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The infrared spectra of crystalline calcium and sodium formates 1964

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THE INFRARED SPECTRA OF CRYSTALLINE CALCIUM AND SODIUM FORMATES by THOMAS LAURIE CHARLTON B.Sc,, U n i v e r s i t y o f B r i t i s h Columbia, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of CHEMISTRY We accept t h i s t h e s i s as conforming to the r e q u i r e d standard September, 1964 In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of • B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study, I further agree that per- mission for extensive copying of this thesis for scholarly pur-poses may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publi- cation of this thesis for f i n a n c i a l gain sh a l l not be allowed without my written permission* Department of The University of B r i t i s h Columbia,, Vancouver 8, Canada ABSTRACT The infrared absorption spectra of single crystals of calcium and sodium formate have been recorded be» tweem IJJOOO and 5 0 6 osf With the aid of polarized radiation i t was possible to carry out an analysis of the spectra* Factor group splitting was observed for the molecular modes of calcium formate. It was possible to assign the origin of the components of any given molecular fundamental to one of the two sets of symmetrically non-equivalent formate ions in the Calcium formate unit c e l l . This assignment was made on the basis of the relative Intensities of the com- ponents of the fundamentals. For the sodium formate single crystal spectrum* a combination of factor group and site group analysis was required to satisfactorily interpret the results. The results obtained in this case helped to resolve the controversy surrounding the assignment of the molecular modes for the formate ion. lattice mode frequencies of 36* 61, 90, 128, 1£IL and 192 cm''1 for calcium formate and $8, 91, 112, 128 •1 and 231 cm for sodium formate are inferred from combinations with molecular modes. The analysis was completed by the assignment of several modes to formate Ion fundamentals and to combination and over- tone modes of molecular fundamentals,, i ACKNOWLEDGEMENT I would l i k e t o thank Dr. K. B. Harvey f o r h i s a d v i c e and a s s i s t a n c e i n c a r r y i n g out t h i s study. I would a l s o l i k e to thank Dr. S. Melzak and Mr. A. MacDonald f o r t h e i r a i d i n the x-ray a n a l y s i s . The th o u g h t f u l n e s s o f Dr. R. M. Hamraaker i n making a v a i l a b l e h i s unpublished r e s u l t s Is a l s o a p p r e c i a t e d . I i i TABLE OP CONTENTS page Acknowledgement I Abstract 11 Table of Contents i l l L i s t of Tables V L i s t of Figures v i i Chapter 1'- Introduction 1 1-1. Preliminary remarks 1 1-2. The problem 4 1- 3. Methods of investigation 9 Chapter 2'- Experimental work 10 2- 1. Materials investigated spectro- s c o p i c a l l y 10 2-2. Growth of single c r y s t a l s 10 2-3. Grinding 18 2-4. Apparatus 28 2- 5. Spectra 29 Chapter 3< Theory 42 3- 1. General theory of vibrations i n cr y s t a l s 42 3-2. Selection rules 45 iv page 3-3. C r y s t a l symmetry 47 3- 4. A n a l y s i s of v i b r a t i o n s i n c r y s t a l s 49 Chapter 4. D i s c u s s i o n 57 4- 1. C r y s t a l s t r u c t u r e s 57 4-2. Calcium formate 58 4-3. I n f r a r e d s p e c t r a o f c a l c i u m formate 62 4-4. 34 r e g i o n (3100 - 2700 cm - 1) 64 4-5. %\ r e g i o n (820 - 760 cm" 1) 67 4-6. r * £ r e g i o n ( . 1 1 0 0 - 1 0 5 0 cm" 1) 71 4-7. 4̂ -.and r e g i o n ( 1 7 0 0 - 1300 cm~x) 73 4-8. L a t t i c e modes 74 4-9. Combinations and overtones. 77 4 - 1 0 . I s o t o p i c s h i f t s 79 4-11. I n f r a r e d spectrum of sodium formate 81 4 - 1 2 . Combination and l a t t i c e modes; i s o t o p i c s h i f t s 88 4 - 1 3 . C o n c l u s i o n 91 B i b l i o g r a p h y 93 V LIST OF TABLES page Tab l e 1. I l l u s t r a t e d I n f r a r e d s p e c t r a 30 2. P a r t i a l summary of i n f r a r e d and Raman f r e q u e n c i e s of c a l c i u m formate and c a l c i u m formate-d n 31 1 3. P a r t i a l summary of the i n f r a r e d and Raman f r e q u e n c i e s of sodium formate and sodium formate-d^ 33 4. The i n f r a r e d a b s o r p t i o n f r e q u e n c i e s of ca l c i u m formate 35 5. The i n f r a r e d a b s o r p t i o n f r e q u e n c i e s o f sodium formate 39 6. Summary of ex p r e s s i o n s f o r c h a r a c t e r s of group o p e r a t i o n s 59 7. C h a r a c t e r t a b l e and f a c t o r group a n a l y s i s f o r c a l c i u m formate 60 8 . Squares o f d i r e c t i o n c o s i n e s o f t o t a l l y symmetric o s c i l l a t i n g d i p o l e c a l c u l a t e d from the c r y s t a l s t r u c t u r e of c a l c i u m formate 66 9. R a t i o s o f c a l c u l a t e d and observed t o t a l r e l a t i v e i n t e n s i t i e s f o r * J 66 10, R a t i o s of c a l c u l a t e d and observed r e l a t i v e i n t e n s i t i e s w i t h v a r i o u s p o l a r i s a t i o n s f o r ̂  67 11, R a t i o s of c a l c u l a t e d and observed r e l a t i v e i n t e n s i t i e s f o r ^3 69 12, R a t i o s of c a l c u l a t e d and observed r e l a t i v e i n t e n s i t i e s w i t h v a r i o u s p o l a r i z a t i o n s f o r ̂  70 v i page T a b l e 13. R a t i o s o f c a l c u l a t e d and observed r e l a t i v e i n t e n s i t i e s w i t h v a r i o u s p o l a r i z a t i o n s f o r 73 14. P o p u l a t i o n of l a t t i c e v i b r a t i o n a l s t a t e s a t 20°C 76 15. Symmetries of combination and over- tone modes f o r c a l c i u m formate 78 16. C h a r a c t e r t a b l e and f a c t o r group a n a l y s i s f o r sodium formate 82 17. S i t e group c o r r e l a t i o n t a b l e f o r sodium formate ' 83 18. C a l c u l a t e d and observed t o t a l i n - t e n s i t y r a t i o s f o r sodium formate 88 19. Symmetries of combination and over- tone modes f o r sodium formate 90 v i i LIST OF FIGURES To follow page Figure 1. Mounting of seed c r y s t a l s 14(on page) 2. Calcium formate single c r y s t a l 20(on page) 3. Infrared spectrum of p o l y c r y s t a l l i n e calcium formate 4 l 4. Infrared spectrum of p o l y c r y s t a l l i n e calcium formate-d^ 4 l 5. Infrared spectrum of p o l y c r y s t a l l i n e sodium formate 4l 6. Infrared spectrum of p o l y c r y s t a l l i n e sodium formate-d^ 4l 7. Polarized infrared spectrum of calcium formate (3020 - 2680 cm"1) 41 8. Polarized i n f r a r e d spectrum of calcium formate (1800 - 1000 cm"*1) 4 l 41 9. Polarized infrared spectrum of calcium formate (840 - 740 cm"1) 10. Polarized i n f r a r e d spectrum of sodium formate (3100 - 2650 cm-1) 4 l 11. Polarized infrared spectrum of sodium formate (1800 - 1000 cm"1) 4l 12. Polarised i n f r a r e d spectrum of sodium formate (1050 - 750 cm*1) 41 13. C r y s t a l structure of calcium formate projected on (001) plane 3nd symmetry elements of calcium formate unit c e l l 4l 14. C r y s t a l structure of sodium formate projected on (010) plane and symmetry elements of sodium formate unit c e l l 4 l - 1 * CHAPTER 3,. IHTROEPCTION 1-1* Preliminary remarks In the infrared region of the electromagnetic spectrum l i e the r o t a t i o n a l and v i b r a t i o n a l spectra of molecules. That i s , the manifestations of the changes i n the molecular r o t a t i o n a l and v i b r a t i o n a l energy that can occur under c e r t a i n conditions by the i n t e r a c t i o n of infrared radiation with matter* The frequencies which thus appear are d i r e c t l y dependent on c e r t a i n molecular constants and a study of these frequencies w i l l give much information concerning the structure of the molecules involved* Up to the present, a vast amount of experi*- mental data has been obtained but very l i t t l e has been analyzed* Although procedures f o r t h i s analysis are w e l l developed (39) j many workers have been deterred because of the amount and complexity of the work required to s a t i s f a c t o r i l y complete the analysis. The major portion of the work that has been done has been concerned with the gas phase where interactions between molecules are l i m i t e d , i t i s only r e l a t i v e l y r e- cently, a f t e r the motions of single molecules were gener- a l l y w e l l understood* that attention has s h i f t e d to the study of molecules i n condensed systems and p a r t i c u l a r l y - 2 - to c r y s t a l s . Although a f a i r amount of work has been done with single c r y s t a l s i n the Haman effect (where comparatively much thicker c r y s t a l s can be used), l i t t l e work has been done i n the Infrared e f f e c t because of the d i f f i c u l t y i n obtaining single c r y s t a l s t h i n enough so as not to be t o t a l l y absorbing throughout most of the spectrum. The development of better techniques f o r obtaining very t h i n single c r y s t a l s w i l l help to Increase the knowledge about interraolecular c r y s t a l l i n e forces. Because of the Interraolecular forces i n condensed systems, the r o t a t i o n a l f i n e structure associated with the spectra of gases i s generally not observed> and i f i t i s , then mainly i n the l i q u i d state. (Rotational f i n e structure has been reported i n some s o l i d ammonium s a l t s . ) Three main differences between the spectra of vapors and s o l i d s occurs (1) usually small changes or s h i f t s i n the frequencies of v i b r a t i o n s , (2) the appearance of new v i b r a t i o n a l frequencies and (3) s p l i t t i n g of the vapor phase frequencies into two or more peaks. The f i r s t e ffect i s found to be temperature depen- dent as w e l l as being a function of the electrostatic i n - teraction of an o s c i l l a t i n g dipole with i t s surroundings. The second effect may be explained by the occurrence of polymers or associated molecules i n the s o l i d state. The - 3 .- t h i r d e f f e c t arises because of a degradation of the symmetry of the molecule concerned when b u i l t into a c r y s t a l . This degradation can bring about the removal of degeneracy as we l l as causing some modes, which are Inactive i n the high symmetry of the free molecule, to be active i n the lower symmetry of one or more subgroups of that molecule. In t h i s work, we are concerned with the t h i r d e f f e c t and a more complete discussion w i l l be given l a t e r . The use of polarized radiation f o r the analysis of the spectra of cr y s t a l s i s desirable. For substances i n non-ordered arrays, the al t e r a t i o n s i n dlpole moments giving r i s e to absorption bands w i l l take place i n a l l possible d i r e c t i o n s . Thus, the e x c i t a t i o n of a l l v i b r a - tions possible i s equally probable. This i s also true f o r p o l y c r y s t a l l l n e samples i n which a l l c r y s t a l l i t e s are randomly oriented* The s i t u a t i o n i s e n t i r e l y d i f f e r e n t with single crystals. Because of the f i x e d orientation of the mqleculejs, the o s c i l l a t i n g dipoles w i l l also be a l l l g h e d . By using polarized r a d i a t i o n , the orientation of dipoles In the c r y s t a l ffiay be determined. I f the c r y s t a l structure of the sample Is known, then a v i b r a t i o n a l assignment may be made by combining the p o l a r i z a t i o n and X-ray r e s u l t s , S i m i l a r l l y , I f the c r y s t a l structure i s - 4 - unknown, the infrared spectrum may a i d i n i t s determina- t i o n . This technique i s p a r t i c u l a r l y useful i n structure determinations involving hydrogen atoms, where X-ray methods are I n s u f f i c i e n t . Halford (1,2) and others have determined c r y s t a l l i n e space groups by studying the i n - frared spectra of single c r y s t a l s . 1-2. The problem A s a l t of the formate ion was chosen f o r t h i s study f o r several reasons. F i r s t , the modes of v i b r a t i o n of XYZ/» type systems have been w e l l characterized (3 ) I second* as a continuation of the work started i n t h i s laboratory by Morrow(4) i n h i s study of barium formateJ and t h i r d , as a further test of the theories of vibrations i n c r y s t a l s developed by Bhagavantum and Venkatarayudu (5*6) and Halford (7) using a dif f e r e n t c r y s t a l system than that used by Morrow* The reasons f o r the choice of therp a r t i c u l a r s a l t s studied w i l l be outlined l a t e r . A considerable amount of previous work involving the spectra of formates has been carr i e d out i n both the Raman and infrared e f f e c t s . In 1936, Gupta(8) obtained the Raman spectra of sodium, barium, and cadmium formates i n the c r y s t a l l i n e state and i n aqueous s o l u t i o n . An attempt was made to explain the res u l t s on the hypothesis - 5 - that the C-H linkage undergoes prototropic change i n solution to y i e l d dihydroxymethylene, The explanation i s based on an Incorrect assumption. The spectra given are rather incomplete with very few frequencies reported. At the same time, Venkateswaran(9) reported more complete spectra f o r sodium, cadmium, and calcium formates In solution and sodium, cadmium, calcium, and lead formates as c r y s t a l s . The occurrence of l i n e s corresponding to the HCO deformation and the CH st r e t c h , point to the existence of the GH group. Lecomte(lOjll) has reported both the infrared and Raman spectra of several formates with incorrect assignments. Ponteyne(l2 ,13) had done a very complete Study of the Raman spectra of aqueous sodium formate and sodium formate-d^. Force constants fo r the formate ion have been calculated using a Urey- Bradley potential function. More recently, Thomas(14), i n an attempt to deter- mine the inplane force constants of the formate i o n , obtained the Infrared spectrum of sodium formate using t h i n f i lms of the s a l t melted between sodium chloride plates. The spectrum obtained i s i n doubt because of the tendency of the formate Ion to decompose at the temperatures required to melt the s a l t * Also, the assign- ments made for the bands observed are in c o r r e c t . Because - 6 - of the u n r e l i a b i l i t y of experimental r e s u l t s , the force constants derived are of s i m i l a r doubtful accuracy. Newman(l5) i n 1952, studied sin g l e c r y s t a l s of sodium formate using polarized r a d i a t i o n provided by a s i l v e r chloride p o l a r i z e r . The l i m i t e d resolving power of his instrument allowed him to report only the funda- mental formate frequencies and two combination bands. He obtained his c r y s t a l by evaporation of aqueous solu- tions of sodium formate. S i g n i f i c a n t differences, such as a large s h i f t to higher frequency of and an apparent reversal of Ponteyne's assignment of ^ 2 and ^ ^ are noted. He suggests that these discrepancies are related to a strong influence (that i s , hydrogen bonding) of the aqueous solvent on the Raman frequencies, although i n t e n s i t y and depolarization measurements of the Raman l i n e s of the solution support Ponteyne's assignment. Ito and Bernstein (16) did an infrared and Raman study of s o l i d sodium formate and a saturated aqueous solution (both H 20 and DgO) of the s a l t * A rather com- plete assignment of the observed peaks confirms Ponteyne's assignments. The s o l i d films of sodium formate were ob- tained by deposition from water, water-methanol, or water- - 7 - acetone solutions on s i l v e r chloride plates. A s l i g h t change i n v i b r a t i o n a l frequencies, depending on the method of sample preparation* i s noted. Newman's band at 1620 cm 1 i s observed only as a weak shoulder. The presence of two l a t t i c e frequencies at 140 and 40 cm""1 i s assumed f o r assignment of combination bands. A-. Harvey, Morrow and Shurvell(l7) did a v i b r a t i o n a l study of several c r y s t a l l i n e formates suspended i n pressed potassium bromide p e l l e t s . Their assignment of fundamental frequencies i s based on Ponteyne's. The occurrence of Fermi resonance i s assumed as an explanation f o r the variations between observed and calculated values of overtone frequencies. The f i n e structure of the absorp- t i o n due to the formate i n t e r n a l vibrations i s ascribed t o ^ c r y s t a l - f i e l d s p l i t t i n g . Schutte and Buijs(l8), as an extension of work on planar tetra-atomic ions, studied the formate s a l t s of calcium, strontium, barium, and lead. I t was found that a second form of calcium formate ( l a b e l l e d ^ -calcium formate) was c r y s t a l l i z e d when a water misclble organic solvent was added to an aqueous solution of normal oc - calcium formate. The spectrum of (3 -calcium formate was quite d i f f e r e n t from the spectrum of ©c -calcium formate. This second phase of calcium formate might give an - 8 - explanation of the variations i n the spectrum of sodium formate obtained by Itoh and Bernstein. Donaldson, Knifton and Ros.s(l9) undertook a study of several formates to determine a rel a t i o n s h i p ( i f any) between the frequency of carbonyl vibrations and various properties of the cations concerned. S h i f t s i n the 0 C 0 stretching frequencies 2 and -zJ ̂  which, I f related to the M-0 bond strength would imply that some form of co-valent bonding e x i s t s . Three types of structure In- volving co-valent bonding have been suggested* H H H I I 1 c c c o o o o O O •J, \ / ^ ^ (A) (B) (C) Type (A) can be distinguished from types (B) and (G) i n that f o r type (A) compounds ^ s ( 0 C 0 ) increases as the M-0 bond strength increases. Three groups of formates were found In which either or both of the f r e - quencies "2^2 and ^ 4 d i f f e r s u b s t a n t i a l l y from those f o r rubidium formate which i s an ioni c compound free from c r y s t a l - f i e l d s p l i t t i n g e f f e c t s . Some covalent character i s thus Inferred f o r some of the formates studied (for example aluminum, g a l l i u m , and t i n ( l V ) formates), although the formates of sodium, potassium, c a l c i u m , s t r o n t i u m , and barium appear to be i o n i c . Itoh(20) and h i s co-workers have c a r r i e d out a study of calcium formate by means of nuclear magnetic resonance. U n l i k e the N.M.R. r e s u l t s f o r barium formate, which i n d i c a t e the reported c r y s t a l s t r u c t u r e t o be i n e r r o r , the N.M.R. r e s u l t s f o r calcium formate i n d i c a t e the r e p o r t e d s t r u c t u r e to be c o r r e c t other than p o s s i b l e s m a l l e r r o r s i n determining the p o s i t i o n s of carbon and oxygen atoms by X-ray a n a l y s i s . 1-3, Methods of i n v e s t i g a t i o n The I n f r a r e d spectrum of s i n g l e c r y s t a l s of sodium and calcium formate were recorded w i t h the a i d of p o l a r i z e d r a d i a t i o n . The i n f r a r e d spectra of p o l y c r y s t a l l i n e sodium and calcium formate and sodium forraate-d^ and c a l - cium formate-d^ suspended i n pressed potassium bromide d i s c B and of these s a l t s i n aqueous s o l u t i o n were a l s o recorded. X-ray r o t a t i o n and Welssenberg photographs were taken i n order t o confirm r e p o r t e d c r y s t a l s t r u c t u r e s and t o unambiguously I d e n t i f y c r y s t a l axes. - 1 0 - CHAPTER 2> EXPERIMENTAL WORK 2-3.* Materials investigated spectroscopicslly The calcium formate used was reagent grade obtained from the B r i t i s h Drug Houses Limited. Sligh t insoluble Impurities were removed from solu t i o n by f i l t r a t i o n through Whatman number 1 f i l t e r paper. The sodium formate used was reagent grade obtained from Baker and Adamson Products, General Chemical D i v i s i o n , A l l i e d Chemical Corporation. Again s l i g h t insoluble impurities were removed by f i l t r a t i o n . The calcium forraate-d^ used was prepared by neutra- l i z a t i o n of formic acid-d^ with calcium carbonate. The product was f i l t e r e d and r'ecrystallized three times from solution. The same method of preparation was used to obtain the sodium forraate-d^i 2 - 2 . Growth of single c r y s t a l s A s a l t of the formate ion was chosen f o r t h i s study as outlined previously. But, because of the complexity of the v i b r a t i o n a l analysis, anything that may add to the d i f f i c u l t y i s undesirable. Thus s a l t s which are hygroscopic at room temperature or are hydrates when c r y s t a l l i z e d from aqugous solution are unwanted. Of - l i - the a l k a l i and a l k a l i n e e a r t h s a l t s of the formate i o n , on l y c a l c i u m and "barium formates do not have these un- d e s i r a b l e p r o p e r t i e s . Lead formate may a l s o be i n c l u d e d i n t h i s group. Strontium and magnesium formates form hydrated s a l t s a lthough a non-hydrated form i s known f o r the s t r o n t i u m s a l t . L i t h i u m , sodium, and potassium formates are h y g r o s c o p i c , the potassium s a l t extremely so. Large s i n g l e c r y s t a l s are most e a s i l y grown from a s a t u r a t e d s o l u t i o n by slow e v a p o r a t i o n of the s o l v e n t . T h i s method e s s e n t i a l l y e l i m i n a t e d l e a d formate from t h i s present work because of i t s low s o l u b i l i t y a t room tempera- t u r e although r a t h e r i m p erfect c r y s t a l s were o b t a i n e d when grown from a l a r g e volume of s o l u t i o n t o a s i z e o f about 3x3x10 mm. over a p e r i o d of s e v e r a l months. With a great d e a l o f c a r e , useable c r y s t a l s c o u l d undoubtedly be grown. Because o f the extreme hygroscopic nature of potassium formate and the u n a v a i l a b i l i t y o f s u f f i c i e n t q u a n t i t i e s of l i t h i u m formate, no attempt was made t o grow c r y s t a l s o f these s a l t s . Of the remaining s a l t s , c a l c i u m , sodium, and s t r o n t i u m formates seemed t o be the most s u i t a b l e . I t was o r i g i n a l l y i n t e n d ed t o c a r r y out a study of s i n g l e c r y s t a l s of sodium formate. The combination o f - 12 - both a d i f f e r e n t c r y s t a l structure and a monovalent cation would provide a good basis of comparison with the work of Morrow. Although sodium formate tends to be s l i g h t l y hygroscopic, i t was f e l t ' t h a t with c a r e f u l handling, the method of grinding (to be described l a t e r ) , and flushing of the instrument with dry nitrogen, most d i f f i c u l t i e s a r i s i n g from t h i s hygroscopic nature could be avoided. To be of any use the c r y s t a l s had to be large enough to cover the s l i t (10x2 ram.) i n the brass plate used f o r mounting the c r y s t a l s . The most obvious way of obtaining c r y s t a l s of the appropriate size was to mount a seed c r y s t a l i n a saturated solution of the desired s a l t and allow growth to proceed with evaporation. According to S e i d e l l ( 2 1 ) , and the International C r i t i c a l Tables ( 2 2 ) , sodium formate c r y s t a l l i s e s as a dihydrate below temperatures of about 28°C. Therefore, i t was f e l t that by keeping a saturated solution of the s a l t at an elevated temperature, suitable c r y s t a l s could be grown. According to the references above, 1 6 . 0 molei of sodium formate should dissolve i n 1000 gr. of water at k0°0. To obtain absolution of t h i s ooncentratidn, an ex- cess of s a l t was added to a volume of water heated to 60-70°C. The supernatant l i q u i d was decanted through a Whatman number 1 f i l t e r paper into a 250 ml. erlenraeyer - 13 - f l a s k and then placed i n a water bath held a t 4 0 ° C . •The erlenmeyer f l a s k was used to reduce the surface area of the solution and thus the rate of evaporation of the solvent at the elevated temperature used. As the solution cooled, seed c r y s t a l s were formed. Two types of seed c r y s t a l s were obtained. Long needle-like c r y s t a l s approximately 0 .5 tarn, square and In various lengths l i m i t e d only by the dimensions of the f l a s k were usually obtained* Several times however, th i n f l a t plates several ram. wide and a few cm. long were obtained. The formation of the two types appeared to be dependent on the rate of cooling of the so l u t i o n , the f l a t plates appearing on very slow cooling. Both types of seeds were used i n attempts to grow larger c r y s t a l s . An attempt was also made to grow single c r y s t a l s from the melt. A s i m i l a r method was used by Buchanan et a l . (23) to grow single c r y s t a l s of normal and deuter- ated lithium hydroxide. The samples they obtained were then cleaved to y i e l d c r y s t a l plates several hundred microns thick. In t h i s case a large sample (about 25 grams) was placed In a sealed glass tube one h a l f inch in;j diameter and drawn out to a c a p i l l a r y at the lower end. This tube was then suspended i n a furnace by a - 14 - system that lowered the tube from the furnace at a slow rate. As the c a p i l l a r y portion of the tube emerged from the furnace, the sodium formate contained inside c r y s t a l - l i z e d and acted as a seed f o r the remaining s a l t . However, t h i s method was discarded a f t e r a tube exploded i n the furnace because of the decomposition of the s a l t at the temperature maintained inside the furnace. The seeds obtained by evaporation were mounted on a small c o l l of plastic-coated copper wire over which a small piece of spaghetti-tubing was placed. The seed was held i n place by a small amount of G.I.L. Household Cement. Figure 1. Mounting of seed c r y s t a l s . The mounted c r y s t a l was then placed i n a 250 ml. erlenmeyer f l a s k f i l l e d with a saturated s o l u t i o n of sodium formate, the f l a s k being held i n a water bath at 40°C. After several seeds had dissolved i n the f l a s k s , i t was r e a l i z e d that the vapor pressure of water over the bath was greater than the vapor pressure over the saturated - 15 - solution* Thus water from the bath was being d i s t i l l e d into the fl a s k s * To correct t h i s s i t u a t i o n , a small vacuum system was constructed, i n i t i a l l y out of glass and then, when the glass proved to be too f r a g i l e , out of copper tubing. I t consisted simply of a manifold from which several T-joints were made, each Joint being connected to the fla s k s containing the seed c r y s t a l s by means of rubber pressure tubing. A stop-cock was inserted i n each piece of rubber tubing so that I n d i v i d u a l f l a s k s could be handled without releasing the 'vacuum on the other f l a s k s * Vacuum was created by means of a water aspirator* When the vacuum was l e f t on continuously i t was found that too rapid evaporation occurred and great masses of c r y s t a l meal were precipitated. An attempt was then made to promote growth by applying vacuum f o r approximately an hour a day. This provided a reasonably low rate of evaporation and also a chance f o r the c r y s t a l to heal. (It i s well-known that an imperfect c r y s t a l l e f t i n a saturated solution w i l l rearrange to form a more perfect c r y s t a l ) * This method proved to be more successful, a l - though the formation of additional c r y s t a l s on the bottom of the f l a s k and on the c r y s t a l mount was encountered* Growth of cr y s t a l s of sodium formate by t h i s method was very slow and the r e s u l t i n g c r y s t a l s were generally • 16 - poorly formed, some of them being twinned. Eventually, (after several weeks), a few c r y s t a l s o f useable q u a l i t y were obtained. These c r y s t a l s were usually about 2 or 3 cm. long and 2 or 3 mm. thick and varying i n width from 2 to 10 mm. depending on the type of seed c r y s t a l used. But, because of the p o s s i b i l i t y of having to abandon the attempt to obtain reasonable c r y s t a l s of sodium formate, an attempt was made to grow single c r y s t a l s of calcium and strontium formates. Seed c r y s t a l s of both strontium and calcium formate were obtained by evaporation of a saturated solution of the s a l t s from an uncovered Petrie dish. Of the seed c r y s t a l s obtained, those of strontium formate were by f a r the best formed, the calcium formate seeds seeming very i r r e g u l a r . Therefore seeds of strontium formate were mounted as above and placed i n a saturated Solution on the s a l t contained i n a 2000 ml. beaker. The uncovered beaker was then placed i n a cupboard where a i r - c i r c u l a t e d dust was at a minimum. The large beaker was used f o r several reasons. F i r s t , although the rate of evaporation was f a s t , the rate of supersaturatlon of the large volume of solution was slow and growth proceeded quite evenly. Seoond, a large number of c r y s t a l s (15-20) could be accomodated i n the one beaker. Third, the amount of handling r e q u i r e d and equipment used was reduced to a minimum. In the course of a few weeks, l a r g e (up t o 1 or 2 cm. In a l l d i r e c t i o n s ) , well-formed though somewhat imperfect c r y s t a l s were obtained* Even though the c r y s t a l s seemed to be cracked, q u i t e s u i t a b l e p o r t i o n s of the l a r g e c r y s t a l s c ould be used f o r f u r t h e r work. However, when ground down, the i n f r a r e d spectrum of the c r y s t a l ex- h i b i t e d a strong band corresponding t o the OH s t r e t c h i n g frequency, o b v i o u s l y a r i s i n g from water of h y d r a t i o n . The c r y s t a l s of strontium formate dihydrate were s t o r e d f o r p o s s i b l e f u t u r e use and an attempt was made to grow c r y s t a l s of calcium formate. The seeds of calcium formate were mounted i n the usual manner and place i n a 2000 ml. beaker p a r t i a l l y f i l l e d w i t h saturated s o l u t i o n and s t o r e d i n a cupboard s i m i l a r to the procedure used f o r s t r o n t i u m formate. In t h i s case, the r a t e of evaporation was slower. As the calcium formate c r y s t a l s grew, I t was r e a d i l y seen t h a t the c r y s t a l s were q u i t e symmetrical. U n l i k e the case w i t h strontium formate, where very few e x t r a c r y s t a l s grew on the beaker bottom, a t h i c k deposit of c r y s t a l meal was formed on the beaker bottom and more Importantly, along the mount. As the c r y s t a l s grew, t h i s c r y s t a l meal - 1 7 - was i n c o r p o r a t e d i n the bottom p o r t i o n of the main c r y s t a l g i v i n g r i s e t o f u r t h e r c e n t e r s of d e p o s i t i o n l e a d i n g to h i g h l y I r r e g u l a r conglomerations. To prevent t h i s , the c r y s t a l s were p e r i o d i c a l l y removed from s o l u t i o n , the c r y s t a l meal was scraped o f f the mount and the c r y s t a l was then r i n s e d In v e r y c o l d water. Because of the temperature of the water and the p h y s i c a l form of the c r y s t a l very l i t t l e d i s o l u t l o n o c c u r r e d . The r i n s e removed the l a s t t r a c e s of c r y s t a l meal from the mount and any t h a t might have adhered t o the c r y s t a l i t s e l f thus p r e v e n t i n g these s m a l l c r y s t a l - l i t e s from a c t i n g as c e n t e r s of c r y s t a l l i z a t i o n . As the r a t e of growth proved too slow, a h e a t i n g tape was p l a c e d around the bottom of the beaker and the temperature of the s o l u t i o n was i n c r e a s e d by s e v e r a l degrees. T h i s i n c r e a s e d the r a t e of e v a p o r a t i o n and thus the r a t e of growth. In t h i s manner, s e v e r a l l a r g e c r y s t a l s were obt a i n e d , i n c l u d i n g one 3,5x3.5x4.0 cm., although the others were stopped a f t e r they had a t t a i n e d a s i z e of about 2 cm. The c r y s t a l s grown by t h i s method g e n e r a l l y had s e v e r a l flaws c o n s i s t i n g of c r a c k s running a c r o s s the c r y s t a l s . These cracks c o u l d have been caused by the removal o f the c r y s t a l s from the warm s o l u t i o n and the subsequent r i n s i n g In c o l d water. They may a l s o have been caused by - 18 - the removal of the c r y s t a l s from the warm solution and the subsequent r i n s i n g i n cold water. They may also have been caused by stresses created i n the c r y s t a l when i t grew around the mount. In any case the pieces obtained from a cracked c r y s t a l provided several conven- ient specimens which could be ground down. 2-3. Grinding In order to study the important portions of the infrared absorption spectrum of a single c r y s t a l , the c r y s t a l must be no thicker than 40 to 50 microns. With t h i s thickness a l l but a few of the very strongly absorb- ing v i b r a t i o n a l modes (for example, the carbonyl regions i n the case of the formate ion) are e a s i l y resolved. In order to resolve these bands, c r y s t a l s of half t h i s t h i c k - ness are required. Different workers have used various methods of obtaining t h i n single c r y s t a l s . Halford(l,2) and h i s co-workers obtained single c r y s t a l s by controlled feeding of allene and cyclopropane into a s p e c i a l l y con- structed potassium bromide c e l l held at l i q u i d nitrogen temperaturesi However, they had no control over the way the c r y s t a l axes would be oriented with respect to the window. - 19 - Winter, Curnette and Whiteomb(2M) obtained s i n g l e c r y s t a l s of ferrocene* These were then attached to a potassium bromide window by means of a wax s e a l . The c r y s t a l was then thinned and p o l i s h e d by rubbing on a c l o t h moistened w i t h acetone. T h e i r c r y s t a l s however, were of the order of 0.1 ram. t h i c k . Bryant:(25,26) grew c r y s t a l s of sodium and potassium axide from saturated aqueous s o l u t i o n s of the s a l t s , then thinned the c r y s t a l s to a t h i c k n e s s of 0.09 mm. at the t h i n n e s t by g r i n d i n g and p o l i s h i n g w i t h a jewelers c l o t h dampened w i t h acetone. Morrow grew l a r g e s i n g l e c r y s t a l s of barium formate which were p a r t l y ground w h i l e the c r y s t a l was hand-held and then I t was attached t o a r o c k s a l t window f o r the f i n a l g r i n d i n g . I t I s Morrow's general method that was used here. I t i s q u i t e easy t o o b t a i n c r y s t a l s i n the order of 50 microns t h i c k , As Morrow has pointed out, t h i n n e r c r y s t a l s would be d e s i r a b l e but are u s u a l l y mechanically u n f e a s i b l e . However, w i t h a great d e a l of ca r e , and patience , c r y s t a l s of the order of 25 microns t h i c k were obtained as described below. As mentioned e a r l i e r , the c r y s t a l s should have a c r o s s - s e c t i o n a l area of at l e a s t 2x10 mm. i n order t o - 2 0 - completely cover the s l i t i n the mounting p l a t e . Thus i t was necessary t o s t a r t w i t h a c r y s t a l of s l i g h t l y l a r g e r dimensions than those of the s l i t . The c r y s t a l fragments obtained when the calcium formate c r y s t a l s cracked were of an I d e a l s i z e . Because of the s i z e of the sodium formate c r y s t a l s , a s l i g h t l y d i f f e r e n t t e c h - nique was r e q u i r e d and w i l l be described l a t e r . With calcium formate, the c r y s t a l was ground perpendicular t o a l l three axes* The axes used were those designated by Q r o t h ( 2 7 ) and shown below. c(ooi^ 1 3(io6) b (010) F i g u r e 2 . Calcium formate s i n g l e c r y s t a l . I n each case the c r y s t a l was ground u n t i l a maximum cro s s s e c t i o n was obtained perpendicular to the d e s i r e d a x i s . The g r i n d i n g proceeded as f o l l o w s . For the f i r s t stage, the c r y s t a l or c r y s t a l fragment was hand-held and, using 150-C grade emery paper, the major p o r t i o n of the g r i n d i n g was done. This rough g r i n d i n g brought the c r y s t a l to w i t h i n about 100 microns of the d e s i r e d plane. Then, using 36O-A grade emery paper, a rough smoothing of the c r y s t a l was obtained by the e l i m i n a t i o n of the deep soratches i n f l i c h e d by the coarser emery paper. F i n a l l y , a course p o l i s h i n g was done by using 4/0 grade emery p o l i s h i n g paper. At this;--point, the f i n a l smoothing and p o l i s h i n g c o u l d be done. For t h i s , a Hocksalt P o l i s h i n g K i t was obtained from the Research and I n d u s t r i a l Instruments Co. This c o n s i s t e d mainly of a ground g l a s s roughing l a p , a ground g l a s s smoothing l a p , and a v e l v e t p o l i s h i n g s t r i p . The c o a r s e l y p o l i s h e d c r y s t a l was f u r t h e r smoothed on the roughing l a p using the a l c o h o l provided i n the k i t as a l u b r i c a n t . However, i t was found t h a t the a l c o h o l evaporated too r a p i d l y l e a v i n g deposits of ground s a l t which hardened and scratched the c r y s t a l . A l s o , i n the second stage of g r i n d i n g * the a l c o h o l d i s s o l v e d the glue h o l d i n g the c r y s t a l t o a r o c k s a l t window. For these r e - asons kerosene was used as a l u b r i c a n t . Kerosene has a f a i r l y low v o l a t i l i t y and a low v i s c o s i t y as w e l l as not - 22 - b e i n g a s o l v e n t f o r e i t h e r the c r y s t a l s , r o c k s a l t win- dows of g l u e used. T h i s made i t an i d e a l l u b r i c a n t f o r t h i s work. When a l l remnants of f i n e s c r a t c h e s had been r e - moved on the roughing l a p , a f i n a l smoothing was done on the smoothing l a p . The c r y s t a l tuas then p o l i s h e d on the v e l v e t p o l i s h i n g s t r i p u s i n g the p o l i s h i n g rouge p r o v i d e d In the k i t and kerosene as a l u b r i c a n t . I t was then g i v e n a f i n a l h i g h p o l i s h on a dry v e l v e t s t r i p . At t h i s stage, the o t h e r h a l f of the c r y s t a l c o u l d be ground. The p o l i s h e d f a c e was g l u e d to a r o c k s a l t win- dow. I n i t i a l l y C.I.L. Household Cement was used but be- cause of the h i g h v i s c o s i t y of the g l u e , a t h i c k l a y e r of g l u e was obtained between the window and the c r y s t a l . T h i s t h i c k l a y e r of g l u e i n t e r f e r e d w i t h the i n f r a r e d spectrum of the c r y s t a l so t h a t I t was d i f f i c u l t t o determine the t h i c k n e s s of the c r y s t a l . F o r t h i s reason, Radio S e r v i c e Cement, d i l u t e d with acetone to reduce i t s v i s - c o s i t y , was used. Every p r e c a u t i o n was taken to ensure t h a t the e n t i r e p o l i s h e d s u r f a c e was cemented down i n an e f f o r t t o reduce the p o s s i b i l i t i e s of c h i p p i n g when the c r y s t a l became very t h i n . However, t h i s was not completely p o s s i b l e because as the g l u e ' s s o l v e n t evaporated, s m a l l a i r pockets were formed i n the g l u e between the c r y s t a l and the window. These a i r pockets i n v a r i a b l y a llowed - 23 - c h i p p i n g t o occur . Thus the glu e was r e p e a t e d l y d i s s o l v e d o f f and the c r y s t a l r e g l u e d t o the window u n t i l a p o r t i o n o f the c r y s t a l o f s u f f i c i e n t c r o s s - s e c t i o n was obtained t h a t was completely cemented t o the window. T h i s g e n e r a l l y r e q u i r e d s e v e r a l attempts. An i d e a l glue f o r t h i s work would be one o f ve r y low v i s c o s i t y and w i t h a very low s o l v e n t c o n t e n t . When the c r y s t a l had been s a t i s f a c t o r i l y glued to the window, the second stage o f g r i n d i n g began. As i n the f i r s t stage, the c r y s t a l was ground t o a t h i c k n e s s o f about 0.2 mm. u s i n g the 1.50-C grade emery paper and the major s c r a t c h e s removed u s i n g the 36O-A grade and 4/0 p o l i s h i n g emery paper. At t h i s p o i n t the c r y s t a l s were estimated t o be between 50 and 100 microns t h i c k . The roughing l a p was then used t o reduce the c r y s t a l i n th i c k n e s s so that the s t r o n g e s t a b s o r p t i o n bands of the formate i o n were r e s o l v a b l e i n the i n f r a r e d ( i g n o r i n g the a b s o r p t i o n due to the cement). The ve r y t h i n s e c t i o n was then reduced f u r t h e r , u s i n g #950 b a u x i t e a b r a s i v e on the v e l v e t p o l i s h i n g s t r i p . Thl3 p a r t o f the g r i n d i n g proceeded v e r y c a u t i o u s l y because i t was at t h i s p o i n t t h a t the m a j o r i t y of c h i p p i n g o c c u r r e d . When f i n e s t r u c - t u r e was observable i n the r e s o l v e d a b s o r p t i o n bands, the c r y s t a l was g i v e n a p o l i s h u s i n g the p o l i s h i n g rouge and - 24 - the dry v e l v e t s t r i p . At t h i s point,, the c r y s t a l s were estimated to be approximately 25 microns t h i c k . The t h i n c r y s t a l s e c t i o n was then removed from the r o c k s a l t window by d i s s o l v i n g the g l u e i n acetone. T h i s g e n e r a l l y r e q u i r e d s e v e r a l hours. No attempt was made to push the p a r t i a l l y glued c r y s t a l from the window as severe s h a t t e r i n g o c c u r r e d . I n s t e a d , the acetone was p e r i o d i c a l l y s t i r r e d , and when the c r y s t a l moved s l i g h t l y over the window i t was t r a n s f e r r e d to the mounting p l a t e . The p l a t e was simply a brass p l a t e with a narrow s i l t 10x2 mm. i n area that would f i t into, a r e c e p t i c l e i n the Instrument. A p o r t i o n o f a paper c l i p was s o l d e r e d to,the p l a t e f o r use as a handle. The c r y s t a l was very g e n t l y pushed from the window to the p l a t e which was h e l d at the same l e v e l as the window. Because of the method of g l u e i n g the c r y s t a l to the window, not much c o n t r o l over the o r i e n t a t i o n o f the c r y s t a l s e c t i o n was o b t a i n e d . Thus no attempt was made to a l i g n any p a r t i c u l a r a x i s a l o n g the s l i t . I n - stead the p l a t e and c r y s t a l were removed from the acetone and the c r y s t a l was glued to the p l a t e u s i n g Radio S e r v i c e Cement. - 25 - The d i r e c t i o n s o f the c r y s t a l axes were checked u s i n g a p o l a r i z i n g microscope and a r e c o r d kept of the o r i e n t a t i o n of the axes w i t h r e s p e c t to the s l i t . I t was noted, u s i n g the p o l a r i z i n g microscope, t h a t the c r y s t a l s were c l e a r and t r a n s p a r e n t and e x t i n g u i s h e d s h a r p l y and u n i f o r m l y . No i n d i c a t i o n was g i v e n t h a t any o t h e r c r y s t a l segments w i t h d i f f e r e n t o r i e n t a t i o n s were on the c r y s t a l . The c r y s t a l s were a l s o examined w i t h the p o l a r i z i n g microscope u s i n g c r o s s e d p o l a r s and s t r o n g l y convergent l i g h t f o r i l l u m i n a t i o n . By v i e w i n g between c r o s s e d p o l a r s not the image of the c r y s t a l , o r o b j e c t image, but another o p t i c a l image formed i n the p r i n c i p a l f o cus of the o b j e c t i v e by the s t r o n g l y convergent beam of l i g h t > i n f o r m a t i o n r e g a r d i n g the d i r e c t i o n of the c r y s t a l o p t i c axes may be o b t a i n e d . T h i s image i s c a l l e d the d i r e c t i o n s image, image In convergent l i g h t , o r the i n t e r f e r e n c e f i g u r e . More d e t a i l e d d i s c u s s i o n s of t h i s phenomenon are g i v e n by Hartshorne and S t u a r t ( 2 8 ) , and by Bunn(29). To determine how c l o s e l y the c r y s t a l s were ground on a plane p e r p e n d i c u l a r to the r e s p e c t i v e axes, the i s o g y r e s of the i n t e r f e r e n c e f i g u r e s were observed. I t was seen t h a t i n a l l cases these Isogyres were c e n t e r e d - 26 - under the c r o s s - h a i r s o f the microscope i n d i c a t i n g t h a t the c r y s t a l s were ground e x a c t l y p e r p e n d i c u l a r t o the c r y s t a l axes, or a t most o n l y one o r two degrees from the p e r p e n d i c u l a r . S e v e r a l c r y s t a l s o f v a r y i n g t h i c k n e s s e s were ground p e r p e n d i c u l a r t o each a x i s . I n t h i s way, the f i n e s t r u c t u r e o f the fundamentals c o u l d be determined w i t h the t h i n c r y s t a l s and the weaker peaks c o u l d be de t e c t e d u s i n g t h i c k e r c r y s t a l s . As mentioned e a r l i e r , the c r y s t a l s o f sodium formate obtained were of s u f f i c i e n t l e n g t h and width along one a x i s . However, they were u s u a l l y f a i r l y t h i n . T h i s not onl y prevented the c r y s t a l from being hand-held i n the i n i t i a l g r i n d i n g stage but a l s o prevented c r y s t a l s from being ground down p e r p e n d i c u l a r to more than one a x i s . T h i s f*as unfortunate but i t was f e l t t h a t some u s e f u l i n f o r m a t i o n c o u l d be obtained from j u s t the one o r i e n t a t i o n . The o n l y d i f f e r e n c e between the g r i n d i n g techniques f o r the c a l c i u m formate and the sodium formate was t h a t f o r the f i r s t stage o f g r i n d i n g , the f l a t c r y s t a l was glued t o the r o c k s a l t window w i t h C.I.L. Household Cement and one f a c e was ground and p o l i s h e d In the manner - 27 - d e s c r i b e d above. The c r y s t a l was then d i s s o l v e d o f f the window and the p o l i s h e d f a c e was r e g l u e d t o the window. The use of kerosene as a l u b r i c a n t was most convenient i n t h i s case. A t h i n f i l m o f kerosene was c o n s t a n t l y c o v e r i n g the c r y s t a l and no e f f e c t o f the hygroscopic nature of the s a l t was noted. Again, c r y s t a l s of v a r y i n g t h i c k n e s s e s were o b t a i n e d , the c r y s t a l s e c t i o n being p a r a l l e l to the plane of the p l a t e . X-ray photographs o f a s m a l l seed c r y s t a l o f c a l c i u m formate were taken i n o r d e r t o unambiguously c o r r e l a t e the e x t e r n a l c r y s t a l axes a s s i g n e d by Groth (27) and the i n t e r n a l u n i t c e l l axes a s s i g n e d by N i t t a and 0sak i (30) . The c r y s t a l was mounted i n a Nonius i n t e g r a t i n g Welssenberg goniometer w i t h the e x t e r n a l a -axis h o r i z o n t a l , and p e r p e n d i c u l a r t o the i n c i d e n t X-ray beam. R o t a t i o n and o s c i l l a t i o n photographs were taken of the a - a x i s , and Welssenberg photographs of the Okland 1 k l planes were taken t o determine the b and c-axi3 l e n g t h s . I t was.confirmed t h a t the axes a, b and c noted by Groth a r e I d e n t i c a l t o the axes x, y and 2 noted by N i t t a and Osakl r e s p e c t i v e l y . A s i m i l a r s e t of X-ray photographs were taken of the sodium formate p l a t e - l i k e seed c r y s t a l s . - 28 - Zachariasen's c r y s t a l s t r u c t u r e was found f o r a n e e d l e - l i k e c r y s t a l and I t was thought t h a t the d i f f e r e n t c r y s t a l m o d i f i c a t i o n might be composed of a d i f f e r e n t u n i t c e l l . However, i t was found t h a t the c r y s t a l s t r u c t u r e of the f l a t p l a t e s i s i d e n t i c a l t o the s t r u c t u r e p r e v i o u s l y r e p o r t e d . 2-4. Apparatus A l l s p e c t r a were reco r d e d on a Perkln-Elmer model 421 d u a l - g r a t i n g spectrophotometer. Two d u a l - g r a t i n g interchanges were a v a i l a b l e which covered the r e g i o n s 4000-650 cm"*1 and 2000-200 cm""1* The instrument was f l u s h e d w i t h d r y n i t r o g e n to reduce atmospheric absorp- t i o n and to prevent a t t a c k o f moist a i r on the hygro- s c o p i c sodium formate. The p o l a r i z e r was c o n s t r u c t e d by mounting two s e t s of t h r e e 0.5 mm. t h i c k s i l v e r c h l o r i d e p l a t e s In the form of a "V". T h i s arrangement Is necessary t o prevent sideways displacement of the beam on p a s s i n g through the s i l v e r c h l o r i d e p l a t e s . The two s e t s of p l a t e s were s e t a t Brewster's angles to each o t h e r . The p o l a r i z e r was. s i m i l a r t o that d e s c r i b e d by Gharney (32). As was p o i n t e d out by Brugel(33), w i t h t h i s type of p o l a r i z e r , the unwanted component of r a d i a t i o n Is - 29 - s t i l l q u i t e l a r g e , but t o obtain.complete p o l a r i z a t i o n would r e q u i r e a p r o h i b i t i v e number o f p l a t e s . Measurements of o p t i c a l d e n s i t y of the p o l a r i z e r on a Beckman DU spectrometer u s i n g r a d i a t i o n o f 6000 $ and a Wollaston prism i n d i c a t e t h a t the component p e r - p e n d i c u l a r t o the d e s i r e d p o l a r i z a t i o n i s g r e a t e r than lc/y but l e s s than k% o f the d e s i r e d component. Measure- ments of the convergence of the sample beam show t h a t there would be an unwanted component of not more than 1% of the d e s i r e d component of p o l a r i z e d r a d i a t i o n due to t h i s convergence. 2-5. Spectra T a b l e 1 l i s t s the i n f r a r e d a b s o r p t i o n s p e c t r a obtained i n t h i s study and i l l u s t r a t e d i n f i g u r e s 2 - 12, I f the spectrum was r e c o r d e d u s i n g p o l a r i z e d r a d i a t i o n , the plane of p o l a r i z a t i o n Is i n d i c a t e d as being p a r a l l e l t o a x i s a_, | or c i n the case of c a l c i u m formate, and as b e i n g p a r a l l e l o r p e r p e n d i c u l a r t o the c - a x i s In the case of sodium formate. S p e c t r a l s l i t widths were c a l c u l a t e d u sing i n f o r m a t i o n o b t a i n e d from the Perkin-Elmer Corporation ( 4 2 ). The average s p e c t r a l s l i t width was found t o be 0.9 cm" 1 w i t h a range from 1.2 - 0.6 c m - 1 depending on the r e g i o n of the spectrum b e i n g s t u d i e d . - 30 - T a b l e 1. I l l u s t r a t e d I n f r a r e d s p e c t r a F i g u r e no. D e s c r i p t i o n 3 C a ( H C 0 2 ) 2 - p o l y c r y s t a l l i n e i n KBr 4 0 a ( D C 0 2 ) 2 - p o l y c r y s t a l l i n e i n KBr 5 NaHC0 2 - p o l y c r y s t a l l i n e i n KBr 6 NaDCOg - p o l y c r y s t a l l i n e i n KBr 7 C a ( H C 0 2 ) 2 - s i n g l e c r y s t a l (3020-2680 cm"*1) p o l a r i z e d 8 " - (1800-1000 cm' 1) p o l a r i z e d 9 " " " (840-740 cm" 1) p o l a r i z e d 10 NaHC0? - " " (3100-2650 cm" 1) • p o l a r i z e d 11 " - " " (1800-1000 cm" 1) p o l a r i z e d 12 " - " " ' (1050-750 cm" 1) p o l a r i z e d T a b les 2 and 3 l i s t some o f the p r e v i o u s l y r e p o r t e d data f o r c a l c i u m and sodium formate and the observed f r e q u e n c i e s and assignments o f c a l c i u m and sodium formate and c a l c i u m and sodium formate-d-^ obtained i n t h i s work f o r p o l y c r y s t a l l i n e samples. T a b l e s 4 and 5 l i s t the observed i n f r a r e d a b s o r p t i o n f r e q u e n c i e s o f c r y s t a l l i n e c a l c i u m and sodium formate, a l o n g w i t h assignments and p o l a r i z a t i o n p r o p e r t i e s . 31 - Table 2* P a r t i a l summary of infrared and Raman frequencies of calcium formate and calcium formate-d^ Author Schutle Harvey This work and Buijs e t . a l * I.R.(s) I.R.(ag.) I.R.(s) I.R.(ag.) assign. I.R.(mull) I »R* (s ) DCOg DGGg 783 778 782 782 ^ 3 789 788 ^ 3 801 796 800 ^ 804 804 1068 1060 1067 ^ 6 1080 1072 1349 1078 1125 1086 1116 ^ 2 2869 2868 1355 1353 1351 ^ 2 2890 289© 1365 1385 1389 1386 1390 1383 1014 *5 1401 1397 1391* 1020 *V 1403 2 ^ 3 1582 1573 ^ 4 1587 1618 1591 1588 1580 ^ 4 1629 1631 1609 1658 1650 1651 1657 ^ 1 ^ 6 2414 ^ 6 2459 2 ^ 2 2697 2694 2700 2695 1328 1020 I574 w 32 - Table 2 - continued Author Schutle Harvey This work and Buijs e t , a l . I . R * ( B ) I.R.(aq.) I.R.(s) I.R.(aq.) assign. I*R»(mull) IiR.(s) DCOj DCO" 2 ^ 2 2718 2731* 2 7 4 7 2 7 ^ 9 2733 2363 2**^5 2 7 6 0 2 7 5 0 ^ 2 + ^ 5 2780 2 ^ 5 2776 2745 2026 2 * / 5 2 7 7 5 "̂ 1 1353 1359 2872 2832 2164 2123 1362 2894 2853 *2 +*4 2 9 4 5 2 9 ^ 2 2 9 5 0 2 9 2 1 ^ 2 ^ 4 2 9 6 5 2 9 6 7 1 3 ^ 4 ^ 5 3000 3000 3008 2961 x - 33 - of sodium formate and sodium formate-d n Author assign. It o and Bernstein i.R.(s) Newman I i R . ( s . c Ponteyne This work R.(aq.) R.(aq.) I . R.(s) I . R.(s) i) HCOg DGOg DGOJ 722 3 772 784 773 757 772 751 3 766 790 840 898 ^ 6-i4o 923 3 ^ 2 x 6 0 958 X013 6 1073 1070 IO69 918 1062 915 ^_-l40 1226 1312(?) 1366 1377 2825 2122 1365 1333 1344 1377 1365 1386 1028 1014 ^ +140 5 1514 1567 1620 1584 1580 1601 1610 1620 1685(?) 1700 ^ 6 2126 1831 2189(2 5 ? ) ^ 3 213k 2096 ^ 6 2435 - 34 * Table 3 - continued Author I t o and Hewman Ponteyne This work Bernstein R*(a.Q») H,(aq.) I.R.(s) I.R.(s) assign. I.R.(s) I,R*(s*c.) HC0 2 DCO" DCOg 2720-2x60 2599 22^ 2 2719 2 * 4 2732 2032 2019 ^ 2 t ^ 5 2720 2750 ^ - 6 0 2791 2074 (?) 2841 2870 1352 1329 2833 2135 ^ V ^ 5 2953 ^ 2 t ^ 2990 2957 2918 2 * ^ - 6 0 3070 2*^+60 3190 3 6 U 2897 (?) s i g n i f i e s uncertain assignments - 35 - fable 4 . The infrared absorption frequencies of calcium formate a-axis b-axis c-axis „i other (cm"1) polarized polarized polarized ealc*(em ; p o s s i b i l i t i e s 3668v.w. 3933v.« . 3666w 394lw 3669W 3958 3677 * 4 3l87v.w.sh. 3157w 3157M 3158w 3196 2 *3 +*ii=3l84 ^ 1 9 2 3092Vi«i 3087 1 + 1 2 8 30G9wsh. 3008w.sh. 3010w.sh. 3001 z^'-*-128-3018 * 5 2990w*shi 2969*9. sh # 2986** <sh. 2995 ^-»90-2985 .III 2970w.sh. 2972M*sh. 2973w.sh. 2963 2951m 2951m 2950m 2945 2̂ '+ 61*2951 294 lm 294Ira 2935 2934w.sh. 2931 ^3/61^2934 2925v *W i 2926 =̂  + 36 2913 2090 1 2894s 2895B 2896s 1 2890s 1 - 2874v.w.sh, 28723 2873s _2840v*w. 2838v.w. 2844 ^136-2837 2791w 2793w 2793^. 2804 ^ x ' -90--2805 5 2780M 2780m 2780m 2784 ^ - 9 0 = 2 7 8 3 ^ ' - 1 2 8 2 * 5 2751m 2770w 2751v,w, 2766v*w. 2767 2752v.w. * / **128«2762 * /l-128=2745 - 36 - Table 4 - continued _^ 1 a-axis b-axis c-axis , other (cm""A) polarized polarized polarized cale.(cm~ x) p o s s i b i l i t i e s ^ - 1 5 4 2739W 2738v.w. 2738v.w. 2741 2722v.w. 2724v.w. 2730 2 * ' 2 2713v.w. 2710 2701ra 2701m 2701W 2464v.w. 2464v.w. 2464v»w. 2471 2421v.w. 24l9w 2421v.w. 2434 3 5 2l85v.w, 2l86v.w. 2l88v,w. 2190 2 ^ 6 2152v.w. 2152w 2151H 2158 2 * 6 2133" 2137W 2135® 2136 ^ 9 0 1667s.sh. l670w.sh. 1670 < + 3 6 1650s 1654 ^ 6 1 1642s 1641 l600s.v.b. 1616s 1618s 1579s 1580s 1560V*w.sh. 1557 ( G 1 3 ) < 1543w.sh. •^'-61 I506v.w. 1523w.sh. 1519 ^2+154 1507^.sh 1509 * 5 1406B ^ 5 l403s.sh. 1401s ^ 5 1399s 1400s 1395s.sh. 1395s " ^ 6 1 = 1 6 7 9 ^+192=1557 2 %"-36-1544 ^'-90--1528 - 37 - Table 4 - continued a-axis b-axis c-axis other (cm"1) polarized polarized polarized calc.(cm""-) possibilities 1392s ^5 (C 1 3) 1388s 1368s 1362s 1358s 1352s ^ (c1 3) ^ 2 -36 ^5+192 -̂ '+192 ^"+128 6 6" ^-154 ^6-128 *3'+ia8 I26OW ll6lv.w. 1080m.sp. 1068wiSp. 1035V.w„ 972v.w. 938v.w* 1388s 1377v,w. 1364s 1369s 1364s 1355s 1352s 1336w.sh, 1336w.sh. 1330V. w.sh•1329v•w.sh. 1273w 1263w 1196v *w. I I56V.w. 1079 m.sp,1079w.sp. 1067v».sp. lo68m.sp. 988v*w. 989V.w. 967V.w. 927v.w. 933v.w. 1329 1271 1260 1196 1032 989 957 940 931 ^ 5 -6l -1333 •^2-90=1275 2^-90=1265 ^ 2 -154 =1201 ^ +192*980 ^192=974 154=942 =%'-154=925 - 38 - Table 4 r continued ; jl a-axis b-axis o-axis , other ^(cm ) polarized polarized polarized ealc.(cm ) p o s s i b i l i t i e s "^g-192 888v.w. 891V.W. 887 90 -893 ^3+90 878v.». 877v»w. 872 ^192=876 ^3+61 865v .w, 865v.w, 864 ^3 +36 844v.w. 839 ^61=849 ^3+-36 817« 818 ^36=824 806m 806v.w«sh. 806s.sh. •*>' 803m 802s 8G3s 3 799s,sh. *3<C 1 3) 792w.sh. 792v.WiSh, 2^3 788s 788w¥sb,, 788s.sh 786s =̂ 3 782s 782s ^'(C 1 3) 777».sh. 777«*sh. 778s.sh. ^3-36 768w 768w 768w 767 ^3-36 76lv.w. 752 ^3-61 742v.w. 742 > / 3 ,-36=746 ^3-61 732v.w. 7 3 2 v 7 2 7 ^3-128 679v.w* 68lv.w. 675 ^3-154 65GV.W. 646v.w. 649 * /*ll28~65U Symbols s=strong, m=medlura, w=weak, v-yery, b y r o a d , sp,-sharp, sh.=shoulder, G ^carbon 13 formate ion. The calculated frequencies are calculated using observed fundamental values. - 39 - Table 5. The Infrared absorpt ion frequencies of sodium formate a s s i g n , po la r i zed 11 to c p o l a r i z e d X t o c e a l c , ^ l " * i 3882V.w. . 3904 369OV.W. 3190vsw. 319GV.W* 231 306lw 3069 30l6w .sh* 3 0 l 8 v .w .sh . 2987».sh. 2990w.sh, 2958m 29^9s 2959 *1 4 91 2926m. s h . 2929 ^+58 2895».sb* 2986 ^ 2838w 2836w.sp, 2799v.w. ^ - 5 8 2776w 2779v.w. 2780 ^ - 9 1 2 7 4 3 » * s h . 2745 2723v .w.sh. 2725m 2735 Ho ^ - 1 2 8 2704w 2710 263OW ^ - 2 3 1 2597v, w. 2607 ^ - • ^ 5 2429W 2427 2344w 2l46w 2156 3 5 1780w.sh. 1746w.sh, I750w.sh. - 40 - Table 5 . - continued a s s i g n , po lar i zed 11 to c po la r i zed JL to c ca lc "tf 128 ^+112 ^ 9 1 ^ U - 5 8 ^ t - 9 1 5 ^ - 2 3 1 ^ 5 ^ 2 •^2-128 ^ 6 + 5 8 ^ 6 ^ - 5 8 * ^ 1 1 2 ^ 3 +l28 ^3+58 1734m 1722w.sh. 1 5 9 7 W f S h . 1542w.sh, 1493m l468w,sh. l450wBsh. 1390w 1377w.sp. 1310w 1279«.sh. 1228m II69W 1130W 1068m*sp. 1007w 96OW 904 839» l692w.sh. I 6 l 0 s * v * b » l460w*sh, l409v.w. 1375S 1359w.sh, 1303W 1277** 1235** 1068w 1013v .w. 964w 913m 1738 1722 1701 1552 1498 1466 1450 1379 1301 1284 1231 1126 1010 956 908 * 41 - Table 5 - continued a s s i g n , po la r i zed 11 to e po la r i zed JL to c c a l c . ^3 780w.sp 780w.sp. 586w 53lw Symbols s=strong, m^ medium, w=weak, v ^ v e r y , b - b r o a d , sp , -^ sharp, s h . a shoulder The ca lcu la ted frequencies are ca lcu la ted using observed fundamental va lues . Figure 3. Infrared spectrum of polycrystalline calcium formate.  Figure 5. Infrared spectrum of polycrystalline sodium formate. 3°25 2500 2 0 0 0 1500 1000 3 0 0 0 2 5 0 0 - 2 0 0 0 : 1500 \ 1000 Figure 6 • Infrared spectrum of polycrystalline sodium formate-d    3100 3000 2900 2800 2 7 0 0 1100 3000 2900 2800" 2 700 FigurelO. Polorired infrared spectrum of sodium formate (3100-2650cm"') . The plane of polarization is indicated as being perpendicular or parallel to the c-oxis.  1000 900 8 0 0 | • , 1 , | _ 1 : : 1 J —i- 1 1000 900 800 Figurei2 Polarized infrared spectrum of sodium formate. ( J 0 5 0 - 7 5 0 c m - 1 ) . The plane of polarization is indicated as being perpendicular or parallel to the c axis. ~ 42 ~ CHAPTER 3. THEORY 3-1. General theory of v i b r a t i o n s i n c r y s t a l s In o r d e r t o study t h e motions of a polyatomic system, a s e t of c o o r d i n a t e s i s needed t o d e s c r i b e the c o n f i g u r a t i o n o f the system. For a system o f N atoms, . 3N c o o r d i n a t e s d e s c r i b e the motion of the system as a whole. Of these 3N c o o r d i n a t e s , t h r e e d e s c r i b e the - t r a n s l a t l o n a l motion and t h r e e more d e s c r i b e the r o - t a t i o n a l motion of the system l e a v i n g 3N - 6 c o o r d i n a t e s t o d e s c r i b e the v i b r a t i o n a l degrees o f freedom of the system, Wilson, Decius and Cross(34) d i s c u s s a g e n e r a l method whereby the equations of motion may be w r i t t e n i n terms of the chosen c o o r d i n a t e system. The set o f equations thus d e r i v e d y i e l d s a s e r i e s of s o l u t i o n s c orresponding t o the normal modes o f v i b r a t i o n o f the system* The 3N x 3N s e c u l a r determinant which must be s o l v e d i n o r d e r t o determine the normal f r e q u e n c i e s can, i n most eases, be s i m p l i f i e d . T h i s s i m p l i f i c a t i o n a r i s e s from the f a c t t h a t the system under i n v e s t i g a t i o n g e n e r a l l y possesses some form of symmetry. I f , i n a molecule, a symmetry o p e r a t i o n i s c a r r i e d out t h a t transforms the molecule i n t o an e q u i v a l e n t p o s i t i o n , the k i n e t i c and p o t e n t i a l e n e r g i e s w i l l be unchanged. The s e t o f symmetry - 43 - operations that a molecule possesses which carry i t into equivalent positions is? 1 known as a group i n the mathe- matical sense* Each symmetry operation may be represented a n a l y t i c a l l y by a. l i n e a r transformation connecting the old coordinates with the coordinates of the molecule i n i t s new position* The set of l i n e a r transformations i s said to be a representation of the group of symmetry operations. The coordinates, i n terms of which the transformations are written, are said t o form a basis of the representation. I t i s usually possible, by choosing a suitable set of coordinates> to reduce the square 3N x 3N transformation matrices to comparatively simple forms. I t i s possible to separate these coordinates into sets which do not mix with each other i n any of the transformations* ' When the coordinate system has been found such that i t i s impossible to break the coordinates down into any smaller non-mixing sets, the representation f o r which these coordinates form a basis i s said to be completely reduced* When I t i s possible to do t h i s , the o r i g i n a l representation i s said to be reducible. The equations involving the members of any one non-mixing set can be considered by themselves as making up transformations which form a representation of the group* Such a representation i s i r r e d u c i b l e and i t i s seen t h a t a completely reduced representation i s made up of a number of i r r e d u c i b l e representations. - 44 - I t I s u s u a l l y p o s s i b l e t o choose s e v e r a l s e t s o f c o o r d i n a t e s t o form a b a s i s f o r the r e p r e s e n t a t i o n s , but i n each case the r e s u l t s would be the same. Any two r e p r e s e n t a t i o n s a r e s a i d t o be e q u i v a l e n t When they d i f f e r o n l y i n the c h o i c e o f b a s i s c o o r d i n a t e s (the b a s i s c o o r d i n a t e s o f one being l i n e a r combinations of the b a s i s c o o r d i n a t e s of the o t h e r ) . The fundamental theorem concerning I r r e d u c i b l e r e p r e s e n t a t i o n s s t a t e s t h a t f o r each p o i n t group t h e r e a r e o n l y a d e f i n i t e s m a l l number of non-equivalent i r r e - d u c i b l e r e p r e s e n t a t i o n s p o s s i b l e . I t i s p o s s i b l e t o show t h a t the number of times an I r r e d u c i b l e r e p r e s e n t a - t i o n appears i n a reduced r e p r e s e n t a t i o n i s where h i s the order o f the group (equal to the number of symmetry o p e r a t i o n s c o n t a i n e d i n the group), 'X^ i s the c h a r a c t e r o f the r e d u c i b l e r e p r e s e n t a t i o n and 9^ Is the c h a r a c t e r o f the * t h i r r e d u c i b l e r e p r e s e n t a t i o n o f the o p e r a t i o n (R . The sum i s taken over a l l the o p e r a t i o n s of the group* The c h a r a c t e r i s d e f i n e d as the sura o f the d i a g o n a l elements of the t r a n s f o r m a t i o n m a t r i x , the char a c - t e r s o f e q u i v a l e n t r e p r e s e n t a t i o n s being i d e n t i c a l . - 45 - The q u a n t i t i e s on the right;; hand s i d e are e a s i l y d e t e r - mined u s i n g a s e t o f simple r u l e s . A s s o c i a t e d w i t h each non-mixing s e t of normal co- o r d i n a t e s i s a s e t of normal modes of v i b r a t i o n , the number o f normal modes being equal t o the number of normal c o o r d i n a t e s i n the s e t . S i n c e each normal c o o r d i n a t e transforms a c c o r d i n g to one of the i r r e d u c i b l e r e p r e s e n t a - t i o n s o f the group, then by u s i n g e q u a t i o n 1, the number of normal modes of v i b r a t i o n b e l o n g i n g t o each I r r e d u c i b l e r e p r e s e n t a t i o n may be determined. 3-2. S e l e c t i o n r u l e s Group th e o r y may be used t o d e r i v e the s e l e c t i o n r u l e s f o r v i b r a t i o n a l t r a n s i t i o n s In the I n f r a r e d e f f e c t . F o r a fundamental t r a n s i t i o n to occur by a b s o r p t i o n o f i n f r a r e d r a d i a t i o n , i t i s necessary t h a t one or more of the i n t e g r a l s f<M^n- A>»v* have a non-zero v a l u e . Here, ^ i i s the v i b r a t i o n a l ground s t a t e , 7^' i s the e x c i t e d s t a t e * an&yU<yA.y and are the components of the e l e c t r i c d l p o l e moment o p e r a t o r . I t may be determined whether the above i n t e g r a l s v a n i s h i f the symmetry p r o p e r t i e s of } ^j, yU.K ^yUy and yJ~^ are known. Si n c e these are d e f i n i t e I n t e g r a l s over the whole c o n f i g u r a t i o n space o f the molecule, they should be - 46 - unchanged by a symmetry o p e r a t i o n , Inasmuch as such an o p e r a t i o n merely produces a t r a n s f o r m a t i o n o f c o - o r d i n a t e s . That i s , the i n t e g r a l s must be t o t a l l y symmetric or the t r i p l e d i r e c t product of the s p e c i e s o f 1^ * , yu, and must c o n t a i n .the t o t a l l y symmetric s p e c i e s . Now a l l wave f u n c t i o n s f o r normal v i b r a t i o n s i n t h e i r ground s t a t e s {"%) are bases f o r the t o t a l l y sym- m e t r i c r e p r e s e n t a t i o n of the symmetry s p e c i e s o f the molecule (35). Thu3 f o r fundamental t r a n s i t i o n s (from the ground s t a t e t o the f i r s t e x c i t e d s t a t e ) , the i n - t e g r a l w i l l be symmetric i f the d i p o l e moment o p e r a t o r and the f i r s t e x c i t e d s t a t e belong to the same s p e c i e s s i n c e the d i r e c t product of a r e p r e s e n t a t i o n w i t h I t s e l f i s symmetric. I t can be shown (34) t h a t the components of the d i p o l e moment op e r a t o r t r a n s f o r m i n the same manner as the t r a n s l a t i o n a l c o o r d i n a t e s Tv, T and T „ . Thus abnormal mode o f v i b r a t i o n w i l l x* v z i be i n f r a r e d a c t i v e i f rj belongs to the same sym- \ • • metry s p e c i e s as one of; t h e - t r a n s l a t i o n a l c o o r d i n a t e s . S i m i l a r symmetry arguments may be a p p l i e d t o determine the a c t i v i t y of overtone and combination type bands. - 4 f - 3-3, C r y s t a l symmetry A i l previous theory has pertained to molecules com- p l e t e l y i s o l a t e d from a l l other molecules* The v i b r a t i o n a l states of the Isolated molecules have been shown to be dependent on the molecular symmetry* However, i n c r y s t a l s (and i n l i q u i d s ) , molecules are i n close proximity to other molecules and.it i s necessary to determine the ex- tent of inter-molecular interactions and the e f f e c t they might have on v i b r a t i o n a l modes and t r a n s i t i o n s . To do t h i s , a knowledge of c r y s t a l symmetry i s required. For an i n f i n i t e c r y s t a l , there i s an i n f i n i t e number of ways of combining into groups the symmetry operations that carry an atom into an equivalent atom i n the c r y s t a l ; These symmetry operations include ro- t a t i o n , inversion-rotation and screw axes; mirror and g l i d e planes; inversion centers; and Simple t r a n s l a t i o n s . I f we define a unit c e l l as the smallest repeating unit i n the c r y s t a l i then the c r y s t a l may be b u i l t up by translations of t h i s unit c e l l an i n f i n i t e number of times. By disregarding simple t r a n s l a t i o n s , i t has been shown that the remaining symmetry operations of the cry- s t a l l i n e state may be combined into 230 d i f f e r e n t combina- tions known as space groups. - 48 - I t i s shown i n the standard works on space group theory that any space group i s the product of an invar- iant subgroup, consisting of the elements corresponding to pure tr a n s l a t i o n s , and a factor group* This factor group i s the same for corresponding i n f i n i t e and f i n i t e space groups. The factor groups are always isomorphous with one of the 32 crystallographic point groups, a l - though some of them may involve subgroups containing other than purely point operations combined with l a t t i c e t r a n s l a t i o n s , that i s , screw rotations or g l i d e r e f l e c - t i o n s . Because of t h i s isomorphism, we can use the character table of the crystallographic point group corresponding to the factor group representations. I t should be noted that the factor group i s necessarily a subgroup of the i n f i n i t e space group. Now a s i t e i s defined by Halford(7) as a point which i s l e f t invariant by some operations of the space group. These operations may be Shown to form a group which may be designated as a s i t e group. Thus a l l points i n a c r y s t a l are s i t e s with at least the t r i v i a l s i t e group involving only the i d e n t i t y operation. Generally, the s i t e group has an order less than the factor group and i s isomorphous with a subgroup of the factor group. In general, a unit c e l l exhibits several d i f f e r e n t kinds of s i t e s and sometimes several d i s t i n c t sets of the same kind of s i t e . 3-4* Analysis of vibrations i n c r y s t a l s When studying the vibrations i n a c r y s t a l , because of the large number of atoms i n the c r y s t a l , i t would seem that a most complicated spectrum would r e s u l t . However, as experimental r e s u l t s have indicated, a spectrum with £ f i n i t e number of observable bands i s obtained. I t i s thus necessary to consider whether a l l 3N vibrations i n the system are independent. Since only a small number of absorption bands are observed In the v i b r a t i o n a l spectra of c r y s t a l s i n the Infrared and Raman effects and since there i s a very close correspondence between the spectra of c r y s t a l s and the spectra of t h e i r molten states, i t would appear that only a small! unit of the c r y s t a l i s needed In order to determine the nature of the normal modes of the c r y s t a l * Bhagavantum and Venkatarayudu(5,6) have considered the k n i t c e l l of the c r y s t a l . Within the c r y s t a l , there are sets of atoms arrayed i n such a manner that each atom i n a set i s both geo- met r i c a l l y and physically equivalent to every other atom - 50 - In the set. Thus fo r any given normal mode of v i b r a t i o n , each atom i n a set should undergo the same displacements. Atoms i n any other unit c e l l i n the c r y s t a l are related to the f i r s t by simple t r a n s l a t i o n . I f each set of atoms i s considered to form a l a t t i c e , that i s , an arrangement i n which only one atom Is located at each l a t t i c e point, then the c r y s t a l may be thought of as a structure i n which a group of two or more atoms Is located at each l a t t i c e point. That i s , the c r y s t a l can be looked upon as being made up of a set of i n t e r - penetrating l a t t i c e s . Thus the unit c e l l of the smallest possible s i z e w i l l contain as many atoms i n i t as there are interpenetrating l a t t i c e s i n the structure. For n atoms i n the unit c e l l there w i l l be 3n normal o s c i l l a - tions of which three w i l l be non-geniune o s c i l l a t i o n s corresponding to t r a n s l a t i o n of the unit c e l l . These 3n possible v i b r a t i o n a l modes may be c l a s s i f i e d as eithe r pure t r a n s l a t i o n s , l a t t i c e o s c i l l a - tions or i n t e r n a l o s c i l l a t i o n s . The l a t t i c e o s c i l l a t i o n s are further c l a s s i f i e d as a r i s i n g from translatory or rotatory motions of the molecules i n the unit c e l l . The forces between one c l a s s i f i c a t i o n and the others are comparatively feeble whereas the forces that exist between members of any one c l a s s i f i c a t i o n are quite strong. The o s c i l l a t i o n s involving a movement of the molecules as 51 - e n t i t i e s w i l l generall exhibit low frequencies and may be termed external or l a t t i c e v i b r a t i o n s . The o s c i l l a t i o n s involving movements of the i n d i v i d u a l members i n each molecule against themselves w i l l generally exhibit high frequencies and may be termed i n t e r n a l v i b r a t i o n s . The factor group analysis as applied to sodium formate and calcium formate w i l l be explained l a t e r . As Halford(7) has pointed out, while a f u l l con- sideration of a l l the consequence*; of the interactions between the motions of molecules In a c r y s t a l l a t t i c e leads to a very complicated picture, p r a c t i c a l l y a l l commonly observed effects of the molecular interactions can be s a t i s f a c t o r i l y treated by a less rigorous but f a r more manageable i d e a l i z a t i o n . I f the v i b r a t i o n a l motions of a molecule are treated as moving i n an' environment of fix e d symmetry, the v i b r a t i o n a l modes w i l l depend on both the symmetry of the molecule and i t s environment. In t h i s case, point symmetry rather than space symmetry may be used i n the anal y s i s . In the s i t e group analysis, the symmetries of the vi b r a t i o n a l modes of a l l equivalent molecules on a given - 52 - set of s i t e s i s the same and selection rules may be derived using the symmetry of the s i t e group as a basis. In a c r y s t a l , the s i t e symmetry can never be such as to contain symmetry elements not belonging to the free molecule and i n general, the s i t e symmetry i 3 lower than the molecular symmetry. Now the center of mass of a molecule i s invariant under the operations of the associated molecular group and i t s equilibrium position i n the c r y s t a l i s c a l l e d the a f f i x of t h i s group. The molecular group being the group of operations describing the symmetry of the mole- cule. Generally, the a f f i x e s of symmetrical molecules are situated on s i t e s , which requires that the s i t e group be a subgroup of the molecular group* A c o r r e l a t i o n of the species of the s i t e and molecular groups y i e l d s an ind i c a t i o n of the a c t i v i t y of the various vibrations i n the c r y s t a l . In the case of ionic c r y s t a l s , i t i s possible that the anions and cations i n the c r y s t a l are situated on s i t e s of di f f e r e n t symmetry and s t i l l belong to the same space group. The s i t e group method neglects the coupling of vibrations between ions of a set and thus gives a rather incomplete picture of the Interactions - 53 - within the unit c e l l . Realizing t h i s led Couture(37) to point out that the s i t e group method cannot be applied to ionic c r y s t a l s even as a f i r B t approximation. I t should be r e a l i z e d that the s i t e group method i s nothing more than a convenient, and often s u f f i c i e n t , f i r s t approximation to c r y s t a l l i n e spectra* Thus, although the s i t e method may y i e l d an adequate interpretation of the grosser d e t a i l s of the spectrum, complete informa- t i o n may only be obtained by resorting to the unit c e l l a nalysis. The a c t i v i t y of molecular modes i n the c r y s t a l may be determined by the c o r r e l a t i o n of the species of the molecular group and the s i t e group. The subsequent co r r e l a t i o n of the s i t e group into the factor group w i l l give the p o l a r i z a t i o n properties of the molecular modes. Winston and Halford ( 36 ) have used a d i f f e r e n t approach to the c l a s s i f i c a t i o n of the motions of a c r y s t a l on the basis of space symmetry* However, they have shown that equivalent r e s u l t s may be obtained using both t h e i r method and the method of Bhagavantum and Venkatarayudu. The former method i s derived from a t r e a t - ment which considers the motions of a c r y s t a l segment composed of an a r b i t r a r y number of unit c e l l s and subject to the Born-Karman boundary conditions. The l a t t e r method considers only one unit c e l l . The re l a t i o n s h i p between the two forms of analysis (site and factor group) may be discussed In terms of the pot e n t i a l energy of the c r y s t a l . For c r y s t a l s containing molecules or complex ions i n which the i n t e r n a l motion i s only s l i g h t l y affected by the c r y s t a l l i n e f i e l d , the complete p o t e n t i a l function may be written i n the form (38) v ^ - x ( v ^ > vc (2) where the l a t t i c e p o t e n t i a l ^ contains terms involving the center of gravity and orientation terms of the mole- cules . V* i s the potential energy function of the j t h free molecule while Vj includes those terms containing both i n t e r n a l and l a t t i c e coordinates of the j t h raole- « cule. The V j ^ represents cross terms between i n t e r n a l II coordinates of d i f f e r e n t molecules and contains cross terms between Internal and l a t t i c e coordinates. I n i t The terms v"j, V\j k, and V L^ are perturbations of the l a t t i c e and free molecule potentials ( 3 9 )* The v l b r a ^ t l o n a l problem can be separated i n t o that of the free molecules and of the l a t t i c e vibrations i f the pertur- bations are ignored. By considering the term Z^i only, the secular determinant could be factored into blocks, each asso- ciated with only one molecule. This i s c a l l e d the - 55 - "oriented gas" model(40). Inclusion of the term S t i l l permits separation of the secular determinant, but the frequencies are a l l a h i f t e d . Since the term has the s i t e symmetry of the j t h molecule, which i s usually lower than that of the free molecule, i t may s p l i t degeneracies and change sel e c t i o n rules* In t h i s approximation, the vibrations may a l l be c l a s s i - f i e d under the s i t e groups* S t a t i c f i e l d effects i n - clude a l l those e f f e c t s a r i s i n g from the fact that V\j + V\j a f o r the molecule i n the c r y s t a l d i f f e r s from the po t e n t i a l energy f o r the corresponding free molecule. These effects are a measure of the influence the surrounding l a t t i c e has on a given molecule* The s p l i t t i n g caused by the term i s thus termed the s t a t i c f i e l d s p l i t t i n g * However, even i f the remaining terms are n e g l i g i b l e and the s i t e symmetry i s adequate f o r cal c u l a t i n g frequencies, the p o l a r i z a t i o n properties of the normal vibrations may not be determined since the s i t e s may be variously oriented i n space, even though they are symmetrically equivalent* Dynamic c r y s t a l effects a r i s e from the term ^^jfe and give a. measure of the effect i n t e r n a l vibrations of other molecules have on the given molecule. I f the in t e r a c t i o n i s small, f i r s t order perturbation theory * 56 - may be used to calculate the e f f e c t , since only couplings between equivalent modes of v i b r a t i o n of the various molecules need be considered, except i n the case of a c c i - dental coincidence of frequencies* Depending on the number of molecules In the unit cell,* each "gas phase" Internal mode w i l l be s p l i t i n the crystal,. S p l i t t i n g s associated with the step from s i t e group to unit c e l l are c o r r e l a t i o n f i e l d or fac t o r group s p l i t t i n g s . After consideration of the above, i t i s seen that no v i b r a t i o n may be active i n the c r y s t a l i f i t i s f o r - bidden by the s i t e approximation. Also* every v i b r a t i o n which i s active under the s i t e group w i l l give r i s e to at least one active component i n the c r y s t a l . S i m i l a r i l y , any v i b r a t i o n which Is degenerate under the s i t e group remains so i n the c r y s t a l . The p o l a r i z a t i o n properties of the molecular modes may thus be determined by the co r r e l a t i o n mapping of the s i t e group in t o the unit c e l l group. - 57 - CHAPTER 4. DISCUSSION 4-1. Crystal structures N l t t a and Osaki have found that c r y s t a l s of calcium formate are orthorhombic with unit c e l l dimensions a -IO.163, b =13,381 and c = 6,271 a* at 18°C. The c r y s t a l belongs to 15 Space group Pcab (Dg n) with eight molecules per unit c e l l . Zachariasen found that c r y s t a l s of sodium formate are monoelinie with unit c e l l dimensions a - 6 . 1 9 , b = 6,72 and c = 6.49 8 with 6 = 121°42'. The c r y s t a l belongs to the 6 space group G2/c (Cg h) with four molecules per unit c e l l . The formate ion has point symmetry G 2 v with a CO bond length of 1.25 t 0.03 % f o r calcium formate and 1*27 8 f o r sodium formate. The 0C0 bond angle Is 125° - 4° f o r calcium formate and 124° f o r sodium formate. In the sodium formate c r y s t a l , sodium, hydrogen and carbon atoms are on two-fold ro t a t i o n axes of the space group. The oxygen atoms are on general positions. For calcium formate, a l l atoms are on general pos i t i o n s , (See figure 13). In both cases, the "free" formate ion has s i x normal modes of v i b r a t i o n , a l l allowed In the in f r a r e d . These modes are d i s t r i b u t e d among the i r r e d u c i b l e Figure 13. Figure 14. Symmetry elements of sodium formate unit cell. •̂58 - representations of C 2 v as follows} 3Ax+ 2B1-*-XB2, Since the closest approximation to a "free" formate ion i s an aqueous solution of the s a l t , the Raman and in f r a r e d spectra of aqueous solutions of the s a l t s may be used as a basis f o r further assignments. The observed frequencies fo r calcium and sodium formate are i n good agreement with previously reported Raman and Infrared data. Because df the r e l a t i v e l y weak forces between ions or molecules i n a c r y s t a l , the molecular fundamentals should exhibit only s l i g h t s h i f t s i n frequency from those observed f o r the corresponding "free" molecular modes. 4-2. Calcium formate There are no n o n - t r i v i a l subgroups of the fa c t o r group Peab, thus each formate ion i n calcium formate i s In a general po s i t i o n of s i t e symmetry G^. This lack of s i t e symmetry has no e f f e c t on the spectrum of the c r y s t a l since a l l molecular modes of the formate ion are allowed under the selection rules of the C 2 v group. There are no degenerate modes i n t h i s group, thus no s p l i t t i n g i s observed due to s t a t i c f i e l d e f f e c t s * Because of t h i s lack of s i t e symmetry* the factor group analysis must be employed f o r the interpretation of the r e s u l t s . The number of vibrations which can occur i n a unit c e l l can be represented by the expression - 59 ~ • n . - J f l W W * ) (3) where Is the number of v i b r a t i o n s , N i s the order of the group, hj i s the order of the j t h class of opera- tions <Rj Kj(&) i s the character of the operation (R i n the i r r e d u c i b l e representation and K(ti£\ i s the character of the operation ^ i n the reducible repre- sentation* A l l terms i n (3) can be obtained from the space group character table except K(<R), A n a l y t i c a l expressions f o r the K (fc) values have been derived by Bhagavantum and 3fenkatarayudu and are summarized below. Table 6 . Summary of expressions f o r characters of group operations V i b r a t i o n a l class Expression f o r character of operation Total number of vibrations K(<Ri)= U ^ ( i 1 + 2 cos <p) Acoustic (T) = ( t l + 2 cos <t> ) Translatory l a t t i c e (T») K(&)* \j^^s) ~ »"3 (** ¥ 2 oostf) Rotatory l a t t i c e (R») K (®\* U^s-v) ( i t 2 cos 4 ) Internal (n 1) J ((R) - Lu« - U ^ ] ( t l + 2 cos^) - (s-v)(l±2 c o s ^ ) Here, LL f t Is the number of atoms invariant under the opera- t i o n ,GK , s i s the sum of the number of groups occupying — 6o •» l a t t i c e points and v Is the number of single atoms occupy- ing l a t t i c e s i t e s * The character table f o r the point group D g h isomor- phous with the factor group Peab i s given below. By a p p l i - cation of equation ( 3 ) , the number of normal modes f o r the unit c e l l associated with each i r r e d u c i b l e representation has been determined* The symmetry operations associated with the factor group are the i d e n t i t y operation E, three mutually perpendicular two-fold r o t a t i o n axes C2> a center of inversion i and three mutually perpendicular mirror planes °~" , Table 7* Character table and factor group analysis f o r calcium formate D 2 h E C 2(z) c 2 ( y ) c 2 ( x ) i cr(xy) <r(zx n T T« R' n l A g 1 1 1 1 1 1 1 1 27 © 9 6 12 B l g 1 1 *l 1 1 -1 -1 27 0 9 6 ; 12 B 2 g 1 -1 l -1 1 -1 1 * l 27 0 9 6 12 ^ g 1 -1 - i 1 1 -1 -1 1 27 G 9 6 12 V 1 1 l 1 -1 -1 -1 - i 27 G 9 6 12 B l g 1 1 r l -1 -.1 -1 1 l 27 1 8 6 12 % g 1 -1 1 -1 -1 1 -1 l 27 1 8 6 12 B 3 u 1 -1 -1 1 -1 1 1 - i 27 1 8 6 12 * 6 l - The calcium formate c r y s t a l has 8 molecules or 72 atoms per unit c e l l giving r i s e to 3 x 72 = 216 degrees of freedom. Of these* three correspond to pure t r a n s l a - t i o n of the unit c e l l and the remaining 213 correspond to vibrations within the unit c e l l . Of the 96 i n t e r n a l molecular vibrations only 36 are allowed i n the inf r a r e d spectrum of the c r y s t a l , The factor group s p l i t t i n g and the o r i g i n of t h i s s p l i t t i n g under the space group D 2 h w i l l now be explained* In the unit c e l l of calcium formate* a given formate ion may be transformed into i t s seven c r y s t a l l o - graphically equivalent counterparts by the symmetry operations of the D 2 h factor group. For any i n t e r n a l molecular mode of v i b r a t i o n , eight d i f f e r e n t combinations are possible i n which the ions vibrate i n phase or 180° out of phase r e l a t i v e to a given ion i n the set. Each one of these combinations corresponds to one of the irr e d u c i b l e representations of the factor group, Thus we have a possible e i g h t - f o l d s p l i t t i n g of each molecular mode. The other o r y s t a l l o g r a p h l c a l l y non-equivalent formate ion also can give r i s e to an e i g h t - f o l d s p l i t t i n g of each molecular mode i n a manner i d e n t i c a l to that above. Even though the two sets are not symmetrically re l a t e d , the d i f f e r e n t combinations of the two Sets are of the same symmetry species. Thus we have a further two-fold s p l i t t i n g of the eight modes described above, and a t o t a l of 16 components f o r each normal mode of - 62 - the formate i o n . These 16 components are divided evenly among the eight Irreducible representations of the factor group, Only vibrations of the symmetry species B l u> Bg u and Bg u are active however and thus the Infrared spectrum of calcium formate should reveal each fundamental s p l i t i n t o s i x components. For polarized r a d i a t i o n along any one of the p r i n c i p a l crystallographic axes only two of the s i x components of any one fundamental w i l l be allowed. These two components w i l l correspond to the two non-equivalent sets of formate ions i n the unit c e l l as explained above* The above paragraphs give the basis f o r the factor group s p l i t t i n g of the calcium formate fundamentals under the factor group Pcab. 4-3 . Infrared spectra of calcium formate As mentioned e a r l i e r , the Raman and infrared spectra of aqueous solutions of calcium formate may be used as a basis of further assignment since these are es s e n t i a l l y the spectra of the free formate ion. How- ever, the reported Raman and infrared spectra of calcium formate are incomplete and the reported Raman and i n f r a - red frequencies f o r aqueous sodium formate w i l l thus be used as a basis f o r further assignment along with the infrared frequencies of aqueous calcium formate* - 63 - The three t o t a l l y symmetric fundamentals of the free formate Ion, ̂  , and ̂ 3 occur at 2 8 2 5 , 1352.and 773 era**1 respectively In the Raman spectra of aqueous sodium formate while * J and occur at 2834 and 1359 cm"*1 i n the infrared spectra of aqueous calcium formate. There has been some confusion with respect to the assignment of ^ and * The assignment i s made here following the convention that f o r vibrations of the same species, the assignment Is made i n the order of decreasing absorption frequencies* For calcium formate, the mode i s s p l i t into a t r i p l e t at 2895* 2890 and 2873 cm*1 i n the single c r y s t a l spectrum. The ^ mode Is s p l i t i n t o a doublet occurring at 1365 and 1355 cm""1 i n the poly- c r y s t a l l i n e spectrum of calcium formate* The ^ region i n the single c r y s t a l spectrum i s quite confused and w i l l be discussed i n greater d e t a i l l a t e r * The ^3 mode appearing at 773 cm"1 i n the Raman spectrum of sodium formate i s s p l i t into a t r i p l e t at 8 0 3 , 788 and 782 em~x i n the single c r y s t a l spectrum. The p o l y c r y s t a l l i n e spectrum of calcium formate indicates four components to ^ at 8 0 4 , 800* 788 and 782 c a r 1 . The two (Ggy) bg modes of the formate ion* and -*) 5 occur at 1584 and 1386 cm"1 In the Raman spectrum of aqueous sodium formate and at 1588 and 1383 cm"1 I n the - 64 - inf r a r e d spectrum of aqueous calcium formate. In the single c r y s t a l spectrum of oalcium formate i t i s d i f f i c u l t to determine any s p l i t t i n g i n the 2^ and ^5 modes because the c r y s t a l s were s t i l l too thick and these modes were almost t o t a l l y absorbing. However, the p o l y c r y s t a l l i n e infrared spectrum of calcium formate indicates that ^5 i s s p l i t into a t r i p l e t at 1403, 1394 and 1390 cm"1 while appears as a single peak at 1591 cm**1. There 1B some ind i c a t i o n that t h i s may be s p l i t In the single c r y s t a l spectrum into components at 1580 and 1617 cm"1. The out-of-plane mode, ^ , which occurs at 1069 cm"* i n the Raman spectrum of aqueous sodium formate i s s p l i t i n t o a doublet at 1079 and 1068 cm"1 i n the single c r y s t a l spectrum. Each fundamental region of the spectrum w i l l now be discussed i n greater d e t a i l . 4-4. ^, region (3100 - 2700 cm"1) The region around the ̂  fundamental i s quite complicated since i t i s In t h i s region that a large number of overtone and combination bands of the other fundamentals occur. As was mentioned, the fundamental appears to be s p l i t into a t r i p l e t occurring at 2895* 2890 and 2873 cm~""% A l l three components, however, are not equally Intense when polarized r a d i a t i o n i s used. As can be seen from - 65 •- figure 7, a i l three components appear as r e l a t i v e l y strong peaks when the d i r e c t i o n of p o l a r i z a t i o n i s p a r a l l e l to the c-axls, (Hereafter, f o r convenience, â , b and el- ective modes w i l l mean modes which are active when the di r e c t i o n of p o l a r i z a t i o n of the incident r a d i a t i o n i s p a r a l l e l to the a, b, or c-axis of the c r y s t a l ) * Only the 2895 and 2873 cm'1 components are b-actlve while -1 the 2895 cm component i s the p r i n c i p a l a-active com- ponent. A very weak a_-actlve component occurs at 287** em"1. The varying i n t e n s i t i e s of the various components may be better understood i f the d i r e c t i o n cosines of the o s c i l l a t i n g dipoles are considered. To a f i r s t approxi- mation, the r a t i o of the r e l a t i v e integrated absorption i n t e n s i t i e s of the various components should be i n the same r a t i o as the squares of the respective cosines. The calculated and observed r a t i o s give a very good q u a l i t a t i v e explanation of the spectra. A more quantita- t i v e explanation may be possible, depending on the accuracy of the reported c r y s t a l structures. The possible errors i n the reported atomic positions giving r i s e to In- accuracies i n the calculated d i r e c t i o n cosines and the rather crude manner i n which i n t e n s i t i e s were calculated are the two major reasons why there i s a r e l a t i v e l y large deviation between observed and calculated i n t e n s i t i e s . - 66 - Table 8 . Squares of d i r e c t i o n cosines of t o t a l l y symmetric ,S °*°illa*lng «p<*» calculated from the c r y s t a l t structure of calcium formate formate ion 1 formate ion 2 t o t a l i 2 0 . 0252 0 .1105 0 .1357 m2 0 ,5079 0 . 2343 0*7422 n 2 0,4666 0 . 6553 1*1219 Jia ra and n refe r to the d i r e c t i o n cosines the t o t a l l y symmetric o s c i l l a t i n g dipole makes with the a> b and c c r y s t a l axes respectively. Table 9* Ratios of calculated and observed t o t a l r e l a t i v e i n t e n s i t i e s f o r ac/( calculated observed i 2 / m̂  = 0 , 183 a / h - 0 . 4 0 3 l\ / n| = 0 .121 a / c = 0*373 m| / nf = 0 .662 b / c - 0*940 here a, b and c re f e r to the observed integrated i n t e n s i t i e s . - 67 - Table 1 0 . Ratios of calculated and observed r e l a t i v e I n t e n s i t i e s with various polarizations f o r ^ pol a r i z a t i o n calculated observed a i f / i | » 4 , 3 8 A / B — 4 . 2 8 b mf / m§ = 0 .46 A / B = 0 . 8 0 £ nf / n| = 1 .40 A / B = 1.84 A and B ref e r to the 2895 and 2873 era"1 components r e - spectively. In the case of the c-actlve components, the 2890 cm**1 component i s included with the 2895 cm"*1 component f o r the purpose of the above ca l c u l a t i o n s . On the basis of the above calculations i t appears that the 2895 cm"1 component of ẑ , arises from what w i l l henceforth be c a l l e d formate ion I , while the 2873 ow component of arises from the c r y s t a l l o g r a p h l c a l l y non-equivalent formate ion I I . This r e s u l t Is what i s expected on the basis of the factor group analysis. There Is no obvious explanation of the occurrence of the 2890 cm*1 c-aetive mode. The remainder of the -z-', region w i l l be discussed l a t e r along with other overtone and combination modes, 4 - 5 . region (820 - 760 cm"1) The region i s another i n t e r e s t i n g and com- pl i c a t e d region. As indicated e a r l i e r , the fundamental - 6 8 - appears to be s p l i t i n t o a t r i p l e t at 8 0 3 , 7 8 8 and 7 8 2 cm"1. A b-active peak occurring at 7 9 9 cm"*1 i n the single c r y s t a l spectrum may be the 8 0 0 cm""1 component observed i n the po l y c r y s t a l l i n e spectrum. As with the region, a l l components are not active along a l l axes* Thus only the 8 0 3 and 7 8 8 cm""1 components are a-active. The a-actlve component at 7 8 8 cm"1 has a further s p l i t t i n g with a peak - 1 i occurring at 7 8 6 cm , This peak, the b-active 7 9 9 cm~A component and the c-actlve 2 8 9 0 cm"*1 peak are not r e a d i l y explained on the basis of the factor group analysis. - 1 There are strong b-active components at 8 0 2 and 7 8 2 em and a weak b-actlve component at 7 8 8 cm**1. S i m i l a r l y , there are c-active components at 8 0 3 and 7 8 2 cm"1 with a c_-active component occurring as a shoulder at 7 8 8 cm*"1. The c_-actlve 7 8 8 cm*"1 and the b-actlve 7 8 8 cm"*1 component may possibly be due to either a_ or b and a_ or _c components respectively a r i s i n g from one or more of the following sources* (a) the c r y s t a l s may not have been ground absolutely perpendicular to the various crystallographic axes, (IS) the p o l a r i z e r used was not 1 0 0 $ e f f i c i e n t , and (c) the Incident r a d i a t i o n was not p a r a l l e l due to the convergence of the l i g h t beam i n the instrument. However, measurements and tests carried out as part of t h i s i nvestigation indicate that — 6Q — none of these possible sources would allow much more than a 1 or 2$ contribution from other components, and ce r t a i n l y less than 5 $ . As with the - ^ l region, a comparison of the ra t i o s of the squares of the d i r e c t i o n cosines of the o s c i l l a t i n g dipole and the r e l a t i v e i n t e n s i t i e s w i l l give an in d i c a t i o n of the o r i g i n of the various peaks. Since -2̂ , and are of the same species, the square of the d i r e c t i o n cosines given In table 8 also r e f e r to "2/3. In the case of , i t appears that the low f r e - quency components (that i s , the a-active 788 cm"*1 com- ponent and the b and c_-actlve 782 cm"1 components) are caused by formate ion I, while the high frequency com- ponents are caused by formate ion I I . This assignment i s made on the basis of calculated and observed Inten- s i t i e s . Table 1 1 , Ratios of calculated and observed t o t a l r e l a t i v e i n t e n s i t i e s f o r ^ calculated observed k\ / m̂  = 0 . 1 8 3 m t / n t ' 0 , 6 6 2 £f / nf = 0 , 121 1 / 1 = G « 3 5 5 b / c - 0 . 5 8 1 a / c 0 . 2 0 6 - 70 - Table 12. Ratios of calculated and observed r e l a t i v e i n t e n s i t i e s with various polarizations f o r pol a r i z a t i o n calculated observed a_ b c S\ / i f •= 4.38 B / A = 3.80 ra| / m| = o.46 B / A - 0.59 n 2 / n| 1.40 B / A - 1.85 A and B refer to the 803 and 782 cm*"1 (788 cm"*1 i n the case of the a-active component) components respectively. The b-active 786 cm""1 component i s Included with the 788 cm component f o r the purposes of the above c a l c u l a t i o n s . I n t u i t i v e l y , i t would be expected that a given set of c r y s t a l l o g r a p h i e s l l y equivalent formate ions would give r i s e to either the higher or lower frequency components of a l l fundamental modes of the same symmetry species. In the case of the ̂  and ̂  modes there appears to be a reversal of t h i s expected form. This author believes that the explanation of t h i s phenomenon l i e s i n the c r y s t a l structure of calcium formate. Because of the orientation of the formate ions i n the unit c e l l , the c r y s t a l f i e l d f o r the various modes w i l l d i f f e r , Thus the mode, which i s e s s e n t i a l l y a symmetric CH stre t c h , w i l l be l i t t l e affected by c r y s t a l - 71 - f i e l d effects since f o r any group of formate ions a l l hydrogen atoms are oriented away from the calcium ions. Each hydrogen atom i s separated from neighbouring hydrogen atoms by a distance larger than the Van der Waals radius. This lack of c r y s t a l f i e l d i s also an explanation f o r the r e l a t i v e l y small s p l i t t i n g of the mode. (Morrow found a s p l i t t i n g of 113 cm*"1 between the high and low frequency components of the a-', mode fo r barium formate), However, i n the case of the ̂  mode which i s es s e n t i a l l y a symmetric 0C0 bond, the c r y s t a l f i e l d e f f e cts would be expected to be quite large. This comes about Since the oxygen atoms of the formate ion are d i r e c t l y neighbouring the calcium ions and any change i n the 0C0 angle would give r i s e to large c r y s t a l . f i e l d e f f e c t s . This large c r y s t a l f i e l d would.also explain the r e l a t i v e l y large s p l i t t i n g observed i n the ^ mode. (Morrow observed a s p l i t t i n g of only 11.5 cm""1 f o r the high and low f r e - quency components of the ^ mode f o r barium formate). 4 - 6 . ^ region (1100 - 1050) The out-of-plane b g (C 2 v) mode i s the least com- pli c a t e d of a l l the fundamental regions of the calcium spectrum. The two components at 1079 and 1068 em*""* are - 72 - active along a l l three axes as would be expected from the factor group analysis. The i n t e n s i t i e s are i n the r a t i o expected except f o r one Instance. To a f i r s t approximation, the d i r e c t i o n of the o s c i l l a t i n g dipole f o r the mode can be taken as being perpendicular to the plane of the formate ion. In the case of the b-active modes, the square of the d i r e c t i o n cosines indicate that the com- ponents due to the two d i f f e r e n t formate ions should be of s i m i l a r r e l a t i v e i n t e n s i t i e s as the c-active modes. Instead the b-actlve components are reversed. Since the ^ mode i s the out-of-plane bend i t would be expected that c r y s t a l f i e l d effects would be r e - l a t i v e l y smaller than f o r the *^ mode. The small s p l i t t i n g observed (11 era"*1) seems to bear t h i s out. Hence i t would be expected that the high frequency components would be due to the formate Ion I s i m i l a r to , and the low frequency components due to formate ion I I . Thus, considering the d i r e c t i o n cosines of the o s c i l l a t i n g dipole I t i s expected that the high f r e - quency component (1079 cm""1) should be the most Intense f o r a-active modes and the low frequency components (1068 cm*"1) should be the most intense f o r b and c_-actlve modes. This i s observed f o r the s_ and e_ components but, as mentioned above, i s reversed f o r the b components. There i s no obvious explanation f o r t h i s anomaly. - 73 - Table 13. Ratios of calculated and observed r e l a t i v e i n t e n s i t i e s with various polarizations f o r -zs& p o l a r i z a t i o n calculated observed a l\ /£\ = 2.397 A / B = 1 * 6 3 6 b mf / m| = O.717 A / B « 4.238 £ n l / n o = ° - 6 l i * A / B — 0.606 4-7 . a^.a^and region (1700 - 1300 cm ) Very l i t t l e information may be obtained from t h i s region of the single c r y s t a l spectrum. The major d i f f i - c u l t y here was the thickness of the samples used. When the c r y s t a l was o r i g i n a l l y ground, i t was considered t h i n enough when the components of the various modes were resolvable. However, when the p o l a r i z e r was i n place, i t reduced the i n t e n s i t y of the s i g n a l of the sample beam by 5 0 $ . This reduction i n signal i n t e n s i t y caused the and -^s modes to be almost t o t a l l y absorbing. Another d i f f i c u l t y a r i s i n g i n the ̂ , and regions was combination and difference modes of l a t t i c e and fundamental v i b r a t i o n s . Combinations .of *"x with a l a t t i c e mode of 36 cm"*1 should appear i n the region, 1400 - 139O cm"1, while difference modes of ̂  with the 36 era"1 l a t t i c e mode should appear i n the ̂  region. - 74 - The assignment of these l a t t i c e modes w i l l be discussed l a t e r . The appearance of these l a t t i c e modes i s suspected since more components than those predicted by the factor group analysis are observed. In the case of ^ 4 and **r there are at least four poorly resolved a-active com- ponents f o r each fundamental. I t should be noted that combination modes are generally r e l a t i v e l y weak, but since the combination modes and. the fundamental com- ponents must be of the same symmetry i n order to be observed together with the same p o l a r i z a t i o n , Fermi resonance can occur and the r e l a t i v e i n t e n s i t i e s of the combination modes, may be greatly increased. A si m i l a r d i f f i c u l t y arises with the ̂  mode where a combination or difference mode of the 3 6 cm l a t t i c e mode with one of the components of y 4 ) should f a l l at almost the same frequency a3 the other component. Be- cause of the d i f f i c u l t i e s noted above, t h i s problem could not be s a t i s f a c t o r i l y resolved. 4 - 8 . L a t t i c e modes The s i x external degrees of freedom of the formate Ion belong to the following i r r e d u c i b l e representations of C 2 v; I A 1 + 1A 2 + 2B X + 2B . These degrees of freedom - 75 - become translatory and rotatory l a t t i c e modes when the ion i s i n a c r y s t a l . The factor group analysis given e a r l i e r gives the number of l a t t i e e modes belonging to each of the ir r e d u c i b l e representations of the factor group Cgk* As indicated, these l a t t i c e modes undergo a s p l i t t i n g s i m i l a r to the Internal fundamental modes of the formate ion. Generally these modes occur In the region below 300 cm"**1* and are inaccessible on the spectro- meter used f o r t h i s study. Tliese l a t t i c e modes, however, , 1 may be observed as combination modes with molecular modes. In the case of calcium foifraate, at leas t 34 frequencies I may be assigned to combination's with l a t t i c e modes. By I determining the frequency separation of the various i fundamentals and the weak pealls of the spectrum i t i s found that s i x differences oecW most often* These, I differences correspond to l a t t i c e modes at 36, 61, 90, - i i 128, 154 and 192 cm . Both cjombination and difference t bands are r e a d i l y observed with l a t t i c e modes. This Is so because the energy separation of the ground and f i r s t excited state of the l a t t i c e mode i s r e l a t i v e l y small and I at room temperature the f i r s t fexcited State may have a large population. Since a difference band requires that one mode of those involved be i n an excited state, the large population of upper v i b r a t i o n a l states of l a t t i c e - 76 - modes make them Ideal f o r the observation of difference bands* Table 14. Population of l a t t i c e v i b r a t i o n a l states at 20°C 36 61 90 128 154 192 793 (̂ 3) n/n Q 0.848 0.741 0.643 0.534 0.477 6.399 0.020 n/n Q i s the r a t i o of the populations of the upper and lower v i b r a t i o n a l states. Table 14 also gives an in d i c a t i o n why difference bands are very seldom observed with molecular modes. Even f o r a r e l a t i v e l y low frequency mode, the population of the f i r s t excited state i s very small. A study of single c r y s t a l spectra carr i e d out at very low temperatures would be very he l p f u l i n the assign- ment of l a t t i c e modes since at the lower temperatures* the upper v i b r a t i o n a l states of the l a t t i c e -modes would have greatly reduced populations and thus the disappearance of difference bands would be expected. Because of the technical d i f f i c u l t i e s Involved, no such investigation i<*as carried out as part of t h i s study. - 77 - ^-9* Combinations and overtones Of the 84 observed frequencies i n the spectrum of the single c r y s t a l of calcium formate, 16 are attr i b u t e d to combination and overtones of molecular modes. Since f i r s t overtones are t o t a l l y symmetric, no overtones would be I n i t i a l l y expected under the factor group s e l e c t i o n r u l e 3 . However, i t must be remembered that each mole- cular fundamental i s s p l i t into 16 components under the Cg h f a c t o r group. Thus any overtone would simply be a combination of one component of the fundamental with another component of the same fundamental. The inf r a r e d active components of the overtone a r i s e from combinations of an Inactive A-^ component with one of the in f r a r e d active B u modes or combinations of an ina c t i v e A^ u mode with Raman active B_ modes. Thus each overtone should consist of 64 components of which only 12 are Infrared a c t i v e . Table 15 gives the symmetries of possible com- bination modes. As can be seen from the table, not a l l combinations of overtones are allowed i n the factor group. Of the ones that are allowed, only some are i n - frared a c t i v e . Since the components from which the overtones ar i s e are generally i n f r a r e d inactive there i s no way of knowing whether these components have the same - 78 - Table 15* Symmetries of combination and overtone modes for calcium formate symmetry of symmetry of fundamental of l a t t i c e modes fundamental A g  B l g B 2 g B 3 g A u B l u B 2 u B 3 u A g g B l g B2g B3g A u B l u f B3u B l g B l g • A l g A3g A2g B l u + A l u A3U A2u B2g B2g A3g A l g A l g B 2 u f A3u A l u A l u B3S B 3 s A2g A l g A l g B3u A2u A l u A l u Au A u *if B2u B3u A g B l g B2g B3g B l u B i u * A l u A3u A2u B l g A l g A3g A2u B2u B2u A3u A l u A l u B2g A3g A i g A l g B3u 3u A2u A l u A l u . 3g 2g A l g A l g The dagger (t) denotes infrared active species. frequencies as the observed active components. For the purpose of ca l c u l a t i n g overtone and combination f r e - quencies, i t was assumed that the inactive components had the same frequency as the active components* I t was also assumed that the overtones observed were eit h e r combinations of two low frequency components or two high frequency components but never a combination of a low frequency component with a high frequency component since the low and high frequency components a r i s e from d i f f e r - ent formate ions. - 79 - I n s e v e r a l cases, the observed peaks were r a t h e r broad and d i f f u s e so t h a t they might p o s s i b l y c o n t a i n more than one component. In these cases, assignments were made assuming an average frequency f o r the funda- mentals as a b a s i s . S i m i l a r arguments may be advanced f o r the a s s i g n - ment of combination modes. In the case of combination modes o n l y one i n s t a n c e was observed where an assignment c o u l d be made on the b a s i s of separate components of the fundamentals i n v o l v e d . In a l l o t h e r cases, an average value f o r the fundamental frequency was assumed. 4-10. I s o t o p i c s h i f t s On the b a s i s of Hammaker3 r e s u l t s ( 4 1 ) (to be p u b l i s h e d s h o r t l y ) from a study of the i n f r a r e d spectrum of 0^3 e n r i c h e d sodium formate, the assignment of modes due to the n a t u r a l abundance of C 1 ^ i n the c a l c i u m f o r - mate may be made q u i t e r e a d i l y . Modes a r i s i n g from C 1 ^ formate.ions are observed f o r a l l but the mode. T h i s i s not unexpected s i n c e the mode i s v e r y weak compared to the other fundamentals. Only i n the case o f the mode i s the C 1 ^ mode of the h i g h frequency com- ponent observed. Because of the r e l a t i v e l y s m a l l I s o t o p i c s h i f t s the C 1 ^ mode of the h i g h frequency component u s u a l l y - 80 •- f a l l s In the immediate v i c i n i t y of the low frequency comp- onent and i s thus obscured. The v a l i d i t y of these assignments may be v e r i f i e d by making use of the product rule (3) which gives a raethematical rela t i o n s h i p between the product of the frequencies of a given symmetry species f o r two i s o t o - p i c a l l y substituted molecules. For the three a^ modes of the formate ion, t h i s rule i s given by the expression where the primed frequencies r e f e r to the C J species, m refers to the mass of the two atoms i s o t o p i c a l l y sub- s t i t u t e d , and M and M' re f e r to the t o t a l mass of the C 1 2 and C 1 3 ions respectively. Taking the low frequency component of the G 1 2 modes (that i s , 2873* 1355 and 782 cm"1) and C 1 3 modes at 1336 and 777 cm"*1, the frequency of the C 1 3 mode f o r ^ may be calculated as occurring at 2844 cm"1. \ The occurrence of a very weak peak at 2839 cm*"1 may thus be taken as the c 1 3 mode f o r ̂  . A s i m i l a r c a l c u l a t i o n involving the low frequency ^ and C 1 2 components (158O and 1388 am'1) and the C 1 3 mode at 1377 cm"1 gives a frequency of 1548 cm"1 f o r the C 1 3 component of ^4. . A weak shoulder i s observed at 1543 cm"1 thus confirming these assignments. - 81 - 4-11. I n f r a r e d spectrum of sodium formate As s t a t e d e a r l i e r , c r y s t a l s of sodium formate a r e monoclinic w i t h f o u r molecules per u n i t c e l l . The 6 v c r y s t a l belongs to the space group C2/c ( C 2 h ) , However, f o r the purposes of the f a c t o r group a n a l y s i s proposed by Bhagavantum and Venkatarayudu, the u n i t c e l l c o i n c i d i n g w i t h the c r y s t a l l o g r a p h i c u n i t c e l l Is not n e c e s s a r i l y the u n i t c e l l o f the s m a l l e s t p o s s i b l e s i z e . The c h o i c e o f the s m a l l e s t p o s s i b l e u n i t c e l l i s most obvious be- cause of the r e d u c t i o n i n the volume of c a l c u l a t i o n s needed t o adequately d e s c r i b e the system. As mentioned e a r l i e r , I f the c r y s t a l i s regarded as a s t r u c t u r e o r a s e t o f i n t e r - p e n e t r a t i n g l a t t i c e s , then the u n i t c e l l of the s m a l l e s t p o s s i b l e s i z e w i l l c o n t a i n as many atoms i n i t as there are i n t e r - p e n e t r a t i n g l a t t i c e s i n the s t r u c - t u r e . These p o i n t s may be r e f e r r e d to as the non- e q u i v a l e n t p o i n t s of the s t r u c t u r e f o r no one of these p o i n t s may be reached from any ot h e r by performing the p r i m i t i v e t r a n s l a t i o n s c h a r a c t e r i s t i c o f the l a t t i c e . As can be seen from f i g u r e 14, one h a l f of the sodium formate u n i t c e l l may be o b t a i n e d from the other h a l f by p r i m i t i v e t r a n s l a t i o n s . Thus the u n i t p e l l of the s m a l l e s t p o s s i b l e s i z e (on the b a s i s o f Bhagavantum's d e f i n i t i o n of a u n i t c e l l ) c o n t a i n s o n l y two molecules. - 82 - However, the f u l l symmetry of the c r y s t a l i s r e t a i n e d and the f a c t o r group i s s t i l l isomorphous w i t h C^. The c h a r a c t e r t a b l e f o r the p o i n t group C 2 h isomor- phous with the space group C 2 / c i s g i v e n below. By a p p l i c a t i o n o f equation (3), the number of normal modes f o r the u n i t c e l l a s s o c i a t e d w i t h each i r r e d u c i b l e r e - p r e s e n t a t i o n has been determined. The symmetry opera- t i o n s a s s o c i a t e d w i t h the f a c t o r group are the i d e n t i t y o p e r a t i o n E, a two f o l d r o t a t i o n a x i s C^, a c e n t e r of i n v e r s i o n I , and a h o r i z o n t a l m i r r o r plane &^ . Tab l e 16. C h a r a c t e r t a b l e and f a c t o r group a n a l y s i s f o r sodium formate c 2 h E C 2 I n i T T« R' t n l A g 1 1 1 1 6 0 2 1 3 B g 1 -1 1 -1 9 0 4 2 3 A u 1 1 -1 -1 6 1 1 1 3 ( T z ) B u 1 -1 -1 1 9 2 2 2 3 ( T x , Y y ) The sodium formate c r y s t a l has two molecules of 10 atoms per u n i t c e l l g i v i n g r i s e to 3 x 10=30 degrees of freedom. Of these, t h r e e correspond to pure t r a n s l a t i o n s - 83 - of the u n i t c e l l and the remaining 27 correspond to the v i b r a t i o n s of the u n i t c e l l . Of the 12 i n t e r n a l v i b r a - t i o n s , o n l y 6 a r e allowed i n the i n f r a r e d spectrum of the c r y s t a l . I n the sodium formate c r y s t a l , each sodium formate molecule i s s i t u a t e d on a s i t e of symmetry C 2 . Thus a c o r r e l a t i o n of the symmetry s p e c i e s of the mole- c u l a r group w i t h the f a c t o r group w i l l a l s o show the b a s i s of the f a c t o r group s p l i t t i n g under the f a c t o r group C2/c. The f o l l o w i n g c o r r e l a t i o n t a b l e shows the r e l a t i o n between molecular and f a c t o r group v i b r a t i o n s . Table 17. S i t e group c o r r e l a t i o n t a b l e f o r sodium formate molecular group s i t e group f a c t o r group C2v C2 °2h Thus the t o t a l l y symmetric molecular a-̂  modes be- come a modes i n the c r y s t a l and the m o l e c u l a r b modes become b modes i n the c r y s t a l . From t h i s i t f o l l o w s t h a t the a, molecular modes must be a c t i v e along one - 84 - a x i s o n l y ( i n t h i s case the b - a x l s ) . S i m i l a r l y the b modes are a c t i v e i n the ac plane o n l y . U n l i k e c a l c i u m formate, where there i s a d i s t i n c t component a c t i v e along each a x i s , the component observed i n the ac plane i s the same component I r r e g a r d l e s s o f the o r i e n t a t i o n o f the plane o f o b s e r v a t i o n . F o l l o w i n g t h i s e x p l a n a t i o n , I t i s seen t h a t the p o l a r i z e d sodium formate spectrum w i l l be q u i t e u n l i k e t h a t found f o r c a l c i u m formate. Whereas the most obvious p o l a r i z a t i o n p r o p e r t y of the c a l c i u m formate s i n g l e c r y s t a l spectrum was an i n t e n s i t y change, the p o l a r i z e d spectrum of the sodium formate s i n g l e c r y s t a l should e x h i b i t I n - a c t i v i t y o f some modes depending on the d i r e c t i o n o f p o l a r i z a t i o n . As was mentioned e a r l i e r , the sodium formate c r y s t a l s were such t h a t they c o u l d be ground p e r p e n d i c u l a r t o one a x i s o n l y . Hence, the c r y s t a l s were ground p a r a l l e l to the ac plane o n l y . Thus i t was expected t h a t only the b molecular modes would be observed i n the i n f r a r e d spectrum. As can be seen from t a b l e 4, some weak peaks have been a s s i g n e d t o the symmetric a m o l e c u l a r modes which were not expected from the f a c t o r group a n a l y s i s . However, the c r y s t a l s used f o r the sodium formate s p e c t r a were much - 85 - t h i c k e r than those used f o r the c a l c i u m formate s p e c t r a and p a r t l y because of t h i s t h i c k n e s s , unexpected compon- ents o f the molecular modes may be observed. As mentioned e a r l i e r , the convergence of the sample beam, the l e s s than 100$ e f f i c i e n c y o f the p o l a r i z e r and the p o s s i b l e m l s g r l n d i n g o f the c r y s t a l , a lthough c o n t r i b u t i n g a very s m a l l unwanted component t o the spectrum, these sources, i n c o n j u n c t i o n w i t h the t h i c k c r y s t a l would g i v e r i s e t o a measurable component. A comparison of the i n t e n s i t i e s of the symmetric modes wit h the i n t e n s i t y o f the a c t i v e b u modes shows the former I n t e n s i t i e s to be much l e s s than 5$ of the l a t t e r i n t e n s i t i e s which i s i n the order of magnitude of unwanted component expected from the above sources. The three t o t a l l y symmetric modes were observed as weak extraneous components at 2837, 1359 and 780 cm"*1. These are to be compared w i t h the values of I t o and B e r n s t e i n f o r the I n f r a r e d spectrum of an aqueous s o l u - -1 t i o n of sodium formate of 2803, 1352 and 773 cm . S i m i l a r l y , Fonteyne's Raman f r e q u e n c i e s o f 2825, 1352 and 773 era"*1 f o r an aqueous s o l u t i o n o f the s a l t are i n good agreement. Newman's f r e q u e n c i e s f o r the p o l a r i z e d i n f r a r e d spectrum of a sodium formate s i n g l e c r y s t a l are 2870, 1377 and 784 cm""1. There i s some doubt as to the - 86 - v a l i d i t y o f Newman's value s s i n c e no other workers have found values f o r the symmetric f r e q u e n c i e s as h i g h as those r e p o r t e d . No p o l a r i z a t i o n p r o p e r t i e s of the â ^ modes can be i n f e r r e d from the peak3 t h a t were observed. The 4̂ and ^ modes, observed a t 1584 and 1380 -1 cm by Ponteyne i n the Raman s p e c t r a o f an aqueous s o l u t i o n and at 1585 and 1383 c m - 1 by I t o and B e r n s t e i n of the i n f r a r e d spectrum o f an aqueous s o l u t i o n are ob- •1 served a t 1610 and 1376 cm In t h i s work. Newman r e - p o r t s values o f 1620 and 1377 cm r e s p e c t i v e l y f o r the p o l a r i z e d i n f r a r e d spectrum o f a sodium formate s i n g l e c r y s t a l . The va l u e of 1610 cm" 1 f o r ^ i s o n l y an approximation. The c r y s t a l was t o t a l l y absorbing over a l a r g e r e g i o n and the band envelope was assumed to be symmetrical so t h a t an average value f o r the frequency c o u l d be determined. T h i s t o t a l l y a b sorbing component was observed o n l y when the plane o f p o l a r i z a t i o n was p e r p e n d i c u l a r t o the c - a x i s . A weak peak at 1597 cm" 1 was observed w i t h the plane o f p o l a r i z a t i o n p a r a l l e l t o the c - a x i s . S i m i l a r l y , a v e r y s t r o n g , almost t o t a l l y a b s orbing peak was found a t 1375 cm" 1 f o r the component of p e r p e n d i c u l a r to the c - a x l s w h i l e o n l y a weak but sharp peak was observed f o r the p a r a l l e l component a t 1377 cm" 1. The =^ mode observed a t 1069 cm" 1 by - 87 - Fonteyne and by I t o and B e r n s t e i n , and a t 1070 era"*1 by Newman i s found a t 1068 era" i n t h i s case. Since the sodium formate c r y s t a l i s m o n o c l l n i c , the angle between two of the axes Is not 90°. T h i s obtuse angle, as d e f i n e d by Z a c h a r i a s e n ( 3 l ) , l i e s i n the ac plane so t h a t the planes o f p o l a r i z a t i o n a t r i g h t angles to each other would not correspond to the c r y s t a l axes, I t i s not necessary, however, to have the planes of p o l a r i z a t i o n p a r a l l e l t o the c r y s t a l axes i n o r d e r t o be a b l e t o i n - t e r p r e t the r e s u l t s . I f the d i r e c t i o n s o f p o l a r i z a t i o n and the d i r e c t i o n c o s i n e s of the o s c i l l a t i n g d i p o l e a r e both known r e l a t i v e to a common c o o r d i n a t e system, the a n a l y s i s of the r e s u l t s may be r e a d i l y c a r r i e d out. As w i t h the c a l c i u m formate s i n g l e c r y s t a l spec- trum, a knowledge o f the r a t i o s o f the d i r e c t i o n c o s i n e s of the o s c i l l a t i n g d i p o l e w i l l g i v e a good approximation of the r a t i o of the i n t e g r a t e d a b s o r p t i o n i n t e n s i t i e s o f the d i f f e r e n t components of one mode. Tab l e 18 g i v e s the c a l c u l a t e d and observed i n t e n s i t y r a t i o s . No va l u e i s g i v e n f o r the observed r a t i o of the ^ i n t e n s i t i e s because an ac c u r a t e v a l u e f o r the i n t e g r a t e d i n t e n s i t y c o u l d not be r e a d i l y o b t a i n e d . - 88 - Ta b l e 18. C a l c u l a t e d and observed t o t a l i n t e n s i t y r a t i o s f o r sodium formate mode c a l c u l a t e d observed l 2 / n 2 « 0.106 l 2 / n 2 = 0.106 l l / 1 « 0.066 l 2 / n 2 = 9.41 11 / l = 7.85 .11 and 1̂  r e f e r to the plane o f p o l a r i z a t i o n b e i n g p a r a l l e l o r p e r p e n d i c u l a r t o the c - a x l s o f the c r y s t a l . 4-12. Combination and l a t t i c e modes; i s o t o p i c s h i f t s As 'with c a l c i u m formate, the e x t e r n a l degrees o f freedom of the formate i o n become t r a n s l a t o r y and r o t a t o r y l a t t i c e modes when the ion. i s In a c r y s t a l . I n t h i s case, at l e a s t 23 of the 47 observed modes may be a s s i g n e d to combination modes between m o l e c u l a r and l a t t i c e v i b r a - t i o n s . The l a t t i c e f r e q u e n c i e s , determined i n the same manner as the c a l c i u m formate l a t t i c e f r e q u e n c i e s , were found to occur at 58, 91, 112, 128 and 231 cm"*1. I t o and B e r n s t e i n assumed l a t t i c e f r e q u e n c i e s of 60 and 140 _1 cm to e x p l a i n t h e i r spectrum of sodium formate. In t h i s case, both sum and d i f f e r e n c e bands were again ob- served. G e n e r a l l y , d i f f e r e n c e bands a r e seen o n l y w i t h the lower frequency l a t t i c e modes as I s expected from the - 89 - r e l a t i v e p o p u l a t i o n s of ground and f i r s t e x c i t e d v i b r a - t i o n a l l e v e l s . U n l i k e the c a l c i u m formate spectrum, f i r s t over- tones of molecular fundamentals sometimes are not allowed under C 2 h s e l e c t i o n ruler?. S i n c e the a^ molecular modes become a^ and a u modes i n the c r y s t a l and b 1 and b 2 molecular modes become b and b„ modes i n the c r y s t a l . g u then f i r s t overtones would have e i t h e r a or a symmetry o n l y . Since the a u modes are a c t i v e along the b - a x i s o n l y (that i s , along the a x i s p e r p e n d i c u l a r to the plane of the c r y s t a l used In t h i s study) they are not observed i n t h i s case. S i m i l a r l y , s e v e r a l combination modes are not observed i n t h i s study. A knowledge of the symmetry requirements e x p l a i n s t h i s phenomenon. T a b l e 19 g i v e s the symmetries of p o s s i b l e combination modes. The most unusual combination mode i s t h a t a s s i g n e d to »4 a t 2949 cm""1. T h i s i s a r e l a t i v e l y s t r o n g a b s o r p t i o n compared to o t h e r combination modes. However, c o n s i d e r i n g the very i n t e n s e -z^ fundamental, the com- b i n a t i o n mode i n t e n s i t y i s understandable. On t h i s b a s i s , the -^^-^ combination would a l s o be expected t o be prominent, but no peak i s observed t h a t c o u l d be a s s i g n e d to the combination. T h i s anomaly i s not r e a d i l y e x p l a i n e d . - 90 - T a b l e 19. Symmetries o f combination and overtone modes f o r sodium formate mode symmetry A g B g A u B u A S A g g V B I" u g g A g B + u A + A u V V flg B g ^ B u V A g The dagger (+) denotes i n f r a r e d a c t i v e symmetries. No bands are observed here t h a t c o u l d be a t t r i - buted t o C 1 3 i s o t o p e s as i n c a l c i u m formate spectrum. 1^ Since ^ ( t z / - and a r e the onl y a c t i v e modes, C i s o t o p i c s h i f t s would be expected f o r these modes o n l y . However, s i n c e the 24 mode i s 30 weak, the absence 1^ of the C peak i s not unexpected. I n the case of the •7^ and modes, i t i s probable t h a t the a b s o r p t i o n Is so s t r o n g t h a t the C 1 3 peaks are absorbed In the broad C 1 2 peak. The assignment of peaks i n the case of the sodium formate c r y s t a l spectrum Is r a t h e r Incomplete compared t o t h a t o f the c a l c i u m formate c r y s t a l . A more complete assignment, and the o b s e r v a t i o n o f the a u modes would be - 91 - p o s s i b l e i f a c r y s t a l c o u l d be grown l a r g e enough to enable a s e c t i o n being o b t a i n e d i n a plane p e r p e n d i c u l a r t o the s e c t i o n a l r e a d y s t u d i e d . Again, a study of the c r y s t a l spectrum at reduced temperatures would be u s e f u l and i n t e r e s t i n g . 4-13. C o n c l u s i o n A p o i n t has been reached where i t can be s t a t e d w i t h a good d e a l of c e r t a i n t y t h a t the I n f r a r e d spectrum o f c a l c i u m formate (and to a l e s s e r degree, sodium formate) has been s a t i s f a c t o r i l y e x p l a i n e d in-view of the symmetry c o n d i t i o n s imposed by the c r y s t a l s t r u c t u r e . I t i s seen t h a t a knowledge of the c r y s t a l s t r u c t u r e can g i v e a q u a n t i t a t i v e d e s c r i p t i o n of the fundamental a b s o r p t i o n i n t e n s i t i e s * I t a l s o becomes obvious t h a t the I n f r a r e d spectrum of a 3ingle c r y s t a l may be used to h e l p d e t e r - mine the o r i e n t a t i o n s of molecules In a c r y s t a l o f un- known s t r u c t u r e . I t i s i n t h i s manner t h a t i n f r a r e d spectroscopy c o u l d be a powerful t o o l f o r the c r y s t a l l o - grapher. I n f r a r e d spectroscopy i s a l s o a v a l u a b l e t o o l f o r c l a r i f y i n g c r y s t a l s t r u c t u r e s , e s p e c i a l l y of hydrogen c o n t a i n i n g compounds. S e v e r a l weak peaks i n both the c a l c i u m and sodium formate s i n g l e c r y s t a l s p e c t r a are s t i l l to be a s s i g n e d . - 92 ~ I h "both cases, t h e s e peaks a r e probably due to combina- t i o n of l a t t i c e modes w i t h m o l e c u l a r fundamentals. A more complete study of the sodium formate s i n g l e c r y s t a l spectrum would undoubtedly a i d i n f u r t h e r assignments as w e l l as p r o v i d i n g a b e t t e r understanding of the r e s t of the spectrum. As mentioned e a r l i e r , a study o f these sp e c t r a a t g r e a t l y reduced temperatures would a i d the assignment of l a t t i c e combination modes. 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