Structural and Quantum-Diffusion Studies by Steven A ... · rotational specific heat capacity of H2 gas (Hund 1927 and ... it is generally taught that homonuclear diatomic molecules

Structural and Quantum-Diffusion Studies in the Near Infrared Spectra of

Solid Hydrogen

by

Steven A. Boggs

A thesis submitted in conformity with the requirements for the degree of

Doctor of Philosophy in the University of Toronto

September, 1972

© Steven A. Boggs 19 72

-Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

TA B LE OF CONTENTS

ABSTRACTPage No.

CHAPTER I - INTRODUCTION

CHAPTER II - EXPERIMENTAL APPARATUS AND METHODS . . 13

CHAPTER Ill-

Optical and Electro-Optical Components Electronics and Signal ProcessingOptical Cryostats ..............Spectrometer Frequency Calibration Reduction of Results .........

THE FUNDAMENTAL AND FIRST OVERTONE OF ORTHO-ENRICHED SOLID H2 ..

SPECTRA

Experimental Detail.. Results and Discussion

CHAPTER IV -

The QiRegions ..............The SjJO) Group..............The S j (1) Group..............The Qj (1) + S^l) Region

THE FIRST AND SECOND OVERTONE BANDS AND NEARLY PURE PARAHYDROGEN

The First Overtone Band-

is

The q 2 (0) Region ......... 72The Q i + Q i Region......... 76The Q 2 (0) + S0 (0) Region * • 79The Qi (0) + Si (0) Band • • 81The Si (0) + Si (0) Doublet • « 83

OF PURE

1516 18 23 25

27

273435 52 59 65

71

72

I

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Page No.

CHAPTER IV - (Cont'd)The Second Overtone B a n d .............. 85

The Q 2 (0) + Qi (0) Line ......... 85The S 2 (0) + Qi (0) and

Q 2 (0) + Si (0) Lines ......... 85The S i (0) + S2 (0) Feature......... 88

CHAPTER V - INFRARED STUDIES OF QUANTUM DIFFUSIONIN NEARLY PURE PARAHYDROGEN.......... 91Experimental Detail .................. 82

The Spectrometer.................. 92Temperature Control ............. 93Sample Preparation................. 94

Experimental Results and Discussion .. 96The t = 0 Absorption Coefficient .. 102The Growth Factor.................. 104The Rate Constant.................. 108

APPENDIX I - QUANTUM DIFFUSION...................... 115Conceptual basis of the microscopic theory due to Oyarzun andVan Kranendonk ................... * .. 115Calculation and tabulation of the growth factor as a function of temperature and impurity concentration ............. 119

APPENDIX II- Reprint: A MICROAMPERE CURRENT SOURCEWITH A MODULATION CAPACITY AND 200 V COMPLIANCE ........................... 126

APPENDIX III- SCHEMATIC DIAGRAMS ............... 12 7Lock-in reference source.............. 128Heater power amplifier................. 129

BIBLIOGRAPHY ....................................... 130

ACKNOWLEDGEMENTS ................................... 133


ABSTRACT

Experimental data on the infrared spectra of the fundamental and first overtone bands of nearly pure solid orthohydrogen, of the fundamental, first, and second overtone bands of nearly pure solid parahydrogen, are applied to analyses of the energy levels of solid hydrogen and quantum diffusion.

Spectra of the fundamental band of solid hydrogen with orthohydrogen concentrations in the range 75% to 99% orthohydrogen were recorded at 1.05 K and ^ 5 K, i.e., above and below the order-disorder phase transition. The zero-phonon features are interpreted in terms of travelling vibrational, rotational, and orientational excitations (vibrons, rotons, and librons). The Q branches (fundamental and overtone) for ^ 100% o-H2 in the ordered state show a structured side-band, shifted by ^ 6 - 26 cm-1 from the appropriate vibrational transition frequency and due to one- and two-libron excitations; in the disordered state, this becomes a Boltzmann-modified band of half-width ^ 12 cm- 1, peaked at the vibrational frequency, and due to predominantly low-energy orientational transitions of interacting o-H2 molecules. The Si(0) group of maxima in the ordered state is interpreted as the superposition of the transitions, Qi (1) + S0(O) and Qi (1) + S0 (0) + libron, where Qi (1) is an o-H2 vibron and S 0(O) is the localized rotational transition


of an impurity p-H2 molecule, the J = 2 level of which is split into three sublevels by the field of the Pa3 crystal structure.

Spectra of the first and second overtone bands of pure and nearly pure solid p-H2 were recorded. A number of features were observed for the first time and are interpreted in terms of the theory of Van Kranendonk. In particular, this work marks the first investigation of the second overtone region where four zero-phonon features were observed, all due to double transitions.

The fundamental band of almost pure p-H2 solid was investigated for evidence of the diffusion of o-H2 impurity molecules. The absorption features in this band which are due to interacting pairs of o-H2 molecules was observed to increase as a function of time. This growth is characterized by a rate constant which has been measured as a function of temperature in the range 1.15 K to 2.10 K and concentration in the range 0.7% to 1.6% o-H2 impurity. The value and concentration dependence of the rate constant are found to be in generally good agreement with the theory of Oyarzun and Van Kranendonk. Measurements have also been made which yield the ratio of the random number of impurity pairs to the equilibrium number as a function of temperature and concentration. These data are compared with a statistical model of the diffusion process.


- 1 -

CHAPTER I

INTRODUCTION

Hydrogen has been the proving ground for many of the basic concepts of quantum physics. Early interest in the hydrogen molecule centered around the application of nuclear spin statistics which predict the existence of ortho and para modifications (Heisenberg 1927). On this basis, explanations were developed for intensity alternation in the rotational spectrum of molecular hydrogen as well as the rotational specific heat capacity of H2 gas (Hund 1927 and Dennison 1927). McLennan and McLeod (1929) , working with the Raman spectrum of liquid hydrogen, documented the stability of the ortho and para species under nonequilibrium conditions. The anomalies in the specific heat of solid hydrogen, first noted by Simon, Mendelssohn, and Ruhemann (1930), were of considerable interest and Pauling (1930) predicted a second order phase transition due to rotational

=j| ordering in the low temperature solid. Free moleculariH rotation, which distinguishes solid hydrogen and its isotopesH from the solid phase of all other diatomic molecules, had

already been observed in the liquid (McLennan and McLeod t 1929) and was later observed in the solid (Allin, Hare, and

MacDonald 1955). The specific heat anomaly in solid hydrogen

■ ' §

i'S E355


was examined more closely by Hill and Ricketson (1954). Hatton and Rollin (1949), working with NMR, were the first to note a hysteresis effect in the temperature variation of the anomaly. This is, in retrospect, the first evidence of a crystal structure change accompanying the second order phase transition. The work of Clouter and Gush (1965) on the infrared spectrum of normal hydrogen above and below the temperature of the specific heat anomaly led to the discovery of a change in crystal structure accompanying the second order phase transition. This change was later identified as being from h.c.p. to f.c.c. (Mills, Schuch and Depatie 1966). The ordered (f.c.c.) phase of solid H2 supports angular momentum waves which have been termed librons. Evidence of these was first observed by Clouter (1968) in the infrared spectrum of 92% o-H2 solid; his results were explained and published by Homma and Matsuda (1968). The most recent phenomenon to be discovered in solid H2 is that of molecular motion of ortho molecules through a lattice of nearly pure p-H2 solid at a rate many times too fast to be explained by classical molecular diffusion. This phenomenon, termed quantum diffusion, was discovered by Amstutz, Thompson, and Meyer (1968) during the examination of the NMR spectrum of o-H2 pairs in p-H2 solid and is due to resonant ortho-oara conversion.


- 3 -

The discussion of the infrared spectrum of solid hydrogen, various aspects of which are examined in this thesis, must seem a bit strange to the uninitiated, since it is generally taught that homonuclear diatomic molecules have no infrared spectrum. However, if one observes a collection of such molecules in gas, liquid, or solid phase, a weak infrared spectrum is present. This spectrum arises from collision-induced dipole moments which, although transitory in nature, are sufficient to provide a rich spectrum. Figure 1.1 offers examples of the fundamental vibrational band of 92% o-H2 solid (5 K) and below (2.1 K) the h.c.p. -*■ f.c.c. phase transition and will serve to illustrate the types of transitions present in solid hydrogen.It is immediately evident that most of the absorption is in the broad features labelled with the subscript "R". These features which involve molecular transitions accompanied by the creation of one or more phonons, will not be considered in this thesis. The remaining sharp and often structured components are the so-called zero-phonon features which do not involve the creation of phonons. As evident from Fig. 1.1, these features come only in the form of Q(AJ=0) and S (AJ=2) transitions as expected from the AJ = 0, ±2 rotational selection rule required by the symmetry of a homonuclear diatomic molecule. There are two types of transitions evident in the fundamental vibrational band, viz.,


</U7|


if) (/)

a a

( iJ ^ 3) 1N3I0LIJ303 NOIldHOSaV


4250

45

00

4750

5

00

0FREQUENCY

(cm-1)

- 5 -

.SfiST’I■ft

single transitions in ortho [Sjd)] and para [S x (0) ] molecules, where all changes in quantum number occur on one molecule, and double transitions such as Qi (1) + Sg(0) or Ql{1) + S o(1) where a vibrational (Qx) transition on one molecule is accompanied by a rotational (S0) transition in a neighboring molecule. Of the two types, the double transitions are generally much the stronger since the near inversion symmetry in the lattice sites of the h.c.p. structure or true inversion symmetry of the f.c.c. structure does not effect any cancellation in the induced moments such as occurs for single transitions. In the low temperature (f.c.c.) phase, single transitions such as Sx(l) are very weak and occur only because the "impurity" para molecules break the inversion symmetry. There is a great deal of structure in the spectrum which cannot be seen on a diagram with so coarse a scale as that of Fig. 1.1. Much of this fine structure has been neither fully explored nor fully explained.

The theory of collision-induced absorption in homonuclear diatomic gases has been treated by Van Kranendonk (1957, 1958) in terms of the "exp-4" model for the induced dipole moment. This model assumes the induced moment can be divided into two parts: a short range part, ui, of the form £exp(-R/p), where R is the intermolecular distance,


- 6 -

£ the strength, and p the range, and a long range part, \iz, which varies as R-lf. Both and U2 are modulated by the vibrational motion of the interacting molecules and therefore lead to vibrational transitions. The short range term, ui, is primarily due to isotropic electron overlap interaction and is not, therefore, modulated by the rotational motion of the molecules. The longer range term, u2/ mainly due to highly anisotropic electric quadrupoler induction and can cause rotational transitions. The exp-4 model has been successfully carried over to solid hydrogen (Van Kranendonk 1959, 1960) where it is found that essentially all the absorption in the zero-phonon features is due toU 2 •

Experimental work on the infrared spectrum of solid hydrogen has progressed through several levels of sophistication, especially with respect to spectral resolution. The earliest work on the infrared spectrum of the solid (Allin, Hare, and McDonald 19 55) was at relatively low resolution ( 15 cm-1) and brought out only the basic features of the spectrum, viz., the regions of the vibrational, vibrational- rotational (single) and vibrational + rotational (double) transitions. The spectrum at various para concentrations between 25% and 100% p-H2 was also examined at this resolution (Hare, Allin, and Welsh 1955) and helped secure the assignment of features to the various possible molecular


- 7 -

excitations. The examination of the spectrum at * 2 cm-1 resolution (Gush, Hare, Allin, and Welsh 1957) confirmed the difference in frequency between vibrational transitions in ortho and para molecules which had already been observed by Allin, Feldman, and Welsh (1956) in the Raman effect.Work is still proceeding at the next level of resolution (0.5 cm-1) where a great deal of structure not evident at 2 cm-1 can be found. In this resolution range, detailed studies of the fundamental band of nearly pure p-H2 solid were carried out by Gush, Hare, Allin, and Welsh (1960) as well as preliminary, for the most part unpublished, studies of the first overtone region (Gush, Hare, and Welsh 1958).

Theoretical work on pure and nearly pure p-H2 solid has paralleled the experimental work. The earliest work (Van Kranendonk 1959) laid the foundation for the understanding of the gross features of the spectrum, the vibrational and rotational energy bands. As the experimental work revealed finer details of the spectrum, the theoretical work followed with more detailed considerations of the interactions within the solid necessary to explain the observed phenomena (Van Kranendonk 1960, Sears and Van Kranendonk 1964) . This progress in experimental and theory culminated in the review article by Van Kranendonk and Karl (1968) which presented a frequency analysis of p-H2 solid, consistent to within ^ 0.1 cm- 1, based on then current understanding of


- 8 -

interactions in the solid as revealed by infrared and Raman effect measurements (Soots, Allin, and Welsh 1965). This analysis has been updated in the light of more recent investigations by Noolandi (1970). Although the gross and much of the fine structure of the infrared and Raman spectra of p-H2 solid is now understood, there is still fine detail which defies the attack of the theoretician and features which beg experimental investigation. Along these lines, a study of the first and second overtones of p-H2 solid was undertaken, the results of which are presented in Chapter IV. There is every reason to believe that future investigation at resolution of ^ 0.01 cm-1 with tunable lasers will lead to a completely new realm of experimental and theoretical investigation.

Before 1958, all work was carried out on H2 samples with o - H 2 concentrations < 75%, since no method had been found to separate orthohydrogen from normal hydrogen. With the basis for a separation method provided by Cunningham, Chapin, and Johnston (1958) and the ingenious apparatus design of Depatie and Mills (1968), it became possible to attain o-H2 concentrations in excess of 99% o-H2. As already mentioned, a band observed by Clouter (1968) in the infrared spectrum of 92% o-H2 which was interpreted by Homma and Matsuda (1968) gave the first experimental evidence of librons. Silvera, Hardy, and McTague (1969) observed more


- 9 -

'-tJvs.M

direct evidence of librons with Raman effect experiments. They found four distinct features displaced between 8 and 30 cm-1 from the exciting line which they attributed to fe = 0 libron modes. This interpretation was later revised (Hardy, Silvera, and McTague 1971) to attribute the highest frequency feature to two-libron processes. Other features involving rotational and vibrational transitions in the infrared spectrum of ortho-enriched hydrogen were reported by Clouter (1968) and Clouter, Gush, and Welsh (1970) and in the Raman spectrum by Silvera, Hardy, and McTague (1969).

Theoretical work on the nature of o-H2 solid below the h.c.p. -*■ f.c.c. phase transition began in earnest after the discovery of the phase transition by Clouter and Gush (1965). The work of James and Raich (1967) led to the realization that the phase transition is caused by the electric quadrupole-quadrupole (EQQ) interaction which, at lower temperatures, is slightly smaller in the f.c.c. structure with long range orientational ordering of the molecular angular momenta than it is in the h.c.p. structure with random orientations. The long range ordering of the angular momenta raised the possibility of angular momentum waves within the solid. This possibility was investigated by Homma, Okada, and Matsuda (1966, 1967), Mertens, Biem, and Hahn (1968) , Ueyama and Matsubara (1967), and Raich and Etters (1968) and stimulated the search for evidence of such


- 1 0 -

excitations which ended with their discovery by Clouter (1968) and Silvera, Hardy, and McTague (1969).

The large change in the infrared spectrum of solid hydrogen between normal hydrogen and the 92% o-H2 sample investigated by Clouter (1968) motivated interest in examining much purer samples. The first experiments undertaken in the course of the work which is reported in this thesis examined the fundamental vibrational band of hydrogen samples with 75% to 99% orthohydrogen, as well as the first overtone region of 99% orthohydrogen. The results of this work, which are reported in Chapter III, have clarified several previously puzzling aspects of the infrared and Raman spectra.

The fifth chapter reports on an investigation of quantum diffusion, a phenomenon which was described in the first paragraph. Already mentioned, the effect was discovered by Amstutz, Thompson and Meyer (1968) during their study of the NMR spectrum of an o-H2 impurity in p-H2 solid. They were able to identify lines due to pairs of o-H2 molecules and a line due to "isolated" o-H2 molecules. In the course of their work, they noted that a sudden change in temperature was followed by a slow change in the intensity of the lines due to single and pairs of o-H2 molecules. They concluded that their observations could only be explained by motion of


-li

the o-H2 impurities through the lattice. They published fairly rough determinations of the rate constant (inverse time constant) at two concentrations and several temperatures between 0.5 K and 4 K. The rate constant and its dependence on temperature and o-H2 concentration is the only handle one has on the dynamics of the quantum diffusion process and accurate determinations of these quantities are essential in establishing a firm understanding of the phenomenon. To date no such understanding has been established, although progress has been made through the theoretical work of Oyarzun and Van Kranendonk (1971, 1972). They have established that the process by which the diffusion takes place is not intermolecular interchange as suggested by Amstutz e-t at. (1968) but rather one of resonant conversion in which a para molecule converts to an ortho simultaneously with a neighboring ortho molecule becoming a para.

Preliminary experiments were undertaken to examine the feasibility of applying either Raman or infrared spectroscopy to the investigation of quantum diffusion. The use of Raman effect proved impractical (Prior 1971); however, experiments using infrared spectroscopy to measure the growth of features in the spectrum which are known to be due to pairs of o-H2 molecules (Gush e-t at. 1960) were encouraging. These experiments were pursued to the point that determinations of the rate constant could be made with


significantly smaller error than those published by Amstutz o.t a.t. (1968) , although over a smaller temperature range. The details of these measurements and the comparison of the results with the current theoretical understanding of the phenomenon are presented in Chapter V.


CHAPTER II

EXPERIMENTAL APPARATUS AND METHODS

The optical arrangement used in these experiments, shown in Fig. 2.1, is an example of a single-beam absorption spectrometer, the basic components of which are the source, of broad band radiation, the monochromator, and the detector, D. The most noteworthy aspect of the present optical arrangement is that the monochromator is between the source and sample (located in the optical cryostat) rather than between the sample and detector as is more usually the case. This is necessary in order to minimize heat input to the sample from the radiation source, which can be a serious problem when temperatures of 1 K are sought. With the setup of Fig. 2.1, when nearly total absorption occurred with the monochromator slits set at values typical of those used in practice, the heating of the sample as a whole was much less than 0.0025 K with a sample temperature of 1.15 K, i.e., there was no detectable change in the temperature of the solid hydrogen sample as measured by a germanium resistance thermometer (sensitivity of ^ 0.4 Mfi/K at 1.15 K) frozen into the sample.

:rn


-14-

Fig. 2.1: Optical Arrangement

S3 Incandescent source S2 Neon Calibration source M x Source selecting mirror M2 Spherical mirror C Chopper L LensM 3 Spherical mirror D PbS detector in liquid nitrogen

cryostat.



- 1 5 -

Optical and Electro-Optical Components

Radiation enters the monochromator from either of two sources, a neon lamp (S2) for wavelength calibration purposes or a 600-W incandescent lamp (General Electric model FFJ) run at about 450-W from a regulated power supply. The mirror Mj can be rotated to select the desired source. The monochromator used was a Perkin-Elmer 12G with a 600 £/mm Bausch and Lomb grating blazed at 48° 36', so that the fundamental band of hydrogen, from 2.5 u to 2 u, is very near the blaze angle in the first order of the grating. Filters used to select the order of interest are contained in the monochromator box. By appropriate selection of filters, one can study the first overtone of hydrogen in the second order of the grating, or the second overtone in third order, in all cases with the grating near its blaze angle.

It was necessary to use two radiation detectors since the cooled PbS cells used for infrared radiation were not sufficiently sensitive to the visible to pick up the neon calibration lines. The second detector is a room temperature CdS cell of unknown manufacture. A mirror coupled to Mi and located in the monochromator box is used to switch between detectors. Two types of optical cryostats were used and these will be detailed in a later section.


-16-

Electronics and Signal Processing

The major improvements to the spectrometer undertaken in the course of this work were in the area of electronics.A PbS photoresistive cell cooled to liquid N2 temperature offers the best detector for use in the 2.5 u to 1.15 u region. Ideally, such a cell should be used with a constant current source and a preamplifier with infinite input impedance, so that as the resistance of the PbS cell changes, the voltage across the cell changes (due to the constant current) and this is amplified by the preamplifier which draws no current itself due to its infinite input impedance. Such a system is normally used with radiation source chopping and lock-in amplification so that the signal is changed from nearly d.c. to about a 1 Hz bandwith around the chopping frequency. A lock-in amplifier is essentially a synchronous rectifier which operates at the chopping frequnecy and has the effect of a highly tuned filter with about a 1 Hz bandwidth around the chopping frequency. Since the electrical noise generated by most components is frequency- dependent, increasing greatly near d.c., it is advisable to operate at as high a chopping frequency as possible.The upper limit is determined by the time constant of the PbS cell.

Work on the fundamental band of ortho-enriched solid

JMs®■m


-17-

hydrogen (Ch. Ill) was carried out with the electronics as inherited from the previous user. These included a Kodak Type P-2 liquid-nitrogen cooled PbS detector with a dark resistance of ^ 106 ohms used in conjunction with a current-determining load resistor and preamplifier each with impedances of ^ 2 * 106 ohms. The Perkin-Elmer lock- in electronics used included a 13 Hz chopper that drives a mechanical switching system which effects the synchronous rectification. This system was designed for use with thermocouples where the long time constant requires a low chopping frequency. The switching system was very noisy and the combination of small load resistor and low preamplifier input resistance wasted a great deal of signal.

Before the work on the o-H2 overtone was undertaken, the spectrometer electronics were completely replaced.The Kodak PbS cell was replaced with an Infrared Industries cell of higher dark resistance 8 * 107 ohms). A 2 uAcurrent source with compliance to 200 volts was designed and constructed (Munnings and Boggs 1970). A schematic is shown in the reprint in Appendix II. With such a current source, the high dark resistance of the PbS cell is of no concern. A new lock-in amplifier was also constructed. The unit is capable of working in the range 30 to 400 Hz and contained a preamplifier with an input resistance of ^ 108 ohms. Because of coupling between the

J


preamplifier and the internal oscillator of the lock-in

to a separate box where it was powered by batteries. The lock-in amplifier as delivered was not ideally suited for spectroscopy since it contained no input for a reference signal from the chopper. However, it was not too difficult to lock the internal oscillator of the lock-in amplifier toa reference signal. The schematic for the reference detector is shown in Appendix III. The optimum chopping frequency for the Infrared Industries PbS detector was determined to be ^ 90 Hz, which is the same frequency as that used by the manufacturer when testing the unit.

The combination of changes described above greatly increased both the quality and quantity of signal so that it was possible to work with much narrower slits than with the old system. For example, the preliminary work on the 0-H2 overtone was done with the old system and the best resolution attained was 'v 4 cm-1. With the new system, resolution of from 0.6 to 0.8 cm-1 was attained.

Optical Cryostats

Three different optical cryostats of two types were used in the course of these experiments. The type used for most of the experiment is shown in Fig. 2.2. Since two separate dewars A and B were used, the radiation must

amplifier, it was necessary to remove the preamplifier board

-m


\ \

- 1 9 -

Fig. 2.2: Optical Cryostat I

Liquid helium dewar Liquid nitrogen dewar Metal jackets Liquid nitrogen liquid helium Sample cell Pyrex windows


ABCDE


- 2 0 -

travel through eight windows (excluding the absorption cell) in traversing the cryostat. Two metal jackets C, attached to the upper section of the helium dewar A, are filled with liquid nitrogen so that the dewar is effectively surrounded with liquid nitrogen without introducing any liquid nitrogen into the light path. The major problem with this arrangement is that the metal jackets are thin and have little volume so that they require frequent filling. They are also difficult to fabricate and have a definite tendency to leak. A second dewar of the same type was constructed where the metal jackets are replaced by a concentric glass cylinder which is fused to the outer vacuum wall of the helium dewar just above the windows. This provides a liquid nitrogen space of much larger volume which requires less frequent filling and cannot leak. The cryostats described above were used when measurements were to be made at temperatures below the X-point of helium where the sample could be immersed in the superfluid which, of course, does not boil.

For experiments above the X-point, a second type of dewar was constructed (Legge, Boggs, and Timsit 1972).This dewar, shown in Fig. 2.3, has five concentric walls, is in one piece, and Dresents only four windows to the light path. It is, however, extremely difficult to fabricate and has considerably greater helium boil-off than the first type. Chamber C 2 forms the liquid nitrogen jacket.


M U

- 2 1 -

Fig. 2.3: Optical Cryostat IIIA Liquid helium transfer hole B Hole between Ci and inner

chamberCj Outer liquid helium chamber C 2 Liquid nitrogen chamberD Liquid helium


r


- 2 2 -

Liquid helium is transferred into chamber Cx through A.When Ci is full, the liquid helium spills into the central chamber through hole B. For use above the X-point, the central chamber is filled to the base of the pedestal and liquid helium is present in all regions indicated by D.The cell, which sits on the pedestal, is then just above the boiling helium and is nearly surrounded by the liquid in Ci, and yet there is no boiling helium in the light path. The boil-off of helium from the central chamber is slow because of the helium in Ci; the boil-off from Ci presents no problem since this chamber can be filled either continuously or intermittently without affecting the sample in the central region. For work below 2.17 K, both the central chamber and Ci can be filled and pumped to give temperatures below the X-point.

The major problem with the design shown in Fig. 2.3 is that the room-temperature outer windows face directly into the low-temperature inner windows. The outer windows each radiate ^ 4.6 * 10-2 W/cm2 with the peak of the black body curve at about 10 u so that essentially all the radiation from the outer windows which reaches the inner windows is absorbed and contributes to the helium boil-off. From the size of the outer windows (2 cm) and the geometry of the dewar, one can estimate that about 0.45 W from the outer windows is striking the inner windows, causing the boil-off of about 600 cm3/hr of liquid helium. A good


- 2 3 -

pair of silvered dewars with intermediate 77 K windows boil off about 150 cm3/hr, so that the heat input from the 300 K windows in the single-piece dewar has greatly increased the liquid helium boil-off. These approximate calculations seem to agree with experimental observations, since, where a good pair of dewars with a liquid helium volume of ^ 3 t will hold liquid helium 15 to 20 hr, a single-piece dewar of similar volume will hold only 3 to 5 hr. However, the single-piece dewar has definite advantages. It provides a minimum of windows to distort, scatter, reflect and absorb radiation and it offers facilities for 90° and 180° Raman scattering.

Spectrometer Frequency Calibration

The Perkin Elmer 12G monochromator used in these experiments has been altered to provide an extremely simple and reproducible grating drive at the cost of losing linearity in wavelength vs. time. In addition, the drive systems of the chart recorder and monochromator have been coupled by a feedback system employing a pair of servo motors so that the speed of the chart recorder is always proportional to that of the monochromator regardless of sDeed variations in the variable soeed d.c. motor which is used to drive the monochromator. The soectrometer is calibrated by measuring the relative positions on the chart paper


- 2 4 -

of .. 20 neon lines of known frequency which fall in thefundamental region of hydrogen when they are measured in 2nd, 3rd, or 4th order of the grating. By making a least squares fit of a 5th degree polynomial to the chart position vs. frequency data, it is possible to create a very accurate calibration table which ailows the frequency of any spectral feature to be determined if its position on the chart paper can be measured relative to that of any one of the standard neon lines. In this way, frequencies can be measured with high precision (v 0.05 cm-1). A second calibration method sometimes used is very similar and involves the use of three or four neon lines near a feature of interest to make a local calibration table for that region by calculating a least squares fit with a 2nd degree polynomial.

A servo coupled monochromator-chart recorder system is ideal for studies of several small regions where features of interest may be some distance apart, since one can scan quickly over uninteresting sections and slow down for more interesting regions while maintaining a unique relationship between frequency and chart paper position relative to the various standard neon lines. There are disadvantages to such a servo system; for example, there is no way of changing the speed of the chart recorder relative to that of the monochromator drive except by changing the gears driving

;fl


- 2 5 -

a servo motor. The recent availability of precision stepping motors suggests the possibility of using stepping motors for both monochromator and chart recorder drives. Such a system would offer the ability to change the speed of the chart recorder drive relative to that of the monochromator by precise and known ratios.

Reduction of Results

The reduction of intensity vs. frequency chart recorder traces to obtain absorption coefficient vs. frequency is accomplished through the assumption of Lambert's law which gives the relationship between radiation incident upon the sample (10), radiation transmitted through the sample (I), and sample thickness (p) at a particular radiation frequency v as

[2.1]

where a(v) is the absorption coefficient. I0 was measured by taking a "background" trace without the sample in the light path. The background and absorption traces were brought into coincidence by superimposing neon lines.The background intensity and intensity transmitted by the sample were coded onto computer cards at regular frequency intervals using a Ruscom Chart Digitizer. A computer is then used to calculate absorption between predetermined


- 2 6 -

frequency limits. The integrated absorption is defined by

/•V-

[2.2] a = ( 1 / v ) a f v ) d v ,vi

It has been the intent of this chapter to describe those facets of experimental setup, technique, and design which are common to all experiments pursued in the course of work leading to this thesis. Those techniques peculiar to a particular set of experiments will be described in the chapter containing the results.


- 2 7 -

CHAPTER III

THE FUNDAMENTAL AND FIRST OVERTONE SPECTRA OF ORTHO-ENRICHED SOLID H 2

In this chapter, the zero-phonon absorption features in the fundamental vibrational region of solid H2 at various o-H2 concentrations between 75% and 99.7% o-H2 and at temperatures above and below the phase transition will be presented and discussed along with absorption features in the first overtone region of nearly pure solid o-H2. The work in the fundamental region above the phase transition complements that of Gush, Hare, Allin and Welsh (1960) who examinedthe infrared spectrum of solid H 2 from 0% to 75% o-H2.Below the phase transition, a preliminary experiment with 92% o-H2 was carried out by Clouter (1968) who gave extensive attention to the zero-phonon Q branch. Further interpretation has been given by Clouter, Gush, and Welsh (1970) and Boggs, Clouter, and Welsh (1972).

Experimental Detail

Probably the most difficult experimental problem in the study of o-H2 is the separation of o-H2 from normal H2. This cannot be effected by means of thermal equilibrium as in the case of p-H2 since at room temperature and above, normal H 2 (i.e., H2 in thermodynamic equilibrium) is 75%


0-H2 as dictated by the nuclear statistical weights of the two species. Below room temperature, equilibrium favors

and at 20 K and below, equilibrium is essentially 100% p-H2. Certain of the common physical properties of o-H2 differ sufficiently from those of p-H2 to allow measurement of relative concentrations; unfortunately, no difference islarge enough to allow easy separation. Among the differing properties are velocity of sound, vapor pressure, and thermal conductivity. The latter is most often used for concentration determination.

It was first noted by Sandler (1954) that o-H2 is preferentially adsorbed on the surface of Ti02 at 90.2 K and 5.5 mm Hg. He determined a separation coefficient,

where N and N are the numbers of para and ortho molecules p orespectively. Cunningham, Chapin, and Johnston (1958) found that a separation coefficient of 16+3 could be achieved with A120 3 at 20.4 K and 55 mm Hg. This is sufficiently large to make efficient separation practicable and has providedthe basis for all successful separators constructed to date. The separator constructed in this laboratory isbased on the design of Depatie and Mills (1968) and has

p-H2 so that equilibrium at 77 K gives 50% o-H2, 50% p-H2

S = 1.67, where S is defined by

s - ' W g a s 7 ' ' W a d s o r b e d

•• '-Kg-


- 2 9 -

been described in detail by Clouter (1972) so that only a brief description of the principles of operation will be given here.

The separator consists of a column about 40" long which is filled with pellets of AI2O 3 . In use, the column is lowered into a dewar which is cooled to 20 K by a Phillips helium cryogenerator. Once the column is cold, normal H 2 gas is forced through the column; a pressure of 50 mm Hg is maintained at the exhaust. The H 2 enters through a tube at the top of the column and exhausts through a second tube at the bottom. Hydrogen is kept flowing through the column until the equilibrium concentration of ^ 92% o-H2 is attained on the surface of the Al20 3 . At this point the normal H 2 flow is stopped and the column is slowly lifted out of the dewar over a period of about 30 min. As the pellets come out of the dewar, they warm up, and the 92% o-H2 gas desorbs, flows over the pellets which are still in the dewar, and exhausts through the tube near the bottom of the column. The gas flowing over the pellets is now 92% o-H2 rather than 75% so that further purification of the gas on the surface of the pellets is effected. This process of continuous purification proceeds to the point that the gas on the bottom foot or so of the pellets will have a concentration of up to 99.8% o—H2. This gas is stored and used for experiments. The separator can produce about 4 t


f-

- 3 0 -

at STP per hour. For experiments in the fundamental region, where path lengths are typically a few millimeters, 8 I will easily suffice. In the overtone region, where path lengths of 5 or 6 cm are required, 12 to 16 I is necessary.

It is much more difficult to solidify acceptable samples of o-H2 than p - H 2 , especially for path lengths > 1 cm. Part of this difficulty may result from the phase transition and change of crystal structure. It is necessary to solidify samples as quickly as possible, since ortho-to- para conversion proceeds at a rate of ^ 2% per hour at high ortho concentrations. The best method of sample solidification discovered to date involves filling the dewar with liquid helium and pumping it below the X-point which creates a very sharp temperature gradient in the region of the gas-liquid interface. In this temperature gradient, a sample may be formed in a few minutes and a complete spectrum of a region can be obtained within 20 min. after the first o - H 2 is liquefied. For measurements above the phase transition and therefore above the X-point at most concentrations of interest, a sample must be formed more slowly in the gas above the liquid helium, since no really sharp temperature gradient will exist.

For studies of various ortho-para concentrations, accurate determination of the ortho concentration proves to be a difficult task. Ortho concentration measurements

iP'


in a thermal conductivity apparatus, similar to that described by Stewart and Squires (1955), were not found to be reproducible for ortho concentrations greater than 95%. A method of concentration determination from the recorded spectrum was therefore devised.

The method adopted was based on measurements of the integrated absorption of the overlapping SÔ) and 0 i(1 ,0 )+ So (0) zero-phonon features of the spectrum (cf. Fig. 1.1). Since these features arise from o-H2 molecules, the absorption is zero for 100% o-H2 and increases in a characteristic manner as conversion proceeds. According to Cremer and Polanyi (1933) the fractional ortho concentration CQ in solid hydrogen changes according to the rate equation

where A is a constant of integration.

The integrated absorption, ci{si (0) + [Qi(l,0) + S 0 (0) ] }= a [S (0)], was obtained from recorder traces taken at various times after solidification, and a least squares fit of a fourth order polynomial to the dt [S (0 ) ] vs. time data was computed, as shown in Fig. 3.1. Values of 5 above and below

[3.1] dCQ/dt = kc£,

or

[3.2] 1/Cq = kt + A


M U

- 3 2 -

Fig. 3.1: The integrated absorption coefficientof the SjCO) group of p-H2 lines as a function of time after solidification of a ■v 100% o-H2 sample.


Eo

<5CO

15001000500TIM E FROM S O L I D I F I C A T I O N ( m i n )

.j


- 3 3 -

the phase transition have essentially the same time dependence. Extrapolated to zero absorption, the polynomial gave a negative time intercept, indicating that some parahydrogen was present at solidification; the time axis was therefore shifted so that a = 0 corresponded to t = 0 .The value of a[S(0)] was then measured for normal hydrogen and the value of t appropriate for conversion from 1 0 0 % to 75% o-H2 determined from the a vs. t graph. From this value of t at C0 = 0.75 and with t = 0 at C0 = 1.0 the constants A and k in eq. 3.2 were determined. A graph of ci [S (0) ] vs. o-H2 concentration could then be constructed which allowed the ortho-para composition of any sample to be determined from its spectrum; for the range 1 £ C0 ^ 0.75, the graph was very nearly linear and could be represented by the equation item"1) = 5.1 * 10-2 (1 - Co). It is interesting to note that the rate constant obtained, k = 1 . 7 5 x 10 “ 2 per hr, is the same as that obtained by Cremer and Polanyi (1933) and is consistent with the recent value of Jarvis, Meyer, and Ramm (1969) of 1.85 ± 0.1 x 10“ 2

ner hr.

The procedure used here has two main sources of error. The S (0) features are superimposed on the high frequency tail of the Q_. phonon spectrum (Fig. 1.1) and the separation" Rof the two on the reduced profiles introduces some error. Secondly, the S(0) absorption increases initially too rapidly because of the presence of paramagnetic impurities (e.g., 0 2)


- 3 4 -

which cause the rapid conversion of adjacent o-H2 molecules. This effect, the theory of which has been given by Motizuki and Nagamiya (1956), varies from sample to sample, however the same calibration graph a[S(0)] vs. CQ was used for all experiments.

The spectra which are presented and discussed in the remainder of this chapter were obtained with the apparatus described in Chapter II. The spectral slit width for spectra in the fundamental region is 0.8 to 1.0 cm-1 and for the overtone region 0.6 - 0.8 cm-1. The difference is primarily due to changes in the spectrometer made between the two sets of experiments, which greatly improved the efficiency of the spectrometer and allowed much narrower slits (25 to 35 y) to be used in the overtone than had been used in the fundamental region (75 y). The changes are described in Chapter II along with methods of frequency calibration.

Results and Discussion

Complete profiles of the fundamental vibration-rotation band obtained in the disordered (5 K) and ordered (2.1 K) phases are compared in Pig. 1.1 for a solid sample with 92% o-H2 concentration (Clouter 1968). The basic molecular transitions and spectral features in this region have already been discussed in Chapter I. Although all of the zero-phonon


features show change in going through the phase transition, the disappearance of the Sjd) feature is of particular interest and was used by Clouter and Gush (1965) as evidence of a change in crystal structure from one without inversion symmetry to one with such symmetry.

The overtone region of 97% o-H2 (Fig. 3.2) shows similar changes in going through the phase transition, including the disappearance of the S2 (1) feature. Most of the features in the overtone region involve double transitions since these are much more likely than single transitions which suffer from the cancellation effect (Van Kranendonk 1960). There is a far richer variety of double transitions to be found in the overtone region than in the fundamental, since double transitions can occur with a Q 2 jump on one molecule and a rotational excitation on the other or with a single quantum vibrational excitation on each of two molecules with or without additional rotational excitations.The remainder of this chapter is devoted to a region-by- region exposition and discussion of the zero-phonon features in the fundamental and overtone of solid orthohydrogen.

The Q. Reg-iont

Figure 3.3 shows the region of the fundamental Q branch above the phase transition for various concentrations of o-H2. These spectra show broad maxima at 4147.0 and 4152.6 cm-1


- 3 6 -


J

(|_UJ0) INdlOldddCO NOIlddOSGV


which can be identified with the Qi(1) and Qi(0) frequencies respectively; these frequencies seem to be independent of concentration from 75% to 99% 0-H2 . The situation is very different from the case of low o-H2 concentration which will be discussed in connection with quantum diffusion. For pure p-H2, the Q branch is entirely absent since the J = 0 «- J = 0 transition is forbidden for quadrupole-induced transitions and induction by overlap forces is inoperative because of the cancellation effect. At finite ortho concentrations para transitions are allowed when accompanied by ortho reorientations since the cancellation effect is inoperative for double transitions.

In the limit of ^ 100% o-H2 (Fig. 3.3) , the band has a single Boltzmann-modified lineshape with a half-width of 12 cm-1. The large width is believed due to the large variety of orientational transitions which are available.In particular, as will be indicated below, the high frequency tail of the band is probably associated with short-range ordering of the o-H2 molecules, since the energies involved are of the same order of magnitude as libronic transitions in the ordered state.

The absorption coefficient of h.c.p. hydrogen at arbitrary o-H2 concentration has been calculated by Sears and Van Kranendonk (19 64) as


- 3 8 -

Fig. 3.3: Absorption profiles in the region of theQ branch for various 0-H2 concentrations in the disordered phase of the solid (5 K) .


— Q, (0)

£o% 0-Hz

LJoLlLlLUOu

99.0

zoI-Q.q :oCD<

92.

84.0

77.8

73.341604140

FREQUENCY (cm-1)

J


- 3 9 -

1 3 . 3 ] a ( Q ) = ( 8 i r 3/ 3 h ) [ ( 3 / 4 ) S ^ y ' + ( 6 / 5 ) S 2 y " 2 ] CQ ,

where S-j and S2 are lattice sums, y' = (Qoict/a4) and y" = CQa o 1 /alf) r in which Q is the quadrupole moment and a the polarizability of the H 2 molecule and a(= 3.75 A) is the lattice spacing in the solid. The factors Q 0i and aoi are the matrix elements of the quadrupole moment and polarizability respectively between the ground and first vibrational states. If one neglects the small dependence of "a" on the ortho-para ratio, eq. 3.3 predicts a linear increase in ct [Q] with CQ , in agreement with experimental data which give a [Q] = 1.7 ± 0.1 * 10-2 CQ for the o-H2 quadrupole moment and polarizability recently calculated by Birnbaum and Poll (1970), we find for the coefficient of C0 in eq. 3.3, 2.26 * 10-2 cm-1. However, the calculation of Sears and Van Kranendonk (1964) is based on a rigid lattice, and it is now known that the roton-phonon interaction in the real crystal reduces the effective quadrupolar coupling constant. According to the calculations of Noolandi (1970) , based on a frequency analysis of the infrared and Raman spectra of solid p-H2, the effective quadrupolar coupling constant is 0.81 times the rigid lattice value. This reduces one calculated value of the constant in eq. 3.3 to 1.56 * 10- 2 cm-1 in good agreement with the experimental value. Since the calculation assumes random orientations of the ortho molecule, the agreement


seems to show that orientational ordering does not set in much above the transition point.

As seen in Fig. 3.2, the lineshape for both the Q 2 (1) and Qi (1) + Q 1 {1) transitions is the same as that for the Qi (1) and is presumably due to the same process of vibrational excitations accompanied by a wide variety of orientational transitions.

In going through the phase transition, the Q branches in both the overtone and fundamental regions undergo remarkable change, an example of which can be seen in Figs. 3.3 and 3.4. For 99% o-H2, the Qj(1) line is weak and fairly sharp; on the high-frequency side of the Qx (1 ) and separated from it by an energy gap, there is a set of features extending from about 6 to 26 cm-1 from the Qi(1 ) which we identify as a vibron-libron combination. Nothing is seen on the low frequency side of the Qi (1 ) since the energy gap is large compared to kT ^ 0.7 cm-1.

The Qj (1) vibron transition is forbidden in the f.c.c. phase in pure o-H2 solid but appears in the 99% o-H2 spectrum and with increasing intensity at lower o-H2 concentrations because of impurities in the lattice which break the inversion symmetry and hence "attenuate" the cancellation effect. Since the transition is due to a lowering of the symmetry of the o - H 2 lattice, the band presumably corresponds to the entire vibron band. The overall width is 3 cm- , which


/</u I- 4 1 -

Fig. 3.4: Absorption profiles in the region of theQ branch for various o-H2 concentrations in the ordered phase of the solid (1.05 K). The histograms show the density of libron states as calculated by Ueyama and Matsubara (1967) (solid line) and by Mertens zt at. (1968) (dashed line).

Reproduced with permissionof the copyright owner. Further reproduction prohibited without permission

AB

SO

RP

TIO

N

CO

EF

FIC

IEN

T

(cm

-1)

rQ.(l ) p Q. (0)

% o - H

78.2

85.6

91.2

94.3

97.3

99,0|41704150

FREQUENCY (cm ')


- 4 2 -

agrees well with the value calculated by Van Kranendonk(1959) for the case of pure p-H2. The shape of the exciton band is not known; however, according to Sears and Van Kranendonk (1964) , the lower frequency side of the line profile for pure p-H2 varies as Av3/2, where Av is the frequency measured from the low frequency end of the band. The observed profile for high o-H2 concentrations appears to be consistent with this shape.

The general situation in the overtone region is very much the same as that in the fundamental except that, since the Q 2 (1) transition is almost completely localized and therefore does not form a band or display any band structure (see Chapter IV), the width of the Q2 (1) line in Fig. 3.5 is a reflection of the spectral slit width of about 0.7 cm-1. The Qx (1) + Qi (1) (Fig. 3.6) is allowed since the cancellation effect does not affect double transitions. The width and shape of this structure must be due to the creation of vibrons with equal and opposite wave vector.

The nature of the excitations giving rise to the spectra on the high frequency side of the Qi (1), Q2 (1), and Ql(1) + Qi (1) lines was revealed by the theoretical work of Homma, Okada, and Matsubara (1967), Raich and Etters (1968), and Mertens, Biem, and Hahn (1968), which showed that travelling orientational excitations (librons), analogous to spin waves in an antiferomagnet, can exist in the ordered


Fig. 3.5: The absorption profile in the region ofthe Q2 (1) feature for 97% orthohydrogen in the ordered phase of the solid (1.05 K).


A BS

OR

PTIO

N C

OE

FFIC

IEN

T (c

m"

80808070 FREQUENCY (cm"')

8060


phase of pure o-H2. The groups of maxima from ^ 4150 to ^ 4170 in the fundamental region and from -v 8064 to ^ 8085 and 8295 to ^ 8315 in the overtone region are due to the combination of vibronic and libronic transitions; this structure is most clearly developed for 99% o-H2 (Fig. 3.4) and for the Q 2 (1) line in the overtone for 97% o-H2 (Fig.3.5). The libron band structure becomes somewhat diffuse as the o-H2 content is lowered (Fig. 3.4), and the emergence of the para Oj (0 ) transition introduces further complications, The vibron-libron spectrum was first obtained by Clouter (1968) in 92% o-H2 and explained by Homma and Matsuda (1968). Subsequently, a libronic Raman spectrum was observed by Hardy, Silvera, and McTague (1969) .

The first libron band (Fig. 3.4 99% o-H2, Fig. 3.5, and Fig. 3.6) extending from 4150 to 4158 cm-1 in the fundamental region is due to vibron + single libron processes, while the second band from 4160 to 4170 cm-1 is believed to arise from vibron + two-libron transitions. Because the librons are seen in combination with vibrons, librons of any wave vector k may take part in the absorption processes as opposed to libronic Raman scattering, where only k = 0 modes are active. The histograms plotted with the 99% o-H2 spectrum in Fig. 3.4 give the density of states as calculated by Ueyama and Matsubara (1967) [solid histogram] and Mertens t t a.1. (1968) [dashed histogram] , but


/</u

- 4 5 -

Re produced with permission of the copyright owner. Further reproduction prohibited without permission.

in the

ordered

phase

of the

solid

(1.05

K).

COT

O'O'(i-WO) 1 N 3 I3 I3 3 3 0 0 N O lldU O SSV


- 4 6 -

using a recently published value of the quadrupolar coupling constant*. Although the calculated libron band is shifted as a whole by ^ 1 cm-1 in frequency, the width of the band is well reproduced in the spectrum and there is some correspondence between the calculated and experimental maxima.The mismatch in absolute frequencies is probably due to the neglect of libron—libron interactions in the calculations. These anharmonic effects have been shown by Nakamura and Miyagi (1970) and Coll &t at.(1970) to be necessary in the calculation of the k = 0 modes in order to reproduce the observed Raman spectrum. The large anharmonicities are also responsible for the high intensities associated with the two-libron transitions in the spectra observed here. The two-libron Raman spectrum calculated by Berlinsky and Harris (1971) shows two bands with widths of ^ 2 cm-1 centered at 16.2 and 20.7 cm-1; the infrared spectra show very similar structure.

*The value of the quadrupolar coupling constantr _ 6 Q 1Q 2 where Oi and Q 2 are matrix elements of the 25 r?12quadrupole moment for molecules 1 and 2 respectively and r 12 is the separation between molecules 1 and 2 , is taken from the work of Hardy tt at. (1971) who arrived at an effective value of r = 0.539 cm-1 = 0.78 K.


- 4 7 -

As seen in Table 3.1, which compares the frequencies of the libron structure in the various spectral regions, the libron structure accompanying the Q 2 (l) and Q:(1 ) lines is identical to within experimental error. By comparing Figs. 3.4 and 3.5, it is evident that small differences do exist, the most striking of which is the reversal of dominance of the 8 .8 and 10.6 cm-1 peaks in the two cases. No certain explanation has been developed for this small difference; however a few comments are probably worth making. The libron structure accompanying the vibrational transitions cannot be expected to reflect precisely the libron density of states since one has no idea of the way in which the transition probability of the combination transition depends on the wave vector of the vibrational excitation. In the case of the Qi (1 ) excitation which possess a significant width due to the lack of localization, one would expect more k dependence than for the case of the Q2 (1) transition which is much better localized. One might, therefore, expect the Q 2 (1) + libron structure to better reflect the libron density of states than the Q x(1) + libron structure. This would, indeed, seem to be the case when one compares the histograms of Fig. 3.4 with the libron structure of Fig. 3.5. The principal peak in the calculated libron density of states seems to fall closer to the 1 0. 6 cm-1 peak, which is the major peak accompanying the Q 2 Cl) transition, than to the8. 8 cm-1 peak which is the larger accompanying the Qi(1)


permission

of the copyright

owner.

Further reproduction

prohibited w

ithout perm

ission.

Table 3.1: The zero-phonon Q branch frequencies for 98% 0 -H 2 and their assignments

Zero-phononfrequencies

Q j branch (cm- 1)

Zero-phonon 0 2 branch frequencies (cm-1)

Zero-phonon Qj frequencies

+Q j branch (cm-1) Assignment Raman frequencies

and assignments*1Absolute Relative Absolute Relative Absolute Relative

4146.13^0.5 0 . 0 8057.3±0.4 0 . 0 8291.9+0.7 8298.910.5

0 . 0

7.0+0.5vibron 6 .53i0 .l[Eg ]

4154.010.2 4154 . 9-tO.l4156.710.2

7 . 9±0.2 8 .8 +0 . 1

1 0 .6 +0 . 1

8065.110.58066.010.58067.810.5

7.810.38.710.3

10.510.38301.810.8 9.910.7

vibron•

+ libron8.2310.

11.29+0.

1 [Tg(1)]

l[Tg(2)]

k = ( ' librc

4161.4i0.2 15.3±0.1 8072.4+1. 15.110.5 8307.5+1. 15.710.84162.810.2 4166.5 0.3

4168.210.2

16.710.2 20.4 0.3

2 2 .1 +0 . 2

8074.2+1.

8084.711.5

16.910.5

27.411.8316. +1.5 24.111.5

vibron + 2 librons

16.8 1 1 .

2 1 . 0 1 2 .

Two-libron ■ processes

aHardy, Silvera and McTague (1971).

100

J

- 4 9 -

transition. This may very well be the explanation of the change in intensities, since the spread of ^ 3 cm-1 in the energies of the vibron band could drag intensity to lower frequencies and cause the change observed. At this point, it should be emphasized that the density of state calculations are not very trustworthy. For example, Raich has calculated a very large temperature dependence for the peak in the density of states. From the experimental work of Clouter (1968) in combination with that reported here, it is quite certain that no such temperature dependence exists.

The structure accompanying the Qi(1) + Qi (1) transition (Fig. 3.6) is entirely different from that accompanying the other two transitions. The most striking feature in the libron spectrum is the relatively sharp peak at 7.0 ± 0.5 cm-1 on the low frequency slope of the libron structure.One possible explanation is that this corresponds to the creation of two vibrons with equal and opposite wave vectors accompanied by a k = 0 libron. There are, however, no visible peaks corresponding to any of the other k = 0 librons.

None of the 8.8 and 10.6 cm-1 structure so promminent with the 0 2 (1) and Qi(1) transitions is visible accompanying the Qi(l) + Qi (1) transition, but this is not surprising since the combination Qi(1) + Qi (1) + libron would smear out such structure. In such a transition only the total wave vector need be conserved and this is possible in many ways


- 5 0 -

and, depending upon the combination, the frequency can undoubtedly vary widely over the libron band. In short, there is nothing like a one-to-one relationship between libron wave vector and transition frequency.

It is interesting to compare the zero-phonon Q-branch region above and below the phase transition (Fig. 3.7).The predominantly low-frequency orientational transitions of orthomolecules in the disordered phase disappear in passing through the phase transition to the ordered state. However, as already pointed out above, the disordered state shows significant intensity in the higher-frequency region where the libron spectrum appears in the ordered state. This may indicate that there is some short-range orientational ordering in the disordered state near the transition point, although the calculation with eq. 3.3 above indicated that the orientations of the ortho molecules must be predominantly random. It would be interesting to record the spectrum of the fundamental Q-branch region from say, the fusion point (14.1 K) through the phase transition to the lowest temperature attainable. Unfortunately, such experiments were not possible with the experimental arrangement used here.


- 5 1 -

Re produced with permission of the copyright owner. Further reproduction prohibited without permission.

(i_u j o ) 1N3I0UJ300 NOIlddOSaV


- 5 2 -

Thz S i (0) Gôap

The variation of the Sj (0) zero-phonon group of absorption features with o-H2 concentration is shown in Pig. 3.8(a) and (b) above and below the phase transition, respectively.In Fig. 3.8(a) only the S 2 (0) line and the broad peaks of the Qi(1) + Sq (0) and Qj(0) + So (0) double transitions can be identified. As the o-H2 concentration increases to 98.8%, the absorption decreases and must, of course, go to zero at 100% o - H 2 concentration.

Below the phase transition [Fig. 3.8(b)], the single transition SÔ) is much weaker and is absent for 98.7% o-H2 because of the site symmetry of a single para molecule in the o-H2 lattice. As the p-H2 concentration increases, clusters of p-H2 molecules produce some Sj (0) absorption.The absorption in 98.7% o-H2 is due to the double transition Qi (1) + So (0) , in which a Qi(1) ortho vibron is created accompanied by the 2 «- 0 rotational transition in the ground vibrational state of a para molecule. As might be expected the same pattern of four lines is observed in the Raman spectrum (Hardy zt al. 1969). The identity of the infrared and Raman patterns is illustrated in Table 3.2, where the positions of the observed maxima are given as absolute frequencies and also as shifts from the Qi(1 )component at 4146.13 cm-1 (Table 3.1); the shifts correspont extremely well with the Raman S0 (0) frequencies, given in column 4 of Table 3.2, as observed recently in a high resolution

with permission of the copyright owner. Further reproduction prohibited without permission.


t3(,.uuo) ± N 3 l3 ld d 3 0 0 NOIldHOSaV

in

O U_

(j_LU3) 1N310133300 NOIlddOSaV


1 tN

(i-UJO) !N3IOId3300 NOIlddOSaV

(,_ujo) iN3IOIdd300 NOIlddOSSV

eo>-OzLlI3OLlICC


- 5 4 -

uoipCOCD•H0CCD3G1<UU4-1

Wqoa0 £ a1ou<uN<djqEh

cmm

<DH&toEh

CO-pqCD

O'•HCOCO3

M•H<d.c-pT3qCOCH

33I0<*>(Ti(Ti

atoCD

■rC0qCD3 COO' O o o rH0) 3 ■P O o o O<p 1 •0 +i +1 1

e H in CO CNy»»X 0 rH C\ rHo • •

O cno in in in

cn co CO CO CO

q3

CC 00 uU XI& •«H•H rHrH +

H CN + rH4-J o Oq 11 II II II IICD e s £ £ £

< < < < <i—i 1—1 c—i i—i i—i

O' y“x y—x y—X y—x•H o o o o o01 X»y» s_ *■—- X—' X—«'0) o O o o o< cn cn C/1 W cn<—i t__j l—p u wt

+ + H~ + H-y—% /—x y—x y—«\ y—*rH rH rH rH rH

*■—* — ’ *— ■'*—4 H y— i

a a a a a

rHH

O rH rH rH CO rH• # • • •

0 O O o o Oy-«x ■P +1 +1 •H +1 +1r-i cn CN rH CN CO1 (D • • • • •e > r - O in r** cnV •rH in in in m

-P CO CO CO CO CO

CO H(D a)•H os0qCD3 mO’ o rH rH CO rH(D a) « • • • •U -P o o o o oCP •h ■H -M -H

i—i0 o ro CN CO01 • • • * •n VO rH ro in

<7\ cn O O o<3* in in m

txa\i-hu0•Wpcu


- 5 5 -

study by Prior (1971). Table 3.2 also lists a fifth poorly defined component observed in a series of special experiments in which the spectrum was recorded as soon as possible after sample solidification. A spectrum from this series of experiments is shown in Fig. 3.9 and corresponds to a calculated 0-H2 concentration of 99.7%.

The Qi(1) + S 0 (0) group of lines at low para concentrations is evidently due to J=2 J=0 transitions of isolated"impurity" para molecules in the crystalline field of the lattice of orthomolecules. For a Pa3 crystal structure, the site symmetry for such a para molecule would be C^, and the J=2 state would split into three sublevels corresponding to m = 0,±l,+2. The S 0 (0) transition would then show three transitions, Am = 0,±1,±2 with spacings in the ratio 1:3 if a crystalline field of the form Y 2 o(ft) is assumed, where n = (9, <(>) is the orientation of the molecule relative to the C 3 axis. This explanation has been advanced by both Silvera tt al. (1969) and-. Clouter tt al. (1970) with, however, different assignments of the three possible transitions to the four observed lines. With the discovery of a fifth line, it is possible to make a new interpretation of the pattern, in which the three lower-frequency maxima are assigned to the Am = 0, ±1, and ±2 components of the Qi(1) + S 0 (0) transition, and the high-frequency double component (Fig. 3.9) to the-triple combination Qi (1) + Sg(0) + libron.


Reproduced

with perm

ission of the

copyright ow

ner. Further

reproduction prohibited

without

permission.

1

j

Fig. 3.9: The S j (0) region at 1.05 K for an 0 -H2

concentration of 99.7%. The high frequency ^CTl

feature is interpreted as a Q 1 ( l ) + S 0 (0)+ 1

libron transition, the dashed line indicates a computed spectrum of this transition.

^ CM Oo' O

( |_UJ0) 1N3IO ldJ300 NO IldHO SaV


FREQUENCY

(cm-1

)

- 5 7 -

The observed spacings of the three maxima are in the ratio 1:2.2 instead of 1:3 as required for a pure Y 2g(0) field. It must therefore be assumed that the field has a significant admixture of a Y^gfn) component. The calculated splitting patterns for Y2o and Y40 are given in Table 3.3 in terms of parameters A and B. By identifying

Table 3.3: Splitting of the J = 2 rotational state in Y2g and Yi*g fields

Relative energy shift

m *20 *4 0

0 + 2A -6B

±1 +A + 4B

±2 -2A -B

the 4494.04 cm-1 maximum with the Am = 0 transition and writing v0 for the Qi(1) + S 0 (0) transition neglecting anisotropic interaction, one has the following three equa

tions :


- 5 8 -

2A + 6B = 4494.04 - v0

A - 4B = 4496.3 - v0

-2A + B = 4501.2 - v0.

The solution of these equations gives A = -1.72, B =-0.054,and vg = 4497.8 cm-1, respectively; the relative magnitudesof these seem reasonable. The largest component of A isthat due to the nearest neighbor quadrupole-quadrupoleinteractions, a calculation of which gives A = -2.44 cm-1for r = 0.54 cm-1, ef f

The diffuse high frequency doublet of the Qx (1) + S0 (0) transition is identified as the Am = ±1 components of triple combination Qi (1) + S <> {1) + libron. The dashed curve in Fig. 3.9 was constructed by convoluting the experimental profiles of the Am = ±1 components of the Qi(1) + Sg(0) transition with a Gaussian lineshape of 1.6 cm-1 halfwidth as a rough representation of the libron spectrum, and locating the convoluted curve at a frequency 9.25 cm-1 higher than the Am = 0 component. With the intensity of the convoluted arbitrarily adjusted, the dashed curve gives a fair representation of the observed spectrum in this region.The Am = ±2 component of the Qi(1) + Sq (0) + libron combination does not appear in the 98.7% o-Hj spectrum in Fig.3.9 because its intensity is too low. It can however be seen as a broad diffuse feature at lower ortho concentrations


- 5 9 -

where the absorption level is higher. Since the combination tone also appears in the rotational Raman effect it can be assumed that it is due to a strong interaction between the libronic states and the J = 2 rotational state of the para molecule.

The S 0 (O) structure occurs twice in the overtone region of nearly pure o-H2. The Q 2 (1) + So(0) structure is visible in Pig. 3.2 and part of the "So(0)" structure can be found in the Qi(1) + Sq (0) feature, shown in Fig. 3.12. In both cases, the structure appears to be identical to that in the fundamental region and adds nothing to the analysis.

The. S i (I) Gtioup

In the disordered phase (Fig. 3.10), the S^l) transition (4703.2±0.2 cm-1) is present with low intensity, as one might expect since the molecular site in the h.c.p. structure is almost, but not quite, an inversion center. Unlike the Si(0) line in the pure d-H2 solid (Gush zt a.1. 1960), the Si(l) line remains broad even at 98.8% o-H2, probably because, although it is a localized excitation, it is accompanied by orientational transitions of neighboring ortho molecules.

The stronger component m this region, with a maximum at 4735.6 0.4 cm-1 for 98.8% o-H2 is the combination Qi(1) + Sg (1) transition. The halfwidth of this vibron


u

- 6 0 -

Fig. 3.10: The Si(1) region for various o-H2concentrations above the phase

"W


ABSO

RPTI

ON

CO

EFF

ICIE

NTS

c

m"

Q.a +Srt(l)

4710 4 7 5 0FREQUENCY (cm'1)


- 6 1 -

roton band is ■v 20 cm-1, i.e., about the same as the Q 2 (0) + Sg (0) in the p-H2 solid discussed in the next chapter. With increasing p-H2 concentration, the maximum shifts somewhat toward higher frequencies since the Qi(0 ) + Sg(l) transition is also present; as shown by Gush &t at.(1960), at low o-H2 concentration, this transition becomes a narrow band at 4739.7 cm-1. In the overtone region above the phase transition, a broad S2 (l) line (Fig. 3.2) analogous to the Sid) line in the fundamental is observed at 8588±2 cm-1. The Q x (1) + Si(0) (8631.9±1 cm"1) and Q2 (1) + Sg (1) (8645.5+1.5) overlap to form a broad structure.

The S^l) region below the phase transition presents the most complex and least understood zero-phonon features. The spectra are shown in Fig. 3.11 and the frequencies of the maxima for 98.2% o-H2 are given in Table 3.4. The single transition Si (1 ) is not present in the ordered state with f.c.c. structure; as already mentioned, the disappearance of this line was used by Clouter and Gush (1965) as evidence for the existence of a phase transition. There are, however, some weak maxima on the high frequency side of the Si(l) frequency when this is taken to be 4703.2 cm-1, its observed value above the phase transition. This region is shown on an expanded intensity scale in the insert of Fig. 3.11. The most probable explanation for this set of lines is that they represent double transitions,

permission of the copyright owner. Further reproduction prohibited without permission.

- 6 2 -

Fig. 3.11: The Sx (1) region for various o-H2concentrations below the phase transition. The insert is plotted on the same horizontal scale but with a much expanded vertical scale. The position of the S^l) frequency indicated is that for 97% o-H2 above the phase transition.


AB

SO

RP

TIO

N

CO

EFF

ICIE

NT

(cm-1)

I

24

H h

S,(l)

% o-H

98.2

93.6

87.6

82.3

76.1

47504720 FREQUENCY (cm-1)


- 6 3 -

Table 3.4: Frequencies of maxima in the Si(1) group for 99% 0-H2

Frequency (cm“ 1)

Assignment

4703.7±0.2 4705.2±0.2 4708.1±0.24712.4±0.2 4714.6±0.2 4716.3±0.2 4718.5+0.2

■ weak S^l) + libron (?)

4732.4±0.24736.2+0.24741.6+0.5

Qi(1) + S0(1)


S^l) + libron, with both the upper and lower rotational states of the Sj(1) transition split by anisotropic interaction. The more intense band in the fundamental region extending from 4725 to 4755 cm-1 is the vibron-roton band,Qi(1) + Sg(l), at high o-H2 concentrations; as the ortho concentration decreases, the Qi (0) + S0 (1) transition also appears. The three maxima in the band for 98.2% o-H2 show a resemblance to the So(l) transition in the Raman spectrum of o-H2 solid observed by Hardy zt at.(1969), although there is a frequency discrepancy of ^ 2 cm-1. The resemblance is surprising since the Raman spectrum should show only k = 0 transitions to the J = 3 roton band, whereas k values ranging over the entire Brillouin zone should contribute to the infrared Qi(1) + S0(l) transition. The k = 0 levels of the J = 3 roton band have been calculated by Lee, Raich and Etters (1970) but show poor agreement with the Raman spectrum. It is noteworthy that the broad high- frequency maximum (4741.6±0.5 cm-1) is shifted 9 cm-1 from the lower frequency maximum (4732.4±0.2 cm- *). This may indicate that, just as for the Qi (1) + Sg (0) case, the higher- frequency component with its long trailing edge toward still higher frequencies may be a roton—libron combination caused by a strong interaction of the rotational and libronic

motions.


The Q 2 (l) + Sq(1) band (Fig. 3.12) in the overtone of nearly pure o-H2 very much resembles the Q x(1) + S0 (1) in the fundamental region except for a few minor details. Aside from the peaks at 8642.9±0.2 and 8646.2±0.2 cm-1, there is a shoulder at 8643.5+0.4 cm-1 which is not seen in the fundamental region. There are two possible reasons for this. First, the experiments in the fundamental region were carried out at somewhat lower resolution than those in the overtone region; however this alone could not explain its absence in the fundamental region. The second possible reason is the width of the Qi(l) band compared with that of the Q2 (1) band. This has been pointed out earlier in connection with the Q branch structures in the fundamental and overtone regions. Since the Sg(l) roton band is seen in combination with a vibron band about 3 cm-1 wide in the fundamental region and with a band less than 0.1 cm-1 in the overtone region, it is quite reasonable that some fine structure should be washed out in the fundamental region and appear in the overtone.

Th& Qi (?) + Si(?) Rzg-ion

There is one region in the overtone for which there is no analogous structure in the fundamental, viz., the Qi(1) + S 1 (1) region, which consists of a series of five lines spread out over a region of ^ 14 cm-1, the frequencies of which are given in Table 3.5 along with those observed


Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

1( |_UJO) 1 N 3 I0 1 3 3 3 0 3 NOIlddOSaV


Table 3.5: QiSi (1)

(1) + S^l) Infrared Raman Frequencies

and

Q (1) + S (1) frequencies S (1) Raman frequencies5-

Absolute Relative to Absolute Relative to

8854.3 4701.658848.5±0.3 -5.8 4197.49±0.3 -4.168851.2±0.4 -3.1 4698.45±0.3 -3.208854.3+0.3 - 4701.65±0.3 -

8856.4+0.4 2.18857.8±0.5 3.5 4706.18±0.5 4.55

^Prior (1971).

in the Raman effect for the S^l) transition (Prior 1971) The infrared Qj(1) + Sj(1) and the Raman Si(1) are shown in Figs.3.13 and 3.14, respectively. The resemblance bet ween the two is striking; however, the differences are great enough to require explanation. The SDlitting of the si(1) structure in the Raman effect is completely explained by a crystalline field splitting of the J = 3


I

ML

- 6 8 -

Fig. 3.13: The Qi(1) + Si(l) feature for 97%o-H2 below the phase transition.


- 6 9 -

state by a field of Y20 form (Prior and Allin 1972). Such a field splits the J = 3 state into four levels with relative positions -4A, -3A, 0, 5A for the m = 0, ±1, ±2, ±3 transitions, respectively. The experimentally determined value for A is ^ 1 cm-1, while the value calculated from nearest neighbor interaction gives A = 1.14 cm-1, using r = 0.54 cm-1.

The Si(1) excitation itself is, of course, well localized since there are no terms of any significance in the intermolecular potential which depend simultaneously on the internuclear separation and the relative orientations of the molecules. In the overtone region, where the Si (1) is observed in combination with a 3 cm-1 wide vibron band, wave vectors ranging over one entire Brilliouin zone of each excitation are allowed subject to the condition that the sum of the wave vectors is zero. Since for the vibron band there is an appreciable spread in frequency with change in wave vector, dependence may shift the basic (1) structure when seen in combination with the Qi (1) band.This would seem to be the most likely explanation for the structure observed in Fig. 3.13.


- 7 0 -

Fig. 3. : The S^l) Raman feature for 98% o-H2below the phase transition (Prior 1971).


RE

LA

TIV

E

INT

EN

SIT

Y

471047004690RAMAN FREQUENCY (cm )


-71-

CHAPTER IV

THE FIRST AND SECOND OVERTONE BANDS OF PURE AND NEARLY PURE PARAHYDROGEN

The study of the overtone spectra in the free molecule yields information concerning the intramolecular potential and, in dense systems such as solid hydrogen, effect of intermolecular forces on that potential. The first experimental study of the overtone region of solid parahydrogen was undertaken by Gush, Hare, and Welsh (1958); however, no comprehensive study of the band was published and the results which were obtained have largely been dispersed over the last 14 years. A few of the features observed were explained by Gush and Van Kranendonk (1962) and by Van Kranendonk and Karl (1968), who made use of the overtone results in their frequency analysis of p-H2 solid.

The major motivation behind the experiments reported here was the possibility of examining the second overtone region. After preliminary experiments indicated that there was sufficient absorption in a 6 cm path length of solid to make such examination practical, it became imperative to re-examine the first overtone region so that the frequencies required to identify double transitions in the second overtone would be available. It was disappointing, although not entirely unexpected, that all of the four features examined in the second overtone region were double transitions involving a first overtone transition


-72-

accompanied by a fundamental transition.

For experiments in the second overtone region, the Infrared Industries PbS cell was replaced by a United Detector Technology PIN silicon photodiode. The photodiode is more sensitive in the second overtone region than the PbS cell and its response goes rapidly to zero for wavelengths longer than 1.1 u. This sharp cutoff eliminates the need for filters to remove the radiation in first and second orders of the grating from the third- order radiation used in these experiments. Fourth and higher orders were removed by a Kodak Wratten 89B filter.

The First Overtone Band

The first overtone band of nearly pure p-H2 solid (Fig. 4.1) includes six zero-phonon features, two of which [q2 (0) and Q x (0) + Qi(1)] are not found in the spectrum of pure p-H2 solid. By far the greater part of the absorption in this region comes from double transitions, since the near inversion symmetry of the h.c.p. lattice tends to greatly reduce the absorption due to single transitions (Van Kranendonk 1960).

The Q?(0) Region

The Q2 (0) line is entirely absent in pure p-H2 solid for reasons of lattice symmetry identical to those which eliminate the Qi (0) feature (Van Kranendonk 1960). As an


- 7 3 -


Q,(0

)+

S,(

0)

( uuo) 1N3DIJJ300 NOIlddOSaV

Reproduced with permission o fth e copyright owner. Further reproduction prohibited without permission

8000

82

00

8400

88

00

8800

9

00

0FR

EQUE

NCY

(cm

'1)

o-H2 impurity is introduced, a feature due to Q2(0) transitions accompanied by orientational transitions in "isolated" ortho molecules becomes evident at 8070.4 ± 0.1 cm-1. When the o-H2 impurity concentration reaches ^ 3%, a feature caused by Q2 (0) transitions accompanied by orientational transitions in pairs of o-H2 molecules appears on each side of the Q2 (0) line, as seen in Fig. 4.2. These features, displaced by ^ 2.2 cm-1 from the Q2 (0) line, are completely analogous to those which accompany the Qi(0) line (see Chapter V).

Since the Q2(0) feature is due to double transitions, vibrational excitations of any wave vector can be created. However, the Q 2 (0) excitation is very well localized and gives rise to a band less than 0.15 cm-1 wide (Prior 1971) . This is of the same order as the width of the orientational band due to isolated o-H2 molecules which is estimated to be 0.1 cm-1 (Oyarzun and Van Kranendonk 1972). One can obtain some idea of the width of the Q2 (0) + orientational band by assuming that the observed half width ( 0.8 cm M is the sum of the bandwidth of the feature and the spectral slit width (0.65 cm'1). This would indicate a bandwidth of ^ 0.15 cm'1 but offers no insight into the relative contributions of the Q2(0) and orientational bands.


- 7 5 -

Fig. 4.2: The Q 2 (0) region for nearly pure5% o-H2) p-H2 solid at 4.2 K.


ABSORPTION

COEFFICIENT

(cm

8070 FREQUENCY (cm-1)


- 7 6 -

fhe Q.1 * Q,i Re.gi.on

With ^ 5% o-H2 impurity, a great deal of structure can be seen in this region (Fig. 4.3). At impurity concentrations < ^ 2%, only the Qi(0) + Qi(1) line is present. A Qi (0) + Qi(0) feature, which one might expect to occur in pure p-H2 solid, is not allowed. One can demonstrate this by examining an expansion in intramolecular displacements of the dipole moment due to a pair of p-H2 molecules. If the intramolecular displacements for molecules 1 and 2 are defined as x* h (ri~re) and x2 = (r2-re) , where re is the equilibrium intramolecular separation, then the dipole momentum, p, can be written

[4.1] p = Pg + P l X i + P2X 2+ P i2 (XiX2 ) +...,

where pg, pj, p2, and pj2 are constants. It is the last term, which depends on the intramolecular displacements of both molecules, that would give rise to the Qx (0) + Ch(0) transition. Since in equilibrium (x = x2 = 0) , a pair has no dipole moment, pg must equal zero. As long as Xj = x2, there can be no induced dipole moment. Then

!4. 2] 0 = (pi+p2)x + Pl2x'

for all x. It follows that u12 = 0, and the Qi(0) + Qi (0) transition is not allowed.


- 7 7 -


c>( UJO) 1N3I0UJ300 NOLLdHOSaV


- 7 8 -

The Qi (1) + Qj (1) transition is allowed since the ground state of the ortho molecule is not spherically symmetric so that even though xi = the difference in orientations of the two molecules can lead to a dipole moment. Absorption due to this transition, evident in Fig. 4.3, is spread over a region of ^ 5 cm-1. This is due to orientational transitions of the pair which can accompany the vibrational transitions. There is a large number of possible transitions since the separation between pair orientational states in the excited state is different from that in the ground state due to the change in the quadrupolar coupling constant. These transitions cause the two peaks at 8290.4±0.2 and 8295.9+0.2 cm-1 along with some of the absorption between them.

The Qi (0) + Qi (1) feature, which dominates the region, has the same asymmetric line shape which has been discussed in connection with the Qi (1) transition in Chapter III.The two peaks on either side of the Qi (0) + Qi (1) feature are due to reorientations in pairs of 0-H2 molecules and are very similar to the lines observed accompanying Qi (0) feature, which are discussed in Chapter V. To within experimental error, the frequency of the Qi (0) + Qi(1) feature (8299.5±0.1 cm-1) is the sum of the QÔ) (4153.0±0.1 cm-1) and Qj(1) (4146.5±0.1 cm"1) frequencies.


- 7 9 -

Thz Q.2 { 0) + -So ( 0) Region

The region, which includes the S2(0) feature as well as the Q2 (0) + S0(0), is entirely analogous to the Qi (0) + S0(O) region in the fundamental band. The latter has been analysed by Van Kranendonk (1959, 1960) and Van Kranendonk and Karl (1968) who made use of both the Sx (0) and S2(0) frequencies in their frequency analysis of solid parahydrogen. In the absence of rotation-vibration interaction, the states corresponding to the Q2(0) + S0(O) and S2(0) excitations would be degenerate. However, because of this interaction, the states where both the rotational and vibrational excitations reside on the same molecule have about 35 cm-1 less energy than those where the two reside on different molecules, as seen in Fig. 4.4. At a semiclassical level, this difference can be attributed to the stretching of the vibrationally excited molecule which increases its moment of inertia, thereby decreasing the rotational energy between the various states of rotational angular momentum.

The S2 (0) "band" is very narrow since there are no terms of any significance in the intermolecular potential which depend simultaneously on the intramolecular displacements and orientations of neighboring molecules. Only the odd k = 0 mode of this band is infrared active so that one would expect an extremely sharp line. To date, the intrinsic widths of the Si(0) and S2 {0) lines are unknown; the line width' seen in Fig. 4.4 at 8387.35±0.05 cm"1 reflects the


M R R


(,-UUO) iN3l3IJd303 NOIlddOSaV


8390

84

00

8410

84

20

8430

FREQ

UENC

Y (c

nrf1)

spectral slit width.

The narrowness of the S2 (0) line is in extreme contrast to the Q2 (0) + Sg (0) band which is due to the creation of rotons of any wave vector accompanied by the creation of Q 2 (0) excitations of equal but opposite wave vector. As already mentioned, the Q2 (0) band is < 0.15 cm-1 wide so that the entire width of the Q2(0) + S0(0) feature

22 cm-1) is due to the roton band. The fine structure of this band remains one of the unsolved mysteries awaiting theoretical explanation.

Van Kranendonk (1960) gives a formula for a[Sx (0)] + a[Qi (0) + Sg (0)] . If one substitutes the appropriate matrix elements of the polarizability and quadrupole moment, this expression becomes applicable to the overtone region. After such substitutions, one calculates ci[S2 (0)] + a[Q2 (0) + Sg (0) ] = 5.45 x 10"4 cm-1. This is in satisfactory agreement with the experimental value of 4.7 x 10"11 cm-1.

The. Q_t ( 0 ) + S t [ 0 } Ba n d

The Qi(0) + Si (0) feature (Fig. 4.5) consists of a rectangular band crowned with three peaks located at 8635.5±0.2, 8637.7±0.2 and 8639.5±0.2 cm"1. The band results from the splitting of the 10-fold degenerate excited state by EOQ interaction, as shown in the energy level diagram accompanying the spectrum (Van Kranendonk 1972). The


Fig. 4.5: The Qi(0) + Sj(0) band for very purep-H2 solid at ^ 1.5 K. The energy level diagram illustrates the calculated splitting of the excited state due to EQQ coupling. The numbers in parentheses indicate degeneracy while the numbers in column "m" indicate magnetic quantum number.


ABSO

RPTI

ON

COEF

FICI

ENT

(cm

0.6

0.5

<4)±2 0 7 rm 0.7 cm(10)0.4

0.3

0.2

0.1

0 86408635 FREQUENCY (cm”')

8630


- 8 3 -

calculated separations seem to be accurately reflected in the three peaks of the Qx (0) + Sx(0) feature. The width and relatively flat top of the band are probably due to broadening of the components by vibrational interaction.The magnitude of this broadening is not known; however, it is expected to be less than the splitting between components.

The Qi(0) + Sx (0) feature is ^ 1 cm-1 lower in frequency than the sum of the frequencies of the Qx (0) and Sx (0) features. This shift may be due to the interaction between the Qx(0) + Si (0) band and the highly degenerate band created by the rotational excitation leaving the vibrationally excited molecules.

Tf te S t ( 0 ) + S^ 10} V o u b l U

The 25-fold degeneracy of the Si(0) + Si(0) excitation is split into four potentially infrared active levels by the EQQ interaction, subject to the condition that only transitions to states for which M = ±1 can be infrared active, where M is the sum of the rotational magnetic quantum numbers (Gush and Van Kranendonk 1962) . Of the four levels satisfying the condition on M, two have wave functions which are even with respect to interchanging of the two molecules and are not, therefore, infrared active. The other two levels, with odd wave functions, result in the observed doublet (Fig. 4.6) with peaks at 8971.18±0.1 and 8973.18±0.1

cm~ 1.


Fig. 4.6: The S x (0) + S2(0) doublet for very


ABSORPTION

COEFFICIENT

(cm-1)

0.15

0.05

0 89758970

FR E Q U E N C Y (cm *1)

with permission of the copyright owner. Further reproduction prohibited without permission

- 8 5 -

The Second Overtone Band

Of the numerous features one might hone to observe in the second overtone band of pure p-H2 solid, only four, as shown in Fig. 4.7, could be observed with a 6 cm path length of solid. The weakest of these, the Qi (0) + Q2(0) feature, has a peak absorption coefficient of ■v 0.002 cm-1, which corresponds to about 1.5% absorption in 6 cm of solid, The strongest, the Qj (0) + S2(0) line, has a peak absorption coefficient of ^ 0.1 cm-1, which gives about 50% absorption.

T h i Q.? I 0 ) + Q.1 ( 0 ) L Z m

Very little can be said of this feature. The absorption is so weak that nothing concerning lineshape, line- width, or integrated absorption can be measured. The observed frequency of 1223.3±0.1 cm-1 is, to within experimental error, the sum of the Q2 (0) and Qj(0) features.

Tf l e S 7 ( 0 ) + ( 0 ) a n d 0 ? ((7) + S i ( f l ' ) L in e .4

These features, seen in Fig. 4.8, provide most of theabsorption in the second overtone region. Each line has a measured halfwidth of ^ 1.8 cm'1, which probably represents the spectral slit width, since the five fold degeneracy ofthe excited state for these transitions is not split by EQQinteraction. The frequencies of both the S2(0) + Qi (0)


gssaaateaawc n m n w rir -M — ■


(\J <->

C\J u_

jjlio) 1N3I0UJ3O0 NOIlddOSaV


CO(DC

- 8 7 -

cn

+

CMOT3CIT]

O+

CMw0)£Fl

in

+iid

■O0 mCMS1o.©u3a>,u©>u0>w

O'•Htl



( wo) 1N3I3I33300 NOIlddOSaV


1254

0 12

550

1256

0 FRE

QUENCY

(cm'1

(12539.45±0.1 cm-1) and the Q2(0) + S x(0) (12556.9±0.2 cm 1) are slightly lower than one would expect from the sum of the frequencies of the transitions involved. The discrepancy is -v, 0.7 cm-1 for the S2 (0) + Qj (0) and * 0.5 cm-1 for the Q2 (0) + SjfO) transition. This shift may be due to interaction between these two transitions which are separated by only 6.8 cm-1 for the free molecule.

The S,(0) + S?[0) Fta.tu.Ae.

This feature, seen in Fig. 4.9, is by far the mostinteresting in the second overtone region. The excitedstate is 50-fold degenerate and differs from the 25-folddegenerate excited state of the Sj (0) + SjfO) transitiondue to the distinguishability of the molecules possessingv = 1 and v = 2 vibrational excitations. The explanationof the Sj (0) + Si (0) feature, presented above, forms thebasis for the explanation of the S2 (0) + SÔ) feature.In the case of the Si (0) + Sj (0) transition, the eigenstatesare written as linear combinations of the eigenstates ofthe individual molecules, |s> = £ 2m>|l, 2n>, where

mn[v, Jm> are the vibrational, rotational, and magnetic quantum numbers, respectively (Gush and Van Kranendonk 1962). In the case of the S2 (0) + Sj (0) transition, the distinguishability of the molecules leads to |12s> =I c® |1,2m>|2,2n> and |21s> I c® |2,2m>[l,2n>. The mn n mn



o(|_ UJO) J.N3IOIdd300 NOI±ddOS8V


1287

0 12

875

1288

0 FR

EQUE

NCY

< cm

"')

- 9 0 -

eigenstates of the system are then 2^[|12s>±|21s>]. One can now find odd wave functions corresponding to all four states allowed for the SÔ) + SÔ) transition. These levels are shifted by vibrational coupling so that the two levels which were infrared active in the case of the Si(0) + Si (0) transition are shifted down by the vibrational coupling constant of 0.5 cm-1, while the newly infrared active states are shifted up by 0.5 cm-1 (Van Kranendonk 1972).

The spectrum which results from this shifting of the levels given by Gush and Van Kranendonk (1962) is indicated by the sticks in Fig. 4.9. The major discrepancy with experiment seems to lie with the higher frequency transitions which are separated by only half of the 2 cm-1 splitting present in the experimental spectrum.

These experiments represent the first to explore the second overtone band of P-H2 solid and demonstrate that there is spectral structure of interest in this band.Future work with spectrometers of higher resolution will probably reveal a great deal more structure and, perhaps, some entirely new features.


- 9 1 -

CHAPTER V

INFRARED STUDIES OF QUANTUM DIFFUSION IN NEARLY PURE PARAHYDROGEN

The term "quantum diffusion", as already mentioned in Chapter I, is used to describe the motion of o-H2 impurities through a lattice of nearly pure parahydrogen. The electric quadrupole interaction between ortho molecules provides the binding necessary to the establishment of a nontrivial thermodynamic equilibrium between the creation and destruction of pairs and higher order clusters of o-H2 impurities. The basis of all methods devised to date for the detection and measurement of quantum diffusion is founded in changes in spectral features which occur as the number of "isolated" or "paired" o-H2 impurities change as a function of time.The infrared experiments detailed in this chapter take advantage of a feature in the Q-branch fundamental vibrational band whose peak absorption coefficient is proportional to the concentration of pairs of o-H2 impurities in the sample. By beginning the experiment with a random distribution of o-H2 impurity in the sample and watching the growth of the spectral feature as the number of o-H2 pairs goes toward equilibrium at a given temperature, it is possible to measure various properties of the quantum diffusion process. The approach toward equilibrium is slow and the change in the spectral feature moderate. Therefore, very


- 9 2 -

accurate measurement of the peak absorption coefficient is necessary and, of course, accurate temperature control is crucial. These requirements made precise experiments difficult and forced careful selection of the apparatus used.

Experimental Detail

Tke. Sp<Lc.tKom<Ltzti

Because of the rather slow nature of the quantumdiffusion process, the spectrometer had to be stable overperiods of about 10 hr. It was also necessary to obtainspectra with excellent signal to noise ratio. For thesereasons, the "home spun" lock-in amplifier described inChapter II was replaced by a PAR JB-4. This model hastuned amplifiers in both signal and reference channels inorder to minimize the problem of noise outside the bandpass. These tuned amplifiers are normally run at a Q of20. In the interest of stability they were reduced to 10 for these experiments. Because these tuned filters donot track, the JB-4 lock-in amplifier cannot be considereda tracking device. It is therefore essential to have astable chopping frequency, and the induction motor whichhad been used previously was replaced by a Gerrard syncro-lab motor. This motor contains both an inductionand a synchronous section on one chassis; it is intended

with permission of the copyright owner. Further reproduction prohibited without

- 9 3 -

to run synchronously under light load or as an induction motor under heavier load. With this motor driving a 7-inch diameter chopper, about 200 volts were necessary to obtain synchronous operation. Once locked, however, the voltage could be lowered to 90 volts with no fear of losing synchronisation.

Tempe/ia-true ContKol

For reasons which will become clear, the study of quantum diffusion requires accurate temperature control of the solid H 2 sample. The purchase of a Scientific Instruments N2C germanium resistance thermometer (GRT) and construction of a current source with ranges from 0.1 yA to 20 pA used in conjunction with a Medistor A75-A potentio- metric voltmeter solved the measurement problem. The control problem was solved by taking the chart recorder output from the Medistor, which is proportional to the deviation of the instrument from null, and amplifying it to drive a heater.The power amplifier and temperature control circuitry were constructed around a Kepco OPS-15BC power operational amplifier. The schematic for the unit constructed is found in Appendix III. The heater was glued to the body of the optical absorption well with varnish and the GRT was placed inside the cell. In use, the cell was immersed in superfluid helium. Rough temperature control of the helium bath


- 9 4 -

was effected through the use of valves to the helium vacuum pump, and the heater was reserved for fine temperature control. With this technique, the control of the sample temperature could be maintained with no more than 15 mW dissipated in the heater, which is sufficiently small that it has no appreciable effect on the hold-time of the cryostat. The accuracy to which the temperature could be controlled was limited only by the stability of the current source, and temperatures could be kept stable and reproduced to well within ± 0.01 K, probably to within 0.001 K.

Sample. ?n.tpaxatlon

Samples of hydrogen with between 0.5% and 2% 0-H2 impurity were prepared by converting normal hydrogen to parahydrogen with an o-H2 impurity of less than 0.01% and mixing this "pure" parahydrogen with normal hydrogen to obtain the desired 0-H2 concentration. Before proceeding with the mixing process, the "pure" parahydrogen was solidified into the optical absorption cell and a spectrum of the Q—branch region was taken. If no absorption could be detected with a 4.2 cm sample of solid, one could be certain that the o-H2 impurity was less than 0.01%, since such a concentration would produce a readily visible Qi(0)

line about 0.5 in. high.


Preparation of p-H2 samples with less than 0.01%o-H2 impurity is not simple. The method used in theseexperiments required the use of two helium cryostatsoperated simultaneously. One was used to maintain liquidhydrogen near its triple point in a bulb containing aparamagnetic conversion catalyst. The liquid was left inthe conversion bulb for 1% to 2 hours. Because of the

*great efficiency of the catalyst , any heating of the sample to get it out of the bulb will cause a very rapid back conversion. The method adopted for removing the parahydrogen from the conversion cell does not, therefore, requireraising the temperature of the liquid. The parahydrogen is slowly drawn from the conversion bulb onto the stem of the optical cell immersed in liquid helium in the second cryostat. This process takes place very slowly, over a period of 2 to 4 hours. After checking the purity of the parahydrogen and mixing it with the amount of normal hydrogen necessary to obtain the desired o-H2 impurity concentration, it is necessary to form a sample of uncracked solid in the 4.2 cm absorption cell. It is preferable that this sample be formed quickly so that there is a well

*Because of the need to store large quantities of liquid hydrogen for the U.S. space program, highly efficient conversion catalysts have been developed and are available

from the Harshaw Chemical Co.


-96-

defined beginning to the quantum diffusion measurements. Clear uncracked samples of nearly pure p-H2 solid in large (2 cm diameter up to 7 cm long) optical absorption cells are most easily and quickly obtained by lowering the cell full of liquid hydrogen into superfluid helium as quickly as the sample will solidify, always keeping the level of the liquid helium at about the level of the liquid-solid interface in the cell.

Although the experiments were carried out over relatively long periods of time 36 hr), ortho-para conversion was not a problem since, as shown by Motizuki and Nagamiya (1956), ortho-para conversion in solid hydrogen is caused by the nuclear magnetic dipole interaction between nearest neighbor ortho molecules. An "isolated" ortho molecule cannot convert, and since at the impurity concentrations used in these experiments most o-H2 molecules are "isolated", no significant conversion should take place.Any conversion taking place would be reflected in the Qi (1) line. Examination of this line yielded no evidence of

conversion.

Experiment, Results, and Discussion

As already mentioned, the existence of bound pairs and higher order clusters of o-H2 molecules provides the means by which quantum diffusion can be examined. Nearest

Reproduced with permission of the copyrightowner. Further reproduction prohibited without permission.

- 9 7 -

neighbor o-H2 molecules are bound by the relatively long range electric quadrupole-quadrupole (EQQ) interaction of the o-H2 ground state, v = 0, J = 1 (Sears and Van Kranendonk 1964) . If one ignores the weak crystalline field interaction, the ground state of an isolated o-H2 molecule is triply degenerate. The ground state of a pair would, therefore, be 9-fold degenerate in the absence of intermolecular interaction. However, the EQQ interaction between two nearest neighbor ortho molecules splits the 9-fold degenerate level into four levels as illustrated in the energy level diagram in Fig. 5.1. At low temperatures (below 3 K), the vast majority of pairs is in the ground state, bound by 2.2 cm"1. Once bound into pairs, the ortho molecules cease to diffuse through the lattice until the pair breaks up. Because the isolated o-H2 molecules are relatively free to diffuse and the pairs and clusters are relatively free to break up due to the small binding energy, the distribution of o-H2 molecules among singles, pairs, and higher order clusters tends toward a thermodynamic equilibrium dependent on both temperature and impurity concentration, rather than, as had been previously assumed, being frozen into the sample at approximately random distributions upon solidification of the sample. Experimental work has taken advantage of the function of temperature and impurity concentration to investigate the phenomenon.


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

Fig. 5.1: The Q branch at 1.15 K for an o-H2 concentration of 1.6% along with an energy level diagram showing the splitting of the rotational level of an o-H2 pair by EQQ interaction. The numbers in parentheses give degene- i

VOracy; energies are given in terms of r , the »quadrupolar coupling constant, as well as cm- 1. Feature A in the spectrum is associated with transition A indicated in the diagram.

}

_ l l i U J_|K > O b

(i_ujo) !N3IOIdJ300 NOIlddOSaV


FREQUENCY

(cm'1)

- 9 9 -

The spectra in Fig. 5.1 show the Q-branch fundamental vibrational region of parahydrogen with ^ 1.6% o-H2 impurity. The lowest frequency feature is identified as the Qx (1) line due to fundamental vibrational excitations in o-H2 molecules. The strength of the line provides a useful indication of the o - H 2 impurity concentration of the sample. From an expression for the integrated absorption of this feature at ortho concentrations less than about 2% given by Sears and Van Kranendonk, one can obtain

[5.1] a[Qi (1)] = 1.0 x lO-3 C0 cm"1,

where CQ is the ortho impurity concentration. Experimental determinations of a TQ i (1)3 are in good agreement with this value. The next major feature, by far the strongest absorption in the region, is due to a fundamental vibrational transition in a p-H2 molecule accompanied by the reorientation of "isolated" ortho molecule with respect to other o - H 2 molecules in the lattice. Because of the symmetry of the h.c.p. lattice, a pure vibrational [Qi (0)] transition is not allowed and in the absence of an o—H2 impurity, there is no absorption in the Q-branch region (Van Kranendonk 1960). The line shape and width of this feature have already been discussed in connection with the Q X(D and Qx (0) + Qi Cl) features in Chapters III and IV, respectively.


- 1 0 0 -

The last major feature in the region, the one near 4155 cm-1, is due to a vibrational transition in a p-H2 molecule accompanied by an orientational transition in a pair of o-H2 molecules from the ground to the first excited state of the pair. The width of this line is found to be independent of time at a given temperature so that the . absorption at the peak is taken to be proportional to the concentration of ortho pairs in the sample. The data reported in this chapter have been obtained by measuring the growth of this feature as a function of time. It is evident from the spectra in Fig. 5.1 that this is the only feature to show appreciable changes with time.

The experiment consisted of solidifying a sample and recording the spectrum of the line due to pairs of ortho molecules 30 to 50 times over a period of 6- to 10-hr period. This procedure was repeated on the same sample of hydrogen at four temperatures, 1.15 K, 1.40 K, 1.77 K and 2.10 K.Before the run at each temperature, the sample was melted and resolidified in order to randomize the impurity.

The first stage of data reduction consisted of calculating the absorption coefficient at the peak of the 30 to 50 traces of the line. This was then plotted as a function of time from sample solidification for each of the four temperatures, as seen for a typical case in Fig. 5.2. A least fit of an equation of the form


/«

- 1 0 1 -

Fig. 5.2: Peak absorption coefficient of the featuredue to pairs of o-H2 molecules as a function of time from solidification as computed from the results of a typical experiment.


PEAK

AB

SO

RPTIO

N

CO

EFFIC

IEN

T

(CM-1

)

T= I.40K0.10

T*2J0K

005


PEAK

AB

SO

RPTIO

N

CO

EFFIC

IEN

T

(CM

')

Ts I.I4K

m

T - I.40K0.10

T*2J0K

005TTV1E (hr)


- 1 0 2 -

[5.2]

was made to the data at each temperature giving the rate constant, r, the t = 0 absorption coefficient, a + b, and the t = » absorption coefficient, a. At first glance, it is evident that the t = 0 absorption coefficient and the growth of the line are dependent on temperature. Slightly closer examination suggests that the rate constant is also temperature dependent.

Tlit t = 0 Ab4oA.ptj.on. Co&îcUznt

The temperature dependence of the t = 0 absorption coefficient is most easily explained. Since the spectral feature measured for Fig. 5.2 is due to a transition from the ground orientational state to the first excited state, the absorption coefficient for this line should be proportional to the fraction of pairs in the ground state. This fraction is easily calculated from the partition function for pairs of ortho molecules which can be written down directly from the energy level diagram, viz.,

where r/k has been taken as 0.78. Table 5.1 gives the fraction of pairs in the four pair states at various tem-

[5.3] Z = 4.0 + 2e P4r/kT + 2e-r/kT + e-6r/kTf

peratures.

with permission of the copyright owner. Further reproduction prohibitedwithout permission.

- 1 0 3 -

Table 5.1: Fraction of o-H2 pairs in each of the pair states as a function of temperature

Temp.(K) Ground state 1st excited 2nd excited 3rd excited

1.0 0.902 0.080 0.018 1.8x10"^1.15 0.857 0.114 0.029 4.9x10-**1.25 0.826 0.136 0.036 8.1x10“**1.40 0.782 0.168 0.048 1.49xl0-31.50 0.754 0.188 0.056 2.08xl0-3

1.77 0.685 0.235 0.076 4.18x10-3

2.10 0.617 0.279 0.096 7.52x10-3

It is found that the change in the ground state population as a function of temperature accounts for the temperature dependence in the t = 0 absorption coefficient to within experimental error which is no more than 5%. In a few cases, a t = 0 absorption point departed significantly from the value which was expected. In such cases changing the time origin for that run by 10 min. sufficed to bring the point into line. The origin of time is not very well defined since the time required to solidify the sample and bring it to the required temperature is about 10 min. Such a change, which has no significant effect on the rate constant, is therefore considered justified.

W * permission „ copyright owner. ^ ^ ^

- 1 0 4 -

As the fraction of ortho pairs in the first excited state becomes significant, a line becomes evident on the low-frequency side of the intense Qi (1) + reorientation line. This line is due to transitions from the first excited state to the ground state. Just a hint of thisfeature, marked by i, can be seen in Fig. 5.1.

T\ie Growth. Factofi

The growth of the line due to pairs can be characterized by the ratio of the t = ® peak absorption coefficient to the t = 0 peak absorption coefficient. The first step in the estimation of the growth to be expected is the calculation of the fraction of ortho impurities with zero, one, and two nearest neighbor ortho molecules for a random distribution of the impurity in the lattice.This is easily accomplished with the result that Pg =(1 - C )12, P = 12C (1 - C )18, while P, is taken to beo p o o t.1 _ p _ p where P , P , and P, are the fractions of p s s p cortho molecules with zero, one, and two nearest neighbors respectively. The second step involves the calculation of the equilibrium fraction of ortho molecules with zero, one and two nearest neighbors. To date, no exact calculation of these quantities has been possible, and it seems unlikely that any such calculation will be forthcoming.An approximate calculation has been carried out by Meyer


(1969), the details of which along with extensive tabulation of the results for temperatures between 0.5 K and 4 K and for o-H2 concentrations between 0.4% and 2%, is given in Appendix I.

Once the equilibrium distribution of o-H2 molecules has been calculated, the growth factor for pairs can be computed as a function of impurity concentration and temperature and compared with the experimentally determined values. The solid lines in Fig. 5.3 show C i.e.,the

htc RJrratio of equilibrium to random pair.concentratioa as.a..function of temperature for several o-H2 impurity concentration. The dots, circles, and triangles give the measured values at the concentrations indicated on the diagram.The qualitative agreement is reasonable; the general pattern of the data points is fairly well reproduced by the curves. Quantitatively, the agreement between data and theory is not good. The points for each concentration measured fall well below the theoretical line for that concentration, the experimental points at each temperature are bunched more than the theoretical calculations would suggest they should be, and the experimental points consistently show a sharp increase in the growth factor at the lowest temperature which is not reflected in the theoretical lines. Meyer (1969) does not make any detailed comparison of growth factor measurements (actually, in his case, the reduction factor for single molecules) and


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

F i g . 5 . 3 : T h e g r o w t h f a c t o r a s a f u n c t i o n o f t e m p e r a

ture for several o-H2 impurity concentrations. The data points represent experimental deter

minations while the curves result from the calculations presented in Appendix I.

-106-

Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

3.0-

'EP:RP

O-HzIMPURITY

0.4%0.7%1.0%

1.3%1.6%

003= EQUILIBRIUM PAIR CONCENTRATION CRP= RANDOM PAIR CONCENTRATION

0-H2 IMPURITY CONCENTRATIONo 0.7%• 1.2% a 1.6%

TEMPERATURE (K)2.0

- 1 0 7 -

comments that the theoretical approach given in Appendix I "should be considered sufficient for giving a qualitative picture of the equilibrium thermodynamic properties". It is easy, therefore, to blame all discrepancies on the appox- imations in the statistical treatment, in particular, the approximation for the configurational probability.

It is probably worthwhile to speculate on a possible explanation for the increase in the growth factor at the lowest-.temperatures. One of the models used by Oyarzun(1971) in his theoretical investigation of quantum diffusion is that of a gas of 0-H2 impurity molecules. It is possible to calculate the rate constant characterizing the formation of pairs (or the disappearance of singles) on the basis of the time between "collisions" of 0 -H2 impurities. If one assumes that all clusters of three molecules arise from the "collision" of a travelling isolated impurity with a stationary pair, then the rate constant is a factor of 2-^ smaller than that for the formation of pairs, since in the latter case both of the impurity species (two isolated impurities) are moving while in the former case only one is moving. At 1.15 K, the formation of triples whould become significant and if this process proceeds more slowly than that for the formation of pairs then it is possible that the peak absorption coefficient for the formation of pairs will reach a maximum and then actually decrease as


- 1 0 8 -

the number of triples reaches equilibrium. The sharp increase in the data at the lowest temperature might reflect this initially rapid growth since observations were not carried out over sufficiently long periods of time to determine the peak height at equilibrium by other than extrapolation, as evident in Fig. 5.2. One might attempt to observe a sample at as low a temperature as possible over a period of ^ 30 hr. to see if there is any evidence that the absorption of the line reaches a maximum and subsequently decreases. Such behavior would have one very unpleasant implication. It would indicate that neither the approach of single ortho impurities nor that of pairs of ortho impurities toward equilibrium would be truly exponential. In all cases measured to date, there is no evidence for an approach other than exponential.

The. Rate. Constant

The primary purpose in undertaking a study of quantum diffusion was to gain further information on the value of the rate constant as a function of temperature and impurity concentration. It is the rate constant which has the most direct connection with the dynamics of the phenomenon and which is, therefore, of greatest interest in testing detailed theories of quantum diffusion. The most detailed microscopic theory of the phenomenon developed to date is

with permission of the copyright owner. Further reproduction prohibited without

- 1 0 9 -

that of Oyarzun (19 71) and Oyarzun and Van Kranendonk(1972), the conceptual bases of which are given in Appendix I.

Amstutz zt a.Z. (196 8) suggested ortho-para intermolecular interchange as the mechanism responsible for quantum diffusion. This is possible due to the slight overlapping of the wave functions of neighboring molecules that results from the large zero-point motion typical of quantum crystals. One of the principal results of the work of Oyarzun (19 71) and Oyarzun and Van Kranendonk (19 72) is the calculation of the ortho jump frequency due to the overlapping of wave functions. These calculations show that, compared to resonant ortho-para conversion, intermolecular interchange is completely negligible as a mechanism for molecular diffusion (see Appendix I).

The experimental determinations of the rate constant at four temperatures and three concentrations are shown in Fig .5.4. As would be expected from the nature of the data in Fig. 5.2, the calculated error in the rate constant increases with increasing temperature and decreasing concentration. However, several aspects of the rate constant are systematic. There is a systematic increase in the rate constant with increasing temperature and impurity concentration. It is possible to obtain a quantitative



0.8)

-

-•— I I- cvi

O<\l

G)

OQI O I • i

(Q

in

hen h i ^ -i

he! I—H

tO

_L_CVJ

ro

CM

_L_No

_L_COo

iino

roo

(pJM) 1NV1SN00 31VU


TEM

PERA

TURE

(K

)

- 1 1 1 -

idea of the concentration dependence of the rate constant by comparing the ratios of the rate constant between various concentrations with the ratio of the concentration taken to various powers. Such computations show the data to be consistent with a c t o c1/6 dependence, probably closer to1/4c than anything else over the three lower temperatures

where errors in the data are not too large. Both the values of and concentration dependence of the rate constant appear to be in fairly good agreement with the theory of Oyarzun and Van Kranendonk (Appendix I) , although the temperature dependence appears to be much larger than that predicted.

In examining Fig. 5.5, which includes the data of Fig. 5.4 along with that of Amstuz zt at. (1968), one must remember that the quantities measured in the two cases differ: Amstutz zt al. measured the rate of disappearance(or creation) of singles while the data represented in Fig. 5.4 results from the creation of pairs. The two are indistinguishable as far as the theory is concerned where the only way in which singles are considered to disappear is through the creation of pairs, and the existence of triples, etc. is ignored. However in practice, as already mentioned, both pairs and singles may disappear through the creation of triples which may proceed with a rate constant different from that for pairs, so that the reaction rate for the


t\

- 1 1 2 -

Fig. 5.5: A compendium of data for the rateconstant as a function of temperature.The experimental values of Amstutz tt a.1. (1968) and theoretical values of Oyarzun (1971) are shown along with those from the present work.

mm


r (hr-1)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

THEORETICAL EXPERIMENTAL

OYARZUN |- o |.6%- • 1.2%- a 0.7%.

PRESENT WORK

' J \ AMSTUTZ et al. □ 0.5% J -----

0.

TEMPERATURE (K)


- 1 1 3 -

creation of pairs may not be directly comparable to that for the disappearance of single excitations, especially at temperatures below 1.5 K where the formation of triples becomes most significant. Qualitatively, the results from the infrared measurements are in substantial agreement with those from NMR work. Quantitatively, the agreement is good, although the large errors (typically ± 20%) in the NMR-determined rate constants in the temperature region from 1 to 2 K would obscure any but the grossest systematic differences which might exist.

There is much more work on quantum diffusion which could be done using infrared spectroscopy. Accurate measurements could probably be made up to 4 K by first letting the sample come close to equilibrium at ^ 1 K and then rapidly warming the sample to 4 K and measuring its approach toward equilibrium. With such an experiment, one should be able to reduce the errors in the determination of the rate constant in the temperature range 2 K to 4 K to about the same level as those determined at 1.4 K, although with increasing temperature the redistribution of absorption among transitions between various energy levels of pairs will be detrimental to accuracy. Such experiments require a dewar which will hold for 20 hr, or one which can be refilled in use without disturbing the temperature of the sample. The construction of more complicated dewars would also allow measurements to be


- 1 1 4 -

made at lower temperatures where NMR experiments by Amstutz zt at. show that the rate constant drops dramatically toward zero. Measurements at such temperatures

0.5 K) where the formation of triples becomes a major consideration, might also show evidence of differing rate constants for the formation of pairs and triples. Investigation of the phenomenon at lower concentrations is, of course, of great interest. Such measurements would require longer path lengths. The fraction of pairs in the sample is roughly proportional to the impurity concentration, so that measurements of a sample of 0.4% might be managed with a path of 8 cm, while 32 cm would be required for 0.1%. While double passing through an 8 cm cell to obtain 16 cm is practicable, paths much longer than this pose difficulties in the growth of a sample and the conversion of sufficient amounts of hydrogen. The investigation of quantum diffusion is time-consuming, and tedious; however the results of further investigation into the phenomenon will probably reveal much and more than justify the necessary effort.


- 1 1 5 -

APPENDIX I

QUANTUM DIFFUSION

A. Conceptual basis of the microscopictheory due to Oyarzun and Van Kranendonk

By far the most detailed microscopic theory of quantum diffusion to date is that of Oyarzun (19 71) and Oyarzun and Van Kranendonk (1972) . The theory is quite complex and no rigorous account of it would be appropriate. A review of the conceptual basis of the theory together with those equations which prove useful in interpreting the data is probably in order. Oyarzun and Van Kranendonk (19 72) found that the only interactions which are important to their theory of quantum diffusion are the EQQ coupling between ortho impurities and the nuclear magnetic dipole interaction between an ortho impurity and a neighboring para molecule. They found that the dipole interaction connects states corresponding to the conversion of an ortho to para accompanied by the simultaneous conversion of an adjacent para to an ortho. This energy-conserving conversion process is called resonant ortho—para conversion in contrast to the nonresonant process of an ortho going to a para accompanied by the creation of two phonons. The latter process, which has been shown to be the dominant one for nonresonant ortho-para conversion in solid H2(Motizuki and Nagamiya, 1956) , is allowed by the nuclear magnetic

oduced with permission of the copyright owner. Further reproduction prohibited withoutpermission.

- 1 1 6 -

coupling between two ortho molecules. Oyarzun and Van Kranendonk (19 72) show that for a single ortho impurity in a para lattice at absolute zero, the resonant conversion process is extremely efficient and, in fact, the average time the ortho excitation spends on any one lattice point is v 2 x 10 _1* sec. The excitation must be viewed as an "orthon" which travels in a wave-like manner. At finite temperatures, the interaction between lattice vibrations and the rotational angular momentum cause the motion of the excitation to become more random walk than coherent. At ortho concentrations of experimental interest (around 1%) , each ortho molecule has a significant EQQ interaction with other ortho molecules. Because of this anisotropic interaction, an ortho impurity cannot, in general, move one lattice spacing by resonant conversion and conserve energy since at its new position, the interaction energy with other ortho impurities will be different from the old interaction energy.The fact that transitions must conserve energy of interaction to within the quantum mechanical uncertainty reduces the jump frequency by a factor of about 1 0 for an ortho concentration of -v 1%, so that the average time an ortho excitation spends on a given site becomes v 30 min. Under such circumstances, the problem becomes entirely random

walk in nature.

After calculating the jump frequency, Oyarzun (19 71)

produced with permission of the copyright owner. Further reproduction prohibited wi'without permission.

- 1 1 7 -

takes two contrasting approaches to the calculation of the rate constant characterizing the exponential approach of the system toward a thermodynamic equilibrium of ortho impurities. One approach approximates the motion of the o - H 2 impurities as diffusive in character and calculates the mean square displacement of an ortho impurity as a function of time. The reaction time constant, t = 1/r, is then calculated by assuming that the reaction is half way when the mean square deviation is equal to half the initial average separation between ortho impurities. The approximations inherent in this method become good in the limit that the ortho impurities make many jumps, on the average, before meeting a second impurity. At impurity concentrations around 1% , the impurities make 5 jumps (Oyarzun, 19 71) and the approximations begin to break down The expression for the rate constant, r, which results from this method is r(3,c) = 61v (3,c) c^^, where 3 = 1/kT and v(c) = 0.41*10-5 sec"1 is the jump frequency inthe high temperature limit (3 -*• 0) . In this limit, r(c) = 25x10~5 c1/6 sec"1, where c is the ortho concentration.

The second method of calculation is based on the kinetic theory of a Lorentz gas. The number of collisions of a molecule with a cloud of randomly distributed scatterers is calculated by multiplying the volume of the cylinder swept out by the cross sectional area of a molecule by the average density of scatterers. In the application of this idea to


- 1 1 8 -

quantum diffusion, Oyarzun (1971) assumed that at each jump, an ortho molecule investigates six new lattice points for ortho impurities and that an ortho impurity will bind with the first impurity it meets. The first assumption is only good if the molecule does not retrace its steps, which becomes an increasingly good approximation as the number of steps taken, on the average before an ortho meets a second ortho, goes to one. This approach gives r(6,c) =144 v (8,c) c sec-1 or in the limit of high temperature (t > 1 K) , r(c) = 59><10~5 c* sec-1. These expressions result from the kinetic approach after the spread in the time between jumps is taken into account. If the assumption is made that jumps occur at precisely uniform intervals, the resulting expression is r(c) = 111 v(b,c) c ' ^ sec-1 or r(c) = 46*10-5 c1/3 sec-1. It is this expression, intermediate between that for the diffusive model and the more exact result from the kinetic model, which Oyarzun (1971) adopts as the best approximation for the value and concentration dependence of the rate constant. Calculations undertaken to determine the temperature dependence of the jump frequency (Oyarzun and Van Kranendonk 19 72) seem to indicate no significant dependence. Oyarzun (1971) calculated the effect on the time constant from the temperature dependence of the reaction cross-section and the rate of formation and dissociation of ortho-ortho pairs. He found that these effects give rise to an ^ 35% increase in the time constant between 4 K and 1 K.


- 1 1 9 -

B. Calculation and tabulation of the growth factor as a function of temperature and impurity concentration

As pointed out in Chapter V, the calculation of the equilibrium fraction of ortho impurities with zero, one, and two nearest neighbor ortho molecules as a function of ortho impurity concentration and temperature is the stumb- ing block to an accurate calculation of the growth of the spectral feature due to o-H2 pairs. The approximate calculation carried out by Meyer (1969) proceeds from an expression for the free energy, F, due to the distribution of ortho molecules among single, pair, and triple configurations. Meyer (1969) obtains

Ns V2 V3exp (-FAT) = Zg5ZoP Z, W,

where is the number of impurity molecules associated with the Ltk configuration, Zg = 3 is the partition function for single or "isolated" ortho impurities, was written in Chapter V, and Z, is written analogously to Z^ except that for convenience the energy levels of only one of the six possible triangular configurations have been used, in this case, those for the triangle with sides 1, 1, '3 which is the most probable triangle for either an h.c.p. on f.c.c. lattice. The energy levels for this configuration, as well as those for the other five possible in an h.c.p. lattice,


have been calculated by Miyagi (196 8). The configurational probability, W, is the stumbling block to an accurate calculation. Meyer (1969) gives an expression for W which becomes exact as the impurity concentration goes to zero. For this limit,

where M is the total number of H2 molecules in the sample.The second and third factors give the number of ways of arranging labelled ortho molecules and pairs. The first term compensates for the indistinguishability of the molecules Meyer (1969) chose a W for use at finite concentrations which reduces to the above as the ortho concentration goes to zero and which gives the equilibrium single, pair, and triple concentrations to be the same as those for a random distribution of the ortho impurity in the limit T -*■ ®. The expression for W then becomes

where N is the total number of impurity molecules and is the fraction of ortho molecules in the Itk configuration for a random distribution. Clusters larger than three are ignored as being insignificant, an assumption which is justified at all temperatures above 1 K and impurity concentrations below 2%. By minimizing the free energy with


- 1 2 1 -

respect to the , one can obtain the equations necessary to calculate the equilibrium fraction of ortho molecules in the various configurations as a function of concentration and temperature. The concentration enters through the expression for a random distribution of ortho impurity, Ps = (1 - CQ) 12, Pp = 12Cq (1 - CQ) 18, while PT is taken tobe 1 - P - P .p s

The tables below are preceded by the program used to create them. This program uses successive approximation to solve the cubic equation for the equilibrium fraction of ortho pairs which results from the above theory. The value of r used in these calculations is 0.78 K = 0.54 cm"1

The first line of each of the tables below gives the ortho concentration. This number is written in an "E" format where, for example, 0 .400 E-02 = 0.4><10"2 or 0.4% ortho concentration. The second line gives the fraction of ortho molecules with 0 (random ortho singles), 1 (random ortho pairs), and 2 (random ortho triplets) nearest neighbors for a random distribution of ortho molecules throughout the lattice. The remainder of the table gives the fraction of ortho molecules with 0, 1, and 2 nearest neighbors for thermodynamic equilibrium as a function of temperature. The column at the far right of each table gives the ratio of equilibrium ortho pairs to random ortho pairs which is the growth factor for a spectral feature


- 1 2 2 -

due to pairs. The numbers in the tables are written to many more places than can possibly be considered significant, especially in view of the approximate nature of the theory from which they result.


- 1 2 3 -

^ *0661• Ip(44I(M t'MI.I0t«Ml*«0t1lMt.<QSIMI.(n»lMI 6*0.1*IS • 1 .000 til »( • ‘ .90.2 I t « «< /<lit nui • *t*o.too >• cco • i./o • co • o.oai ♦ o.Mm.cn tot • 11.0 - C0>a*12 • ii«cn*u-<R>«ati •ot ••t-oni-ao*00 to Ll < M lIp • «. B •it* «*o *f»*i-o.*i*a6/iitiii •

e *(IM-7. ||*«6/tlLTl» • f IC *(1*1-3.(40*6/tlltlt ♦ •»*«-l.4||«fc/ Hl I •e • • «i*i-i.*u«-./'ainc *(i#l-1.0*'«*6/Til»lH • t«oi-l.0f4*C*l»Utllc • <*•! o. 11***.* tic t • •IT*IT O.U}*wyrictM • M*i 0.|l'*«iufll

C «|i»| Mli'C/tiUM • **•!t *(«fi j.o«i«*/rn,tu • <»•«c *««#» *.it»«c/*ani * t*»«C «(ui i,««i*cMiitii • (1*1 i.ni*wnunc .*«•! ».**4*r./TltT»l» ■ ir««o»«ts**ot/i(t««oiio* ius**asi**twii*Mo*i• • • U ( S * * n i * « i t / t iT « 4 Q r > o * tj.*o-*«a7»2i.T( • •••(/«. • »••»/!». *‘10 l t l . U . 0 1 60 TO >0 60 TO 40

M 4-iu ii.iu ii tieri.co toot l?NMI0l.'TM 00 l«(l UH (0013 *C* I0S*.M**T •'*(' C'C •♦•612.9)

« * • 0K • **«*<*t.«c * t _• ■tl * KCiK • .IF I U ^ M . 1 60 to I N I ( M i • I . losicti • «.(OSttTI • (i<Slt»i-l(MaTI-«Ct0l/2* -00 9000 JJ • I.WCO*ec • *qsn»i*«> • **imuTi«*i*o*iosiiti-«ai • KC/K10 I I X . 6 I . * . ! c n 10 10 CO 10 II

tl (o> • *n«X • KC 1014 • T O M ’.TI t u t i m • f . ’ i i m u w f l i

I ) f o n c t i • ( M e n • M iM i n * « « 4 i / j .■c* • iKtiiI# uc».it.:.ooi i fJi io 14 60 iu 1000 |« «* • » -tI* *• M l TO 1001woe corn..-*1001 (HSitU • mi* • (nuctn/l.loxin • u - losim • /ToT«(ninTi»«i/tts»40Ji«»i loticn • i*»<'iicti-4.i*n.n■OrtKtl • lT04ILtl-0.l*(‘1IH.fll/l»0*-0.l««P1| tO C04T|<4>#C U • CCO/7-2 10 I l C S . M l . c U i C O >0 19 CO TO U ts »4iT( (4.i::ji

1001 >04*41 ct-tl U CO»l iMiC•4|T( (».|0:ot C0.«OS.40*.40T.lTllTI.(aSltTl.C0*IL1>.(OtllTl. C40TTIt.ll. CT • 2.191 10 C94TI6UCtOOO >04*4Tt //ll.’CtlM] C3NC(4TtlTI0N •*.e«.)./.tl.'t449Q» > T K J 3I*CC

CIS •••*4.4,31.•41900* 0*1*4) >4t*J • •.-*•«• It. ' I « r * WtT-O n t >c u t s • ••>».*./ . u . ,Ti#*-*iru»i*.i».’‘5i.a:*tiu* K N C t t s ’.i*.C * (6 U iltB t | i • » « l t V . 4 1 . , ( a i | l I 4 4 |U * T tJ rv J 1 J « .J « . '(0 *» « « *» ./. C l l . fC I0 .J > * t .6 l3 .1 Q .9 t» C l* . l3 > 9 l« 6 l0 . l3 .9 t . * l2 * 9 l t S10F (40

(tit) ((KhTllTIIK *0 .*0P*-37 iitm o*t**i si9tti *0.4*10 •*•■<*'l(M(tlTUt( |OUlel4« I"* 1|46lt90.1001.001.10t.«0t . * 01.(01.001.10l .« 41.401.11

4.104.444.404.001.00

| . t * 4 * 4 * T 0 M 0.49*000 Toro 0.4*2 TO*I0?0 o.N«4ji)eco 0.444110*004 0.44TtT4*CCr 0.40414**060 0.010441 >000 0.024*910000 0.4901*42000 0.49*0419000 0.41*1012300 0.41*417000 0.4*1291*000 0.4*20*01000 0.44*0*12000 0.4*909*40*0 0.4414*21000 1.4*4*2**309 0.44*2991000 0.4*71211000 0.4*4211*000 4.4*09 * 14400

TtMftttuit 0.400( .ooo * 0.000 t.00 '

1*20 t . * 4 1.40 1.20 2.00 2.20 2.4*1.2*J.4*1 .M

huiiiitti» si*a(9 O.I2T044IOCO 0.*1917*1000 0*039124*600 0.7*721*9000 002*0717000 0.0*27*740<90 0.04*12*1000 0.I4I4IIT3OC 0.40*14*0090 0.4tl»*4*000 ‘0.0109*09060 0.422210*400 0.4210101000 0.42T2I944C0 0.1240*4*040 0.9304ll*4'*O 0.411*171 COO 0.4920X0100 (.411*04 VMM |.49**|Tf300 0.4191901*00 0.419*1**000 0.41*124*000 0.494*1*4000

•» O ltM i 4*1*1 • # .0 **7 > U M ll l* * l« * 4*141 O.I-fl-lUOnO

0.*l»«O II-IO C *.2*»t)»»:oo0.11*4119700 3.12*021*040 0.107107*060o.4**o*Tooae>>o iO.loWHCPOO-Ot 0 .*4 < 4 a2« '00*»0t 0 .4 *0 *9 * 1POOt>0!0 .*1 *9 12 1000--01 0 .4 **2 9 1 4 0 0 0 (^1 0 .9 *F *a74-03a-C l 0 .997*04X 001-01 0 .91 (1 *0*000*-01 0 .92*714*0001-0t o .9 i* 9 * o iro e « -o i o .9 ie > * 2 io o c i-o i 0 .90*1021000**01 o .* *> « i7 7 e o :( -o i J .*4 **4 1*3 00 *-0 1 (.*4007416801-91 0.*«*«2t1000>-01 *.*0392290001-01

t«4f*lft OOTlM) I* (Ocii I*** i*m» 0.24*3*4**00 o.wtO’ivM 1 .4 J 4 4 /^ M U T -V l 0.2*l)763uijOC-Ol (. 19(01 » * w »-01 0.10TI«4Uww<it-OI 9.407M31OV4J.-O2 0.**99‘»''HN1t -<*l l,1.W,C.IN0(-42 0.**4C’»O2«»*i*-OJ 0. * it T2-cw.ua -02 0. 1**11***M*U(-02 0.1T0*O**Oow<-02 0. 390*9 »*t*i04-O2 0.31*1)1* U0v4-02 0.)71'l**««0(-C2 0.1l0»a**wuuk-4l2 0. J0l***4vwt-92 0.2***0«*Ouu(-O20.2441*4*000()20.207*074000*-«2 0.2 T40!4*(M«i(-O2 0.2 Tla24*uuo*-«2 0 .2 *0 3*99 00 00 ^2

Q*T«1 »»1*1 • 09*0(OUII 104 (<!• 441*1 0.9. *.**1000 0.*192**9000 0.240*119600 0.200)44 VI00 *.190*41(000 0.121*1*100 0.101*429000 0.421*19 *1007-01 O.(*4)T*70CS(-OI 0.7401*4tOO*-01 O.T*(*4*IC30>-01 0.71121*9130*-016.*«ie*47oeo*-ot 0.41*1711003*-01 0.*1«21)033C(-Ol 9.912**400*-01 ,4 ).42444O0*-01 t211tM0M*(l *.*74*434 - 0.411*1*3900*-01 >..4.44 1*000* -01 t,#OOT'»20000*-Ol c.94*129 *000-01 0.94711**000*1

|4iffS ■ 0.0071|(<4.«U»

l l . « * l

* . • * • * 2.71*1 I.41U 1.1U42 t . - * O l I .*91* 1.2119 1.1171 I.2H* 1.7194 1.2017 I.U49 1.1*01

1.004*1.0022

00760 Cf*<(*!4< (lUOi 91>*1 II K « « ( i*m « (0.tOOo.»oo0.000

1.901.20! . * •IU O1.(0 2.00 2.20 U 40 2 .*0 2.04 10 *3.203.401UO3.00V 40*.2 0

rw>m.l**l<** 1I*M.II 0.1190*1 ICO) 0.147*1970900«*0l***1ooo0.77*42T*C3O o .2 4 *T i* tr- :o 0.0)4*<t||0.'0 o . o t iM io o e i 0.07**0*009 0.*40tO*2C<*0 Q.«*T* 7.90CO 0.40)2(4 IC30 0.4079472069 O.*10(9*l«0 0.4ll*2t*X3 0.4114104003 0.411127*073 0.41(742 7060 0.470011*360 0.4204(44030 0.42ll***3 TO (.42244(4060 0.42)104)060 (.421414.030 0.42**O74C00

((MOO* til 11 >»T*1 • 0.0***fO’tCKCU* **K1 0.9l-***)C30 0.*4 **•91*3*3 0.911 i4**eeo 0.221»11-6X o . i» 4 i i ) ic : o0.1)11*99000 0.114*7*1600 0.10702*9600o . a* ) * e n 6 r o * -o i0.47171*06337-01 ( . (1 * 4 )4 IC639-01 0 .(|t4 40 2C 3 ''*-O l 0 .» U 1*976001-61 0 .7 « **t4 4 6 6 6 (-0 l 0 .71 t2977C ::> -9 l 0.19*411 tC63>-0l O .T * . * l* 1653*-01 0.11*01240306-01 0 .12 9 (1 )70 :5 » -0 l 0.111*1716067-01 0.711221(3307- < t 0.13*1177633*-01 0 .1 0 0 * I903CI-O I 0 .44*7 )9 )0 06 1^1

IIMVa 041*41 Ttml70U U I— !••* 1 4 1*01 30. ).-2'l«000 0.1721-7-4WO.(«40)7 90w6*-4l

0 . *4 *9 * t7 g C *( -4 l 0.10917* 9«007 -01 0.219l'2*»‘'6t-<l 0.1*9l672uuf*-tO .J )» *» » t« IA l-6 l0.119 124-001*1 —4)3.1014 rvtuuO-gt 0.4 )**2 ••OwK -07 0.19*7 T**wwJ(-02 0.*30*27*«'*39-62 O.T1***7«vtiO--v2 0.7)9* I *9co0(-07 0.*4**03-<JUC>—42 0.*7*17*«9t'69 -42 0.*17*7 *«uUC*-42 0.4*22* **WC4-070.*2t4!*eiiU9i-420•(1741-OUVCf-02 0. *0 T*>4* UV9(—42 0.9*4)*—1JuC(-42 0.941(962006*-42

91>X C9<l‘ («MOO* OtTw T(4«((AfU IC

( .*0 0 0.400 0.(00 1 ^ 0 1.20 *

I(*<900 OBTX TOIBKtS • 003* fouic >** iu to m « i s w -b m oO .M )*1 **4 4 0 0.1*417*71100 7.71*7

0.70 *l*-4 v> 0« t-0 1 9.76770. J7W16KJ0OC-OI 3**1070.22421 IMMQf-41 2.79 79(.I974**0000«-Ol 2.7009 (0.114*2 • mwt(-*lt 1.449* [•**0. 4 71 7 Tf-J 0V0> -07 U*774(.•2**09l«n»0(-07 1.91*4 . •o. 72141*<*ow#-0/ U*)*4 ‘-Ja.*4*72"0*i«O*-O2 r't*0.*O»J-'7OWi«-42 1.3094 j *-*•0.9*7M*9gu0t-o2 U7*70 ••0.91721C7U«o(-O2 1.72*4 »♦1.M1U--MCI 1 • 1 I J-JJ0.*4I«9*206U»-4J2 1.17*1 »•|ak)l9l**4U«>*47 1.19#* •0. *7 ***0404-47 I.1J4* j

j-« . i* . - * 4 v* * - 07 l - t “ »B..19*e*94Ktu(«02 1.1077 |a.*2*7 l-/UW*-42 1.0497 ”0.*720r*lMXt-O7 I.* 0 74 4-00*.*|*74*94vO«-42 1.0749 , 90

■TKTtnN «0. ( 9 «0IK I U I '

J .1990*2340 (.19141-23:9 (.9*474)7091 0.1Q09ltOC00 (.71*909-600 0.0177149360 (.J**I24"0:0 (.0*1*41*630 0.071*24 73:3 0.0*2171*630 0.(4*)472090 o.04)2a(oe60 0.»4*447t330 0.(44441*630 0.4023112000 0.40*7*41600 0.409414)000 (.407*101' 60 (.40(44«a039 0.4044141660 0.4l()*4.C4O 0.4ttl077664 0.4IK 1*7040 0.4UU24000

0.*2***27C300.*4l*2**660 0.)2 77*4*604 0.7)4 7*00060 O.II**47*6M 0.19*417*660(.11*1*9*030 o . 120*2*7eoo 4.!!li*ate000.10* M*—600 0.44120*9656*-410.*9 )**I*C53(“61 0.47 1*1*20:3*-01 ( . . - • r i i i e - w * - 4 l0.*T44t94';6C*-fll0.>*)4|**OC3>-01 0«**4<0*l3OW-OI 0. ■ It 704*663* -41 o .o i* 4 7 i* e o :* - o i O.*7340l*63e*-0l 3.(1 )*9079*3*-4l 0.(07*7I*SM> -01 (.(OWOMXe*-*!(.<47)1370000-01

i M n * r«r>e x i> 9 • o .o i*« 11*6619 fgutil-dL.- ooios nee* o**-o Ttimn ICultl**!'** 1(1*1(19

*.*11o . i t o : - '< « jo 0.14)0177000 0.1024*9*«WJ 0.4410*410000-41 0 . ) ( • * ! <1UOL*—610.217*1-lOOOe-'ll 0.2192*710060-41 0.17111**0060-61 0.i1779Uooo«—•» (.131I142WI00-01 0.122*21 »wo6*-«t 0.11)7)47003*-41 0.10*7*210WO* —41 o . io t )2i>woe«-oi

• 3.00—

1.1071

0.4*441 :>-%20.41»2|4|OOCa-«2 0.404*2* IUGbO -070.*«tl?44<MVt-*2o. (*0**T - UW4* -*'2(.«»)** r*ww6> -/,20.07*41*9 0*04-02 0.(119*7*046*-42 0 . (4** 4*'• 40 « -42 a.74*)«2*uoO*—>2 1.0*191.07*1


o.J 2 0 *oo«u m <~o i

• m o o * o»T*fl t i t n i T t • • • o m(VlilUMhi* fIMCLCS

o.«rtoa;ti*ec-<n

o.2*172i*m»-«i4 .JT«M 4«bG 6*-0 l

' 0.lll»»OC39


- 1 2 5 -

'• -4 1 •4*00* MM II.M O

U NU NU NU N

U NU NU NU Nk .MU NUNU NU N

u ^ i t i 9 . * l ia l7 ; M 0 » N I ' •.32»it*i:*o

• . ) * * • « * « : 00

• » N *))4 * *P 1 0.**3 t**-.*90• .•4 *21 1**30 •.72*ioe«6oa •.»* luwco • .rV N « « .* c c U T M N T .'Q N •.77*702.*co0.7«U 72%JfO •.taa*3t**90o . i i t i v m s• .7442«4«V*0• .N lC *« rN O• .•QN***300 0.109a I It-OO 0.*07***«310 O .M IN t 'X O t.H O O )K<M O• .tt lU U 'Q O Q

M l» t

• .114|*4«9.40 •.37**<4|093 0.2*)**)*490 0 .2 2 l* t '0 0 « 0o.icmiccca•.li*i i2*K8 0.|a44*»tC90 U U V M IO C O 0.1 7*0*1 76M •.17632)1900 0.t**1l3*CCO •.1*1***«900o.i*im*»ooo o.miuceoa o.n«oatecono.ti**4Q*6ooo.nwocio0.14**4*193Qo.idmic::o( . I V i t W M•.1121004900

u « * la nai«o r u m l o t ' l l ! • • iu« t a ia i H I I .W O u r in O t •.)4922*A»v4 •.22****4wuft•.12101*2*40

•.•0|7l29*VCt-0t 0.*«N«6*vuC(H)| U*la*4)*.w4»-4I • . I * } * ! * ! * * ' * - * ! O.ITUal-gOCt-OI 0.*42«9*l«<;'.4| ••**l3*0t dwOt *41 9 . • * • * > • tv u e i-o i •.4)11 7410007-«l 0.4232a*».6CtNl O.N44|02oO««-9l 0.*0944*»go;l-0t 0. l«10l«agu0tNI o.i«*4»*;uocf-oi ••M97IN3«M»-0| 0.)>1’«%«ogC<Nl o. j f i« o r iw o e t* o t ••)*7T2)tiMQt-Ol

tl • •.*' (•»<«. 1.1*11 i.«;m I.vm|.«*4II . M I I 1.4214 i . t i n i . » * i» i.2**»i . t i* *I . N k l 1.1 I t * 1.144* l . l t * * i.iiii 1.10** u o * i * 1.0*1* 1.0*00 1.01)1 1.0*74 1.0*31 1.047)

o * t N e o * c i * t * tlUOOi nil* n t ( * * ( ( t tU « ( 0 .N 0

U N O • .»00

I . 00 U N U N U N U 90!.0C

2 .MI . N1.00U NU NU NU 004.004 . NU NU NU IO1 .M

no* o.i*<»-<niwuii.** ii • M*n«io . v . u t i r r o » - o i9.2U7***.*<*0

o . ) : : i u » c fi* o .*« :? »w>v> q O.UklM VCO o * » i;» ) * « 'c o o . * i i» o P ir? r• ♦•••Ukp'V'O 0.tt*ll*4?C*0. r )•*«*« to0.1** 130*330 o . r * ) )« r * * e o O.trCHU'OO• .7*H*0-310 • .7 H i* 4 ) : r * c 0.1 !«]• 1*3*3 0.77t44l):60 0.ttCt«3*C00 t . m m t c c c o . n ^ j u - o o 0.7ft U*«9CC Q.7«l«lt2?99 0.7*«I4*06CO 0.7402)2*030

(*WW1* AltMl Mt*t • I l l ' l l l«< III* l 0. *3*. ***i*09 0.4)*•**|30C 0.1*01*3*000 0. 13*11**300 0.3*«00t>900 0.3***1)1000 0.3 *1111 r?Q0 0.3tN***3(M 0.3C*4*03nOO o.i**3m«:oo 0.1*11*4*000 0.t***«l*390 O.l»241*2?90 O.t 1*1*11330 0.1 7*171 .'COO 0.17*0«»*900 9.l77l»4a*20 o.i7o*4t*coo0.l**13**JO O

0.t* l*0« C 33 00. la * 1**7300 0 .1*1 **0 *00 0 0.1*10*1*300 0 .1***13 *00 0

• • ••mu. 0*1*0 Ut*ltI0MI l*a «•# >• la'll tl O.M* I**|b0u 0.)4)»2!Nw0 o w * i* * ;< a u 4 9.lll*M*uyO 0 .1 )* * * *K m 0 O.ll2a2**gvo 9.*1t71*luuO*NIO .u o e tn .g g o r -o i0.J***O3;O»HJt Q.M210**(M*Jr-OI 0.t)1 '0A*w(l|H)| 0.144t40*gv0t-01 O.lIO** Wvwl-01 o.)4«*triMO«-oi 0.)M3*3«ggO(-4t e . ) M i* * iv o c « - o i O.1O2k2Mgt.0f-«|0.4*33 l*»:3Ct-Ot 0.MK HTuiQtNI O.*714)4lgg<3t-Ol0.4*4*4 7 luoOONI0.4*2*00* OW94-01 0.4*7«a01ggo(-OI o .4 ) ) u a iM o a - o i

I(«*(f*lli4| 0.400 U N O 0.100 I.00

U N

!6u!U«a:.» 114.11 o.)*4tt'ioee*-ci0 .3 1 *4 314)00 9.143411*? 30 •.101034*000 O.H**1'WCO 0 . * 4 * ) ) i :3 0 0• . * • • • •1 *9 6 0• . 7 t l * ) W K '0 • .7 )1 0 **4 0 0 0 0 .7 *4 *42 '0 90 • .t1 « * )4 i0 0 Q• . ? * * * 1*1000 •.rrt**T4coo ••TT tN tlO O O •.> •11 *4 *00 0 « . r a i3 4 iN N 0.30*4314000o . ta o *a * io o o t.nilaTMCO •• >011)44000 o.m**44oee• . » • • » * * w• .TN4>90000 •.•00*34*to o

U N O * 0 *1 *0 n t H • 0.1*4f6»ti!a*i:<* aaias • . 4 2 4 « i i t : r o

0 . * ) * * |4 * 0 0 0 0 .)*9 0 *4 lC ? 0• .122*2*1690 9.2t«4744CC0 0 .3**11*1900 0.33*713*000 ••21221*7000 0.30 0* to * eco 0.14311*0330 0 .1 *4 tc o *e ioo . i» o * o i* c ? o 0.11443)1000 0.17101*1009 0.17013*4000• • 1 * N * * N 0 0 0.14*014 7010 0 .14*4*2*909 0 .l* )1 0 3 3 ? 9 0 0.1*1*41*090 0.1*0144*000 0.|1a*7*3C9Oo . i i * e a i t : o o « . iia t4 4 io o o

0.41Q71U0C0•.W1042UQQ• .2)7M**gOO •.l>11*2)6w0 O .IIO N IU U O 0.1944400wO0.tTiaiicj40*-oi0.>*|4|«4guCt-Ol0.**0*C72ugC(-Ol0.*323*0*wuCI*010 . 1 ’ »1**>ggC |-0 t0.9*4122*og0t-010.911 >l*)vu02-01 0 .4 1 7a*34gwOI - 0 1 0 *4# 0a4*«uoN -« l0. 4«M44 >*00*N!• .*441*««WCt-Ol 0.4*1420-awCrf-Ot 0.4)7I44),'|-01 0 .4 700*4*4117 3*-OI 0.*2NOM*iiue<-01 0.«|IM*3«W*-1I 0.*l)aaa*wW01-01o.4*a>i*ttowQ(-oi

3.44C*3.»1)2.*m2.0*421.02241.412)1.4470t.)44)1.12721.2>3*1.2)041.14721.17111.1*421.1)201.11741.104*1.04*11.0*4*1.01741.0704

OiTMO CCK(*r«otlOa •O.HCI-41,M W 001*4) l l w . l l ^ . 7 i * * a**ooo ooTaQ o o io i tfaonoruac (uiii:mi»* itacifi (CiMtixtu*

••4 00 o . i4 « o it7 c : :> -o t o . * 1*4331000••N O 0.39M a2l»3C 0.*12170*200* • ! # • l . l 1 « ) l l t 4 <S 0.)oeia229001.00 0.*>110)7SC? 0.)3002*1?00UN Q.)*24)ik?}o o.2«aj*4e:eo

1 .n o .* i* i< > ii9 .*a o .2 4 4 tu i:o oU N t . i t l l i l H O ) 0 .2)< 1*3 ;e?0U N 0.***4 «2 *c?9 0.33)4*019002.N 0.70a* )**c 00 0.213444*eiQ2 . N 0.77tl7179?0 0.20)40003002.N 0.7)12*1*3*0 0.147*01*0002 . N 0.7*71)44900 0.t42)*Q7C0O2 .00 O .tkam iC O O 0.1**1110090U N 0.79134**900 0.1**40*0300U N 0.7441)41000 0.1*21*74000U N 0 .7 * ) 4 * l 19*0 0.17411*1300U N 0.7»«*)72?90 0.17 >4011300U N 0 .7 a 4 7 * * . ;c i 0.17472139OO♦ • N 0.7771**2140 0.17*714*0004 J 0 0 . 7 l4 l ) * 4 ? T 0.17 )4 )0 *990U N 0.7744 70*.?,Ml 0 .1 724043<1004 « N 0 . 777117)^90 0 . 17)4 I I tOOOu « 0 0.?7«40010M o . l 707*12990U N O.7Mt20MCO 0.1700012000

• o . i * i4 a t to n p o a iN t a i n r t s ■ o .o **3 » * ia j i o u K i t * i u * la iP K T ) ic # /* c *Q.IKl 7*2wOO

0 .)* * 7 4 » !M 0 2.42040.2941241000 3.24*10 .1« 2 )4 t*4b 9 1.4*42O.I*4||*3u<7Q 1.7)440.1212940000 1.17)70.102*7**w00 !.* 9 )40.»44*^1CtiW0<-01 I . ) * * *o .« 0 4 t:o 2 b 0 0 t- i i t . ) 0 t *Q .7 * U * * * l .0 0 !^ l 1.241)O.*41*4»90UO«-«l U T tJ IO .t lU 2 7 2 W 0 t-< l 1 .1 *1 )0.*2M 477g<.0*-0) 1.14**0 .*C 0 **4 1 « i0 7 -« l 1.11*4O . ia il lT tO w v tN I I.1 2 ) t0 .1 *9 9 |4 7 o u » |-9 t 1.10*10 .1 4 t4 *4 tg g g f-« l I.9M Q0 .4 N *0 1 2 0 v Q (N l . I.O*aO0.1)01*040C O t-9 l 1.07410.123 *0119*31—I t 1.672k0.11 12 *9 Ib b flf-01 1.0*440 .4 0 4 0 7 **g c n f-n 1.0*010.4O)4*A20bO*-Ol U 041*O .N B fia a u N k N I 1.0410

M T N C M C fa ra a tim * 0 J90 7-0 1li« 0 0 4 C N N S l* a .f1 *0 .7*47 • • * « ) • 0 * T N *a t« 4t ( M ( a i i u i t f U l t U * 4 t l» 4 1*6414

U N O 0.4 2 4 2 4 4 2 9 M fN l 0 .*10007-000• M O 9 .1 4 * .* * *6 0 0 0 .*2 « *a 7*600U N O • ,>*4212M Ô 0.1»C17HC9O

(N O • .**1 )7 *7 9 9 0 0.1)60441300UN 0 .1 4 0 1 **WOO ••3 41*42 **90 . 'UN •.*6162*1990 0.2 k ))**tC 0 OUN • • * * 4 * 1 V9<*0 0 .2« 12 0**:901.00 «7*1)oacM • .2 2 —244:002.00 0^4*4147090 0 .2 1 * * * * 7;90I.N 0.71 02 ** *090 0 .2 0 4 1 )* 7300UN •.722311)000 0 .202*004:002.N •.7 )1 17 0*9 60 0.1474*4*690UN •.7)«)72*OCO •.14UD1 :ooUN •.74*11*2900 •.1*91714600)*N •%7M| l**000 « . u n i * i : NUN •.n)t*»eeo •.11930*7390UN «.79*1*12000 o.iiui*o:ooUN l . 7 1 « h l t< ! ( 0 o.iaumrooUN •.7010161090 o . ia c 2 2 2 * :9 0UN t.7 *)449*0< '0 o.17*on*:ooUN •.7 *171*4090 0.177*790:90UN • .7 *7 )0 1 *0 0 0 •.1 7 7 0 4IKMUN •.r*«7*«io*)0 •.I7*i*ie3ooUN «.7*4««40000 a. 179*** UN

I aaNQ* MtN tamffl ICUtltaaii;* HIHItl0.4)71*** 900 0.)7**))*4MO 0.370***)g90 00242 7*990 0.l50)>47u00 0.124*24N93 0.1101*1tuwO0.17044*»wg9(-0l0.0 711*129427-01 O.N4N4*wwJI-Ot O.DlONlVgOCNl 0.7121f*4gwC(-01 O.NOl 1*09*37-010.*4472l"U03t-01 0.*))* 1NO0C7-01 M1MI44«0K1 0.N22243UO47-OI 0.9NNMtf*4f-«l 0.474**»*tt00t-0t 0.170 7*12wwul-01 O.IOWO 1*0097-010.144)04 70*01 N1 0.110**** UOC7-01 0.141*447W04(Nl

• 0.0*-4 t o * '* ! - ' 2.1444 2.*1*0 2.1744 1.412* 1.701* 1.4*47 !.*») 1.1403 l.2<4| 1.2*1) 1.20*7

1.114*I . I D )1 .1 )1 *1.11*11.104)1.04)4 1.0*4* 1.01*0 t.0700 t.0*)4 1.01471.04)4 1.0*44

OatK] CONI*l«*tli a .0.2707-01 **400* C*T-<1 II t|a*|4*7u*f

O IN O C 8 < f4 t* * 7 t* 4 •0 .2 1 0 IN 1 0M0QO 0*T«O SI4U.IS *0.7142O.NO 0 ^ 0 0

U N 1 *N U N U N U 00 2 NO 2 » N U N U N 2 .N U N U N

if t u a i * « iu * ) i * a i s i.m«mi roaoi-oi o.t4o*4tmo 0.114** 70040 0147«0D0N •.11I4)1!0N •.14*41*1090• .*•***04090 •.**24112000 ••** 7*411490• .*440*91090 •.71040*40*0 •.72010*1000 •.727 742*920 •.71)7)41040 0.7M4U40N ••7*37*7)000• .7**ialK»C• .7*411**000 •.fllaUKCO •.71|74)NN 0.741 70*1000 •.71724170N• .744 71*1000 •.7144 742NO

11*004 0*TN MU1 fOutltMIU* O.N2D4NOO 0.*2*)431900 0.17437**900 o.Dtai 71400 0.2***« 7*700 0.2»7t42470O• 47*)««?00 4.2 )24*t)?00 0.222)174200 4.21*40*1000 «.20774li:M 0.202*17*000 0.1«t>*aoooO 4.14434*^900 o . i4 2 * * * « :o o 0.|4O)0***N 0.1*4*24*300 O.I»«7* 7*100 0.1*1702*200 o.ta * i4 c*4Q O 0. |*M 1*1700o . i m i t i v i o• .I4117**NCa . ia

a 0.17W • 1*00* 0*7N HIHftS • 0-0434Mil) (OUlllNU* talPUH nwittv0.1444 *4 **M UMM14NN • J1112IVM 4.21I1NN •.||71*MMO 0.11NN4UW0 o.ttoo7otroo 0.10420DOUO 0.4*7712*N0l -01 o • • *«7 *o 7 «mo*-o i •.•tl*4l7aNC-«t 0.7 70Hll*«*f0(NI 0.71*4*2 *<M9C-41 0. 70444*0114 94-010N07)404v*»9tNI 0.**43)74V«M-0I0.*4 )*• * 40bA* -01o.*4iai4t*-*ot 0 . * to 14440b21-41 0.*7044*0*Nt-4l MU'IktkNHil0.N4 l*»ln<*’4t 0.4«4|»4 rwoMl 0.14)4412 UNONI

2.004)3.11*02.10*11.4430l . * M lU1I** 1.41)1 1.1)11 1.7773 1.2)17 t•147* 2.1*43 1.1*4* UITN 1.11*4 1.1011 1.0411 1.0*20 1.47)* 1.0*77 1.04)7 1.01*1 1.0*1*

0 .N O U N O O.MO

1.00U NU NU Nuoa2.002 . N2 .N2 . N2U 01.00).301 .N IN O U Nu o oU N

aaN C * n a tN M | i { < Ib U l l 17* 1u* 11*0.71 lO i ' l t l * a n

« .*4 7 *4 **? ? C t-9 l0.1*4»*?««»??o.l)«l*4*cr: 0.44**7l*C99 0.17*1)4 *090 0.1*241720 9 3 0.*22777JO<-3 0.*1111**000 0^77210*099 0.*a7*17’?C9 o . te r i* *7 ? c o 0.704)**3900 0.7171)!1C?Cf l .7 2 i i i* to e e0.72D**00C0 0.712111*000 0.71*917*999 0.7141017700 C.74|M*t09? 0.7*)(*1*030 0.7*4**ICO00 0.7* IN* JO?*) O.laNlOkON a. 71021 MOO

O.ltuiO 1000 0.4702777020 0.)7»14*1020 0.1)2*1)7CCO 0.3*4*71*000 0.210*1 >0090 0.2H** 0*000 0.2) 771*0090 0.7211* 12900 0.71 *76* >200 0.21227N9N 9.207*20*900 0.291*1*7200 0.20007*1000 0.1471711000 . 0.1441*77000 0.14)2)71000 0.1411)1*000 0.1101192090 0.ia«414)090 0.10744)1900 Q.|aT92*IQOO 0.11*22*1000 I.ID -1 U O O O

i a i d * o a iN ta nf O l i l l la a l t * I * 1*11 f 0.14*3) 1IJU9 0.14**4 Hw4 0 J41411UN 0.2221*44 bgO 0.t7k#«|*.g0 0.1**421*aN 0.1217* I'-uO 0.111*012 wO0.l0l.*0*0jv0 0.4)31 T47ggOO-Ot t.H144lM0Ht •.•)01*4*wu0t-0t 0 .74 4 4 4 1 7 M 3 tN I 0.7**1244<mO(-O1 0.743* )1«gC0tNt 0.72)14)2 0001-01 0.70 764*00(1Of-01 0.*4)*20>0u64-0l• . • • 2e io * -g o (H ) i0.*71*44)000*-01 9.M394«**.0(N1 0.*14**21 wOOf-Ol O.*44C6O*U0t-01 •.**12*12 NOt-Ol

1.47**2.77*4 2.01*1 I . *14)

I.)*** 1.)712 1.24*0 U77)) 1.1*41 t.l*)* 1.1*21 I.17N 1.IC*# t.9**t 1.3*77 1.01*4 1.97101.04*1 1.04*7 • 1.01*11.3107

oattao C0NI*ra* •1400* c a r « i ) ; T IN ia iT V I f

0 .N O U N O U N O1.00U NU NU N1 .N 7 . N 7.202 .N I . N 2.10

I . NI . NU NU N

.71(4 NJlOf-41 •H 11 •O .H ** *1400* M t N **1 *4(vntit)i!<•* SlNlfS «0>ai»aii'«* a.»l4l**06C3(Nl O.1742N70C0

U J2 1 1 N N 0 00^1*177*693•.11*41*10^00.171*2*7099• 11)4)1009 •.*N22*20CC 0 ^ * I ) * * N 0 9 •.*772144993 B .N •1*4*099 0.*44t»l2C?3 0.70«a74*O09 • . I l l lO iT N O •.7113)12991• .722401*009• .7 2*9 *)I99 3 •.7246*4*961 •.7111419099a . t i n ia T o n•.7114)1)001• .7)l**9i:69• .tiaiOTON•.7 *4*4 )4000

0.)41002*090U41S*))I0MU)l*l4*a«60 9.11)44*2000 Q.244|9**0M 0.371412N0O •.21*«l1*090 0.2*10472009 9.2)07271000 0.227*41)000 0.31*41)1630 0.211*420000 0.2077*4)000 0.20*4)10000 0.201*711090 0.14**121000 0.147*102000 9. I**t491900 3.1*. 71*1010 0.1414441900 0.1429*10090 9.1*1* ***990 0,14Q*a44C90 0.110I4**OM

• 0.111* IMW*. MTN 7*m*T1 • e.0*21**1*1 (Ouliiaaiu* 7«tai(Ti K.*«*ua O.)*l*202<i00 1.40*7• .N * * )a 'S v O O.Ml«227btfO •«2)22771W0 •.1*1*44*ggO ••11***2W m O 0.1))**O30U0 0.ll*ll*4oN 0.107*2**3000.441*411.3(-0)0.4)a*21*wwu«-«l 0.a*1147KwN-<l 0.04 |79**bOO#-OI •.a3iia*U«NNl•.t404l**MM-4l•.77a**|7'A«l»-«l |.7*1**0**491NI 0.7*704 IU-CI -01 0. 714?**7«.w0a-Ol •.72*)/4 )w«* -41 0.7141 74*wuO*Nl• ,70T|4»TW>*1 0.700<:44|gM»-41• .*4)4*0kgu0t-0l

2.1*0*

1.1770 1.1070 1 .2 **) 1.21*1 i . i * i a I.()** t . M * l 1.1144 1.1097 1.34*1 l.3<**7

1.34 2*


- 1 2 6 -

APPENDIX IIR eprin ted fro m T in : Ncvo-w Si ik n t ik u : I n s t k i' m i- v r> .

rrinlcti in U. S. A.\ «»l. 41. \ t i . 0, l.^p'Kl.170, September 1970

A Microampere Current Source with a Modulation Capability and

200 V Compliance*

R. H. Muxxisrs and S. A. liounst

Department of Physics, Unircrsity o f Voronin,Toroitlo 5, Ontario, Canada

(Received 20 April 1970; and in final form, 27 -May 1970)

' I 'H IS unit was designed to provide dc bias for an unusually high resistance I’bS cell (100 M l!) used

as the detector of mechanically chopped radiation in an ir spectrometer. The unit will be of use in many applications where a constant current of between 1 and 100 #»A is required and a potential of up to 200 V is necessary. In some less critical applications, the large compliance range can be used to reduce the need to control background radiation level at the detector.

Figure 1 shows a schematic of the circuit used. The component values shown are those used in the 1 yik unit. The circuit can be adapted to provide an adjustable output anywhere in the range from 1 to 100 pA. The 500 kfi potentiometer can be used to adjust the output by ±20% . Larger changes in output current are best made by changing the total resistance R, connected in series with the source lead of the field effect transistor (FET). The output current is proportional to 1/2?,. A 10 pA current source would require that R , be changed from 15 to 1.5 MQ. While operating from a 250 V supply this circuit has been used to provide 1 /s.\ with a measured minimum output impedance of 2X10’ 9. or 7 /i\ with a measured minimum output impedance of 2 X 10s 12 throughout a load voltage range of 0-200 V.

The rms noise component of the output current is less than 1 part in 105 of the total output, e.g., 1X10-11 A in a 500 H z bandwidth when the source is used to provide 1X10-4 A. This noise component does not appreciably degrade the signal-to-noise (S, X ) ratio of the detector.

A facility is included for modulating the output current to a known degree. This permits a direct measurement by comparison of the detector equivalent noise current.

An F E T rather than a bipolar transistor is used in

-230

awOUTPUT

io lufIMCOUIATION

INPUT

Fig. 1. C ircu it diagram ut a I u A current source.

order to avoid the leakage current that would be associated with the reverse biased collector-base junction of the latter. The 2X5543 is manufactured by Texas Instruments and sells for S5. I t is designed for high voltage use and is characterized by a 300 V minimum drain gate breakdown voltage. The output resistance of an FET, connected as this one is with a resistance R, in series with the source lead, is

R,= (l+R£f,)/G,„where R0 is the output resistance, R, is the total resistance connected in series with the source in ohms, Gy, is the forward transconductance in mhos, and G 0, is the output admittance in mhos.

These G parameters are not identical to the small signal y parameters used in the manufacturers’ specification sheets. However, when the corresponding y parameters are substituted into the equation they can provide a useful approximation of the performance to be expected of the device. When this substitution is made for the 2X5543 the calculation indicates that the output resistance will be 1X107 9. at 100 fi\ output current. We can reasonably assume that the ratio of Gy, to G „ will be constant down to 1 jiA and that the required value of R, will be inversely proportional to the output current. This being the case we conclude that the output resistance will be 1X103 9. at 1 /iA- The measured values of the output resistance show good agreement with these estimates. This indicates that no new or unexplained mechanism is responsible for the performance of the device and that we therefore should expect it to be reliable.

Since this is a high voltage resistance circuit, reasonable care should be exercised in the selection and use of the materials used in its construction. This would involve the use of Teflon insulated wire, a glass epoxy board for the reference source, and a Tellon-insuiated output connector. I f the 250 V supply used is not well filtered the output noise current will be increased. The authors found that, in such a case, a 20 kfl resistor and a 40 fc i 450 wv electrolytic capacitor functioned effectively when used as shown by the dashed lines in Fig. 1.

* T h is ivurk was sup|«irtcil in (a r t In’ a i;nuu from the .National Research l'<»unctl <»l t.unada.

f Holder of a I ’rm uice ut Ontario tira .iuatc fdluvvshifi.


APPENDIX III

SCHEMATIC DIAGRAMS

A. Reference detector

When radiation source chopping is used with lock-in amplification it is necessary to provide the lock-in amplifier with a reference signal. This signal is obtained by placing a small light on-one side of the chopper and the reference detector, which uses a phototransistor, on the opposite side. The circuit shown below has the advantages of fast response time, high sensitivity, and extremely low current drain when no light is incident on the phototransistor.

B. Heater power amplifier

The manner in which this power amplifier was used has been described in Chapter V. The amplifier has a maximum gain of 'v 103 and provides a choice of ten time constants varying from 0.002 sec to 2 sec. A voltage bias control is used to obtain approximately the correct heater power and the power amplifier controls around this preset bias.The maximum output is 1.5 amperes at 15 volts. Switches Si and S2 are located on the rear pannel and are used during the adjustment of voltage and current offset potentiometers. These switches are shown in their normal positions. The voltmeter circuit, which is provided to measure the potential across the output, is not shown.


- 128-

N O

Q


-129-

fawHfaH

faofafawrt!was

H ! M PHlh— «H ih —f

- t t l - S

— I?— ;

s:4-


- 1 3 0 -

BIBLIOGRAPHY

Allin, E.J., Feldman, T., and Welsh, H.L. 1956. J. Chem.Phys. 5, 1116.

Allin, E.J., Hare, W.F.J., and MacDonald, R.E. 1955 . Phys.Rev. 9J3, 55 4.

Amstutz, L.I., Thompson, J.R., and Meyer, H. 1968. Phys.Rev. Letters 2_1, 1175.

Berlinsky, A.J. and Harris, A.B. 1971. Phys. Rev. B4, 2808.Brinbaum, A. and Poll, J.D. 1969 . J. Atmos. Sci. 26_, 946 .Boggs, S.A., Clouter, M.J. and Welsh, H.L. 1972. Can. J.

Phys. Accepted for publication.Clouter, M.J. 1968. Ph.D. Thesis, University of Toronto.Clouter, M.J. 19 72. To be published.Clouter, M.J. and Gush, H.P. 1965 . Phys. Rev. Letters 15_, 200 .Clouter, M.J., Gush, H.P. and Welsh, H.L. 1970. Can. J.

Phys. £8, 2 37.Coll, C.F., III, Harris, A.B., and Berlinsky, A.J. 1970.

Phys. Rev. Letters 25_, 85 8.Cremer, E. and Polanyi, M. 19 33. Z. Phys. Chem. B21, 459.Cunningham, C.M., Chapin, P.S. and Johnston, H.L. 1958. J.

Am. Chem. Soc. 80_, 23 82.Dennison, D. 192 7. Proc. Roy. Soc. London 115, 483.Depatie, D.A. and Mills, R.L. 1968. Rev. Sci. Inst. 39_, 105.Gush, H.P., Hare, W.F.J., Allin, E.J., and Welsh, H.L. 1957.

Phys. Rev. 106, 1101.Gush, H.P., Hare, W.F.J., Allin, E.J., and Welsh, H.L. 1960.

Can. J. Phys. 3_8, 176.Gush, H.P., Hare, W.F.J. and Welsh, H.L. 1958. Private

communication.


- 1 3 1 -

Hardy, W.N., Silvera, I.F., and McTague, J.P. 1969. Phys. Rev. Letters 22_, 29 7.

Hardy, W.N., Silvera, I.F., and McTague, J.P. 1971. Phys. Rev. Letters 26_, 129.

Hare, W.F.J., Allin, E.J., and Welsh, H.L. 1955. Phys. Rev. 99 , 18 87.

Hatton, J. and Rollin, B.V. 1949. Proc. Roy. Soc. London A199, 222.

Heisenberg, W. 1927. Z.F. Phys. 41, 239.Hill, R.W. and Ricketson, W.A. 1954. Phil. Mag. 45_, 277.Homma, S. and Matsuda, H. 196 8. Prog. Theor. Phys. 40,

1191.Homma, S., Okada, K., and Matsuda, H. 1966. Prog. Theor.

Phys. 36, 1310.Homma, S., Okada, K. and Matsuda, H. 196 7. Prog. Theor.

Phys. 3j[, 76 7.Hund, F. 192 7. Z.F. Phys. 42, 9 3.James, H.M. and Raich, J.C. 196 7. Phys. Rev. 162, 6 49.Jarvis, J.F., Meyer, H. and Ramm, D. 1969. Phys. Rev.

178, 1461.Legge, J., Boggs, S.A., and Timsit, R. 19 72. Proceedings

of the 17th Symposium on the Art of Glass Blowing, American Soc. of Scientific Glass Blowers. Accepted for publication.

McLennan, J.C. and McLeod, J.H. 1929. Nature 123_, 160.Mertens, F.G., Biem, W. and Hahn, H. 1968. Z.F. Phys.

213, 33.Meyer, H. 1969. Phys. Rev. 187, 1173.Mills, R.L., Schuch, D.A. and Depatie, D.A. 1966. Phys.

Rev. Letters 11_, 1131.Motizuki, K. and Nagamiya, T. 1956. J. Phys. Soc. Japan

11, 93.


-1 3 2 -

Munnings, R.H. and Boggs, S.A. 19 70. Rev. Sci. Inst. 41,1369. —

Nakamura, T. and Miyagi, H. 1970. Prog. Theor. Phys. 44,883. —

Noolandi, J. 19 70. Can. J. Phys. 48, 20 32.Oyarzun, R. 1971. Ph.D. Thesis, University of Toronto.Oyarzun, R. and Van Kranendonk, J. 19 71. Phys. Rev. Letters

26, 6 46 .Oyarzun, R. and Van Kranendonk, J. 19 72 . Can. J. Phys. 50,

1494. —Pauling, L. 1930 . Phys. Rev. 36_, 430 .Prior, W.R.C. 1971. Ph.D. Thesis, University of Toronto. Prior, W.R.C. and Allin, E.J. 1972. To be published.Raich, J.C. and Etters, R.D. 196 8. Phys. Rev. 16 8, 425 .Sears, V.F. and Van Kranendonk, J. 196 4. Can. J. Phys.

42, 9 80.Silvera, I.F., Hardy, W.N., and McTague, J.P. 1969.

Discussions Farady Soc. 4J3, 5 4.Simon, F., Mendelssohn, K., and Ruheman, M. 1930. Naturwiss

18, 34.Soots, V., Allin, E.J., and Welsh, H.L. 1965 . Can. J. Phys.

43, 19 85.Stewart, A.T. and Squires, G.L. 1955. J. Sci. Inst. 32,

26.Ueyama, H. and Matsubara, T. 196 7. Prog. Theor. Phys. 38,

784.Van Kranendonk, J. 195 7. Physica 23_, 825 .Van Kranendonk, J. 195 8. Physica 2_4, 34 7.Van Kranendonk, J. 1959 . Physica 25_, 10 80 .Van Kranendonk, J. 1960. Can. J. Phys. £8, 2 40.Van Kranendonk, J. 1972. Private CommunicationVan Kranendonk, J. and Karl, G. 1968 . Rev. Mod. Phys. £0, 531.


- 1 3 3 -

ACKNOWLE DGEMENTS

The research reported in this thesis benefitted greatly from the financial, moral, and scholarly support of the author's supervisory committee, Professors E.J. Allin, J. Van Kranendonk, and H.L. Welsh.

The author is especially indebted to Professor Welsh for his help in the preparation of this thesis.

The patient assistance and miraculous glass blowing of Mr. J.E. Legge was of great help in the course of this research.

The award of University of Toronto Open Fellowships and Province of Ontario Graduate Fellowships are gratefully acknowledged.


Documents

Structural and Quantum-Diffusion Studies by Steven A ... · rotational specific heat capacity of H2 gas (Hund 1927 and ... it is generally taught that homonuclear diatomic molecules