computer generated structure of K+TPB- a) Tetrahedral arrangement of the four phenyl rings; b) Cross-section showing K on the phenyl rings. .................................... 189 7Li NMR spectra of Li+TPB- at UL: 69.95 MHz. a) Proton coupled static; b) Proton decoupled static; c) Proton coupled MAS; d) Proton decoupled MAS. ....................................... 191 01 m Figure Figure 56 23Na NMR spectra of Na+TPB- at ”1: 47.61 MHz. a) Proton decoupled static; b) Proton decoupled MAS. Observed (———), Simulated ( -~). --------------- 192 o + .— Figure 57 39K spin echo NMR spectrum of K TPB a) at UL = 18.67 MHz and B7Rb spin echo NMR spectrum of Rb+TPB— b) at uL = 130.93 MHz. Observed (———), Simulated ('H', - - -). xiv Figure Figure Figure 58 Prgton_coupled static 133Cs NMR spectrum of Cs TPB at UL: 52.482 MHz. ObserVed (_) , Simulated (. . . . ) . .................. 194 Figure 59 Magnetic susceptibility of Li:C211'e-. a) C = 0.0; b) C = 7.43 x 10 . .................... 199 XV Sc molecul properi bonding covalel number electr these c°mPlei methyl alkali electr F Naiczz Csins been i and Ch includ maEnet phOtol CHAPTER I INTRODUCTION Solids are composed of basic units such as atoms, molecules, and ions held together' by' chemical bonds. The properties of solids are very much dependent on the type of bonding. There are five main types of bonding: ionic, covalent, Van der Waals, hydrogen, and metallic. Recently a number of new ionic crystals known as alkalides and electrides have been synthesized and characterized [1,2]. In these solids the cations (M+Ln) are alkali metal cations complexed by macrocyclic or macrobicyclic polyethers or fully methylated aza-crown ethers [Figure 1] and the anions are alkali metal anions (N-) in alkalides or solvent free trapped electrons in electrides. Following the synthesis of the first alkalide Na+CZ22'Na_ [3] and the first crystalline electride Cs+(18cs)Z-e‘ [4], about 30 alkalides and 7 electrides have been isolated. Much work has been focussed on the synthesis and characterization of new compounds by numerous techniques including optical spectroscopy, Chemical analysis, NMR, EPR, magnetic susceptibility, and conductivity. Recently photoluminescence [5], photoelectron emission and thermionic 1 (N N\ «rows 0 O \___J a) Cryptand 222 (C222) b) 18-Crown-6 (1806) c) Hexamethyl hexacyclen (HMHCY) Figure 1 Representative complexants. emissic structi compour contaii been 0 ether, succes: single and ti have p electr fully stable T [11] 5 study Spinni and C: were c not p, exteng The n they SPher; quest ident 3 emission [6]. have been investigated to study the band- structures of these materials. However, the study of these compounds has been hindered by decomposition of the solution containing M+Ln, N-, and solvated electrons. This problem has been overcome by using aprotic solvents such as dimethyl ether, ‘trimethyl amine» and. diethyl ether' [7]. Since the successful development of methods to grow and handle good single crystals [8,9], the crystal structures of 15 alkalides and three electrides have been determined. The structures have provided a better understanding of the nature of trapped. electrons and alkali metal anions in these materials. With fully methylated aza-crown ethers, relatively thermally stable compounds have been synthesized [10]. The first identification of alkali metal anions by NMR [11] established this technique as an excellent tool for the study of alkalides and electrides. Magic angle sample spinning (MAS) NMR was used to identify Li+,'Na+, Na-, Cs+, and Cs- in these materials [12-14]. The species K- and Rb- were observed in potassides and rubidides [15,16], but it was not possible to study complexes of these cations because of extensive quadrupolar line broadening and low sensitivity. The narrow NMR lines of alkali metal anions indicate that they are almost unperturbed ions with two electrons in a spherically symmetric ns orbital. However there remains a question whether these relatively narrow lines are broadened by quadrupolar interactions. In addition to the identification of species, NMR has been used to study some phys quad obta The know low the cati inte sta1 tha‘ amo1 elk; oth com Phe tem K+c bel [22 cla [25 ant ex; St} a physical properties of_ these compounds. For example, a quadrupolar coupling constant of 1.2 i 0.1 MHz for Na+ was obtained by simulating the MAS spectrum of Na+C222-Na- [17]. The temperature dependence of the 133Cs Knight shifts and the [known magnetic susceptibility of Cs+(18C6)2-e- yielded a very low fraction of cesium s-orbital character, 3.3 X 10-4, for the trapped electron [14]. This indicates that the cesium cations are well screened from the unpaired electrons by interaction with the crown ethers. Summaries of the solid state NMR results from previous studies are listed in Tables 1, 2, 3, and 4. The ns2 electronic configuration of alkalides predicts that they are diamagnetic. Although the presence of small amounts of trapped electrons gives EPR signals [18,21], alkalides are indeed diamagnetic [22]. Electrides, on the other hand, show a wide range of magnetic properties. The compounds Li+C211°e- and K+C222-e- exhibit spin pairing phenomena below their transition temperatures. The transition temperature (temperature of nmximum susceptibility) of K+0222-e- has not been observed since the compound decomposes below its critical temperature. Cs+(1806)2-e- is paramagnetic [22], while CS+(15C5)2'e- [8] and Li+PMPCY'e- [23] behave as classical«antiferromagnetics. Li+0211-e- [24] and K+0222-e- [25] might be classified as limiting cases of antiferromagnetics since the magnetic behavior may be explained by invoking only small interactions'between more strongly’ coupled electron jpairs. 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Uncertainty of the last digit given in parentheses. Probably contain K and Rb as indicated by rubidium XANES. Table 3 87Rb MAS NMR results [19] Compound 6, ppma Aul/z’ Hz KRb(1505)2 -191(1) 370(30) Rb+(1505)2-Rb' -191(1) 460(30) Cs+(15C5)2-Rb_ -189(2) 490(30) Cs+(1806)2-Rb" -194(1) 650(30) KRblSCG no signal + . a . 3Chemical shifts are referenced to Rb (aq? at inflnlte dilution. Uncertainty of the last digit given in parentheses. anu WHHZWNH fizz WQZ moan. 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At this time it is not clear 'whether these teractions depend primarily on the distance between the ctrons or on the presenceof channels that interconnect pped electron sites in the structure. In this thesis research, the theory and experimental ults of single crystal NMR will be described to obtain atively accurate parameters associated with the local ironment of the nuclei of interest. With the background of position of NMR lines in single crystals, the powder terns of static and variable angle sample spinning (VAS) will be described. In order to understand the NMR eshapes of alkalides and electrides, complexed alkali al salts have been used as models to investigate static magic angle sample spinning (MAS) NMR techniques. Also, modification of the Oguchi theory of magnetism by a-Nishiguchi (based on the Heisenberg Hamiltonian) has 1 applied to explain the magnetic susceptibility of 3211°e-. gen: wid' by int che ani to how obt ste SUC am tel run in‘ to CHAPTER II THEORY OF SOLID STATE NMR It is well known that the NMR spectrum of a liquid erally consists of numerous sharp lines, typically with the of less than a few Hz. These lines are characterized the isotropic chemical shifts and scalar spin-spin eractions. All possible anisotropic interactions, namely, mical shift ,anisotropy, dipole-dipole interactions, and sotr0pic quadrupolar interactions are averaged to zero due the rapid isotropic molecular motion. In the solid state, Fever, all these anisotroPic interactions are retained and more or less bell shaped, structureless line may be ained. In this sense, the goal of high resolution solid :e NMR can be described as utilizing or designing methods, I as decoupling, magic angle sample spinning, and mensional NMR, to observe the interactions to be studied to suppress or separate unwanted interactions. The nuclear spin Hamiltonian consists of a number of that describe physically different interactiOns of the ar spins. Some of these terms are related to the action of the nuclear spin with the apparatus, and some he PhYSical properties of the sample. These terms 10 desc the: inte high II. whe in1 Zee mag ext pro B( can nuc 3111] "ht 11 scribe the shifts and broadening of the NMR lines. It is erefore necessary to understand basic nuclear spin teractions in solids to design experiments that‘ achieve. gh resolution solid state NMR. . A. Nuclear Spin Interactions in Solids In this work most experiments have been done in a regime re the magnitude of lo and 31 is much larger than that of e (not always $1 > £6). The terms %0 and 31 represent the man. interactions with the external static and dynamic netic fields, 80 and 31(t), respectively. The resulting ernal Hamiltonian, 3 depends, apart from nuclear ext’ bperties, only on external parameters (external fields 80’ ft)), which are under the control of the experimenter. one 1 use this control to distinguish interactions between lear spins and external fields and internal fields. In mary then: ”int +geext (2.1) e = 760 + 31. The nuclear spin Hamiltonian flint may be written as: : a H + 3688 + 27618 + 360 + 768" + 968R + RL (2.2) whel indl spit bot] cas: and (ch all all fre int ten VQC 1'2 ere ”11’ ”88’ and ”IS represent the direct as well 'as direct interactions among I spins, S spins and between I ins and S spins, respectively. Note that one can consider th the homonuclear case (0I = ms) and the heteronuclear 3e (wI ¢ ms). 30 is the quadrupolar Hamiltonian of the I :l S spins, while 385" includes all shift interactions aemical shift and Knight shift) of the I and S spins, RSR the spin rotation interaction Hamiltonian and ”L contains L spin lattice interactions. It is convenient to express L Hamiltonians in tensorial form and to use angular equency units throughout. One can assume that all spin :eractions can be expressed by second rank Cartesian ISOPS. For example, if one lets I and S represent general :tors and A a general Cartesian tensor, one has _xx xy xz Sx ' ' - 2.3 1A8— (Ix, I,’ I2) AN AN Ayz Sy ( ) s zx zy 22 z The Zeeman interaction of spin I with the external field also be written in this form as: .. 2.4 12130 ( ) = I'Z‘Bi(t) (2.5) and The pre tr: ra: ruQ 1i: 8“ 13 J: (Bx’By’Bz)’ 81(t) = 2(B1x(t)’Biy(t)’Blz(t))c°sut ad 2 = -111 with the unit matrix 1. 1e 21 + 1 Zeeman energy levels are separated by wL in the OOH 0H0 l—IOO 'esence of the external magnetic field 80' Magnetic dipole 'ansitions between these levels can be induced by a .dio—frequency field 81 transverse to Bo. The selection .les are Am = :1; hence there is a single magnetic resonance he at the Larmor frequency, w comprised of 21 L, perimposed components. One can express the coupling between I and S spins as = in Is (2.6) e tensor DIS covers both direct spin-spin interactions ipolar interaction) as well as indirect spin-spin teractions (scalar coupling). Although not generally so, in a case of the dipolar interaction, D is symmetric and aceless. In this case DaB (a, 3 = X. Y, Z) may be written g (D’ (D) ”Q ) (2.7) «B R2 are R is the: distance between the spins and 6015 is the Kron« dege: secc the Sinc the grat ten and not fro per des hue sit int Sat int C0! 14 Lecker delta. The quadrupolar Hamiltonian Uta can be considered as a nerate case of ”IS with I'Q'I (2.8) e Q = [eQ/ZI(ZI-1)h]VV with V = {VEB} (a, B = x, y, z) rhich Q is the nuclear quadrupole moment and V is the trio field gradient tensor. One may interpret V as the nd derivative of the potential due to charges external to nucleus, i.e. VZV = 0. Hence, the trace of V vanishes. e the quadrupolar interaction is the interaction between nuclear electric quadrupole moment and electric field lent around the nucleus. The electric field gradient 3r V is very sensitive to the electronic configuration :oordination of the surrounding atoms. It is important to that the quadrupolar interaction vanishes if’ll arises a spherically symmetric charge distribution or if it has act cubic symmetry. When imperfections or impurities my the local symmetry at the site of each resonant :us, electric field gradients which vary from site to arise. The NMR line is then broadened by quadrupolar 'actions, the line shape being determined by the 'ibution of the electric field gradient. Only the ,lites are broadened in first order by quadrupolar 'actions. If the second order broadening of the central inent is large, indicating a large distribution of field arm 00“ the The bei 15 ients, the satellites may be too broad to be observed. The shift Hamiltonian of spin I can be given as the ling of the spin I with the static magnetic field Bo via shift tensor S: I-S-Bo. (2.9) tensor is no longer the unit matrix, since the coupling established via the electronic surroundings rather than L8 direct. In the case of the chemical shift, xx xy xz 0) 070': (2.10) vx vy yz lg H Oe - (m - iwz‘: + here m indicates a transition between m and m - 1. Fitting n orientation plot with these equations gives at least three f six tensor elements. Therefore one can obtain all six atrix elements from two or more rotation plots. The matrix an then be diagonalized to yield three principal components. 1 addition, three Eulerian angles can be calculated from the iagonalization matrix by comparison with R(a,8,7). Although Cartesian tensors lead to a direct physical iderstanding in terms of direction cosines, the use of the rreducible tensor operator is a very powerful way to ascribe the rotation transformation since the change in the rientation of the crystal is the simple rotation of a sometrical factor without changing its magnitude. The nine imponents A1 (i, j = x, y, z) of a Cartesian tensor of J rcond rank can be decomposed into their irreducible . + instituents with respect to the full 3d rotation group, 03: =A° +.A.1 +112 (2'21) is the isotropic constituent R is a tracele with t: (2.21) with : delta. transf zero, the 0 terms or 25 = 1/3 Tr{ R }-1 = R-l. (2.22) is a scalar and 1 is the unit dyadic. The term A1 is a iceless antisymmetric tensor of first rank: = 1/2 (Aij- A..) C (2.23) i J 1 :h three independent components. The third term in Equation 2 21) is the traceless symmetric second rank tensor A : = 1/2 (Aij+ Aji) - Raij (2.24) h five independent components, where 5H is Kronecker . . 0 1 2 ta. The components of the three quantities A , A , and A nsform in the same way as the spherical harmonics of order 0: one, and two, respectively. With the spherical unit vectors = e ; e = x1//‘§'(éx s Léy) (2.25) components of a first rank irreduc1ble tensor A19 in ns of the Cartesian components (Ax,Ay,Az) are given by = . - - ‘ + Aé (2°26) e1q A — é1q (Axex + Ayey z z) = A: ; Aiti: -1//2(Ax- LAy). To def Cartes. obtain« rule 1 [29]: ékq: [ (2k where The 11 Cartes tW0 ir = (2k+ as 25 define the irreducible tensor components in terms of the rtesian components, first the appropriate unit vectors are tained. The spherical unit vectors are constructed from the Le for the product of two irreducible tensor operators 9]: 1 1 = e xe I [ lkq (2k + 1)"2 Eq (-1)'q[ 1 l k ]e1q1.e1q2 (k 0. 1. 2). 2 q1 q1 qz-q (2.27) re 1 1 k is the Wigner 3-j symbol [26]. q,-q2 q = irreducible spherical tensor components in terms of the ctesian tensor components are obtained as the product of > irreducible tensor operators: 2k+1)"2 2 (-1)q[ 1 1 k ]A A (2.28) ‘lq‘l 1q2’ re 2 é1'éjA (i, j = x, y. 2) In exp ‘1 an, additi Produc the c Hamilt Here inter, iflVol, coll1130: perm types 27‘ D 22 = -1//3 (A + A + A ) XX yy = -t//§ (Ax - A“) 3 y n: -1/2 [Azx- szi «(Azy- An” = 1//§ [3Azz— (Axx+ Ayy+ AZ )] ' z :1: :Fl/Z [sz+ Azxt L(Ayz+ Azy)] :2: 1/2 [Axx- Ayyi «(Axy+ Ayx)]' expressing the components of'a tensor, the subscripts, +,' and :I: belong to the second subscript and do not mean ition or subtraction. The Hamiltonians are usually expressed as scalar ducts of corresponding irreducible tensors A“ and Th with components Akq and qu, respectively, leading to a iltonian of the form +q _ q . 2.29 X ( 1) Amman,” ). ( ) II-q 3 Ah contains physical constants appropriate to the q eractions including the magnetogyric ratios of the species >lved. Au and TR are irreducible spherical tensor q -q >onents of the tensor of rank k, which express the lattice emeters and spin ‘variables respectivelyu Two different as of time dependences are involved in Equation (2.29). The or static ANN what dominz rotat: time indep« tens: anti: cont] whie} elem. Prin is c axis 28 The one imposed on T;?q(t’) stems from spin precession in the static magnetic field 80, whereas the time dependence of 4kq(t) results from the sample spinning. In nearly all of what follows the Zeeman interaction will be overwhelmingly iominant. Tensor T can be transformed to a coordinate system rotating (in spin space) at a Larmor frequency w to remove L :ime dependences in T . For static NMR A is time k-q kq .ndependent; the Hamiltonian can then be rewritten as + q ._ _ q ‘- kg—q( 1) Aqu‘I-q' (2'30) ’ables 5 and 6 enable us to express all internal Hamiltonians n terms of irreducible tensor operators (Akq,T ). One may k-q mit interactions requiring tensors of higher than second ank, e.g. magnetic octapole, electric hexadecapole, so k = 1 1, 2 only. In the case of dipolar and quadrupolar nteractions A00 and A1q vanish, and only the five terms with = 2 will be nonzero. Chemical shift and Knight shift ensors, however, do have non-zero traces and, of course, ntisymmetric parts in general, i.e. A00 and A1q do ontribute. The principal axis system (X, Y, Z) is the frame in hich the symmetric part (AZq) is diagonal. The diagonal lements in the principal axis system (PAS) are called A . The following convention AYY’ zz rincipal components, Axx’ 3 chosen to label the principal axes of one of the principal {is systems: Table 5 C t Interacti Dipole-d J coupli: STin rot Quadrupo \ 29 rle 5 Connection between Cartesian tensors and spherical tensors for spin interactions [27] Cartesian Spherical tensor eraction Aa8(0t,3=x,y,z) Akq mical shift yfiaB A00 = -1//§ (Axx+Ayy+Azz) ole-dipole DhB A10 = -i//§ (Axy-Ayx) oupling J03 A111= -1/2 [Azx-szii(Azy-Ayz)] n rotation CaB A20 = l//§ [3Azz-(Axx+Ayy+Azz)] eQ A _ i - :1/2 [Axx-Ayyii(A +A )1 T drupole 21(21-1)hvh8 211 xy YX r¥ 30 N N N N NHN HHHHM «a HH Hm Hfl HmHHH o s N N N N N N N N N N N N o N HHN A HHHJH HTH» A PHJH. HI...» H muH+um Haw..." H maH+um CH...» mafia” 5 mx a mx s s mx s s mx s .HH+H.H-sHe.|I as..- msHm.|I Am._- m Hmvul.xm..- m Heart on Hm \ ems s. H H H H m\ N N W\N N N M\N o N HHH my A dflHlflb HVIJW. A mHHlflm HVIMI. o QRHNI E as be a o A+mIHIIm+HvMHI A+WIHIIW+HVMHI o o h .e. as a e. as e . Isl- a... ..Hu- e H Isl maomsapszo soHuauos swam msHHasoo e bHoaHe-oHosHo seHsm HeoHsoso HNNH msoumaodo swmm mo coeueucmmoemoa women» Heowamzam m oHndB |Azz ' where Tao If the angles I Princi Then 1 axis 5 :z - AlsoI Z IAxx ' Aieol z IAYx - Alaol . (2°31) are = 1/3 tr{ A } = 1/3 (Axx + A + A ). IO '1"! 22 the principal axis systems are noncoincident, Eulerian gles between one axis system and another must be defined. It is often convenient to introduce instead of the three Incipal components three new parameters, one of which is , and the other two, 5, and n, are defined by i 0 :A22 - Aiao’ (2.32) (A A,,)/(Azz— A. ) = (A,,- Axx)/5' (2.33) yr. :80 en the spherical tensor A can be expressed in the principal .3 system as = -/5 A. 180 followi em, ([ "0 x [Aztz where ycsm ks ) respec‘ in ter ConVen sPheri 0f ran the ir Alq: R Akp: Where fl I',I° and 1212: -1/2 n6. (2.34) The observed spectral frequency is determined by the 'ollowing equation: CSA ”CSA "0 _ mL(1 -1//§ A00 + /2/3 A 20 ) - (m - 1/2)/§ A 20+ 2 /wL ([24m(m - 1) - 4a + 9][A;:1] - [6m(m - 1) -4a + 3] [A;:2]2) (2.35) ”0 here the quadrupolar and chemical shift terms (Akq and ZESA) are correct through second order and first order, espectively. In order to express Azq in the laboratory frame n terms of principal tensor components Akq via Akq’ it is onvenient to use a spherical representation. In this pherical representation the irreducible spherical tensor Ak f rank k with 2k + 1 components Akq transforms according to k . 1e irreducible representation D of the rotation group +k . , - , k .q= R(¢,9,w)Aqu 1(¢,e,w) = k§_,AkPth(¢’e’W) (2.36) - k (2.37) IP_ 2 Alelp(a’B’Y)’ rere -' ’ vim? . 8 ,Im(a.B.r) = e ‘m “ d:.(B) e (2 3 ) Id (1:45) = I The obtained Equation A20: ”2 isinBsin24 A212: ”2 IZCOSHSin and Alzlo: 1/2 ”1an + Aim: ”/5 ”in” + “rose” WT) . 33 .15) = Z (- -1)“ U" + “0'13 - mLLk + m )2(k - m )1 (k - m .- n)! (k + m - n)!(n + m’ - m)!n! B)2k+n-n'-2n x (COS E B)n'-m+2n (-sin 2 The geometrical parts Al'w and Azq of the operator are tained by inserting Equation (2.40) into Equation (2.36)and uation (2.37): 0: 1/2 {3/2 6 (3coszB - 1 - nsinZBcosza) :1: t ’3/2 5 (V3/ZsinBcosfi + W//Ei[sin8cos3c032a i inBsinZal) ex" t2: 1/2 few 5 (/3/2sin2)3 — rWE [(1 + cos23)cos2a i= xzyt :osBsinZa])e (2.39) ,= 1/2 {3/2 6 ( 8(3cosze - 1) - /Ssin6cose[bcos(1 + ¢) + Ln(7 + ¢)] + {3/Zsin26[db032(1 + ¢) + esin2(7 + ¢)]) ._. ”/57? 5 [1/2 masineeose - (25in29 - 1)[bcos() + o) + .n(1 + ¢)] - sin6cose[dCO52(1 + ¢) - esin2(7 + ¢)] 3 3039[bsin(1 + ¢) _ 0003(7 + ¢)] + sin9[dsin2(¥ + ¢) + 32(7 + ¢)])]em / Azt27 csin(1 ((sinell ecosZ(T 6: 'II) The ff Produc orient of th chemic the ex around °°ntra intera ‘ 1) e spam 34 V3/2 5 [(3/8 asinze + sin9cose[bcos(1 + ¢) + . t2? :in(1 + ¢)] +1/2 (1 + cosze)[dc032(1 + ¢) - esin2(7 f ¢)] : 3in9[bsin(1 + ¢) - 0005(7 + ¢)] + cosG[dsin2(7 + ¢) + oszu + ¢)])]e‘2"" _ (2.40) 1/2 (3coszfi - 1 - nsinZBcosza) V3/ZsinBcos8 + n//E sinBcosBcosZa -n//§ sinBsinZa 1/2 [V3/23inZB - n//§ (1 + coszfl)c03261 -n//€ cosBsin2a. a first order terms of the chemical shift interaction aduce the same shift for all transitions at a certain ientation of the crystal, while small second order effects the chemical shift interaction can be ignored. The smical shift in frequency units is directly proportional to a external magnetic field since the induced magnetic field )und a nuclear spin depends on the applied field. In rtrast the two resonance lines due to quadrupolar Leractions (transitions between m and m - 1 and between -(m >) and -m), which are called satellites, lie almost equally iced on opposite sides of the center of gravity of each pair. Th 21 comp between 1), and units). constant the qu. satellit first ( multiplz‘ remains since t] In secc satelli‘ proport: satelli‘ quadrup anisotr fundame progra researc crysta T inform depend second powder 35' pair. The magnetic resonance line is therefore split into its 2I components. The relative intensity of the transition between m and m - 1 is very nearly equal to I(I + 1) - m(m - 1), and the positions are field independent (in frequency units). The quadrupolar frequency w or quadrupolar coupling Q constant (QCC), x, is a convenient measure of the strength of the quadrupolar interaction. The distance between two satellites has exactly the same orientation dependence as the first order chemical shift interaction except for the nultiplicative constants. For half-integral spins there remains a central component that is unshifted in first order since the first order term contains the multiplier (m - 1/2). In. second order, however, the centred component and all satellites are shifted and these ShiftS‘ are inversely >roportional to the field. It is important to note that the satellite center of gravity is unshifted by the first order [uadrupolar interaction and is only shifted by chemical shift Inisotropy and higher order quadrupolar interactions. These ’undamental relationships are illustrated in Figure 4. The )rogram XTAL (Appendix A), written as part of this thesis esearch, can calculate the resonance positions for.a single rystal. The program is based on Equations (2.35) and (2.40). The NMR spectrum of a powdered sample provides some nformation about anisotropic interactions since the angular ependences of the chemical shift, and the first order and econd order quadrupolar interactions are different. The owder pattern results from the sum of all individual single b) Figure 4 36 .. u : -3/2 ' '——— ......... __ ' -1/2 a) ‘s" ’ 1/2 3/2 80= 0 30;: 0 30¢ 0 Boat 0 x - o x ¢ 0 x at o (1) (2) E2 E0 E0 10 Zeeman transition .3 4 3 b) 1 l 1 First Order Splitting I: . I o 4 u a 32 I 33 LE l E i5 Second Order Shift 0 Figure 4 Quadrupolar splitting of the magnetic resonance of spin 3/2 in a single crystal at an arbitrary orientation. a) Energy level diagrams; b) Theoretical NMR spectra without linebroadening. The numerals indicate relative intensities. , crystal li fields prov quadrupolar technique ( used to stu continuous the powder all the nuc line is tel II. C. 1. E In a written as between t1“- invariant Thus the I angle I- coordinate system 31“ respec’cive .7 rystal lines. In addition the NMR spectra. at different ields provide more information about both chemical shift and uadrupolar interactions. The variable angle sample spinning echnique (including magic angle sample spinning) can also be sed to study anisotropic interactions in solids. I. C. Powder Patterns The random distribution of orientations in a lecrystalline or powdered specimen gives rise to a )ntinuous distribution of frequencies. The NMR spectrum of re powder sample is a superposition of the NMR lines from ll the nuclei of all the grains of the sample. The resulting Lne is termed a powder pattern. I. C. 1. Static Powder Patterns In a single crystal, the resonance position can be dtten as a function of a set of Eulerian angles (a, B, 1) (tween the field and the principal axis systems and is (variant under the sample rotation around the field axis. .us the NMR line position does not depend on the Eulerian (gle 1. In addition, it should be noted that the polar ordinates ¢ and 6 of ZH with respect to the principal axis stem are identical with the Eulerian angles or, and B, spectively. weal I The norms. number of + (it). PIT) = (4" where d!) resonance Equation T principle1 condition be resdilt on both I possible) computer transitiO? Equation 38 u = ”(093) = Q(¢,9), or (s) = Nam). I1 = c058. 7 (2°41) The normalized lineshape function P(w) is proportional to the number of nuclei whose NMR lines fall into the interval Q ~ m + d0. _1 _1 w+dw P(o) = (4n) do I dQ (2.42) w . where Cl!) is an element of solid angle (d9 = -duda). The resonance condition (Equation (2.35)) is inserted into Equation (2.42) and the powder pattern P(w) can, at least in principle, be calculated. It turns out that if the resonance condition is only a function of p and not of a, then P(w) can be readily calculated in closed form. However, when m depends on both )1 and a, a closed form for P(w) is not generally possible, and one must calculate the powder pattern with a computer [30]. The resonance peak position w” for a transition from level m to In - 1 can be obtained from Equation (2.35): SA mm = m [1 + 5CSA + 5CSA(3coszB - 1 - 0C sinchoSZa)] L 130 - (m — 1/2)wQ(3coszfi - 1 - nosinZBcosza) + (”i/lzwtl X [24mm - + nocosza + - [6u(m - 1) value of cos TM space w powder patt +dw; every , correspondin space. Up to interactions considered. theoretical absorption I G (M 61"“ where! TI“ ' I due t° 39 02 02 ( [[24m(m - 1) - 4a + 9][% (1 - cos22a) + coszB(% - Hg 02 02 . t nQCOSZa + g cos226) + cos4B(- % - nocos2a - fig coszza)] ' Q 02 - [6m(m - 1) - 2a + 3][§ - fl cosZa + 2— c0522a + coszB X 8 4 24 02 02 0 Q2 -éfl..-n__2 ‘33 [La ( 4 + 6 12 cos 2a) + cos B(8 + 4 cos2a + 24 cos 26)]] (2.43) vhere a = I(I 4-21). In a powder sample, all orientations of the principal axis system are equally probable, i.e. the value of cosB and a in Equation (2.43) are distributed in the 1-0! space with equal probability. Thus one can obtain the powder pattern by counting P(w) for um lying between m and w t d0; every combination of values of u and a should be chosen :orresponding to the lattice points of an equally divided p—a space. i Up to this point the effects of dipole-dipole .nteractions and (the spin-spin relaxation have not been :onsidered. These effects will broaden and smooth out the :heoretical powder pattern. Mathematically, the true Lbsorption pattern I(w) will be given by Q .(Q)=I p(w’)F(w - e')db', (2.44) --0 'here F(w - w’) is a function that describes the broadening ue to dipole-dipole interaction or/and the spin-spin relaxation. multispin sy: as Gaussian sufficiently assumption t [31], FW - (0') = [ The peak-to- e(lual to 20 width". Figu Which are ca This Prograr Part of thi Gaussiam cor Static and II Program ass QUadrupolar all tPatnsit; (Equation ( Res and th with respem may have t: fr°m Six pc the‘ mo te Pattern. 40' relaxation. 'This process is known as convolution. In a. multispin system, the distribution of spins may be considered as Gaussian. In most cases considered in this work, sufficiently accurate results are obtained with the assumption that F(w - w’) is a normalized Gaussian function [31], F>>eeeeez ..L o se>eeee>> O seeeeeeee s>>eeeee> _ O 20 - N O - 0 ppm ppm PF") 8 '0 'o 3 l ppm 025 GS 035 to J ll 0 Figure 8 Computer simulated VAS second order quadrupolar lineshapes for the central transition of a quadrupolar nucleus. II. D. Magne A numb: width of a 1 of the appli can be redu magnetic di broadening broaden the second orde: lattice re] P0pulation ' the lifetim: resonance 1 Uncertainty In thi dipole cou; the Zeeman will be ign Small magne nuclei. Th magnetic Inc 48 II. D. Magnetic Dipolar Broadening A number of physical phenomena may contribute to the width of a resonance line. One of them is the inhomogeneity of the applied field. This is not a serious problem since it can be reduced to a few milligauss out of 104 Gauss. The magnetic dipolar interaction is the dominant source of broadening in solids. Also the quadrupolar interaction can broaden the observed line due to first order splittings or second order shift as already discussed. The fact that spin lattice relaxation (T1) processes produce an equilibrium population by balancing rates of transitions puts a limit on the lifetime of Zeeman states, which effectively broadens the resonance lines by an energy of the order of h/T1 due to the Uncertainty Principle. In this section, however, the contribution of magnetic dipole coupling between the various nuclei to the width of the Zeeman transition will be considered, and other effects will be ignored. Magnetically active nuclei (I ¢ 0) behave as Small magnets and create magnetic fields at the surrounding nuclei. The classical interaction energy E between two magnetic moments H1 and #2 is A A A A .R 3 5 R12 R12 where R12 is the radius vector from #1 to H2 and R12 15 the magnitude c Hamiltonian usual: {1371111 izqshS. The dipolar 3i: '3 1113121 St A180 the "diPOIar a1 B: '1/4(j :1/2 (1 49 magnitude of the vector R12. For a quantum mechanical Hamiltonian one simply substitutes operators for M1 and #2 as usual: A p1:11hl A The dipolar Hamiltonian then becomes (2.49) 3(I'RIS)(S'RIS)]. 2 R18 -3 Z ‘ 7175R15h[l's - Also the dipolar Hamiltonian can be expanded into the "dipolar alphabet" [36] rlvsh IS 3 [A + B + c + D E ] I R s where 2 A = (1 - 3cos 6)Izsz B = -1/4 (1 - 3cos26)(1*s' - I's*) = 1/2 (1 - BooszeHIZSZ ’ I-S) -ao -3/2 sinecoseuzs+ + I+Sz)e C) II . . - ‘ £¢ C* = -3/2 sinecose(IZS + I Szle C1 H E: -3/4 Sir. F: 13* = '3/ Wavefunctior numbers mI term A, whi diagonal an 0n the 0th connects or usually saj the other L rePresented States, wh defienerate. determined Terms (3 an both Spins are of f-di zem‘order adm'le'Au'e Methods. 1 transition em“ Peal disregards terns C, . to drop C so -3/4 sin26 I‘s‘e‘2‘¢ [11 II a: _ _ . E = -3/4 sin29 I S effub '21 ll Wavefunctions can be given in terms of the individual quantum numbers mI and ms, which are eigenvalues of I2 and 32‘ The term A, which is proportional to 1282’ is clearly completely diagonal and it connects the state Imlms) with state (mlmsl. On the other hand, B is proportional to I‘S- + 1-8+, and or to (mI It is -1mS+1|‘ connects only Imlms) to ("Mums-1| usually said that B simultaneously flips one spin down and the other up. B has no diagonal matrix elements for the mlmS representation, but it has off-diagonal elements between two states, which become important when the states are nearly degenerate . When the proper zero-order functions are determined, B turns out to have diagonal matrix elements. Terms C and D each flip one spin only. Finally E and F flip both spins up or both spins down. The terms C, D, E, and F are off-diagonal. The effect of these four terms is to admix zero-order states into the exact states. The amount of the admixture can be computed by second-order perturbation methods. A consequence of the admixture of states is that transitions at o) = 200 and near (a) = 0 can be induced. The extra peaks at 0 and 290 are very weak and may usually be disregarded. Since they are the principal effects of the terms 0, D, E, and F, it will be an excellent approximation to drop C, D, E, and F from the Hamiltonian. Therefore the truncated dipolar Hamiltonian becomes : - 41 2‘IS 1/ For va becomes simi one can eas: zero by spi chemical sh interaction this term 1 Spins can 3 time averag the sample magnetic f important interactior In on Samples, 0 Hamiltonia] Slim 1128 each Other V1°°k [37; compute th eigenstate (300526 jk 3 (SIszk2 - Ij'Sk) (2.51) 3w xis = '1/4 7178h 2 R For variable angle sample spinning this expression becomes similar to the first order equation of CSA. Of course one can easily see that this interaction will be averaged to zero by spinning the sample at the magic angle just as with chemical shift anisotropy and the first order quadrupolar interaction. It is clear that one can theoretically average this term to zero if 3cos29j — 1 = 0. Although only a few k spins can satisfy this condition under static conditions, the time averaged coszejk can be made to equal 1/3 by spinning the sample at the magic angle with respect to the applied magnetic field B0. The dipolar Hamiltonian WIS is very important in considering homogeneous and inhomogeneous interactions in solids. In order to estimate the dipolar broadening of static samples, one has to obtain the eigenvalues of the total Hamiltonian and it is a formidable task to solve this problem since Iszkz and Ij-Sk operators do not commute with each other. In fact, there is a clever technique by Van VleCk [37], called moment analysis, which enables one to COmPute the dipolar width without solving explicitly for the eigenstates and eigenvalues of energy. The nth moment of f(w) is defined by the equations or") = -— and i2” (66“) = — Equation (2 expressions single cry: f°r intere R“ is th. angle beta The Nose reapecth,‘E rePlaced t For t 52 n I: w"f(w)dw ' = m , (2.52) I0 f((0)d0) ' and I: (w - )“r(o)du = 00 (2-53) \[ f(m)dw 0 Equation (2.53) for n = 2 is called the "second moment". The expressions for the second moment of nuclei of type I in a single crystal are 2 2 (1 - Scos 91k) 2 _ g 4 2 II - 4 11h I(I+1)§ R6 (2.54) jk for interactions between like spins and (1 - 3cos29 )2 2 _ l’ 2 2 2 jk (AwI>IS - 3 ylysh S(S+1) § R6 (2.55) jk for interactions between unlike spins. In these expressions R“ is the distance between nuclei 3' and k and 9“ is the angle between the Rjkvector and the external field direction. The subscripts II and IS indicate like and unlike spins, respectively. For a powder pattern Xkfl - 3cosZ9-ik)2 can be rePlaced by its average value 4/5. For a Gaussian line shape, the second moment is given by or AU1 = 2.355% (2.56) 2 _ 1 (A0 > - 0.721("2‘ A0.) /2 (A02), ‘1/2) where 401/2 5 The a calculations and Ollom [ nuclei. Thre spins in idr inequivalent modification involves si (2.54) by ar In case (c) the same a Unsiplit 11,, H‘ E- Homo Inte AS mer Shifted by Shapes, ho result for A hem 53 where A01 is full width at half-height. /2 The appropriate modifications of Van Vleck’s calculations have been carried out in first order by Kambe and. Ollom [38] for the central component of quadrupolar nuclei. Three cases are considered:. (a) broadening by like spins in identical sites, (b) broadening by like nuclei in inequivalent sites, (c) broadening by unlike nuclei. The modification for dipolar broadening in cases (a) and (b) involves simply the replacement of 3I(I + 1) in Equation (2.54) by an appropriate factor Fx (x = a, b). Fa = % I(I + 1) + [ 212(1 + 1)2 + 3I(I + 1) + %3 ]/2(21 + 1). -51 l _1 3 Fb - 3 I(I + 1) + 2 (21 + 1) + 32 (21 + 1) . (2.57) In case (c), the second moment of the central component is he same as Equation (2.55) given by Van Vleck for the nsplit line. I. E. Homogeneous, Inhomogeneous, and Heterogeneous Interactions As mentioned before, every single line is broadened or hifted by some mechanism. Three types of spectral line hapes, homogeneous, inhomogeneous and heterogeneous will esult for a powder sample. A homogeneous line is a sum of individual lines all having the and no shif comes from but are fluc with a spin spin system absorption all the spi with one a interaction "indistingu fliP-flop I also be tru frequency ( Since ener; another. If th. “mimetic f remarkably "1°59 Spi conditiox), in this s interactim have to be different overall re ‘54 having the same line width (i.e. the same broadening 1/T2*) and no shift with respect to each other. If the line width comes from some mechanism which are external to the system but are fluctuating rapidly compared with the time associated with a spin transition, then the thermal equilibrium of the spin system will be maintained at all times. Therefore, the absorption or emission of the energy will be distributed to all the spins. In a liquid where all the spins are coupled with one another' or' in. a solid, where the dipole-dipole interaction among the like spins makes the spins "indistinguishable" in the frequency domain and where spin flip-flop processes relax all 11MB spins equally, this will also be true. Therefore selective irradiation at a particular frequency of the line causes saturation of the whole line since energy will be quickly transferred from one spin to another. If the line width arises from variations in the local magnetic fields, the physical response of the system is remarkably different. Energy will be transferred only to those spins whose local fields satisfy the resonance condition. Further, the processes for spin-spin interactions in this system will be slow as compared to the direct interaction of the spins with the lattice because energy will have to be transferred to the lattice in order for spins in different local fields to come to equilibrium. Then the overall response of the spin system will be a superposition of the individual responses of the spin packets. Therefore an inhomogeneoi lines will by a distri individual packet, no irradiation pattern) ca since coupl sources of (6.8- qua< anisotropie inhomogenei A beta Packets, w) t0 each 0th in NMR is 1' case dilute "sea" of I to the pa interactim other 8 Sp interactio] different proceSSes spinning, 1"" 0f t) other hand 55 inhomogeneous line which is a superposition of individual. lines will be observed and the lineshape will be determined by a distribution of the shifts. Although there may be some individual lifetime broadening connected with each spectral packet, no coupling between the packets exists. Selective irradiation at a particular frequency of the line (powder pattern) causes a saturation of only that part of the line since coupling to other spectral elements is excluded. Some sources of inhomogeneous broadening are anisotropy broadening (e.g. quadrupolar interactions, chemical shift enui g anisotropies), dipolar interactions between unlike spins, and inhomogeneities in the applied magnetic fields. A heterogeneous line is made up of individual spectral packets, which are distributed in frequency and are coupled to each other. The typical case where this behavior is found in NMR is in the case of heteronuclear spin coupling. In this case dilute S spins are coupled to a more or less isotropic "sea" of I spins via a few I spins which are strongly coupled to the part of the abundant I spins via dipole-dipole interaction. Some of the other I spins are in turn coupled to other S spins. The flip-flop processes of this dipole-dipole interaction among the I spins cause exchange between the different spectral parts of the S spins. If the flip-flop processes are quenched, which may be achieved by magic angle spinning, decoupling, or multiple pulse experiments, the NMR line of the S spin system become inhomogeneous. If on the other hand the flip-flop rate can be enhanced a homogeneous line will rt line will result. 56 The fa from the S; of electron dynamic prc While magne Preperties. There mailletic pr and ferromg the Simple metals) e) paramaSneti that Par 5 connected . important thESe mate: CHAPTER III MAGNETIC SUSCEPTIBILITY The fascinating magnetic properties of materials arise from the spin of electrons or nuclei and the orbital motion of electrons. Magnetic resonance spectroscopy deals with the dynamic properties of the spins in a static magnetic field, while magnetic susceptibility is the study of static magnetic properties. There are three main classes of solids as far as static magnetic properties are concerned: diamagnetic; paramagnetic; and ferromagnetic substances. Most insulators and about half the simple metals are diamagnetic. Some other insulators and metals, except for a few ferromagnetic substances, are paramagnetic. The ferromagnetic materials become paramagnetic when heated to sufficiently high temperatures, a fact showing that paramagnetism and ferromagnetism are intimately connected. The magnetic properties of electrides are very important to study the electron—electron interactions in these materials. 57 III. A. Defi The re: characteriz: x u DU": where M is per unit vo symbol xv Parallel an tensor. It on a gram gram susce] susceptibii x: a, denSlt: In the Cgs Ionic) sus “Ma. 58 III. A. Definitions of Terms and Units The response of materials in a static magnetic field is characterized by the susceptibility X, through the relation (3.1) x u an: where M is the magnetization, defined as the magnetic moment per unit volume, and B is the applied field. Quite often the symbol )(v is used for X. In isotropic media B and M are parallel and X is a scalar; for anisotropic materials, x is a tensor. It is more convenient to express the susceptibility on a gram (x) rather than on a unit volume (X) basis. The gram susceptibility is also called the mass or the specific susceptibility. It is related to the volume susceptibility by )C : density (3'2) . 3 - In the cgs system the unit for X 18 cm g 1. The atomic (or ionic) susceptibility (XA) and the molar susceptibility (XM) , o o 3 - 1 a expressed in cgs units of cm mole are given by >< ll X X atomic weight (3.3) X ll X X molecular weight (3.4) III. B. Dian The ap motion of magnetizatir Lenz’s law. temperature diamagnetic be attribu structure 0 gram-atom ( distributic Present on: SuSceptibii l=-E A 2 one where N is around the electron r the Perman 0f electr SuSceptibi SuSCeptibi 59 III. B. Diamagnetism The application. of EHI external field ix) the orbital motion of an electron around 21 nucleus induces a magnetization in.iflue opposite direction, in accordance with Lenz’s law. Diamagnetic susceptibility is independent of both temperature and the applied field. Any significant changes in diamagnetic susceptibility with temperature in most cases may be attributed to a change in the physical or chemical structure of the material. The diamagnetic susceptibility per gram-atom (XA) depends primarily on the average radial charge distribution , taken perpendicular to the field and is present only when an external field is present. The atomic susceptibility XA is given by the Langevin equation: 2 1A = - Z (3.5) i where N is Avogadro’s number, 2 is the number of electrons around the nucleus, e and m are the charge and mass of the electron respectively, and c is the speed of light. However, the permanent magnetic moment arising from the unpaired spins 0f electrons which gives rise to a high paramagnetic susceptibility masks the very small diamagnetic susceptibility. c. Parama: When a the respon: Classically with the f produced by which in t} than the I magnetic p< finch each field in randomizat Susceptibi the Curie x I. OHIO Where C is When Imrmit th! dig“ the of known "m01ecu1a; is the s molchlar Simplest 60 C. Paramagnetism, Ferromagnetism and Antiferromagnetism When a magnetic field is applied to noninteracting spins the response of the spin system is remarkably different. Classically, the tendency of the magnetic dipoles to orient with the field is counteracted by the randomizing effects produced by thermal energy kT (k is the Boltzmann constant), which in the room temperature region is several times greater than the magnetic energy of the dipoles. Hence very weak magnetic polarizations are observed in many paramagnetics, in which each dipole is able to-orient itself parallel to the field independently of others. Experimentally the randomization effect. is reflected in 21 decrease in susceptibility with increasing temperature. This is known as the Curie law: -9 X - T (3.6) where C is the Curie constant. When the distance between the spins is short enough to permit them.tx> interact, the internal interaction tends to align the spins. First Weiss proposed that all interactions of known or unknown origins could be replaced by a single "molecular field" Bm, such that the total force on each spin is ‘the sum of an externally applied field B and of the molecular field Bm. In the mean field approximation the simplest assumption for 21 molecular field ii; that .B is m proportiona From the calculated: T3? The susce; temperatur’ magnetizat Curie-Weis susceptibi Curie Poir. X“\ (T- at temper; Heis: may be 81 of Want magnetiZa momenta, magnetic 61 proportional to the magnetization M: XM (3.7) tn u From the molecular field the susceptibility can be calculated: x = C - e = c). (3.8) T - 9’ ' The susceptibility has a singularity at T = 9. At this temperature and below there exists a spontaneous magnetization. This expression, which. is called the Curie-Weiss law, describes fairly well the. observed susceptibility variation in the paramagnetic region above the Curie point. Detailed calculations [39] predict X a . , (3.9) at temperatures close to Tc. Heisenberg [40] first showed that the Weiss local field may be given a direct and simple explanation in the language of quantum theory. His model suggests that spontaneous magnetization arises from a coupling of the spin angular momenta, S, rather than the total angular momenta. The magnetic atoms are assumed to interact in pairs and to be subjected to an applied field. This interaction is not solely the V magneti< magnetic fi another spi] general too interaction which is mechanical Hamiltonian =-22J. i O and to the lack of a phase transition. 6'5 a) b) ' ' 'bility b) of ' 1f1c heat a) and Suscepti Figure 9 Sfizfdimensional Ising ferromagnet [42]. The t) antiferroma is exactly for the sp net in the L. Onsager M=(1-x' where J1 a particular energy U(T UiT) : ‘JC With x = elliptic j Kt”) : Jaw When the the S'Peci found as exPension fOr the m 66 The two-dimensional Ising model of ferromagnetism or antiferromagnetism is one of the rare many-body problems that is exactly soluble and shows a phase transition. The formula for the spontaneous magnetization of the rectangular Ising net in the absence of an external magnetic field was given by L. Onsager [42]: ZJ 2J -2 1/8 . _ . 1 . 2 M - (1 — x ) With x - SlnhET—SlnhET- (3.15) where J3 anui J2 are the two exchange interactions. For the particular case of the square net, J1 = J2 = J, the internal energy U(T) is derived as (2tanh2%% - 1)K(X)) (3.16) 3 .. 3.2.3. U(T) - -JcothkT (1 + n with X = Zsinh(2J/kT)/cosh2(ZJ/kT) defined as the complete elliptic integral of the first kind, -1/2 le) = In/2(1 - Xzsin2¢) d¢. 0 When the internal energy is differentiated in turn to obtain the specific heat” a .logarithmically' singular ‘behavior is found as shown in Figure 10. On the basis of exact series expansion methods, Fisher [44] has obtained the following law for the magnetic susceptibility: Figure 10 Fisure 3 kT/J Figure 10 Exact specific heat C/k versus kT/J for isotropic nearest-neighbor, square-lattice Ising model [45]. O£> (15 )1) L5 2x) "A: Figure 11 Specific heat C/k versus kT/J for isotropic, nearest-neighbor, fcc Ising model. Results of series methods exact everywhere except near Tc I46]. maisz - 1 if kT ( In contrast not valid w? Despit physicists, has not yet expansions indicated two-dimensi singularit: shown in three-dime) 2 2 :M lch( Which is ( the result T0 in Spin-me Slater [4: the ProbL different A q quantizat The e l eme Ngzu2 Tc 1 75 xsz (1 ’1‘“) ' TzTc (3.17) ' C In contrast to the Curie-Weiss law, Equation (3.17) is evidently not valid when T » TC. Despite intensive efforts by theoretical chemists, physicists, and mathematicians, the three-dimensional problem has not yet been solved. However, very highly accurate series expansions of the three-dimensional i Ising model have indicated results similar to many features of the exact two-dimensional solution. For example, there is a logarithmic singularity in the specific heat. The specific heat curve is shown in Figure 11. Near the .Curie temperature, the three-dimensional susceptibility obeys the law T ard-1°25 TBT (3.18) x = 33:32} 1 - kT Whioh is closer in) the molecular field Curie—Weiss law than the result of the two-dimensional Ising model. To improve Heisenberg’s treatment of ferromagnetism the Spin-wave theory was developed by Bloch [47] and extended by Slater [48]. Although this treatment casts a new light upon the problem of ferromagnetism, the results are not radically different from those of Heisenberg’s theory. A quantized spin wave is called a magnon. The Quantization can be interpreted in terms of spin reversal. The elementary excitations of a spin system have a wavelike form. SP1 orientation of magneti Heisenberg‘ which all 1 atoms, eac] the one-e1 atomic fun 130 = N(e0 Where £0 interactic exchange ; ““13, am in are de1 the Spin These‘N f, of magnet B°hr magi, 69 form. Spin waves are oscillations in the relative orientations of spins on a lattice. Bloch constructed a set of magnetic wavefunctions that bear the same relation to Heisenberg’s atomic wave functions and chose the state To in which all electron spins are parallel. Consider a system of N atoms, each of which has one valence electron and assume that the one-electron wave functions m(r - r(n)) = 1P“ are like atomic functions. The energy of the state To is E0 = N(e:0 + c - -12-Jz) (3.19) where 80 is the energy of a free atom, NC is the Coulomb interaction energy of the system, J is the Heisenberg exchange integral involving the wn for pairs of neighboring atoms, and z is the number of nearest neighbors. The states ¢n are determinants of functions that differ from ¢o in that the spin of an electron on the nth atom has been reversed. These N functions have the same energy and have a 2 component of magnetic moment equal to (N - 2)Be, in which 89 is the Bohr magneton. The spin waves ¢k are ¢k = ak 2 exp(2flik-r(n)¢n) ' (3.20) and have the energy Eu = Eo + 2J 2 (1 — exp(2fiikfp)) (3-21) p where P is nearest ne: small comp: in which ’0] is E(k --~,k where E(k1,) = Ek The Partil by ll/Z f=2e: ixo "here .3 i temPEratu may be re EW) 2 J Using th M Satisf 70 where p is summed over the vectors joining an atom with its nearest neighbors. As long as the number of spin waves is small compared with N, the energy of the crystal in a state in which there are i spin waves of wave number k1, k2,---’,k£ is 1 E(k1,----,ki) = E0 + Z £(kv) (3.22) 081 where €(kv) = Ek - E0. (3.23) The partition function f for the system of electrons is given by u/z EO - 393(N - 2i) v-i.".i £(ku) f = 2 exp(- ) 2 n exp(- ) . kT kT 1.0 k1,..’kl (3.24) where -B°B(N - 2i) is the field interaction term. At low temperatures, when only the lowest levels are excited, 5(k) may be replaced by the value €(k) 2‘ J 2 (p-k)2. - (3.25) Using this approximation, Bloch found that the magnetization M satisfies the equation M.— : 1 '(%) Mo) where 9 is on the cry: Antifl constant .1 them antip on whether state is l are paral singlet.) and spin ) Heisenber Periodici an arbit Obtain 1 Propertie We acct There is three di simple interact theory c down abc exchan86 5° that . Cemplet‘ 71 2;. = 1 -6593” (3.26) G where 9 is the approximate Curie temperature, which depends on the crystal structure. Antiferromagnetism arises when the exchange interaction constant J is negative between neighbors, tending to align them antiparallel. (Since J is positive or negative depending on whether the energy of the triplet state or of the singlet state is lower, this simply restates the fact that the spins are parallel in the triplet state and antiparallel in the singlet.) One can use Heisenberg’s model, the Ising model, and spin wave theory to interpret antiferromagnetism. However Heisenberg’s model has the following weaknesses. First the periodicity of the lattice is not taken into account. Second an arbitrary approximation. (Equation (3.11)) is used to obtain the distribution of levels. Since the thermal prOperties are strongly dependent upon this distribution, a more accurate description is required for better calculation.' There is at present no known simple derivation based on the three dimensional Ising model. Also real systems are not as simple as the Ising model since there are some other interactions besides the pairwise interactions. The spin wave theory can consider the periodicity of a crystal, but breaks down above Tc. The molecular field theory replaces all of the exchange interactions in the crystal by an effective field, so that certain properties of the Sp-Sjinteraction are completely lost. If all of the exchange interactions are considered, obvious co: crystal, t exactly, remainder model [4! antiferrom model modi For a given by'I interactir nonequival i-spin. 1 neighbors nearest n OSUChi mg the j spi Spin and (With [J Hamiltoni "= as "here th. i‘Spin e the 1.3p 72 considered, the problem cannot be manageable. Therefore an obvious compromise is to consider some small section of the crystal, treat the exchange interactions within the section exactly, and assume the section to be coupled to the remainder of the crystal by an effective field. The Oguchi model [49] and the modified Oguchi model [50] of antiferromagnetism are based on this idea. Here the Oguchi model modified by Ohya-Nishiguchi will be described. For a collection of spins, the Heisenberg Hamiltonian is given by Equation (3.10). Consider a pair of spins i and j interacting antiferromagnetically' with. each. other and nonequivalently interacting with z nearest neighbors of the i-spin. The j-spin 'is excluded in counting the nearest neighbors. It is important to note that the number z of nearest neighbors in this model is equal to z - 1 in the Oguchi model where z is the coordination number, including the j spin. In addition to the interaction, J, between the i spin and the j spin, the inter-pair exchange interaction, J' (with IJ'ISIJI), is introduced in the system. Then the Hamiltonian can be written as 33 = — . _ . _ ’ . . guB (31 + SJ.) 2.15118j 23 ( § 3). Si + :5 31 SJ.) (3.27) where the sums k and 1 go over the nearest neighbors of the i-spin except for the j-spin, and of the j-spin except for the i~spin, respectively. When J' is equal to J, this model reduces to the Oguchi model. The terms Sn and 31 are replaced by their t the z axis below the field, the and S 1 van g is iso reasonably electrides axis and ' the Hamil1 l: -2Js.- 1 Where a: -ZXJ( b: ‘2XJ(' The term With the and (1M: EQUation matrix T 3‘J(- 73 by their thermal mean expectation values, S. If one chooses the z axis (easy axis), along which the spins would line up below the Néel temperature in the absence of an external field, then the x and y components of the mean values of SR and S1 vanish. For the sake of simplicity, it is assumed that g is isotropic and S is 1/2. This assumption may be reasonably good for the trapped electrons in solids such as electrides. By applying an external field B along the easy axis and using two new parameters, x = zJ'/J and h = guB/J, the Hamiltonian can be rewritten as a = -2JSi-Sj + a5: + b5: (3.28) where a = -2xJ(4§ + 8§z) - hJ 0" H -2xJ(§ + as!) - hJ The term 5S2 is the component added by the external field. With the basis functions, “10‘3" (1//2—)(0‘i8j + Biaj). 8:8," and (1//§)(ai3j - Biaj), the Hamiltonian matrix obtained from Equation (3.28) can be diagonalized by a transformation matrix T to yield the eigenvalues: t!) u 1 J(-1 — 4x8§z- 2h)/2 [:1 ll 2 3(1 - 2n)/2 M II 3 J(-1 + 4x5§z+ 2h)/2 where R = (1 + 4: The transf OC)C>I)-A Where Sin external dEflned a has: Since th eXPand t order te on! u / :0 A (D x whEre .j is 74 E4 = J(1 + 2R)/2 (3.29) where R = (1 + 4x2§2)1/2 The transformation is 1 o o o ‘ T = 0 cose 0 -sin9 (3.30) 0 O l O L 0 sine 0 cose ‘. where sin29 = ZXS/R and c0326 = 1/R. The magnetization in the external magnetic field along the z-axis is self-consistently, defined as g + &§z= Tr{Szexp(-BW)}/Tr{exp(’BW)}- (3.31) Since the terms containing h and 5-8-2 are small, one can expand the right hand side of Equation (3.31). The zeroth order term which leads to the non-zero solution of S is g —x§sinh(jR) (3.32) - R(exp(j) + cosh(jR)) where j = J/kT. At the limit of R = 1, the Néel temperature is From the 1 susceptibi By a Perpendic relation 26? = Tr Where th, . 3‘ = ~2Js :][+: and 75 ll NIH ._. :3 r——\ x x + I 03H h.) (3.33) From the first order term of Equation (3.31) the parallel susceptibility is obtained as —Z x. - Ngu[aaf ] BB 2 2 :1 NS I1 3 (3.34) 2 -jx + 1 +exp(-j)cosh(jR) By applying the external field along the x axis, the perpendicular susceptibility is obtained. The self-consistent relation becomes 26%" = Tr{(S: + 8:)exp(-B7£)}/Tr{exp(-E7€)} (3.35) where the Hamiltonian is 36 = -2JSiSj + 2.1258: - 2Jx§s’j‘ - (2J><<‘5§x +suB”)(S: + 8:) = + H (3.36) and + ZJXS8: - 2JX§Sx a = - 0 2JSiS j J it = - (ZJX' as: + The eigenv functions (3.30): Eoz : J“ E03 ‘J/E According °°mPonent {T exrl-B exm +_ Where 76 a = — (2Jx5§x +ngx)(s: + 3:) X §(Si + Sj) The eigenvalues of 380 are obtained by using the same basis functions and the transformation matrix given by Equation (3.30): E01 = -J/2 E02 = J(1 - 2R)/2 E03 = -J/2 E04 = J(1 + 2R)/2 (3.37) According to Karplus and Schwinger’s method [51], the component of the transformed matrix can be given by ‘l {T exp[-B(7£O + 31)] 1?. }ij = exp(-3on)5ij eXP<‘BE .) - exp(-BE .) _ + 01 E OJ {T RiTliJ1 (3.38) E01 _ Oj Where y 0 cose 0 sine 1 T H1T'1 = _§ cose 0 cose 0 /§ 0 ' c039 0 sine- Lsine 0 sine 0 ‘° One can obt Tr(eXP(-B(() and the nu Tr{(S: + S From Equa suscertit x1 = NSZP Where In the j coincids Wu» '1‘ ) 77 One can obtain the denominator of Equation (3.35) Tr{exp(-flfl)} = Tr{exp(-B%0)} = 2exp(j/2)[1 + exp(-j)cosh(jR)] (3.39) and the numerator of Equation (3.35) exp(-BE ) - exp(-BE ) 2§( :1 °2 01 2 cos 9 T{S’.‘+S’.‘ 4336} r ( . J)exp( ) _ E02 exp(-BE01) - exp(-BEO4) E01 ' E04 + Sinze) . (3.40) From Equations (3.35), (3.39), and (3.40), the perpendicular susceptibility is derived as x = NgZIJzC/(-2Jx§ + 2312(1 - R2)[1 + exp(-,j)cosh(jR)]) (3.41) .L where C = (1 - R2)exp(—j)cosh(jR) + 2R - (1 + R2)€XP[-J(1 - R)] In the paramagnetic region (S = 0), Equation (3.41) for x” coincides with Equation (3.34) for xi. At high temperatures (TN» T ), the susceptibilities take on a simple form: In the ant with a d constant ‘ is given ‘ --§( x10 7 The prog] thesis r1 electridq shown ir Critical 3 x < 1’ there i X maxi ( temperat the 3118 and the 78 _ Ngzuz, 1 X” = x; ' 4k T - J(x + 1)/2k' (3'42) Then the Weiss temperature 6 becomes 9 = J(x + 1)/2k. (3.43) In the antiferromagnetic region (R > 1), x” decreases rapidly with a decrease in temperature, while 7‘1 gives a nearly constant value x10 near absolute zero. The susceptibility x10 is given by X10 = - Ng2p2(: ‘ 11. (3.44) 4JX The program ANTIMAG (Appendix C) was written as part of this thesis research and was used to fit susceptibilities of some electrides. Some calculated magnetic susceptibilities are shown in Figure 12. At temperatures comparable to J/k, X critically depends on the molecular field parameter, X. For 0 5 K < l, the susceptibility curves show a broad maximum and there is no phase transition. The maximum susceptibility, KM”, occurs at a temperature higher than the Néel temperature in the region where 1 S x < ~2. When X increases, the susceptibility at iflue Néel temperature approaches X... and the maximum on the susceptibility curve becomes sharp. 79 .voma 0:» ca o>uso on» so causewccfi one ensuenooloa Hooz one .0.» on o.o u x you ousueuooloa voosvou msmpm> huwdanwpooomsm cevocmaa couscou poadasodeo «H ahauam _H. _ \Bx 95932.88. goose-m 3 o u + u o . . _ _ tr r p . p a 0.0 I «a land m.~ I Ow O.N I m 18.0 mh.~ I o m.u I h . mN._. I o 1.80 c.- I m mh.o I v .139 m.o I m mN.o I N 1.00.0 0.6 I w x lead 180° 13.9 [mad stud 1u6 =6 ‘2”ZSN)/|rlx Karrrqrsdeosns peonpea IV. A. Syn IV. A. 1. Stoi X' = 01" 1806, 022 ants) a‘ eVaPoratf used fc IV. A! 2 Abc Small be added s: heated the rei had for cryStal CHAPTER IV EXPERIMENTAL METHODS IV. A. Synthesis and Recrystallization of Model Salts IV. A. 1. Method 1 Na+, K+, Rb+, Cs+; Stoichiometric amounts of M+X- (M x“ = 01', Br’, 1’, SCN’, B(CGHS)4’) and L (L = 1204, 1505, 1806, C222) were dissolved in methanol (acetone for B(C6H5)‘- salts) at room temperature. Crystals formed upon very slow evaporation of the methanol. The crystals were collected and used for recrystallization and for solid state NMR measurements IV. A. 2. Method 2 + - . About 0.5 ml of saturated solution of M X was made in a small beaker at about 90°C. A slight excess of-complexant was added so that some crystals were formed. Then the beaker was heated to dissolve the crystals after which it was kept in the refrigerater to permit crystallization. After crystals had formed, the residual water was filtered off. Slightly wet CrYstals were obtained, squeezed between several dry filter 80 papers and IV. A. 3. Polyc recrystali ketone, 0 was cover the paraf solvent. listed 1 Prepare t IV. Bo S} E: IV. B. 1 81 papers and then dried in air. IV. A. 3. Recrystallization Polycrystalline samples were redissolved in a recrystallization solvent (mainly acetone, methyl ethyl ketone, or methanol) in a 50 ml Erlenmeyer flask. The flask was covered with a piece of parafilm and a small portion of the parafilm was lifted to allow very slow evaporation of the solvent. The model salts synthesized by these methods are listed in Table 7 together with the conditions used to prepare them. IV. B. Synthesis and Recrystallization of Alkalides and Electrides IV. B. 1. Synthesis The purification procedure for solvents and metals [53] and the synthetic methods used to prepare alkalides and electrides [7] have been decribed in detail. However it was more complex and tedious to synthesize alkalides and electrides by using a synthesis apparatus called the "cow". The synthetic procedures have been improved by using a Simpler apparatus called a K-cell (or Kim’s cell). The K-cell shown in Figure 13 was designed by combining the advantages Of a commercial H-cell [Kontes] and the cow. The evacuated K-cell and a metal ampoule [53] were taken into the Table 7 Sy Method 1 Method 2 Other Mchods RecrYStz 11l2ati< I SamPle bet al, sample [18], 82 Table 7 Synthesis and recrystallization of model salts Solvent Compound Na:15c5-sgN‘, §a+1805-gi', Na+18C6-Br- Na+18C6°I_, Na $222°Cl , Na 0222-Br _ methanol Na 0222-1 , Cs (18C6)2'I , Cs (1505)2-I Cs:18C6-SCN’, gs+c2g2-SCN‘ 1 Cs:C222-I- _ Rb (1505)Z SCN ,Rb C222-Br , K 0222 SCN . K+18C6-I- Method 1 + _ + _ + _ Na+18C6-SCN , Na cgzz TPB , Cs_18C6-TPB acetone Cs (1806)2-TPB ,Cs (15C5)2'TPB , Cs+C222-TPB-, methanol Cs+18C6'I- Toluene toluene Na+(12C4)2 Ci", Na+(12c4)2-Br” Na+(12C4)2-I_, Na+(12c4)2-SCN' Method 2 water Ns+1505-Br', Na:15C5 I“, Na+15g5-scn’ _ K (12C4)2-I , K (1204)2 SCN, K (1505)2 I K+(1505)2-SCN', Li+TPB-a Other + -a + -a + -a methods THE/water K TPB , Rb TPB , Cs TPB 1-propanol Cs+(1806)2-SCN-b, Cs+18C6'I-b methanol Rb+(15C5)2'SCN-,Cs+18C6'SCN-, Recrysta- acetone Cs+(15C5)2-I-, Na+C222-Br-, Na+C222-TPB- llization K+$PB', Rb+TPB- Cs+TPB-, Cs+C222-SCN- I. , — Cs 18C6°TPB , Cs+(18C6)2-TPB methyl + _ ethyl Cs (18C6)2°I ketone a SamPle prepared according to the directions of Bhattacharyya bet al. [52]. ' €i§ple prepared according to the directions of Ellaboudy 83 ._Haoo -.x_ mooanpooao use mopwamxao mo memosuchm 050 how msuuuooo< «H ousmwm helium-f1 through t to which sealed me arm. The unions ( removed containi1 small De The Ultr vacuum s been mad K-cell a dimethyl Cooled c Vacuum J and i801 301utio) from ch several metal m when t] Poured trimeth crystal keeping _ _ Mezo h 84 helium-filled box. The purified complexant [53] was loaded through the short side arm in the crystallization chamber A, to which the fingers for crystal harvest were attached. The sealed metal ampoule was broken and loaded in the long side arm. The side arms were sealed with two 3/8" Ultra-torr unions (Cajon) and closed end glass caps. The K-cell was removed from the inert atmosphere box, the chamber A containing the complexant was cooled by liquid nitrogen in a small Dewar. The cell was then evacuated to 2 X 10'5 torr. The Ultra-torr unions and end caps were removed by making a vacuum seal-off of the side arms. After a metal mirror had been made by distillation, the side arm was removed and the K-cell allowed to thermally equilibrate. About 15 ml of dried dimethyl ether (MeZO) was distilled onto the liquid nitrogen cooled complexant. The closed apparatus was removed from the vacuum line and placed in a cold bath that contained dry ice and isopropanol. When the complexant had been dissolved, the solution was poured onto the metal mirror. MeZO was distilled from chamber B to chamber A and then poured back and forth several times to dissolve the complexant completely. The metal mirror was dissolved by gentle agitation of the K-cell. When the reaction was completed, the entire solution was Poured into the crystallization chamber A. Diethyl ether, trimethylamine, or pentane was added as a co-solvent for crystallization. MeZO was distilled into a waste bottle,while k("v‘ePing a temperature gradient of about 10°C. When most. of M620 had been removed, the co-solvent was added one more time. The decanted co-solven the cryst Finally t bottle or cell was chamber After di: were imm and stor IV. B. 2 A s mmole 01 (either in a ct NESLAB j Was sca At the bath, 1 distill Vacuum. Na+0222 refrig, 85 time. The solution then became light blue and the solvent was decanted into chamber. B. If washing was necessary, the co-solvent was redistilled onto the crystals. After washing the crystals, the co-solvent was again decanted to chamber B. Finally the solvent was distilled from chamber B to the waste bottle or to a liquid nitrogen trap under dynamic vacuum. The cell was removed from the vacuum line and the crystallization chamber side including "fingers" was cooled with dry ice. After distributing the crystals into the fingers, the fingers were immersed into liquid nitrogen in a Dewar, sealed off, and stored at —80°C or at liquid nitrogen temperatures. IV. B. 2. Recrystallization A saturated methylamine solution containing about 1 or 2 mmole of Na+C222-Na- was prepared by addition of a co-solvent (either trimethylamine or diethyl ether) in.a cell at --20° C in a constant bath. The cell was placed in a programmable NESLAB LT-Q bath at -20° C and then the crystal growing bath was scanned from -20° C to -70° C over a period of 48 hours. At the end of the scan the cell was placed in a ~70° C cold bath, followed tw'ti decant. The decanted mother liquor was distilled off into a liquid nitrogen trap under dynamic vacuum. Some large single crystals and multiple crystals of Na+0222-Na- were 'harvested and stored in a vial in the refrigerator in the helium-filled dry box. IV. 0' NM] Samp rotors wi 00.. For and end tightenec coming a Instrume type of rotor we rest to centrift procedu1 rates a and 91 atmOSph while i Probe. or a h With E 00min After Nicole the‘ol 1 NMR d, 86 IV. C. NMR Experiments Samples were loaded into cylindrical Delrin. or A1203 rotors with Kel-F turbines and end caps from Doty Scientific 00.. For lxnv temperature experiments special Kel-F turbines and end caps with center holes (Doty Scientific Co.) were tightened with Kel-F bolts and nuts to prevent them from coming apart. Also ZrO2 rotors and Kel-F caps from Bruker Instruments, Inc. were used for the Bruker VAS probe. Each type of rotor was filled with fine powder and capped. The rotor was inserted into a stator and then brought slowly from. rest to spinning at rates of 1 to 5 KHz. This allowed for centrifugal packing of the sample within the rotor. This procedure was repeated to achieve stable and high spinning rates and.1x> enhance the signal to noise ratios. Alkalides and electrides were loaded under a cold dry nitrogen atmosphere. The samples were transferred to the Spectrometer while in liquid nitrogen and inserted into the precooled NMR probe. Single crystals were mounted in a nitrogen glove bag or a helium-filled dry box onto Acrylic holders [Figure 14] with markers every 5°. Apiezon high vacuum grease or Dow Corning 'vacuum. grease was employed for crystal mounting. After NMR experiments the crystal holder was mounted on a Nicolet P3F single crystal x-ray diffractometer to determine the orientation of the crystal with respect to the holder. The program KINFIT [54] was used to fit single crystal NMR data. The program XTAL was also used to calculate (or to 87 21' Figure 14 Single crystal holder and cap used in the single crystal NMR studies. fit) resc fitting W with the The Bruker A! or on it 1180 com] mm o.d. MHz and capacito to lower 23.61 ME carried home-bu; Obtaine. static Univerg Spectrc 0f 111i cold w Extern, determ Stahda f°r a: SimPle durini 88 fit) resonance frequencies in order» to confirm that the fitting was correct. MAS and static spectra were simulated with the program VMASS (Appendix B). The studies of Na+ and Cs... salts were carried out on a Bruker AM 400 spectrometer equipped with a Bruker VAS probe or on a Bruker WH 180 spectrometer equipped with a Nicolet 1180 computer system and a Doty Scientific MAS probe using 7 mm o.d. rotors. This probe has a tunable range of 45 to 72 MHz and proton decoupling capabilities at 180 MHz. Additional capacitors were used with the tuning and matching capacitors to lower the probe tuning range for measurements of 13305 at 23.61 MHz and 39K at 8.4 MHz. Single crystal NMR studies were carried out on the Bruker WH 180 spectrometer equipped with a home-built static probe. The 39K and‘ 87Rb NMR spectra were. obtained on a Bruker AM 400 spectrometer with a home—built static probe in the Department of Radiology at Michigan State University Clinical Center' and an 11.7 Tesla. home built spectrometer in the Department of Chemistry at the University of Illinois. A dewared NMR probe was used to keep the samples cold without cooling the electronic circuits or the magnet. External standards were used as secondary references to determine the chemical shifts; the chemical shifts of the standards were determined with respect to a value of 0.0 ppm for an infinitely dilute. aqueous solution of the cation. Simple one pulse experiments with and without decoupling during acquisition were used for 23Na and 13305 NMR studies. 87 In the case of 39K and Rb NMR, very weak free inductior time, aft the prob« with unw distorti eliminat echo pul destruct construc sequence RD - P1 where R acquisi (D H Where Shifts times aCquis x! XI v.4 as fol 89 induction decay signals (FID) decayed within a very short time, after the strong observing pulse and before recovery of the probe circuits and receiver. Therefore the FID was mixed with unwanted extraneous signals, which produce a baseline distortion after Fourier transformation. In order to eliminate this probe ringdown, a 16 step phase cycling spin echo pulse sequence [55] was used. This phase cycling causes destructive interference of the noise signals and constructive interference of the echo signals. The pulse sequence is based on the Hahn spin echo which is given by: RD - P1(61) - D1 - P2(92) - D2 - ACQ(93) (2.1) where RD is a relaxation delay and ACQ is the time needed for acquisition. The phases of the pulses and the receiver were: 91=xxxxyyyyxxxxyyyy 92 = x y § 5 x y E‘; x y E § x y §'§ (2.2) 93=§y§y§x§X§y§y§x3Ex Where x, y, E, y represent 0°, 90°, 180°, and 2700 phase shifts respectively. The intervals D1 and D2 are the delay times for echo formation between the two pulses and acquisition. Since the phases of the receiver were fixed (x x i E y y § §), the phases of the two pulses have been modified as follows: 6w}: The pulse pulse for for soli solution. sequence quadrupo [56]. th and this Fortunat used. It circuits Pulse t group, 1 enough, echoes elimina Procedu lasts differ: echo e: 0“ the a horn buried 15 b) y y y y x x x x y y y y x x x x (2.3) CD ll 6 = y § x E y E x E y 5 x E y § x E. The pulse widths, P1 and P2 were used either a 45°or a 900 pulse for quadrupolar nuclei in solids. Usually a 900 pulse for solids was taken as half of a 90° pulse for aqueous solution. Spectra were obtained by using both a 450-900 pulse sequence and a 90o-900pulse sequence. Since the 900 pulse for quadrupolar nuclei depends on the QCC and nuclear spin I [56], the 900 pulse should be determined for every compound and this is impossible for compounds containing these nuclei. Fortunately, it was not important that accurate 900 pulses be used. Instead, a relatively good quality factor of the probe circuits was required. It seems that the shorter the 900 pulse the better. According to the experience of the Dye group, 900 pulses for liquid samples less than 14 us are good enough. One desires the minimum delay time for formation of echoes that are free of ringing, since longer delays can eliminate the ringing but reduce signal intensity. The best procedure is to determine how long the probe ringdown signal lasts with an empty probe. Figure 15 illustrates the difference between a normal one pulse experiment and a spin 39K NMR and the effect of the delay times echo experiment for on the spin echo spectra. Figure 15 a) shows a spectrum from a normal one pulse experiment, in which the KSCN peak is buried under the distorted base line. The spectrum in Figure 15 b) shows the partial removal of the baseline distortion 91 .mmoaop acofiofimusm nufl3 acosfinomxo osoo swam an Esauommm mzz ovum GOwwaoumflo A0 ”whoamp powwowmmsmcw an“: womanhomxo 0500 L t. 0 O avl o b ? D ' . ’ L } o m.moo. / oo< Ich cflmm an newpaOpmMp mafiaomon mo Ho>oson Acevedo An mucoswuomxo omasm mamcfim Sufi: mafia mzz pouuoumwo Ad .oswfisnowv osoo :flmm an Hd>osou :OMHHOpmmp mafiaomom ma ouswwm ~ m a — 4 A1 - a |._ QT: c In. or! om Em i a .o o .... o F 0000 Old, lf 3 _ n ... oo< 1— .1... oz 9 J 9m A0 3 Ad with tWO spectrum times we: 92 with two short delay times. However the most distortion free spectrum shown in Figure 15 c) was obtained when long echo times were used. Thi alkali 1 model 1 studies discuss Chapter v. AOA V.A,1 CPYStaI 00. MA result Spin 5 the p] those Hz. y Signit Surrm CHAPTER V RESULTS AND DISCUSSIONS This chapter is divided into two areas. The first is alkali metal'NMR studies of alkalides, electrides and related model compounds. The second is magnetic susceptibility studies of electrides. The experimental data are analyzed and discussed based on the theories which. were described in Chapters II and III and on the structures of the compounds. V. A. Akali Metal NMR v. A. 1. A Single Crystal NMR Study of Na+0222-Br- Figure 16 shows. the 23Na INMR spectra. of the single crystal Na+0222-Br-, obtained by using the Doty Scientific Co. MAS- probe on a Bruker WH 180 spectrometer. ‘The spectra result from three transitions of a quadrupolar nucleus with Spin 3/2. The linewidth A01 of the central transition of /2 the proton coupled spectrum is about 2600 :t 200 Hz, while those of the proton decoupled spectra vary from 200 Hz to 500 Hz. This linewidth reduction suggests that there are significant interactions between the sodium nucleus and its surrounding protons. More discussions about dipolar 93 a) b) Figure 94 8) b) #1114111IgulALAAILIALILLJJJLALIJLIALIJL 200 0 -200 ~400 PP"! Figure 16 Single crystal 23Na NMR spectra of Na+C222-Br- at ”L = 47.61 MHz. a) Proton coupled; b) Proton decoupled. interact The satellit a distri relaxati oriental magnetic orienta' satelli Because proport central is only This 1 relaxai intera. on the F Plot 1 of ti envirc °rien1 Systel Value of th by u: eshat 95 interactions will be included in other sections. The different linewidths of the central transition and satellites at various orientations indicate that there may be a distribution of quadrupolar coupling constants or that the relaxation time T2* of a single crystal varies with the orientation of the crystal with respect to the external magnetic field. However, it is very difficult to study this orientation dependent linewidth since only a few spectra show satellites within the sweep width of the spectrometer. Because the peak positions of the satellites are directly proportional to the QCC they are much broader than the central transition. The linewidth of the central transition is only slightly dependent on the orientation of the crystal. This implies that the effects of orientation dependent relaxation times, T2 and T1, orientation dependent dipolar interactions, or the distribution of electric field gradients on the linewidth of the central transition is small. Figure 17 shows that the periodicity of the orientation Plot is not H for a tilted goniometer. Although the structure of this compound is not known, an axially symmetric environment of Na+ was assumed in order to fit the orientation plot. Without including CSA there is a slight systematic deviation between the experimental values and values calculated by using only the second order expression 0f the quadrupolar Hamiltonian. An excellent fit is obtained by using KINFIT [54] and XTAL (Appendix A), in which the equations include both first order CSA terms and second order 96 .-em.~NNo+az Haemsao unease a as ezn~ mo cofluflwsoup Hopucoo on» mo mocopsomop uoasmc< ha ousmwm 3338.0 + cote-no a 70.305 ne:-«en «o 33 0+ [3+ r. O“... I on! 1. 5... r. on... T an! 1 3.. a.&. l «MI I «a! ... an... on! 3... I ha. r. 3... T 3.. T *«l T 3.. 1. flu... I 31 I 31 .I “I. (wad) ”ms norm-Ito quadrupo determin 622: -14 0.605 ME = 73.15 since t symmetr: will be salts Because of th:‘ correla cannot CSA tel three-: A 1. no, basis Obtain pOWdeI alkal Was ll anion 97 quadrupolar terms. The chemical shift parameters thus determined are 5130: -16.25 ppm, 5xx = 5YY= -16.92 ppm, and 62 z = -14 . 92 ppm . The quadrupolar coupling constant , x, is 0.605 MHz , and the Eulerian angle angles are 8 = 48.31oand 7 = 73.15o. The Eulerian angle a can be chosen arbitrarily since the resonance position of a nucleus with axially symmetric coupling tensors is independent of the angle a. As will be discussed later, the powder patterns of most Na+C222 salts support an axially symmetric environment for Na+. Because both the orientation of the crystal and the structure of this compound are unknown, information about the correlation between these tensors and the crystal structure cannot be obtained. Presumably, the principal Z axis of the CSA tensor and the quadrupolar coupling tensor lie along the three-fold axis. A single crystal NMR study is the best way to determine X. 7P. and. the chemical shift jparameters, and jprovides a basis for understanding powder patterns. The parameters obtained will be used in order to calculate static and MAS powder patterns. V. A. 2. A Single Crystal NMR Study of Na+0222-Na- This is the first single crystal NMR study of an alkalide or electride. As mentioned in the Introduction, it was unknown whether the narrow single peaks of alkali metal anions are broadened by quadrupolar interactions. The single crystal 1 the real satellitc with I = Fig of the respecti frequenc spectrul decoupl for a p a singl 12.55) differ: and t1 ration. arbitr made. intere to us Chemi. cryst temps Cent; cI‘Ysi addi. 98 crystal NMR spectrum shown in Figure 18 clearly proves that. the peaks originate from a central transition and two satellite transitions of the half integer spin 23Na nucleus with I = 3/2. Figure 18 a and 18 b are the single crystal NMR spectra of the compound without and with proton decoupling, respectively, at an arbitrary orientation at the resonance frequency, 47.61 MHz. The linewidth of the proton coupled spectrum is about 2270 :1: 100 Hz and that of the proton decoupled one is about 400 :1: 50 Hz. A calculated linewidth for a powder sample of Na+0222-Na- [17] is about 3670 Hz. For a single crystal, the term 2 (1 - 3cos29jk)2 in Equation (2.55) cannot be replaced by its average value,.4/5. Thus a difference between the value observed for the single crystal and the value calculated for a powdered sample can be rationalized. The linewidth for a single crystal at an arbitrary orientation can be calculated but no attempt was made. The NMR line was broadened by strong dipolar interactions from many surrounding protons. It was necessary to use a proton decoupler in order to accurately measure chemical shifts and to improve the signal-to-noise ratio. Figure 19 illustrates the spectra of the bronze single Crystal of the compound at arbitrary orientations at a temperature of about -50 (IL The chemical shifts. of the central transitions for Na+ vary with the orientation of the CPYstal due to the second order quadrupolar shift. In addition to the shift, the linewidth of Na+ strongly depends FiEUr. 99 a) 1000 500 0 '500 4000 ppnl b) AIIALLJ LumlanLJlenhmn nlL141 1 114 LleLLmlmn LJILLLAL “b0 0 '5“ 40m PD") 23 Figure 18 Na NMR spectra of a single crystal Na+0222'Na- at UL = 47.61 MHz. a) Proton coupled; b) Proton decoupled. FiSUI 100 W? “\N _J_1Llllll.__llllLHLlJLlLLL+LJLllLL1LL41lJLLlLLLLllLuLl 1000 500 0 -500 ‘1000 p p m Figure 19 Single crystal 23Na NMR spectra of Na+C222-Na- at three different orientations, at Vt = 47.61 MHz. on the external relaxati the full effectiv 1/2' with (T2 )' Where 1 second linewi. 3 Spi1 fluctu that ; depend impurj and c Orien shape MAS s Patte anisc S°dil 101 on the orientation of the crystal with respect to the external field because the encapsulated Na+ has a different relaxation times T2 at different orientations. Frequently, the full width at half-height, A0) is used to define an 1/2’ effective relaxation time, T by means of the relation 2 A -2'r*’ (51) Q1/2_ /2 ' with * -1 -1 -1 (T2 ) - T2 + (2T1) where the first term is the spin-Spin relaxation rate and the second term is the field inhomogeneity contribution to the linewidth. The T2 rate depends on the lifetimes of the a and 3 spin states, the T1 rate is a contribution due to fluctuations in the energy difference between the two levels, that is, the 2 components of the local field. Orientation dependent relaxation times T2 and T1 due to paramagnetic impurities» dipolar' interactions, quadrupolar' interactions, and changes in these interactions due to defects can cause orientation dependent linewidths. Therefore the powder line shape ofNa+ of this compound might be distorted. However the MAS spectrum shown in Figure 20 is a good second order powder pattern due to the elimination of the orientation dependent anisotropy broadening by MAS. Satellite transitions of the sodium cation have not been seen in this experiment (a total 102 I I T I '40 ‘20 '30 “40’ ppm from Na+(oq.) ()4 Figure 20 MAS 2314a NMR spectrum of NchzzoNa' at vL = 52.94 MHz [17]. of 120 buried sweepwic Na' is the pos orienta interac T1 black 1 the re: freque cation crysta room Partic const. MHz.‘ Chang local cause Studf Probj thos blac difi Sher 103 of 120 orientations). Presumably broad satellites must be. buried under the baseline or are beyond the spectrometer sweepwidth. While the frequency of the central transition of Na- is nearly independent of the orientation of the crystal, the positions of the satellite transitions'vary with the orientation of the crystal due to the first order quadrupolar interactions. The spectra shown in Figure 21 were obtained from a black Na+C222:Na- crystal at room temperature. In contrast to the result with the bronze crystal at lower temperatures, the frequency of the central transition of the cryptated sodium cation is almost independent of the orientation of the crystal and the satellites of Na+ in the black crystal at room temperature have occasionally been observed at particular orientations. Presumably, the quadrupolar coupling constant of Na+ has been reduced to the range of 0.3 ~ 0.6 MHz. The reason for this change is not clear but a structural change with temperature (possibly a phase transition or a local structural change due to thermal motion of atoms) might cause the change. Both black and bronze crystals should be studied at different temperatures in order to solve 'this problem. However the peaks of Na- are almost the same as those of tflue bronze crystal. This fact indicates that the black crystal still contained sodide ions. The single crystal NMR spectra of two bronze crystals at different temperatures are shown in Figure 22. It clearly shows that the line is narrower at higher temperature, Lu L l 14 14 L1 1*ng LA LJ ‘ L Ll LL 14 Li L LJ Ll L 14 LL‘ LLLLLLALLJ 1000 500 o -500 .1000 ppm Figure 21 Three orientation dependent 23Na NMR spectra at UL = 17.61 MHz for a black single crystal of Na C222-Na at room temperature. 8) 105 ‘b) 4:": :7. ~‘ "‘3: :5, - - C) _Llj_lllIJIJLAllell_l_llliLlllJ_1_LLlL_JLl|LLLIIJLILIllllll 1000 $00 0 -500 «one Ppfll o 23 a Figure 22 Ma NMR spectra of two Single crystals of Na 0222'Na at different temperatures, at u = l. 47.61MHz. o a) Crystal 1 at ~-50°C.; . o b) Crystal 2 at ~-30 C.; c) Crystal 2 at ~-20 C. probabl cryptan -4 °c I of 0.0! powder F single around edge t the c orient satell to t trans chemi a de\ i.e. the meth« usin 1.26 PPm Chen are 1801 Wer. Val 106 probably due to line narrowing by the thermal motion of the cryptand. Later, an Na- powder pattern of Cs+(18C6)2-Na_ at -4 °C [57] was observed and a quadrupolar coupling constant of 0.090 MHz was obtained from the two singularities of the powder pattern of the satellites. Figure 23 shows the orientation plots of one of the single crystals of Na+C222-Na_. The crystals were rotated around three orthogonal axes, one of which is parallel to an edge of the crystal. Figure 23a is the orientation plot of the central transitions for Na+, while‘Figure 23b is the orientation plot of half of the distance between the two satellite transitions for Na-. These transitions correspond to the first order quadrupolar shifts for either the transition (3/2,1/2) or the transition (~1/2,-3/2). The chemical shifts at0o and 1800 are slightly different due to a deviation of the principal axes from the goniometer axis, i.e. none of the pmdncipal axes are parallel to an edge of the single crystal. In this case, analysis by the Volkoff method [28] is very difficult. However KINFIT [54] analysis using XTAL (Appendix A) yielded a veryr accurate QCC. x = 1.268 i .008 MHz, isotropic chemical shift, 5i8° = -5.5 i 0.6 ppm for Na+ and QCC, x = 0.1763 1: 0.0003 MHz, isotropic Chemical shift, 5130 = —61.8 i 0.9 ppm for Na“. The results are summarized in Table 8. A QCC, x = 1.2 a 0.1 MHz and an isotropic chemical shift, 6130: -6.5 i 0.5 ppm for Na+ [17] were obtained by simulation of the MAS spectrum and the values agreed well with the above results within experimental 8) b) Chunk-l lhlfl (ppm) (ppm) Iputuu (Thouoando) Var-t Order 107 1m -n‘ -‘q .u «- -o 0+ 0+ +0 04 -uj «:4 T l I _f l r l U 1U 1D “0 I” I” mun-thumaghuu) calculated 0." 07-1 “-1 ”-1 “-4 u- ”-1 ..1" .-—+ 4n- 4u~ «us 4nd an 40 4n- and -9- +0 9 I fTfi I r fi;’—_¥——T_T‘—'t -1.8 T I in I” “0 I” l” mu a mun (Degree) 9- We Figure 23 Angular dependence of he 23Na_NMR transitions in a single crystal of Na C222-Na . a) Chemical shift variation of the central transition of Na 0 9 b)_Ha1f the separation between the satellites of Na ' . Table 8 108 Table 8 The results obtained from a single crystal 23Na NMR study of Na C222-Na axis X. (MHz) 5(ppm) 3(0) 1(0) 1 ' 1.274ro.014 -6.2io.98 60.83 38.27 Na+ 2 1.268ro.013 -5.1r0.92 56.39 148.87 3 1.261:0.012 -5.3so.81 48.38 42 62 1,2,3 1.268:o.ooa* -5.5ro.6* 1 0.176410.0004 -61.8io.9 60.83 38.27 Na” 2 0.1763io.0005 -61.8so.9 ' 56.39 148.87 3 0.1762ro.0009 —61.8t0.9 48.38 42.62 * 1,2,3 0.176310.0003 * Results from multiple data set KINFIT analysis error. both I Presum: the 58 are d chemic tensor experi and orien‘ sodiu elect This unit 1.1 Catie Pseu. Plan and indi req. axis Dos Wit tem to an 109 error. Principal axes and quadrupolar coupling tensors for both Na+ and Na- (are parallel and [axial respectively. Presumably the local symmetries for both Na+ and Na- are of the same type, even though the magnitudes of the interactions are different. There may be a small anisotropy of the chemical shift tensor associated with Na+. However, the tensor was not extractable from data obtained in this experiment since the NMR peaks for both Na+ and Na_ overlap' and the NMR peaks for Na+ are broadened at these orientations. The small quadrupolar coupling constant of the sodium anion implies that there is little perturbation of the electron cloud of Na- from surrounding protons and charges. This compound is rhombohedral, with space group R32 [58]. The unit cell is hexagonal (Z = 3) with a = 8.83 A and c = 29.96 A. It can be described as closest-packed.sodium cryptate cations with the sodium anions occupying the pseudo-octahedral holes. The sodium anions form parallel planes perpendicular to the three-fold axis. Also, both Na+ and Na- have a local three-fold ‘symmetry axis, which indicates that they have axially symmetric tensors. This requires that one of the principal axes (the principal Z axis) is along the three fold axis. Unfortunately it was not possible to determine the absolute orientation of the crystal with respect to the goniometer since it is so air- and temperature-sensitive. In addition, the crystal was too large to handle conveniently in the X-ray diffraetometer. There was an attempt to determine the absolute orientation of a crystal before crystal gave di Th is one study has be broade satell deteru Also ‘ orient coupl [31. Pres insu obta 'unce 0bs< Prh Sin dif 110 before 'measuring the single crystal. NMR. spectra, 'but the. crystal turned black after the X-ray study, and as a result gave different spectra as mentioned previously. This section shows that single crystal NMR spectroscopy is one of the most powerful techniques available for the study of alkali metals in alkalides and electrides. Also it has been shown that the NMR line of the sodide anion is broadened by quadrupolar interactions. The location of the satellites as a function of orientation provided an accurate determination of the quadrupolar coupling constant of Na-. Also the variation of the frequency of the Na+ signal with orientation yielded an accurate value of its quadrupolar coupling constant. v. A. 3. A Single Crystal 1330s NMR Study and a Powder Lineshape of Cs+(15C5)2'I- The title compound was studied as a powder by our group [8]. but the results differ from those obtained in the present study. The reasons for these differences are insufficient data collection, and the fact that information obtained from powder patterns contains relatively large ‘uncertainties since the singularities of a powder pattern are obscured by line broadening. In order to obtain the six Principal tensor values and the three Eulerian angles from a Single crystal study, at least five] to eight spectra at different orientations are required for an axially symmetric orvnon- the fac paramei shift ' Q.C.C. previo 62.0 1 and no Figure proce< corre. trans Progr obtai diste tensc Also yiel. as t Both The coul ‘VYl tet 13. 111 or non-axially symmetric case respectively. This is (hue to the fact that the line positions are functions of the unknown Q parameters, x, n , Sxx’ 5YY, 622, a, B, and 7. The chemical shift parameters 5180 = 29 ppm, SCSA = 52 ppm, ncsA = 0, and Q.C.C. x = 0.427 MHz and no = 0 were obtained from the previous study. In this study the tensor parameters 6xx = 62.0 ppm, 5Y1! = 46.1 ppm, 522 = -28.4 ppm, x = 0.5810 MHz, and no = 0.4755 were extracted by fitting the data shown in Figures 24 and 25 by Equation (2.18) with KINFIT [54]. This procedure is known as the Volkoff method [28]. In order to correct for the second order quadrupolar shift of the central transition (or the center of gravity of the satellites), the program XTAL (Appendix A) was used with the parameters obtained by fitting the pure quadrupolar splittings (the distances between transitions (3/2,1/2) and (-1/2,-3/2)). The tensors of 13305 and direction cosines are given in Table 9. Also, the least-squares fit of the data by Equation (2.35) yields three principal components of the CSA tensor as well as the quadrupolar coupling tensor and three Eulerian angles. Both methods gave the same results within experimental error. The principal axes of both the CSA and the quadrupolar 2 Coupling tensors are coincident; in addition IV IV 22' XX' From the previously determined crystal structure [8] the IVYYI and A522" 5130' 2 Iaxx- 5isol 2 |5YY- also tetragonal space group, I4, the unit cell dimensions, a = b = 13.173 A, c = 16.645 A, and z = 4 were obtained with an R factor of 8.6 % and an Ru factor of 11.6 %. Because of the 112 70 6 7 a had: a e 6 i . . . . . . a ‘ s . 0 . O ‘ . ' ii i “d I U 3 9 _ s ”'l ‘ aux!- I g a- o 9 8) . * ‘ I 5 1" 9 cash g o ‘ L o - - 9 9 1. v I g 0 -...- 8 * a ' I * I -u t j T _' I fij f I j I fi 'fi I T r 0 a 0. C C I. I. I“ I. I. Madman-(Degree) 7. - 7 “$8 . - a .m. a a . g B 9 I A .1 ' c . i I l ‘ I 49- I g s . , g '9‘ bath ‘ V b) n '0' 2 3 V a 10" a . , é o + . at .. - g 8 a " emu v ..a- 3 . ‘ A O ’ , o o I 1 I 8 * -N r fiT I I T I r f7 w I firfirfi 1—1 0 . ‘0 . . a! I. ta 8. 3. mama-(noun) x .9 n ovumnd A v 9' sandman Figure 24 Angular dependence of the entral transitions of Cs in a single crystal Cs (15C5)2'I . a) Site A; b) site B. +80 3° (Z/e- uaanieq 3: MW 01 091 px “I Inunnw: + A V ‘8 9118 (q 5V 3118 (18 ‘ I-Z(gogt) so Isisxio eTBurs s u; so so (Z/E-‘Z/I:) pus (Z/f‘Z/s) ‘suornrsuexs on: uaanieq aoueisrp eqi gfeq JO aouepuadep JetnSuv M0 (“unimmwnsnwhv 00! 1 L 1 0 X 8'7- D)! N 04 Th - 9-1.. .. r3... - 3- -flk ~rw -fik ~tk 9 ~81! en -"fl ~0‘0 1-I r-I'! I"!- t ...- r- 9'!- - 3'1... - g- - r0- - 9%- .. 9'9- .. 2‘0- w" -91 *- 0‘0 - I - 2‘! 9'! EII sz axnsra JO’JQ . ..I 'l (q (opuucnoqg) .‘fl0fl‘. (‘1‘) (B 3.38831‘0 4.’30 8.335 (Otto-30“.) (undid) Table occ‘- 114 Table 9 133Cs chemical shift, quadrupolar coupling tensors and direction cosines (ppm) Eigen Vectors E. Angles(°) 6xx 62.010.7 0.0227 -0.0017 0.9997 a 1.3 'A 5YY 47.2iO.7 0.9860 0.1655 -0.0221 3 89.7 CSA‘ 5xx 61.6t0.7 0.0191 -0.0758 0.9969 a 4.5 B 6YY 46.210.7 -0.1669 0.9829 0.0779 3 89.6 22 -28.7i0.7 0.9858 0.1679 -0.0061 7 170.3 Axx -1298.i2 0.0053. 0.0041 1.0000 c -0.4 A AYY -469.i10 0.9859 0.1674 -0.0059 3 90.2 All 1757.:5 -0.1674 0.9859 -0.0032 7 80.4 Qcc‘ Axx -1297.i2 0.0072 -0.0187 0.9998 a 1.1 B AYY -454.110 -0.1779 0.9838 0.0196 3 90.2 A22 l768.t10 0.9840 0.1780 -0.0038 7 169.7 (ppm) Eulerian Angle (0) Site A Site B sxx 62.0:0.7 a 0. 0. CSA" 6YY 46.1iO.7 B 90. 90. 22 -28.4t0.7 Y 80. 170. Axx -1297.12 a 0. 0. Qcc“ AYY -461.i2 B 90. 90. A22 1758.i2 7 80. 170. x1” = 0.5810 MHz nQ"= 0.4755 A22 = 3x/[21(21 - 1)] :Results obtained by the Volkoff method Results from multiple data set KINFIT analysis with XTAL E. Angles = Eulerian Angles poor R Nicole solvir symmet of th' infor crown succe cryst orier envii 8X80A sing the crys show frou 0P1! ext. (3/ gen Sit SEV tr: Wit 115 poor R factors, the diffraction data were redetermined with a Nicolet P3F diffractometer. So far, we have not succeeded in solving the structure of this compound due to the very high symmetry of the crystal. However, its space group must be one of the several space groups belonging to I4 or 14: From this information, one can obtain some structural information. The crown ether sandwich unit lies on a screw axis which requires successive complexed cations along the c-direction in the crystal to be rotated by 900 about the c-axis. Thus, two orientations of the crown sandwich units exist. The environments of these different crown sandwich, units are exactly the same chemically, but different magnetically. Figures 26a and 26b show the“ 1330s NMR spectra of a single crystal of Cs+(15C5)2-I- with proton decoupling when the static magnetic field is directed along the crystallographic a and c axes, respectively. The central peak shown in Figure 26a is assigned to the (1/2,—1/2) transitions from the two magnetically' different sites which. at this orientation have the same environments with respect to the external magnetic field. The two peaks from the transition (3/2,1/2) of both sites also coalesce at this orientation. In general, however, two distinct sets of peaks from the two sites were observed as shown in Figures 26b and 260. All seven transitions (2I = 7) from one site (A) and three transitions from the other (B) were detected within the sweep width of the spectrometer [Figures 26b and 26c]. The principal axes of both the CSA and the quadrupolar Figure 26 116 13:05 NMR spectra of a single crystal Cs (15C5)2-I at'vL = 23.61 MHz. a) B0 is along the crystallographic a axis; b) B0 is along the crystallographic c axis; c) At an arbitrary orientation. tensors ar plots in 1 chemical 1 the cryst site A a! and the respectiv of 10° r. related t ions are quadrupol shift a: Smeetri tensors and 1' 1 interact be Very an n-fol Which n the Carl the loc Th the pc freQUer Static tr30811 patter] 118 tensors are coincident as can be noticed from the orientation plots in Figures 24 and 25. The X principal axes of both the chemical shift and the quadrupolar coupling tensor are along the crystallographic c—axis. The Y'znui Z principal axes at site A are about 10° away from the crystallographic a-axis and the b—axis in the counterclockwise direction, respectively. The Z principal axis at site B makes an angle of 10° from the b-axis towards the -a-axis, where site B is related to that at site A by the crystal symmetry. If the I- ions are strongly related to the chemical shift and quadrupolar interactions, one might expect both the chemical shift and the quadrupolar coupling tensors to be axially symmetric since the structure is tetragonal. However, both tensors are nonaxially symmetric. If the distance between Cs+ and I- is large enough to diminish the effect of I- on these interactions, the local environment of the complexed Cs+ may be very important. It is probably impossible for Cs+ to have an n-fold rotation or an n-fold rotation reflection axis for which n > 2. Even if there is a local symmetry axis, often the carbon atoms in the crown ethers are disordered [6]. Thus the local environment of Cs+ may be highly asymmetric. The spectra shown in Figure 27 are a MAS spectrum and the powder patterns of Cs+(1505)2°I_ at the resonance frequency, 52.482 MHz. Figure 27b shows an experimental static powder spectrum. The powder pattern of the central transition is very similar to an axially symmetric CSA powder pattern but it is nonaxially symmetric. Combination of the Figure 27 119 13303 NMR spectra of a powdered sample of 05"(15c5)2 I at ”t = 52.482 MHz. a) MAS spectrum with Mr ~ 4 kHz; b) Observed static powder pattern; c) Computer simulated powder pattern. a) I 32 l 120 8) WWW b) WW {‘VTI'Y'lfiI'T'I‘TfiTT—fil'fTfiI'W—‘Ifiiffr1**r' .0000 53°00 20000 0.0 -30000 -60000 -90000 new ao'oo' 7i 'sdof' [Tdf' [i-Joo'Tr'qhoo ppm isotropic and the, pattern tensor sf extract 1 some otl differen‘ simulate- crystal fits ve transiti contribt broadeni satellit transit: Visible Order q Pattern CSA wit Fi differ. MHz. pJ intera, nuclei than intera SpeCtr 121 isotropic chemical shift from a MAS experiment (Figure 27a) and the chemical shifts at singularities of the powder pattern ‘can provide evidence for a nonaxially symmetric tensor since 5‘80 = 1/3 (6xx+ 6YY+ 5 ). It is dangerous to 22 extract parameters from only a smatic powder pattern since some other experiments such as_ MAS and ‘VAS can provide different information. The spectrum shown in Figure 27c is simulated by using the parameters obtained from the single crystal NMR study of Cs+(1505)2°I-. The simulated spectrum fits very well to the experimental one. The central transition is broadened mainly by CSA and a small contribution from dipolar interactions. Quadrupolar broadening is negligible for the central transition. The satellite transitions are very weak compared to the central transition. However, most of the singularities are clearly visible. The satellites are broadened primarily by first order quadrupolar interactions and the symmetry of the powder pattern of corresponding satellites is destroyed by mixing CSA with the quadrupolar contribution. Figure 28 shows the 133Cs MAS NMR spectra obtained with different spinning speeds at a resonance frequency of 23.61 MHz. Proton decoupling was used to minimize the Cs-H dipolar interactions. Because the dipolar interactions between cesium nuclei and protons are about 1 kHz, spinning speeds slower than 1 kHz cannot completely average out the dipolar interactions. Thus at low spinning speeds, broad bumpy Spectra due to partial averaging of the dipolar interactions 122 b) ~ .1... Arie. __A__Aj_‘lJ_JJ_ALlL_LL_lJ_L#J_lL_LJL lllllkL+LLALLLI 200 I00 o -100 DP"! 13:Cs MAS NMR spectra of a powdered sample of Cs (15C5)2°I_ with decoupling at UL = 23.61 MHz. a) wr~ 470 Hz; b) wr~ 650 Hz. Figure 28 were ob81 proton 0 under sl inhomogel spectra . Acc interact and dip interact spins ( studied (i.e., being s corresp NMR 111 term 5‘ Spin 1 Propor1 be obs will b Strong from 1 CouPle Where, SYSte: ideas aroun 123 were observed instead of narrow spinning sidebands without. proton decoupling. The proton decoupled spectra obtained under slow spinning conditions are clearly broadened by 133Cs NMR inhomogeneous interactions. Also the proton coupled spectra at slow spinning speed behave inhomogeneously. According to Maricq, et al. [59], "when two or more interactions are present simultaneously (e.g., chemical shift and dipolar) the result is homogeneous if any of the interactions is homogeneous." In other words, if abundant spins (for example proton. spins) around the nuclei to 'be studied (for example Cs) are strongly coupled to each other (i.e., homogeneous), then the NMR line of the spin system being studied (Cs) becomes homogeneous. This usage does not correspond exactly to that in the section II. E. since the NMR line of the spin system behaves inhomogeneously if the (ij‘°)(t) (in equation 26 in reference [59]) for each term 5 spin in an array of spins with dipolar interactions is proportional to one another; However, "hole burning" cannot be observed in the resonance of 'such a system since energy will be quickly transferred from one spin to another due to strong dipolar interactions. In addition this is different from heterogeneous broadening. Only a few spin packets are coupled to each other in a heterogeneously broadened line, whereas all spin packets interact with each other in the spin SYstem. Complexed cesium compounds are good models to test ideas about these interactions since the charge distribution around the cesium cation causes a relatively large chemical anisotrop: If the ‘3 origin, 8 instead 1 broadened obtained shift par pattern. nearly 1:! for the broadeni: 4.7 kHz 0971505 may not line of will be MAS exp. inhomog. is obse when t1 eXPerim BiVen i 124 anisotropy and the organic complexant provides many protons. If the 13303 in Cs+(1505)2°1- were completely homogeneous in origin, a single Lorentzian (or Gaussian) line would result instead of a normal powder pattern. However, a Gaussian broadened CSA powder pattern for the central transition was obtained for this compound and the simulation with chemical shift parameters yielded a matching static theoretical powder pattern. The lineshape of a proton decoupled spectrum is nearly the same as that of a proton coupled spectrum except for the extra broadening due to protons. The dipolar broadening is smaller than the width of CSA broadening (about 4.7 kHz at DL = 52.482 MHz). Therefore, the lineshape of Cs+(15C5)2-I— appears to involve inhomogeneous broadening and may not agree with the Maricq predictions. Whether the NMR line of the cesium compound follows the Maricq predictions will be clearly solved by a hole burning experiment or slow MAS experiments at higher field. If the line broadening is inhomogeneous, the FID with a train of rotational spin echos is observed (or spinning sidebands after Fourier transform) when the sample is slowly spun at the magic angle. More experimental evidence for inhomogeneous broadening will be given in Section V. A. 5. V0 A0 4' fl) In 1 yielded ‘ It was : shifted ‘ chemical crystals easily quantite pattern. complex: Qu: (for qu (mainly listed Static Some re 30, 31 Second T] Symmet; Na+czz contai Smeet betwee 125 V. A. 4. Lineshape Analysis of 23Na NMR Spectra of Model Salts that contain Na+ In Section, V. A. 1., an NMR study of a single crystal yielded very accurate parameters associated with the nuclei. It was shown that the central transition of Na+ is mainly shifted by second order quadrupolar interactions and that the chemical shift effect is negligible. However, good single crystals are not always available, while powdered samples are easily synthesized. Although the accuracy is limited, quantitative information can often be obtained from a powder pattern. In this section, 23Na NMR spectra of various complexed sodium salts will be discussed. Quadrupolar coupling constants, asymmetry parameters (for quadrupolar interactions) and chemical shift parameters (mainly isotropic chemical shifts) for Na+ model salts are Vlisted in Table 10. These values were obtained by simulating static and MAS spectra with or without proton decoupling. Some representative 23Na NMR spectra are shown in Figures 29, 30, 31, and 32. ‘These spectra. have lineshapes typical of second order quadrupolar powder patterns. The sodium NMR spectra of C222 compounds are axially SYmmetric except for Na+0222°SCN-. The structure of N8+0222-I- [60] shows that the individual twin component Contains two molecular species in a trigonal unit cell with Symmetry P31c. Average distances abetween Na+ and O and between Na+ and N are 2.574 A and 2.752 A. Axially symmetric Table 10 v 00 1 MHz Na Na Na Na Na Na Na Na N: N: N: N. N. N. N N \ N 105.84 N 1 N A \ ‘5 130’ X DNEProto FiprOto 126 Table 10 NMR parameters obtained by simulations of the static and MAS 23Na NMR spectra at two different fields QGaussian broadening, Hz UL Compound 5i80 QCC 0 Static Mass MHz ppm MHz DF DN DN Naiczzz—Cl:a -16 0.65 0 1200 150 55 Na+C222iBr -16.2 0.605 0 1200 200 60 Na+C222-I _ -9 1.05 o 1200 400 150 Na+C222'SCN_ -10 1.0 0.5 1300 300 160 Na c222 TPB -8 0.5 0 1200 200 150 Na+(1ZC4)2 Cl‘ -3 o 700 250 150 Na+(12C4)2 Br” -3 0 800 350 150 Na+(12C4)2-I- -3 0 650 200 120 47.61 Ne+(1zc4)2-SCN" 0 1.2 0 1100 350 100 NailSC5-Br_ -5 1.1 0.4 500 250 60 Na+15C5-I _ -5 0.81 0.4 750 170 90 Na 15C5 SCN -1 0.8 1 900 500 300 NaIlSCG-Cl: -10 2.2 0 800 350 --- Na+18C6-Br -9 2.05 0 650 300 --- Na+1806°I - -7 3.0 0.4 650 —-- --- Na+18C6-SCN_ -9 1.05 0.8 700 210 140 Na 18C6-TPB -6 1.72 0.16 1000 600 250 NailBCG-Cl: —10 2.2 0 800 --- 200 105.84 Na+18C6-Br -9 2.1 0 650 —-- 200 Na 18C6-I -7 3.0 0.4 1050 --- 200 a i8°, X,and 00 obtained by a single NMR study DN;Proton decoupled DF;Proton coupled Figure 29 127 23Na NMR spectra of Na+C222-I- a), b), and c). and Na 0222-SCN d) at UL = 47.61 MHz. a) Proton coupled static spectrum; b) Proton decoupled static spectrum; c)Proton decoupled MAS spectrum; d) Proton decoupled static spectrum. Observed (-——)and simulated (-- ). 128 wv VV "w 129 ....-’.-.---‘-A .. b) A 50 0 ~50 '1“ Figure 30 Proton decoupled static 23Na NMR spectra of Na (1204)2-1 a) and Na (1204)2.Tp3' b) at "1 = 47.61 MHz. Observed (--) and simulated (-°-). 130 I I I I I I I I I I I -‘o .20 .” 0‘0 -50 um. .70 a” a” -5” so no 70 no so 40 so 20 10 0 PPM Proton decoupled sta ic 23Na'yMR spectra of Na 15C5-I a) and Na 15C5-SCN b) at UL = 47.61 MHz. Observed (-——) and simulated (-~). Figure 31 131 Figure 32 23we was spectra of Na+1806 Br“ a), Na+18C6°I- b). c), Na 18C6-SCN d) and e). a) Proton decoupled static, UL: 47.61 MHz; b); d) Proton coupled static, DL= 105.482 MHz; c) Proton coupled MAS, UL: 105.482 MHz; e) Proton decoupled MAS, UL = 47.61 MHz. Observed (-——), Simulated (----). 132 8) 'IW 'm “n b) C) If OICO '7! ". ‘ I i 8" ll Q.CO' I 400 Ilnl PHI 133 electric field gradient tensors were obtained from these. compounds as expected from the local structure around Na+. In general, any n-fold rotation or n-fold rotation reflection axis for which n > 2 requires n = 0 [61]. In Na+C222-SCN- a stick-shaped SCN- ion might cause deformation of a cryptand strand, which could cause a nonaxially symmetric electric field gradient. Presumably, changes in the structures due to different anions are responsible for the different QCC values observed. Axially symmetric powder patterns of most Na+(1204)2 ions with QCC of about 1.2 MHz imply that Na+(12C4)2 ions are in very similar environments in different salts. The lineshapes agree well with those expected from their structures. For example, Na+(12C4)2 has an approximate D4 symmetry with two polyether rings, each of which obeys approximate C4 symmetry [62]. Two crown ether rings form a square-antiprism arrangement [Figure 33] . The average distance between Na+ and o is 2.497 A. Sodium cations in Na+(12C4)z are relatively well shielded from anions or intercrystalline water. The distinctly nonzero asymmetry parameters of 15C5 compounds that contain Na+ indicate asymmetric electronic environments around the sodium cations. The complexant 15C5 can form 1:1 compounds with sodium salts and thus cannot completely shield the Na+ ion. Hence, anions or water molecules may interact with Na+. In addition there may be no n-fold axis (r1 > 2) around Na+ because of the asymmetry of 134 Figure 33 Square-antiprism arrangement of oxygen atoms in Na (12C4)2-C1 -5H20. The inter-oxygen distances and the crystallographic two fold axis are shown. 135 1505. Na+ ions in the cage of 18C6 have relatively large quadrupolar coupling constants except for N8718CGISCN_. The NMR spectra of Na+18C6-Cl- and Na+18C6-Br- are nearly axially symmetric while those of Na+18C6-I-, Na+1806°SCN-, and Na+18C6'TPB- are nonaxially symmetric. The structure of Na+18C6-SCN- has been determined [62]. The six ether oxygens have a highly irregular conformation in which five approximately coplanar atoms surround Na+. A water molecule and the remaining oxygen of the crown ether are bonded to form a distorted pentagonal bipyramidal coordination of the cation [Figure 34]. The distances between Na+ and 0 range from 2.321 A to 2.623 A. The water oxygen is the shortest distance away from Na+. The average distance between Na+ and O is 2.516 A. An asymmetry parameter of 0.8 for this compound is easily explained by the highly distorted coordination structure. Although Na+ is present in a highly. distorted crown ether molecule, it has a relatively small QCC. Structural asymmetry gives rise to big asymmetry parameters without having large QCC values. It seems that 1806 is too large for Na+ but that the flexibility of the crown ether molecule allows it to form different coordination structures. The 23Na NMR spectra of most compounds obtained at a Larmor frequency of 47.61 MHz show the second order quadrupolar powder pattern. It was not possible to obtain MAS POWder patterns of some 1806 compounds with about 4kHz Spinning speed at the resonance frequency of 47.61 MHz due to Figure 34 Local structure of Na+18C6-SCN-. a) View along a direction in the mean plane; b) View in a direction normal to the mean plane. 137 large quadrupolar interactions. Except for Na+1806-Cl-, Na+1806°Br-, Na+1806-I-and Na+1806-TPB-, most compounds at a 23Na Larmor frequency cf 105.842 MHz gave broad structureless lineshapes due to small quadrupolar and large dipolar interactions. At this frequency, static and MAS powder patterns for sodium 18C6 halides and tetraphenylborate were obtained. These spectra are much narrower than the spectra obtained at a resonance frequency of 47.61 MHz. This demonstrates that the quadrupolar linewidth for a second order powder pattern are inversely proportional to the external field, and hence the resonance frequency. Although. there are many’ protons, inhomogeneously broadened quadrupolar 23Na NMR lines were observed. The Gaussian broadening in Table 10 is almost half the full width at half height. Although the values of the Gaussian broadening contributions have relatively large errors, rough estimation about the extent of dipolar interactions can be obtained. In general, 0222 compounds have the largest proton dipolar interactions since the linewidths are greatly reduced by proton decoupling. The structure of Na+0222-I— is known but the coordinates for the protons are not available. However the calculated proton dipolar contribution to the static linewidth (full width at half height) of Na+ in Na+0222-Na- is 3340 Hz [17]. Excellent agreement between the observed width (3700 Hz below -20°C) and the calculated value (3670 Hz) considering both, dipolar and quadrupolar contributions was obtained. However, the static linewidth of 138 most 0222 Compounds is about 2400 Hz at room temperature.- Significant motional narrowing at room temperature might cause the linewidth reduction. A recent study of the 23Na relaxation time in Cs+(1806)2fiNa— showed that T2* increases with temperatures and changes dramatically above ~0°C [57]. The calCulated dipolar contributions from protons within a 6 A radius from Na+ in Na+(12c4)2-Cl‘-5nzo and Na+1806-SCN'-Hzo are .3800 Hz and 3950 Hz, reSpectively, using Van Vleck equations for a powder sample. Linewidths for 1204 and 1806 compounds vary from ~1000 Hz to 2000 Hz, depending on the compound. Linewidth reduction of these crown ether compounds by decoupling is much smallercthan that of 0222 compounds. The dipolar interactions in the crown compounds are weaker than those in the 0222 salts as shown by data obtained from the static proton coupled spectra. This implies that motion of the crown ring is much easier than that of a cryptand. The crown ethers might have rotational motion about an axis normal to the ring plane as well as vibrational motion. Only limited vibrational motions are allowed for the cryptands since their ether chains are bonded to the two end nitrogens. With. proton. decoupling, the residual dipolar interactions vary from ~150 Hz to ~600 Hz in static experiments. MAS studies further reduce the linewidths to about 100 ~ 350 Hz. Without proton decoupling, the NMR lines were additionally broadened by proton dipolar interactions such that the singularities of the quadrupolar powder pattern were masked. Second order quadrupolar powder patterns of complexed 139 sodium salts were discussed in this section. It is very important to examine samples both with and without proton decoupling and with both static and with MAS techniques at different fields in order to extract useful NMR parameters. V. A. 5. Lineshape Analysis of 133Cs NMR Spectra of Cesium Model Salts A variety of complexed cesium salts were very carefully reexamined in order to investigate the anisotropic interactions of Cs+ with the surrounding environment. Complexed cesium cations provide very good examples of both chemical shift anisotropy' and quadrupolar interactions in addition tx> dipolar effects. These interactions vary significantly with the coordination structures. Figures 35, 36, and 37 show the 13303 NMR spectra of three cesium 0222 salts. Cryptated cesium cations have suprisingly large quadrupolar coupling constants of about 1 MHz. No appreciable spinning sidebands in comparison with other Cs compounds and no powder patterns of satellites were observed from MAS and static experiments. Pure second order quadrupolar powder patterns were obtained from MAS exPeriments at a frequency of 23.61 MHz since MAS eliminated the CSA contribution, and different Cs+0222 compounds showed different MAS powder patterns. With proton decoupling at the same field, unusual static powder patterns were obtained because of the combination of CSA Iand quadrupolar 140 Figure 35 13303 NMR spectra of Cs+0222-I-. a) Proton coupled static, UL: 52.482 MHz; b) Proton decoupled static, UL: 23.61 MHz; c) Proton decoupled MAS, U = 23.61 MHz. L Observed ( ), Simulated ("--). 141 '1 '—'T r‘fifjfi' TU1 '1—1—‘711 ' V ‘rY—Y— JAAAL+1L1+141LILHL AALILI4L_14_A_LAJLIJ;_L+¥ zoo 250 ppm zoo ISO ..i l l 1 ° AAA—14+ LAL__L WA ALLLJJLLA LILA—AJLLAA A+LL‘- zoo 250 p p m 25 230 Figure 36 133CS NMR a) Proton b) Proton c) Proton d) Proton 142 spectra of Cs+0222-SON—. coupled static, UL: 52.482 MHz; coupled MAS, ”L: 52.482 MHz; decoupled static, UL: 23.61 MHz; decoupled MAS, UL: 23.61 MHz. Observed (———),Simulated ('-~, ----- ). PPH AAAL.ALALlAA+AlAALLIAAgLJAAAAlALAALALAAlLA d) Figure 37 144 133Cs NMR spectra of Cs+0222°TPB-. a) Proton coupled MAS, UL: 52.482 MHz; b) Proton decoupled static, UL: 23.61 MHz; c) Proton decoupled MAS, UL: 23.61 MHz. Observed (-——), Simulated (-~-). all 1' till . 1 100 Ak—A. . L no I l - 220 210 ppm 44 l ALkA l LAA AL-‘AJ ‘ 240 146 interactions. The structural changes due to different anions causes significant changes in lineshapes. At a frequency of 52.482 MHz, where quadrupolar broadening is smaller, the powder patterns primarily reflected CSA and proton dipolar coupling. A proton decoupler was not available at this frequency. Proton dipolar interactions broadened NMR lines and. hindered this study» Fortunatelyg Cs+0222-I-' showed. a single powder pattern corresponding to a single site in the structure. However two salts, Cs+0222'TPB- and Cs+0222-SCN- showed two powder patterns corresponding to two magnetically different sites. It was therefore extremely difficult to simulate static powder patterns from these samples since the two powder patterns, each broadened by CSA and quadrupolar interactions, overlap. Figure 38 shows the 13305 NMR spectra of three' sandwiched compounds, Cs+(1806)z'I-, Cs+(1806)2-TPB-, and Cs+(1806)2-SCN-. These yielded good powder patterns for all transitions, indicating small quadrupolar coupling constants. The NMR lineshape of Cs+(1806)2:I- is nearly axially symmetric, while that of Cs+(1806)2-TPB- is very asymmetric. Their local symmetries must be greatly different. In contrast, the one-to-one complexes in Cs+1806-I_, Cs+18C6-TPB-, Cs+1806-SCN_, and Cs+1806'I—-toluene showed spectra [Figures 39 and 40], which consist of a large CSA broadened central transition and quadrupolar broadened satellites that also show CSA broadening. Highly asymmetric local structures of the Cs+1806 moiety are responsible for 147 Figure 38 Proton coupled_static 133Cs NMR spectra of Cs (1806)2-TPB a), Cs+(1806)2-SCN- b), and Cs+(1806)2 I- c) at “t: 52.482MHz. Observed (-—-), Simulated (°'°°). 148 ' V 'm V fifi Ti I—fi' .. '— ' ' I. fit a .. -I _ . - 600 600‘ 200 ' '0 '200 '100' 3600 . .3800 P W" o ........ .... ...-.0 ....“" _ -_...-” .. 149 Figure 39 Proton coupled static 1330;. NMR spectr of _ Cs 1806-I a), Cs 1806-TPB b), and Cs 1806-SCN C) at UL: 52.482 MHz. a ) WWI ' w i "‘ A I lath VI ; ppm ' 4an ' b) . WW V I ' ["fi' '1 V ' 'I I "'5' U '1 """""" 'vvvv‘vvvv' --- 600 . 400 200 0 -200 "600~ - 00 .800 7 PPm 0) fi—f '. T‘ I ' ' 'VI ' ' '7‘--- ' V'J" " " ' ' I'I 1&00 , 1000 $00 .500 4000 PP" 151 Figure 40 Proton coupled static 13303 NMR spectra of Cs 18C6-1‘ Toluene at ”t: 52.482 MHz. a) Observed; b). c). d), e), f) and g) Simulated. 152 a) . r V'ryy rf'*r *' *1 I'T'IVT' 500 400 200 75 .200 -400 "m x . 0.33 MHz n e 0.14 g I! 6x): 5” 5221mm) Li b) 145 100 200 c) 100 145 200 I i d) 100 200 145 h 9 c) 145 200 100 V E r) 200 145 100 III \I , 9) 200 100 145 "n / 0 H I.\\ l ’1' I. ’*(\ ‘s \ ~--"’\ z b) ’---—" ~..'[ I \‘ / a”l \“'\"'-""~- ----r-I /\\ XI \ I ‘ /\' \~-_,'~--__ .e J \ '\ W "\ »‘__ _' ‘— a».. /\_j /\\J [fl’, (*_’,/\~ ~._____. 8) -..J L___—/\\_ rfr'rrfirr' TTrrT'VTrfirwyrri 800 600 400 200 0 -200 -400 - 00 ppm 153 these large anisotropic broadenings. The structures of two salts, Cs+1806-SCN--HZO [64] and Cs+1806-I--toluene [65] were determined and they show that the two highly asymmetric complexed cesium cations are bridged by either SCN- or I- in these compounds [Figure 41]. MAS and static spectra of Cs+(1505)2'TPB- are shown in Figure 42. The lineshape of the central transition does not correspond to a nucleus in an axially symmetric environment like that of Cs+(1505)2-I-. The isotropic chemical shift is 39.9 ppm from a MAS experiment and two principal CSA tensor elements, ~-10 ppm and ~90 ppm, can be obtained from two singularities of the static powder pattern. From this information the hidden chemical shift tensor element can be calculated as ~40 ppm by using 5i8°= 1/3 (5xx+ 5YY+‘5ZZ). Some 13303 MAS NMR spectra at slow spinning speed are shown in Figure 43. Spinning sidebands cover the spectral range of 11m: static powder patterns. This clearly indicates that all 1330s NMR spectra are inhomogeneously broadened, even in the presence of strong proton-proton interactions. The results from simulations or estimations are listed in Table 11. The range of the isotropic chemical shifts of Cs+0222 is from ~226 ppm to ~270 ppm with an average value of about 250 ppm. The chemical shifts of inclusive complexes of Cs+ with 0222 in various solutions occur at ~240 ppm._ Thus the chemical shifts of 0222 model salts are quite sensitive to the minor changes in local structures of 03+. The large 156 a) b) 1. a B 1 155 a) b) [ sv— f v—f YiV—I" fi—‘rw fifv—w I f‘ vfri ”1 Fl Ioco soc o eseo clocc -Isoo PHI Figure 42 Proton coupled_13305 NMR spectra of Cs (1505)2-TPB at ”1‘ 52.482 MHz. 8) Static; b) MAS. 156 Figure 43 133Cs MAS NMR spectra at vL = 52.482 MHz under slow spinning conditions. b) Cs+0222°TPB-, or~1 kHz; c) Cs+0222°TPB-, “.7 2 kHz; a) Cs+1806°TPB-, ”.7 1 kHz. 157 mg t5... ' If ‘r *1 ' T v I f 7 r 1 280 260 240 220 200 180 160 PPM C) r v ‘r v “T— ’V V "V I ‘r V U 1 I i V I I ' $00 0 -500 . \PPH Table 11 A summary of the 158 1 3305 NMR results xxx 0 Compound 5xx 5YY 522 SHAS QCC 0 (ppm) (MHz) Cs+(1505)2-TPB"‘-10 40 90 39.9 0.625 0.4 It Cs+18C6 I’ 130 150 240 158.4 —-- --- 0s+1806-I’-¢CH; 145 100 200 144.1 0.33 0.14 Cs+1806-SCN"‘ 35 68 110 69 --- --- Cs+1806-TPB"' -85 -51 7 -41 0.5 0 Cs+(1806)2-1“ -65 -65 -40 -52.5 0.17 0 Cs+(1866)2-SCN““—-- -—- --- -61 -—- --- Cs+(1806)2-TPB“‘-75 -45 -30 —51.1 --- —-- Cs+c222-1" 270 258 227 245.8 1. 0.8 Cs+0222-SCN" 265 265 265 265.2 1. 0.3 270 270 270 270.3 1. 0.3 03+0222-Tp8”' 225 225 225 223.3 1. 0.4 238 238 238 236.8 1. 0.4 :Earameters obtained by simulations. uEstimated parameters. Chemical shift at maximum height, v = 52.482 MHz. 1. 159 paramagnetic shifts for Cs+0222 compounds are considered to + due result from electron donation from ether oxygens to Cs to overlap of the electron wavefunctions of the cesium with nonbonding p orbital of the oxygen and nitrogen atoms of 0222. Encapsulation of 03+ in a cage that is "too small" enhances the effect. The 18C6 sandwich compounds show fairly constant lisotropic chemical shifts of -52.5 ppm, -51.1 ppm, and ~61 ppm for 1-, tetraphenylborate, and SCN— respectively. These chemical shifts are fairly close to the chemical shifts observed for Cs+ at high 1806 concentrations in various solvents. This invariance to the anionic environment and solvent implies that Cs+ is well screened by the crown ethers from interactions with the anion or solvent. The one-to-one salts, Cs+1806-I- and Cs+18CS-I--toluene have isotropic chemical shifts of about 158.4 ppm and 144.1 ppm. The isotropic chemical shifts 'of ’Cs+1806-SCN- and Cs+1806-TPB- are 69 ppm and ~41 ppm. The structure of the Cs+18CG-I--toluene compound shows that the toluene molecules are trapped in a cage of the protons of the crown ethers and that two Cs+ ions and two 1‘ ions form an ion contact square which is screened by two crowns from the toluene molecules [Figure 41]. Thus, nearly the same isotropic chemical shifts can be expected for the two iodide salts. Although the structure of Cs+1806-SCN-'HZO is similar to that of Cs+1806-I--toluene, the isotropic chemical shift of the SCN- salt is very much different from that of the I- salt. The 160 smaller paramagnetic shift of the SCN_ salt can be attributed to lower electron donicity of a SCN- group.. The most diamagnetically shifted Cs+ of Cs+1806-TPB- among one-to-one compounds may result from a diamagnetic shielding ring current effect of the phenyl groups. This ring current effect is supported by the structure of the compound [65]. The NMR study of Cs+(1505)2-I_ has been described in an earlier section. Its isotropic chemical shift of 23 ppm is very close to those' of *Cs+(1505)2'Na-, Cs+11505)2-K- and Cs+(1505)24uf- [8]. However, the isotr0pic chemical shift, 51... = 40.0 ppm, of Cs+(1505)2-TPB_ differs by 17 ppm from that of the iodide salt. It is difficult to discuss the reason for this behavior without ‘crystal structures. In general, the isotropic chemical shifts of the 1505 compounds are more paramagnetically shifted than those of the 1806 sandwich compounds. This could result from shorter mean interatomic distances. Table 12 lists mean interatomic distances between Cs+ and 0. As listed in Table 11, the C222 salts show different asymmetry parameters for different compounds which are very sensitive to the local structures of these compounds. Even if two sites in the crystal are chemically equivalent, they are magnetically nonequivalent in some compounds and have different chemical shifts. This indicates that NMR can probe Very small changes in the local structure. The principal axes for only the quadrupolar coupling tensor are labeled according to Equation (2.31) and the 161 Table 12 Mean interatomic distances between Cs+ and O Compound Distance (A) Reference 0s+(1565)2-e' 3.154 t 0.09 80 Cs+(1505)2-K' 3.102 t 0.07 6 Cs+(1505)2°I- 3.108 t 0.148 8 Cs(18C6)2-e‘ 3.352 1 0.105 8 Cs(18C6)2-Na- 3.357 i 0.111 8 0s(1806)2-0s' 3.314 i 0.202 6 030222 Cs- 2.958 t 0.07 6 Cs6222 03" 2.986 t 0.098 6 030222-90N’ 2.966 1 0.007 20 Cs0222 scu“ 2.983 i 0.065 20 ‘Cs-N interatomic distances included. 162 coincidence of both CSA and quadrupolar principal axes was assumed in order to simulate powder patterns. Even with this simplification there are six combinations between the elements of the two tensors and there is no preferential order to use for the elements of CSA tensor once the quadrupolar coupling tensor has been defined. Thus, no varies CSA 0 O 0 from zero to one but n can be zero, negative or p031t1ve if nCSA 5xx)/(5zz- 5. o). In a real spin system the (YY- 18 chemical shift tensor is fixed relative to the quadrupolar coupling tensor. In fact it is very difficult to assign tensor elements without a single crystal NMR study or simulations for the six possible cases. Six simulated spectra and a spectrum of Cs+18C6-I-°toluene are shown in Figure 40. They show exactly the same CSA powder pattern for the central transition but different powder patterns for the satellites. Only one of them (Figure 40b) is very similar to the real spectrum. The two tensors are 1 -00188 0 0 0 -0.142 0 0 0 0.33 163 for a chemical shift tensor in ppm. 'If one chooses the. following convention for both tensors: IR33- R. 180 12 IR“- R. 180 I 2 IRZZ- Risol’ the quadrupolar coupling tensor is the same as with the other convention but the chemical shift tensor becomes 100 0 0 0 145 O and appropriate Eulerian angles between the two axis systems would be required to calculate the powder pattern. In this way both no and negA can be in the range between 0 and 1 since n = (R22- R11)/(R33- Rico). However this makes it very complicated, to calculate the powder pattern. In contrast + _ either convention yielded the same tensors for Cs (15CS)2-I (section V. A. 3.). This section has shown powder patterns due to CSA, first order and second order quadrupolar effects, and a mixture of both. Isotropic chemical shifts and anisotropies were discussed with their structures. In addition the importance of the convention for choosing principal axes was described. All of the 133Cs NMR spectra observed here are inhomogeneously broadened. 164 39 V. A. 6. A study of K and 7Rb NMR spectra by the spin echo method. The spectrum of KSCN shown in Figure 44 was obtained by using the phase cycled spin echo pulse train described in the previous chapter- It has been expanded to demonstrate the second order static powder pattern. Simulation, of this spectrum yielded a QCC, x, = 0.65 MHz and an asymmetry parameter, n: 0.8. Figures 45 and 46 show the observed 39K NMR spectra of the iodide and thiocyanate salts of K+(12C4)2, K+(1505)2, K+1806, and K+0222 respectively. The sharp NMR line of K+(1204)2-I_ implies that the coordination structure of the complexed cation may be very similar to a cubic structure which has a QCC, x N 0 MHz. The NMR lines of the 1505 salts are broader than those of the 12C4 salts but don’t show any satellite powder pattern. The range of the estimated QCC is probably from 0.3 to 0.6 MHz. Although the two 1505 molecules may form a quite symmetrical coordination environment around K+, it may not be as symmetrical as that of the 1204 salt. The nonaxially symmetric environment of the corresponding Cs salt has been discussed in the previous section. However the SCN- salt of 1204 showed only a very broad line at a resonance frequency of 23.32 MHz and was not detectable at a + .— resonance frequency of 18.673 MHz. K (15CS)2°SCN had an axially symmetric second order powder pattern With QCC. X 1.75 MHz Unlike K+(1204)2-I-, the electronic environment of 165 550 050 350 250 150 45 -450' -260 -§60 -350 -§Bo "a . 39 . - — Flgure 44 K spin echo NMR spectra of K+SCN at uL = 18.67 MHz. Observed (-——), Simulated (-~-). 166 a) ‘l ' Y I v I am am 2000 I m .1 I u 1 I i O i b) T T ' 1 I - 1 am 30“ 2000 1m 0 4000 -2.00 4000 oo‘oo - so'oo 5 zo'oo , :o'oo - "in mice V noon fl -3300 - «500 Figure 45 39K spin echo NMR spectra of K+(12C4)2°I- a), x+(1505)2-1’ b), and 8*1806 I“ c) at vL = 18.67 MHz. Observed (-—-), Simulated (-~-). 167 Figure 46 .39K spin echo NMR spectra of K+(1ZC4)2-SCN- a), K+(1505)2-SCN' b), K+18CG-SCN- c), at ”L = 23.32 MHz and K+C222-SCN-d) at uL = 18 67 MHz. Observed (-—), Simulated (----). 168 8) f 1 I l l I I I I 0' T—‘ 2000 99... '200 1 fiU'UUUU‘I'.'I"'I '2000 d) V I." 0 ppm I ppm 08“ ' 0“ 169 K+(12C4)2-SCN- might be distorted by forming a staggered. sandwich compound or by the intrusion of the SCN- or water of crystallization into or between the crown rings. Surprisingly the electronic environment of K+(15C5)2-SCN- is not distorted as much as that of K+(12C4)2-SCN-. In contrast to the two classes of sandwich compounds, the complexed K+ has only a single crown ether ring and thus cannot be shielded by the l8-crown-6 molecule. This allows the potassium cation to interact with I_, SON-or protons [66]. Hence, a quite asymmetric environment can be expected. Simulation of the spectrum of K+18C6-I- yields a QCC, x = 2.5 MHz, and an asymmetry parameter, Q = 0.3. Surprisingly, very strong quadrupolar interactions have been observed with all 0222 salts. The several spectra of the C222 salts are doubtful because of the distorted lineshapes or bad signal to noise ratio. In general the results with 87Rb NMR measurements are less successful than those with 39Klrflfl2 measurements. This can be explained by the large quadrupole moment and 87 Sternheimer antishielding factor of Rb [67]. + _. Figure 47 shows the spectra of Rb 18C6-Cl and Rb+18C6-SCN-. It seems that the lineshapes are mainly The estimated QCC and the x~6MHz,n°~ Quadrupolar mixed with some CSA. asymmetry parameter of these compounds are; 0.75, x s 4.5 MHz, and no N 1, respectively. + .- Recently, the 87Rb spectrum of Rh 0222-Br [68] was observed with a VXR—3OO NMR spectrometer with 2 MHz 1 1 1 W W 1 1 1 1 r r fi— IOO. 0 ‘30. 0 400. c 200. 0 0. 0 *390. l P" L 1,11---,vvv~~ Vssnsssssvvsstss - swas,w-vs-esssnsssv 303.0 200.0 N” 0.0 -IO°.O -m P" Figure 47 e7§b spin egho NMR spectra of Rb+18C6 Cl- a) and Rb 1866-80N b) at vL = 130.93 MHz. 171 sweepwidth capability. It is shown in Figure 48, and has a linewidth of about 520 kHz. In this case the strength of the quadrupolar interaction is comparable to that of the Zeeman interaction. This explains why the 87Rb and 39K NMR signals of some Rb+ and K+ salts could not be detected by using a spectrometer with ~150 kHz sweepwidth capability. Fortunately the 39K and B7Rb spin echo NMR spectra of alkalides and electrides are much better than those of ,the model salts. Figure 49 shows the 39K NMR spectra for K+(15C5)2-e-, K+(15CS)2 K”, K+(15CS)2 Na’, and K+18CB K‘. The peak of K+(15CS)2 e‘ at 47 ppm clearly indicates that it is for K+ and this compound is a genuine electride, while K+(15C5)z-K- shows only a single sharp peak at -101 ppm, which is a proof of K-. The potassium cation in K+(15C5)2-K- may be in a very unsymmetrical environment unlike that of the corresponding electride. However, K+(1505)2-Na- has a peak at -89 ppm. This could be either K- or Ki, but this peak was assigned for the latter since 23Na NMR showed clearly an Na- NMR peak [19]. However, the possibility of a mixture of the sodide and the potasside cannot be excluded. The total linewidth of the NMR spectrum of K+18C6-K- is almost the same as that of K318C6oSCN’, indicating that it may be a KI NMR spectrum with a QCC of ~1.7 MHz and an asymmetry parameter of of dimers or chains such as in the 1. The formation ... + .. compounds, K+0222 K” [69], Rb+C222-Rb [69], and Rh 1806 Rb [59] might distort the NMR lineshape; however, the triangular . + -, Shaped peak of K+18C6-K— could result from e1ther K or K , 172 Figure 48 87Rb spin echo NMR spectra of Rb+C222-Br— at UL = 98.16 MHz. Figure 49 173 39K spin echo NMR spectra of alkalides and elec rides containing crown ethers. a) K (15C5)2-e , DL = 23.32 MHz; b) K+(15c5)2-K", ”L = 18.67 MHz; c) K+(15C5)2-Na_, uL = 18.67 MHz; d) K+1866-K'. uL = 23.32 MHz. Observed (———), Simulated ('-“). 174 I ] jfi T j I o -2000 PPM c) Lew v v—v fv rvvvv—v r . w (y r r' fit 1T I r 1 er—Vfi. 'fivv . W” 3000 2500 1000 1500 30°C 5°C '3' ~53: 4030 4350 '20“ 92506 -3330 ' [iv ‘vvvvawrvvvvwvv‘vvv'v v 'v 'vvfi'vlfiifi v ' 2000 0 -2000 175 more likely K+. The spectra shown in Figure 50 are .obtained from x+0222-e'. K+C222°K-, and K+0222°Na-, respectively. The NMR line of K+0222-Na- is an almost typical quadrupolar powder pattern with a QCC, X = 2.65 MHz and an asymmetry parameter, f1:= 0.25. However, the distorted lineshapes of 16+ for the corresponding electride and potasside have been observed. The field dependence of these spectra showed that they are mainly broadened by the second order quadrupolar interaction. The reason for the lineshape distortion is unclear. The K+ spectra for both K+0222°e- and K+0222'K- show almost the same linewidth and lineshape as expected from their similar packing structure [6]. Even though their detailed structures are different, the packing of the cations and the anions (or anionic sites) are very similar* in these compounds. The estimated QCC for K+ in both cases is ~2.7 MHz. In contrast the structure of K+C222‘Na- [6] is completely different from those of the potasside and electride. In the sodide, K+0222'Na-, the anions are isolated in the structure, whereas pairs of K- and e- are very close to one another in K+C222'K- and K+C222-e- respectively. The cationic NMR spectra indicate that the nature of the counteranion (K_ or e_) ‘is less important than the nature of the packing and that K+ in the potasside is very similar to that in the electride. The quadrupolar lineshape for K_ in K+0222°K- is consistent with the dimer structure of K- and indicates a substantial symmetry. This is the first Perturbation from spherical :::‘~.Oo .88. ". b) fi V V V j I f' 7 j—' Y'T Y Y _" j V "rj *1 V V V I 2000 1000 0 -1000 -2000 mm C) 4:00 t 3H v IOTOO “V‘— JJOO Ti 0 V 4600 Va. -2600 v .3660 v -C;00 PP! Figure 50 39K spin echo NMR spectra of alkalides and electrides containing 0222 at vL = 18.67 MHz. a) x*0222 Na’; b) K+C222 x’; c) K+C222 e”. Observed (-—-), Simulated (-°~). 177 second order quadrupolar powder pattern (QCC = 0.8 MHz and U = 0) obtained for an alkali metal anion. When the electron density distribution is distorted from spherical shape, the higher energy state can be mixed into the lower states according to the calculation of Sternheimer [70]. A "bond" between two akali metal anions may be formed by mixing the ns orbitals with low lying excited states such as (n - 1)d orbitals. Figure 51 shows the 87Rb NMR spectra of Rb+(1505)2-Rb- and Rb+(15C5)2'e-. It is difficult to extract information from the 'broad, noisy spectrum of Rb+(1505)2'e—, but an approximate QCC, x N 6 MHz is estimated. Only the Rb- peak has been observed from Rb+(1505)2-Rb—. This is similar to the result obtained from the 39 K NMR spectrum of K+(1505)2-K- and indicates that the complexed cations of both the rubidide and the potasside may have very similar environments. The 87Rb NMR spectra of two rubidium 18C6 alkalides are shown in Figure 52. The lineshape of Rb+18CS-Na-' is not characteristic of the second order quadrupolar pattern, but yields an approximate QCC, x ~ 6.5 MHz. The spectrum of Rb+18C6-Rb- at the resonance frequency of 163.6 MHz shows both Rb+ and Rb- peaks on a distorted baseline. Unfortunately, the exPeriment could not be repeated. Another synthesis of this compound did not yield good spectra at the resonance frequency of 130.9 MHz, even though many attempts were made. The fact that the Rbf spectra for both Rb+1806-Na- and Rb+1806-Rb- show similar powder patterns may indicate 1.78 ., JL r I 1 1'7 1 t 1— —r f fl 1 ' 1 «0.0 000.0 ‘00. I . a... 0. 0 ~500. O "I I f v 1 r . 1 —V I T fi I f 7' 000. 0 005.0 000.0 200.0 0.9 -3”. 0 "I Figure 51 87Rb spin echo spectra of Rb+(15C5)2'Rb- a) and 86*(1565)2~e' b) at ”t = 130.93 MHz. 179 IlwfiI1IfiTlIIITIIl—i 400 200 O -200 ~HOO PPM b) A f I '1’— 1 T T '—_T r fl 1 fi 000.0 609.0 000.0 200.2 0.0 -200.0 .0N.0 Figure 52 871215 spin echo spectra of Rb+13C6’Rb- 3’ at ”t = 163.7 MHz and 86*1806-Na' b) at uL a 130.93 MHz. 180 that their structures are very similar. An extremely broad NMR spectrum of Rb+C222-Rb_ is shown in Figure 53. The structure of this compound is known and it is isostructural with the corresponding potassium potasside, K+C222fiK-. The calculated QCC’s are about 18.6 MHz and 3.7 MHz for Rb+ and Rb- in Rb+C222-Rb- respectively. (1 + VRb)QRb] (5 2) an = XK'[ (1 + 1K)QK It was assumed that the electric field gradients for both Rb+ and Rb- in Rb+C222'Rb- are the same as those for the corresponding potassium ions in K+0222'K- and Sternheimer antishielding factors were taken into account. Table 13 is a list of the values for quadrupole moments [71] and Sternheimer antishielding factors [72] used for calculation. The observed spectrum is probably that of Rb- since the peak [of Rb+ in this compound may be extremely broadened by strong quadrupolar interactions as indicated by calculated results. A Ramsey’s paramagnetic term might shift Siso for the homonuclear rubidide as a result of the formation of a dimer. In contrast, the isotropic chemical shift (which could not be determined accurately) is quite far from the sharp Rb- peak eXpected at ~-195 ppm and a spectrum for Rb- could be broadened unexpectedly. Thus it could be assigned to the Spectra for Rb+. However it is not possible to assign the spectrum of either Rb+ or Rb- with certainty because of the 181 7 ”T T 7 F 1 400.0 200.0 0.0 '300.0 PP” I 1 1— 000.0 800.0 Figure 53 87Rb spin echo spectrum of Rb+C222-Rb- at uL = 130.93 MHz. 182 Table 13 Nuclear quadrupole moments and Sternheimer antishielding factors for K and Rb Q(x 10.28m2)a Antishielding factorb 39K+ 0.049 -18.82 39x’ 0.049 -46.35 B7Rb+ 0.13 -47.2 8786‘ 0.13 -79.05 :See Reference 71. See Reference 72. 183 Table 14 39K and 87Rb NMR parameters Compound QCC(MHZ) no 5 (ppm) KSCN‘ 0.65 0.8 10 K+(1264)2-I“‘ ~0 K+(15c5)2-I"‘ 0.3 0.6 K+(1505)2-SCN" 1.75 0 0 + —x K 1806 I 2.5 0.3 + —x K 1806 SCN. 2.1 0.5 0 Rb+18C6-01“ 6. 0.75 17 Rb+1866-SCN"‘ 4.5 1. 40 K+(1505)2-e"‘ 0.3~0.6 + —:x K (15C5)2-Na 0~0.6 + . —xx K (1505)2 K ~0 + -x K 18C6 K 1.8 1. —100 K+C222 Na" 2.65 0.25 0 K+0222 K" 0.8 0. 0 K+c222-K"‘ 2.7 ? K+0222-e‘** 2.7 2 + _ Rb (15C5)2-e " 6 3 + --x:: Rb 18C6 Na 6.5 I Itfarameters obtained by simulations. Estimated parameters. 184 large linewidth and distorted lineshape. The results with 39K and B7Rb model salts are summarized in Table 14. There are some plausible guesses for the causes of the lineshape distortion: (1) quadrupolar interaction with large CSA; (2) quality of the instrument such as the quality of an excitation pulse B1; (3) the orientation dependent relaxation time T2*; and (4) validity of the second order perturbation theory when applied to strong quadrupolar interactions. The first may be ruled out for 39K NMR because the effects are mainly quadrupolar as discussed above, but it seems that the Rb+ NMR spectra are broadened by both quadrupolar and chemical shift interactions. The second may also be eliminated since the lineshapes of other salts such as K+18C6-I— and K+C222 Na- obtained under the same experimental condition are correct. The orientation dependent relaxation times (T2 and T1), orientation dependent dipolar interactions, or distribution of orientation dependent quadrupolar coupling constants due to crystal defects could be responsible for this distortion . Large CSA effects might be a good factor for the distortion of rubidium signals. Another [factor is that the second order perturbation calculation might be invalid because the magnitude of the Quadrupolar interaction is so large. This section has shown that the spin echo technique is a very useful tool for the study of the local environments of 39 the K and 87Rb nuclei. However, an NMR spectrometer with a 2 MHz sweepwidth capability would be required for some 185 compounds. Also, pure nuclear quadrupole resonance studies of these compounds might yield more information. V. A. 7. Chemical Shifts of Alkali Metal Tetraphenylborates Unusual chemical shifts of alkali metal tetraphenylborates (M+TPB-, M = Li, Na, K, Rb, Cs) which have large anions were observed when their NMR spectra were measured as model compounds for the NMR study of alkalides and electrides. Alkali metal borohydrides were also measured in order to demonstrate that these unusual chemical shifts of tetraphenylborate salts are related to 1flne ring current of the phenyl groups. I The origin of the chemical shift is the induced magnetic moment of electrons due to the‘ orbital motions of the electrons. It is well known that the ring current of organic aromatic compounds [73] shield or deshield. protons around aromatic rings depending on the locations of the protons. Recently, the proton NMR results and structures of 2,8,17-Trithia[45’12][9]metacyclophane [74], its ‘trisulfone and [34’10][7]Metacyclophane [75] showed the effect of aromatic ring current on the chemical shifts of their methine Protons, -1.68 ppm, -2.79 ppm (2.16~2.21 A from the ring) and “4.03 ppm (1.78 A from the ring) respectively. Not many studies of the effect of ring currents have been made for nuclei other than protons. Table 15 shows the chemical shifts of simple alkali metal halides, alkali metal 186 Table 15 MAS NMR chemical shifts of alkali metal nuclei in some alkali metal salts ' Anion Li+ Na+ K+ Rb+ Cs+ (ppm) 61' -2.8 7.7 46.7 125 228 8r" --—- 5.6 54.5 179 260 1‘ ---- —2.9 ---— ---- 284 TPB- —7.3 -45‘ -92“ -175“ —261.4 BH4_ —0.4 -7.1 24.5 91.5 205.8 TPB-a —26. -26.7 -25.7 -25.5 -25.2 BHq-a —60.1 —61.7 ~57.4 -54.3 -47.6 311 B NMR chemical shifts, The chemical shifts are referenced to saturated boric acid solution. The chemical shift obtained by simulations of the static and MAS spectra. , The chemical shifts obtained by simulations of the static spectra. 187 anions, alkali metal borohydrides,' and alkali metal tetraphenylborates. The M+ chemical shifts of the halides and the borohydrides from those in their gaseous state increase with atomic number and this can be explained by Ramsey’s "paramagnetic" terms [76]. 13K: "paramagnetic" contribution depends on the energy of appropriate excited states and the degree of overlap with neighboring atoms or ions. It vanishes for electrons in s orbitals which have zero angular momentum. The calculated or observed chemical shifts of 11’9“, Mg“, and M.gas are very close together (within a few ppm) and have no "paramagnetic" shift [77]. However, in condensed phases there may be considerable contribution to a shift if low—lying excited states are available. For the same metal, the heavier salt has the more "paramagnetic" shift for M+ in M’x' (x = 01, Br, I, etc., and M = K, Rb, Cs). This can be explained by the electron donicity of the halides to the empty orbitals of'the cations. This is not the case for Na+X-; the paramagnetic contribution to the chemical shift of Na+ may be small, so that the effect of the small diamagnetic contribution from the halide ions can be seen. According to the chemical shift change of the alkali metal borohydrides, there must be a "paramagnetic" contributions due to the electron donation by BH4-, which may not therefore be considered as a hard sphere anion. The 11B chemical shifts of 9H4- in these salts support the existence of electron donation; boron nuclei are deshielded in going from the lithium salt to the cesium salt. This is very surprising 188 because there are no nonbonding electrons in BH4-. It seems that the electron donicity of BH4- is smaller than that of Cl- according to their chemical shifts and probably depends on the gap between the ground state and the excited states of the cation. If there is bonding character between M+ and phenyl rings, it will cause a paramagnetic shift rather than a diamagnetic shift. Therefore the large diamagnetic shifts observed in the alkali metal NMR of M+TPB_ in comparison with alkali metal halides and borohydrides must to be interpreted as the result of the ring current. This conclusion is strongly supported by the structure of these salts. They are very similar in structure and belong to a tetragonal space group [78,79]. For example, K+ is sitting on the four phenyl rings of the two nearest tetraphenylboron anions [Figure 54a], which form a trigonal pyramid around the K+ ion. Figure 54b shows that K+ is almost touching the n electron cloud of the phenyl rings. Presumably, the effect of the ring current can be divided into two parts, i.e. direct and indirect. The direct effect in ppm might have to be constant for each nucleus if their structures are exactly the same, although in practice they would be different slightly. On the other hand, the indirect effect should be proportional to the number of the electrons around the nucleus; if the nuclei feel local fields from phenyl rings due to the ring current, then the electrons must also feel the local fields. Hence this secondary effect might perturb the chemical shifts of alkali metal tetraphenylborates in the observed way. However, it is a) b) Figure 54 Computer generated structure of K+TPB- a) Tetrahedral arrangement of the four phenyl rings; b) Cross-section showing K+ on the phenyl rings. 190 difficult to theoretically interpret these unusal chemical shifts and the proposed explanation is only qualitative. A single sharp NMR peak was obtained from MAS experiments with M+BH4- due to its cubic structure [80]. Figure 55 shows the 7Li NMR spectra of Li+TPB-. The static spectrum is broadended by a large dipolar interaction with protons. The proton decoupled spectrum may be broadened by residual dipolar interactions or unresolved quadrupolar interactions. A MAS experiment with proton decoupling averages out most of the residual broadening, yielding a single narrow peak at -7.3 ppm. The small quadrupole moment and Sternheimer antishielding factor of 7L1 make identification of Li easy but Li+’ NMR spectra provide no further information because there are no appreciable anisotropic interactions. Figures 56, 57, and 58 show 23Na, 39K, 8"Rb, and 133C3 NMR spectra of M+TPB-. Simulation of these spectra yielded QCC and chemical shift parameters shown in Table 16. Cs and Rb have relatively large chemical shift anisotropies, while Na has only a small chemical shift anisotropy. From the 3SK NMR spectrum it was impossible to obtain information about CSA due to the noisy spectrum. It was very difficult to observe 133Cs NMR from Cs+TPB- and 39K NMR from K+TPB— because of an extremely long spin lattice relaxation time and very low sensitivity, respectively. In this study, the relaxation delays for Cs+ were 10 to 20 9 . seconds. The results from the 3 K spin echo NMR spectrum were confirmed by two different chemical shifts for a single 191 “ij \m/V will) Lilith 41-1.1ngnliFALL14LQIJQL-J-- iqlnnnll 200 H” 0 400 '3!” PPM Figure 55 7L1 NMR spectra of Li‘TpB' at ”1‘ 69.95 MHz. a) Proton coupled static;- b) Proton decoupled static; c) Proton coupled MAS; d) Proton decoupled MAS. 192 ......00' ‘-¢.....- J 14 A A__A_L l4_4 L Li A L_A__A l kl L_A_~l l ;__L L 1*; A L_J_lJ.__A_A LLLJ 3m 0 °|m '2“ PD?“ 1 l I I I 3 I 0 I : . .' o . 0 I. . l o a. '. 0' 'o I ’o 0. ~. .0 .0 . O O' " I». I I I . | I ' L ' ". -00.... no.-. ) “fzgmtggm"" ‘ l LLALIJAJ Ll LALLIUQLLLLLIJLALJLA LA LJ_LJ_ALLLA_LILJ_AL1 100 0 .lm -zoo -300 2 . 3 - Figure 56 Na NMR spectra of Na+TPB at UL: 47.61 MHz. a) Proton decoupled static; b) Proton decoupled MAS. Observed (———), Simulated (-~-).’ 193 8) ' ' v ' I ' I ' I " v~ .v aw. 400 200 o 73'? «’00 «'00 4'00 0 o' i . ' .0. i! \. 0. ' . ’[. ! . z ./ I I e 'o . A. / ': ‘. /' t .' \ /' I b ) : '\ /' I l '0’ a». .\ | 0 . 0 .‘ .1 \. ----- I \, .u./ x". 400 a #20070 vvvvv 0'0 -205.0 7" 7:400.ov -sob.o nu Figure 57 39K spin echo NMR spectrum of K+TPB- a) at.»L = 18.67 MHz and 87Rb spin echo NMR spectrum of Rb’TPB’ b) at ”t = 130.93 MHz. Observed (—-—), Simulated (-~°, -----), l9lo 0 , l o 0” 0 . g o .' 0 e .' '. -‘. O o ......oe of. ...e-DO... .0 ....... 0.... Vfi'vvv‘vtvav “hwfivhvvvijYV'fiVV'VYV—VIYfiWrVWVTI—VVfiW'r—YVY[YYVV'fiYV'W'Yfi'vww—v 000. 0 400. 0 200.0 0.0 -300. 0 -500. 0 ~700.0 400.!) PP” Figure 58 Pr ton coupled static 13308 NMR spectrum of Ca TPB- at ”1‘ 52.482 MHz. Observed (-—-), Simulated (-”’). 195 Table 16 Alkali metal NMR parameters for M+TPB- obtained by simulations 6xx= 5YY(ppm) 522(ppm) QCC (MHZ) NaTPB -40 -55 1.23 KTPB -92 -92 1.26 RbTPB' —150 —225 6.5 RthB" -160 -160 7.0 CSTPB -235 -310 0.34 :‘See Figure 57b (--- -). See Figure 57b ( °°°° ). 196 crystal of K+TPB- at two different orientations within the chemical shift range of the powder pattern. In this section, the chemical Ishifts of simple alkali metal salts and alkali metal tetraphenylborate salts were qualitatively described. Theoretical questions about highly shielded alkali metal cations remain unresolved. V. B. Magnetic Properties of Some Electrides V. B. 1. Magnetic Properties and Structures In tflua previous chapter the exchange interactions were shown to be very important in describing the magnetic behavior of antiferromagnetic or ferromagnetic samples. The behavior depends on the overlap of the wavefunctions of the electrons and decreases rapidly with the distance between electrons. Since it arises from quantum mechanical exchange of the electrons it is known as the direct exchange. It is possible that the magnetic ions (electrons in electrides) have their magnetic interaction mediated by the electrons in their common nonmagnetic neighbors. This type of magnetic interaction is called superexchange. In this case the charges of the magnetic ions do not overlap directly, but they have overlap with the same non-magnetic ion. This is known as indirect exchange, in which a magnetic interaction is mediated by interactions with the intervening electrons. Often, conduction electrons are involved. Perhaps direct and indirect exchanges are important in electrides. Therefore, 197 the environments of the trapped electrons in the edectride salts are of crucial importance Sui determining the interactions between the electrons. The structures of three electrides, Cs+(18C6)2-e_, Cs+(1505)2-e-, and K+0222-e_ have been recently solved. First, the localized electride, Cs+(1806)2°e- behaves as a Curie-Weiss paramagnet [22]. It is monoclinic with Z = 4 and the cell parameters are: a = 13.075 A, b = 15.840'A, c = 17.359 A, 8 = 92.30°, and v = 3592.3 A3 [81]. Second, Cs+(15C5)2-e- is triclinic with one molecule per unit cell [82]. The'distances between the nearest neighbor electrons are the same as the cell parameters: a = 8.60 A, b = 8.89 A, c = 9.94 A, a = 102.91°, B = 90.06°. 7 = 97.74°, and v = 733.1 A3. It behaves as 21 classical antiferromagnet [8]. Finally, the structure of K+C222°e- is monoclinic CZ/c with a = 12.129 A, b = 20.692 A, c = 21.519 A, v = 5378 A3, and z = 8 [25]. The susceptibilities and the structure of this compound indicate that there is a strong coupling between pairs of electrons with weak interactions from neighbor pairs [25]. The distance between electrons, the channels in the structures, or formation of magnetic superlattice might cause different magnetic behavior of the electrides, but the reason for the susceptibility behavior is unclear. From the macroscopic point of view, trapped electron density through the material might be related. The electron densities are 1.113 x 10'3 X3, 1.364 x 10'3 A”, 1.488 x 10'3 A'3 for Cs+(1806) -e-, Cs+(15C5) 'e—, K+C222-e- respectively. This 2 2 198 reflects the strength of magnetic interactions in these material. More structures of electrides such as K+(1505)2°e-, Rb+(1505)2-e-, and Li+PMPCY°e_, etc. may provide information in 'understanding magnetic behavior of electrides microscopically. V. B. 2. Application of the Modified Oguchi Theory of Magnetism Although there were several attempts [83] to fit the experimental magnetic susceptibility results of Li+C211-e_ over a wide range of temperatures using a Curie-Weiss equation cur Wojciechowski’s equation [84], these expression did not fit the experimental susceptibility data. Figure 59 shows the magnetic susceptibility data for Li+C211°e-. The experimental values shown in Figure 59 were provided by J. Skowyra and measured on an S.H.E. variable temperature susceptometer with a Superconducting Quantum Interference Device (SQUID). Calculated values were obtained by ‘using KINFIT [54] and ANTIMAG (Appendix C). The magnetic susceptibility of a powdered sample can be written as: Xav, = 1/3 (X‘L + X”) (5'3) Where x" and Xl are given by Equations (3 . 34) and (3 . 4 1) respectively. Practically, the susceptibility data were fit to the following equation: a) b) Figure 59 Magnetic susceptibility of Li 7.43 x 10 '0.“ I ueeepubtluy lueee’u butt, 199 0.000 -‘ 0.“ 1 0.“ - 0.0040 - 0.006 - 0.0030 - 0.” -' 0.“ - 0.” '- lfltl - m1 0.“ am- 0.01:- 0.0000 '1 0.” "‘ 0.00:- 0.“ " EN“ 0 '2 II efla *3”. 7+!!!) ++ I I I I r I fl 0 100 Temperature“) Obeerved + *9 +1: +0 +1: + B +1: ’9 a0 '9 l I l I I I N 00 100 Temperature“) Oblerved #- 0.0; b) C = "9 fl 9' 2‘ Hard. 3+ q. a a" r ".lo'lo +0211-e‘. 4 200 X I calc av Fx + C (5.4) where F and C are correction factors. The term F can account for loss of magnetization due to impurities such as M+C-M- or decomposition products. Diamagnetic impurities and the uncompensated background can be accounted for in the C term. The parameters from the KINFIT analysis [54] are listed in Table 17. Regardless of the correction factor F, average values of J/k, the Weiss temperature and a molecular field parameter are about -13.9 t 1.3 K, -20.6 t 1.5 K and 1.96 t 1.3, respectively. The parameter J’[k can be calculated as -13.6 K, -9.1 K, -6.8 K, -5.5 K, and -4.5 K for z = 2, 3, 4, 5, and 6, respectively, since the structure of this compound is not known. If the number of the nearest neighbors is 3, the strength of the interaction between pairs is appreciable compared with that of the interaction between an i-j pair. Even if z = 6, it is 1/3 of J. Therefore the molecular field is quite strong as the temperature at the maximum susceptibility indicates. A Neel temperature of 17.0 I{ was obtained from these fittings. The magnetic susceptibility of Li+C211-e— was explained by the theory of magnetism by Ohya-Nishiguchi [49]. It is a Promising theory to interpret the magnetic behavior of some other electrides. 201 Table 17 Parameters obtained by KINFIT analysis of Li Cle-e magnetic susceptibility data J/k (K) 9 (K) TN (K) x F (%) C -13.6i0.5 -19.3:0.7 15.5 1.96 62.3i3.3 7.43 X 10.4 -14.2i0.8 -22.010.8 18.5 2.09 82.4:2.4 O. 202 CHAPTER VI CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK VI. A. Conclusions Three salts, Na+0222-Br', Na+C222 Na", and Cs+(1505)2 I‘ were studied by a single crystal NMR method. Single crystal NMR studies yielded accurate values of the parameters associated with nuclear spin interactions. From 23Na and 39K NMR studies of alkalides, it was found that the NMR spectra of two alkali metal anions, Na- and K- are broadened by quadrupolar interactions. Numerous complexed alkali metal salts were investigated by static and MAS experiments with or without proton decoupling at different fields. A phase cycled spin echo technique was used to study complexed alkali metal salts that contained K and Rb. Distorted powder patterns from some K and Rb salts were obtained and further investigation is needed for these compounds. Sodium—23 NMR spectra of complexed salts showed that their powder patterns are mainly Quadrupolar in origin. Also, single crystal 23Na NMR studies 0f Na+C222-Br_ and Na+C222-Na— proved that the central transition of Na+ is shifted by second order quadrupolar interactions and that CSA is almost negligible. For the 203 heavier nuclei, more CSA was observed. Strong quadrupolar interactions for 133Cs in cesium C222 salts were observed and second order quadrupolar powder patterns were observed by MAS at a Larmor frequency of 23.61 MHz. Unusal static powder patterns result from the simultaneous presence of chemical shift and quadrupolar interactions. Static CSA powder patterns were obtained from most complexed cesium salts at higher field (uL = 52.482 MHz). Most dipolar and chemical shift interactions were averaged by spinning samples at an angle of 54.74°, leaving a single sharp peak at the isotroPic chemical shift or a second order quadrupolar powder pattern. First order quadrupolar interactions are theoretically averaged to zero; however, numerous spinning sidebands resulted, since the spinning speed is slow compared to the magnitude of the first order quadrupolar interactions. An NMR study of alkali metal tetraphenylborates showed that they are the most diamagnetically shielded alkali metal cations known to date. Inhomogeneously broadened NMR lines were observed from complexed alkali metal salts in spite of strong proton-proton interactions. In summary, the external magnetic field dependence of some interactions on NMR lines is described in Table 18. This study has proven that solid state NMR spectroscopy is a very powerful technique for the identification of. species in solids and for the (study of anisotropic interactions between a nucleus and its surrounding environment. Single crystal NMR is the best way 204 Table 18 External magnetic field effect on NMR lineshapes Interaction ppm Hz Dipolar G l/B0 independent CSA independent C 80 First order G'l/Bo independent quadrupolar Second order a 1/33 on 1/80 quadrupolar L's—l 205 to obtain the elements of anisotropic coupling tensors. Information from a static powder pattern is not enough to obtain parameters associated with anisotropic interactions. Therefore, both static and MAS (or VAS) experiments with or without proton decoupling at different fields are strongly recommended. A study of the magnetic susceptibility of Li+0211-e- demonstrated that Oguchi’s theory of magnetism as modified by Ohya-Nishiguchi is a very promising way to investigate the' magnetic susceptibilities of electrides. VI. B. Suggestions.for Future Work (1) Since quadrupolar broadened NMR spectra for alkali metal anions have been observed for three alkalides but not much besides MAS-NMR studies of other compounds have been attempted, the NMR behavior of various alkalides and electrides should be reexamined in detail. (2) More single crystal NMR studies can be done with a good single crystal NMR probe with proton decoupling and low temperature capabilities. One of the best single crystal NMR (or EPR) candidates is K+0222-Na- because of its orthorhombic structure and stability at room temperature. (3) 13C and proton NMR studies of alkalides and electrides by cross-polarization, multipulse, two dimensional NMR, etc. may provide new information. (4) Interactions other than quadrupolar interactions 206 may be separated by two dimensional NMR studies of alkali metal nuclei. 7 (5) Theoretical studies of unusal lineshapes of some alkalides and electrides such as K+0222-K7 and K+C222-e_ are challenging. (6) Single crystal magnetic susceptibility studies of electrides, K+C222-e-, Cs+(l5CS)Z-e-, and Li+PMPCY-e-, etc. will provide more information. about electron-electron interactions. APPENDICES APPENDICES APPENDIX A PROGRAM XTAL AUTHOR JINEUN KIM JAMES L. DYE* *DEPARTMENT OF CHEMISTRY MICHIGAN STATE UNIVERSITY EAST LANSING, MI 48824 U.S.A. THE PROGRAM "XTAL.FOR" COMPUTES CHEMICAL SHIFTS CORRESPONDING TO NUCLEAR MAGNETIC RESONANCE TRANSITIONS WHICH RESULT FROM THE SPIN HAMILTONIAN WITH NUCLEAR ZEEMAN, QUADRUPOLE AND CHEMICAL SHIFT TERMS. THE QUADRUPOLE AND CHEMICAL SHIFT TERMS ARE TREATED AS PERTURBATIONS ON THE ZEEMAN TERM, AND ARE CORRECT THROUGH SECOND ORDER AND FIRST ORDER RESPECTIVELY. THE SINGLE CRYSTAL NMR PEAK POSITIONS ARE CALCULATED BY SETTING THE EULERIAN ANGLES, ALPHA, BETA, AND GAMMA, WHICH ARE DEFINED WITH RESPECT TO THE GONIOMETER AXIS SYSTEM. (A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press) IF BOTH TENSORS ARE AXIALLY SYMMETRIC, THE EULERIAN ANGLE ALPHA IS MEANINGLESS. IT CAN BE AN ARBITRARY NUMBER. THE ANGLE VAN IS DEFINED AS AN ANGLE BETWEEN MAGNETIC FIELD DIRECTION AND THE ROTATION AXIS. EACH TRANSITION INTENSITY IS PROPORTIONAL TO I(I+1)-M(M-1). TOTAL TRANSITIONS TO BE CALCULATED ARE 2I*TOTAL ANGLE OF ROTATION(ANG)/ANGLE OF ROTATION(XI). THE PRINCIPAL AXIS SYSTEM OF THE QUADRUPOLAR COUPLING TENSOR (Xq,Yq,Zq) IS DEFINED SO THAT THE ABSOLUTE MAGNITUDE OF sz AND Vyy CAN BE THE LARGEST AND THE SMALLEST RESPECTIVELY. ETA IS DEFINED AS: ETA = (Vyy - Vxx)/sz. THE ELEMENTS (SSX,SSY,SSZ) OF THE CHEMICAL SHIFT TENSOR ARE CHOSEN IN THE SAME PRINCIPAL AXIS SYSTEM AS THE QUADRUPOLAR COUPLING TENSOR, SPECIFICALLY, ************************************************ * Xq // Xcs, Yq // ch, AND Zq // ch. * ************************************************ SSX, SSY, AND SSZ ARE IN PPM SCALE. THERE IS NO PREFERENTIAL ORDER-OF THE CSA TENSOR ELEMENTS. THERE ARE OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 207 ‘¥— '208 6 COMBINATIONS WITH THE QUADRUPOLAR COUPLING TENSOR. THIS PROGRAM CANNOT CALCULATE NMR TRANSITION FREQUENCIES OF THE NUCLEUS, WHICH HAS NONCOINCIDENT PRINCIPAL AXIS SYSTEMS. INPUT TITLE OF RUN 18A4 SPIN,VL,VAN,TRAN FREE FORMAT SPIN NUCLEAR SPIN VL LARMOR FREQUENCY IN MHZ VAN VARIABLE ANGLE IN DEGREE TRAN SELECTS TRANSITIONS TO BE CALCULATED EQQ,ETA,SSX,SSY,SSZ FREE FORMAT EQQ QUADRUPOLAR COUPLING CONSTANT IN MHZ ETA ASYMMETRIC PARAMETER OF ELECTRIC FIELD GRADIENT AT NUCLEAR SITE SSX,SSY,SSZ CHEMICAL SHIFT PARAMETERS IN PPM ALPA,BETA,GAMA,XI,ANG FREE FORMAT ALPA,BETA,GAMA: EULERIAN ANGLES IN DEGREE XI: ANGLE INCREMENT OF EACH ROTATION IN DEGREE(CONSTANT) ANG: MAXIMUM(TOTAL) ANGLE OF ROTATION OUTPUT XL,CHEM,DMAG,FIRS,MULTI,SECO,FNN XL XL*XI = SUM OF THE ANGLE ROTATED CHEM 'CHEMICAL SHIFT ANISOTROPY(PPM) DMAG M , FIRS FIRST ORDER SATELLITE TRANSITION(PPM) MULTI I(I+1)-M(M~1) SECO SECOND ORDER SHIFT FOR EACH TRANSITION(PPM) FNN SUM OF CSA, FIRST ORDER, AND SECOND ORDER SHIFT(PPM). . ************************************************************ XTAL.DAT (AN EXAMPLE OF DATA FILE) TEST CS(1505)2+ 3.5,23.61,90,3.5 .15,.5,60,40,-30 O,90,170,10,180 ************************************************************ COO000000000OOOOOOOOOOOOOOOOOOOOOOOOOOQOO IMPLICIT REAL*8 (A-H,O-Z) CHARACTER*20 F20,F21 DIMENSION TITLE(18),S(10) DIMENSION SGM(100),CGM(100),SZG(100),C2G(100) DIMENSION A(10),B(10),C(10),INT(10),D(10) C c******************** OPEN FILES ************************* C 7 . WRITE(*,20) READ(*,40,ERR=7) F20 OPEN(UNIT=1,FILE=F20,STATUS=’OLD’) 8 WRITE(*,30) READ(*,40,ERR=8) F21 l“ 209 OPEN(UNIT=2,FILE=F21,STATUS=’NEW’) C - . C************************ FORMAT ************************** C . 10 FORMAT(/) 20 FORMAT(’ ENTER INPUT FILE NAME’) 30 FORMAT(’ ENTER OUTPUT FILE NAME’) 40 FORMAT(A) 50 FORMAT(9H ALPHA =,F7.3,1X,8H BETA =,F7.3,1X,9H GAMMA =,F7.3, *1X,17H VARIABLE ANGLE =,F7.3) 60 FORMAT(9H SPIN =,F4.1,5X,14HLAMROR FREQ. =,F7.3, *5X,21HQUAD. COUPL. CONST. =,F7.3,5X,13HASYMM. PAR. =,F7.4) 7O FORMAT(ZQH CHEM. SHIFT PARAMETERS;SSX=,F9.3,1X,4HSSY=, *4HSSZ=,F9.3) 80 FORMAT(19H ISO. CHEM. SHIFT=,F9.3,1X,18HCHEM. SHIFT DELTA=,F9.3, - *1X,16HCHEM. SHIFT ETA=,F5.3) 9o FORMAT(1X,31HDELTA M = 1 TRANSITIONS BETWEEN,F5.1,4H AND,F5.1, *5H ONLY) 100 FORMAT(3X,’NO’,5X,’CSA’,6X,’M’,2X,’SATELLITE’,2X, ’MULTI’, *2X,’SECOND’,2X,’PEAK PPM’) 110 FORMAT(1X,F5.1,1X,F9.3,1X,F4.1,1X,F9.3,1X,I4,1X,F9.3, 1X,F9.3) 120 FORMAT(1H1) 130 FORMAT(18A4)' 140 FORMAT(ZX,18A4) 150 FORMAT(’ ENTER TITLE’) 160 FORMAT(’ ENTER SPIN (3/2 = 1.5)’) 170 FORMAT(’ LARMOR FREQUENCY IN MHZ’) 180 FORMAT(’ VARIABLE ANGLE VAN’) 190 FORMAT(’ TRANSITIONS TO BE CALCULATED’) 200 FORMAT(’ SPIN,VL,VAN,TRAN’) 210 FORMAT(’ ENTER QUADRUPOLE COUPLING CONSTANT’) 220 FORMAT(’ ASYMMETRY PARAMETER FOR QUADRUPOLE TERMS’) 230 FORMAT(’ 3 CSA TENSOR PRINCIPAL VALUES IN PPM’) 24o FORMAT(’ EQQ,ETA,SSX,SSY,SSZ’) 250 FORMAT(’ ENTER 3 EULERIAN ANGLES;ALPHA,BETA,GAMMA’) 260 FORMAT(’ ANGLE INCREMENT OF EACH ROTATION;XI’) 270 FORMAT(’ TOTAL ANGLE OF ROTATION;ANG’) 280 FORMAT(’ ALPA,BETA,GAMA,XI,ANG’) C********************************************************** 10:2 C c*********************** INPUT DATA *********************** C C WRITE(*,150) C WRITE(*,160) C WRITE(*,170) 00000000000000 210 WRITE(*,180) WRITE(*,190) WRITE(*,200) READ(*,*)SPIN,VL,VAN,TRAN WRITE(*,210) WRITE(*,220) WRITE(*,230) WRITE(*,240) READ(*,*)EQQ,ETA,SSX,SSY,SSZ WRITE(*,250) WRITE(*,260) WRITE(*,270) WRITE(*,280) READ(*,*)ALPA,BETA,GAMA,XI,ANG READ(1,130)TITLE READ(1,*)SPIN,VL,VAN,TRAN READ(1,*)EQQ.ETA,SSX,SSY,SS2 READ(1,*)ALPA,BETA,GAMA,XI,ANG C*********************************************************** C C C VARIABLE ANGLE THE=3.141593*VAN/180. SAL=SIN(THE) CAL=COS(THE) SCA=SAL*CAL SSA=SAL*SAL SSB=2.*SSA-1. CCA=(1.+CAL*CAL)/2. DCB=-SSB , TCT=(3.*CAL*CAL-1.)/2. EULERIAN ANGLE ALPHA ALP=3.141593*ALPA/90. STP=SIN(ALP) CTP=COS(ALP) EULERIAN ANGLE BETA BTA=3.141593*BETA/180. SBT=SIN(BTA) CBT=COS(BTA) CSB=CBT*SBT CCB=CBT*CBT EULERIAN ANGLE GAMMA J=ANG/XI+1 DO 300 I=1,J YI=I-l. ROT=3.141593*(GAMA+YI*XI)/180. SG=SIN(ROT) . CG=COS(ROT) CSG=2.*SG*CG CCG=2.*CG*CG-l. SGM(I)=SG CGM(I)=CG SZG(I)=CSG C2G(I)=CCG 300 350 400 405 410 420 430 440 445 C CONTINUE CSIGMA=(SSX+SSY+SSZ)/3. CDEL=Ssz-CSIGMA IF (CDEL.EQ.O.) GO TO 350 CETA=(SSY-SSX)/CDEL GO TO 400 CETA=O. CSISO=CSIGMA/1.E6 CSDEL=CDEL/1.E6 S(1)=SSX S(2)=SSY S(3)=SSZ CONTINUE DO 420 K=1,2 LA=K+1 DO 410 L=LA,3 IF(S(K).LE.S(L)) GO TO 420 SK=S(K) S(K)=S(L) S(L)=SK CONTINUE CONTINUE IF(S(2).LT.S(1)) GO TO 405 IF(ABS(S(1)—CSIGMA).LE.ABS(S(3)-CSIGMA)) GO TO 430 IF(ABS(S(1)-CSIGMA).GT.ABS(S(3)-CSIGMA)) 51:3(1) S(1)=S(3) S(3)=Sl CSDELTA=S(3)-CSIGMA IF(CSDELTA.EQ.0) GO TO 440 CSETA=(S(2)-S(1))/CSDELTA GO TO 445 CSETA=O VQ=EQQ TRAX=-TRAN SIX=SQRT(6.) c****************** OUTPUT FOR PARAMETERS ******************* C C******************* WRITE(IO,120) WRITE(IO,140) TITLE WRITE(IO,10) WRITE(IO,60) SPIN,VL,VQ,ETA WRITE(IO,10) WRITE(IO,70) SSX,SSY,SSZ WRITE(IO,10) WRITE(IO,80) CSIGMA,CSDELTA,CSETA WRITE(IO,10) WRITE(IO,50) ALPA,BETA,GAMA,VAN WRITE(IO,10) WRITE(IO,90) TRAN,TRAX WRITE(IO,10) WRITE IO 100) ( " ********x*x***************************** ETA=-ETA 450 500 550 600 650 212 CSETA=-CETA SSl=SPIN*(SPIN+1.) IF(SPIN — 0.5) 450,450,500 VQQ = 0.0 GO TO 550 VQQ=3-*VQ/(2.*SPIN*(2.*SPIN-l.))/VL FACT=VQQ*VQQ/12. XMAG=-TRAN+1. MAG=1 CONTINUE XX1=XMAG*(XMAG-1.) A(MAG)=24.*XX1-4.*SSl+9. B(MAG)=6.*XX1-2.*SSl+3. C(MAG)=—VQQ*(XMAG—.5) D(MAG)=XMAG INT(MAG)=SSl—XX1 IF(XMAG.GE.TRAN) GO TO 550 XMAG=XMAG+1. MAG=MAG+1 GO TO 600 CONTINUE MGMX=MAG ECP=ETA*CTP CON=(3.-ECP)*CSB/SIX SON=ETA*STP*SBT/SIX CTW=((3.+ECP)+(-3.+ECP)*CCB)*.5/SIX STW=ETA*STP*CBT/SIX FR=.5*(-1.+ECP+(3.-ECP)*CCB) ANI=.5*(-1.+CSETA*CTP+(3.-CSETA*CTP)*CCB) CCO=(3.-CSETA*CTP)*CSB/SIX CSO=CSETA*STP*SBT/SIX CCT=(3.+CSETA*CTP+(-3.+CSETA*CTP)*CCB)*.5/SIX CST=CSETA*STP*CBT/SIX DO 700 L=1,J DO 700 M=1,MGMX XL=L—1. CSlL=CON*CGM(L)-SON*SGM(L) CSZL=CON*SGM(L)-SON*CGM(L) CSBL=CTW*CZG(L)-STW*SZG(L) CS4L=CON*CGM(L)+SON*SGM(L) CS5L=CTW*CZG(L)+STW*SZG(L) CSSL=CTW*SZG(L)+STW*C2G(L) SAT=TCT*FR+.5*SIX*SSA*CSSL-SIX*SCA*CS4L F1R=-SCA*CS3L+DCB*CS4L+.5*SIX*SCA*FR F11=SAL*CSSL-CAL*CSZL F2R=CCA*CS3L+SCA*CS4L+.25*SIX*SSA*FR FZI=CAL*CS6L+SAL*CSZL CAN=CSDEL*(TCT*ANI+.5*SIX*SSA*(CCT*C2G(L)+CST*SZG(L)) *-SIX*SCA*(CCO*CGM(L)+CSO*SGM(L))) DMAG=D(M) - MULTI=INT(M) ‘VLCC=CSISO+CAN FST=C(M)*SAT SND=FACT*((F1R*F1R+FII*F1I)*A(M)-(F2R*F2R+FZI*FZI)* 213 B(M)) CHEM=VLCC*1.E6 FIRS=FST*1.E6 SECO=SND*1.E6 FNN=(VLCC+FST+SND)*1.E6 WRITE(IO,110) XL,CHEM,DMAG,FIRS,MULTI,SECO,FNN 700 CONTINUE STOP END 214 APPENDIX B 0000 0000000000000000000000000000000000000000000000 PROGRAM VMASS JINEUN KIM J. L. DYE* *DEPARTMENT OF CHEMISTRY MICHIGAN STATE UNIVERSITY EAST LANSING MI 48824 U. S. A. PROGRAM POWPAT QCPE PROGRAM 154 POWDER PATTERNS AND SPECTRA CORRESPONDING TO NMR TRANSITIONS ***POWPAT PROGRAM*** H.S. STORY AND D. KLINE DEPARTMENT OF PHYSICS STATE UNIVERSITY OF NEW YORK AT ALBANY POWPAT PROGRAM COMPUTES AND PLOTS POWDER PATTERNS AND SPECTRA CORRESPONDING TO NUCLEAR MAGNETIC RESONANCE TRANSITIONS WHICH RESULT FROM THE SPIN HAMILTONIAN WITH NUCLEAR ZEEMAN, QUADRUPOLE AND CHEMICAL SHIFT TERMS, SEE, FOR EXAMPLE, NARITA, UMEDA AND KUSUMOTO, J. CHEM. PHYS.44,2719 (1966). OR BAUGHER, TAYLOR, OJA AND BRAY UNPUBLISHED BROWN UNIVERSITY REPORT. THE QUADRUPOLAR AND CHEMICAL SHIFT TERMS ARE TREATED AS PERTURBATIONS ON THE ZEEMAN TERM, AND ARE CORRECT THROUGH SECOND ORDER AND FIRST ORDER RESPECTIVELY. THE POWDER PATTERN IS SIMULATED BY SETTING THE CRYSTAL AT A LARGE NUMBER OF ANGLES, THETA AND PHI ON A REGULAR GRID IN COSINE THETA SPACE. THE FREQUENCIES ARE THEN COMPUTED AND A DENSITY FUNCTION OR POWDER PATTERN CONSTRUCTED. THE POWDER PATTERN IS THEN CONVOLUTED WITH A GAUSSIAN (OR GAUSSIAN DERIVATIVE TO PRODUCE A SPECTRUM SIMULATING THAT OBTAINED FROM A WIDE-LINED SPECTROMETER (DERIVATIVE MODE)). THIS PROGRAM ASSUMES THAT ALL TRANSITIONS HAVE THE SAME LINEWIDTH. NOTE--- IF THE POWDER PATTERN EXTENDS BEYOND THE RANGE OF PPMIN OR PPMAX, THE RESULTING STEP WILL PRODUCE A RESPONSE IN THE CONVOLUTED SPECTRUM. THUS PEAKS AT THE ENDS OF THE SPECTRA ARE TO BE DISREGARDED. *** DATA ENTRY FORMAT *** - TITLE OF RUN 18A4 SPIN,XM,PS,VL,PPMIN,PPMAX,ALP,DUMMY,NORM FREE EQQ.ETA,SSX,SSY,SSZ,SIGHZ,WEIGHT,TRAN FREE DATA LIST SPIN NUCLEAR SPIN XM NUMBER OF POINTS OF SPHERE DIVISION PS NUMBER OF FREQUENCIES AT WHICH POWDER PATTERN AND.SPECTRUM WILL BE COMPUTED. VL LARMOR FREQUENCY PPMIN,PPMAX MINIMUM AND MAXIMUM CHEM. SHIFTS IN C C C C C C C C C C C C C C C C C C C C C C C . C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C 215 POWDER PATTERN AND SPECTRUM. ALP VARIABLE ANGLE DUMMY NUMBER TO ADD A BACKGROUND SPECTRUM NORM NORMALIZE INTENSITY, IF NORM = 1 EQQ NUCLEAR QUADRUPOLE COUPLING CONSTANT ETA ASYMMETRY PARAMETER OF ELECTRIC FIELD GRADIENT AT NUCLEAR SITE SSX,SSY,SSZ CHEMICAL SHIFT PARAMETERS(NOT SHELDING) SIGHZ GAUSSIAN BROADENING IN HZ WEIGHT RELATIVE WEIGHT TO BE ASSIGNED TO SITE TRAN SELECTS TRANSITIONS TO BE INCLUDED A GRID IS ESTABLISHED IN COSINE THETA, PHI SPACE WITH EACH COORDINATE HAVING XM INCREMENTS, SO THAT THE GRID IS XM*XM, IF THE QUADRUPOLAR COUPLING AND CHEMICAL SHIFT TENSORS ARE GENERAL. IF BOTH ARE AXIAL, THE PHI ROTATION IS NOT DONE. VL,PMIN, PMAX, AND EQQ ARE IN MEGAHERTZ. ETA AND WEIGHT ARE DIMENSIONLESS. THE PRINCIPAL AXIS SYSTEM OF THE QUADRUPOLAR COUPLING TENSOR (Xq, Yq, Zq) IS DEFINED SO THAT THE ABSOLUTE MAGNITUDE OF sz AND Vyy CAN BE THE LARGEST AND THE SMALLEST RESPECTIVELY. ETA IS DEFINED AS: ETA = (Vyy - Vxx)/sz. THE ELEMENTS (SSX, SSY, 882) OF THE CHEMICAL SHIFT TENSOR ARE CHOSEN IN THE SAME PRINCIPAL AXIS SYSTEM AS QUADRUPOLAR COUPLING TENSOR, SPECIFICALLY, INN“!**************************************** ** Xq // Xcs, Yq // ch, AND Zq // ch. ** ******************************************** SSX, SSY, AND 882 ARE IN PPM SCALE. THERE IS NO PREFERENTIAL ORDER OF THE CSA TENSOR ELEMENTS. IN GENERAL, THERE ARE 6 COMBINATIONS WITH THE QUADRUPOLAR COUPLING TENSOR. IF THE PRINCIPAL AXES OF THE QUADRUPOLAR COUPLING TENSOR AND THE CSA TENSOR ARE NONCOINCIDENT, THIS PROGRAM HAS TO BE MODIFIED IN ORDER TO SET EULERIAN ANGLES BETWEEN TWO PRINCIPAL AXIS SYSTEMS. SIG IS THE GAUSSIAN HALFWIDTH EXPRESSED AS A FRACTION OF THE TOTAL RANGE PMAX-PMIN. THE PATTERNS WILL INCLUDE THE INTENSITY FROM ALL ADJACENT LEVEL TRANSITIONS BETWEEN THE TRAN AND -TRAN. FOR EXAMPLE, IF TRAN IS SET EQUAL TO .5 FOR HALF ODD-INTEGRAL SPIN, THEN ONLY THE CENTRAL TRANSITION WILL BE DONE. IF TRAN IS SET T0 1.5 THEN THE CENTRAL TRANSITION PLUS TWO ADJACENT SATELLITES WILL BE INCLUDED. THE WEIGHTING FACTOR I(I+1)-M(M-1) FOR THE SATELLITE INTENSITIES IS INCLUDED. I IS THE SPIN AND M IS THE MAGNETIC QUANTUM NUMBER. THIS PROGRAM SIMULATES VARIABLE ANGLE SAMPLE SPINNING NMR SPECTRA (NOT SPINNING SIDEBANDS).IF FAST SPINNING CAN AVERAGE FIRST ORDER SATELLITES TO ZERO, THE SIMULATION PROGRAM WHICH USES AVERAGE HAMILTONIAN THEORY WILL BE APPROPRIATE (M. M. MARICQ AND J. S. WAUGH, J. CHEM. PHYS. 70, 3300(1977). STATIC SPECTRA CAN BE SIMULATED BY SETTING THE VARIABLE I‘— o()O()OC)O C)O(3C)O<1C)O(3()O<3()O(D()O(7C)O<1()O(1C)O(dC)O(3C)O<3C)O(1C)OC§()OC)C)O()6)O 216 ANGLE ALP = 0. * DUMMY = FACTOR TO DETERMINE IF SIMULATION Is TO BE NORMAL(ANY NUMBER WHICH IS NOT EQUAL TO 2) OR IF SIMULATION IS TO ADD IN A BACKGROUND SPECTRUM FROM A FILE(SPECTRUM.DAT). THE VL,PPMIN,PPMAX, AND PS MUST BE THE SAME TO ADD. THE RELATIVE RATIOS BETWEEN THE SITES SHOULD BE USED. NORM SHOULD BE 0. AN ARRAY P(K) STORES FREQUENCIES. THE UNIT 2 OUTPUT FILE ’ F21 ’ KEEPS PARAMETERS. THE UNIT 1 OUTPUT FILE ’ F22 ’ HAS CHEMICAL SHIFTS AND CORRESPONDING FREQUENCIES. THE UNIT 4 FILE IS A DATA FILE TO EXECUTE THIS PROGRAM. VMASSI.DAT *TESTI *3.5,23.61,1000,1000,-1000,1000,0,0,0 *0.1,0.1,-12,-12,24,100,1,3.5 VMASSZ.DAT *TESTZ *1.5,47.61,360,100,-100,100,0,0,0 *1,0,0,0,0,200,1,.5 VMASSS.DAT *TEST3 *105,470613360,100,-1003100’54o7356,0,0 *1.3,0,10,10,10,100,1,.5 VMASS4.DAT *TEST4 *3.5,23.61,360,100,-50,50,0,0,0 *0,0,35,15,-30,50,1,.5 VMASS5.DAT *TESTS *1.5,47.61,360,100,-100,100,0,0,0 *1.3,0,10,10,10,150,1,.5 VMASS6.DAT *TEST6 *1.5,47.61,360,100,-100,100,0,2,0 *1.3,0,10,10,10,150,25,.5 1.THE RESULT OF TESTl IS A Cs SPECTRUM AT 180 MHz WHICH SHOWS 7 TRANSITIONS AND CHEMICAL SHIFT ANISOTROPY(1 hr. at IBM PC XT WITH MATH. COPROCESSOR). 2.THE TESTZ YIELDS CENTRAL TRANSITION OF Na AT 180 MHz WHICH IS PURE QUADRUPOLAR SECOND ORDER PATTERN (WITHIN 1 min. at IBM PC XT WITH MATH. COPROCESSOR). 3.THE TEST3 DEMONSTRATES A MAS SPECTRUM OF TEST5 WITH ISOTROPIC CHEMICAL SHIFT 10 PPM. 4.THE TEST4 SIMULATES PURE CHEMICAL SHIFT ANISOTROPY POWDER PATTERN. 5.THE TESTS YIELDS SECOND ORDER QUADRUPOLE POWDER 217 PATTERN WITH ISOTROPIC CHEMICAL SHIFT 10PPM. 6. IF THIS COMPOUND HAS 80% TESTZ SITES AND 20% TEST5 SITES THE SUM OF THE SPECTRA CAN BE GENERATED BY USING VMASSG.DAT AND SPECTRUM.DAT WHICH IS RENAMED FROM THE OUTPUT OF VMASSZ.DAT. DIMENSIONS OF PARAMETERS WHICH AGREE WITH EACH OTHER ARE COUPLED AND SHOULD BE RESET ACCORDINGLY, IF MODIFICATION IS NECESSARY. IMPLICIT REAL*8 (A- -H, 0- -Z) CHARACTER*20 F20, F21, F22 DIMENSION TITLE(18),G(1000),GT(1000),KG(1000), ST(1000) DIMENSION COD(1000), CXD(1000), COSTP(1000) DIMENSION HT(1000),P(1000),PPM(1000),INT(10) DIMENSION B01(10),B02(10),BO3(10),BO4(10) DIMENSION B05(10),BO6(10),BO7(10),S(10) 0000000000 OPEN FILES- 4000 WRITE(*,76) READ(*,78,ERR=7) F20 OPEN(UNIT=4,FILE=F20,STATUS=’OLD’) 8 WRITE(*,72) READ(*,78,ERR=8) F21 , OPEN(UNIT=1,FILE=F21,STATUS=’NEW’) 9 WRITE(*,74) . READ(*,78,ERR=9) F22 OPEN(UNIT=2,FILE=F22,STATUS=’NEW’) C c************************ FORMAT ************************* C 12 FORMAT(/) 14 FORMAT(9H SPIN =,F4.1,5X,21HQUAD. COUPL. CONST. =, F7.3 *,5X,13HLARMOR FRQ. =,F7.3,5X,13HASYMM. PAR. =,F7.4) 16 FORMAT(16H SPHERE DIV. =,F5.0,1X,26HNO. POINTS TO BE PLOTTED =, *F5.0,1X,16HMIN CHEM SHIFT =,F8.2,1X,15HMAX CHEM SHIFT =,F8.2) 18 FORMAT(28H CHEM. SHIFT PARAMETERS;SSX=,F7.2,1X,4HSSY=,F7.2,1X, *4HSSZ=,F7.2) 20FORMAT(’ ISO CHEM SHIFT=’,F7.2,1X,’CHEM SHIFT DELTA=’,F8.3,1X, *’CHEM SHIFT ETA=’,F7.2) 22 FORMAT(zzHGAUSSIAN BROADENNING =,F7.5,1X,20HGAUSSIAN LINEWIDTH =, *FlO. 5, 4H KHZ) 24FORMAT(IX, 31HDELTA M = 1 TRANSITIONS BETWEEN,F5.1,4H AND, F5. 1, *5H ONLY) 26 ’FORMAT(1H1) 28 FORMAT(18A4) 30 FORMAT(ZX,18A4) l‘ 218 32‘ FORMAT(ZX,17H VARIABLE ANGLE =,F10.7) 34 FORMAT(ZX, 36 FORMAT(ZX, 38 FORMAT(8X, 40 FORMAT(8X, 42 FORMAT(8X, 44 FORMAT(8X, 46 FORMAT(8X, 48 FORMAT(8X, ’ENTER TITLE’) ’ENTER SPIN (FOR EXAMPLE I=7/2--->3.5)’) ’LARMOR FREQUENCY IN MHZ’) ’NUMBER OF POINTS FOR ANGLES (<1001)’) ’NUMBER OF POINTS FOR PLOT (<1001)’) ’PLOT MINIMUM IN PPM (FOR EXAMPLE -100)’) ’PLOT MAXIMUM IN PPM (FOR EXAMPLE 100)’) ’VARIABLE ANGLE (54.7356 FOR MASS, 0 FOR STATIC)’) 50 FORMAT(8X,’DUMMY (0 FOR NORM. SIM., 2 TO ADD A SPECTRUM)’) 52 FORMAT(36H SPIN,VL,XM,PS,PPMIN,PPMAX,ALP,DUMMY) 54 FORMAT(ZX,’ENTER QUADRUPOLE COUPLING CONSTANT IN MHZ’) 56 FORMAT(8X,’ASYMMETRY PARAMETER OF EFG’) 58 FORMAT(8X,’3 CSA TENSOR PRINCIPAL COMPONENTS IN PPM’) 60 FORMAT(8X,’GAUSSIAN BROADENING IN HZ’) 62 FORMAT(8X,’RELATIVE INTENSITY’) 64 FORMAT(8X,’TRANSITIONS TO BE CALCULATED(.5 FOR CENTRAL *TRANSITION)’) 66 FORMAT(38H EQQ,ETA,SSX,SSY,ssz,SIGHz,WEIGHT,TRAN) 68 FORMAT(2E12.4) 70 FORMAT(’*** THE SPECTRUM.DAT FILE HAS BEEN ADDED. * ADD THE PARAMETER FILE ***’) 72 FORMAT(’ ENTER OUTPUT FILE NAME’) 74 FORMAT(’ ENTER OUTPUT PARAMETER FILE NAME’) 76 FORMAT(’ ENTER INPUT PARAMETER FILE NAME’) 78 FORMAT(A) 80 FORMAT(’ WEIGHT = ’,1F10.6) 82 FORMAT(8X,’1 FOR NORMALIZATION’) IO=2 **************** INPUT PARAMETER ****************** WRITE(*,34) READ(*,28)TITLE WRITE(*,36) WRITE(*,38) WRITE(*,40) WRITE(*,42) WRITE(*,44) WRITE(*,46) WRITE(*,48) WRITE(*,50) WRITE(*,53; WRITE * 8 READ(£,S)SPIN,VL,XM,PS,PPMIN,PPMAX,ALP,DUMMY,NORM WRITE(*,54) WRITE(*,56) WRITE(*,58) WRITE(*,60) WRITE(*,62) WRITE(*,64) WRITE(*,66) 00000000000000000000000 l‘- C C************************************************************ 100 110 120 130 131 I READ(4,*)SPIN,VL,XM,PS,PPMIN,PPMAX,ALP,DUMMY,NORM 219 READ(*,*)EQQ,ETA,SSX SSY SSZ SIGHZ WEIGHT TRAN READ(4,28)TITLE ’ ’ ’ ’ ’ READ(4,*)EQQ,ETA,SSX,SSY,SSZ,SIGHZ,WEIGHT,TRAN IF(XM.GT.1000.) XM=1000. IF(PS.GT.1000.) PS=1000. IF(XM.LT.1.) XM=360. IF(PS.LT.1.) PS=100. M=XM NS=PS PMIN=VL+(PPMIN*VL/1.0E6) PMAX=VL+(PPMAX*VL/1.0E6) PR=PMAX-PMIN GIP=(PS-1.)/PR ANG=3.141593*ALP/180. CAL=COS(ANG) CXC=CAL*CAL CTC=CXC*CXC THC=3.*CXC-1. TTC=.25*THC DO 100 I=1,M XI=I U=(XI-.5)/XM UXU=U*U COD(I)=UXU CXD(I)=UXU*UXU TPHI=3.141593*(XI-.5)/XM COSTP(I)=COS(TPHI) CONTINUE . D0 110 K=1,NS G(K)=0. GT(X)=0. KG(K)=0. ST(K)=0. XK=K P(K)=PMIN+(XK-1.)*PR/(PS-1.) ppM(K)=(P(K)-VL)*1.0E6/VL CONTINUE SIGKC=SIGHZ/1000. SIG=SIGHZ/PR*1.E-6 CSIGMA=(SSX+SSY+Ssz)/3. CSDEL=ssz-CSIGMA IF(CSDEL.EQ.O.) GO TO 120 CETA=(SSY-SSX)/CSDEL GO TO 130 CETA=0. CISO=CSIGMA/1.E6 CDEL=CSDEL/1.E6 S(1)=SSX S(2)=SSY s(3)=ssz CONTINUE D0 134 K=192 132 134 136 138 139 C 220 LA=K+1 DO 132 L=LA,3 IF(S(K).LE.S(L)) GO TO 132 SK=S(K) S(K)=S(L) S(L)=SK CONTINUE CONTINUE ' IF(S(2).LT.S(1)) GO TO 131 IF(ABS(S(1)-CSIGMA).LE.ABS(S(3)-CSIGMA)) GO TO 136 IF(ABS(S(1)-CSIGMA).GT.ABS(S(3)-CSIGMA)) SI=S(1) S(1)=S(3) S(3)=Sl 'CSDELTA=S(3)-CSIGMA IF(CSDELTA.EQ.O) GO TO 138 CSETA=(S(2)-S(1))/CSDELTA GO TO 139 CSETA=O VQ=EQQ TRAX=-TRAN c************** OUTPUT FOR INPUT PARAMETER ***************** C C************************** 140 150 160 WRITE(IO,26) WRITE(IO,30)TITLE WRITE(IO,12) WRITE(IO,16)XM,PS,PPMIN,PPMAX WRITE(IO,12) WRITE(IO,14)SPIN,VQ,VL,ETA WRITE(IO,12) . WRITE(IO,18)SSX,SSY,SSZ WRITE(IO,20)CSIGMA,CSDELTA,CSETA WRITE(IO,12) WRITE(IO,24)TRAN,TRAX WRITE(IO,12) WRITE(IO,32)ALP WRITE(IO,12) WRITE(IO,22)SIG,SIGKC WRITE(IO,12) WRITE(IO,80)WEIGHT ********************************* CSETA=-CETA ETA=—ETA EE=ETA*ETA EE6=EE/6. SSl=SPIN*(SPIN+1.) IF(SPIN-0.5) 140,140,150 VQQ=0.0 GO TO 160 VQQ=3.*VQ/(2.*SPIN*(2.*SPIN-1.)) FACT=VQQ*VQQ/(12o*VL) VLC=VL*CDEL VCC=VLC*CSETA VIS=VL*(1.+CISO—.25*THC*CDEL) 170 221 FATA=FACT*ETA XMAG=-TRAN+1. MAG=1 CONTINUE XX1=XMAG*(XMAG-1.) A=24.*XX1-4.*SSI+9. B=6.*XX1-2.*SSl+3. C=-VQQ*(XMAG-.5) VMA=.5*A*(1.-CTC)-.125*B*(1.+6.*CXC+CTC) VMB=.5*A*(1.-3.*CXC+4.*CTC)-.5*B*(1.TCTC) VMC=1.5*A*(CXC-CTC)-.375*B*(1.-2.*CXC+CTC) VCEC=VCC+ETA*C VLCC=VLC+C BOl(MAG)=-TTC*C+FACT*(.375*VMA+EE6*VMB+.25*VMC) BOZ(MAG)=TTC*VCEC+FATA*(.25*VMA-.5*VMC) BO3(MAG)=FACT*EE*(VMA/24.-VMB/6.+.25*VMC) BO4(MAG)=3.*TTC*VLCC+FACT*((EE6-.75)*VMA+(1.5-EE6)*VMB- 1.5*VMC) 180 190 200 210 B05(MAG)=-TTC*VCEC+FATA*(-VMB+2.*VMC) B06(MAG)=FACT*(.375*VMA-1.5*VMB+2.25*VMC) BO7(MAG)=FATA*(4.25*VMA+VMB-1.5*VMC) INT(MAG)=SSl-XX1 IF(XMAG.GE.TRAN) GO TO 190 XMAG=XMAG+1 MAG=MAG+1 GO TO 170 CONTINUE MGMX=MAG MM=M IF(SSX.EQ.SSY.AND.ETA.EQ.O.) MM=1 AMP=WEIGHT/MM DO 200 J=1,MM CTP=COSTP(J) CXP=CTP*CTP DO 200 MAG=1,MGMX BOC=B03(MAG)*CXP EUN1=VIS+B01(MAG)+B02(MAG)*CTP+BOC EUN2=BO4(MAG)+B05(MAG)*CTP-2.*BOC EUN3=BO6(MAG)+BO7(MAG)*CTP+BOC MULTI=INT(MAG) DO 200 I=1,M FNN=EUN1+EUN2*COD(I)+EUN3*CXD(I)-PMIN NN=FNN*GIP+1.5 IF(NN.LT.2.0R.NN.GT.NS) GO TO 200 KG(NN)=KG(NN)+MULTI CONTINUE DO 210 LL=1,NS GTLL=KG(LL)*AMP GT(LL)=GTLL CONTINUE - CALL SHIFT(NS,GT) ~CALL CONVOL(NS,SIG,GT,O,ST) IF(DUMMY.NE.2) GO TO 230 ’ OPEN(UNIT=3,FILE=’SPECTRUM.DAT’,STATUS=’OLD ) 220 230 222 READ(3,68)(PPM(K),HT(K),K=1,PS) WRITE(*,70) DO 220 N=1,PS ST(N)=ST(N)+HT(N) CONTINUE CONTINUE IF(NORM.NE.1) GO TO 240 CALL UNITY(NS,ST) C********************** OUTPUT *************************** 240 00 0000000 26 29 27 28 20 23 24 22 30 33 WRITE(1,68)(PPM(K),ST(K),K=1,PS) STOP END SUBROUTINE CONVOL(NS,SIGMA,G,NDERIV,SA) CONVOL DOES A CONVOLUTION OF THE NTH GAUSSIAN DERIVATIVE WITH AN ARBITRARY FUNCTION. N IS 0,1,2, OR 3. ENTRY IS NS POINTS, SIGMA THE GAUSSIAN HALF-WIDTH, G CONTAINING THE FUNCTION TO BE CONVOLUTED AND NDERIV THE VALUE OF N. THE OUTPUT IS IN SA. IMPLICIT REAL*8 (A-H,O-Z) DIMENSION G(1000),SA(1000),XSTORE(1000) IF(SIGMA.LE.0.) RETURN RX=1./(2.*(SIGMA*NS)**2) SQRX=RX**.5 DO 28 L=1,NS SA(L)=0. XL=L-1 POW=XL*XL*RX . IF(POW—30.) 26,27,27 XSTORE(L)=EXP(-POW) GO TO 28 XSTORE(L)=O. CONTINUE IF(NDERIV.LT.0.0R.NDERIV.GT.3) NDERIV=1 MOVE=NDERIV+1 GO TO (20,30,40,50) MOVE CONTINUE DO 22 I=1,NS DO 22 J=1,NS XXX=I-J IF(XXX) 23,24,24 L=J-I+1 GO TO 22 L=I-J+1 SA(J)=SA(J)+G(I)*XSTORE(L) RETURN CONTINUE DO 32 I=1,NS DO 32 J=1,NS XXX=I-J IF(XXX) 33.34.34 L=J-I+1 34 32 40 43 44 42 50 53 54 52 00 00000 223 GO TO 32 L=I-J+1 SA(J)=SA(J)+G(I)*XSTORE(L)*XXX*SQRX RETURN CONTINUE DO 42 I=1,NS DO 42 J=1,NS XXX=I—J IF(XXX) 43,44,44 L=J—I+1 GO TO 42 L=I-J+1 SA(J)=SA(J)+G(I)*XSTORE(L)*(1.-2.*XXX*XXX*RX) RETURN CONTINUE DO 52 I=1,NS DO 52 J=1,NS XXX=I—J IF(XXX) 53,54,54 L=J—I+1 GO TO 52 L=I-J+1 SA(J)=SA(J)-G(I)*XSTORE(L)*(—6.*XXX*SQRX *+4.*XXX*XXX*XXX*RX*SQRX) RETURN END SUBROUTINE SHIFT(N,G) SHIFT SUBTRACTS A FIXED AMOUNT FROM THE POWDER PATTERN TO MINIMIZE THE EFFECTS OF THE CONVOLUTION AT THE ENDS OF THE SIMULATED SPECTRUM. IMPLICIT REAL*8 (A-H,O-Z) DIMENSION G(1000) GMIN=G(1) DO 1 I=2,N IF(GMIN.LE.G(I)) GO TO 1 GMIN=G(1) CONTINUE DO 2 I=1,N G(I)=G(I)-GMIN CONTINUE RETURN END SUBROUTINE UNITY(N,ST) UNITY NORMALIZES INTENSITY AFTER CONVOLUTION (MAXIMUM PEAK HEIGHT = 1.). IMPLICIT REAL*8 (A-H,O-Z) DIMENSION ST(1000) STMAX=ST(1) DO 1 I=1,N IF(STMAX.GE.ST(I)) GO TO 1 224 STMAX=ST(I) CONTINUE DO 2 I=1,N ST(I)=ST(I)/STMAX CONTINUE RETURN END 225 APENDIX C 00000000000000000000000000000000000000000000000000 PROGRAM ANTIMAG JINEUN KIM JAMES L. DYE* *DEPARTMENT OF CHEMISTRY MICHIGAN STATE UNIVERSITY EAST LANSING MI 48824 U.S.A. THIS IS THE PROGRAM TO FIND THE ROOTS OF THE ZEROTH ORDER SUSCEPTIBILITIES AND CALCULATE THE MAGNETIC SUSCEPTIBILITIES. REF;BULL.CHEM.SOC.JPN. V52 P3480 1979, PROG.THEOR.PHYS. V13 P148 1955 NUMERICAL RECIPES;THE ART OF SCIENTIFIC COMPUTING, WILLIAM PRESS ET.AL. CAMBRIDGE UNIV. PRESS, CHAPTER 9. CALC=THEORETICAL SUSCEPTIBILITY SUS,SLJ,SRJ;;SUSCEPTIBILITIES SRED,SRLJ,SRRJ:: REDUCED SUSCEPTIBILITIES. TR=REDUCED TEMPERATURE MOLECULAR FIELD PARAMETER AK=(Z-1)*J’/J INPUT _ ENTER 444. IN THE ANTIMAG.TEM;FOR THE NEEL TEMPERATURE ENTER 999. IN THE ANTIMAG.TEM;FOR PROGRAM STOP ANTIMAG.TEM (AN EXAMPLE INPUT FILE) 2 4 6 8 10 13 16 20 25 30 35 40 45 50 60 70 80 90 100 120 140 160 180 200 250 300 C)0(3C)0(3C)0(7 226 444 999 THE INTERACTIVE INPUTS ARE AJ,WT,EFF. (AN EXAMPLE) COUPLING CONSTANT AJ=-J/k (14.2) FREE FORMAT WEISS TEMPERATURE WT (-22 ) FREE FORMAT EFFECTIVE SPIN NUMBER(%) EFF (82.5) FREE FORMAT IMPLICIT REAL*8 (A-H,O-Z) CHARACTER*20 F20,F21 WRITE(*,IO) READ(*,30,ERR=7) F20 OPEN (UNIT=1,FILE=F20,STATUS=’OLD’) WRITE(*,20) READ(*,30,ERR=8) F21 OPEN (UNIT=2,FILE=F21,STATUS=’NEW’) c************************ FORMAT **************************** 10 20 30 40 50 60 70 80 90 100 110 TEM, 120 130 FORMAT(’ ENTER INPUT FILE NAME : TEMPERATURE’) FORMAT(’ ENTER OUTPUT FILE NAME’) FORMAT(A) FORMAT(29H COUPLING CONSTANT -J/k = AJ=) FORMAT(19H WEISS TEMPERATURE=) FORMAT(ZOH WEISS TEMPERATURE=,F10.5,2H K) FORMAT(25H EFFECTIVE SPIN #(%) EFF=) FORMAT(27H COUPLING CONSTANT AJ=-J/k,F10.5,2H K) FORMAT(31H MOLECULAR FIELD PARAMETER AK=,F10.5) FORMAT(23H EFFECTIVE SPIN # EFF=,F10.5,2H %) FORMAT(5X,3H TR,6X,5H SRED,8X,5H SRLJ,6X,5H SRRJ,6X,4H *4X,5H CALC,5X,4H SLJ,4X,4H SRJ,8X,2H R,6X,4H SUS) FORMAT(1X,F9.3,1X,E10.4,1X,E10.4,1X,E10.4,1X,F10.4, *1X,F8.6,1X,F8.6,1X,F8.6,1X,F10.5,1X,E10.4) FORMAT(45X,7H T NEEL) C************************************************************ 200 NN=100 WRITE(*,40) READ(*,*)AJ WRITE(*,50) READ(*,*)WT WRITE(2,60) WT WRITE(*,70) READ(*,*)EFF WRITE (2,80) AJ AK=-2.*WT/AJ-1. WRITE (2,90) AK WRITE (2,100) EFF WRITE (2,110) READ (1,*) TEM IF ((TEM.EQ.444.).AND.(AK.GT.1.)) GO TO 444 IF (TEM.GE.999.) GO TO 999 AKB=1.38054E-23 AJJ=AJ*AKB BJ=-AJ/TEM 300 350 444 400 450 500 550 999 227 TR=-1./BJ IF (AK.LE.1.) GO TO 400 BN=LOG((AK-1.)/(AK+3.))/2 IF (BJ.GE.BN) GO TO 400 ABA=ABS(BJ*AK) XI (LOG(ABA)+ABA.GT.85.) GO TO 300 X2=AK+1.E-6 CALL 2BRAX(FM,BJ,AK,X1,X2,NN,XB1,XB2) XACC=1.E-10*(XBl+XBZ)/2 ROOT=RTSAFE(BJ,AK,XB1,XB2,XACC) RJ=ROOT IF (ABS(BJ*(1.+RJ)).GT.85.) GO TO 300 EBJ=EXP(-BJ) CBR=COSH(BJ*RJ) CHA=1.+EBJ*CBR CHB=(1.-RJ)**2*EBJ*CBR+2*RJ-(RJ**2+1.)*EBJ*EXP(BJ*RJ) CAB=-AK*CHB+RJ*(1.-RJ**2)*CHA SRLJ=AJ/2./TEM/(CHA-AK*BJ) SRRJ=-CHB/2./CAB GO TO 350 SRLJ=0. SRRJ=.25*(AK-1.)/AK**2 SRED=(SRLJ+2.*SRRJ)/3. CALC=2.07616E—23/AJJ*SRED SLJ=2.07616E-23/AJJ*SRLJ SRJ=2.07616E—23/AJJ*SRRJ SUS=CALC*EFF/100. GO TO 550 BJ=BN TR=-1./BJ TEM=-AJ/BJ WRITE (2,130) GO TO 400 RJ=1 IF (ABS(2.*BJ).GT.87.) GO TO 450 CHC=1.-BJ*AK+EXP(-BJ)*COSH(BJ) SRED=AJ/2./TEM/CHC GO TO 500 SRED=0. SRLJ=SRED SRRJ=SRED CALC=2.07616E-23/AJJ*SRED SLJ=CALC SRJ=CALC SUS=CALC*EFF/100. GO TO 550 WRITE (2,120)TR,SRED,SRLJ,SRRJ,TEM,CALC,SLJ,SRJ,RJ,SUS GO TO 200 STOP END FUNCTION ZESUS(BJ,AK,X) 10 20 228 IMPLICIT REAL*8 (A-H,O-Z) ZESUS=X*EXP(BJ)+X*COSH(BJ*X)+AK*SINH(BJ*X) RETURN ' END SUBROUTINE DFUNC(BJ,AK,X,FM,DF) IMPLICIT REAL*8 (A-H,o-Z) FM=ZESUS(BJ,AK,X) DZESUS=EXP(BJ)+(1.+BJ*AK)*COSH(BJ*X)+BJ*X*SINH(BJ*X) DF=DZESUS RETURN END SUBROUTINE ZBRAK(FX,BJ,AK,X1,X2,NN,XB1,XB2) IMPLICIT REAL*8 (A-H,O-Z) X=X1 DX=(X2-X1)/NN FP=ZESUS(BJ,AK,X) DO 10 I=1,100 X=X+DX FX=ZESUS(BJ,AK,X) IF (FX/ABS(FX)*FP/ABS(FP).GT.0.) GO TO 7 XB1=X-DX XB2=X RETURN FP=FX CONTINUE RETURN END FUNCTION RTSAFE(BJ,AK,X1,X2,XACC) IMPLICIT REAL*8 (A-H,O-Z) MAXIT=100 CALL DFUNC(BJ,AK,X1,FL,DF) CALL DFUNC(BJ,AK,X2,FH,DF) IF (FL.GE.0.) GO TO 10 XL=X1 XH=X2 GO TO 20 XH=X1 XL=X2 SWAP=FL FL=FH FH=SWAP DO 60 J=1,MAXIT RTSAFE=0.5*(XL+XH) DXOLD=ABS(XH-XL) DX=DXOLD CALL DFUNC(BJ,AK,RTSAFE,F,DF) DFF=(DF-F)/ABS(DF-F). 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