INVESTIGATION OF THE LIQUID VAPOUR SHIFT IN THE INFRA-RED ABSORPTION FREQUENCY OF DIATOMIC MOLECULES by William LeBreton Ross A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of DOCTOR OF PHILOSOPHY. Members of the Department of Physics THE UNIVERSITY OF BRITISH COLUMBIA July, 1953 ABSTRACT The changes i n the infra-red absorption frequency of three diatomic molecules:, HC1, HBr .and CO, on the transition between the gaseous and the solution phase have been investigated experimentally. The magnitude of this shift has been predicted using a purely electrostatic interaction.. The experimental values and predicted values for HC1 agree within 17$. The experimental and predicted values for HBr and CO agree within an order of magnitude. The investigation provides a method for independently checking the variation of the dipole moment of these molecules with internuclear distance against values previously determined by infra-red intensity measurements. i THE U N I V E R S I T Y OF B R I T I S H COLUMBIA F a c u l t y of Graduate Studies PROGRAMME FINAL ORAL OF OF EXAMINATION DOCTOR OF THE FOR THE DEGREE PHILOSOPHY of WILLIAM LeBRETON B.A., M.A., ROSS ( B r i t . Col.) WEDNESDAY, AUGUST 1 2 t h , 1953 a t 2:30 P. IN ROOM 301 PHYSICS B U I L D I N G COMMITTEE I N CHARGE ,H. F..Angus, Chairman A. M. Crooker W. A. Bryce F. Kaempffer H. B. Hawthorn 0... Theimer R. D. James G. M. Shrum B..Savery E x t e r n a l Examiner - IL J . B e r n s t e i n T H E S I S INVESTIGATION-OF THE L I Q U I D VAPOUR S H I F T I N THE INFRARED ABSORPTION FREQUENCY OF DIATOMIC MOLECULES The change i n the i n f r a r e d absorption frequency of three:diatomic molecules, H C 1 , H B r , and CO, on the t r a n s i t i o n : b e t w e e n the gaseous and the s o l u t i o n phase have been i n v e s t i g a t e d e x p e r i m e n t a l l y . T h i s change depends-.on the, d i e l e c t r i c c o n s t a n t o f t h e p a r t i c u l a r s o l v e n t i n . w h i e h t h e gas i s d i s s o l v e d . . The. magnitude of t h i s e f f e c t has been p r e d i c t e d by using: a purely e l e c t r o s t a t i c : m o d e l . The experimental and p r e d i c t e d ' v a l u e s f o r .K-Gl agree , within. 17%. The agreement between e x p e r i m e n t a l and t h e o r e t i c a l values f o r H-Br.and CO. are w i t h i n an. order of magnitude. : 1 T h i s i n v e s t i g a t i o n has shown t h a t • t h e major c o n t r i b u t i o n t o the change i n the i n f r a r e d absorpt i o n f r e q u e n c y a r i s e s from e l e c t r o s t a t i c i n t e r a c t i o n . :In a d d i t i o n , i t p r o v i d e s a m e t h o d . f o r independently checking'the v a r i a t i o n of the d i p o l e moment of these'molecules:with-internuclear distance a g a i n s t v a l u e s p r e v i o u s l y determined b y • i n f r a r e d i n t e n s i t y measurements. GRADUATE STUDIES Field of Study: Physics Theory, of D i e l e c t r i c and Magnetic S u s c e p t i b i l i t i e s - - :A. J . Dekker Molecular Spectra -- A- M. Crooker Theory of S o l i d State r - II-.: Koppe Quantum-.Theory of R a d i a t i o n -- p. Kaempffer S t a t i s t i c a l Mechanics -- W. Opechowski Other Studies: Theory o f Functions of a Complex V a r i a b l e -- B. Moyls D i f f e r e n t i a l Equations ••- T. E. H u l l JGKNOwIEGMENTS I wish to thank Dr. A.M. Crooker for the assistance and encouragement he has generously offered. I am greatly indebted to Mr. H. Buckmaster for the assistance and discussion he has given on the Morse potential and the solution of the integrals; i n Appendix I I . In addition,, I wish to acknowledge the financial assistance of the National Research Council and the B.C. Telephone Company which have made this work possible. CONTENTS Pag Chapter I Introduction . . . . . . . . . . . Chapter II Electrostatic Interaction of. a Polar Molecule i n a Dielectric Medium 6 Electrostatic Model ' The Reaction. Field 6 . 9 ' Energy of a Dipole i n i t s Reaction. Field Interaction of the Reaction Field with an Anharmonic Oscillator Chapter III Experimental Procedure '. 11 12 ............... Calibration ' 17 17 Preparation of Gases, Solvents and Solutions Chapter IV i 20 Interpretation of Results . . . . . . . . . . . . . .26 Hydrogen Chloride .29 Hydrogen Bromide 30 Carbon Monoxide 31 Conclusion 34. 36 Appendix I Appendix II . o » . . . . . . . . . . . . . . « . « « . « . . » . . .4-0 Appendix III 4B Bibliography 51 o ILLUSTRATIONS, , To follow page Figure 1 6 Figure 2 Figure 3 Figure 4 . 17 ...... • Figure 5 Figure 6 17 . 1 • • • ^ . . . . . . . . Figure 7 19 -22 Figure 8 29 . Figure 9 2 Figure 10 9 30 Figure 11 . . . . . . . ,'30 Figure 12 Figure 13 7 31 • • 32 CHAPTER I INTRODUCTION The transition from the vapour to the liquid phase has two effects on the observed infra-red rotation-vibration spectra. The f i r s t effect i s the disappearance of the rotational-fine structure of the rotation-vibration band vrith a resulting sharpening of the overall appearance of the band. This effect i s easily understood since the classical frequency of rotation i s of the same order of the frequency of inter-molecular collisions i n the liquid phase. For- example, i n liquid hydrogen chloride, 10^ sec~^ and '^rotation ~ 10"^ sec"^" . collision — The only notable excep- tion, i s observed i n the Raman spectra of liquid hydrogen where rotational fine structure has been observed (13). On the other hand, quantised vibrational states exist i n the liquid state, but their energy i s so altered that the observed absorption spectrum i s shifted toward longer wavelength. This shift i s usually of the order from 1% to. 5% of the fundamental frequency, i f the molecule has a permanent dipole moment (see Herzberg (10)). In molecules with a centre of inversion, there i s only a very small shift i n the Raman active vibra^.". tions. McLennan and McLeod (I4) have measured the shift i n oxygen, nitrogen and hydrogen; i n these liquids the observed shift toward the red i s about 0,2% of the' fundamental frequency or less. The previous investigations of this phenomena that are of special interest have been carried out by West and his collaborateurs (2U, 25, 26) , Breit and Salant (3), Bauer and Magat ( l ) , Leberknight and Ord (12), and Plyler and Williams (21). These investigators have studied hydrogen chloride and hydrogen bromide dissolved i n a variety of polar solvents and a few non-polar solvents. Apparently, no systematic attempt has been made to obtain a series of measurements i n non-polar solvents* West and Edi^ards (26) have found two absorption bands for hydrogen chloride i n certain solvents. The low frequency band i s shifted to the red of the frequency of the gas band. The high frequency band i s usually shifted to the blue from the gas band value. They have shown that the separation, of the two bands rules out their interpretation as P and R branches and they have implied that the high frequency band may be a combination of the lower . vibrational frequency with a torsional oscillation of the hydrogen chloride molecule i n the solvent. There are three separate effects which could contribute to the observed shift i n the vibrational spectrum. 1) These effects- are Lorentz-Lorenz shift 2) Electronic interaction 3) Electrostatic interactionThe magnitude of the Lorentz-Lorenz shift has been investigated by Breit and Salant (3). The expected shift i s where n e 1 m = number of molecules per unit volume = 'effective charge' of the oscillator =• mass of the oscillator For liquid hydrogen chloride, the expected shift would be -©,003 cm" 1 0,001$ of the fundamental absorption, frequency. or This; mechanism must be discounted since the observed shift i s much larger and since i t also depends, on. the value of the permanent dipole moment. •Cremer and Polyani (5) have attempted to calculate the magnitude of the frequency shift arising from the electronic interaction of neighbouring molecules according to London's theory. Unfortunately the calculation could be made only for hydrogen chloride i n the solid state since the interatomic distances must be known accurately. The calculated shift i s about 0,3% of the fundamental frequency or 9 cm" 1 toward the red However for hydrogen chloride ^ gas == 2886 cm" 1 liquid == 2785 cm" 1 solid == 2768 cm" 1 It appears that this mechanism contributes very l i t t l e to the observed shift i n hydrogen chloride. However i t i s possible that i t would account for the observed shifts i n non-polar gases such as oxygen, nitrogen and hydrogen. The electrostatic interaction has been discussed by Kirkwood* and by Bauer and Magat ( l ) . + West and Edwards have checked Kirkwood's quoted by West and Edwards ( 2 6 ) e formula, (see Chapter IV, page $2. ). (2a) as a (A. 3 (2b) y cj^ I where a = radius of cavity containing HC1 molecule = I permanent dipole moment i n Debye units =~ moment of inertia i n atomic units and dielectric constant of the solvents a i s the static They have found an empirical value = 0.62.2 for HC1 which i s unreasonably low. .That i s , i f a larger value i s used for the radius, of the cavity, the expected shift according to this model w i l l be too small. In this work, a similar model has. been set up> to specifically test the magnitude of the frequency shift i n non-polar solutions. This model treats: a mechanically and electrically anharmonic oscillator , surrounded by a continuous dielectric medium. In order to check this model, the experimental investigation has been limited to diatomic molecules dissolved i n non-polar solvents. This choice of solvents eliminates the necessity of accounting for dipole-dipole interaction. CHAPTER II THE ELECTROSTATIC INTERACTION. OF A POLAR MOLECULE "IN A DIELECTRIC MEDIUM Consider a point dipole sphere of radius situated at the centre of a a. The dielectric constant of the homogeneous medium inside the sphere i s €- = c n? whereas on the outside i t i s £ c « Figure 1. The solution of LaPlace's equation i n cylindrical coordinates w i l l be /),.A.\. 2 ^ ) T^(c^6) (3) 7 The potential inside the sphere must be 0 B £<• ^ *=«> and outside <$ 0 = y i±- P^(^B) (5) At the boundary between the two media, the potential and the normal component of the electric displacement must be continuous, that i s , for r and 0 £ © < and If tfi = €i ) 0, = #c €. (6a) (6b) 5* These conditions are satisfied i f - 2.1 Sm' (7a) = a 8 Or' and AQ = A ze *e 0 2 (7b) ( = . . . Aj. = B Q = B 2 =v . .B r = 0 (7c) Therefore (8) - f - ^ +s v W ( 9 > It should be noted that the potential outside the sphere arises from a dipole moment of magnitude — w h i c h i s smaller than the magnitude of the hypothetical point dipole moment at the centre of the sphere. .When - 1, we have the vacuum dipole moment which is. commonly referred to as the 'dipole moment of the molecule'. 9 The Reaction Field The concept of the reaction f i e l d , which was f i r s t discussed by Onsager (17), arises, as follows! the dielectric outside the sphere i s polarised by the presence of the dipole ^>&. <T ' ; l e t i n the medium C this polarised state be fixed and the dipole and i t s surrounding medium <T ' 4 be removed* The reaction, f i e l d i s the f i e l d remaining i n the cavity.. We have % where = ~T£777~ (7b) £± ^ 1, but from equations. (7a) and (7b) we obtain ff, ' — ( i i ) but Bj i s fixed and then the dipole and surrounding dielectric removed. Setting <f- = 1 we have t Q. 3 (2 (12b) are 10 The reaction potential i s (13) where r ^ a. Using the relations: R - #X - (14a) c^-i & - R A>^~e = & 7^ - K^^-i^.e .+ K a s (Hb) (34c) we obtain 7^* = 0 (15a) Z(£o~/) 7? hence " " a 3 . (15c) It has been shown by Frood (8) that equation (15c) s t i l l holds i n the case of an extended dipole which has an arbitrary position inside the sphere. 11 In the model being discussed, a rigid dipole i s embedded i n a dielectric £V . I f the molecule i s spherical, this model closely approximates an actual molecule with a dipole moment *-^tr and with electronic polarisability _ g c - / e t a. c where £. •= <=< . Thus, i f e oc_ (16) c a 3 r? , then equation (15c) i s identical with where . ^/£o-/) ( 1 S ) Eottcher (2) has shown that this result i s valid for a poiarisable point dipole at the centre of the sphere, but that i t does not hold exactly for an extended dipole. Energy of Dipole i n i t s Reaction Field When the molecule i s dissolved i n the solvent, a dipole with a magnitude ^<^- i s placed i n a reaction f i e l d is. Using equations (17) and (18), we obtain R , the change i n energy 12 (20) This formula i s convenient i n this form since dipole moment of the molecule and °< e i s ustially called the i s practically independent of the physical state of the molecule. Interaction of Anharmonic Oscillator i t h the Reaction Field w The vibrational potential of diatomic molecules can be represented quite accurately by the Morse potential function (15) which i s U = D c (e - J 3e (21) On substitution of this potential function i n the Schrodinger equation, a solution may be obtained directly i f j radial eigenfunctions are A. = 0 (see Appendix I ) . The where (22b) 'A (22c) -(3 (A-A,) (22d) fc = k - * ^ (22e) - J. The eigenvalues are Si -1 cm (23a) * (23b) where c **~' . 14 The expectation values of the perturbing potential given i n equation. (20) may be calculated with the following assumptions: 1) A l l quantities i n equation (20) are independent of the internuclear distance with the exception of the dipole moment which may be represented by the expansion. where £ 2) = - ^ e (24b)- Equation- (22a) provides, a satisfactory zero order represen- tation of the molecule i n the dissolved phase,,. In other words, we may treat the molecule as non-rotating oscillator with non-degenerate energy states. Then. The energy states: of the molecule i n solution are Consequently, the shift i n the absorption, frequency between the gaseous: phase and the solution phase is. 15 where v > v . The expected shift i n the fundamental absorption, band 1 os a diatomic molecule would be Ac * 1 C A -) t I, (23) H where (29) These integrals have been evaluated i n Appendix I I . In review, i t should-be emphasised that this model depends on the following assumptionss l) The major contributions to the dielectric constant of the solvent must occur at frequencies greater than the absorption frequency of the diatomic, molecule.. This assumption rules out polar solvents, however one might expect that replacing the dielectric constant by n should give results that are reasonably correct providing the dipole moment of the solvent i s small and that the absorption band associated with this dipole moment l i e s to the red of the absorption band i n the diatomic molecule.- 16 Z) The dissolved molecule i s spherical. The particular form of the expression for the reaction f i e l d given i n equation (20) depends on the existence of. a spherical cavity.. 3) The radius of the cavity. This i s discussed i n some detail i n Chapter 17. 4.) The solvent may be treated as a continuous dielectric media. This assumption i s certainly justified from a macroscopic point of view., but from the microscopic point of view i t i s rather crude.. However, even, this; assumption seems: justified since we are dealing with the time average of the energy levels of the molecule i n the solution. The best possible approximation.to this assumption would certainly be the case i n which the solvent molecules are spherical and •non-polar. 17 CHAPTER III EXPERIMENTAL PROCEDURE The spectrum of hydrogen chloride, hydrogen bromide and carbon monoxide were examined i n the gas phase and the solution phase with a Model 12B; single beam Perkin Elmer infra-red spectrometer equipped with D.C. recording system.. A l l measurements. were made with a LiF prism since' this prism gives the maximum resolving power and dispersion i n this range, that i s i n region of 5.0 to 1..8 microns, or 2000 to 5600 cm""*" „ Calibration. o The spectrometer was calibrated by two separate methods. In the region of the fundamental absorption bands of. HC1, HBr:- and CO at 2886, 2559 and 2L43 cm"-*- respectively, there i s sufficient resolution to completely separate the individual rotational lines in,the vibration-rotation bands. Furthermore, these line systems extend about 150 cm"l on both sides of the band centre. Therefore the spectrometer was calibrated graphically i n these regions. Values, for these absorption, lines; were taken from the Perkin Elmer Manual (1950). Smoothed curveS:Jhave been plotted through these calibration : points, which f i t a l l the points within 0.5 cm"! (see figures 2, 3 and 4)» In. the region of the f i r s t overtones, that is, 4000 or 5600 cm"*-*-, the resolving power and the dispersion have decreased sufficiently so that HCI of G as Calibration L i F Prism 40L 9 201 2720 80 FIGURE 2. 2900 . y (Cm ) -1 FIGURE 3. Fl GURE 4 . Lift row Mirrow Position 18 the rotational lines cannot be resolved; furthermore there are comparitively few sharp absorption lines available for calibration, therefore the spectrometer was calibrated analytically; according to the method described by Ross and L i t t l e (22). Table I contains the constants for the calibration formula T-To where x 10 3 i s the frequency i n wave numbers, and T x 10 3 i s the reading on_the Littrow micrometer of the spectrometer. Table I -1.09602 a T o -6.98571 1.76976 Table II i s composed of the calibration points available i n this region. Calculated values of T and V ere included i n the table to show the calibration equation. The values of AT a r e consistent with the estimated precision i n the measurement of line positions (see following page). The trichloro-benzene calibration points come from a table of calibration standards set up by Plyler and Peters (20). The H 0 calibra2 tion points; are taken from the Perkin Elmer Manual. FIGURE 5- 19 •Table II Substance V (stnd) T (meas) V Av (calc) B^O vapor 3839.0 cm* 1328.2 div. 3837.3 cm" trichlorobenzene A101.-6 1380.2 trichlorobenzene 4160.3 trichlorobenzene- A T •-1.7 «4 4103.2 1.6 -.3 1390.5 4162.1 1.8 -.3 4644.2 1461.4 4643.1 -1.1 .1 B^P vapor 5346.0 1534.1 5343.0 -3.0 .3 trichlorobenzene 6020.3 1582.8 6022.4 2.1 -.1 1 1 The probable error i n the calibration i n this range i s 1.5 cm . Figure 5 shows the dispersion of the spectrometer plotted against frequency. The dispersion i s given i n cm'Vdivision. During the calibration measurements one division on the Littrow micrometer corresponded to 0.08 inches on the recorded chart i n the Brown recording potentiometer.. The position of a sharp symmetric line can be estimated to about 0.01 inches which corresponds to aboiit 0.2 divisions on the Littrow micrometer.. This figure i s consistent with the values for A T i n the above table. Figure 6 shows the resolving power of the spectrometer fitted \-jith the LiF prism. In this spectrometer the resolving power i s limited by the sensitivity of the detector i n that the s l i t width must be increased beyond the optimum width i n order to obtain a significant 20 signal, on. the detector. Several curves are shovn corresponding to the s l i t widths commonly used during the experimental work. It i s apparent from figures, 5 and 6 that considerable errors may be involved i n measuring the position of broad absorption, bands i n the region of 4OOO to 5600 cm"*. 1 Preparation of Solvents. Gases and Solutions Dry hydrogen chloride gas was prepared by dropping concentrated hydrochloric acid on concentrated sulphuric acid., The gas was passed through phosphorous pentoxide and then through a spiral glass tube immersed i n alcohol and solid carbon dioxide i n order to remove a l l traces of x^ater vapour. Hydrogen bromide gas was prepared by dropping concentrated sulphuric acid on sodium bromide. The gas M&S dried i n the same manner as the hydrogen chloride, except that the temperature of the dry-ice bath, was raised by adding water u n t i l the bath temperature was above the boiling point of hydrogen bromide which i s -67°C. This reaction produces, both bromine and sulphur dioxide as by-products, but the dry-ice bath removed both of these compounds successfully. Carbon monoxide was prepared by dropping formic acid on concentrated sulphuric acid. The gas was dried i n the same manner as described above. Water vapour i s an impurity common to a l l three methods of preparation. I t was found that this drying procedure removed i t very successfully. This was checked by examining large samples of the prepared gases i n the region: of strong water vapour absorption, bands. 21 Three conditions determine the choice of solvents., They must be transparent i n the particular region of investigation.. They should be nonpolar and they must not react chemically with the gas.. The solvents used are shown.in Table III. Table III n Solvent Benzene. C C 1 4 2 C 1 C C S * 2 Br 2 SnCl TiCl 4 4 SiCl 2.29 2.26 2.18 2.14 2.46 2.27 2.60 2.66 3.18 2.76* 3.2 ? 2.25 — 2.4 ? 4 2 2.60 2.02 The f i r s t four solvents, were dried with anhydrous sodium sulphate and d i s t i l l e d in. a three foot glass-packed vacuum insulated column;, The last four solvents are extremely d i f f i c u l t to handle and no attempt was made to purify them; C.P.. reagents, were used instead. * 5893 n = probably increases 1.-661, however Br^ absorbs strongly i n the red.. This £ ^ to 3»18. 22 Saturated solutions were prepared by rapidly bubbling the gas through the solvent.. The spectrum of the gas i n solution was examined Immediately after preparation, since the hydrogen chloride and hydrogen bromide solutions became contaminated with water very rapidly. In recording the spectra of the dissolved gases, the c e l l was f i r s t f i l l e d with solvent and the spectrum of the solvent recorded. Then the c e l l was f i l l e d with the solution and the record of i t s spectrum was superimposed on the same chart. A typical band i s shown i n figure 7. Immediately before or after each measurement was made, the calibration was checked i n the appropriate region by remeasuring a few calibration, points and then correcting, i f necessary, to the original calibration data. The frequency of the absorption, line was determined by measuring the point of maximum absorption. Since the bands have broad shallow maxima and are asymmetric, their position could not be determined as precisely as could the individual rotational lines which were used for calibration. absorption lines are shown i n Tables IV, V and VI. The position of the The estimated probable error i n calibration and the srror i n estimating the position of maximum absorption of the band, but i t does not include any possible error arising from measuring the position of maximum absorption rathei'* than the centre of the area of the band. 2700 HCI 2800 2900 bond in CCI HCI bond in C CI 2 7 • n 4 4 and CCI and 3000 4 C CI ? : 4 -1 background background ' FIGURE cm ' 7. : " 23 Table IV HC1 Solvent SiCl = 2886 em" 1 cci 4 SnCl GS"2 Br 2 C6H6 V = 5668. cm" 1 CC1 GS 2 4 4 4 Cell. Length V cm" 1 l 3*0 mm 294-5 2848 3.0 mm 2933 2831 + 5.0 •7 + 4.8 % 1 -V 1 g 59 -38 .8 47 -55 .4 mm 2930 + 10.0 2829 + .•7 44 -57 .4 mm. 2940 ± 12.0 2826 ± .7 54 -60 3.0 mm 2927 ± 2.0 2826 + 1.3 41 -60 • 5 mm 2798 + 4ol -SO 3.0 mm 2787 + l 5 «99 Ml a cm" .2 mm 2747 t 2.3 -139 12.0 cm 5692 ± 5558 t 7.5 7.5 24 -110 12.0 cm 5601 + 26.0 -67 24 Table V HBr Solvent CCI4 C Cl a V = 2559 cm- 1 g V = 5028 g cm"* C e l l Length 3 . 0 mm 4 1.0 mm Vl cm" 1 2518 i 1.5 opaque % -V cm" 11 g -41 — CS2< 3 . 0 mm 2487 ± 1.0 -72 Brz 3 . 0 mm 2476 i 1.5 -83 3 . 0 mm 2448 ± 1.5 -111 cci4 1 2 . 0 cm 4932 + 10*5 -96 cs 2 1 2 . 0 cm 4 8 3 2 + 10.5 -196 1 Table ¥1 CO Solvent V = 2143 cm" 1 g C e l l Length % Vl - V cm* 1 C6H6 1 . 0 cm 2134 ± 1.5 -9 G2CI4 3 . 0 mm 2134 ± 1.9 -9 CCI4 3 . 0 mm 2133 ± cs 2 . 2 mm .9 cm" 1 g -10 opaque Carbon monoxide and hydrogen bromide could not be used i n the other solvents: shown i n Table III since they react chemically with them. 25 Very few observations at the harmonic have been, successful since a l l the solvents except Br2 have absorption- spectra which interfere to some: extent,, I t was found that the spectrum of the solvent i n the 12 cm, path length, was usually too strong to allow measurement of the f i r s t overtone. In the case of carbon, monoxide, the l s - t overtone was not observed. This was apparently due to the limited solubility of the gas i n these solvents. Unfortunately 12 cm., was the maximum c e l l length, that could be used i n the spectrometer. The effect of concentration was investigated with C.S2, CCl^, C 2 4 C 1 a n t i benzene. Successive dilutions of 1 ; 2, and 1. s 4 were tried; there was no change i n the position of the absorption'bands., I t appeared that the saturated solutions were sufficiently dilute that the observed shift was independent of the concentration of the dissolved gas. Only i n the case of hydrogen chloride were double bands, observed. In general, the higher frequency component i s much weaker and more diffuse than the low. frequency component. The high frequency component did not appear i n B r 2 , CS2 and benzene. I f a similar, intensity ratio existed between: the two bands: i n hydrogen bromide, the second component would have been observed. Ia. the case of carbon, monoxide, the absorption: spectrum of the gas i n solution was so weak that the high frequency component could very easily have been missed. 26 CHAPTER IV INTERPRETATION, OF RESULTS' In order to correlate this data with the electrostatic model, the polarisability of the molecule, the dielectric constant of the solvent, the radius of the cavity and the dipole moment of the molecule must be known.. The polarisability of hydrogen chloride, hydrogen bromide and carbon monoxide are shown i n Table VII. They have been calculated from the refractive index:, of the gases according to the Claussius-Mossotti relation .£-1 4 nr A/o 3 V £ +Z where £ = (3D <>< xS-y that i s (32) for gases at low density.. The values of the refractive index of the gases are taken from International C r i t i c a l Tables. Table. VII n (N.T.P. A= 6708 1) x 10 ^ 2 HCI I.OOO444 2.,63 HBr 1.000608 3.60 CO 1.000333 1.97 cm 3 27 These values of »< do hot include the atomic polarisability since has been used instead of. n.o© y however since «* » e n^Qg (See Van. Vleck (23) )> these values are satisfactory.. The dielectric constant and the square of the refractive index, are shown i n Table III.. The relation £ n 2 = n 2 holds f a i r l y well, except for SnCl^ and SiCl^. were used instead of € The values of for these two molecules. The dimension of the cavity containing the solute molecule i s very uncertain., It i s apparent that i t depends: on the particular solvent as well as the solute.. While i t i s relatively simple to estimate the average volume of a molecule in,a pure liquid, i t i s d i f f i c u l t to e s t i mate the effective volume of a molecule dissolved i n a solvent.. As a f i r s t approximation..a value of CL has been selected which i s charac- t e r i s t i c of the solute molecule only. Assuming that the molecule i s * composed of two spherical atoms, the minimum estimate of the cavity diameter i s twice the internucleur distance, that i s a = r . e On the other hand, i f the molecule i s approximately spherical, the minimum estimate of the cavity radius would be a < r . e Other possible estimates are the sum of the covalent radii, the sum of the ionic radii and the molecular volume defined by the c r i t i c a l density of the gas. These values are shown i n Table VIII... The values of the covalent and ionic radii are given by Pauling (19). It was found empirically (see figure 8), that the best value for hydrogen chloride i s a = 1.34 A, this corresponding closely to r + I 1 which i s the average internucleur distance of the molecule i n the second excited state., Since the 0—• 2 transition for both HCI and HBr are observed, this value has been used as the minimum, value of the radius of the cavity., It would be extremely interesting to know whether . or not higher overtones may be observed i n the solution, phase. 23 Such observations might enable one to f i x the maximum radius of the cavity. Table VI[II ™ cm x 10 8 r Covalent e Ionic CO 1.128 1.25 HCl 1.275 1.29 1.81 HBr 1.414- . 1*44 1.95 r e • 22 L 1.53 1.151 1.51 1.354 1.496 Intensity measurements by Dunham (7)., Penner and W ber (18) and e Dennison (6) permit estimates of the variation of the dipole moment with the internucleQr distance., These are ^HCl x. 1 0 ^ x IO 1 8 1.04- [ 1.06 f + (0 or + 2.28)f ] (7) a. , CO 1 8 .118 i ^ * HBr x 10 [ 3.55f + (.165 or 9.79)f [ + J (13) 1 8 .78 ± .30 § ] (6) The ambiguity i n the coefficients arises from the solution of a quadratic equation which i s obtained by comparing the relative intensity of the fundamental and the f i r s t overtone.. Dennison (6) had not measured the intensity ratio of the fundamental and the 1 no estimate of the quadratic term i s possible. for i s low.. s t overtone, therefore I t seems that his value 29 Hydrogen Chloride Molecule Substitution of the following values i n equation (28) gives the following equation for the frequency shift of the low frequency component i n the fundamental absorption, band of HCl. a = 1.34 x IO" cm 8 =• 2.63 x 10" + cm 2/ 3 = 1.04 x 10" esu cm 18 ^ / = 1.06 x 10"* esu cm 18 ~ (fo- 1) 10 ^ 2(1..0Z.)(1.06) * 10" (31>3-» 1Q" ) ... 1.275 • 10" 2 36 2.40(2^+.1) - 2.(2.63) (^- 1) 1.12.<. 10." (14.0 .10~ ) 36 + 19 1.63 • 10" 11 d (5.036 • lO ^) cm 1 -1 (28b) 16 where l/hc = 5.036 x lO-l^cm'^/erg.. Equation (28b) i s plotted i n Figure 8 for two values of a, i . e . .a =• 1„275 and a = 1.34. that the term containing I t i s apparent from the numerical values i n (28b) (I - I 1 1 ) controls the sign of the whole expression, therefore i t i s reasonable to infer that the positive root for »-^/ must be used.. That i s , the dipole moment increases with internucleqr separation. Furthermore, i t i s apparent that -^a. = 0 w i l l provide the best f i t with the experimental data. Figure 9 shows the variation, of 1.0 1-5 2.0 2.5 FIGURE 8. 3.0 Dielectric 3.5 Constant Dielectric FIGURE 9. .Co.nst.aAt 30 •A 4^with. G .. I t i s apparent from figures 8 and 10 that benzene does not 0 correspond at a l l with the other solvents. At present there i s no satisfactory explanation. The observed value of -67 cm" f o r ^ ^ i n CS2 i s 1 0 much too small. It i s quite possible that the observation i s quite incorrect since the band i s very weak and diffuse and i t i s badly obscured by atmospheric water vapour absorption. Hydrogen Bromide Molecule Figures 10 and 11 show £1/ and AV plotted against £ c for the values. a = 1^50 x 10" cm 8 °<e = M - l 3.60 x l O " ^ cm 2 3 = .78 x 1 0 " = .30 x 10 " e s u cm 18 esu cm. 18 In addition, curves are plotted for the empirical value = 1.39 x 1 0 " 18 esu cm.. This value was selected by f i t t i n g equation (28) to the observed shift for CCl^., In view of the fact that bromine l i e s lower on the electronegativity scale than chlorine, this value i s possibly too large. As; i n the case of hydrogen chloride, the positive value for ^ / must be selected unless sU x has a very large value. and HBr suggests that this i s unlikely. The similarity between HCI 3.0 FIGURE 10. Dielectric 3.5 Constant i—i 1 -i—i T—i—n—r r -i—i—i—r 1—r l 1 T 3 001 Observed and Shift of HBr in Predicted for V zo Solution E o CO o N>CP 20 0 > o to II = + 1.3 9 * 10 18 o TV 100 I o j .0 i i _l_ L 1.5 I I L J I 1 J L 2.5 2.0 FIGURE I J 3.0 L 1 3.5 Dielectric Constant 31 Carbon Monoxide Molecule Figure 12 shows, the expected shift i n the 0 —*• 1 transition for carbon monoxide for the values. a — 1,15 x IO" cm °<<r = 8 1.97 x 10-24 cm3 ^ = .118 f (3.55 § A*!/ f ) x 10" esu cm 6 5 + 9.79 18 ^ Since one root f o r - - * ^ i s very large, i t i s not possible to uniquely choose the sign for This i s illustrated i n able IX. T Table IX x to esu cm y 9 ( , • * • / # -*-<u>, x/0 39 l .118 +3.55f+ .165| + 6.0x10-39 + 24.2x10-39 0 x 10-39 30.2 2; .118-3..55f- .I65f" - 6.0 + 24.0 0 18.0 3 .118+3.551+ 9.79 f' +6.0 + 28.5 + .5 35.0 4 .118- 3..55f+ 9.-79 f -6.0 + 19.7 + .5 14.0 Since the four curves shown i n figure 12 are well above the observed values for A.V /e , i t i s impossible to assign the appropriate coefficients. Mulliken (16) has pointed out that the coefficient of f negative. i s probably This argument i s based on the assumption that carbon monoxide + - has three resonating structures (see Pauling (19))j a) C - 0, b) C = 0, 32 - _ + o c) C = 0 with equalibrium distances 1.25, 1.13 and 1.10 A. respectively. ; Structures a) and c) have large opposing moments. Mulliken suggests that the dipole moment reverses sign as the internuclear separation increases, since the stability of structure a) increases at the expense of structure c). This would suggest that curve (2) or (4) i n figure 12 i s the correct one. The large deviation from the observed value i s probably due to an. unreasonable assumption i n the radius of the cavity or that carbon monoxide cannot be treated as,ia spherical molecule. In order to compare equation (28) with Kirkwood's equation, A Y = - k £4-1 ( ) 2A a resume of the observed shifts for hydrogen chloride i n polar and polar solvents i s shown i n Table XI., non- In figure 13, the best straight line i s drawn through these points (exclusive of benzene). It i s apparent the equation w i l l not f i t liquids with high refractivities such as C^and Br . 2 A. value -for the radius of the cavity, a = .85 2 has been calculated from equation. (2b) and the straight line i n figure 13. nuclear distance for HCI i s 1.275 too small. Since the inter- this value must be rejected as being 33 Table X. Solvent n SIC1(-65°C) 4 2.5 .250 K1 (-65°C) 4.2 .340 CHG1 (-65°C) 3 HCI (-90°C) 7...0 8..S5 3 Observer 2 AV Raman West & Aurthur 2860 -26 Raman West & Aurthur 2828 -58 .400 Raman West & Aurthur 2826 -60 .415 Raman West & Aurthur 2800 -86 • S0 (-90°C.).' 22.4 .467 Raman West & Aurthur 2798 . -88 C H B r (-90°C) 25 14.0' .448 Raman West & Aurthur 2797 -89 CH. C0C1 (-90°C) 20.3 .465 Raman West & Aurthur 2804 -82 CC1,,(20°C) .220 I.R. West & Edwards 2833 -53 I-.R.- West & Edwards 2780 -106 2753 -133 2 3 4 HCI (-100°C) • *? • 2.14 • : C H (20°C) 2.29 .231 2.26 C H CI (20°C) 65 SnGl (20°C) 5.94 .384 2.26 I.R. Leberknight & Ord 2783 -103 3.2. .298 6 6 I.R. Leberknight & Ord I.R. Leberknight & Ord 2833 -53 CC1 (20°C) I.R. Leberknight & Ord 2833 -53 C.H (20°C) oo I.R. this work 2747 -139 CC1^(20°C) I.R. this work 2829 -57 SnCl^(20°C) I.R. this work 2826- -60 4 2.25 4 / C C1^(20°C) 2.46 .247 2.27 I.R. this work 2826 -60 SIC1 (20°C) 2.4 .242 2.00 I.R.. this work 2848 -38 .258 2.60 I.R. this work 2831 -55 2 4 TiCl^(20°C) — cs(20°c); 2.63 .263 2.66, I.R. this work 2798 -88 Br (20°C) 3.18 .296 2.76 this xrork 2787 -99 2 2 I.R. 34 Conclusion In.spite of the fact that the agreement between the observed and predicted shifts i s not conclusive, i t seems quite reasonable to conclude that the major cause of the shift in. the vibrational energy levels of the molecule i n solution i s electrostatic interaction. In the three cases studied, the average absolute differences between the observed and predicted shifts are 17$ and 31% for HCI and HBr respectively. For CO, the agreement i s within an order of magnitude.. The assumption that the electrostatic interaction arises through the reaction f i e l d appears to be justified. The assumption that the cavity size i s dependent only on the dissolved molecule i s not f u l l y justified. The failure of this assumption would provide the only explanation that i s consistent with this model for the erratic variation of the observed Shift with the dielectric constant. It should be pointed out that the values used for the radius of the cavity.are slightly too small. values used are For instance, the a = 1.34, 1.50 and 1.15 & for HCI, HBr and CO respectively, 3/ O but *V° < e = 1.38, 1.53 and 1.25 A for these molecules. In order to satisfy the Clausius-Mossotti relation which has been used to obtain equation (17), the following relation must hold a > This means the predicted shifts are too highj however, the contribution from the electronic interaction has been neglected. Cremer and Polanyi's work (5) indicates this effect would give contributions of the same sign. 3 5 The most serious objection to the electrostatic model proposed is that i t f a i l s to explain the presence of the high frequency component. At present, there i s no satisfactory•explanation for this phenomena f i r s t noted by West and confirmed i n this work for HCl. 36 . APPENDIX. I WAVE. FUNCTIONS OF THE ANHARMONIC OSCILLATOR WITH MORSE POTENTIAL The radial equation of the Schrodinger equation i s JR X R + J(J»0 + y± (E - Uc*))R=o (i) where o (2) In.order to solve this equation directly, the second term i n this equation, must be neglected, that i s , j =0. Setting R + £ . .+ g# \ £ - K y = e" ^ ( ~* e^ • * V T 7 ? < y ) «<> r (3) Terms: in.y" and y " are removed by setting 1 2 z = 2dy (4) B (5) =e ^ and adjusting the parameters d and b such that r 37 .2_ 2 2 _ = ^ D * () 6 ( ? ) Equation (3) becomes +. ( • / -3) f * U</-6-/J -0 (s) The solution of equation (8) w i l l be a f i n i t e polynomial i f (2d - b - l)/2 is. a positive integer, /if b a Setting 2d - 2v - 1 = b > 0 (9) defines a finite series of eigenvalues E J ^ - ^ ^ J J ^ t**!f) ~ T^rr^t.ergs do) where D* i s i n ergs, or the spectroscopic term values Qfo) = a UkA- (tr+'/J - ^t$ X (v*of- cm-i (11) 33 where D i s i n cm".. The coefficient of (v + l/2) i s 1 e 0 = 4>« [*2h+. , therefore cm" (12) 1 Furthermore, from equation. (6), we obtain. - 2cJ = 1 X (13) -4$ I f (2d - b - l)/2: i s a positive integer v, then the polynomial (14.) satisfies equation. (8) where r < r *M - (v+hXv*b-i) (v+b-(v-Jt-/)) The radial eigen-functions are where z; = 2de~ # ( " e ) . r r The normalisation constant i s defined by (15) 39 Substituting equation. ( 1 6 ) i n (17) ve obtain •l J L b-t r b 1* (18) Using the relation (19) and equation (I4) and interchanging the order of summation, and integration, equation (18) may be put i n the following forms Integration of (20) by parts •f |3 v times, we obtain .a. b v! (21) 4,0 APPENDIX II In order to evaluate equation (28), the following integrals must be solved; I fdr * 1 K i (1) n. = 1, 2, 3, 4- and v = 0, 1, 2 where (2a) b f /^W-l/ (2b) z = ke- & (r - r ) b = k - 2v - 1 $ I For n; = 2, v (2c) e ir/ */)J b > 0 (2d) (2e) = 2 0Ae (3) 41 In order to perform the integration, the range of integration, must be extended from. 0 ^ z < ke^ e to 0 * z<°o r s This i s not f u l l y justified since the Morse potential function i s very large, but finite at r = 0, i.e. the range z = ke & e . Dunham (7) has shown the contribution i n r ke P e * z<«*» , i..e. -o° < r i r 0 i s negligible for the integral For the integral I , the error incurred i n extending the limit of integration, i s , s= \ ^ (C* ®) U. M*k)J} r ; (4a) Re 3'- but z » l 8 * k- > b > bu v! 1 -lr V " ' air H. (4b) (4c) 5 (A-av-/)v/ tktrt-3. 42 for a l l n; such that z' > k + n 1 ^ k^n.-2 < (k +• n. - 2) (k + n. - 3) < -2 ^ Z £< (k-2*r-l)v! f-i tfy**~*th+ -i,\ r(h-*)p* { ' ** , I At g' / /-3' (fe-g«r-/)<r/ / » g r(k-ir)(Z \ 1 e , s (5b) \ K For HCl, ko-57, £ = 1.7 x 10 For n. = 4- , the fractional error w i l l be v = 2 and (5a) n s ,r = 1.8 x IO" therefore z* = 522. 8 e -^-aa. The approximation i s certainly justified for the values of v and n used here. The integrals may now be evaluated with the aid of the following identity J^Hz) _ 41 / e V ' ' J « = f^-'^uJu (6) 43 Equation: (3) can be expressed as a sum of terms i n /(h). and These functions may be evaluated numerically by using the di-gamma function f a ) and i t s derivatives i n the following relations: ra) r'h)= y^-/) • yf(b-i) ra) rib) rj6)= r(i>) v ' u - i ) * ir'h-dtti-oWa-/)f> w*(b->) Tabulated values of TfV*)are given i n Jahnke and Emde ( l l ) as i?ell as the expansion. Since this series i s uniformly convergent for x > l , i t may be differentiated term by term to obtain values for equations (7). For the values used ( x > 50 ), the convergence of the series i s sufficiently rapid that only the f i r s t three or four terms are required. The integrals i n equation (l) cannot be expressed i n a- convenient closed form for n^-3 since the recurrence relations for the- second and third derivatives of the di-gamma function do not lead to any simplification i n the expression. and ( r - r However, the integrals containing ( r - r e ) ) can be evaluated i n the general case by a method suggested 2 e by Buckmaster (4)* The method i s as follows. For n = 1, equation (l) becomes 44 oo A = -~ 6 Wl^yMkfe? <? }Uln 0 P(AT+bH) (B - In k«C) ^ (9a) (9b) where b tr! J B= y 'ui. (»t<£~) 6 (10) (n) Using the relations 4 we obtain B = i»2.(-rW^? -D (12) where D = 2 ^tirJ* > )J y (13a) 45 Equation (13) must be treated in. two separate cases, for ^ = 0 and for 0, For-4* = 0, (13b) Integration by parts v times gives vhere i-l Integration of (I4) gives > 0, For Integrating (13c) by parts J D = i-t)~'U-l) ! E where - 1 times, note that b > v , we obtain (16) JS. (17) ~_ Integrating (17) by parts (y -? 1. + l ) times, we have (18) Substituting (18) i n (16), we have (19) Combining (19) and (.15) and then substituting i n equation. (12) gives (20) Therefore (21) w The solution of the integral for n. = similar manner. The result i s 2 i s obtained i n a 47 rib*) «=/ ' <=/ J_ These results have been checked for the special cases, (22) v = ° 0 , 1, 2. Numerical values for these integrals are tabulated i n Appendix III, 48 APPENDIX III The spectroscopic constants U ) ^ t ^ e ^ t t ^ * / * r t a n d a r e e taken from a compilation of spectroscopic data for diatomic molecules by Herzberg (9). The value of the potential minimum has been calculated from the formula De = 4u*x (1). - - c This formula i s based on the assumption.that only the f i r s t anharmonicity coefficient, ^ ) « ^ e i s different from zero. Actually, the ^second coefficient, <">e He i s also different from zero, - 10 « 3 I f the cubic term i s included i n the term formula, we obtain. Z_ 3 then For HCI, the second term increases D by about 3%, On the other hand, & D e = D 0 + G(0) where D i s the dissociation, energy and the zero point energy i s Q (4) 49 G(6) = - +- ^ J / r 5. -9- Herzberg gives values of. D D (H ), D ( 2)> o ( 2 ) C1 0 2 D B r a n d 0 Q - - - (5) ? for HCI and HBr which are calculated from heats of dissociation of HBr and HCI. values from (4) are about 15$ lower than the values from (l)., These Equation (l) was used since a parabolic term formula very accurately represents the f i r s t three vibrational states which we are dealing with. The values @ Q = /.in where *>J and D e e and 7 £J k e are I d±E can" (6) 1 are i n wave number units and i s i n atomic units k = +nt / zsc z> () g where D* i s i n ergs, and e 7 i s i n grams, The values for the integrals shown i n Appendix II- have been calculated from the above values of @ and k.. Since the evaluation, involves the calculation of small differences between large terms, the value of k has. been treated as exact.. This i s permissible since the terms contain only functions of (k - j) where J - 0, 1, 2, 3, 4. The number of significant figures tabulated i s consistent with the number of significant figures given for £<)? • 50 NUMERICAL VALUES HCI.. ^J. atomic units p 1 2939.74 2649.67 52.05 48.21 cm-1 cm" 1 1 0 4 cm" x.10" 1 8 k. I cm. x 1 0 1 00) 1 1 CO 6.35841 2170.21 13.461 .0308 .056 « • D cm"' x 10" e .99558 .979389 cm" t+Y* HBr 1.27460 I.4138 1.1231 4;293 3.882 3.7471 1.739 1.634 2.3400 57.43 58.60 15.22 15.87 161.20 3.994 , 1^ cm x 1 0 1 1 46.53 48.50 12.064 I cm x 1 0 1 1 79.21 82.53 20.254 6.14 6.81 1.160 I 1 22 cm x 1 0 2 2 oo 1 9 I ^ 2 cm- x: 1 0 19 20.11 22.87 3.590 I 2 cm x: 1 0 19 36.82 40.69 6.193 I 2 2 22. cm x 1 0 3 3 oo 2 7 l3 m3 x 1 0 11 14 io35 oo 2 7 C • — — .017 . — — .027 .000 x 1^ cm x 1 0 4 3 5 — .000 51 BIBLIOGRAPHY Bauer, E. and Magat, M., Jour, de Phys. et Rad. VII 2, 319 (1938) Bottcher, G., Physica 6, 59 (1939) Breit, G. and Salant, 0., Phys. Rev. 1 6 , 871 (1930) Buckmaster, H., Can. Journal of Physics 1 0 , 3 H (1952) Cremer, E. and Pblanyi. M., Z e i t s . f phys. CheMe Bodenstein Festband 770 (1931* Dennison, D.M., P h i l . Mag. 1, 195 (1926) Dunham, J.L.., Phys. Rev. 2£, 438 (1929) Frood, D.G., M.A. Thesis, University of B r i t i s h Columbia (1951) Herzberg, G. Spectra of Diatomic Molecules (D. van Nostrand, 1950) second e d i t i o n p 502 f f Herzberg, G. Infrared and Raman Spectra of Polyatomic Molecules (D. van Nostrand, 1949) p 535 Jahnke, E. and Emde, F., Tables of Functions (Dover Publications (1945) 4th e d i t i o n - • Leberknight, C.E. and Ord, J.A., Phys. Rev. j>l, 430 (1937) McLennan, J.C. and McLeod, J.H., Trans. Roy. Soc. I l l , 22, 413 (1928), 22, 19 (1929) McLennan, J.C. and McLeod, J.H., Nature 123. 160 (1929) Morse, P.M., Phys. Rev. 1 4 , 57 (1929) Mulliken, R.S., J.C.P. 2, 400 (1934) o Onsager, L., J . Am. Chem. S . £8, i486 (1936) o c Penner, S.S. and Weber, D., J.C.P. l g , 807 (1951) Pauling, L., Nature of the Chemical Bond (Cornell University Press, 1948) P l y l e r , E.K. and Peters, C.W., J . Nat. Bur. Standards, 45, 462 (1950) P l y l e r , E.K-. and Williams, D., Phys. Rev. A9., 215 (1936 52 Ross, W.L. and L i t t l e , D.E., J.O.S.A. ^1, 1006 (1951) t van Vleck, J.H.., Theory of Magnetic and Electric Susceptibilities (Clarendon Press, 1932) p 45ff West, W., J.C..P. 2, 795 (1939) West, W. and Aurthur, P., J.C.P. 2, 215 (1934) West, W. and Edwards, R.T., J.C.P. j>, 14, (1937)
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Investigation of the liquid vapour shift in the infra-red absorption frequency of diatomic molecules Ross, William LeBreton 1953
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Title | Investigation of the liquid vapour shift in the infra-red absorption frequency of diatomic molecules |
Creator |
Ross, William LeBreton |
Publisher | University of British Columbia |
Date Issued | 1953 |
Description | The changes in the infra-red absorption frequency of three diatomic molecules, HCl, HBr and CO, on the transition between the gaseous and the solution phase have been investigated experimentally. The magnitude of this shift has been predicted using a purely electrostatic interaction. The experimental values and predicted values for HCl agree within 17%. The experimental and predicted values for HBr and CO agree within an order of magnitude. The investigation provides a method for independently checking the variation of the dipole moment of these molecules with internuclear distance against values previously determined by infra-red intensity measurements. |
Subject |
Diatoms |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-02-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085483 |
URI | http://hdl.handle.net/2429/40802 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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