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Vibrational spectra of the ammonium halides and the alkali-metal borohydrides McQuaker, Neil Robert 1970

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VIBRATIONAL SPECTRA OF THE AMMONIUM HALIDES AND THE ALKALI-METAL BOROHYDRIDES hy NEIL ROBERT McQUAKER B.Sc. (Hons.)> University of B r i t i s h Columbia, 1965 M.Sc, University of B r i t i s h Columbia, 1966 • A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF. THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of . . CHEMISTRY accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1970 In presenting th is thesis in par t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f r ee l y ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes is for scho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date ^ OCT 7 Q i i ABSTRACT Using both infrared and Raman techniques the v i b r a t i o n a l spectra of selected polymorphs of the ammonium halides; NH4C1, ND4C1, NH^Br, ND^r, NH^I and ND 4I, have been recorded i n the spectral region 4000 - 50 cm - 1. The Raman spectra of NH^F and ND4F were also recorded. Although a l l the foregoing crystals have been the subject of previous spectroscopic invest-igations some important new features are observed. These include ( i ) Raman active longitudinal modes and ( i i ) previously unreported internal and ex-ternal modes. In addition more meaningful assignments are made for some previously reported spectral l i n e s . Included are assignments involving (i) the non-degenerate NH 4 + stretching mode i n combination with acoustical modes and ( i i ) the transverse and longitudinal components of the t r i p l y degenerate NH 4 + bending mode i n combination with the l i b r a t i o n a l mode. A study analagous to that involving the ammonium halides was made of the following alkali-metal borohydrides: LiBH 4, LiBD 4, NaBH4> NaBD4, KBH4, KBD4, RbBH4 and CsBH4. From the v i b r a t i o n a l spectrum of potassium borohydride recorded at 10°K i t i s possible to in f e r that the ordered phase associated with this s a l t has a cubic structure compatible with the 2 space group T^ . The vi b r a t i o n a l spectrum of the ordered tetragonal phase of sodium borohydride allows the placing of the seven BH4 (E^j) internal v i b r a t i o n a l modes. In addition a mode of translatory o r i g i n i s observed and a mode of rotatory o r i g i n i s inferred from a l i n e assigned as the second overtone of a l i b r a t i o n a l mode. In the case of lithium borohydride seven of the nine BH ~(C ) modes are observed. Six modes of translatory o r i g i n i i i appear i n the infrared and a mode of rotatory o r i g i n i s inferred from a l i n e assigned as the second overtone of a l i b r a t i o n a l mode. The structure of potassium borohydride at room temperature i s discussed and evidence i n support of a structure intermediate between the ordered 2 5 cubic T_^  phase and the disordered cubic 0^ phase of sodium borohydride i s given. F i n a l l y , the F matrix associated with an undistorted XY^(T^) (XY^ = NH^+, BH^ ) ion and the G matrix associated with the same ion which has undergone a s l i g h t angular d i s t o r t i o n to give an XY^fD^^) ion i s used to calculate the contribution of the k i n e t i c energy to the spectrum of the XY^(D2(j) ion. It i s found that the order of appearance i n the spectrum of the B^ and E components associated with,the two vibrations i s correctly predicted. iv TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION 1-1 Historical Background 1 1- 2 Crystallographic Data 13 II. EXPERIMENTAL 2- 1 Materials 26 2-2 Technique 26 2- 3 Instrumentation 30 III. THEORY 3- 1 The Vibrations of Molecules 31 3-2 Symmetry and Selection Rules 38 3-3 Crystal Symmetry and Solid State Spectra 39 3- 4 Phonons and Lattice Vibrations 41 IV. THE VIBRATIONS OF THE AMMONIUM HALIDES 4- 1 The Phase IV Ammonium Halides 58 4-2 The Phase III Ammonium Halides 77 4-3 The Phase II Ammonium Halides , 93 4-4 Ammonium Fluoride 109 4-5 The Barrier to Rotation 114 4-6 The Force Field for Crystalline NH 4 + 118 4- 7 The Effects of a D O J Distortion 125 V. THE VIBRATIONS OF THE ALKALI METAL BOROHYDRIDES 5- 1 The Phase III Alkali Metal Borohydrides 133 5-2 The Phase II Alkali Metal Borohydrides 145 5-3 The Phase I Alkali Metal Borohydrides 154 5-4 Lithium Borohydrides 161 5-5 The Barrier to Rotation 169 V CHAPTER PAGE 5-6 The Force F i e l d f o r C r y s t a l l i n e BH4~ 171 5-7 The E f f e c t s of a D n, D i s t o r t i o n 176 Za VI. CONCLUSION 183 REFERENCES 1 8 7 v i LIST OF FIGURES FIGURE PAGE 1-1 The Crystal Structure of the Phase IV and the Phase II I Ammonium Halides 18 1-2 The Crystal Structure of NaBH^ - Phase II 23 1-3 The Crystal Structure of KBH^ - Phase I I I 25 4-1 The Infrared Spectrum of NH4C1CIV) 67 4-2 The Raman Spectrum of NH 4(IV) 68 4-3 The Infrared Spectra of ND4C1(IV) and ND 4Br(IV) 69 4-4 The Raman Spectra of ND4C1(IV) and ND 4Br(IV) 70 4-5 The Infrared Spectra of NH 4Br(IH) and NH 4I(III) 84 4-6 The Raman Spectra of NH 4Br(III) and NH 4I(III) 85 4-7 The Infrared Spectra of ND 4Br(III) and ND 4I(III) 86 4-8 The Raman Spectra of ND 4Br(III) and ND 4I(III) 87 4-9 The Temperature Dependence of v 4 - NH^Cl and ND4C1 99 4-10 The Temperature Dependence of 11444 - NH^Cl 100 4-11 The Temperature Dependence of v 4 - NH4Br and ND4Br 103 4-12 The Temperature Dependence of the I ( v 4 ) - NH4Br 104 4-13 The Temperature Dependence of the I ( v 4 ) - ND4Br 105 4-14 Barrier Heights as a Function of the Lattice Parameter 117 4-15 The Symmetry Force Constants F3 and F4 as a Function of the Mixing Parameter-NH4Br(IV) 121 4-16 The Symmetry Force Constants F 3 £ t as a Function of the Mixing Parameter-NH4Br(IV) 122 4-17 Calculated Frequencies for and v 4 as a Function of the Mixing Parameter-NH4Br (IV)-*ND4BrClV) 123 4-18 Predicted S p l i t t i n g of as a Function of the Mixing Parameter- NH.Br(T,)+NH.Br(D,,) 128 I v i i FIGURE PAGE 4-19 Predicted S p l i t t i n g of v 4 as a Function of the Mixing Parameter-NH 4BrCT d)-vNH 4Br(D 2 d) 129 4-20 S p l i t t i n g of and v 4 as a Function of Angular D i s t o r t i o n (X= -0.235) - NH^Br(TjHNHjBr(D 2 d) 131 4- 21 S p l i t t i n g s of v 2 as a Function of Angular Dis-t o r t i o n - NH.Br-*NH.Br(D » ,) 132 4 4 v 2d' 5- 1 The Infrared Spectra of NaBH 4(I), KBH 4(I), NaBH 4(II) and KBH(III) ' 138 5-2 The Raman Spectra of NaBH 4(I), KBH 4(I), NaBH 4(II) and KBH 4(80°K) 139 5-3 . The Infrared Spectra of NaBD^I), KBD 4(I), Na'BD4(II) and KBD 4(III) 140 5-4 The Raman Spectra of NaBD 4(I), KBD 4(I), NaBD 4(II) and KBD 4(80°K) 141 5-5 The Infrared Spectra of NaBD 4(II), NaBH 4(II), KBD 4(III) and KBH 4(III) 142 5-6 The Infrared Spectra of RbBH 4(I), CsBH 4(I), RbBH 4(III) and CsBH 4(III) 143 5-7 The Raman Spectra of RbBH 4(I), CsBH 4(I), RbBH 4(80°K) and CsBH 4(80°K) 144 5-8 The Infrared Spectra of LiBH 4 and LiBD 4 164 5-9 The Raman Spectra of LiBH 4 and LiBD 4 165 5-10 The Symmetry Force Constants F3 and F^ as a Function of the Mixing Parameter-KBH 4(III) 5-11 The Symmetry Force Constant F31+ as a Function of the Mixing Parameter-KBH 4(III) 5-12 Calculated Frequencies f o r and v 4 as a Func-t i o n of the Mixing Parameter-KBH 4(III)->KBD 4(III) 5-13 Predicted S p l i t t i n g of v 3 as a Function of the Mixing Parameter-NaBH 4 (T d)+NaBH 4 (D 2 d) 172 173 174 178 Predicted S p l i t t i n g of as a Function of the Mixing Parameter-NaBH 4 (Tp+NaBH^ CD2d) S p l i t t i n g of and as a Function of Angular D i s t o r t i o n C*= -0.48) - NaBH 4(T d)+NaBH 4(D 2 d) S p l i t t i n g of v 2 as a Function of Angular Dis-tortion-NaBH 4 (Td)-*NaBH4 (D 2 d) ix LIST OF TABLES TABLE PAGE 1-1 The Phases of the Ammonium Halides 5 1-2 The Phases of the Al k a l i - M e t a l Borohydrides 11 1-3 The C r y s t a l Structure of the Phase II Ammonium Halides 15 1-4 The C r y s t a l Structure of the Phase III Ammonium Halides 16 1-5 The C r y s t a l Structure of the Phase IV Ammonium Halides 17 1-6 The C r y s t a l Structure of the Phase I A l k a l i Metal Borohydrides 21 1-7 The Cr y s t a l Structure of the Phase II A l k a l i Metal Borohydrides 22 1-8 The C r y s t a l Structure of the Phase III A l k a l i Metal Borohydrides 24 4-1 Assignments for NH 4 .ci(IV) and ND 4C1(IV) 65 4-2 Assignments for NH 4Br(IV) and ND 4Br(IV) 66 4-3 Assignments for NH 4Br(III) and ND 4Br(III) 82 4-4 Assignments for NH 4I(III) and ND 4'I(III) 83 4-5 The External Vibrations of the Ammonium Halides 88-89 4-6 Assignments for NH 4C1(II) and ND 4C1(II) 107 4-7 Assignments for NH 4Br(II) and ND 4Br(II) 108 4-8 Assignments for NH4F and ND4F 113 4-9 L i b r a t i o n a l Frequencies and Barrier Heights 116 4-10 Calculated and Observed Spectra - NH 4 +(T^) 124 4- 11 Calculated and Observed Spectra - NH 4 +(D 2 d) 130 5- 1 Assignments for KBH 4(III) and KBD 4(III) 136 5-2 Assignments for RbBH 4(III) and CsBH 4(III) 137 5-3 Assignments for NaBH (II) and NaBD.fll) 152 X TABLE PAGE 5-4 Assignments f o r NaBH^(I) and NaBD^(I) 158 5-5 Assignments f o r KBH 4(I) and KBD 4(I) 159 5-6 Assignments f o r RbBH 4(I) and CsBH 4(I) 160 5-7 Assignments f o r LiBH 4 and LiBD 4 163 5-8 Calculated and Observed Spectra - BH 4~(T^) I 7 5 5-9 Calculated and Observed Spectra - BH. _(D 9,) I 8 0 1 CHAPTER I INTRODUCTION 1-1 H i s t o r i c a l Background Ever since Ewald's (1) discovery i n 1914 of an anomaly i n the heat capacity of ammonium chloride the ammonium halides have continuously attracted the attention of physicists and chemists. As a result an enormous amount of information about these substances has accumulated and from this information has emerged the fact that the structures of; NH^Cl, NH^Br and NH^I (at one atmosphere and over the temperature in t e r -val 0°K-450°K) are distributed among four d i s t i n c t c r y s t a l l i n e phases. A l l three of these ammonium halides exist i n the high temperature modification known as phase I. Both x-ray and neutron d i f f r a c t i o n data (2) indicate a face centered cubic phase I structure (Z=4) compatible with the space group 0^(Fm3m). Under th i s space group a number of possible structural models are allowed. However, for each model the overall c r y s t a l symmetry must be achieved s t a t i s t i c a l l y by randomness involving the hydrogen atoms thus giving r i s e to a disordered structure. Levy and Peterson's neutron d i f f r a c t i o n data (2) narrows the choice of models to three. One of the three models allows unaxial rotation of the NH^+ ion and the remaining two models, while they do not allow for either unaxial or free rotation, do allow for a NH^+ ion which i s r e l a t i v e l y free. Recent neutron i n e l a s t i c scattering studies by Venkataraman and his co-workers (3) show no torsional peak i n the scattering patterns for NH^Ifl), instead a broad d i s t r i b u t i o n i s observed. This result is i n agreement with an e a r l i e r infrared study (4) which shows a very broad 2 d i f f u s e absorption a r i s i n g from the combination of the t o r s i o n a l mode with the NH^+ bending mode. However, whereas Plumb and Hornig (4) interpreted t h e i r i n f r a r e d r e s u l t i n favor of a model involving unaxial r o t a t i o n the more recent work of Venkataraman and h i s co-workers (3) shows that form factor measurements tend to preclude the p o s s i b i l i t y of unaxial r o t a t i o n - thus favoring a model i n which the NH^+ ion i s r e l a t i v e l y free. On t h i s basis the choice i s between the two models which Levy and Peterson denote as the double approach model and the t r i p l e approach model. In the former two of the hydrogen-halide distances associated with each NH^+ ion are minimized and i n the l a t e r three are minimized. The model maximizing the hydrogen-halide distances i s excluded by the . neutron d i f f r a c t i o n data. Such a model would place 32/2 H atoms on the 3m positions under the space group 0^(Fm3m) and, as w i l l be indicated l a t e r , t h i s i s the structure of the phase I a l k a l i metal borohydrides. It i s found that the second highest temperature modification of the ammonium halides i s also a disordered cubic structure and that the three s a l t s , NH^Cl, NH^Br and NH^I,exist i n t h i s phase, known as phase II. The phase II structure was established by Levy and Peterson (4-6). Their neutron d i f f r a c t i o n data gave a body centered cubic structure (Z=l), compatible with the space group 0^(Pm3m) and having 8/2 H atoms occupying the 3m p o s i t i o n s . Thus, the o v e r a l l c r y s t a l symmetry required under the space group i s achieved by a random d i s t r i b u t i o n of the NH^+ tetrahedra between the two possible orientations i n the unit c e l l . It i s c l e a r then, that the phase t r a n s i t i o n , II -*• I i s of the disorder-disorder type and heat capacity data (7-9) show i t to be of the f i r s t order. The remaining two phases, III and IV are the low temperature poly-3 morphs and both possess ordered structures which have been determined by X-ray (10) and neutron d i f f r a c t i o n (2) methods. Of NF^Cl, NH^Br and NH^I only NH^Br i s found to exist i n both low temperature phases - the lowest temperature phase f o r NH^Br being phase IV. For NH4C1 the low temperature phase i s IV and for NH^I i t i s I I I . Phase IV has a body centered cubic structure (Z=l) under the space group T d(P43m). The 4 H atoms l i e on the 3m po s i t i o n s . This compares with the 8/2 H atoms on the 3m positions under the space group 0^(Fm3m) for the disordered phase II structure. These configurations f o r the hydrogen atoms allow for the second order phase change IV ->• I I . The expected excess entropy of t r a n s i t i o n of R In 2 = 1.36 e.u. f o r such an order-disorder phase change i s indeed observed, within experimental error, f o r ammonium chloride (8). The phase III structure which can be thought of as a d i s t o r t e d body centered arrangement has a tetragonal structure (Z=2) compatible with the 7 space group D4^(P4/nmm). In t h i s structure the nitrogen atoms occupy the 42m positions; (0,0,0) and (h,H,0) and the halide ions occupy the 4mm pos i t i o n s , (0,^,u) and (%,o,u). The parameter, u, i s a measure of the displacement of the halide ions from the corresponding cubic po s i t i o n s which would be (o,%,%) and (%,0,%). Also, i n order to have a true cubic structure the condition, a = c/^fT"would have to be s a t i s f i e d . The 8 H atoms associated with the two nitrogen atoms i n the unit c e l l w i l l occupy 7 the m(i) positions of the space group (P4/nmm). This configuration f o r the hydrogen atoms allows f o r a III -*• II phase transformation which i s of the second order. For ammonium bromide the reported excess entropy of t r a n s i t i o n (7) i s i n agreement with the expected value of R In 2 f o r an order-disorder transformation. A corresponding experimental r e s u l t i s 4 not a v a i l a b l e f o r ammonium iodide. The f i n a l phase change to be considered i s the order-order trans-formation IV III f o r ammonium bromide. Here the hydrogen atoms are also involved i n the phase transformation, since i n phase IV the NH^+ ions are arranged i n p a r a l l e l layers and i n phase III they are i n a n t i -p a r a l l e l layers. Because there i s no change i n order f o r the t r a n s i t i o n s , the entropy change as expected i s small (7,9) corresponding to a small increase i n volume. It has been postulated (7) that the phase change i s of the f i r s t order type. Most of the above information i s summarized i n Table 1-1. Also included are recent values f o r l a t t i c e parameters (11-16), t r a n s i t i o n temperatures (17-19) and the percent volume changes (12-16, 19,20) experienced by the s a l t s as they go through t h e i r respective phase t r a n s i t i o n s . The f i n a l ammonium halide to be mentioned i s ammonium f l u o r i d e . Not unexpectedly t h i s s a l t i s not isomorphous with any of the ammonium halides already discussed. Instead, the structure i s hexagonal (Z=2) 4 and i s compatible with the space group C ^ ^ ( P 6 3 m c ) (21). Under t h i s space group both the nitrogen atoms and f l u o r i d e ions w i l l be on the 3 m positions; (1/3, 2/3, z) and (2 / 3 , 1/3, 1/2 + z). For the nitrogen atoms z = 0 and for the f l u o r i d e atoms z = u. If the parameter u which fi x e s the positions of the f l u o r i d e ions i s 3 / 8 and i f the a x i a l r a t i o c/a i s ( 8 / 3 ) 2 then the s i t e symmetry for the NH^ ions w i l l be rai s e d from C 3 v to an e f f e c t i v e symmetry of T^ and, indeed, the X-ray studies reported by Plumb and Hornig (21) do show these conditions f o r a t e t r a -hedral arrangement of NH.+ ions to be s a t i s f i e d . Only one modification TABLE 1-1. The Phases of the Ammonium Halides NH4C1 ND4C1 NH4Br ND .Br 4 NH 4I ND^I 4 REFERENCES PHASE IV T d 1(P43m) a(A°) 8 8 ° 3.820 8 2 ° 4.009 11, 12, IV -*• III T(°K) AV(%) AS(e.u.) 108.5° +2.20% 0.3 166.7° +2.23% 0.3 7,9 12,12 7,9 PHASE III D4^(P4/nmm) a (A 0) c(A°) 1 5 7 ° 5.718 4.060 1 5 7 ° 6.105 4.338 12, 13 IV,III -* II T(°K) 242.2° 249.7° 234.57° 215.05° 231.0° 224° 8, 8, 7, 9, 7,19 AV(%) +0.48% +0.39% -0.43% -0.73% -0.28% -0.21% 20,20,12,12,13,19 AS(e.u.) 1.3±0.2 1.3±0.2 1.3±0.2 1.3±0.2 8, 8, 7, 9 PHASE II 0*(Pm3m) a(A°) 3.8756 o 29 9 4.0594 2i+8° 4.333 11, 11, 13 II -+ I T(°K) 456.3° 448° 410.4° 405° 257° 254° 17,18,17,18, 7,19 AV(%) +19.43% +18.3% +16.96% 14, 15, 16 PHASE I ojj(Fm3nO a(A°) o 1*73 6.6004 if 5 3 ° 6.8725 0 282 7.256 14, 15, 16 6 of ammonium f l u o r i d e i s known to e x i s t at one atmosphere (21). In view of the foregoing o u t l i n e of the polymorphism associated with the ammonium halides i t i s easy to understand why i t was early recognized by spectroscopists that the ammonium halides would provide for a most i n t e r e s t i n g study, p a r t i c u l a r l y i f such a study involved temperature dependence so that the spectra of d i f f e r e n t phases could be compared. It was Wagner and Hornig (22,23) who were the f i r s t to under-take a comprehensive i n f r a r e d study of ammonium chloride (22) and ammonium bromide (23). Their published work included s p e c t r a l r e s u l t s f o r : N H ^ C l f l l ) , ND 4C1(II), NH 4C1(IV), ND 4C1(IV), NH 4Br(II), ND 4Br(II), NH 4Br(III), ND 4Br(III) and ND 4Br(IV). It was not u n t i l 1968 (24) that the i n f r a r e d spectrum of NH 4Br(IV) was published. The i n f r a r e d spectrum of NH 4I(III) was included i n Bovey's paper (25) of 1950. Plumb and Hornig (21) i n 1954 published spectra for NH4F and ND4F. In the above works (21-25) i t i s found that the i n f r a r e d spectral l i n e showing the greatest s e n s i t i v i t y to the phase changes, III -*• I I , IV •+ II and IV •> III i s the l i n e a r i s i n g from the v 4 bending mode. In phase IV the NH 4 + ion l i e s on a s i t e of T^ symmetry and the expected s i n g l e t i s observed f o r the t r i p l y degenerate v i b r a t i o n , v 4 ( F 2 ) . In phase III the degeneracy i s p a r t i a l l y l i f t e d - here the NH 4 + ion l i e s on a s i t e of D^^ symmetry and as expected a doublet appears corresponding to v 4(B2) and v 4 ( E ) . As we go to phase II, which has a cubic structure, v 4 unexpectedly appears as a doublet. Wagner and Hornig (22) i n t h e i r o r i g i n a l work associated the anomalous component with the disorder i n the l a t t i c e . It i s now well known (26,27) that for disordered systems there can be d e n s i t i e s of states with K ^ 0 associated with the fundamental 7 vibra t i o n s predicted i n the zero wave-vector l i m i t and recently Garland and Schumaker (28) have shown that the anomalous phase II component of must be a t t r i b u t e d to a density of states with jC / 0. Disordered systems i n the ammonium halides w i l l be discussed at greater length i n Chapter IV - the important point to note now i s that the appearance of the doublet at high temperatures serves to help i d e n t i f y phase II spectra. F i n a l l y , the in f r a r e d spectrum of ammonium f l u o r i d e (21) shows a singl e sharp l i n e f or \>4, thus supporting a tetrahedral configuration of NH^+ ions as indicated by the x-ray data (21). Turning now to the Raman spectra of the ammonium halides we f i n d that the most comprehensive studies were those c a r r i e d out by Couture-Mathieu and Mathieu (29) i n 1952. E a r l i e r studies were made by Krishnan (30,31) i n 1947 and 1948. Couture-Mathieu and Mathieu recorded spectra of NH 4C1(II), NH 4C1(IV), NH 4Br(II) and NH 4Br(III). The Raman spectrum of NH 4Br(IV) was reported by Schumaker (24) i n 1968. The Raman studies are a very important complement to the in f r a r e d work because they place the i n f r a r e d i n a c t i v e NH 4 + bending and stretching modes. In addition Raman spectra were the f i r s t to place c r y s t a l modes of tr a n s l a t o r y o r i g i n for the ammonium halides (29 - 31). Consideration of the foregoing o u t l i n e of previous i n f r a r e d and Raman studies indicates an extensive set of spectroscopic data. However, i t should be noted that the studies mentioned have not included the in f r a r e d spectrum of ND 4I(III), nor have they included Raman spectra f o r ND 4C1(II), ND 4C1(IV), ND 4Br(II), N D ^ r C l I I ) , ND 4Br(IV), NH 4I(III), ND 4I(III), NH4F and ND4F. Also, most of the work on the ammonium halides was c a r r i e d out i n the 1950's and since then there has been considerable s o p h i s t i c a t i o n 8 i n instrumentation. Thus, witji the recent a v a i l a b i l i t y of laser Raman spectrophotometers as well as grating spectrophotometers that can scan the infrared down to about 30 cm * considerable data can now be obtained which w i l l complement that obtained by e a r l i e r workers. The study of the ammonium halides was carried out, then, with a view to placing those l a t t i c e modes of rotatory and translatory o r i g i n which had not been previously observed. It was hoped, too, that assignments for the internal fundamentals of the various sal t s could be made which would be more complete than those of previous workers. Also, e a r l i e r workers (22,23,29) had observed lines i n the low temper-ature phase spectra which were not obvious combinations or overtones of the fundamental modes. Since a clear one to one correspondence between the spectra of the protonated and deuterated samples was not established for these lines i t was hoped that such a one to one correspondence could be established. This would then aid i n making meaningful assignments. F i n a l l y , i t was believed that i t would be very worthwhile to compare a study of the ammonium halides with an analogous study of the a l k a l i metal borohydrides - about which considerably less i s known. The high temperature modification of NaBH^ and KBH^, known as phase I (32), has a face centered cubic structure (Z=4) under the space group 0^CFm3m) (33-35). There are 32/2 H atoms lyi n g on the 3m positions thus giving r i s e to a disordered structure. Heat capacity data (36,37) for both sodium and potassium borohydride indicate a disorder-order trans-formation at low temperature. For sodium borohydride t h i s transformation results i n a contraction along what becomes the £ axis of a body centered tetragonal structure (38) and i f a simple order-disorder transformation 9 involving the orientation of the borohydride ions i s assumed the low temperature structure (z=2) must be compatible with the space group 9 _ 9 _ D2^(I4m2). A tetragonal structure under the D2£j(I4m2) space group places 8 H atoms on the m(i) positions and allows for an order-disorder trans-formation analogous to that of the ammonium halides. The heat capacity data (37) for potassium borohydride suggest that the disorder-order transformation takes place over a f a i r l y large temper-ature i n t e r v a l f i n a l l y undergoing completion at 77.2°K where there i s a pronounced anomaly i n the heat capacity curve. No X-ray d i f f r a c t i o n data have been reported for potassium borohydride below 90°K. However, i t i s known that the structure does remain cubic down to 90°K (40) and the infrared results of th i s work recorded at 10°K are also compatible with a cubic structure and can hardly be interpreted on any other basis. For the infrared spectrum of KBD^ the v 4 half-height l i n e width i s only 3.5 cm * at 10°K and since the observed s p l i t t i n g s for sodium boro-hydride are about 20 cm ^ any d i s t o r t i o n from true cubic symmetry would have to be very small indeed; especially when i t i s considered that the probable angular distortions from a tetrahedral configuration of BH^" ions i n sodium borohydride are only +2.22°and -1.10° for the two different types of H-B-H angles (see section 1-2). Thus, i t would appear that the disorder-order transformation for KBH^ e f f e c t i v e l y d i f f e r s from that of NaBH4 i n that there i s no a x i a l contraction. This gives a face centered cubic structure (Z=4) under the space group T^(F43m) and places 16 H atoms on the 3m positions. Since there are 32/2 H atoms on the 3m positions of phase I a second order phase transformation involving hydrogen atoms i s allowed for. The heat capacity data (36) show that the anomaly 10 at 77.2°K accounts for 0.70 e.u. of the 1.36 e.u. expected for the excess entropy of t r a n s i t i o n . However, as previously indicated, the data suggest that the o r i g i n a l onset of the t r a n s i t i o n i s at a much higher temperature with the missing excess entropy of t r a n s i t i o n extending over a very broad temperature i n t e r v a l which l i e s above 77.2°K. Schutte (32) has called the low temperature modification of NaBH^ phase II and th i s designation i s followed here. In addition, the low temperature phase of KBH^ w i l l be cal l e d phase I I I . Thus far no mention has been made of RbBH. and CsBH.. It has been 4 4 found (38) that the high temperature modifications of these sal t s are also face centered cubic, so they very l i k e l y possess the phase I structure. Cooling and warming curves for rubidium and cesium boro-hydride respectively show breaks at 44 and 27°K (41) and i t i s l i k e l y that each of these temperatures represents the completion of a disorder-order phase transformation, which l i k e that of potassium borohydride takes place over a wide temperature i n t e r v a l . Most of the foregoing information r e l a t i n g to the a l k a l i metal borohydrides i s summarized i n Table 1-2. F i n a l l y , we come to the remaining a l k a l i metal borohydride - LiBH^, whose structure i s not isomorphous with any of the phases common to the other a l k a l i metal borohydrides. The structure for lithium borohydride given by the sole reference i n the l i t e r a t u r e (42) i s compatible with the space group D^fPc-mn). Since there are four molecules per unit c e l l the boron atoms must l i e on m positions giving a BH^ ion of C g s i t e symmetry. The lithium ions must also l i e on m positions i f they are to be surrounded by the distorted tetrahedron of H atoms suggested i n reference 42. There TABLE 1-2. Phases of the A l k a l i Metal Borohydrides . NaBH. 4 KBH4 RbBH, 4 CsBH. 4 REFS. Phase I I I T d 2(F43m) a (A 0) 0 9 0 6.636 39 II I ->• 1 T(°K) AS(e .uO 77.2°' 0.70 44 27 36,41 Phase II D2^(I4m2) a (A 0) c(A°) 7 8 ° 4.354 5.907 37 II + 1 T(°K) 189.9° 35 Phase I 0h5(Fm3m) a (A 6) 2 9 6 ° 6.1635 2 9 6 ° 6.7272 7.029 7.419 37 12 w i l l be 8 H atoms on two different m positions and the remaining 8 H atoms w i l l l i e on the general positions. Heat capacity data (43) indicates that lithium borohydride undergoes no transitions i n the temperature i n t e r v a l 15-300°K. Consideration of the foregoing structural information r e l a t i n g to the a l k a l i metal borohydrides suggests some very worthwhile studies for v i b r a t i o n a l spectroscopists. However, i n actual f a c t , very few of these studies appear to have been pursued. No reported Raman studies have been made of any of the c r y s t a l l i n e phases of the borohydrides and the only infrared studies that have been previously reported are the infrared studies of NaBH^(I) and NaBH 4(II) by Schutte (32). The present work includes both infrared and Raman spectra of LiBH^, LiBD^, NaBH^fl), NaBH 4(II), NaBD 4(I), NaBD 4(II), KBH 4(I), KBH 4(III), KBD 4(I) and KBD 4(III). Raman and infrared spectra of RbBH4 and CsBH4 were also recorded. The studies were carried out with the following objectives; ( i ) the placing of cry s t a l modes of translatory and rotatory o r i g i n , ( i i ) assigning for both NaBH 4(II) and NaBD 4(II) the 7 internal fundamentals predicted by the BH4 D 2 d s i t e symmetry, ( i i i ) assigning for both LiBH 4 and LiBD 4 the 9 non-degenerate fundamentals predicted by the BH4 C g s i t e symmetry, and deducing, i f possible, the structure of the low temperature phase of potassium borohydride. The result of this f i n a l objective has already been given. This allows for a more ordered presentation of the detailed crystallographic information which follows. In keeping with the order of the discussion thus far the information r e l a t i n g to the ammonium halides i s presented f i r s t . 13 1-2 Crystallographic Data Table 1-3, 1-4 and 1-5 respectively, give detailed c r y s t a l l o -graphic data for phases I I , I I I and IV of the ammonium halides. In a l l cases an N-H bond length of 1.03 A° determined by Levy and Peterson (2,5,6,44) was used to f i x the hydrogen positions. P a r t i c u l a r attention should be paid to phase I I I . This structure 7 i s compatible with the space group D4h(P4/nmm) and has the hydrogen atoms lying on the m(i) positions. Under the symmetry requirements of the space group the (X,0,Z) nitrogen atom associated with the (0,0,0) nitrogen atom may be placed anywhere i n the XZ plane bounded by the unit c e l l edges. Once th i s (X,0,Z) hydrogen atom has been placed the symmetry elements of the space group w i l l generate the remaining hydrogen atoms. Levy and Peterson (2) carried out a neutron d i f f r a c t i o n study of an ammonium bromide powder i n 1952. Their results were compatible with the hydrogen atoms lying on the face diagonals. However, th e i r results do not meaning-f u l l y exclude the p o s s i b i l i t y that the hydrogen atoms are s l i g h t l y re-moved from these X = Z positions. Since the H/2 atoms point toward the nearest bromide ions i n phase I I , the p o s s i b i l i t y that the phase I I I H atoms point d i r e c t l y toward the nearest bromide ions should also be considered. The above two configurations for the NH^+ ions give two different situations for the angular d i s t o r t i o n of the NH^+ ions. I f the hydrogen atoms are on the face diagonals then the angular d i s t o r t i o n from a t e t r a -hedral configuration for the NH^+ ions w i l l r e s u l t i n two H-N-H angles less than the tetrahedral angle (-0.23°) and four H-N-H angles greater 14 than the tetrahedral angle (+0.11°). I f the hydrogen atoms are pointing d i r e c t l y toward the nearest bromide ions the reverse s i t u a t i o n w i l l hold; i.e. there w i l l be two H-N-H angles greater than the tetrahedral angle (+3.09°) and four H-N-H angles less than the tetrahedral angle (-1.52°). It i s th i s l a s t s i t u a t i o n for the angular d i s t o r t i o n which the BH^ ions i n NaBH^CII) almost cer t a i n l y experience. (This i s discussed l a t e r on i n this section). Now, the spectral results of th i s work (see sections 4-2 and 5-2) show that the v 3 / v 4 s p l i t t i n g patterns for NH 4Br(III) and NaBH 4(II) are of the same type; i . e . the following results are obtained: v(cm ) 38 31 v,(B 2) v 3(E) 28 26 -NH 4Br(III) •NaBH 4(il) v 4(B 2) Therefore, i t i s very l i k e l y that the si t u a t i o n for the D^ ^ distortions are also of the same type and the implication i s that i n NH 4Br(III) the hydrogen atoms are probably pointed d i r e c t l y toward the nearest bromide ions. This means that the parameter, u, which fixes the positions of the halide ions w i l l e f f e c t i v e l y determine the angular distortions for the NH 4 + ions. Ketalaar (10) has determined these parameters for NH 4Br(III) i l i TABLE 1-3. C r y s t a l Structure of Phase II Ammonium Halides Space Group: O^1(Pm3m) Atomic Positions 1 X Atom at (%,%,%) 1 N Atom at (0,0,0) 8/2 H Atoms at (X,X,X), (X,X,X), (X,X,X), (X,X,X) (X,X,X), (X,X,X), (X,X,X), (X,X,X) (i) NH4C1 a = 3.8756 A° X = 0.153 ( i i ) NH 4Br a = 4.0594 A° X = 0.146 ( i i i ) NH 4I a = 4.333 A° X = 0.137 Ammonium Ion Parameters - NH 4 +(T^) /(H-N-H) = 109.47°, r(N-H) = 1.03 A° TABLE 1-4. C r y s t a l Structure of Phase III Ammonium Halides 7 Space Group: D^CTM/nmm) Atomic Positions 2 X Atoms at (0,%,u), (%,0,u) 2 N Atoms at (0,0,0), (%,%,0) 8 H Atoms at (0,x,z), (X,0,Z), 0s,%+X,Z), (%+XA,Z) (0,X,Z), (X,0,Z), ft,J2-X,Z), (%-X,%,Z) (i) NH 4Br a = 5.718 A° c = 4.060 A° u = 0.47, X = 0.150, Z = 0.141 ( i i ) NH 4I a = 6.105 A° c = 4.338 A° u = 0.49, X = 0.138, Z = 0.137 Ammonium Ion Parameters - NH 4 +(D2 d) (i) 4/(H-N-H) = 107.95°, 2/jH-N-H)= 112.56°, r ( N-H) = 1.03 A° ( i i ) 4/WN-H) = 109.06°, 2/^H-N-H) = 110.30°, r ( N-H) = 1.03 A° TABLE 1-5. Crystal Structures of Phase IV Ammonium Halides Space Group: T*(P43m) Atomic Positions 1 X Atom at (h,k,k) 1 N Atom at (0,0,0) 4 H Atoms at (X,X,X), (X,X,X), (X,X,X), (X,X,X) a = 3.820 A° X = 0.156 a = 4.009 X = 0.148 Ammonium Ion Parameters - NH^+(Td) /(H-N-H) = 109.47°, r(N-H) = 1.03 A° (i) NH4C1 (i i ) NH4Br 18 F I G U R E I— I T H E C R Y S T A L S T R U C T U R E O F T H E P H A S E I V A N D P H A S E III A M M O N I U M H A L I D E S P R O J E C T I O N O N ( O O I ) P L A N E P H A S E I V 19 and NH 4I(III) to be 0.47 and 0.49 respectively. The value of 0.47 for ammonium bromide was subsequently confirmed by Levy and Peterson (2). Figure 1-1 shows the ordered structures for both phases I I I and IV and thus complements the structural data given i n Tables 1-4 and 1-5. We turn now to the alkali-metal borohydrides. Tables 1-6, 1-7 and 1-8 respectively give detailed crystallographic data for phases I, II and I I I . In a l l cases a B-H bond distance of 1.26 A° determined by Peterson (34) was used to f i x the hydrogen positions. Special attention should be given to phase II which has a t e t r a -9 gonal structure under the space group D2d(I4m2). The hydrogen atoms l i e on the m(i) positions. In direct analogy to the case of the phase I I I ammonium halides, the symmetry requirements of the space group allow the (X,0,Z) hydrogen atom associated with the (0,0,0) boron atom to be placed anywhere i n the XZ plane bounded by the unit c e l l edges. Again, once t h i s (X,0,Z) hydrogen atom has been placed, the symmetry elements of the space group w i l l generate the remaining hydrogen atoms. Neutron d i f f r a c t i o n studies could aid i n placing the hydrogen atoms; however, these studies have not been carried out. One p o s s i b i l i t y i s that the hydrogen atoms are pointed toward the second nearest neighbour sodium ions. This p o s s i b i l i t y i s seen to be reasonable when we consider the phase change which yields the low temperature tetragonal modification of sodium borohydride. In the high temperature disordered cubic structure the H/2 atoms point toward the second nearest neighbour sodium ions. The ordering 20 process results i n a contraction along what becomes the c_ axis of the tetragonal structure, This contraction could very conceivably res u l t i n a fla t t e n i n g of the BH^ tetrahedra with the H atoms s t i l l pointed toward the second nearest neighbour sodium ions. Such a si t u a t i o n gives the two angles lying i n £he XZ and XY mirror planes a positive d i s t o r t i o n of +2.22°. The remaining four angles have a negative * angular d i s t o r t i o n of -1.10°. Figures 1-2 and 1-3 which respectively show the phase II and phase III structures complement the data contained i n Tables 1-7 and 1-8. * When the contraction along the C_ axis i s considered for NaBH^ any d i s t o r t i o n of the BH^ ions w i l l almost c e r t a i n l y r e s u l t i n two angles greater than 109.47° and four angles less than 109.47°. The si t u a t i o n cited here i s probably the most l i k e l y . 21 TABLE 1-6. C r y s t a l Structure of Phase I A l k a l i Metal Borohydrides 5 Space Group: 0k(Fm3m) Atomic Positions 4 X Atoms at (%,%,%) 4 B Atoms at (0,0,0) 32/2 H Atoms at (X,X,X), (X,X,X), (X,X,X), (X,X,X) (X,X,X), (X,X,X), (X,X,X), (X,X,X) (i) NaBH4 a = 6.1635 A° X = 0.118 ( i i ) KBH4 a = 6.7272 A° X = 0.108 ( i i i ) RbBH4 a = 7.029 A° X = 0.103 (iv) CsBH 4 a = 7.419 A° X = 0.098 Borohydride Ion Parameters - BH 4(T d) ^H-B-H) = 109.47°, r(B-H) = 1.26 A° 22 TABLE 1-7. C r y s t a l Structure of Phase II A l k a l i Metal Borohydrides g — Space Group: D 2 d ^ I 4 m 2 J Atomic Positions 2 Na Atoms at (0,0,%) 2 B Atoms at (0,0,0) 8 H Atoms at (X,0,Z), (X,0,Z), (0,X,Z), (0,X,Z) . a = 4.353 A° c = 5.907 A° X = 0.240, Z = 0.120 Borohydride Ion Parameters - BH^ ( D 2 c l - ' 4^(H-B-H) = 108.37°, 2^(H-B-H) = 111.69°, r(B-H) = 1.26 A° F I G U R E I—2 T H E C R Y S T A L S T R U C T U R E O F N a B H — P R O J E C T I O N O N (OOI ) P L A N E 23 P H A S E II TABLE 1-8. C r y s t a l Structure of Phase III A l k a l i Metal Borohydrid 2 — Space Group: T, (F43m) Atomic Positions 4 K Atoms at (H,h,H) 4 B Atoms at (0,0,0) 16 H Atoms at (X,X,X), (X,X,X,), (X,X,X), (X,X,X) a = 6.636 A° X = 0.110 Borohydride Ion Parameters - BH^ (T^) /(H-B-H) = 109.47°, r(N-H) = 1.26 A° 25 F I G U R E 1 — 3 T H E C R Y S T A L S T R U C T U R E O F KBH^— P H A S E P R O J E C T I O N O N (OOI) P L A N E Z = — z = o 26 CHAPTER II EXPERIMENTAL 2-1 Materials A l l compounds used i n th i s work, with the exception of ND^I and ND^F, were obtained commercially. The alkali-metal borohydrides, LiBH^, NaBHj and KBH^ were obtained from Metal Hydrides Inc. and the remaining two salt s i n this series, RbBH^ and CsBH^ were obtained from Gallard Schlesinger Ltd.. B r i t i s h Drug Houses Ltd. supplied the ammonium halides -this included the fl u o r i d e , chloride, bromide and iodide. The f u l l y deuterated s a l t s ; LiBD^, NaBD^, KBD^, ND4C1 and ND^Br were obtained from Merck, Sharpe and Dohme (Canada) Ltd.. A l l of the above salt s were of reagent grade, with an assay of not less than 98%. The f u l l y deuterated analogs of NH^I and NH^ F were prepared by f i v e successive r e c r y s t a l l i z a -tions of the appropriate protonated compound from D20. The isotopic pur-i t i e s of the resulting salts were estimated spectroscopically to be i n the range 90 ^95%. 2-2 Technique The preparation of samples for the infrared studies involved three different techniques; the choice of technique depended on the compound under study and also on the region of the infrared to be scanned. The f i r s t technique consisted of mulling f i n e l y divided c r y s t a l l i n e powder with nujol and then mechanically depositing a t h i n uniform layer of the re-sulti n g slurry on a suitable support. For our work the supports used with this technique were either cesium iodide or polyethylene. 27 The second technique allowed for mechanically depositing a th i n uniform f i l m of c r y s t a l l i n e powder on either a sodium chloride or potassium bromide support. This was accomplished by f i r s t grinding one surface of the polished support with a very f i n e emery paper (Carbimet 600 G r i t ) . A small mound of very f i n e l y divided c r y s t a l l i n e powder was then placed on the surface of a ground glass plate. Carbon tetrachloride was then added to the surface of t h i s ground glass plate i n such a manner so as to completely surround and moisten the mound of c r y s t a l l i n e powder without dispersing i t . The ground side of the support was then placed d i r e c t l y on top of the mound of moistened c r y s t a l l i n e powder and then rubbed, using small c i r c u l a r motions, against the surface of the ground glass plate. The support was s l i d o f f the ground glass plate before a s u f f i c i e n t amount of the carbon tetrachloride had evaporated to cause " s t i c k i n g " . After s l i d i n g the support o f f the ground glass plate and also allowing for s u f f i c i e n t "drying time" the excess c r y s t a l powder was ca r e f u l l y brushed off the ground surface of the support; a small, f i n e , h a i r brush was used for this purpose. Depending on the s a l t the above procedure was repeated up to three successive times i n order to obtain a s u f f i c i e n t l y thick f i l m . It was found that films prepared i n such a manner can give excellent cryst a l spectra. The f i n a l technique involved the preparation of a polyethylene p e l l e t . The f i r s t step, i n the preparation of the p e l l e t , was to intimately mix a small amount of very f i n e l y divided c r y s t a l l i n e powder with a larger amount of polyethylene powder.(The optimum ra t i o s of c r y s t a l l i n e powder and polyethylene powder for the different s a l t s i 28 could only be found by t r i a l and error). After the mixing process was complete an optimum portion of the mixture was placed on the surface of a 1%" x 1%" x V glass plate and then spread evenly over the surface so as to give a t h i n , roughly c i r c u l a r , layer. The glass plate supporting the mixture and a second glass plate of equal dimensions were then placed on a hot plate heated to about 300°C. As soon as the polyethylene was observed to melt enough to flow s l i g h t l y the second glass plate was placed d i r e c t l y on top of the glass plate supporting the sample. The two plates were then firmly pressed together by hand and then immediately transferred to a mechanical press which allowed a uniform pressure to be maintained u n t i l the sample had cooled. P e l l e t s prepared i n t h i s manner were of uniform opaqueness and gave good cr y s t a l spectra. They were, however, useful for only room temperature work as the thinness of the pe l l e t s did not allow for good thermal contact i n our low temperature c e l l s . I t should be noted that for a l l three techniques, the handling of the c r y s t a l l i n e compounds was carried out i n a dry box containing a dry nitrogen atmosphere. A l l spectra, with the exception of the polyethylene p e l l e t spectra, were recorded with.the samples mounted either i n a l i q u i d nitrogen, Hornig type, c e l l or a l i q u i d helium c e l l . In each of these c e l l s provision was made for the sample supports to be f i t t e d t i g h t l y into a cold block which was i n direct contact with the coolant. A copper-cons tantan thermocouple attached to the cold blocks allowed for monitoring the temperature of the cold blocks. When using the l i q u i d nitrogen c e l l i t was possible to bring the c e l l into the dry box and thus mount 29 the samples i n a dry atmosphere. However, because of i t s size this was not possible with the l i q u i d helium c e l l . In t h i s case the sample was removed from the dry box i n a sealed container. Once th i s container was opened the sample was f i t t e d , under a stream of dry nitrogen, as quickly as possible and evacuation of the c e l l started. The spectra of the ammonium halides recorded i n the spectral region, 4000-500 cm 1 were recorded using the l i q u i d nitrogen c e l l and samples prepared using the th i n f i l m technique. With the exception of the lithium s a l t a l l the spectra taken of the alkali-metal borohydrides i n the spectral region 4000-500 cn; * were also recorded using samples prepared as thin c r y s t a l l i n e f ilms. However, for these studies the l i q u i d helium c e l l was used. The spectra taken of lithium borohydride were recorded using nujol mull samples mounted i n the l i q u i d nitrogen c e l l . F i n a l l y , for spectra recorded i n the 600-50 cm * region poly-ethylene p e l l e t s were used at room temperature and for lower temperatures samples prepared by the nujol mull technique were mounted i n either the l i q u i d nitrogen or l i q u i d helium c e l l . Thus f a r , the discussion of sampling techniques has related to the infrared experiments. We turn now to the preparation of samples for Raman experiments. . In order to prepare the sample a l l that was necessary was to t i g h t l y compress f i n e l y divided c r y s t a l l i n e powder into a conical cavity at the end of a brass rod. Once the sample had been prepared i n this way, the brass rod could then be f i t t e d into a s p e c i a l l y designed low temperature c e l l . (The c e l l used was a modification of the one described by G.L. Carlson (45) which was designed for coaxial viewing , using the Cary 81 spectrophotometer f i t t e d with a He-Ne las e r ) . In the 30 c e l l which was used for t h i s work the non sample end of the brass rod f i t t e d precisely into a c y l i n d r i c a l cavity i n the cold block and i n so doing butted against a spring placed i n this cavity. When the c e l l was f i t t e d together t h i s spring caused the sample end of the brass rod to butt t i g h t l y against one end of a pyrex l i g h t pipe. When the c e l l was correctly positioned the opposite end of the pyrex l i g h t pipe was in o p t i c a l contact with the hemispherical lens of the spectrophotometer. Again, the samples were prepared and the c e l l assembled i n a dry box containing a dry nitrogen atmosphere. 2-3 Instrumentation The infrared spectra were recorded on two complementary double beam grating spectrophotometers, the Perkin Elmer 421 and the Perkin Elmer 301. The 421 was used i n the spectral region, 4000-500 cm - 1 and the 301 i n the region 600-50 cm The wave number scale of both these spectrophotometers was calibrated with the atmospheric water vapor spectrum and the uncertainty i n the measured frequencies i s ± 2 cm A l l spectra recorded on both instruments were recorded with the o p t i c a l paths sealed from the atmosphere. This allowed for the use of an a i r dryer and also allowed for the purging of atmospheric CO^ with dry nitrogen. The Raman spectra were recorded with the Cary 81 Raman spectro-photometer equipped with the Spectra-Physics He-Ne laser. The instrument was calibrated with emission lines from a neon lamp over the spectral range 0 - 4000 cm * and the uncertainty i n the measured frequencies i s ± 2 cm - 1. 31 CHAPTER I I I THEORY 3-1 The Vibrations of Molecules The c l a s s i c a l treatment of molecular v i b r a t i o n a l motion, yielding solutions for the vi b r a t i o n a l frequencies, i s well known (46). However, since both computed vib r a t i o n a l frequencies and force constants form an important part of the discussion i n both Chapters IV and V - i t w i l l be helpful to give ah outline of the c l a s s i c a l treatment leading to the formation of the vib r a t i o n a l secular equations. The solution of these equations w i l l then be b r i e f l y discussed according to the method of Green and Harvey (47). C l a s s i c a l l y , the molecular model that i s used to describe v i b r a t i o n a l motion i s that of N e l a s t i c a l l y coupled point masses (or n u c l e i ) . Such a system of N nuclei has 3N degrees of freedom of which s i x (only the non-linear case i s considered) account for the translations and rotations of the molecule as a whole. I f the 3N mass weighted Cartesian displacement co-ordinates, {q}, are introduced, where, q. = (m.^-Ax., ( i = 1,3N) (1) then the k i n e t i c energy T, of the molecule i n matrix notation i s given by: 2T = (2) The potential energy, V, can be expressed as a Taylor series expanded about the displacement co-ordinates {q} giving: 32 2V = 2V0 + 2T\z±. } q. + E | r - r - , q,q, + (3) where the sums are over the 3N co-ordinates. Now, the equilibrium con-figuration i s associated with a minimum i n the potential energy surface and when i t i s defined to have zero energy the f i r s t two terms i n the above expression vanish. Thus, neglecting higher order terms we have: 2 V = M m , , 1 q-q- = Z f..q.q., or (4) 2V = q* F q ( 5) The dependence of the potential energy only on second order terms implies harmonic motions of the atoms; i f the displacements are large this approximation i s not v a l i d and higher order terms begin to become important. The motions of the system are governed by Newton's equations of motion, which are, i n the mass weighted Cartesian system and i n Lagrangian form: atl£ "ax a ° . <i - ^ ^ where, L, the Lagrangian function i s given by L = T-V. Since T i s a function of the q^ and V of the q^, substitution of equations 1 and 4 into equation 6 gives the set of 3N homogeneous d i f f e r e n t i a l equations: q + Z .f.±.q. =0, ( i = 1,3N) (7) One possible solution i s : q± = q i° s i n (tX** + 6) (8) 33 wKere i s the amplitude of the motion, 6 i s a phase factor and X i s relat e d to the v i b r a t i o n a l frequency. Substi t u t i o n of equation 8 into equation 7 gives r i s e to a set of 3N l i n e a r homogeneous equations which have a n o n - t r i v i a l s o l u t i o n only i f the secular determinant equals zero; i . e. | f . . - X 6. . | = 0 (9) Six of the 3N values of X s a t i s f y i n g equation 9 are always found to be zero; these correspond to the three molecular rotations and to the three t r a n s l a t i o n s . The remaining 3N-6 values of X are r e l a t e d to the — -1 2—2 2 normal frequencies of v i b r a t i o n , v, ( i n cm ) by X = 411 v c ; s u b s t i t u t i o n of these values of X back into equation 8 shows how each of the co-ordinates q^ varies with time. The motion of the nu c l e i corresponding to each normal frequency i s known as a normal mode of v i b r a t i o n . The normal frequencies and the normal modes of v i b r a t i o n are, of course, independent of the co-ordinate system used. The analysis of the v i b r a t i o n a l problem would be s i m p l i f i e d i f a co-ordinate system could be found f o r which a l l cross terms between co-ordinates i n both the p o t e n t i a l and k i n e t i c energy expressions were zero; (the mass weighted Cartesians have t h i s property f o r only the k i n e t i c energy). It i s found, though, that such a co-ordinate system can be defined and that a sin g l e displacement co-ordinate i n t h i s system describes the motion executed by a l l the atoms when the molecule under-goes a normal v i b r a t i o n . I f the orthogonal transformation r e l a t i n g the 3N Cartesian displacement co-ordinates and the 3N normal co-ordinates, 34 {Q}, i s written as: £ = X Q UO) where X i s the transformation matrix; then the expressions for the k i n e t i c and potential energies may be written as: .t . 2T = Q Q, and (11) 2V = QlA Q (12) respectively. The diagonal matrix A i s defined by: 4 = X 1 F X (13) Since i t i s usual to treat the k i n e t i c energy i n terms of internal co-ordinates a further transformation must be introduced. It i s : q = T S (14) Where T i s the matrix r e l a t i n g the 3N Cartesian co-ordinates, {q}, with the internal co-ordinates, {S}. (In the absence of redundancy there w i l l be 3N-6 internal co-ordinates) Now, the matrix T i s not readily available from the molecular geometry and i t i s convenient to introduce the G matrix elements of Wilson (36). From the molecular geometry the matrix B for the inverse transformation S = B q (15) can be calculated. Since {S}and {q} generally have different dimension, B i s not square and thus cannot be inverted to give T. However, BB^ i s square and i f the matrix G i s defined as: G = BB 1 (16) 35 then i t can be shown (reference 46, appendix 7) that the k i n e t i c energy-i s given by: 2T = S* G" 1 S U 7 ) Also, i n terms of i n t e r n a l co-ordinates the p o t e n t i a l energy may be written: 2V = S l F S CIS) where F i s the force constant matrix f o r the i n t e r n a l co-ordinates. The normal co-ordinates are l i n e a r l y r e l a t e d to the i n t e r n a l co-ordinates by the transformation: S = L Q (19) Substituting equation 19 into 17 and 18 y i e l d s : 2T = Q* L* G" 1 L Q = if I Q , and C20) 2V = l} F L Q = Q l A Q (21) From the equations 20 and 21 we obtain the v i b r a t i o n a l secular equations: F L a A- , and (22) 36 By s u b s t i t u t i n g equations 17 and 18 into equation 6 the v i b r a t i o n a l secular equations can be cast into the form: |G E - X . I| =0 ( 2 4 ) However, G F i s not symmetrical and a symmetric matrix formulation of the v i b r a t i o n a l secular equation i s c l e a r l y desirable since the eigen-values and eigenvectors of a symmetric matrix are more e a s i l y evaluated than those of a non-symmetric matrix. Green and Harvey (47) have recently cast the v i b r a t i o n a l secular equations into a symmetric matrix formulation. In t h e i r treatment they set L = U r^2 P (25) where the matrices U and P are orthogonal. Thus, we have the set of r e l a t i o n s : L = U T 2 P r - l t -h „t L = p r 2 y , •t L " = U r 2 P, and , t t U „t h = p r 2 u . Substitution of equation 25 into 23 gives p r 2 y G y r 2 P = I (26) The matrix U i s -now defined as the orthogonal matrix diagonalizing the G matrix, according to: 37 U* G U = T (27) and I becomes the diagonal matrix giving the eigenvalues of G. The orthogonal matrix U i s then composed of the eigenvectors of G. Thus, J' -1 - t L = U T 2 P, s a t i s f i e s L G L =1 and also retains the appropriate freedom to s a t i s f y L 1 F L = A i. e . we can write: P* \±Z I U r 2 E = A (28) The matrix T 2 U F U T 2 i s symmetric and i t s eigenvalues are shown by Green and Harvey to be the same as those of G F; i t s eigenvectors are orthogonal and thus comprise the orthogonal matrix P. Equation 28 can be rearranged to give an expression for the force constants which i s F = U T~% P A P 1 uZ (29) It i s shown by Green and Harvey (47) that P possesses the important property that the elements P„ of P equal zero unless the belong to the same symmetry class. This arises from the fact that the orthogonal matrix U can be written as: U = $ U (30) where $ i s the usual orthogonal matrix r e l a t i n g internal and symmetry co-ordinates, Vi i s an orthogonal matrix defined by equation 30. It should 3S be noted that the symmetry co-ordinates are j u s t the l i n e a r combinations of the i n t e r n a l co-ordinates formed by projecting one member of each symmetrically equivalent set of i n t e r n a l co-ordinates into the point group, of the molecule. This allows f o r maximum symmetry f a c t o r i z a t i o n since no i n t e r a c t i o n terms i n the F or G matrices w i l l occur between two co-ordinates of d i f f e r e n t symmetry. The force constants calculated using symmetry co-ordinates are known as symmetry force constants. 3-2 Symmetry and S e l e c t i o n Rules ' By the use of group theory i t i s poss i b l e . t o show how the 3N normal modes of v i b r a t i o n (including t r a n s l a t i o n s and rotations) are d i s t r i b u t e d among the symmetry types associated with the molecular point group appropriate to the molecule being considered (46). The r e s u l t i s summarized by the equation: n Y = i Zh X,(R)xJ(R) (1) i N j J J J Y where n' i s the number of times the i r r e d u c i b l e representation I\ i s contained i n the reducible representation rT ; Xj (R) an<i xJ(R) a r e the characters under the operation R of I\ and fT r e s p e c t i v e l y ; N i s the order of the group and h.. i s the number o f group operations f a l l i n g under the c l a s s . A l l terms i n equation 1 with the exception of X j(R) may be obtained from the appropriate point group. The expression fo r xJ(R) i s : Xj(R) = w R(±l + cos + R) . (2) 39 where to^ i s the number of atoms invariant under the operation R and <j>R i s the angle of r o t a t i o n associated with the operation R. The +ve signs are taken for proper rotations and the -ve signs f o r improper ro t a t i o n s . Group theory may also be used to derive the t r a n s i t i o n moment selec-t i o n rules f o r both i n f r a r e d and Raman experiments. For a fundamental trans-i t i o n to occur by absorption of in f r a r e d r a d i a t i o n the i n t e g r a l , < I|K |u|i|>j >, must have a non-zero value. In the i n t e g r a l I/K i s the t o t a l l y symmetric v i b r a t i o n l e s s ground state, I|K i s the excited state and u i s the dipole moment operator. Now, the Cartesian components of u w i l l transform l i k e the Cartesian displacement co-ordinates: T , T and T . Therefore, \i>. r x y z 3 must transform l i k e e ither T , T or T i n order that the integrand x y z for the above i n t e g r a l have a t o t a l l y symmetric component and so be non-zero. For a t r a n s i t i o n to be Raman active one of the elements of the three by three symmetric tensor, [< i|>.|a|ij>. >] must have a non-zero value. Again I|K i s the t o t a l l y symmetric ground state and IJK i s the excited state; a represents the p o l a r i z a b i l i t y tensor. I f the t r a n s i t i o n i s to appear i n the Raman e f f e c t , \b. must transform l i k e at le a s t one of the nine elements of the p o l a r i z a b i l i t y tensor. Methods for determining the transformation properties of the p o l a r i z a b i l i t y tensor under the various point groups are discussed by Herzberg (48). 3-3 Cry s t a l Symmetry and S o l i d State Spectra It i s found that a l l of the symmetry operations of a c r y s t a l form I 40 what i s known as the f i n i t e space group f o r the c r y s t a l . The space group consists of two subgroups: the t r a n s l a t i o n operations form one subgroup and the remaining operations form the second subgroup c a l l e d the factor group. The f a c t o r group or un i t c e l l group i s isomorphous with one of the 32 point groups possible f o r c r y s t a l s . It i s convenient also to mention the s i t e group - t h i s group i s the group of a l l symmetry operations acting through any point, or s i t e , i n the c r y s t a l . When the s i t e coincides with a molecular p o s i t i o n i n the c r y s t a l , the s i t e i s then a subgroup of the molecular point group as well as the factor group. We have already seen that the v i b r a t i o n a l analysis f o r free molecules analyzes the molecular motion under the appropriate molecular point group. In c r y s t a l s the motion of the unit c e l l i s analysed under the appropriate factor group. I f there are N atoms i n the p r i m i t i v e u n i t c e l l then there w i l l be 3N c r y s t a l or l a t t i c e v i b r a t i o n s . Three of these, known as ac o u s t i c a l modes, correspond to pure t r a n s l a t i o n s and i n the zero wavevector l i m i t w i l l have zero frequency. The remaining 3N-3 vibrations are known as o p t i c a l modes. The three a c o u s t i c a l and 3N-3. o p t i c a l modes may be further c l a s s i f i e d as either i n t e r n a l or external modes. The external modes always include the three a c o u s t i c a l modes and a r i s e from t r a n s l a t o r y or rotatory motions of the molecules i n the unit c e l l . The i n t e r n a l v i b r a t i o n s are those which a r i s e from motions of the atoms associated with the i n d i v i d u a l molecules i n the unit c e l l . By considering the group of N non-equivalent points corresponding to the N non-equivalent atoms contained i n the p r i m i t i v e u n i t c e l l and 41 applying the p r i n c i p l e s of group theory i t i s possible to obtain an expression f or n Y; the number of times a p a r t i c u l a r i r r e d u c i b l e representation I \ i s contained i n the reducible representation, rj. The desired expression corresponds to equation 1 of section 3-2. Again, Xj W a n d Xj 0*) a r e t n e characters under the operation R of r. and rT res p e c t i v e l y ; N i s the order of the group and h. i s the J J J number of the group operations f a l l i n g under the j t h c l a s s . A l l terms except xj0*) c a n be obtained from the appropriate f a c t o r group. Ana-l y t i c a l expressions f o r the xJ(R) have been devised by Bhagavantam and Venkatarayudu (49). By a s u i t a b l e choice of the reducible representation rT and by u t i l i z i n g the characters, Xj( R)> appropriate to i t , i t i s possible to determine how the i n t e r n a l , external t r a n s l a t o r y , external rotatory and aco u s t i c a l modes are d i s t r i b u t e d among the various symmetry species of the factor group. It i s to be emphasized that the f a c t o r group analysis considers only a s i n g l e unit c e l l and the assumption i s made that a l l equivalent atoms i n adjacent u n i t c e l l s move i n phase. In the following sections i t w i l l be seen that t h i s assumption may f a i l i n c e r t a i n cases. 3-4 Phonons and L a t t i c e Vibrations The energy i n a l a t t i c e v i b r a t i o n or e l a s t i c wave i s quantized -giving r i s e to quanta of energy known as phonons. This i s analogous to the case of electromagnetic waves, where the quanta are known as photons. It i s found that phonons of wavevector, K p h o n o n "C | JC_| = 2 T T / A } , and photons of wavevector, p h o t o n ^ Ijil = 2TT/A}, i n t e r a c t 42 as i f they possessed momenta h.Kph o n o n and hK^oton' The momentum con-servation law for c r y s t a l s , which acts as a selection rule for o p t i c a l and acoustical t r a n s i t i o n s , may therefore be expressed i n terms of wave-vectors and for this reason the wavevector i s of s i g n i f i c a n t physical importance. A photon-phonon interaction may resu l t i n either the creation or absorption of a phonon, with the photon being scattered i n the process; as a result both the photon wavevector and frequency w i l l change. For the creation of a phonon the conservation of wavevector requires that: K. = K + K (1) —1 —s — v where K^, and IK are the wavevectors of the incident l i g h t , scattered l i g h t and phonon respectively. S i m i l a r l y by conservation of energy oi . = oi + oi (2) l s ° v ' where oi^, o» and w o respectively are the angular frequencies of the i n -cident l i g h t , scattered l i g h t and phonon. Equation 2 may be rewritten as: c|K.| = c l K j + v|K| (3) where, v, the v e l o c i t y of the phonon (soundwave) i s much less than, c, the v e l o c i t y of the photon ( l i g h t wave). I f K (phonon) and (photon) are of comparable magnitude OK (photon) w i l l be much greater than oi^ (phonon) Thus, the percentage difference^ between OK and w s i s small and i t follows that |K^ | and are not very di f f e r e n t . In a vector momentum diagram with |K^ | = |KSI> IKI, w i l l form the base of an isoceles triangle and we can write: 43 K = 2 K. sin%<f> ( 4 J where <j> i s the scattering angle. For Raman experiments using right angle viewing (cb = 90°) the phonon wavevector w i l l be 2 KL; so i f the incident o Raman exciting l i g h t has a wavelength of about 6000 A the phonon wavevector w i l l be of order 1.5 x 10 5 cm-'''. In the case of infrared absorption experiments wavevector and energy-conservation lead to: to. = u)6 ( 5 ) K. = K (6) — l — where OK and jK^ are angular frequency and wavevector respectively for the incident l i g h t and co0 and K are the angular frequency and wavevector of the phonon. For absorption at about 1000 cm * the phonon wavevector w i l l be 3 -1 of order 6 x 10 cm Since i t i s possible for the phonon wavevector to take on any value lying i n the B r i l l o u i n zone - the maximum value being of order n/a where a i s the l a t t i c e constant - i t i s evident that the phonon wavevectors associated with both Raman scattering and infrared absorption experiments 8 -1 are small compared to the maximum order of about 10 cm . The r e l a t i v e smallness of the phonon wavevectors has important consequences since t h i s implies that the phonons w i l l have wavelenghts very long compared to the l a t t i c e constant. In the case of infrared inactive phonons the frequencies are determined mainly by short range forces i n the l a t t i c e ; 44 phonons with wavelength long compared to the lattice constant are not significantly influenced by the dispersive effects of these forces and have essentially the same frequency as inf i n i t e wavelength phonons. The Raman shifts thus measure the phonon frequencies at K = 0. The infrared active phonons produce an electric dipole moment in the lattice and the accompanying long range (macroscopic) electric fields may extend over many lattice c e l l s . Thus the motion of the dipoles may be influenced by these long range electric forces; i.e. the motion of the vibrating dipoles through the presence of the macroscopic electric f i e l d may couple with the motion of the electromagnetic radiation in the lattice. The result is that for each infrared active phonon there w i l l be a frequency difference between different directions of phonon propa-gation relative to the electric f i e l d , E_. For non cubic crystals where the polarization, P_, is no longer parallel to Fi , the magnitude of the frequency difference between transverse (P J_ IC) and longitudinal (P | | K) phonons wil l depend on the orientation of the crystal with respect to the electric f i e l d . In order to see how the above results are arrived at the motion of i both the electromagnetic radiation and the vibrating dipoles w i l l be con-sidered in a classical context. Maxwell's equations can be used to describe the propagation of the electromagnetic waves in homogeneous media. Four vector quantities are involved. These are, E_, the electric f i e l d , H, the magnetic f i e l d , D_, the electric displacement and, J3, the magnetic induction. They are related by the two equations: 45 P_ = §1 (7) B = UH (8) where e and y are r e a l symmetric tensors of second order. Since magnetic eff e c t s can usually be ignored, most c r y s t a l s w i l l behave as i f y was a unit tensor and so B = H. For a p a r t i c u l a r c r y s t a l the elements of e are determined by measuring i t s d i e l e c t r i c properties. The p o l a r i z a t i o n , P_, which represents the dipole moment per u n i t volume i s re l a t e d to E_ and D_ by the equation D = E + 4IIP (9) According to the theory of Maxwell the above vector quantities are r e l a t e d by the following four fundamental equations: V • H = 0 (10) V • D = 0 (11) V x H = 16 (12) V x E = - i H - (13) If the electromagnetic waves as they propagate i n the l a t t i c e are assumed to have plane form with s p a t i a l and time dependence exp i (K-r_ - ait) then Maxwell's equations impose the requirement (50) that 46 E = - 4IIK (K • P) - to 2/c 2 P ? 2 7 1 ( 1 4 ) K co / c If K > > to/c for the phonons which i s the case i n Raman experiments then equation 14 becomes The above r e s u l t f o r E_ i s i d e n t i c a l to that obtained when Maxwell's equations are replaced by the equations of e l e c t r o s t a t i c s : V. • D = Z . x ! = 0 (16) Thus, equation 15 i s the s o l u t i o n of: K • D = 0 (17) K x E = 0 (18) Born and Huang (51) have shown that i n the harmonic and adiabatic approximation the p o t e n t i a l energy density f o r a c r y s t a l may be expressed as follows: <!> = yj^LU - U t QE - ^ X E (19) Here U i s a 3n dimensional vector whose components represent the d i s -placements of the n d i f f e r e n t types of atoms, i d e n t i f i e d by k = l,2,....n. The displacements are represented by a super p o s i t i o n of t r a v e l l i n g waves so that i n a wave of wavevector K and frequency co the displacement of the 47 atoms at a point r w i l l be U exp i (K • r_ - u>t) and the f i e l d w i l l be E exp i (K-r - wt). The 3n x 3n matrix L has elements L' (k k 1) which — = a p ' depend on the force constants between atoms of type k and k' (a,3, and y are Cartesian co-ordinates). Q i s a 3n x 3 matrix and has elements Q ag(k) which depend on the charges residing i n atoms of type k. F i n a l l y the elements, xa^> of the matrix x are the components of what i s known as the electronic s u s c e p t i b i l i t y tensor. According to equation 19 the potential energy density, <j>, arises from (i) the displacement of the 3n atoms from t h e i r equilibrium positions, ( i i ) the r e l a t i v e displacement of the atoms due to the presence of a macroscopic e l e c t r i c f i e l d and ( i i i ) the displacement of the electrons i n the atoms r e l a t i v e to the nuclei. The equations of motion for atoms moving i n such a potential are: p U = - L U + Q E (20) where p i s the diagonal 3n x 3n mass density matrix for which the elements m (k k) = m, for a l l a. When the periodic solutions for E and U are substituted into equation 20 i t y i e l d s : o j 2 p U = L U - Q E ' (21) We see that i n equation 21 two terms contribute to the restoring forces. These are the local e l a s t i c restoring forces and the long range e l e c t r i c forces. The pola r i z a t i o n , P_ = - 3<f>/8E_, (Reference 51 - Appendix 5) i s given by: 48 P = Q^U + X E (22) I f the matrix p i s eliminated by def i n i n g : W a ( k ) = (Pk)2ua(k) (23) (24) Z a 3 ( k ) = Q a 3 ( k ) / ( p k ^ ' (25) then equations 21 and 22 become: OJ 2W = N I - Z E (26) P = (27) and we r e c a l l that f o r K > >ui/c the macroscopic e l e c t r i c f i e l d i s given by equation 15; i . e . , Equations 26 and 27 completely define the motion of the v i b r a t i n g dipoles i n the presence of a macroscopic f i e l d and so are applicable whenever conditions are uniform over regions containing many l a t t i c e c e l l s . The expression f o r the macroscopic f i e l d , E_, has been obtained by consid-ering the motion of electromagnetic r a d i a t i o n i n homogeneous media. Since the solutions of equation 26 are dependent on E_, these solutions w i l l r e l a t e 49 the motion of the dipoles to the motion of the electromagnetic radiation. Using the above three equations we w i l l consider two cases after Cochran and Cowley (52). Case I; E_ = 0 In t h i s case the macroscopic e l e c t r i c f i e l d i s suppressed and the 3n eigen frequencies, fi^ (j = 1,2,... 3n), w i l l be given by the secular equations: |N - Q2. I| = 0 (28) = J = Three of these frequencies w i l l correspond to acoustical modes for which Q. = 0, say j = 1,2,3. If the rows and columns of the singular matrix N which involve the k atom are deleted, then the resulting minor, N, i s non singular and of order 3n-3. Its determinant i s independent of the choice of k and 3n > 2 3 Det N = II ti. (29) j=4 An effective inverse N~l may be formed by taking the inverse of N and adding zeros i n those rows and columns which are eliminated from N to give N. The matrix N~* i s of importance i n the next case. Case I I , E f 0 When the c r y s t a l i s i n equilibrium i n a static f i e l d 8<j>/9U = 0 and thus NW = ZE. Using this r e s u l t to eliminate W from equation 27 gives a r e l a t i o n between P_ and E_ which, with the help of equations 7 and 9, leads to the following expression for the s t a t i c d i e l e c t r i c constant. 50 = 6 a +' 4n(Z t N" 1 Z + x) a3 a3 = = = A a 3 (30) The o p t i c a l d i e l e c t r i c constant i s then: = 6 Q + 4 n ( X ) 0 a3 a3 = a3 (31) ( o ) ( oo ) The elements of | v ' and e ' may be determined experimentally. Those ( o o ) of r are determined at a r e l a t i v e l y high frequency such that the i n -e r t i a of the atoms eliminates the contribution due to t h e i r r e l a t i v e displacement. In considering the eigenvalue problem i t i s convenient to choose axes so that K i s p a r a l l e l to a d i r e c t i o n a, which may be taken as being com-p l e t e l y a r b i t r a r y . Thus, from equations 15 and 17; E a = -4IIP with Eg = Ey = 0 and Pg = Py =0. This r e s u l t can be used to eliminate E_ from equation 26. The secular equations f o r long wavelength v i b r a t i o n s with K i n the a d i r e c t i o n then become: I M 411 £ ( oo ) z i f - il . aa 1 0 0 where X = 0 0 0 0 0 0 (32) Cochran and Cowley (52) show that f o r the 3n-3 o p t i c a l modes: 3n II ui..(a) j - 4 J" = Det N , A 411 t M - l 7 . 1 + ~r^r ( z N Z) ( oo ) v = = =Jaa aa (33) 51 The important result from cases I and II i s that equations 29,30,31 and 33 lead to: Equation 34 can be sim p l i f i e d i f some of the frequencies OK (a) are the same as the frequencies Qy This can be so only i f the pol a r i z a t i o n i n the a direc t i o n vanishes and the macroscopic f i e l d , E_, vanishes. (When a l l the Maxwell equations are considered (51) the macroscopic f i e l d w i l l be suppressed but E_ cannot vanish i d e n t i c a l l y ) . I t can be shown (52) that the polarization of each of the modes whose frequencies are the i s along one of the crystallographic axes for crystals of orthorhombic or higher symmetry. Therefore from equation 17 we can say that for modes sa t i s f y i n g UK (a) = Sly where a_ specifies the dir e c t i o n of a crystallographic axes we w i l l have K J_ (P || a). Such modes are known as transverse modes. The other solutions are the longitudinal modes of case II for which K i s p a r a l l e l to a- Here a instead of specifying a general cry s t a l direction w i l l now be taken to be the oi associated with the transverse modes. In this case K | | (P | | a) • Remembering that there w i l l be n-1 modes associated with each crystallographic axis we can rewrite equation 32 as: (34) (35) 52 where the u ^ and the coT are res p e c t i v e l y the frequencies of modes of longitudinal and transverse p o l a r i z a t i o n . For a p a r t i c u l a r dipole the frequency of the transverse mode w i l l l i e lower than the lon g i t u d i n a l frequency since i n a transverse mode Ji vanishes and the v i b r a t i o n a l f r e -quency i s determined s o l e l y by the l o c a l e l a s t i c r e s t o r i n g forces (see equation 26). For a longi t u d i n a l mode there i s an e l e c t r i c f i e l d , E, which through the Q matrix contributes an addit i o n a l r e s t o r i n g force. I f a mode has no p o l a r i z a t i o n associated with i t ( i . e . i f i t i s i n f r a r e d inactive) then tii and co^ , w i l l be equal for that p a r t i c u l a r mode. Equation 35 i s a generalized form of the LST r e l a t i o n which was f i r s t derived for diatomic cubic c r y s t a l s by Lyddane, Sachs and T e l l e r (53) . It should be emphasised that equation 35 holds f o r long waves. Thus, jC i s small. However, i n order f o r these lon g i t u d i n a l and transverse waves to propagate i n the l a t t i c e they must be of wavelength smaller than the dimension, d, of the c r y s t a l ( i . e . i t i s necessary to have K <d). The d i f f e r e n c e , then, between a mode for which jC = 0 and a mode f o r which K 0 l i e s not i n the value of K but i n the absence of a macro-scopic f i e l d i n the f i r s t case and i t s possible presence i n the second. Born and Huang show that the <4^  w i l l be the i n f r a r e d absorption f r e -quencies. It may be noted also that the i n f r a r e d active phonons must be those for which K = 0 since the p o l a r i z a t i o n associated with the modes w i l l vanish otherwise (51). Alternately, conservation of wavevector (equation 6) t e l l s us that the phonon wavevector w i l l be small compared to the dimensions of the Bullouin zone and therefore an i n f r a r e d absorption 53 measures the phonon frequency at K = 0. From the foregoing discussion we see that the f i r s t order Raman and infrared spectra give information about the phonon dispersions only where K is effectively zero. More complete phonon spectra may be obtained from neutron inelastic scattering experiments. Recently Brockhouse and his co-workers (54) have used neutron inelastic scattering techniques to obtain the phonon dispersions for the translatory external modes of a crystal of interest in this work - ND^CIQV). Below is a sketch of their spectra of the longitudinal optical (LO), transverse optical (TO), longi-tudinal acoustical (LA) and transverse acoustical (TA) phonons; JC is in the (^,0,0) direction. (The dispersions w i l l , of course, be different when K is in other (symmetry) directions such as (.h,h,0) or (h,h,h)) • 257 calc. 215 (209) 175 (150) 137 (142) 100 (102) T(0,0,0) X(%,0,0) K ND.Cl(IV) phonon dispersions - ref. 54 54 The symbols r(0,0,0) and X(h,Q,0) appearing i n the foregoing diagram represent symmetry points i n the r e c i p r o c a l l a t t i c e . They occur at the zone edges and t h e i r number i s determined by l a t t i c e symmetry. The simplest model which we may use to p r e d i c t o p t i c a l and ac o u s t i c a l phonon dispersions i s the l i n e a r diatomic chain - only nearest neighbour in t e r a c t i o n s are considered. Such a model may be r e l a t e d to a simple cubic l a t t i c e (55) where atoms of mass l i e on the odd numbered planes and atoms of mass M 2 l i e on the even numbered planes. The spacing between the planes i s a and the l a t t i c e constant i s 2a. I f f i s the force constant connecting nearest neighbour planes (e.g. planes 2s and 2s + 1) then the equations of motion become: M 1 ^ 2 S + 1 - f { C X 2 s + 2 " X 2 s + 1 > " ( X 2 s + l - X 2 s » at M 2 7 T 2 S " £ { < X 2 s + l " X2s> " < X 2 s - X 2 s - l » <36> at Convenient t r i a l solutions i n the form of t r a v e l l i n g waves are: X 2 s + 1 = A exp i {(2s+l)Ka - cot} X 2 s = B exp i {2s Ka - cot} (37) New equations of motion are obtained by s u b s t i t u t i n g equations 37 into equations 36. They are found to be: - co2 M^A = (2fcosKa)B - 2fA - to2 M2B = (2fcosKa)A - 2fB (38) 55 There w i l l be a non t r i v i a l solution only i f the determinant of the co-e f f i c i e n t s of the two unknown amplitudes, A and B, vanishes. This re-quirement leads to the dispersion rela t i o n s : ui (optical) = f ui (acoustical) 1 1 M l + M 2 1 1 +? M, + MT - r 1 2 4sin Ka MXM2 4sin Ka M 1 M 2 (39) The force constant f may be related to either transverse or longitudinal polarizations. It w i l l , i n general, be different i n each case. Substi-tuting equations 39 into equations 38 gives (at jC = 0) A/B = 1 for the acoustical branch and A/B = -M^/M^ for the o p t i c a l branch. These relations show that i n the K = 0 l i m i t adjacent atoms move together for acoustical modes and that for the o p t i c a l modes they vibrate against each other with t h e i r center of mass fixed. 2 At K = 0 the roots of the dispersion relations 39 are; ^ (optical) = 2 f + 1/M2) and w (acoustical) = 0. At the zone edge = ±n/2a. 2 Here the roots of equation 39 (M1 > M2) are OJ (optical) = 2f(l/M 2) and 2 co (acoustical) = 2f(l/M^). In the case of ND4C1(IV) we can use this simple model to estimate the zone edge LO, TO, LA and TA frequencies (We associate the mass of the ammonium ion with M 2 and the mass of the chloride ion with Mj). The results are shown on the diagram on page 53. Considering the s i m p l i c i t y of the model the agreement i s remarkable. A knowledge of phonon dispersions throughout the B r i l l o u i n zone (and p a r t i c u l a r l y at the symmetry points where the densities of states 56 possess c r i t i c a l points) i s helpful i n interpreting the second order Raman and infrared spectra. This i s because the phonon wavevectors may-range throughout the entire B r i l l o u i n zone. The only requirement imposed by conservation of wavevector i s that the wavevectors of the two phonons should e f f e c t i v e l y be equal and opposite. That this i s so i s seen by considering the conservation relations 1 s 1 ~ z (Raman) &± = K! + K 2 (infrared) (40) In these relations the symbols have their usual meaning and the subscripts 1 and 2 refer to the phonons giving r i s e to either a combination or over-tone mode. In the case of both infrared and Raman experiments the photon wavevectors are negligible compared to the B r i l l o u i n zone dimensions and hence wavevector conservation requires that the phonon wavevectors are ef f e c t i v e l y equal and opposite. The frequency d i s t r i b u t i o n for two phonon processes i s thus proportional to a weighted density of l a t t i c e states i n which two phonons of equal and opposite wavevectors are present. It i s found that there are par t i c u l a r values for the K vector (e.g. at the symmetry points) where there w i l l be discontinuities i n the frequency d i s t r i b u t i o n . Therefore i t i s important to know the selection rules at part i c u l a r K vectors for the pairs of phonon branches which can contribute to the second order spectrum. The selection rules are simplest to calculate for processes due to i 57 phonons with zero wavevector. In t h i s case the re q u i s i t e group theory and i t s application are well known (49)• Selection rules for two phonon processes due to phonons with non zero wavevector are i n p r i n c i p l e calculated the same way as for phonons with jC = 0. However, even though the mechanics of the calculations are b a s i c a l l y the same as for K_ = 0 phonons, the actual manipulations are more complex due to the higher dimensionality of the representations involved. Because of the complexity involved, detailed selection rules have been worked out for only a very few simple crystals. These include the rock s a l t , zinc blend and diamond structures (56,57). It i s interesting to note that acoustical phonons may take part i n two phonon processes. This, of course, i s not the case f o r f i r s t order spectra where the selection rules forbid t h e i r appearance. 58 CHAPTER IV VIBRATIONS OF THE AMMONIUM HALIDES 4-1 The Phase IV Ammonium Halides The phase IV ammonium halides possess an ordered cubic structure under the space group Td(P43m). The factor group analysis i s carried out under the T^ factor group and the irreducible representations for the int e r n a l , o p t i c a l translatory (OT), o p t i c a l l i b r a t i o n a l (OL) and acoustical-translatory (AT) vi b r a t i o n a l modes are: r(internal) = A : + E + 2F 2 r(OT) = F 2 r(OL) = Fx r(AT) = F 2 In the infrared only the t r i p l y degenerate modes of F 2 symmetry may appear, whereas i n the Raman effect modes of Aj and E as well as F 2 symmetries may be active. We r e c a l l from section 3-4 that for dipole allowed vibra-t i o n a l modes - i n the present case modes of F 2 symmetry type - there w i l l be both transverse and longitudinal components. Since the phase IV ammonium halides are not centrosymmetric the transverse modes may be both infrared and Raman active. The longitudinal modes can appear only i n the Raman effect. Turning now to the NH4C1(IV) results obtained by e a r l i e r workers (3,22,29,30) we fi n d that the fundamental modes v ^ A i ) , v 2 ( E ) , v 3 ( F 2 ) trans-verse, v^(F 2) transverse, T^(F 2) transverse and Lj(F^) have a l l been ob-served and assigned. The translatory mode, T j ( F 2 ) , has been observed i n the Raman spectrum (29,30) and the l i b r a t o r y mode, L..(Fi), which i s both infrared i 59 and Raman inactive has been observed using neutron i n e l a s t i c scattering techniques (3). From the results of th i s work assignments have been made for v 3 (longitudinal) and (longitudinal). Reference to the Raman spectrum of NH4C1(IV) (see Figure 4-2) shows a d i s t i n c t doublet associated with v 4 ( F 2 ) . The low and high wave number components appear at 1401 and 1421 cm * respectively. The low wave number component compares with the infrared absorption at 1400 cm * and can be assigned as (transverse). The high wave number component can reasonably be assigned as (longitudinal) since; (i) there are no obvious combinations or overtones to which i t , may be assigned and ( i i ) i t s a t i s f i e s the requirement that the longitudinal component w i l l be at higher frequency than the transverse component. There are two Raman scatterings which appear at higher wave number than the (transverse) scattering at 3127 cm They are found at 3141 and 3164 cm Since the 3141 cm * scattering can be s a t i s f a c t o r i l y assigned as a combin-ation mode (this assignment i s discussed later) we assign the 3164 cm"''' scattering as v ^ l o n g i t u d i n a l ) . Unfortunately the longitudinal component of T^(F 2) i s not observed. However, recent infrared r e f l e c t i o n measurements by Lowdnes and Perry (58) have placed T^(longitudinal) at 275 cm * and: Tj(transverse) at 188 cm The la t e r frequency compares with the Raman value of 183 cm 1. The f i r s t order Raman spectra of the remaining phase IV ammonium halides; i.e. NH 4Br(IV), ND4C1(IV) and ND 4Br(IV) very closely p a r a l l e l the spectrum observed for NH 4C1(IV). For NH 4Br(IV) Schumaker (24) does not report a l i n e corresponding to (longitudinal); he does however report a l i n e which may be assigned as v.(longitudinal) (see Table 4-2). The Raman 60 spectra of the two deuterated s a l t s (see Figure 4-4) c l e a r l y show the appearance of the transverse and l o n g i t u d i n a l components f o r both v 3 and v 4 ; i . e . i n each case there i s only one higher wave number scattering i n the near v i c i n i t y of the transverse components. Infrared r e f l e c t i o n measurements (58) have placed 7^(longitudinal) f o r NH^BrflV) and neutron i n e l a s t i c s c a t t e r i n g r e s u l t s (54) have placed t h i s mode i n the case of ND 4C1(IV). The r e s u l t s f o r the transverse and lon g i t u d i n a l s p l i t t i n g s are summarized below ( d i e l e c t r i c constant data (58) are also included). NH 4C1(IV) ND 4C1(IV) NH 4Br(IV) ND 4Br(IV) v 3(L) 3164 2371 (3140) 2370 v 3(T) 3127 2333.5 3126 2332 v 4 ( D 1421. 1069.5 , 1418 1065 v 4CO 1401 1063.5 1402 1061 T ^ L ) 275 257 215 (197) T 2(T) 188 175 160 147 6.0 . 5.4 e C-) 2.7 2.9 The bracketted frequencies have been computed using the product r u l e : 2 _ £ 0 0 eC«0 Since there i s a.complete set of s p l i t t i n g s f o r the ammonium chlorides the r a t i o s e ^ / c ^ c a n be calculated using the product r u l e . The r e s u l t s are: 61 E ( ° ) / E H (calc.) e ( ° ) / e ( » ) (expt.) NH4C1(IV) 2.25 2.22 ND4C1(IV) 2.25 2.22 The agreement i s seen to be excellent. Since the internal fundamentals are at much higher frequencies than the components of T^(F 2) t h e i r small s p l i t t i n g s have l i t t l e effect on the calculated e C ° ) / e C c o ) r a t i o s . This i s seen from calculations below where only the s p l i t t i n g of the trans-latory mode has been considered. We note, however, that the agreement between calculated and experimental ratios does improve when the f r e -quencies of the internal modes are included. NH4C1(IV) ND4C1(IV) NH 4Br(IV) ND 4Br(IV) e ( ° ) / e H (calc.) 2.14 2.16 1.81 1.80 (external modes only) e ( ° ) / e W (expt.) 2.22 2.22 1.86 1.86 In addition to the product rule there are sum rules which may be derived from the eigenvalue problem of section 3-4. When the polariza-t i o n i s along a crystallographic axis (a) of a cr y s t a l of orthorhombic symmetry or higher the eigenvalue matrix for the longitudinal modes may be written as: u ) . 2 = Wt(N + Z X Zt)W =L = »•= £ ( o o ) = = = •> = aa t 4TI t t = W NW + W Z X Z W 62 which leads to 2 2 4n t t <4 - Sr = S 1 I I I (2) E a a We note that there may be no more than n-1 non-vanishing elements assoc-2 2 iated with the diagonal 3n x 3n matrix {wL - u }. This is because we have taken the polarization to be along a crystallographic axis. If we take the trace of the matrices on both sides of equation 2 we have: 0 7 4TT t + Trace {y. - y T } = -^-^ Trace {W Z X Z W} (3) which gives the sum rule: e(") p 2 a a _ T _ a (4) 4TI ~* i 2 2 n-1 u>L - to T where P ais the polarization experienced by a transverse mode of fre-quency co^ ,, (If to^ and to^ , coincide the polarization vanishes). Since 2 is proportional to the (infrared) intensity of the transverse mode, the 2 2 sum rule t e l l s us that the sum of the ratios, intensity/(co^ - ui^, ) is an isotopic invariant. We may note that Haas and Hornig (59) have incorrectly used the equivalent of equation 4 with n-1 = 1 to compute absolute intensities for a number of crystals having n > 2. In the case of the ammonium halides i t would be necessary to experimentally obtain a complete set of relative intensities before the sum rule could be used to compute absolute intensities. t 3^  Remembering that W W = I and that Z (k) = Q (k )/(p ) 2 we can write — — Otp Ot p K a second sum rule: 63 N 411 qaa(^) _ _ 2 2 EC°°) k = 1 m k n-1 J aa The q (k) are related to the charges on atoms of mass m. and N i s the naa • k number of unit c e l l s i n a unit volume.(When the pol a r i z a t i o n i s along a crystallographic axis the elements QagCO with a £ 3 vanish). Because the phase IV ammonium halides are isotro p i c there are only three unknown q a a ^ ' •' T K e s e a r e ^aa^ ' a n d V t ^ ' I s o t o P i c d a t a w i n g i v e us two equations i n three unknowns. This can be reduced to two unknowns by remembering that 2Q ag(k) = 0- The results for ammonium chloride are: q H = +1.92e q N = + 8.63e q c l = -16.31e These calculated charges appear to be unreasonably high - results of simi l a r calculations could not be found for purposes of comparison. However, i t i s to be remembered that the charges are those associated with the atoms i n the presence of a transverse wave. When the d i s t o r t i o n of the electronic charge d i s t r i b u t i o n due to such a perturbation i s removed, the new effective charges may he lower. This i s found to be the case for diatomic cubic crystals ( 6 p ) . S z i g e t i (60) has calculated the effec-t i v e charge for atoms i n a diatomic cubic l a t t i c e . When the external f i e l d i s removed, E_ i s determined by the depolarization f i e l d . In the case of atoms lying on cubic sites i n a spherical c r y s t a l the depolar-4n i z a t i o n f i e l d according to the Lorentz r e l a t i o n i s - y P. Substituting t h i s result into: P_ = ( ZtW) + x E f6-) —a v = —•'a aa—a v"J 64 gives the pol a r i z a t i o n of a spherically shaped sample i n the absence of l an external f i e l d as: P = 3 f z » l -« -r^ 1= - J a (7) This result can be interpreted as giving effective charges of 3/(e^+2) q (k) aa The atoms, however, must be oi} cubic s i t e s . This means that this r e s u l t cannot be applied i n a rigorous sense to the phase IV ammonium halides since the hydrogen atoms occupy C^v s i t e s . It can only be applied i f we associate mass m with the NH^ ions and mass with the halide ions. Since n i s now e f f e c t i v e l y two the sum on the righ t hand side of equation 6 has only one term: This w i l l correspond to the external l a t t i c e vibra-t i o n . The calculated values of q and { 3 / ( e ^ + 2)"j- q for the ions of ammonium chloride and ammonium bromide are: NH4C1(IV) ND4C1(IV) NH 4Br(IV) ND 4Br(IV) q 1.21e 1.21e 1.06e 1.075e |3/(e(°)+2)j q 0.77e 0.77e 0.65e 0.66e For a completely ionic s a l t i t i s expected that ^3/ ( e ^ * 2 ) J q/e w i l l be unity. Deviations from unity have been attributed to overlap of neighbouring ions (60). Tables 4-1 and 4-2 include assignments for the fundamental modes of a l l the phase IV ammonium halides. The corresponding phase IV spectra are found i n Figures 4-1, 4-2, 4-3 and 4-4. In regard to the phase IV spectra and also to the phase I I I and II spectra which are presented l a t e r i t i s to be emphasized that a large 65 TABLE 4-1. Assignments f o r NHXI (IV) and ND Cl (IV); cm - 1 Assignment NH 4C1(IV) - THIS WORK ND 4C1(IV) - THIS WORK See also r e f s . See also r e f s .  29,30 22,24 22,24 R. I.R. R. I.R. 3049 2215 v 2 1719 1227 v 3 ( t ) 3127 3126 2333.5 2332 v 3 ( D 3164 2371 v ^ t ) 1401 1400 1063.5 1061 v-(£) 1421 1069.5 L i (390)* (280) T i 183 175 v A(K^O) (92) v A(K/0) (51) (62) V A(K^O) (25.) (33) V i + T i 3223 V l + V A 3141: v 1 + v A 3100: 3098 2277 2275 v 1 + v A 3074 2248 2247 (F 2) 3050 2260 2259 2 V i , ( F 2 ) 2840. 2828 2145 2139 2v l t(Ai) 2119 v 2+Li 2022 2025 1458 1460 v ^ + L i 1813 1813 1344 1348 Vi» (t)+Li 1791. 1791 2Li 748 546 * the bracketed frequencies have been i n f e r r e d from Raman active combinations; the frequencies are inf e r r e d assuming r e l a t i v e l y f l a t v 1 phonon dispersion. r 66 TABLE 4-2. Assignments f or NH.Br(IV) and ND.Br(IV); cm"1 Assignment NH 4Br(IV) - SCHUMAKER See Ref. 24 R. I.R. ND 4Br(IV) - THIS WORK See also r e f s . R. 23,24 I.R. V i v 2 va(t) v 3 U ) Vi»(t) v - U ) L i T i 3047 1699 3126 1402 1418 (349)* 160 3124 1398 2211 1211 2332 2370 1061 1065(sh) (251) 146 2330 1060 v A(K#0) V A(K*0) V # 0 ) Vi+Ti V l + V A V l + V A V i + v A (F 2) 2V 4(F 2) 2v 4(Ai) V 2 + L 1 Vi»(£)+Li V l t(t)+Li 2Li (28) 3200 3075 3040 2805 1970 1765 1747 672 (48) (23) 2259 2234 2246 2131 2110 1415 1312 2258 2232 2246 2127 1413 1312 488 the bracketed frequencies have been i n f e r r e d from Raman active combinations; the frequencies are inf e r r e d assuming r e l a t i v e l y f l a t v. phonon dispersions. 30NVGcdOSaV F i g u r e 4 - 2 The R a m a n Spect rum of N H 4 C l ( i v ) NH Cl 0 V ) - 8 O ° K 4 3 2 0 0 3 0 0 0 1600 ~i 1 r~7//~i—1—1—i— 1 4 0 0 2 0 0 lOO WAVENUMBER CM-1 ON 00 69 Figure 4 - 3 The I. R. spectra of N D 4 C l ( i v ) and N D 4 B r ( i v ) 0.8-1 .0"f ' r—1 ' 1 ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1- r-2 3 0 0 2 1 0 0 1 9 0 0 1 7 0 0 1 5 0 0 1 3 0 0 1100 WAVENUMBER C M - 1 70 Figure 4 - 4 The Raman Spec t ra of ND 4 CL ( i v ) and N D 4 B r a v ) WAVENUMBER C M - 1 71 portion of the work undertaken, p a r t i c u l a r l y where the i n f r a r e d work i s concerned, i s not o r i g i n a l and, indeed, for the most part serves only to v e r i f y the findings of e a r l i e r workers. When r e s u l t s are presented which duplicate those of e a r l i e r workers t h i s i s indicated i n the tables and reference i s made to the o r i g i n a l work. There are f i v e combinations and overtones appearing i n the v i b r a t i o n a l spectra of the phase IV ammonium halides which are r e a d i l y assigned. These are, i n order of increasing frequency, 2Li(A + E + F 2 ) , + L^(A 2 + E + Fi + F 2 ) , 2v 4(Ai + E + F 2 ) , v 2 + v 4 ( F x + F 2) and \>1 + T 1 ( F 2 ) . Of these f i v e , 2Li and + T^ are not seen i n the Raman e f f e c t ; a l l the re s t appear i n both the Raman and i n f r a r e d spectra. Perhaps the most important combination i s v 4 + (or v 4 + v^) . Its importance arises from the f a c t that h i s t o r i c a l l y (22) i t has been used to place and, indeed, the frequencies obtained from' the Raman observed v4 + VR s c & , t t e r i n 6 s f ° r & H the ammonium halides are found to d i f f e r by only ±2 cm 1 from the corresponding frequencies obtained from neutron i n e l a s t i c s c a t t e r i n g measurements (3) (see also Table 4-8). In both the infr a r e d and Raman spectra of NH 4C1(IV) and NH 4Br(IV) v 4 + appears as a doublet. In a l l cases the observed s p l i t t i n g i s comparable to the s p l i t t i n g associated with the transverse and lon g i t u d i n a l components of v 4- This suggests that the previously anomalous component (22) of component v 4 + be assigned as v 4 ( £ ) + L^. The low wave number^will be v 4 ( t ) + L^. The deuterated analogs do not show a v 4 + s p l i t t i n g - t h i s i s not t unexpected, however, since i n t h e i r case the s p l i t t i n g associated with the transverse and lo n g i t u d i n a l components of v. i s about 5 cm 1 compared 72 to about 20 cm"1 f o r NHXUIV) and NH 4Br(IV). For ND 4C1(IV) and ND 4Br(IV) the overtone of v 4 > 2v 4(A x + E + F 2) i s s p l i t into two d i s t i n c t components i n the Raman spectra. This i s most l i k e l y due to the appearance of a component of A} symmetry type and also a component containing both E and F 2 symmetry types. The v^A,) funda-mental l i e s near 2v 4 at higher wave number (see Tables 4-1 and 4-2) and i f v 1 ( A 1 ) undergoes Fermi resonance with 2v^(k\ + E + F 2) the 2v 4(Ai) component i s expected to appear on the low wave number side of 2v 4(E + F 2 ) . On t h i s b a s i s , then, the sharp symmetric l i n e appearing at 2119 cm"1 i n the Raman spectrum of ND 4C1(IV) becomes 2v 4(Aj) and the broader asymmetric l i n e at 2144 cm 1 becomes 2v 4(E + F 2) (see Figure 4-4). This l a t e r l i n e compares with the l i n e 2 v 4 ( F 2 ) which appears i n the i n f r a r e d spectrum at 2139 cm 1 . Similar 2v 4 l i n e s are observed f o r ND 4Br(IV). The Raman spectra of the protonated phase IV s a l t s do not show a s p l i t t i n g f o r 2v 4 -most l i k e l y because, i n t h i s case, 2v 4 i s about 200 wave numbers removed from as compared to about 1Q0 wave numbers f o r the deuterated s a l t s . There i s one f i n a l point to be noted i n r e l a t i o n to 2v 4 for the phase IV ammonium halides, and t h i s i s that the observed i n f r a r e d frequencies, 2 v 4 ( F 2 ) j a n d the observed Raman frequencies, 2v 4(E + F 2 ) ^ a r e c o n s i s t e n t l y higher than those predicted from about 10 to 40 cm Fermi resonance cannot be invoked to explain t h i s s i t u a t i o n since there are no l i n e s of F 2 symmetry type l y i n g nearby at lower wave number. Since the predicted frequencies are those predicted i n the zero wavevector l i m i t i t would appear that f o r the two phonon process giving r i s e to 2v 4 that the phonons involved must o r i g i n a t e from points i n K space where K j= 0. Such a i I :! 73 si t u a t i o n would explain the discrepancy between the observed and predicted 2v^ frequencies. For the phase IV ammonium halides v 2 + v ^ F j + F 2) l i e s some 75 cm - 1 below v^CF ) and can therefore be expected to undergo Fermi resonance with v 3 ( F 2 ) . The enhanced intensity observed for v 2 + especially i n the infrared indicatesthat Fermi resonance does, indeed, take place. A further indication of Fermi resonance i s that the observed v 2 + frequencies are considerably lower than those predicted; for NH^CICIV), NH 4Br(IV), ND4C1(IV) and ND 4Br(IV) the \>2 + frequency discrepancies are 69,57,29 and 25 cm 1 respectively. Evidently + T^CFj) which appears only i n the infrared spectra of the protonated s a l t s where i t l i e s some 75-100 cm 1 above v,j(F 2) does not undergo Fermi resonance since the observed frequencies are about 8 cm 1 lower than those predicted. In the infrared spectra of the deuterated s a l t s the presence of + T^  i s very l i k e l y masked by the intense absorptions. The f i n a l l i n e appearing i n the second order spectra which i s r e a d i l y assigned i s 2L^(A^ + E + F2)• This l i n e appears only i n the infrared spectra. For the series of s a l t s ; NHjClflV), NH 4Br(IV), ND4C1(IV) and ND 4Br(IV) the respective anharmonicities are 16, 13, 7 and 7 cm 1. In addition to the f i v e combinations and overtones discussed above there are four lines appearing i n the vi b r a t i o n a l spectrum of the phase IV ammonium halides which are not obvious combinations or overtones of 74 the fundamentals. With reference to NHX1CIV) the f i r s t of these we s h a l l discuss i s the l i n e appearing at 2025 cm - 1 i n the i n f r a r e d . H i s t o r i c a l l y (22) t h i s l i n e has been assigned to \>2 + L-^  (Fi + F 2) • As the following discussion w i l l show i t turns out that t h i s i s the most acceptable assignment but not f o r obvious reasons. Assuming \>2 + i s the correct assignment i t i s found that the frequencies pre-dicted f o r v 2 + i n the N H X l f l V ) spectrum and i t s counterparts i n the spectra of NHjBrflV), NH 4Br(III) and NH 4I(III) are r e s p e c t i v e l y 84, 78, 55 and 37 cm 1 higher than those observed. For the same serie s of deuter-ated s a l t s these values are 47, 49, 31 and 20 cm This s i t u a t i o n can-not a r i s e from Fermi resonance since there are no l i n e s of F 2 symmetry lyi n g nearby at higher wave number. Also, recent studies (24) by Schumaker involving t h i n s i n g l e c r y s t a l s l i c e s have shown that the absorption i n question i s structured f o r both N H X l f l V ) and NH^BrflV) and possesses four d i s t i n c t maxima. These appear at 2015, 2040, 2085 and 2100 cm 1 for NH4C1(IV) and at 1970, 1985, 2Q20 and 2040 cm - 1 for NH 4Br(IV). Since the observations mentioned above are not compatible with the observations for the v 4 + (or v 4 + v R) which indicate unstructured li n e s whose frequencies coincide (within experimental error) with- those predicted, the assignments + Lj (or v 2 + v R ) may be supposed to be inc o r r e c t . However, there are no other obvious combinations or overtones which w i l l place these l i n e s i n the desired spectral region. Also, the frequency r a t i o s of the l i n e s appearing i n the spectra of the deuterated and protonated s a l t s f o r the serie s NHXUIV), NH 4Br(IV), NH 4Br(III) and NH 4I(III) are compatible with the assignment v 2 + v R ; i . e . the frequency r a t i o s are 0.719 ± 0.002. (This value corresponds to corresponding v 2 and v R r a t i o s of 0.715 ± 0.002 and 0.717 + 0.002.) Furthermore, Schumaker found the temperature dependence of the v 0 + v D l i n e s to be very s i m i l a r to that of the + v R l i n e s . Therefore, we have frequency r a t i o and temperature dependence r e s u l t s which are compatible with the assignment v 2 + and predicted frequency and band contour r e s u l t s which seemingly are not. I f the assignment v 2 + L ^ f F i + F 2) i s made only a l i n e of F 2 symmetry type can contribute to the observed i n t e n s i t y i n both the in f r a r e d and Raman spectra. This means that the anomalous components ( i . e . the four maxima observed f o r NH^CICIV) and NH^BrCIV)) must involve phonons with K 4 0. Such a s i t u a t i o n could also account for the discrepancies between the observed and predicted frequencies. On t h i s basis the assignment v 2 + i s not an unreasonable choice. The remaining three l i n e s which are not obvious overtones or com-binations of the fundamentals w i l l be discussed as a group. In the case of NH^CICIV) these l i n e s , as observed i n the Raman spectrum, appear at 3141, 3100 and 3074 cm"1 (see Table 4-1). Of these three l i n e s only the 3100 cm 1 l i n e appears i n the in f r a r e d ; the appearance of the 3141 and 3074 cm 1 l i n e s i n the i n f r a r e d i s probably masked by the intense and + absorptions which appear at 3126 and 3050 cm 1 r e s p e c t i v e l y Schumaker (24) does not report the corresponding l i n e s f o r the Raman spectrum of NH 4Br(IV). However, the N H ^ B r f l l l ) Raman l i n e s of s i m i l a r o r i g i n appear at 3123, 3098 and 3068 cm - 1 (see Table 4-3). Of these three l i n e s only the 3068 cm 1 l i n e i s observed i n the i n f r a r e d . The remaining two li n e s are very l i k e l y hidden by the intense v 3 absorption at 3115 cm 1 In the spectrum of ND.Cl(IV) there are anomalous Raman l i n e s at 2277 and 76 2248 cm 1 . The corresponding i n f r a r e d l i n e s appear at 2275 and 2247 cm - 1. For ND 4Br(IV) the same set of l i n e s appears at 2259 and 2234 cm - 1 i n the Raman and at 2258 and 2232 cm - 1 i n the i n f r a r e d . The spectra of the f u l l y deuterated s a l t s do not give evidence f o r the t h i r d high wave number l i n e which i s observed f o r NHX1CIV). However, i t s appearance could e a s i l y be masked by the very intense li n e s observed for v^. The only fundamentals that these heretofore unassigned l i n e s can be s a t i s f a c t o r i l y r e l a t e d to i s i n combination with acoustical modes. This means that these l i n e s l i k e those f o r both v 2 + ^ and 2v^ must involve phonons with 0. If we assume a one dimensional type l a t t i c e as discussed i n section 3-4, the calculated zone edge transverse a c o u s t i c a l modes for N H X l f l V ) , NDX1CIV), NH 4Br(IV) and ND 4Br(IV) l i e at 104, 102, 71 and 70 cm 1 r e s p e c t i v e l y . Providing we assume r e l a t i v e l y f l a t phonon despersions, the assignment + acoustical modes (v^ + y^) for the l i n e s under discussion gives a c o u s t i c a l modes at 92, 51 and 25 cm 1 for NH 4Cl(IVj; 62 and 33 cm"1 for ND 4C1(IV); 85, 60 and 30 cm"1 f o r NH 4Br(III) and 48 and 28 cm"1 f o r ND 4Br(IV). The experimentally i n f e r r e d a coustical mode at 92 cm 1 for NH 4C1(IV) compares with the calculated zone edge frequency of 104 cm 1 . In the case of NH 4Br(III) we can compare the i n f e r r e d a c o u s t i c a l modes at 85 and 60 cm 1 with the predicted NH 4Br(IV) zone edge frequency of 71 cm 1 . It i s to be emphasized that the calculated zone edge frequencies are only a rough estimate of the upper frequency l i m i t s ( i . e . at maximum wavevector) for the a c o u s t i c a l modes. However, since the i n f e r r e d frequencies do tend to f a l l within these l i m i t s the most meaningful assignment for these previously unassigned 77 l i n e s i s i n combination with ac o u s t i c a l modes. A complete set of assignments f o r the phase IV combinations and overtones which have been discussed above are included i n Tables 4-1 and 4-2. 4-2 The Phase III Ammonium Halides 7 For the phase III ammonium halides we have the space group, D^CP^nmm) Under the factor group analysis the four fundamental modes associated with the free ammonium ion become the precursors of 14 c r y s t a l modes; i . e . Vi b r a t i o n a l Mode Free Ion Symmetry T„ S i t e Group Symmetry -D2d Factor Group Symmetry -4h A i , Bx B 2, E B 2, E A , B, l g l u lg lu lg l u B2g> A2u' Eg' E u B , A , E , E 2g' 2U' g' u From the above c o r r e l a t i o n diagram i t i s seen that the 14 c r y s t a l modes ar i s e from a s p l i t t i n g of the degenerate bending and stretching modes by the s t a t i c f i e l d ( s i t e group) and also from a further s p l i t t i n g by the c o r r e l a t i o n f i e l d (factor group). This l a t t e r s p l i t t i n g a r ises from the fact that there are two molecules per un i t c e l l and hence they may execute the same normal v i b r a t i o n either i n phase or out of phase to give r i s e to c r y s t a l modes of g and u symmetries., Summarizing the i r r e d u c i b l e representation f o r the 14 i n t e r n a l fundamentals gives: 78 T (internal) = 2A, + B, + 2B + 2E !g lg 2g g + A, + 2Ao + 2B, + 2E I U 2U I U u Under the factor group a l l of the above gerade symmetry species contain elements of the p o l a r i z a b i l i t y tensor and hence are associated with Raman active vibrations, whereas only the A„ and E vibrations are ' 2u U infrared active. The irreducible representations for the three types of external modes are: T(OT) + A, +B + 2 E + A +E lg 2g g 2" u r(OL) = A. + E + B„ + E 2g g 2u u r (AT) = A + E 2u u We may note that the 7 infrared active modes can be expected to possess longitudinal components which w i l l be Raman active. The only free ion fundamental mode for which s i t e group components have been assigned i s (23,29). In the case of NH^Brflll) both the infrared C^ CA.^ ), v^(E u)) and the Raman (v4(B2g)» v^ CE^ )) assignments have been made. From the results obtained i n th i s work the NH^Br^II) frequencies are: infrared Raman 1405 cm"1 1402 cm"1 1433 1422 We see that the magnitude of the infrared s p l i t t i n g i s 28 cm 1 and that 1 79 of the Raman s p l i t t i n g only 20 cm"1. For ND 4Br(III) the in f r a r e d spec-trum gives a s p l i t t i n g of 23 cm 1 . Unexpectedly no Raman s p l i t t i n g i s observed. There are two p o s s i b i l i t i e s : (i) the high wave number NH 4Br(III) s c a t t e r i n g i s indeed due to v 4 ( B 2 g ) and i n the case of ND 4Br(III) i t j u s t does not appear with observable i n t e n s i t y or ( i i ) the assign-ment ^(Bgg) of previous workers i s in c o r r e c t and the high wave number v 4 s c a t t e r i n g i s an expected l o n g i t u d i n a l component. I f we assume t h i s l a t t e r s i t u a t i o n there i s a v 4 longitudinal-transverse s p l i t t i n g of 17 cm 1 . This compares with the 16 cm 1 NH 4Br(IV) s p l i t t i n g of s i m i l a r o r i g i n . The corresponding ND 4Br(IV) s p l i t t i n g i s only 4 cm 1 . Since i t i s determined by assigning a shoulder i t i s not unexpected that a v 4 ND 4Br(III) Raman s p l i t t i n g would not be resolved. Of the two p o s s i -b i l i t i e s f o r the assignment of the v 4 high wave number sc a t t e r i n g v 4 (longitudinal) i s the most acceptable since i t i s compatible with the observed NH 4Br(III) v 4 s p l i t t i n g s and also suggests that the ND 4Br(III) v 4 Raman s p l i t t i n g w i l l not be resolved. The v 4 r e s u l t s obtained f o r the ammonium iodides are s i m i l a r to those obtained f o r the ammonium bromides (see Table 4-4). With respect to the phase III assignments previous workers (23,29) have only been able to i d e n t i f y the more intense doubly degenerate com-ponents. In th i s work i t i s found that there i s one exception. The Raman spectrum of NH 4Br(III) allows us to place v^(^2^) at 3079 cm 1 ; V g(Eg) i s placed at 3117 cm 1 . (This l a t t e r frequency compares with the v 3 ( E u ) i n f r a r e d frequency of 3115 cm 1.) The Raman s p l i t t i n g and the infr a r e d v 4 s p l i t t i n g thus give e f f e c t i v e v 3 / v 4 s i t e group s p l i t t i n g s of 80 if 38 and 28 cm 1 . The corresponding s p l i t t i n g s f o r NaBH 4(II) which has a comparable angular d i s t o r t i o n are 31 and 26 cm 1 (see Table 5-3). None of the Raman spectra of the phase III ammonium halides possessed s c a t t e r -ings which could be i d e n t i f i e d with a lo n g i t u d i n a l component. Unexpectedly no v 2 s i t e group s p l i t t i n g was found i n the Raman spectra of the phase III ammonium ha l i d e s . This has also been the f i n d i n g of previous workers (29,30). Without exception the v 2 Raman l i n e s are very sharp. They are found to have a half-height width of about 5 cm 1 and give no evidence of the presence of a second component. It i s i n t e r -esting to note here that the Raman spectra of phase II sodium borohydride (see Table 5-3) shows a d i s t i n c t doublet f o r v 2 and only one component f o r My This i s ju s t the reverse of the s i t u a t i o n observed f o r phase III ammonium bromide. One of the i n t e r e s t i n g features of phase III spectra i s that the p a r t i a l l i f t i n g of the degeneracy associated with the L^(F 1) l i b r a t i o n a l mode of the phase IV ammonium halides allows f o r the appearance of an E l i b r a t i o n a l mode i n the Raman and an E l i b r a t i o n a l mode i n the in f r a r e d , g u Altogether there are four possible l i b r a t i o n a l modes which may be distinguished according to the scheme: L^Ag ), L 2 ( E g ) , L 3 ( B 2 u ) and L 4 ( E u ) . The L-^A^) and L 3 ( B 2 u ) modes are neither Raman nor in f r a r e d active. It i s found that the L 2(Eg) modes do appear, a l b e i t weakly, i n the Raman spectra of NH 4Br(III) and NH 4I(III). The inf r a r e d active mode, L 4(E u),does not appear with observable i n t e n s i t y i n phase III spectra. The remaining external fundamentals to be discussed are those of trans l a t o r y o r i g i n . There are s i x such fundamental modes and they may 81 be distinguished according to the following scheme: C B 2 )> T (E ), T^A2u)' W' T 5 C A l g ) a n d VV' T h e v i b r a t i o n s V T 2 ' T 3 a n d T4 ar i s e from t r a n s l a t i o n a l motions i n which the motions of the ha l i d e and ammonium ions are out of phase. The T 5 and T^ vibr a t i o n s involve halide and ammonium ion motions which are i n phase. Of the four t r a n s l a t o r y modes predicted to be Raman active only T i ( B ) and T (E ) were observed i n the phase III spectra. In the Raman spectrum o f NH^BrCIII) these l i n e s which a r i s e from the two non-equivalent ° -1 N-Br distances of 2.41 and 2.55 A are found at 183 and 137 cm . The early s i n g l e c r y s t a l studies of Couture-Mathieu and Mathieu (29) suggest the respective assignments T (B ) and T„(E ) for the low and high wave i 2g Z g number scatterings. The i n f r a r e d spectrum of N H ^ B r f l l l ) shows only a s i n g l e intense absorption at 172 cm 1 which almost c e r t a i n l y derives i t s maximum i n t e n s i t y from T ^ ( E u ) . In the case of NH 4I(III) where the o two non-equivalent N-I distances are 2.69 and 2.74 A the T ^ f B ^ ) and T 2(Eg) modes are observed at 124 and 158 cm The T^(E u) mode appears i n the inf r a r e d at 151 cm At t h i s point i t i s to be noted that very recently two papers (58,61) appeared simultaneously which report the r e s u l t s of independent studies of the external v i b r a t i o n s of the ammonium ha l i d e s . In general, the r e s u l t s of these studies are i n good agreement with the r e s u l t s of t h i s work (see Table 4-5). One very i n t e r e s t i n g feature of both studies i s the assignment of the Raman active T^(A^) and T^(E^) v i b r a t i o n s associated TABLE 4-3. Assignments f or NH Br(III) and ND.Br(III); cm"1 NH 4Br(III) - THIS WORK ND 4Br(III) - THIS WORK See also r e f s . See also r e f s .  Assignment 29,32 23 23 R. I.R. ' R. I.R. Vi(Aig) 3037.5 2209 v 2 ( A i g / B i g ) 1694 1209.5 v 3 ( B 2 g , A 2 u ) 3079 v 3(E g,E u) 3117 3115 2339 2336 v 4 ( B 2 g , A 2 u ) 1433 1089 Vi»(E g,E u) 1402 1405 1063 1066 v 4(E u)£ 1422 L 2 ( E g ) , L i | ( E u ) 336 (241) T i ( B l g ) , T 3 ( A 2 u ) 137 124.5 T 2 ( E g ) , T l t ( E u ) 187 172 164 VA(UO) (85)* VA(K^O) (60) V A ( K f O ) (30) 3195 Vi + V A 3123 v 1 + v A 3098 Vi+V A 3068 3068 V 2+V 4(E g,E u) _ 3037 2243 2v„(E g,E u) 2817 2805 2127 2122 2v4(Aig) 2105 V 2 + V R 1975 1959 1420 V1+ + VR 1737 1302 1296 2V R 648 470 the bracketed frequencies have been i n f e r r e d from Raman active combinations; the v A frequencies are i n f e r r e d assuming Vi phonon dispersions which are r e l a t i v e l y f l a t . 83 TABLE 4-4. Assignments f or NHjICIII) and ND 4 I ( I I I ) ; c m - l NH 4I(III) - THIS WORK ND 4I(III) - THIS WORK See also r e f s .  Assignment 25 R. I.R. R. I.R. V i ( A i g ) 3035 2202 v 2 ( A i g / B i g ) 1667 1194. .5 V3 ( B 2 g,A 2 U) V 3 ( E g , E u ) 3105 3105 2320 2316 v 4 (B2 g,A 2u) 1421 1077 Vit (E g,E u) 1398 1401 1059 1062 1409 L 2 (Eg) , Li» (E u) 291 208 T i ( B l g ) , T 3 ( A 2 u ) 124 113 T 2 ( E g ) , T l t ( E u ) 158 151 146 vi+v T 3166 V2+Vi» (E g,E u) 3025.5 3023 2233 2229 2v 1 +(E g,E u) 2798 2791 2121. ,5 2118 2 v 4 ( A i g ) 2101. 5 v 2+v R 1921 1912 1385 1383 Vi*+VR 1686 1681 1268 1265 2v R 568 410 84 Figure 4 - 5 The I.R. spect rq of N H 4 B r ( i i i ) and N H 4 I ( i i i ) o.o-0.2 0 . 4 H 0 . 6 0.2 0 . 4 O. 6 0.8-I . O N H B r ( i n ) - 8 0 0 K 4 1 1 I r ' 1 1 7/ 1 1 1 1 1 r , 1 , , , 1 r-N H 4 I (lll)-80°K -i i i i i — i — r -3 2 0 0 3 0 0 0 T l I i r — T / / , 1 1 1 1 , , , , , , , ( r 2 8 0 0 2 6 0 0 1 8 0 0 1 6 0 0 1 4 0 0 WAVENUMBER C M " 1 85 F i g u r e 4 - 6 The Raman Spectra of N H 4 B r ( i n ) and NH 4I( I I D NH 4Br(lll)-80 0K W A V E N U M B E R C M - 1 86 Figure 4 - 7 The I .R .spectra of ND 4 Br ( i i n and ND 41 (in) 0.6H 0.8-1 .OH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 r 2 2 0 0 2 Q O O 1 8 0 0 1 6 0 0 1 4 0 0 1 2 0 0 WAVENUMBER C M " 1 87 F i g u r e 4 - 8 T h e R a m a n S p e c t r a of ND 4 Br( i i i ) and ND 4 1 ( I I D N D B r (III)- I 9 5 ° K ND 4 I ( IM )-80 °K | ' ' ! 1 ' ' ' 1— 1 1 1 ' 1 1 1 1 1 //-> ' — i — I — 2 4 0 0 2 2 0 0 2 0 0 0 1 4 0 0 1 2 0 0 1 0 0 0 3 0 0 100 W A V E N U M B E R C M ~ 1 88 TABLE 4-5 The External Vibrations of the Ammonium Halides; cm PHASE IV PHASE II THIS WORK REF. 58 REF. 58 THIS WORK REF. 58 R R I.R. I.R. I.R. NHXicrp 183 185 188 177 172 ND ^ i c r p 175 177 177 170 166 NH 4Br(T 2) 159 157 154 147 ND 4Br(T 1) 146 147 149 139 138 THIS WORK REF.58 REF.61 NH 4Br(III) 336 300 331 w 137 133 132 183 179 179 T 4(Eu) 172 153 -TS<V 62 63 75 76.5 ND 4Br(III) - - -W 124.5 122 119 T 0 ( E ) 164 165 163 T 4(Eu) - 145 -T 5t A lg) 63 62 T 6 ^ g ^ 75 72 TABLE 4-5: continued THIS WORK REF. 58 REF. 61 NH 4I(III) L 0 ( E ) 291 285 287 2 V z T 1 ( B l g ) 124 123 123 T_(E ) 158 155 155 2 V gJ T 4(Eu) 151 151 147 TrdK ) 45 45 5^ lgJT,(E ) 57 56 6' gJ ND 4I(III) L 2 ( E g ) 208 T. (B. ) 113 115 1 l g T„(E ) 146 145 T 4(Eu) W 4 3 T 6 ( E g ) 55 90 with the phase III ammonium halides. In the Raman spectrum of NH^BrCIII) Durig and Antion (61) placed T 5 ( A l g ) at 63 cm - 1 and T 6 ( E g ) at 76.5 cm - 1. The corresponding frequencies as observed by Perry and Lowndes (58) are 62 and 75 cm The reason these Raman active l i n e s were not observed i n t h i s work i s apparently due to f a i l u r e to scan s u f f i c i e n t l y close tp the e x c i t i n g l i n e . The combinations and overtones observed i n the spectra of the phase III ammonium halides are s i m i l a r i n o r i g i n to those observed for the phase IV s a l t s . Thus, we have the following "combinations" and "over-tones": 2v R, v 4 + v R , v 2 + v R , 2v 4, v 2 + v 4 , + v T and + v A > where the subscripts associated with v R , Vj and r e f e r to rotatory ( l i b r a t i o n a l ) , t r anslatory and a c o u s t i c a l modes re s p e c t i v e l y . C l e a r l y each of the above "overtones" and "combinations" may have a number of components c o n t r i -buting to i t s i n t e n s i t y . However, i t i s convenient to i n i t i a l l y make only the gross d i s t i n c t i o n s between the various combinations and overtones ; which have been indicated. As the discussion progresses more e x p l i c i t assignments can be made as they are needed. The only phase III ammonium halide f or which acoustical modes i n comb ina t i o n with appear i s NH 4Br(III). These l i n e s which l i e at 3074, 3100 and 3141 cm have previously been discussed. For the remaining phase III s a l t s i t i s l i k e l y that band overlapping and/or i n s u f f i c i e n t i n t e n s i t y obscures appearance of the li n e s + v^. In both the i n f r a r e d and Raman spectra of the phase III s a l t s bands with s i n g l e maxima are observed for both v 4 + v R and v 2 + v R (see Tables 4-3 and 4-4). These bands may however contain contributions from several nearly a c c i d e n t a l l y 91 degenerate components. It i s to be noted that there i s a s i g n i f i c a n t i n f r a r e d -Raman frequency discrepancy f o r both v 4 + v R and + v R-For example for NH 4Br(III) v 4 + v R appears at 1737 cm - 1 i n the Raman and at 1728 cm 1 i n the i n f r a r e d . For + v R the corresponding f r e -quencies are 1975 and 1959 cm The most l i k e l y explanation for t h i s type of behaviour i s a s i g n i f i c a n t g-u type s p l i t t i n g . The l i n e s associated with both 2v R and v-^  + Vj appear only i n the inf r a r e d . The phase III assignments f o r the 2v D i n f r a r e d l i n e s are: L_(E ) + L.(E ) = 2v D(A, ) 2 V gJ 4 u R 2ir U (A ) + L.(E ) = 2v D(E ) 1 g 4 u"^  R v u' L2<V + W = 2 W Since 2v D shows only one maximum i t i s l i k e l y that the three allowed components are a l l contained i n the one band envelope. A s i m i l a r s i t u a t i o n occurs f o r + v^ , where the four i n f r a r e d active components; • + T 3 ( A 2 U ) = V l + V A2U> W + W = v l + W W + VV = V l + V T ( A 2 u ^ VBlu> + T 2 ( V = Vl + W give only a si n g l e maximum. If i t i s assumed that the maximum i n t e n s i t y i s due to the combination ^(A^g) + T 4 ( E U ) then the observed frequencies for both NH 4Br(III) and NH 4I(III) are about 30 cm"1 lower than those 92 predicted. The overtone 2v 4 for the phase III s a l t s appears i n the i n f r a r e d as an absorption with a si n g l e maximum. The predicted components of t h i s absorption are: VV + v 4 ( A 2 u ) = 2W w + w - 2VV W + v 4 ( E u ) = 2 v 4 ( A 2 u ) In the zero wavevector l i m i t the 2 v 4 ( A 2 u ) components f o r a l l the phase III spectra have predicted frequencies which are higher than those observed. Therefore the maximum i n t e n s i t y of 2v 4 i n the in f r a r e d i s most l i k e l y derived from the two combinations of E u symmetry type. In the Raman spectra only components of gerade symmetry type may appear - there are si x such 2v 4 components. For the protonated s a l t s only one Raman peak for 2v 4 appears and on the basis of frequencies predicted i n the zero wavevector l i m i t the most reasonable assignments are: w + v 4 ( v = 2 vv v 4 ( A 2 u ) + v 4 ( E u ) = 2 v 4 ( E g ) The deuterated phase III s a l t s show two d i s t i n c t 2v 4 Raman peaks. The p r i n c i p a l components of the high frequency peak are very l i k e l y the two E components and the p r i n c i p a l components of the low frequency peak w i l l then be the overtones of the E u and E g modes which are both of symmetry 2v.(A + B, + B. ). Since these components 93 contain the symmetry type A i t may well be that they undergo Fermi resonance with v i ( A ). The f i n a l combination to be discussed i s v 2 + v 4- In phase III spectra v 2 + appears at e s s e n t i a l l y the same frequency and without any observable s p l i t t i n g i n both the i n f r a r e d and Raman. Also, v 2 + v 4 shows enhanced i n t e n s i t y . This may be at t r i b u t e d to Fermi resonance with v,(E ) and v,(E ) i n the i n f r a r e d and Raman res p e c t i v e l y . Thus, i t i s o U j g very l i k e l y that the l i n e s observed f o r v 2 + v 4 a r i s e from a combination of one p a r t i c u l a r component of v 2 with v 4 ( E g ) and v 4 ( E u ) . I f the v 2 component involved i s the one observed i n the Raman spectra the observed v2 + v4^ Eu^ f r e 9 u e n c i e s f o r NH 4Br(III) and NH 4I(III) are 62 and 42 cm - 1 lower than those predicted. These values f o r the corresponding deuterated s a l t s are 33 and 34 cm The large differences between the observed and predicted frequencies are a further i n d i c a t i o n of the eff e c t s of Fermi resonance. A complete set of assignments f o r the fundamental, combination and overtone modes observed f o r the phase III ammonium halides i s found i n Tables 4-3 and 4-4. The corresponding spectra are reproduced i n Figures 4-5, 4-6, 4-7 and 4-8. 4-3 The Phase II Ammonium Halides The v i b r a t i o n a l analysis f o r the phase II ammonium halides i s com-p l i c a t e d by the disorder introduced into the l a t t i c e v i a the randomness associated with the hydrogen atoms. Both L i f s h i t z (26,62) and more recently Whalley and Bertie (27) have shown that when t r a n s l a t i o n a l p e r i o d i c i t y 94 i n a l a t t i c e i s destroyed the wavevector s e l e c t i o n rules breakdown. L i f s h i t z considers the o p t i c a l behaviour i n the i n f r a r e d of a cubic c r y s t a l containing a random mixture of two isotopes. For t h i s simple model three important r e s u l t s are obtained; (i) f o r a p a r t i c u l a r p o l a r i z a t i o n whether long i t u d i n a l or transverse t r a n s i t i o n s with K •/ 0 w i l l become allowed, ( i i ) the absorption band due to the t r a n s i t i o n s having K ^ 0 w i l l be s h i f t e d i n frequency r e l a t i v e to the K = 0 absorption and ( i i i ) the i n t e n s i t y of the K / 0 band i s proportional to c ( l - c ) where c i s the mole f r a c t i o n of the impurity isotope. The treatment by Whalley and Bertie considers a c r y s t a l having molecules arranged on a regular l a t t i c e . Disorder i s introduced into the l a t t i c e by allowing the molecules (or dipoles) to take up random orientations. The r e s u l t i s that a l l external t r a n s l a t i o n a l l a t t i c e v i b rations whether o p t i c a l , a c o u s t i c a l , transverse or lo n g i t u d i n a l w i l l become ac t i v e i n both the inf r a r e d and Raman spectra. In each case there w i l l be densities of states having K. ^  0. Whalley and Bertie preface t h e i r treatment of a three dimensional c r y s t a l by considering the simple case of a monatomic chain which has masses, M. The equation of motion f o r the s**1 member of the chain may be written as: d 2X s = 2 f p ( X s + p - X s) (1) dt p In the summation p runs over a l l p o s i t i v e and negative integers such th that a l l i n t e r a c t i o n s of the s atom with i t s neighbours are considered; th th f D i s the force constant connecting the s atom to i t s p neighbour. 95 If we associate the integers s and p with planes i n a monatomic l a t t i c e we may have longi t u d i n a l or transverse p o l a r i z a t i o n s ( 5 5 ) . The fp w i l l , i n general, be d i f f e r e n t i n each case. We look f o r solutions of equation 1 i n the form of t r a v e l l i n g waves. Convenient t r i a l solutions are: X g = A exp i{ska - wt) X g + p = A exp i{(s+p)ka - ait} (2) where a i s the l a t t i c e constant. When the t r i a l solutions are substituted into equation 1 we have the new equation of motion: u^ 2 = 1 Z_ f n ( l - c o s pka) (3)  - f l - cp>0 P r th The displacement X v of the s plane during the v i b r a t i o n k w i l l be: S j K X g k = A exp { i ska} • exp {-i ^ t } = L s , k Q k <4> th where i s the normal co-ordinate f o r the k v i b r a t i o n . The displacements X , and X v are r e l a t e d to the dipole moment S , K S+p,K d e r i v a t i v e f o r the l a t t i c e ; i . e . we may write the dipole moment deriv-a t i v e as: 3y _ 3y 3 ( X s + p " V 3Qk s,p 3 ( X s + p - X s) 8Q k (S) 96 This sum i s i d e n t i c a l l y zero unless K = 0 and f o r our monatomic l a t t i c e i t i s always zero since the K - 0 v i b r a t i o n i s a pure t r a n s l a t i o n . Disorder may be introduced i n t o the l a t t i c e by randomly placing p o s i t i v e or negative charges on the atoms. Thus, the 9u/3(X - X ) = q w i l l ~ s+p s' Mp be randomly p o s i t i v e or negative f o r a l l pa i r s of atoms. Whalley and Bertie show that under these conditions the most probable value of 2 (9U/3QJP i s non-vanishing and we have: (W 9 Q k ) 2 = Z q p 2 ( L s + p > k - L ^ , ) 2 (6) S,P 2 The i n f r a r e d absorption i n t e n s i t y i s proportional to (9u/8QjJ and the implication i s that f o r the a c o u s t i c a l modes, whether transverse or l o n g i t u d i n a l , absorption bands may appear with d e n s i t i e s of states having K i= 0 present. The i n t e n s i t y of Raman sca t t e r i n g i s given by a s i m i l a r treatment except that the derivatives of the p o l a r i z a b i l i t y with respect to the normal co-ordinate replace the d e r i v a t i v e of the dipole moment. If a diatomic chain i s considered the r e s u l t s may be extended to include o p t i c a l modes. The two d i f f e r e n t treatments of disorder; i . e . by L i f s h i t z and by Whalley and Bertie are r e a l l y complementary. In the L i f s h i t z model the force constants between the atoms are assumed to remain f i x e d and only the masses of the atoms are allowed to change; whereas i n Whalley and Bertie's model the masses of the atoms remain the same and only the orientations of neighbouring dipoles change - t h i s o r i e n t a t i o n a l d i -order may be associated with a change i n force constants. L i f s h i t z has 97 shown (27) that when these two complementary treatments are considered i n a quantum mechanical context they w i l l lead to perturbations which are completely analogous. Since the phase II ammonium halides possess o r i e n t a t i o n a l disorder ( i . e . the NH^+ ions may randomly take up two alternate orientations) the r e s u l t s we have discussed w i l l apply. If one assumes that only i n t e r -actions between nearest neighbour NH^+ ions change the p r e d i c t i o n of the L i f s h i t z theory f o r NHX1 i s that the i n t e n s i t i e s of K £ 0 bands w i l l be proportional to p(l-£) where p i s the p r o b a b i l i t y that a given nearest neighbour NH^+ - NH^+ pair are oriented " p a r a l l e l " to each other. In the case of NH^Br p w i l l be the p r o b a b i l i t y that a given nearest neighs bqur NH^+ - NH^+ p a i r are oriented " a n t i p a r a l l e l " to each other. The q u a l i f i c a t i o n s of " p a r a l l e l " or " a n t i p a r a l l e l " correspond to the arrangement of the NH^+ ions i n the ordered phase IV and phase III structures. For the completely disordered state p = % and for the ordered state p = 1. , Among the t r a n s l a t i o n a l vibrations of the ammonium halides the only evidence f o r bands appearing as a r e s u l t of disorder i s i n the Raman spectra of N H X l ( I I ) and N H ^ B r f l l ) . The low wavenumber scatterings of i n t e r e s t have been observed by Couture-Mathieu and Mathieu (1952) and by Perry and Lowndes (1969). In the e a r l i e r work assignments were not made and i n the more recent work of Perry and Lowndes attempts to assign the scatterings to two phonon sum or d i f f e r e n c e bands were inconclusive. Evidently the e f f e c t s of disorder were not considered. If disorder i s considered the observed scatterings may a l l be accounted f o r . Be-low are the frequencies of the low wave number phase II scatterings 98 observed by previous workers. The assignments belong to t h i s work. NH 4C1(II) NH 4Br(II) REF. 58 REF. 29 REF. 58 REF. 29 TA(K r 0) 95 88 55 56 LA(K £ 0) 140 143 135 136 T0(K = 0) 170 (172) 170 - (147) 147 T0(K £ 0) 195 197 185 -190 The bracketted frequencies are observed i n the i n f r a r e d and are associated with the K = 0 phonons. In the case of NH 4C1(II) the 170 cm"1 K = 0 scattering s h i f t s on cooling to the phase IV 183 cm 1 K = 0 s c a t t e r i n g . The highest wave number scatterings may be associated with T0(K ^ 0) phonons since the L i f s h i t z theory predicts that they may be s h i f t e d i n frequency r e l a t i v e to the allowed K = 0 mode. In the case of the TA(K ^ 0) and LA(j( ^  0) scatterings the frequency separation i s about 50 cm"1 f o r NH 4C1(II) and about 80 cm 1 for NH 4Br(II). This compares with a zone edge frequency separation of about 75 cm 1 f o r ND 4C1(IV). The ND 4C1(IV) zone edge TA phonon l i e s at 100 cm 1 and compares with the NH 4C1(II) scattering at 95 cm 1 . Recently Garland and Schumaker (28) have applied the predictions of the L i f s h i t z theory to the anomalous high wave number component of \>^ . This anomalous component i s present i n a l l i n f r a r e d phase II spectra (see for example Figure 4-9). Garland and Schumaker have considered ammonium chloride i n p a r t i c u l a r . By using an Ising model to approximate the disorder-order t r a n s i t i o n they were able to r e l a t e the heat capacity F I G U R E 4 — 9 THE TEMPERATURE DEPENDENCE OF Z/ — NH Cl AND ND CL 4 4 A o z < CD or o co CD Phase II 2 9 5 ° K O NH4CL 7 A T P h a s e IV 8 0 ° K T 1 5 0 0 I 4 0 0 1 5 0 0 1 4 0 0 WAVENUMBER CM"' P h a s e II 2 9 5 ° K (0 O ND. Cl 4 (0 O - / A P h a s e 8 0 0 K U3 IOO IOOO I IOO IOOO WAVENUMBER CM - 1 100 F I G U R E 4 - I O THE TEMPERATURE DEPENDENCE OF I l r 1 : 1 120 180 240 300 TEMPERATURE (°K) 101 data f o r ammonium chloride to the configurational energy of an Ising model and so cal c u l a t e p(l-p) as a function of temperature. By r e f e r r i n g to Figure 4-10 which reproduces Garland and Schumaker's r e s u l t s i t i s seen that the predicted and observed i n t e n s i t i e s of 121+ for the d i s -order-order process are i n excellent agreement. At the t r a n s i t i o n temperature of 242.2°K there i s seen to be a pronounced d i s c o n t i n u i t y i n the i n t e n s i t y curve f o r 11 i+j+it and the progress of the ensuing ordering process i s evidenced by a progressive loss i n i n t e n s i t y f o r I11^4. When the ordering process i s complete there i s zero i n t e n s i t y . The behaviour of Intuit f o r the disorder-order process may be summarized as: 11 i+t+1+ {^ 4(K £())}->• zero i n t e n s i t y The fact that the i n t e n s i t y behaviour of Iii+i+it so c l o s e l y p a r a l l e l s that predicted by the L i f s h i t z theory suggests that s i m i l a r considerations may apply to i n t e r n a l modes such as v 4 as well as external modes of trans1atory o r i g i n . Figure 4-12 shows the temperature dependence of for ammonium bromide as observed i n th i s work. The components appearing i n phase II are I1399 {^(K = 0)} and 1^27 {^ Qi^  0)}. At the t r a n s i t i o n tempera-ture of 234.57°K there i s a d i s c o n t i n u i t y i n the i n t e n s i t y curves f o r both I1399 and Iit+27' As the ordering process progresses, I1399 over a temperature i n t e r v a l of about 25° gradually loses i n t e n s i t y and also gradually s h i f t s to 11 i+o5• This behaviour may be summarized as: I 1 3 9 9 { v 4(K=0)} + Ii405^ 4CE u)}. 102 For Iji+27 there i s an apparent gradual increase i n i n t e n s i t y and a l s o a gradual s h i f t to 111+33 associated w i t h the di s o r d e r - o r d e r process. This behaviour i s summarized as f o l l o w s : 111+27^4(^0)} zero i n t e n s i t y I139 9{v 4(K= 0 ) } -> I 1 1 + 3 3 { v 4 ( A 2 u ) } Thus, the appearance o f Ii43 3 ( v 4 ( A 2 u } obscures the temperature dependence °f 11 t+27"tv4(K^O)}. The i n t e n s i t y curves f o r ammonium bromide do not show experimental r e s u l t s f o r the order-order t r a n s i t i o n , I I I IV. This i s because t h i s t r a n s i t i o n occurs at about 78°K on c o o l i n g (7) which i s j u s t below the low temperature l i m i t of the l i q u i d n i t r o g e n c e l l used f o r these experiments. Reference to Figure 4-13, however, w i l l show the r e s u l t s f o r the I I I -> IV t r a n s i t i o n f o r the f u l l y deuterated s a l t . I t i s seen that both 11066^4( E u)} a n c * ^1083^^(.^2U^ show marked d i s c o n t i n u i t i e s at the t r a n s i t i o n temperature of 166.7°K. At t h i s temper-ature the i n t e n s i t y o f Ii089 begins to ab r u p t l y decrease to zero, w h i l e the i n t e n s i t y of I1066 shows an abrupt increase which i s accompanied by an abrupt frequency s h i f t to 1060 cm Thus we have: Il08 9 { v 4(A. 2 u)} -*• zero i n t e n s i t y Il066<VE u)} + I 1 0 6 0 { v 4 ( F 2 ) } The discontinuous i n t e n s i t y changes and discontinuous frequency s h i f t s a s s o c i a t e d w i t h the I I I -»• IV order-order t r a n s i t i o n c ontrast sharply with the r e s u l t s f o r the I I -> I I I d i s o r d e r - o r d e r t r a n s i t i o n which i s F I G U R E 4 — 11 THE TEMPERATURE DEPENDENCE OF z*-NH 4Br AND ND4 Br CD Phase iv P |09°K I 1 / / — I 1 1 1 — I I O O I O O O II 10 1 0 0 0 1100 1 0 0 0 WAVENUMBER CM"' Phase in I09°K in o I 1 // i 1 1 5 0 0 1 4 0 0 1 5 0 0 1 4 0 0 WAVENUMBER CM-' . O H if) h 0.9 Z D >- 0.8 < h 0.7 m o r < Q . 3 H >—.> F I G U R E 4 — 12 TEMPERATURE DEPENDENCE OF THE I -O—Q—O— Q Q Q - C L X T " 1 4 0 5 UJ h o.i H o.o o-"1433 1 . 4 0 5 S H , F T S T O I , 3 9 9 I I 4 3 3 S H I F T S T O ^ 4 2 7 IN T H I S T E M P . I N T E R V A L _ O IOO 150 200 *4 NH 4 Br 3 9 9 1427 250 TEMPERATURE (°K) F I G U R E 4 — 1 3 TEMPERATURE DEPENDENCE OF THE l7y—ND. Br IOO ISO 200 TEMPERATURE (°K) 106 accompanied by gradual i n t e n s i t y changes and a gradual frequency s h i f t . These r e s u l t s , then, provide s t r i k i n g spectroscopic evidence f o r the two d i f f e r e n t types of phase t r a n s i t i o n s which occur i n ammonium bromide. Figures 4-9 and 4-11 show the in f r a r e d spectra of f o r the various phases of both ammonium chloride and ammonium bromide. It may be noted that the anomalous components of did not appear with observable i n t e n s i t y i n the Raman spectra. The t r i p l y degenerate v i b r a t i o n , v^, as observed i n phase II spectra shows no evidence of d i s t i n c t components either i n the i n f r a r e d or Raman which may be assigned to states with K £ 0. However, as observed i n both the in f r a r e d and Raman i s a very intense broad band. In the i n f r a -red the half-height band width i s some 140 cm 1 for the protonated s a l t s and some 120 cm 1 f o r the deuterated s a l t s . These values compare with corresponding half-height band widths of some 60 and 75 cm 1 for the ordered phases of the protonated and deuterated s a l t s r e s p e c t i v e l y . It i s found that there i s pronounced band sharpening as the disorder-order t r a n s i t i o n progresses and thus i t i s very l i k e l y that some of the in t e n s i t y of i n phase II spectra i s due to the presence of a density of states with K / 0. Also, exhibits anomalous behaviour i n that i t s h i f t s s i g n i f i c a n t l y to lower frequency as a r e s u l t of the disorder-order transformation. For the ammonium chlorides and ammonium bromides t h i s s h i f t i s 12±2 cm 1 . Such behaviour i s consistent with the presence of a density of states with JC ^  0 on the high wave number side of a band due to an allowed K = 0 t r a n s i t i o n . However, i t i s to be remembered 107 TABLE 4-6. Assignments f or NH C l ( I I ) and ND . C l ( I I ) ; cm N H X l f l l ) - THIS WORK ND 4C1(II)- - THIS WORK See also r e f s . See also r e f s . Assignment 29,31 22 22 R. I.R. R. I.R. V l 3050 2218 v 2 1709.5 1220 v3 3138 2348 2346 v 4 1403 1401 1064 1061 v ^ O ) 1444 1087(sh) Li (359)* (263) Ti 177 170 v 2 + V £ t 3044 2248 2\>n 2818 2810 2140 2128 2vu .:' 2116 * a neutron i n e l a s t i c s c a t t e r i n g r e s u l t - Ref. 3. 108 TABLE 4-7. Assignments f o r NH Br(II) and ND Br(II); cm"1 NH 4Br(II) - THIS WORK ND 4Br(II) - THIS WORK See also r e f s . ; See also r e f s .  Assignment 29,31 23 23 R. I.R. R. I.R. Vi 3039 2215 v 2 1685.5 1205 V 3 3128 3128 2347 2345 vk 1401 1399 1063 1060 v^K+O) 1427 1083 (sh) Li (311) (233)* T i 154 139 V 2+v 4 3028 2240 2vh 2803 2797 2125 2119 2vk 2104 * a neutron i n e l a s t i c s c attering r e s u l t - Ref. 3. 109 that such behaviour could also be r e l a t e d to an increase i n hydrogen bonding as a r e s u l t of the phase change. A summary of the assignments f o r the fundamental modes associated with the phase II ammonium halides i s given i n Tables 4-6 and 4-7. It w i l l be noted that included i n these two tables are assignments for 2v 4 and v 2 + v^; the only l i n e s which appear i n the second order spectra of the phase II s a l t s . 4-4 Ammonium Fluoride Ammonium f l u o r i d e has a c r y s t a l structure compatible with the 4 space group C^ v(P62mc). The v i b r a t i o n a l analysis i s c a r r i e d out under the C^ v f a c t o r group - r e s u l t s f o r the various types of vib r a t i o n s are summarized below. r ( i n t e r n a l ) = 3AX + 3B X + 3E! + 3E 2 T(OT) = Al + 2B : + Ei + 2E 2 T(OL) = A 2 + B 2 + E : E 2 r(AT) = Aj + Ei The four fundamental modes associated with the free ammonium ion become the precursors of the 12 i n t e r n a l c r y s t a l modes which a r i s e from both s t a t i c f i e l d ( s i t e group) and c o r r e l a t i o n f i e l d (factor group) s p l i t t i n g . This i s i l l u s t r a t e d by the c o r r e l a t i o n diagram which follows. 110 Fundamental Free Ion S i t e Group Factor Group Mode Symmetry-Tj Symmetry-C 3 v Symmetry-C 6 v v ! A l . "* Ai A i , Bi v 2 E -> E -> E i , E 2 v- F 2 A j , E -»• A i , B i , E i , E 2 3 u4 F 2 + A i , E -> A i , B i , E i , E 2 The symmetry species A j , E i and E 2 contain elements of the p o l a r i z a -b i l i t y tensor and hence w i l l be associated with Raman act i v e v i b r a t i o n s . Only vi b r a t i o n s of Ai and Ei symmetry species are i n f r a r e d a c t i v e . In section 1-1 the point was made that although the NH^+ ions i n ammonium f l u o r i d e l i e on s i t e s of C^ v symmetry, the c r y s t a l geometry i s such that i t i s compatible with a tetrahedral configuration f o r the NH^+ ions. It was further noted that a tetrahedral configuration i s also supported by the fact that Plumb and Hornig (21) found no observ-able s p l i t t i n g f o r i n the i n f r a r e d spectrum of c r y s t a l l i n e ammonium fl u o r i d e . From the i n f r a r e d spectra Plumb and Hornig were able to assign v 3 ( A 1 } Ei) and v4(kx, Ei) f o r both NH^F and ND^F. The Raman spectra recorded i n t h i s work of both NH^F and ND^F show only three l i n e s with observable i n t e n s i t y and these have been assigned to v ^ ( A i ) , v ^ f A j , E i , E 2) and v,p(Ai, E j , E 2) . It was found that there was no s i g n i f i c a n t frequency discrepancy between v^CAi, E i ) as observed i n the in f r a r e d and v ^ ( A i , E i , E 2) as observed i n the Raman. Unfortunately v 2 ( E i , E 2) was not observed i n the Raman spectrum of either NH^F or ND^F. I f the frequency of v 2 i s i n f e r r e d from v 2 + as observed I l l i n the inf r a r e d the frequencies of the i n t e r n a l fundamentals for NH4F are 2874, 1728, 2815 and 1494 cm 1 f o r V j , \>2> v 3 and v 4 r e s p e c t i v e l y . For ND^F the corresponding frequencies are 2117, 1240, 2148 and 1120 cm - 1 (see also Table 4-8). As noted by Plumb and Hornig the most s t r i k i n g feature associated with these ammonium f l u o r i d e frequencies i s the r e l a t i v e l y low frequencies of the stretching modes as compared to the corresponding frequencies of, say, the phase IV ammonium chlorides. Comparing NH^F and NH 4C1(IV) i t i s found that su f f e r s a s h i f t of 175 cm 1 and the corresponding s h i f t f o r i s 311 cm 1 . For the deuterated analogs the corresponding and s h i f t s are 98 and 184 cm respectively. This behaviour i s i n d i c a t i v e of greatly enhanced hydrogen bonding i n ammonium f l u o r i d e . The obvious combinations and overtones appearing i n the i n f r a r e d spectra of the ammonium f l u o r i d e s which were observed and assigned by Plumb and Hornig are v 4 + v R, + v R , 2v 4 and v 2 + v 4- In' addition to the above absorptions the in f r a r e d spectrum of both NH4F and ND4F possessestwo absorptions of moderate to strong i n t e n s i t y which are not r e a d i l y assigned. For NH4F the absorptions i n question appear at 3082 and 3010 cm ^. Plumb and Hornig have assigned these two l i n e s to v 4 + 3v R and a second component of 2v 4 r e s p e c t i v e l y . The corresponding anomalous l i n e s f o r ND4F appear at 2323 and 2251 cm They have been assigned r e s p e c t i v e l y to + v^ , and v 4 + 3v R. These assignments are not altogether s a t i s f a c t o r y and, indeed, Plumb and Hornig (21) make the following comment on the assignments v. + 3v R. "The biggest problem 112 (in making assignments) i s the strong band at 3082 cm - 1 i n NH^F and 2251 cm 1 i n ND^F. It has been assigned to + 3v^(v^ + 3v R), but i t i s s t a r t l i n g that such a high combination should appear with such inten-s i t y . Furthermore, the observed frequency i s some 40 c m 1 higher than predicted, and th i s can only be correct i f the frequency i s raised by Fermi resonance with some of the lower frequencies. At present the situa t i o n on th i s peak cannot be considered satisfactory." Also, the assignment of the 3010 l i n e i n the spectrum of NH^F i s not r e a l i s t i c for the following reasons: no such second component i s observed for ND^F and the frequency i s some 20 cm 1 higher than that predicted by the frequency of as observed i n the infrared. I t would appear that the assignment of the 2323 cm 1 l i n e i n the spectrum of ND^ F to + i s correct. Such an assignment gives a frequency 31 cm 1 below that predicted providing y^ , i s taken as the Raman observed frequency of 237 cm 1. Now, i f the o r i g i n a l assignment of the 3082 cm 1 NH^ F absorption to + 3v R i s changed to + the predicted-observed frequency discrepancy i s -41 cm 1 providing has a frequency equal to the Raman frequency of 247 cm - 1. We are now l e f t with the 3010 cm"1 NH^F l i n e and the 2251 cm"1 ND^F l i n e . A possible assignment i s i n combination with acoustical modes. Such a si t u a t i o n would compare with that already inferred for ; the phase IV and phase III ammonium halides. Assuming the assignment vl + VA 2 ^ v e s ac°ustical modes at 136 and 134 cm 1 respectively for NH^F and ND^F. Using the approximation of a linear l a t t i c e zone edge acoustical modes for NH.F and ND.F are calculated to l i e at 179 and 176 cm 1. Thus, 4 4 the inferred modes are within the frequency l i m i t s predicted for the 113 TABLE 4-8. Assignments f o r NH.F and ND F; cm NH .F ND„F Assignment THIS WORK Ref. 11 THIS WORK Ref. 11 R_. I J ^ FL I.R. Vi 2874 2117 V 2 (1728)* (1240) V 3 2818 2815 2145 2148 V,, 1494 1120 V R (523) (376) V T 247 237 V A(K*0) (136) (134) Vi+VT 3082 2323 Vi+VA 3010 2251 Vz+Vk 3222 2360 2\>h 2974 2233 v 2+v R 2277 1633 V^+VR 2017 1496 * the bracketed frequencies have been i n f e r r e d from i n f r a r e d a c t i v e combinations; .the v A frequencies are i n f e r r e d assuming r e l a t i v e l y f l a t Vi phonon dispersions. 114 acoustical modes. 4-5 The Barrier to Rotation The b a r r i e r to r o t a t i o n f o r NH^+ ions i n a CsCl type l a t t i c e has been treated by Nagamiya (63). He assumed a p a r t i c u l a r charge d i s t r i b u -t i o n for the tetrahedral NH^+ ion by placing a charge of +<5e on each hydrogen atom. The p o t e n t i a l energy of one of the hydrogen atoms at an a r b i t r a r y point, (X^, Y^, Z^) i n the unit c e l l with respect to a l l charges outside i t s own can then be written as: / 4 4 4 \ A, 1 rr + r rT\4 xi + y. + z. 3 \ , * = - [Co + C 4(-) ( 1 J j . 1 _ - - U ] ( 1 ) where a i s the l a t t i c e parameter and N i s the N-H distance i n the NH^+ ion. The second term of t h i s expression represents the angular dependence of the p o t e n t i a l energy. Hence the p o t e n t i a l appropriate for r o t a t i o n of the ammonium ion i n a CsCl type l a t t i c e i s : In t h i s expression <f>o i s d i r e c t l y r e l a t e d to the b a r r i e r to r o t a t i o n . For r o t a t i o n about a four f o l d axis the b a r r i e r height is(9/4)<|>° (28). Gutowsky, Pake and Bersohn (64) use equation 2 to consider the l i b r a t i o n a l modes of the ammonium ion. R e s t r i c t i n g t h e i r treatment to the simple case i n which the ammonium ion executes only small v i b r a t i o n s about the cubic (x, y, z) axis of the unit c e l l , they developed an approximate p o t e n t i a l which may be written: 115 Vo (12 o f 2 2 2 3 2 2 2 2 2 2 2 2 2 2 f,-, - 8 ( ? n + ? C + ' T i C ) + > 1 J and where £, n and ? are the angles of r o t a t i o n about the cubic (X, Y, Z) axes. C l e a r l y , the assumption of small o s c i l l a t i o n s i m p l i c i t i n the model y i e l d i n g the above f o u r - f o l d cosine type p o t e n t i a l i s appropriate for the ordered cubic phases of the ammonium hal i d e s . With the use of f i r s t order perturbation theory Gutowsky, Pake and Bersohn were able to a r r i v e at an expression f o r the energy le v e l s of the anharmonic o s c i l l a t o r described by t h e i r model. The r e s u l t they obtained i s : E n i , n 2 , n 3 = 4 V° + { n i + n2 + n3 + |K> l r 2 2 2 -yg-J2ni + 2n 2 + 2n3 +• n i n 2 + nin3 + n 2n3 +3ni + 3n 2 + 3n 3 + |j Cnco) 2/Vo ( 4 2 J" + where (fin) = (1670 11 /2I) 2 and I i s the moment of i n e r t i a of the NH^ ion. Using equation 4 i t i s possible to c a l c u l a t e Vo when the value of (E100 " Eooo) i s known from the l i b r a t i o n a l frequency, i . e . Vo (h. + 51i 2/2I) 2 ( 5 ) 1i /2I The expression f o r V i defined by equation 5 although derived for the ordered cubic phase can reasonably be applied to the ordered tetragonal phase since the d i s t o r t i o n from cubic symmetry i s s l i g h t . The computed values f o r Vo(G.P.B.) using equation 5 are included i n Table 4-9. Also included are the b a r r i e r heights, Vo(N.M.R.), which 116 TABLE 4-9. L i b r a t i o n a l Frequencies and Ba r r i e r Heights; (cm- )* NH4C1(.IV) NH 4Br(IV) NH 4Br(III) NH 4I(III) V R(N.S.) 389 — 335 293 v R(R) 336 291 v R ( v , + v R ) 390 349 337 290 V 0 (G.P.B.) 1861 1515 1420 1080 V0(N.M.R.) 1650 1400 1110 ND 4C1(IV) ND 4Br(IV) ND 4Br(III) ND 4I(III) V R(N.S.) 280 254 V R(R) _ 208 V R ( v 4 + v R ) ?80 251 241 208 V0(G.P.B.) 1836 1492 1382 1049 the frequencies V (N.S.), VRCR) a°d vR(VI*+VR) have been r e s p e c t i v e l y obtained from neutron s c a t t e r i n g ( 3 ) , Raman and Vi»+vR Raman r e s u l t s . The underlined frequencies have been used to ca l c u l a t e the V 0(G.P.B.). The V0(N.M.R.) are found i n references 65,66 and 67. 117 F I G U R E 4 - 1 4 BARRIER HEIGHTS AS A FUNCTION OF THE LATTICE PARAMETER —I 1 1 1 — 1 1— 0.7 0.8 0.9 LO I.I 1.2 1 0 - 3 7 ( - ^ ) f c m - 5 118 have been obtained from N.M.R. measurements (65,66,67). It i s seen that the Vo(G.P.B.) and the Vo(N.M.R.) compare favorably. Gutowsky, Pake and Bersohn (64) have shown that i f the NH^+ ion i s assumed e l e c t r i c a l l y to be a system of point charges of + e/4 at the v e r t i c e s of a r i g i d regular tetrahedron then the r e l a t i o n Vo a r^/a^ holds. Figure 4-14 shows that a p l o t of Vo(G.P.B.) versus a " 5 ( { V/2~} " 5 for the tetragonal s a l t s ) gives a l i n e a r r e l a t i o n as predicted by the point charge model. 4-6 The Force F i e l d of C r y s t a l l i n e NH 4 + Since the frequencies of the fundamental modes of v i b r a t i o n of c r y s t a l l i n e NH^+ are well known we may use these frequencies to compute the force f i e l d f o r the c r y s t a l l i n e ammonium ion. The computations may be performed using the simple mixing program of Green and Harvey (47). The equation which i s basic to t h i s program i s equation 29 of section 3-1; i . e . F = y r - J sp A ^T'SJ1 In the case of tetrahedral NH^+ there i s one redundant symmetry co-ordin-ate and so the dimension of a l l the matrices appearing i n the above equation w i l l be..-10 x 10. Once the v i b r a t i o n a l frequencies and the molecular geometry of the basis molecule are made a v a i l a b l e to the program, the G matrix and also the orthogonal matrix U diagonalizing G are constructed. This y i e l d s the three matrices defined i n equation 27 of section 3-1; i . e . U t G U = r 119 The v i b r a t i o n a l frequencies d i r e c t l y y i e l d the matrix f\. The one re-maining matrix required i s the orthogonal matrix, P. It was noted i n section 3-1 that t h i s matrix has the important property that the elements P^ .. of P equal zero unless and \^ belong to the same symmetry c l a s s . For our cal c u l a t i o n s a sui t a b l e orthogonal matrix P i s : where: 1/71+X2 -x/7l+X2-120 Only the one parameter, X, appears and the force f i e l d problem i s one dimensional. X has the e f f e c t of mixing the computed (stretching), F^ (bending) and F 3 4 (stretch-bend) symmetry force constants and may be r e f e r r e d to as the mixing parameter. There w i l l be a family of acceptable force constants; each set i n the family corresponding to a p a r t i c u l a r value of the mixing parameter. Inspection of the orthogonal matrix P shows that the F^ (stretching) and F^ (bending) force constants do not depend on X. Calculations have been c a r r i e d out with the NH^+ ion of NH^BrQV) chosen as the basis molecule. Reference to Figures 4-15 and 4-16 shows the v a r i a t i o n of F^, F^ and F^^ with the mixing parameter on the i n t e r v a l -1.2 £ X < 1.2. Calculated ND^+ a n d frequencies using the NH^+ force constants are shown i n Figure 4-17. It i s found that the two values of, X which r e s p e c t i v e l y reproduce the observed ND^+ and frequencies do not coincide; so i t i s convenient to choose t h e i r mean as the value of X which gives the "best" computed spectrum. In Table 4-10 which shows calculated and observed spectra t h i s mean value of X i s denoted as X^. , When NH^+ i s chosen as the basis molecule the average absolute deviation of computed ND^+ frequencies from observed frequencies i s 23 cm The NH^+ force constants used to compute the ND^+ spectrum are as follows: Force Constant • (10 s Dynes/cm.) Fj. = 5.51026 F 2 = 0.60485 F 3 = 5.28516 F. = 0.52378 4 F 3 4 = 0.07670 121 to l i ! 2.2 5 u 2.6-if) UJ Z 3.0-> Q m o 3.4-h z < 3.8-h 0) z o 4.2-o LU u 4.6-o LL 5.0 5.4H F I G U R E 4 — 1 5 T H E S Y M M E T R Y F O R C E C O N S T A N T S F3 A N D F 4 A S A F U N C T I O N O F T H E M IX ING P A R A M E T E R — N H Br (iv) 1 r— -1.2 -0.8 -0.4 O.O 0.4 0.8 MIXING PARAMETER 122 u if) LU Z > Q m o Z £ if) Z o o LU u O LL F I G U R E 4 — 1 6 THE SYMMETRY FORCE CONSTANT, AS A FUNCTION OF THE MIXING PARAMETER 0 . 5 H O.TA OB-NH 4Br (iv) 1.3" 5H .7-2 . H 0.8 -0.4 O.O 0.4 0.8 MIXING PARAMETER F I G U R E 4 — 1 7 123 2 4 0 C H C A L C U L A T E D F R E Q U E N C I E S FOR ^ 3 A N D 14 A S A F U N C T I O N O F T H E MIX ING P A R A M E T E R — NH^B rOv ) — N D ^ B r d v ) H I 0 9 0 H O 8 O H I 0 7 0 H O 6 O •I050 H 0 4 0 H 0 3 0 H 0 2 0 h l O l O h l O O O •O.8 -0 .4 - O . O 0 . 4 0.8 MIXING PARAMETER 124 TABLE 4-10. Calculated and Observed Spectra - NH^ +(Td). (i) NHjBrtIV) + ND 4BrClV) X1 = -0.36, V 3 = 2330 X 2 = -0.11, v 4 = 1060 X 3 = -0.235 V l V2 V3 V4 Calculated Spectrum x = x 3 2155.4 1201.8 2305.7 1048.3 Observed Spectrum 2211 1211 2330 1060 Deviation -45.6 -9.2 -24.3 -11.7 ( i i ) ND 4Br(IV) •*• NH 4Br(IV) X1 = 0.54, V 3 = 3124 X. = -0.28, V„ = 1398 2 4 X 3 = -0.36 V l V2 V3 V4 Calculated Spectrum x = x 3 3125.6 1712.0 3169.9 1407.9 Observed Spectrum 3047 1699 3124 1398 Deviation +78.6 + 13.0 + 25.9 +9.9 125 No other NH^  force constants could be found for purposes of comparison. 4-7 The Effects of a D_, Distortion 2d As an XY^ molecule suffers a distortion from a Tj to a configur-ation both the kinetic and potential energies and thus G and F w i l l change. The spectrum, {X^}, of the XY^D,^) molecule may be computed providing both G and F are known. When the geometry of an XY^CD^) molecule is known G(D 2 d) is readily obtained. If we do not have a prior knowledge of the D2tj spectrum we cannot compute F(D 2 d). A f i r s t approximation would be to use the F(T d) matrix of the XY' (Tj) molecule which undergoes a slight angular distortion to give the XY^CD^) molecule. The computed D 2 d spectrum, {A^}, is then the solution of: l£(Dd) • £(T d) " \ l \ = 0 (1) where F(T^) may be calculated according to: F ( T d ) = yr'^P A (2) Here U is the orthogonal matrix diagonalizing G(T^) to give r . P is the orthogonal matrix of the previous section. The matrix G(D 2 d) appearing in the secular equations is computed from the D 2 d molecular geometry. When we use the above approximation to predict the kinetic energy contribution to the spectrum of the crystalline NH 4 +(D 2 d) ion, the NH^+(Td) ion is a hypothetical ion and there is necessarily an element of arbi t r a r i -ness introduced when selecting i t s unperturbed frequencies. We may choose the v 1( D2d)» v2^ D2d^' v3^ Eu^ a n c* v4^ Eu^ frequencies (recall that only the one 126 component of v 2 i s observed). When performing the actual c a l c u l a t i o n s i t was found that frequency s h i f t s of up to about 50 cm * for the unper-turbed frequencies, whether s i n g l y or i n combination, had l i t t l e quantita-t i v e e f f e c t on the predicted s p l i t t i n g pattern. The computed v ^ / v ^ s p l i t t i n g w i l l be a function of the mixing parameter, X, which appears i n P (see Figures 4-18 and 4-19). At X = 0 the ammonium bromide s p l i t t i n g pattern predicted f or small angular d i s t o r t i o n s involving two angles greater than 109.47° and four angles less than 109.47° i s the following: i v 3 ( B 2 ) V4 (B 2 ) ' V3 (E) j l V3 V2 V4 This r e s u l t agrees with the observed spectral r e s u l t s and thus supports a structure for NH^BrfUI) where the NH 4 + ions have two angles greater than 109.47° and four angles less than 109.47°. We may note that when the s i t u -a tion for the d i s t o r t i o n s i s reversed the order of the predicted B 2, E s p l i t t i n g s at X = 0 reverses (see Figure 4-20). 127 Table 4-11 reports the maximum v^/v^ s p l i t t i n g s predicted on the i n t e r v a l -1.2 < X < 1.2 when the angular d i s t o r t i o n i s +3.09° and -1.52°. In the case of NH^Br X =-0.35 maximizes the predicted s p l i t t i n g s . They are 15.8 and 7.2 cm * for and r e s p e c t i v e l y . These values compare with the observed s p l i t t i n g s of 38 and 28 cm 1 which a r i s e from the probable angular d i s t o r t i o n of +3.09° and -1.52°. Figures 4-18 and 4-19 show that at X = +0.47 which corresponds to the unperturbed vg/ v4 frequencies zero s p l i t t i n g i s predicted. The s p l i t t i n g pattern reverses as we go from X < 0.47 to X > 0.47. We have favoured a predicted s p l i t t i n g pattern with X < 0.47 since the r e s u l t s of numerous force constant c a l c u l a t i o n s f o r many small molecules have shown that when the force constant solutions -J« t -J-are of the form F =ur PAP r 2u the most acceptable values of the mixing parameters are i n v a r i a b l y less than zero (68), e.g. for NH^BrfT^) ND^Br(T^) the value of X giving optimum force constant t r a n s f e r a b i l i t y i s -0.235 (see Table 4-10) If we f i x the mixing parameter at say X = -0.235 and vary the NH^*^^) geometry we may use: I 9 C D 2 d ) F ( T d ) - X.I| = 0 to compute the D^^ spectrum as a function of angular d i s t o r t i o n (see Figures 4-20 and 4-21). We see that.the observed s p l i t t i n g requires an angular d i s t o r t i o n of -3.7° and +7.8°. For the required angular d i s t o r t i o n i s -6.0° and+12.8°. These values compare with the probable d i s t o r t i o n of -1.5° and +3.1°. i 128 FIGURE 4 — 18 3I22H 3I2CH 31 18 H 31 I6H CO 3II4H 31 \z4 31 lOH 3I08H 3106 3104 P R E D I C T E D S P L I T T I N G O F 7S3 A S A F U N C T I O N O F T H E M IX ING P A R A M E T E R N H 4 B r ( T H ) N H 4 B r (D 2 d ) -1.2 T T •0.8 -0.4 O.O 0.4 0.8 MIXING PARAMETER 129 FIGURE 4 — 1 9 P R E D I C T E D S P L I T T I N G O F z/4 A S A F U N C T I O N O F T H E M IX ING P A R A M E T E R — i 1 1 1 1 1 1— -1.2 -0.8 -0.4 O.O 0.4 0.8 1.2 MIXING PARAMETER TABLE 4- 11. Calculated and Observed Spectra (i) NH 4Br(T d) -> NH 4Br(D 2 d); X 4 = -0.35 V i ( A O ' v 2 ( A i ) v 2 ( B i ) V 3 ( B 2 ) V 3(E) V-(B 2) V 4(E) : OBS. 3037.5 1694 3079 3117 1433 1405 A 38 28 CALC. 3037.5 1710.2 1677.5 3104.2 3120.2 1409.8 1402.6 A 32.7 15.8 7.2 ( i i ) ND 4Br(T d) - ND 4Br(D 2 d); X. = -0.45 4 v 2 ( A i ) v 2 ( B i ) v 3 ( B 2 ) v 3(E) Vii (B 2) V.*(E) OBS. 2209 1209.5 2339 1089 1066 A 23 CALC. 2209 1221.1 1197.7 2321.6 2343.1 1072.6 1062.7 A 23.4 21.5 9.9 131 3I40H 31 2CH F I G U R E 4 - 2 0 SPLITTING OF Z£ AND ^ AS A FUNCTION OF ANGULAR DISTORTION (X=-0.235) ,z/(B ) NH^Br (Td) •NH 4Br(D 2 d) 1390H i 1— 4/ - 6.0 -4.0 T -2.0 O.O + 2.0 + 4.0 2/ -1-12.8 +8.4 +4.1 O.O -3.9 -7.7 ANGULAR DISTORTION 0 +6.0 -11.4 i 1 3 2 u ill QQ D Z UJ > 1 7 7 0 H 1 7 5 0 1 7 3 0 171 O H 1 6 9 0 1 6 7 0 H 1 6 5 0 H 1 6 3 0 i F I G U R E 4 - 2 1 S P L I T T I N G O F i / 2 A S A F U N C T I O N O F A N G U L A R DISTORTION N H A B r (T d ) NH^Br ( D 2 d ) - 6 . 0 2 / +12.8 - 4 . 0 + 8 . 4 - 2 D +4.1 — I — O.O O.O 1 1 — + 2 . 0 + 4 . 0 - 3 . 9 7.7 + 6 . 0 - 1 1 .4 ANGULAR DISTORTION 0 133 CHAPTER V THE VIBRATIONS OF THE ALKALI METAL BOROHYDRIDES 5-1 The Phase III A l k a l i Metal Borohydrides The c r y s t a l structure of the phase III a l k a l i metal borohydrides i s 2 compatible with the space group T^ (F43m). When the f a c t o r group analysis i s c a r r i e d out the following v i b r a t i o n s are obtained: r ( i n t e r n a l ) = A L + E + 2F 2 T(OT) = F 2 T(OL) = F1 r(AT) = F 2 In the Raman spectrum the modes of F 2 symmetry can be expected to s p l i t into transverse and lon g i t u d i n a l components. From the i n f r a r e d spectra recorded at 10°K (see Figure 5-1) v 3 ( F 2 ) and v 4 ( F 2 ) are r e a d i l y assigned f o r both KBH 4(III) and KBD 4(III). The Raman spectra were recorded at 80°K (see Figure 5-2) and since the phase transformation II III i s thought to undergo completion at 77.2°K the Raman spectra do not show the true and phase III frequencies. How-ever, for and v 4 there i s a n e g l i g i b l e frequency s h i f t i n the observed frequencies as the temperature i s lowered from 80°K to 10°K. Therefore i n Table 5-1 we have included the v^(Ai) and v 2(E) frequencies as observed at 80°K i n the phase III r e s u l t s . On t h i s basis the observed frequencies f o r the i n t e r n a l fundamental modes for KBH 4(III) become 2314.5, 1251.5, 2281 and 1121 cm"1 r e s p e c t i v e l y for v ^ A x ) , v 2 ( E ) , v 3 ( F 2 ) and v 4 ( F 2 ) . The corresponding frequencies f o r KBD.(III) are 1599.2, 892, 1724 and 855 cm 134 As expectedjlines f o r v,, a n ^ v ^ a r e observed i n the Raman as well as the in f r a r e d spectra. This, however, i s not the case for T.^(F2) which i s observed only i n the i n f r a r e d . The T j ( F 2 ) frequencies are 178 cm"1 f o r KBH 4(III) and 172 cm"1 f o r KBD^CIII). The Raman spectra gave no evidence for the appearance of the lon g i t u d i n a l components of and v^. 10 11 Since the B - B isotopes occur with a natural abundance of about 10 11 39 1+0 1:4 we may expect to see; (i) the ef f e c t s of B - B (and K - K) 10 11 disorder a f t e r the treatment of L i f s h i t z (62) and ( i i ) a B - B i s o -topic s p l i t t i n g f o r v^, and T^ . Evidently any frequency s h i f t s due to disorder are too small to be observed. The i s o t o p i c s h i f t s f o r and are, however, observed. The in f r a r e d spectrum of K B D ^ f l l l ) , where the l i n e widths at half-height are about 5 cm \ gives a s t r i k i n g example of i s o t o p i c 10 11 B - B s p l i t t i n g which i s f u l l y resolved (see Figure 5-5). In the spectrum 10 10 i of KBD 4(III), v 3 C p 2 ) a n d V4CF 2) are observed at 1739 and 864 cm . These frequencies give respective i s o t o p i c s h i f t s of 15 and 9 cm 1 . The correspond-ing s h i f t s f o r KBH 4(III) are 16 and 10 cm"1. In the second order spectrum of the phase III potassium borohydrides only three p r i n c i p a l l i n e s are observed; these are 2v 4(Aj + E + F 2 ) , v2 + v 4 ^ ! + a n < 1 2 v 2 ^ A l + E ) ' ^ e o v e r t o n e i s observed only i n the Raman e f f e c t but 2v 4 and v 2 + v 4 are common to both the in f r a r e d and Raman spectra. 2 v 4 ( F 2 ) i s strongly affected by Fermi resonance with v ^ ( F 2 ) ; i . e . i n the i n f r a r e d 2 v 4 ( F 2 ) has considerably enhanced i n t e n s i t y and the observed KBH 4(III) and KBD 4(III) frequencies are 25 and 28 cm"1 lower than those predicted. In the Raman spectrum of KBH.(Ill) 2v 4 appears as a doublet with 135 frequencies of 2185.5 and 2224 cm"1. Since both VJ(AJ) and v 3 ( F 2 ) l i e nearby at 2314.5 and 2291 cm"1 i t i s l i k e l y that 2v 4(A x + E + F 2) undergoes Fermi resonance with both these fundamentals. On t h i s basis the low wave number component of 2v^ can be assigned as 2v 4(Ai) and the high wave number component to 2 v 4 ( F 2 ) . Both of these components are observed as sharp l i n e s and i t i s u n l i k e l y that either component could contain any appreciable i n t e n s i t y contribution from 2v 4(E). The Raman l i n e appearing -1 1 0 at 2238 cm can be assigned as 2 v^(F ). This compares with an i n f r a r e d frequency of 2235 cm 1 which gives an observed-predicted frequency discrep--1 1 0 ancy of 27 cm . In the spectrum of K B D ^ f l l l ) 2 v^(F ) appears i n the i n f r a r e d at 1697 cm ^; the r e s u l t i n g frequency discrepancy i s 30 cm 1 . The predicted v 2 + v 4 ( F 2 ) frequencies are 2374 cm 1 f o r KBH^CIII) and 1747 cm"1 f o r K B D ^ f l l l ) . The corresponding i n f r a r e d frequencies are 2386 and 1757 cm 1 . Since these observed frequencies are about 10 cm 1 higher than those predicted + must undergo Fermi resonance with v 3 ( F 2 ) . 10 10 In the i n f r a r e d spectrum of KB D ^ f l l l ) + v 4 ( F 2 ) i s also observed. It appears at 1767 cm 1 . F i n a l l y we come to 2v 2(Ai + E). This l i n e appears with frequencies of 2497 and 1782 r e s p e c t i v e l y i n the spectra of KBH 4(III) and KBD 4(III). The respective anharmonicities are 7 and 2 cm 1 . The assignments f o r the phase III s a l t s are found i n Tables 5-1 and 5-2. The corresponding spectra are included i n Figures 5-1, 5-2, 5-3, 5-4, 5-6 and 5-8. The Raman r e s u l t s obtained f o r rubidium and cesium borohydride at 80°K are included i n the phase III r e s u l t s . Since the 136 TABLE 5-1. Assignments f o r KBH 4(III) and KBD (III) ; cm" ASSIGNMENT KBH 4(III) - THIS WORK KBD (III) - THIS WORK R. I.R. R. I.R. V l v 2 1 0 v 3 1 0 Vit T i v 3 Vi* 2314.5 1251.5 2291 1122 2297(sh) 2281 1131 1121 178 1599.5 892 1737 1723.5 856 1739 1724 864 855 172 2v 2 v 2+ 1 0v^(F 2) V 2+V l t(F 2) 2 1 0vaF 2) 2v. t(F 2) 2 1 0V. t(Ai) 2497 2386 2238 2224 2384 2235 2217 1782 1770 1757 1698 1682 1767 1757 1697 1682 2vit(Ai) 2185.5 137 TABLE 5-2. Assignments f or RbBH 4(III) and CsBH 4(III); cm"1 ASSIGNMENT RbBH 4(III) - THIS WORK CsBH 4(III) - THIS WORK R. I.R. I.R. V l v 2 1 0 v 3 1 0 v 3 V4 2v 2 v 2+ I 0v^(F 2 D v 2+Vi t(F 2) 2 1 0 v . ( F 2 ) 2v,(F 2) 2 1 0v 1 +(A 1) 2 1 V ( A 1 ) 2301 1239 2278 1119 2477 2365 2224 2209 2183 2174(sh) 2277 1116 2361 2218(sh) 2207 2281.5 1222 2255 1104.5 2443 2338 2202 2187.5 2156 2166 2268(sh) 2252 1113 1103 2345 2336 2201 2185 138 F i g u r e 5—. i T h e I . R . s p e c t r a of N a B H 4 ( i ) , K B H 4 ( i ) NaBH 4 ( i i ) and K B H 4 ( i n ) 0 . 0 1 W A V E N U M B E R C M - 1 139 Figure 5 - 2 The Raman Spect ra of N a B H ( i ) , K B H 4 ( i ) NaBH 4 ( i i ) and K B H 4 ( 8 0 0 K ) - i — i 1—i 1—i—7/~~\—i 1 — i — r 2 4 0 0 2 2 0 0 " 1300 1100 2 5 0 0 2 3 0 0 " 1 3 5 0 1150 W A V E N U M B E R C M ~ 1 140 F i g u r e 5-3 I he I. R. spectra of N a B L ^ d ) , K B D 4 ( i ) NaBD 4 ( i i ) and K B D 4 ( i i i ) c o -l l i O Z < 0.2-QQ cr o CO m 0.4-< 0.6-0.8-I .O N a B D (D-295°K 4 ~1 1 T " -r-—I 1 1 r-111 o z < CQ rr o CO DQ < o.o 0 . 2 0 . 4 O. 6 O. 8 I . O ~ ~ 1 N a B D 4 (ii)-80°K -i 1 1 r-K B D (I'll )-IO°K T ' <—/ An ' r 1 9 0 0 1700 IOOO 8 0 0 1 9 0 0 1TOO 1 0 0 0 8 0 0 W A V E N U M B E R C M 141 F i g u r e 5-4 T h e Raman Spect ra of N a B D 4 ( i ) , K B D 4 ( i ) N a B D 4 ( i i ) and K B D 4 ( 8 0 ° K ) V N a B D 4 ( l l ) - 8 0 0 K i — i — | — r -1 8 0 0 — i — i — / / — i — i 1 — i — i — I 1 6 0 0 IOOO 8 0 0 1 6 0 0 IOOO 8 0 0 W A V E N U M B E R C M " 1 142 Figure 5 - 5 The I.R. s p e c t r a of N a B D ( n ) , N a B H (n ) 4 4 KBD 4 (n i ) and K B H 4 ( i n ) 1760 1720 1680 890 850 1150 I MO W A V E N U M B E R C M - 1 143 F igure 5 - 6 The I .R. spect ra of R b B H ( i ) f CsBH ( i ) , RbBH 4 ( i i i ) and C s B H / m ) o.o-111 (J z < Q . 2 -DQ LT O CO GQ 0 . 4 -< 0 . 6 -0 . 8 -1 .o-o.o-LU U Z < 0 . 2 -DQ LT O CO DQ 0 . 4 -< 0 . 6 -0 . 8 : 1 . O -R b B H (I )-295°K 4 T I 1 1 1 1 r -C s B H (lll)-10°K 4 ~r—1—1—'—r 2 4 5 0 2 2 5 0 1250 1 0 5 0 2 4 0 0 2 2 0 0 W A V E N U M B E R C M ~ 1 V A - T—i—i—p 1 2 5 0 I 0 5 0 144 F i g u r e 5-7 T h e Rqman S p e c t r a of R b B H ^ d ) , C s B H 4 ( i ) , R b B H 4 ( 8 o ° K ) and C S B H 4 ( S O ° K ) C S B H 4 ( I ) - 2 9 5 ° K ,,f< - , 1 1 1 1 — / / T 2 4 5 0 2 2 5 0 1 3 0 0 1100 2 4 5 0 i 1 1 1 — 7 / ~ i ' 1 1 r 2 2 5 0 1 3 0 0 1100 W A V E N U M B E R C M - 1 145 frequencies i n the i n f r a r e d spectra of these s a l t s showed a n e g l i g i b l e s h i f t as the temperature was lowered from 80° to 10° K, i t can be assumed that the Raman spectra as observed at 80°K w i l l c l o s e l y approximate the true phase III spectra. 5-2 The Phase II A l k a l i Metal Borohydrides Phase II sodium borohydride has a tetragonal structure under the space • 9 group D 2 d(I4m2). The r e s u l t s of the fa c t o r group analysis under the Y)^ factor group can be summarized as follows: r (internal) = 2AX + Bj + 2\. + 2E r(0T) = B 2 + E r(0L) = A 2 + E I* (AT) = B 2 + E For the i n t e r n a l vibrations associated with the BH^ ion the s i t e group and fa c t o r group coincide; hence, there are no c o r r e l a t i o n f i e l d e f f e c t s and the four v i b r a t i o n s associated with the free BH^ ion become the pre-cursors of 7 c r y s t a l modes which a r i s e from a s p l i t t i n g of the degenerate modes by the s t a t i c f i e l d as shown by the following c o r r e l a t i o n diagram. V i b r a t i o n a l Free Ion Site/Factor Group Mode Symmetry-Tj Symmetry-D^^ : v1 Ai -> Ai • v 2 E •> A i , Bi v 3 F 2 + B 2, E v 4 F 2 -> B 2, E 146 For the phase III a l k a l i metal borohydrides i t i s r e l a t i v e l y easy 10 11 to sort out the modes a r i s i n g from B - B i s o t o p i c s p l i t t i n g . However, for the phase II sodium s a l t s where the s i t e group s p l i t t i n g i s of the same 10 11 order of magnitude as the B - B i s o t o p i c s p l i t t i n g the s i t u a t i o n i s much more complicated. Nevertheless i t has been possible to make a complete set of v i b r a t i o n a l assignments f o r both NaBH^CH) and NaBD^CII). Since the v i b r a t i o n a l spectrum of NaBD^ClI) i s characterized by sharp well resolved l i n e s assignments f o r t h i s s a l t w i l l be made f i r s t . The Raman spectrum of NaBD^(II)(see Figure 5-4) allows us to place V j ( A i ) at 1622 cm The two expected components of v 2 also appear; the l i n e appearing at 916 cm * i s a r b i t r a r i l y assigned to v 2 ( A i ) and the l i n e -1 ' appearing at 904 cm i s then assigned to v ^ B j ) . (It w i l l be seen sh o r t l y that these assignments can be confirmed when the possible combinations for + are considered). Of the four D 2^ l i n e s associated with ,and only one, VgfE) appears i n the Raman spectrum of NaBD^fll), (the same s i t u a t i o n occurs f o r NaBH^(II)) therefore i t i s convenient to consider the in f r a r e d spectrum (see Figures 5-3 and 5-5). The i n f r a r e d bending region shows four l i n e s . These appear at 854.5, 865, 870 and 877 cm 1 . C l e a r l y , the most intense l i n e appearing at 854.5 cm can be assigned to v^(E). Now, for K B D ^ f l l l ) the i s o t o p i c s p l i t t i n g i s 9 cm *; therefore i t i s reasonable to assign the 865 cm * l i n e as v^(E). This leaves two l i n e s either one of which could be assigned as v 4 ( B 2 ) . The most reasonable choice i s the 870 cm ^ l i n e since as w i l l be seen s h o r t l y , i t predicts 2v^ and V2 + V4 o v e r t o n e s a n c * combinations which are compatible with the experi-147 mental r e s u l t s . The anomalous l i n e at 877 cm - 1 has been assigned to the second overtone of a mode of rotatory o r i g i n . A further comment on t h i s assignment i s made l a t e r . The stretching region of the i n f r a r e d spectrum i s very r i c h i n spectral information. A convenient l i n e to s t a r t with i s the 1735.5 cm - 1 l i n e ; because of i t s i n t e n s i t y i t i s r e a d i l y assigned as v^CE). A second l i n e possessing s t r i k i n g i n t e n s i t y i s found at 1692 cm"1 and i t must be assigned to a component of 2v^. Since i t i s observed with such enhanced i n t e n s i t y i t can be assumed to be undergoing Fermi resonance with v ^ f E ) . This immed-i a t e l y selects E type symmetry f o r t h i s l i n e and hence i t must o r i g i n a t e from the combination v 4(E) + v 4 ( B 2 ) = 2v 4(E) which has a predicted frequency of 1725 cm 1 . The discrepancy of 33 cm 1 between the predicted and observed frequencies i s further evidence of Fermi resonance. The l i n e appearing at _1 10 1715 cm on the high frequency side of 2v 4(E) can be assigned to 2 v 4 ( E ) ; 10' _ j v 4 ( B 2 ) has a predicted frequency of 880 cm and therefore the predicted 10 _! frequency of 2 v 4(E) i s 1745 cm which gives a predicted-observed f r e -quency discrepancy of 30 cm 1 . Besides 2v 4(E) there are two other possible 2v 4 components. These are \>4(B2) + v 4 ( B 2 ) = 2v 4(Ai) and v 4(E) + v 4(E) = (Ai + Bi + B 2 ) . This l a t t e r component i s i n f r a r e d a c t i v e and therefore expected to appear i n the i n f r a r e d with a predicted frequency of 1709 cm 1 . However, no evidence was found f o r the appearance of t h i s l i n e . Having -1 1 0 assigned the l i n e s at 1692 and 1715 cm r e s p e c t i v e l y to 2v 4(E) and 2 v 4(E) the remaining l i n e on the low frequency side of \>g(E) at 1724 cm 1 can be assigned to v.j(B 2). On the high wave number side of V g ( E ) a l i n e appears at 148 _1 10 10 11 1755 cm which must be assigned to V 3 ( E ) - T n e r e s u l t i n g B - B i s o -topic s p l i t t i n g of 20 cm 1 compares with the i s o t o p i c s p l i t t i n g of 15 cm"1 observed f o r KBD 4(III). The remaining l i n e s appearing at 1764, 1773, 1789 and 1798 cm 1 must be assigned to the combinations v 2 + v^. There are 8 possible predicted combinations. These are: v 2 ( B l ) + v 4(E) = 1759(E) 10 10 v 2 ( B 1 ) + v 4(E) = 1769(E) v 2CAi) + v 4(E) = 1771(E) 10 10 V A l ) + V E ) 8 8 • 1781(E) v 2 ( B i ) + v 4 ( B 2 ) = 1774 (A 2) 1 0 v 2 ( B ! ) + 1 0 v 4 ( B 2 ) = 1784(A 2) v 2 ( A x ) + v 4 ( B 2 ) = 1786(B 2) 1 ° v 2 ( A 1 ) + 1 0 v 4 ( B 2 ) = 1796 (B 2) The combinations of E and B 2 symmetry types w i l l be both i n f r a r e d and Raman act i v e , while the combinations of A 2 symmetry type w i l l be i n a c t i v e . Since combinations of E type symmetry can undergo Fermi resonance with v^CE) and those of B symmetry can in t e r a c t with v , ( B 2 ) , i t can be expected that the 2 "5 ' observed combinations may have enhanced i n t e n s i t i e s and may also l i e at frequencies s l i g h t l y higher than those predicted. On t h i s basis the observed l i n e s at 1764, 1773, 1789 and 1798 cm * may r e s p e c t i v e l y be assigned to: 10 10 v 2 ( B i ) + v 4 ( E ) , v 2 ( A 1 ) + v 4 ( E ) , v 2 ( A i ) + v 4 ( B 2 ) and v ^ A j ) + v 4 ( B 2 ) . This choice of assignments i s the only reasonable one and allows us to remove the a r b i t r a r i n e s s associated with the assignments v 2 ( A i ) , v,,^) and v 4 ( B 2 ) . 149 For the Raman observed l i n e s 2v^ and + the second order spectrum c l o s e l y p a r a l l e l s the i n f r a r e d spectrum. The Raman spectrum shows a band for which i s absent i n the i n f r a r e d . It i s observed at 1821 cm - 1 and very l i k e l y the three predicted components f o r 2v 2 a l l contribute to i t s i n t e n s i t y . The foregoing assignments f o r NaBD 4(II) are summarized i n Table 5-3. We turn next to the spectrum of NaBH 4(II). The Raman spectrum (see F i g -ure 5-2) allows us to place v^(A 1) at 2341 cm 1 and v 2 appears as a r e l a t i v e l y broad band at 1280 cm 1 which undoubtedly contains the two expected v 2 components. Therefore the only assignment which can be made i s v 2 ( A l 5 B j) = 1280 cm"1. Just as f o r NaBD^fll) four l i n e s appear i n the i n f r a r e d bending region of the spectrum (see Figures 5-1 and 5-5). These l i n e s appear at 1122, 1134, 1148 and 1153 cm 1 . The most intense l i n e i s the 1122 cm 1 l i n e and hence 10 11 can be assigned as v^f E ) . For KBH 4(III) the B - B i s o t o p i c s p l i t t i n g -1 1 0 -1 i s 10 cm . This indicates the assignment v^CE) f o r the 1134 cm l i n e . Either of the the l i n e s appearing at 1148 and 1153 cm 1 could reasonably be assigned to v 4 ( B 2 ) and, while a choice of eit h e r one w i l l involve some degree of a r b i t r a r i n e s s , i t i s found that on the basis of expected pre^ dicted-observed frequency discrepancies f o r the overtones and combinations the 1148 c m 1 l i n e appears to be the best choice. The 1153 cm 1 l i n e has been assigned to the second overtone of a mode of rotatory o r i g i n . This assignment i s commented on l a t e r . The infrared, stretching region of the NaBH.(II) spectrum i s not nearly 150 as well resolved as the corresponding NaBD^ClI) s p e c t r a l region. However, much valuable information i s present. The intense l i n e appearing at 2303 cm 1 can be assigned to v^fE) and the intense l i n e appearing at 2236 cm - 1 which may be assigned to a component of 2v^ must c e r t a i n l y undergo Fermi resonance with \>3(E). This f i x e s E type symmetry f o r the 2236 cm - 1 l i n e and i n analogy with the s i t u a t i o n f o r NaBD^ClI) we have the assignment V 4 ( B 2 ) + V4(E) = 2v 4(E). The predicted-observed frequency discrepancy i s 34 cm * which compares with a corresponding value of 33 cm * for NaBD^CII) Just as f o r NaBD^fll) the expected absorption, v^(E) + v 4(E) = 2v 4(Ai + H\ + B 2) does not appear i n the i n f r a r e d spectrum. The l i n e appearing at 2256 -1 1 0 cm can be assigned to 2 v ^ f E ) , and the shoulder which appears at 2272 ^ ( E ) must be assigned as v^l cm * on the low frequency side of v,( , ( B 2 ) . The shoulder appearing on the high frequency side of v 3 ( E ) at 2334 cm * can be 10 10 11 • _i assigned to V g ( E ) . This gives a B - B i s o t o p i c s p l i t t i n g of 31 cm and compares with a corresponding i s o t o p i c s p l i t t i n g of 30 cm * f o r KBH^CIII). The remaining l i n e appearing i n the i n f r a r e d stretching region appears at 2404 cm * and can be assigned as v 2 ( A i , Bj) + v^CE) which has a predicted frequency of 2402 cm The lines appearing i n the second order Raman spectrum of NaBH^CII) may be assigned to.2v^, v 2 + and 2v 2 > For + and 2v^ weak bands, which most l i k e l y contain more than one component, are observed at 2408 and 2560 cm * re s p e c t i v e l y . A doublet i s observed f o r 2v^; the components appear at 2210 and 2244 cm The low wave number component l i k e l y o r i g i n -ates from the overtone v^fE) + v^fE) = 2v^(Ai + Bj + B 2) and'Since i t very 151 l i k e l y undergoes Fermi resonance with v ^ ^ ) the correct assignment i s probably 2v^(A 1). For the highest wave number component the most l i k e l y assignment i s v^(B 2) + v^fE) = 2v 4(E). A complete summary of the assignments for NaBH 4(II) and NaBD 4(II) are found i n Table 5-3. Unfortunately the Raman spectra of the phase II sodium borohydrides give a dearth of information f o r the modes of tra n s l a t o r y and rotatory o r i g i n and i n the in f r a r e d only one absorption appears which may be associated with the external fundamentals. In the spectrum of NaBH 4(II) i t appears at 172 cm 1 and i n the spectrum of NaBD 4(II) at 167 cm 1 . The assignment i s v^B^, E). As has previously been indicated, an anomalous l i n e appears i n the spectra of the phase II sodium borohydrides which has t e n t a t i v e l y been assigned to 3v R. In the spectrum of NaBH 4(II) i t appears at 1153 cm 1 and i n the spectrum of NaBD 4(II) i t appears at 877 cm Assum-ing the 3v D assignment, modes of rotatory o r i g i n , without co r r e c t i o n f o r anharmonicity, are placed at 384 cm 1 for NaBH 4(II) and at 292 cm 1 f o r NaBD 4(II). The frequency r a t i o i s 0.76 which compares with an expected r a t i o of 0.71. There i s only one piece of experimental information with which to check the 3v R assignment. Stockmayer and Stephenson (35) using heat capacity data have estimated that v R for NaBH 4(II) should l i e at about 350 cm While t h i s r e s u l t does not exclude the 3v assignment, i t does not give exceptionally strong support for i t either and hence the 3v assignment must be regarded as only t e n t a t i v e . F i n a l l y i t i s appropriate to compare the e a r l i e r i n f r a r e d r e s u l t s of Schutte (32) for NaBH. with those obtained i n t h i s work. The two sets of TABLE 5-3. Assignments f o r NaBH (II) and NaBD (II); cm - 1 ASSIGNMENT NaBH 4(II) - THIS WORK NaBD 4(II) - THIS WORK R. I.R. R. I.R. V i(Aj) 2341 1622 v 2 ( A O 916 v 2 ( B i ) 1280 904 V 3(B 2) 2272(sh) 1740.5 1724 v 3(E) 2322 2303 1735 1 0 v 3 ( E ) 2334(sh) 1757 •1755 Vi*(B2) 1148(sh) 870 v 4(E) 1122 854 : V ( E ) 1134 865 V T(B 2,E) 172 167 \ ( A 2 ) E ) (384)* ( 2 9 2 ; 2v 2 2560 1821 v 2+ 1 0v. t(B 2) 1800 1798 v2+Vi, (B 2) 1791 1789 v 2+v 4(E) 1775 1773 V2+Vi»(E) 2408 2404 1766 1764 2 1 0 V. t(E) 2257 2256 1720 1715 2V 4(E) 2244 2236 1701 1692 2v l t(A 1) 2210 3 VR 1153 877 * bracketed frequencies f o r v are i n f e r r e d from 3V 153 re s u l t s are summarized below. Assignment. This Work Schutte - REF. 32 v 4(E) 1122 1123 l 0 v 4 ( E ) 1134 1135 v.w. 3v_ 1148(sh) K v 4 ( B 2 ) 1153 1152 w. 2197(sh) w. 2223 s. 2v 4(E) 2236 2238 s. 1 0 2 v 4 ( E ) 2256 2256 m. v 3 ( B 2 ) 2272 2287 v.s. v 3(E) 2303 2305 v.s. 1 0 v 7 ( E ) 2334(sh) v 2+v 4 2404 2404 w. From the above comparison i t i s seen that Schutte reports two l i n e s f o r 2v 4 which are not compatible with the r e s u l t s of t h i s work. It i s possible that the 2223 cm 1 absorption could be assigned to v 4(E) + v 4(E) = 2v 4(B 2) which has a predicted frequency of 2244 cm - 1. However, i f t h i s i s the case such a l i n e should c e r t a i n l y also be observed i n the spectrum of NaBD 4(II) where the l i n e s .are exceptionally well resolved. The shoulder at 2197 cm"1 appears to be completely spurious. Also, the 2287 cm - 1 l i n e reported by 154 Schutte does not appear i n the spectrum recorded i n t h i s work and since i t is reported as a strong d i s t i n c t absorption i t i s d i f f i c u l t to associate i t with the shoulder we observed at 2272 cm . The remaining discrepancies involve the absorptions we observe as shoulders at 1148 and 2334 cm - 1. These may have been missed i n the e a r l i e r work because of t h e i r low i n t e n s i t y . There i s a possible explanation for the discrepancies between the r e s u l t s . Schutte's samples may have been prepared while exposed to the atmosphere. E a r l i e r work (39) done i n t h i s laboratory has shown that such exposure to the atmosphere can allow for the occurrence of s l i g h t decomposition and perhaps more important the formation of NaBH^ » 1^0 (69). Such introduction of impurities into the sample could e a s i l y explain the discrepancies. The point to be emphasized though i s that Schutte's r e s u l t s , l i k e those of t h i s work, show an unambiguous s p l i t t i n g of which may be associated with the disorder-order phase transformation. 5-3 The Phase: I A l k a l i Metal Borohydrides The i n t e r p r e t a t i o n of the phase I spectra of the a l k a l i metal boro-hydrides must.take into account the disorder introduced into the l a t t i c e v i a the randomness associated with the hydrogen atoms. However, since the heat capacity, data (36, 37) give evidence of d i f f e r e n t s i t u a t i o n s f o r the ordering processes, NaBH^fl) -> NaBH 4(II) and KBH 4(I) -> KBH 4QlI), i t i s appropriate to review t h i s data before attempting to i n t e r p r e t the experi-mental r e s u l t s . It w i l l be r e c a l l e d from Chapter I that the heat capacity data (36) for sodium borohydride indicate a well defined disorder-order phase trans-155 formation, with the anomaly at 189.9°K i n the heat capacity curve accounting for the excess entropy of t r a n s i t i o n expected f o r a disorder-order process. For potassium borohydride the s i t u a t i o n i s somewhat d i f f e r e n t (37). In th i s case the heat capacity anomaly at 77.2°K accounts f o r only 0.70 of the expected 1.36 e.u. with the heat capacity measurements giving evidence for the o r i g i n a l onset of the disorder-order transformation at a much higher temperature. It i s possible that a simple s t e r i c argument can be used to explain the dif f e r e n c e between the sit u a t i o n s f o r sodium borohydride and potassium borohydride. The e f f e c t i v e i o n i c radius of the BH^ ion has been o ^ given (38) as 2.03 A and the i o n i c r a d i i of the Na and K ions may be taken o as 0.98 and 1.33 A re s p e c t i v e l y (55). Using these r a d i i the predicted o B-Na and B-K distances are 3.01 and 3.36 A re s p e c t i v e l y . These predicted o distances compare with the respective c r y s t a l distances of 3.08 and 3.36 A at 298°K. I f these predicted and observed i n t e r - i o n i c distances are at a l l r e l i a b l e f o r purposes of comparison i t i s immediately evident that at 298°K r e l a t i v e l y t i g h t e r c r y s t a l packing occurs i n KBH^ than i n NaBH^. Pursuing t h i s idea further we can extend the comparison of predicted and observed i n t e r - i o n i c distances to include the remaining a l k a l i metal boro-hydrides as well as the phase II ammonium halides. Using an e f f e c t i v e + - ° NH^ i o n i c radius of 1.48 A (55) and a l k a l i metal ion and halide ion r a d i i found i n K i t t e l . (55) the r e s u l t s may be summarized as follows: 156 r(B-X) r(B-X) r(N-X) r(N-X) Predicted Observed Difference Predicted Observed Difference NaBH. 4 3.01 3.08 0.07 NH.C1 4 3.29 3. ,36 0.07 KBH4 3.36 3.36 0.00 NH.Br 4 3.44 3. ,52 0.08 RbBH4 3.51 3.51 0.00 NH.I 4 3.67 3. ,75 0.08 CsBH 4 3.70 3.71 0.01 It i s c l e a r l y seen that two groups of s a l t s emerge; those with a di f f e r e n c e o of about 0.07 A between the predicted and observed i n t e r - i o n i c distances which e s s e n t i a l l y and those with predicted and observed i n t e r - i o n i c d i s t ances^ coincide. The f i r s t group of s a l t s are known to undergo well defined disorder-order phase transformations and with respect to the second group i t i s known that the disorder-order transformation for KBH4 i s poorly defined. It i s possible that these two d i f f e r e n t s i t u a t i o n s are indeed r e l a t e d to differences i n the r e l a t i v e looseness of the c r y s t a l packing. It i s to be noted also that over the temperature i n t e r v a l 293-90°K (40) NaBH4 suffers an a x i a l contrac-t i o n of 4.02% whereas KBH4 suffers an a x i a l contraction of only 1.32%. . Turning now to the spectra of the a l k a l i metal borohydrides we can see that they too are i n d i c a t i v e of the two d i f f e r e n t s i t u a t i o n s being considered. For the serie s of s a l t s , NaBH4, KBH 4» RbBH4 and CsBH 4 the frequencies of the tran s l a t o r y modes at 295°K are r e s p e c t i v e l y 160, 173, 164 and 155 cm - 1 (see also Tables 5-4, 5-5 and 5-6). The anomalously low trans l a t o r y frequency observed for NaBH4 may be at t r i b u t e d to a r e l a t i v e l y looser c r y s t a l packing f o r t h i s s a l t . Furthermore, a comparison of the 157 half-height band width r a t i o s , v 3(10°K)/v 3(295°K) f o r the same serie s of s a l t s shows that the r a t i o f o r NaBH4 i s anomalously low. The r e s u l t s which show t h i s to be so are presented below. The half-height band widths shown are f o r band heights of 9.5 ± 0.5 cm. Half-height band widths Ratios v 3(295°K) v 3(10°K) v 3(10°K)/v 3 (cm-1) (cm-1) NaBH. 4 80 38 0.47 KBH. 4 45 28 0.62 RbBH, 4 45 36 0.80 CsBH. 4 45 25 0.56 NaBD. 4 47.5 14 0.26 KBD, 4 15 6 0.47 If we assume that the contribution to band width at 295°K due to the population of v i b r a t i o n a l l e v e l s l y i n g above the ground state i s constant f o r the series of s a l t s , then the discrepancies i n the r a t i o s are l a r g e l y due to contributions to. the high temperature i n t e n s i t i e s from de n s i t i e s of states with K £ 0. Therefore, i f we assume complete disorder f o r NaBH^ at 295°K the observed half-height band width r a t i o s indicate that at 295°K complete disorder i s perhaps not associated with KBH^, RbBH^ and CsBH^ and that i f t h i s i s so the r e l a t i v e disorder f o r these three s a l t s as indicated by the observed r a t i o s i s CsBH^ > KBH^ > RbBH^. Although the neutron d i f f r a c t i o n r e s u l t s obtained by Peterson (34) were interpreted as being compatible; with a completely disordered KBH. structure at 298°K, the heat capacity data 158 TABLE 5-4. Assignments f or NaBH^(1) and NaBD^fl) ; cm"1 ASSIGNMENT NaBH^(1) - THIS WORK NaBD^(1) - THIS WORK R. I.R. R. I.R. Vi 2335 1618 V 2 1278 907.5 v 3 2297 1728 1729 vk 1119 853 160 155 2v 2 v2+vk 2403(sh) 2393 1772.5 1767 2v 4 2229 2222 1695 1690 2Vn 2197.5 159 TABLE 5-5. Assignments f or KBH 4(I) and KBD 4(I)*; cm ASSIGNMENT KBH 4(I) - THIS WORK KBD 4(I) - THIS WORK R. I^R. R. I.R. Vi ' 2310 1599 v 2 1248.5 890.5 1 0 v 3 1733(sh) 1732 v 3 2279 1717 1717 I 0 V i » 858(sh) v 4 1118 1117 851 850 Ti 173 168 2v 2 2495 1782 v 2 + I 0 V i t 1762(sh) v 2+v 4 2380 2377 1752.5 1752 210Vn 1694 1693 2v^ 2217 2213 1679 1677 2v 4 2181 The assignments are probably f o r a pseudo Phase I structure. This point i s amplified i n the text. 160 TABLE 5-6. Assignments f o r RbBH^(I) and CsBH 4(I)*; cm 1 ASSIGNMENT RbBH^(I) - THIS WORK CsBH 4(I) - THIS WORK R. I.R. R. I.R. Vi 2297 2281 V 2 ' 1237.5 1222 v 3 2276(sh) 2273 2256(sh) 2251 vk 1113 1112 1102.5 1100 v^ 164 155 2v 2 2472 2440 V 2+V 4 2360 2361 2335(sh) 2330 2\)h 2202 2202 2182 2181 2v 4 2167 2151 ' The assignments are probably f o r a pseudo Phase I structure. This point i s amplified i n the text. 161 (37) which show a broad t r a n s i t i o n from about 200 to 450°K were i n t e r -preted as giving evidence that the o r i g i n a l onset of the disorder-order transformation was at about 450°K. This l a t t e r s i t u a t i o n i s c l e a r l y compatible with the spectroscopic results and also with s t e r i c considerations. Therefore, i t i s very l i k e l y that the structures of KBH^, RbBH^ and CsBH 4 > at 295°K possess considerable disorder but not complete disorder. The infrared and Raman spectra together of NaBH^, KBH^, RbBH^ and CsBH4 at 295°K (see Figures 5-1 - 5-8) possess bands which may be assigned as V l ' V 2 ' V 3 ' V4' ^ 1 ' 2 v4* V2 + V4 a n c* 2 v 2 ' ^ complete summary of assignments i s contained i n Tables 5-4, 5-5 and 5-6. 5-4 Lithium Borohydride The crystal structure of lithium borohydride i s compatible with the space group D^fPcmn) and hence the factor group analysis i s carried out under the factor group. The results for the various types of vibrations may be summarized as follows: T(internal)= 6A + 3B, + 6B„ + 3B„ + 3A + 6B. + 3B„ + 6B3 v g lg 2g 3g U lu 2u 3u r(0T)= 4A + 2B, + 4B„ + 2B„ + 2A + 3B, + B„ + 3B„ v g lg 2g 3g u lu 2u 3u T(0L)= A + 2B1 + B„ + 2B0 + 2A + B. + 2B0 + B. g lg 2g 3g u lu 2u 3u T(AT)= B, + B~ + B 0  v •* lu 2u 3u I t i s found that the four fundamental vibrations associated with the free BH4~ ion become the precursors of 36 cr y s t a l modes. The correlation diagram for s i t e group s p l i t t i n g follows. 162 V i b r a t i o n a l Free Ion S i t e Group Mode Symmetry - Symmetry - C g Ai A 1 v 2 E -> A', A" v 3 F 2 -> A 1, A', A" v 4 F 2 •> A', A 1, A" Under the f a c t o r group each A' mode w i l l give four c r y s t a l modes of symmetries A , B. , B, and B„ and each A" mode w i l l give four c r y s t a l g 2g lu 3u to / modes of symmetries Bj , B , A y and B^. Modes of gerade symmetry a l l possess elements of the p o l a r i z a b i l i t y tensor and therefore w i l l be Raman active. Only modes of B , B 2 u and B 3 u symmetries are i n f r a r e d a c t i v e . Since i t i s found that s p l i t t i n g f o r the i n t e r n a l fundamentals a r i s i n g from c o r r e l a t i o n f i e l d e f f e c t s i s not experimentally observable, i t i s convenient to make the i n f r a r e d assignments f o r the i n t e r n a l fundamentals according to the following scheme: V B l u + B3u) V B l u + B3u>. V<B2u> V B l u + B3U>. V< Blu + B3u>' V<B2u> v.(B + B ), v ' (B. + B. ), v "(B. ) 4^ lu 3u" 4 v lu 3u'' 4 2u' For the Raman assignments the corresponding scheme i s : TABLE 5-7. Assignments f or LiBH and LiBD ; ASSIGNMENT LiBH, - THIS WORK LiBD, - THIS WORK R. I.R. I.R. R. V l 2300 1605.5 1606 v 2 1287 1284 923 919 1 v 2 1325 1323 937 933 v 3 2274 2277 1721 1722 v 3' 2309 (sh) 2307 1739 1736 v 3 " 2350 1757 1089 828 1 0 v , 1099(sh) 837 VR (418) (319) T i 391 340 T 2 324 276 T 3 274 248 T 4 232 215 Ts 175.5 T e 162.5 149.! i V 2 +Vi» 2423 1785(: V 2 + V i * 2387(sh) 2 1 0 V , 2197 1679 2v^ 2176 1654 3 VR 1253 1254 959 957 * bracketed frequencies f o r v have been i n f e r r e d from 3v 164 F igu re 5 - 8 The I.R. spectra of L i B H . and L i B D . 2 4 0 0 2 2 0 0 1400 1 2 0 0 6 0 0 4 0 0 2 0 0 1 1 1 • 1 1 i 1 ' 1 1 1 / / ' 1 1 1 1 1 1 1 ' 1 1—r 1 8 0 0 1 6 0 0 1 0 0 0 8 0 0 5 0 0 3 0 0 1 0 0 W A V E N U M B E R C M " 1 165 F i g u r e 5-9 The Raman S p e c t r a of L i B h L and L i B Q n 1 1 1 1 r 1 8 0 0 1 6 0 0 1 4 0 0 W A V E N U M B E R C M - 1 166 It should be noted that f or n e g l i g i b l e f a c t o r group s p l i t t i n g s that the frequencies of corresponding assignments f o r the two schemes w i l l es-s e n t i a l l y coincide. The Raman spectra (see Figure 5-9) of the lithium borohydrides allow us to place ^ ( A l g + B 2 g) f o r LiBH^ at 2300 cm - 1 and V ; L f o r LiBD^ at 1605.5 cm 1 . Turning now to the i n f r a r e d spectra (see Figure 5-8) we f i n d that f o r both LiBH^ and LiBD^ four l i n e s appear i n the bending region o f the spectra. In both cases the absorption l y i n g at lowest wave number must be assigned to a component of v^; very unexpectedly no other nearby l i n e s are found which may be assigned to the two remaining predicted components. In the spectrum of LiBD^ the absorption associated with v 4 appears at 828 cm It i s very sharp and has a half-height band width of 6 cm *. Therefore, i t i s u n l i k e l y that more than one component of v 4 contributes to i t s i n t e n s i t y and thus i t i s l i k e l y that the remaining components just do not appear. Since none of the components of v 4 appear i n the Raman spectrum only the one l i n e can be assigned. In the spectrum of LiBH^ i t appears at 1089 cm 1 . Of the three remaining li n e s appearing i n the bending region of the spectrum of each of the lithium borohydrides, two must be assigned to the two predicted v 2 components. In the spectrum of LiBH^ we may associate the two sharpest li n e s of the t r i p l e t with v 2 and i f we further choose the most intense l i n e of the v 9 doublet as v 9 ( B + B ) then the following assignments can be made: v 2 ^ i u + ^3u^ = 1284 cm 1 and v 2 ' ( B 2 u ) = 1323 cm"1. The corresponding assignments f o r LiBD 4 are v„(B. + B q ) = 919 cm - 1 and v-'(B„ ) = 933 cm"1. The mean 2 lu ^u 2 2u v,2 frequencies f o r LiBH^ and LiBD 4 are 1304 and 926 cm 1 and the 167 frequency r a t i o i s 0.711 which compares with a predicted frequency r a t i o of 0.707 f o r the free ion. A f t e r assigning the components of v 2 there remains an anomalous l i n e at 1254 cm 1 i n the spectrum of LiBH 4 and an anomalous l i n e at 957 cm - 1 i n the spectrum of LiBD^. It i s very l i k e l y that these l i n e s have the same o r i g i n as the anomalous l i n e s observed i n the bending regions of the spectra of the phase II sodium borohydrides. It w i l l be r e c a l l e d that f or the sodium borohydrides these l i n e s were t e n t a t i v e l y assigned as 3v_. If the same assignment i s made for the lithium borohydrides, modes of rotatory o r i g i n , without correction f or anharmonicity, are placed at 418 and 319 cm 1 for LiBH^ and LiBD^ r e s p e c t i v e l y . The frequency r a t i o i s 0.76 which compares with an i d e n t i c a l frequency r a t i o f o r the corresponding l i n e s as observed i n the sodium borohydrides. The predicted frequency r a t i o i s 0.71. Turning now to the stretching region of the i n f r a r e d spectrum of LiBH^ we f i n d that the most prominent feature i s a t r i p l e t of intense li n e s which appear at 2277, 2307 and 2350 cm These l i n e s may be associated with v^. The two most intense lowest l y i n g l i n e s may be assigned as ^ ( B ^ + B„ ) = 2277 cm - 1 and v '(B. + B. ) = 2307 cm"1. The highest wave number 3uJ 3 l u 3u' 6 absorption i s then assigned as v 3 " ( B 2 u ) = 2350 cm 1 . It i s found that the two low wave number components also appear i n the Raman spectrum where the respective frequencies are 2274 and 2309 cm 1 . The two absorptions appear-ing on the high frequency side of v^1' at 2387Csh) and 2423 cm 1 can be assigned as v 2 + and v2* + v4* ^ v4 * s t a ^ e n a s 10°>9 cm 1 then the predicted frequencies for + and v 2 ' + are r e s p e c t i v e l y 14 and 11 cm 1 lower than those observed. This s i t u a t i o n can be a t t r i b u t e d to Fermi 168 resonance with a component of Vy The two remaining l i n e s appearing i n the stretching region of the spectrum of LiBH^ can be assigned as 2v 4 and 10 _i 10 2 v 4- They appear at 2176 and 2188 cm r e s p e c t i v e l y and i f and v 4 are taken to be the observed i n f r a r e d frequencies then the respective anharmonicities are two wave numbers and one wave number. The only l i n e s th appearing i n the second order Raman spectrum of LiBH^ are associated wi 2v 4 > A. doublet appears with frequencies of 2169 and 2173 cm"1. The high wave number component can be associated with the 2176 cm 1 absorption observed i n the i n f r a r e d . The low wave number component, no doubt, involves d i f f e r e n t components of v 4 and may have an i n t e n s i t y which i s enhanced by Fermi resonance with v,(A + B 2 ) which appears at 2300 cm 1 . The i n f r a r e d stretching region of the spectrum of LiBD 4 has features which c l o s e l y p a r a l l e l those observed f o r L i B H 4 > One important diffe r e n c e i s that i n the i n f r a r e d spectrum of LiBD 4 v j ( B l u + Bg u) i s observed. In the i n f r a r e d spectrum of LiBH 4 i t s appearance i s obscured by the l i n e s associated with v^. The v ^ ( B l u + B g u) frequency f o r LiBD 4 i s 1606 cm 1 . This compares with the corresponding Raman l i n e at 1605.5 cm The l i n e s associated with i n the LiBD 4 i n f r a r e d spectrum may be assigned as v^CB^ + B„ ) = 1722 cm"1, v *(B + B, ) = 1736 cm"1 and v "(B, ) = 1757 cm"1. The 3U-* ' 3 iu 3iT 3 ^ 2u' two low wave number absorptions compare with l i n e s observed i n the Raman spectrum at 1721 and 1739 cm 1 . On the high wave number side of v^" an absorption appears at 1785 (sh) cm"1. It can be assigned as + v 4 . If -1 -1 v 4 i s taken as 828 cm the predicted frequency i s 24 cm lower than that observed. This s i t u a t i o n can be at t r i b u t e d to Fermi resonance with a 169 component of \>y The two remaining l i n e s appearing i n the i n f r a r e d s t r e t c h -ing region of the LiBD^ spectrum have frequencies of 1654 and 1672 cm - 1. 10 These l i n e s can be assigned as 2v 4 and 2 re s p e c t i v e l y . In the v i b r a t i o n a l Raman spectra of the lithium borohydrides none of the predicted modes of either rotatory or tran s l a t o r y o r i g i n appear. How-ever, the long wave length i n f r a r e d spectra are very r i c h i n s p e c t r a l . i n -formation. No modes were observed which could be assigned to rotatory o s c i l l a t i o n s but 6 LiBH^ modes and 5 LiBD^ modes of tran s l a t o r y o r i g i n were 6 7 10 11 observed. We may note that the L i - L i and B- B natural abundances, of about 1:12 and 1:4 w i l l introduce disorder into the l a t t i c e which w i l l destroy true t r a n s l a t i o n a l p e r i o d i c i t y . Therefore some of the observed modes may or i g i n a t e from states with K ± 0. In the zero wavevector l i m i t 7 i n f r a r e d active t r a n s l a t o r y modes are predicted. Meaningful assignments f o r the translatory modes with respect to symmetry type are impossible without undertaking a study involving s i n g l e c r y s t a l s and polarized r a d i a t i o n and therefore no attempt at making such assignments w i l l be made. The assignments f o r the lithium borohydrides are summarized i n Table 5-7. 5-5 The Barrier to Rotation In Chapter. IV i t was seen that b a r r i e r height, Vo(N.M.R.) as obtained from N.M.R. experiments provided good estimates of the b a r r i e r to r o t a t i o n for the ordered phases of the ammonium halides. It i s very l i k e l y that a s i m i l a r s i t u a t i o n holds f o r the ordered phases of the a l k a l i metal boro-hydrides. However, the four f o l d cosine p o t e n t i a l used to cal c u l a t e the 170 b a r r i e r heights, Vo(G.P.B.), f o r the ammonium halides w i l l not be r e l i a b l e for the a l k a l i metal borohydrides. For the phase III and IV ammonium halides r o t a t i o n of the NH^+ ion about the 4 axis gives four e s s e n t i a l l y equivalent positions of minimum po t e n t i a l energy. A s i m i l a r s i t u a t i o n holds f o r the phase II and III a l k a l i metal borohydrides but with the important diffe r e n c e that intermediate between the four equivalent positions of minimum po t e n t i a l energy there are four a d d i t i o n a l equivalent positions which w i l l probably l i e close i n energy to the positions of minimum po t e n t i a l energy. This w i l l have the e f f e c t of lowering the b a r r i e r height from that predicted by the four f o l d cosine p o t e n t i a l . That t h i s i s so i s indicated by considering the N.M.R. r e s u l t s (70) f o r the b a r r i e r heights of the a l k a l i metal boro-hydrides. For NaBH 4(II) Vo(N.M.R.) i s 1230 ± 70 cm"1. The r e s u l t s f o r the to r s i o n a l frequency of NaBH^fll) as in f e r r e d from heat capacity data (35) and the second overtone, 3v D, are 350 and 384 cm 1 r e s p e c t i v e l y . For these t o r s i o n a l frequencies the f o u r - f o l d cosine p o t e n t i a l predicts respec-t i v e b a r r i e r heights, Vo(G.P.B.), of 2160 and 2850 cm"1. These r e s u l t s for the b a r r i e r height are not within the l i m i t s • predicted by Vo(N.M.R.) and indicate that a four f o l d cosine p o t e n t i a l cannot be meaningfully rel a t e d to the o s c i l l a t i o n s of the BH^ ions i n the ordered phases of the a l k a l i metal borohydrides. The b a r r i e r height Vo(N.M.R.) f o r potassium borohydride as determined by Tsang and Farrar (70) i s 1240 ± 40 cm 1 . These same authors in t e r p r e t t h e i r N.M.R. r e s u l t s f o r lithium borohydride as giving evidence f o r two c r y s t a l l o g r a p h i c a l l y non-equivalent BH^ ions i n the unit c e l l which have b a r r i e r s to r o t a t i o n of 1650 ± 100 and 1330 ± 100 cm 1 . This i n t e r p r e t a t i o n i s probably subject to review since both the 171 previous X-ray r e s u l t s (42) and the i n f r a r e d r e s u l t s of t h i s work unambig-uously ind i c a t e that each of the four BH4~ ions i n the u n i t c e l l has i d e n t i c a l C s s i t e symmetry; i . e . once a s i n g l e BH^ ion of s p e c i f i c C g symmetry i s appropriately placed i n the u n i t c e l l the symmetry operations of the space group generate three more BH4" ions of i d e n t i c a l symmetry at positions i n the unit c e l l which s a t i s f y the symmetry requirements of the space group. 5-6 The Force F i e l d f o r C r y s t a l l i n e BH4" The fundamental frequencies of the tetrahedral borohydride ion as observed i n the v i b r a t i o n a l spectrum of KBH 4(III) have been used to compute the force f i e l d of the c r y s t a l l i n e BH 4 ion. The force constants calculated at X^ = -0.48 are as follows Force Constant (1Q5 Dynes/cm) F : 3.17937 F 2 0.49194 F 3 2.76021 F 4 0.48640 F 3 4 0.06611 When these force constants are used to compute the spectrum of the c r y s t a l -l i n e BD4~ ion i n KBD 4(III) the average absolute deviation i s 18 cm"1 (see Table 5-8). Figures 5-10 and 5-11 show the dependence of F 3 > F 4 and F 3 4 on the mixing parameter i n the i n t e r v a l -1.2 i X < 1.2. The dependence of v, and v. on the mixing parameter i n t h i s same i n t e r v a l i s shown i n Figure 172 F I G U R E 5 — IO T H E S Y M M E T R Y F O R C E C O N S T A N T S I 1 1 1 1 1 1— - 1.2 -0.8 -OA O.O OA 0.8 1.2 MIXING PARAMETER 173 0.4-F I G U R E 5 — 1 1 T H E S Y M M E T R Y F O R C E C O N S T A N T F3 4. A S A F U N C T I O N O F T H E M IX ING P A R A M E T E R K B H 4 ( i i i ) ^ 0.6-U \ LJ a 8 " Z > Q in o H Z CO z o u LU o o r o I.O-1.2-.4-1.6-.8-2.0--1.2 -0.8 -0.4 O.O 0.4 0.8 MIXING PARAMETER 1.2 174 u o r UJ 00 Z) Z LU 10 I 7 8 0 H 1 7 6 0 I 7 4 0 H I 7 2 0 H I 7 0 0 H I 6 8 0 H I 6 6 0 -I 6 4 0 -I 6 2 0 H F I G U R E 5 - 1 2 C A L C U L A T E D F R E Q U E N C I E S F O R z/3 A N D ^  A S A F U N C T I O N O F T H E MIXING P A R A M E T E R K B H ( III ) K B D 4 (III) 8 9 0 h 8 8 0 h 8 7 0 h 8 6 0 h 8 5 0 h 8 4 0 r - 8 3 0 h 8 2 0 h 8 I O , 1 1 1 1— - 1 . 2 - 0 . 8 - 0 . 4 O.O 0 . 4 0 . 8 MIXING PARAMETER 175 TABLE 5-8. Calculated and Observed Spectra - BH~(T d) (i) KBH 4(III) + KBD 4(III) X x = -0.60 v 3 = 1724 X„ = -0.36 v. = 855 2 4 X 3 = -0.48 V l V2 V3 V4 Calculated Spectrum X = X 3 1637.2 885.3 1704.6 845.1 Observed Spectrum 1599.5 892 1724 855 Deviation +37.7 -6.7 -19.4 -9.9 ( i i ) KBD 4(III) KBH 4(III) Xj = -0.89 v 3 = 2281 X 2 = -0.59 V 4 = 1121 X 3 = -0.74 V l V2 V 3 V4 Calculated Spectrum X = X 3 2261.2 1261.0 2306.0 1134.6 Observed Spectrum 2314.5 1251.5 2281 1121 Deviation -53.3 +9.5 +25.0 +13.6 176 5-7 The Ef f e c t s of a D_, D i s t o r t i o n on BH~ 2d 4 The "unperturbed" v i b r a t i o n a l frequencies of the hypothetical BH^"(T^) ion used to ca l c u l a t e the contribution of the k i n e t i c energy to the spectrum of the BH 4 (D 2 d) ion were chosen according to the following scheme: v 1 = V j C A O v 2 = [v 2 ( A ! ) + v 2(B!)]/2 v 3 = v 3 C E ) - 1/3A v 4 = v 4(E) + 1/3A Here A i s the magnitude of the s p l i t t i n g associated with the appropriate t r i p l y degenerate T^ v i b r a t i o n . It i s e a s i l y seen that the above scheme i s based on the s p l i t t i n g pattern predicted i n section 4-7. We r e c a l l that t h i s s p l i t t i n g pattern was f o r small D 2 d angular d i s t o r t i o n s involving two angles greater than the tetrahedral angle and four angles less than the tetrahedral angle. This s i t u a t i o n i s compatible with the geometry of the phase II BH 4 ion given i n Table 1-8; i . e . there are two angles with a d i s t o r t i o n of +2.22° and four angles with a d i s t o r t i o n of -1.10°. Figures 5-13 and 5-14 show the NaBH 4(II) v^/v^ s p l i t t i n g s as a func-t i o n of the mixing parameter f o r an angular d i s t o r t i o n of +2.22° and -1.10°. When the mixing parameter i s chosen so as to maximize the v 3 / v 4 s p l i t t i n g on the i n t e r v a l -1.2 i X < 1.2 the predicted s p l i t t i n g s are those of Table 5-9. In the case of NaBD4 there i s a complete set of experimentally ob-served s p l i t t i n g s which can be compared with the predicted s p l i t t i n g s . For v 2 > v 3 and the observed s p l i t t i n g s are 12.0, 11.0 and 15.5 cm 1 . These compare with respective predicted s p l i t t i n g s of 12.6, 13.9 and 8.9 cm"'''. 177 Figures 5-15 and 5-16 show the r e s u l t s when F(T d)(X =-0.48) i s fi x e d and G(D 2 d) ^ s varied (X = -0.48 gives optimum t r a n s f e r a b i l i t y of force constants f o r KBH^ (T^) KBD^ (T^) - see Table 5-8). We see that the observed s p l i t t i n g of 31 cm 1 requires an angular d i s t o r t i o n of -3.3° and +6.9°. The 26 cm"1 s p l i t t i n g requires a d i s t o r t i o n of -5.6° and+11.90°. These values compare with the probable d i s t o r t i o n of -l.l Cand+2.20? 178 o 2 3 0 0 H 2 2 9 6 H 2 2 9 4 H o r LU 2 2 9 2 DQ D Z 2 2 9 0 UJ > 2 2 8 8 H 2 2 8 6 4 FIGURE 5-13 PREDICTED SPLITTING OF z/3 AS A FUNCTION OF THE MIXING PARAMETER NaBH 4 (Td ) NaBH 4( D 2 d) 2 2 8 4 H "T— 1.2 -1.2 -0 .8 - 0 . 4 O.O' 0.4 0.8 MIX ING P A R A M E T E R 179 F I G U R E 5 - 1 4 P R E D I C T E D S P L I T T I N G O F A S A F U N C T I O N O F T H E M IX ING P A R A M E T E R -N a B H (Td ) — - N a B H ( D ) -1.2 -0.8 -0.4 O.O 0.4 0.8 1.2 MIX ING P A R A M E T E R TABLE 5-9. Calculated and Observed Spectra - BH.(D ,) NaBH 4(T d) •> NaBH 4(D 2 d); X. = -0. 4 60 V i(Ai) v 2 ( A O v 2 ( B i ) V 3(B 2) v 3 CE) V 4(B 2) V*(E) OBS. 2341 1280 2272 2303 1148 1122 A 31 .0 26. 0 CALC, 2341 1288.8 1271.1 2285.7 2296.2 1134.1 1129.0 A 17.7 10 .5 5. 1 ( I D NaBD 4(T d) NaBD 4(D 2 d); X„ = -0. 4 70 V i(AO V 2 ( A i ) V 2 ( B i ) v 3 ( B 2 ) v 3 ( E ) Vi,(B2) V- (E) OBS. 1622 916 904 1724.5 1735.5 870 854.5 A 12.0 11.0 15 .5 CALC. 1622 916.3 903.7 1722.5 1736.4 864.3 857.4 A 12.6 13.9 8.9 181 F I G U R E 5 - 1 5 SPLITTING OF Z£ AND ^ AS A FUNCTION OF ANGULAR DISTORTION (X=-0.48) — i 1 1 1 1 1 1— 4/ -6.0 -4.0 -2.0 O.O + 2.0 +4.0 + 6.0 2/ + I2.8 +8.4 +4.1 O.O -3.9 -7.7 -11.4 ANGULAR D ISTORTION 0 1 8 2 F I G U R E 5 - 1 6 S P L I T T I N G O F ^ A S A F U N C T I O N O F A N G U L A R D I S T O R T I O N N a B H 4 ( T d ) — N a B H 4 ( D g d ) ' — i 1 1 1 1 1 r~ 4/ - 6.0 . -4.0 -2.0 O.O +2.0 + 4.0 +6.0 2/ +12.8 +8.4 +4.1 O.O -3.9 -7.7 -11.4 A N G U L A R DISTORTION 0 183 CHAPTER VI CONCLUSION It i s apparent from the r e s u l t s of t h i s work that for c r y s t a l l i n e tetrahedral ions an average angular d i s t o r t i o n of about two degrees from a tetrahedral configuration can r e s u l t i n s p l i t t i n g s of the degenerate bending and s t r e t c h i n g v i b r a t i o n s of up to some 30 cm"1. This s e n s i t i v i t y of the spectrum to angular d i s t o r t i o n s associated with the absorbing or sc a t t e r i n g molecule has important implications. For example, both N.M.R. (71) and X-ray (42) r e s u l t s f o r lithium borohydride have been reported as being compatible with a BH^ ion i n the l a t t i c e which has an e f f e c t i v e tetrahedral symmetry but the s i t e group s p l i t t i n g observed i n t h i s work i s c l e a r l y i n d i c a t i v e of a d i s t o r t e d BH^ ion; i . e . the v 2 s p l i t t i n g of 14 cm 1 observed i n the spectrum of LiBD^ compares with a v 2 s p l i t t i n g of 12 cm 1 for NaBD^fll) and thus indicates an average angular d i s t o r t i o n of at least one or two degrees for the c r y s t a l l i n e borohydride ion i n the lithium borohydride l a t t i c e . Attempts to c a l c u l a t e the spectrum of the XY^(D 2 d)(XY^ = NH 4 +, BH 4 ) ion using the F matrix associated with an XY^Crp ion and the G matrix associated with the same ion which has under-gone a s l i g h t angular d i s t o r t i o n to give the XY^(D 2 d) ion were successful i n that the order of appearance i n the spectrum of the B 2 and E components associated with the two F 2 v i b r a t i o n s of the XY^(T^) ion were c o r r e c t l y predicted. This r e s u l t indicates that s i m i l a r c a l c u l a t i o n s could be used i n analysing the v i b r a t i o n a l spectra of other systems providing the spectrum of an appropriate basis molecule i s known. 184 Further interesting features of t h i s work are the assignments associ-ated with the second order spectrum of the ammounium halides which involve phonons with 0. Of pa r t i c u l a r interest are the assignments involving v, i n combination with acoustical modes. These combinations indicate a valuable source of information about the phonon dispersions and also the densities of states associated with the acoustical phonons. A most important area of interest with respect to t h i s work i s the effect of the phase transitions on the vib r a t i o n a l spectra of the various crystals studied. For the v i b r a t i o n a l spectra of the ammonium halides i t i s found that the int e n s i t y versus temperature curves for the absorptions associated with the t r i p l y degenerate free NH^+ ion bending vibration give evidence of the two types of transitions which occur; i . e . the d i s -order-order transitions are characterized by gradual i n t e n s i t y changes and the order-order transitions are characterized by abrupt intensity changes. I t i s further found that the f i r s t order spectra of the ordered phases can be s a t i s f a c t o r i l y interpreted on the basis of fundamental modes predicted i n the zero wavevector l i m i t , but that for the lowest temperature disordered cubic phase this i s no longer the case, since there i s d i s t i n c t evidence of densities of states with K f 0 appearing i n the f i r s t order spectrum. This s i t u a t i o n i s most pronouncedfor the t r i p l y degenerate bending mode. In t h i s case the intensity dependence of the density of states with K f 0 on the ordering process has s a t i s f a c t o r i l y been explained by Garland and Schumaker (28). Since Garland and Schumaker*s work shows that the disorder associated with the phase II ammonium halide 185 l a t t i c e can appreciably contribute to the intensity of the fundamental vibrations v i a densities of states with K=f 0, i t i s very l i k e l y that this same si t u a t i o n also occurs i n other disordered systems. An example would be the ices 1^ and I which possess disorder (72) involving half-hydrogens. Whalley and Bertie (73) and Whalley and Labbe (74) have interpreted the spectrum of the ice 1^ and I translatory modes by allowing for the presence of K f 0 absorptions. The results for the ammonium halides which indicate the presence of K f 0 absorptions i n the spectrum of the internal funda-mentals suggest that the band shape of the very broad v^/v^ ice absorptions may also be influenced by the presence of JC j- 0 absorptions. Turning now to the a l k a l i metal borohydrides i t i s found that the spectroscopic results for sodium borohydride at 295°K and 10°K are com-patible with a disorder-order phase transformation which involves a trans-i t i o n from a disordered cubic phase to an ordered tetragonal phase. In the case of potassium borohydride the low temperature spectrum i s interpreted as indicating a low temperature ordered cubic phase which has a 2 structure compatible with the space group T^ and the spectrum recorded at 295°K i s interpreted as indicating a cubic strucutre intermediate 2 5 between the ordered T, structure and the disordered 0, cubic structure of d h sodium borohydride. A neutron d i f f r a c t i o n study of potassium borohydride 2 should be able to determine i f the T^ structure i s , indeed, correct. An X-ray determination of the l a t t i c e parameters of KBH^ over the temper-ature i n t e r v a l 0-500°K would also be useful, since any phase transitions are very l i k e l y to be characterized by a discontinuous percent volume 186 change for the unit c e l l just as in the case of the ammonium halides. Such a study would be an important adjunct to the heat capacity data (37) and perhaps could more accurately f i x the temperature at which the onset of the disorder-order transition begins. There is one fi n a l item that w i l l be commented on and this is the assignment 3v R associated with the spectra of both sodium and lithium borohydride. In the spectrum of sodium borohydride this absorption appears with observable intensity only in the ordered tetragonal phase. This situation is comparable with that observed for the ammonium halides where lines associated with v D appearing in the second order spectra suffer a striking loss in intensity as a result of a transition from an ordered phase to a disordered phase. Also, with respect to the members of the series KBH 4(III), NaBH4(II) and LiBH 4, where the BH4~ ion suffers a progressive distortion from T^ to to C g site symmetry, i t is found that the intensity of 3v D starting with zero observable intensity for KBH 4(II), is progressively enhanced. This is evidently a result of the fact that for the members of the series KBH 4(III), NaBH4(II) and LiBH 4 there is a progressively greater number of possible components which can contribute to the intensity of 3v D. A recent communication (75) from T.C. Farrar indicates that neutron inelastic studies of the a l k a l i metal borohydrides wi l l be undertaken at the National Bureau of Standards; Laboratories in the near future. 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