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1
Novel Far-Infrared Application to
Spectroscopic Techniques and Their the^Stu&y-of- Zeolite Chemistry
Simon Robert Gibbon
A Thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of
Imperial College
Department of Electrical Engineering, Imperial College of Science and Technology, South Kensington,London SW7
November 1987
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To Kate
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Abstract
The use of untuned resonators has been developed so as to allow absorptions of zeolites to be studied below 100cm“l.
Computer simulations of the radiation density within various geometries of untuned resonator were used to determine the best design for untuned resonators to study solid samples. Several different resonators were used in order to test the predictions of the simulations.
An untuned resonator was used to study the absorptions of sodium exchanged zeolites X, Y, A and L.No sharp absorptions were observed below 100cm“ .̂ Silicalite, a high silica zeolite, was also studied and found to have weak absorptions below 40cm"l when dehydrated. These absorptions were seen to disappear on rehydration of the zeolite.
The vibrational spectra below 400cm~l of a series of zeolites were obtained, in order to gain insight into the interactions of adsorbates with cations within the zeolites.
Monovalent cation forms of hydrated zeolite X were studied both at 303K and 108K. Spectra at low temperatures show localisation of cations about the most energetically favourable site. This has revealed information on the mobility of different cations and their degree of solvation within the zeolites.
A series of monovalent-divalent mixed cation exchanged zeolite A have been investigated and their spectral properties related to ion-exchange properties.
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The spectra of a large number of cation exchanged zeolite L samples have been determined and related to the results obtained from Szilard-Chalmers recoil studies.
These results have been used to obtain a clearer picture of the environment of cations, both solvated and dehydrated, within zeolites.
Acknowledgements 5
I am greatly indebted to Professors J.C. Anderson,H.A. Gebbie and L.V.C. Rees for their friendly supervision and guidance.
I would like to thank all my friends in both the Electrical Engineering and Chemistry Departments for many fruitful discussions, encouragement and generosity.
Financial support was provided by the Science and Engineering Research Council in the form of a Research Studentship during the course of this work.
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CONTENTS1 Introduction ................................... 141.1 Zeolites ..................................... 141.2 Major Uses of Zeolites ....................... 141.2.1 Sorption ................................... 151.2.2 Molecular Sieving .......................... 151.2.3 Ion Exchange ............................... 161.2.4 Catalysis .................................. 161.2.5 Curious Uses ............................... 171.2.6 Future Uses ................................ 181.3 Zeolite Structure ............................ 181.3.1 Zeolite L .................................. 191.3.2 Zeolite X .................................. 211.3.3 Zeolite A .................................. 221.3.4 Mordenite .................................. 231.3.5 Silicalite-1 ............................... 251.4 Infrared Spectroscopy for the Study of ZeoliteStructure .............. 261.5 Magic Angle-Spinning Solid State NuclearMagnetic Resonance ............................... 291.6 Far-Infrared Spectroscopy .................... 301.7 Far-Infrared Spectroscopy of Zeolites ........ 311.8 The Untuned Resonator Technique .............. 391.9 Objectives of the Work ....................... 402 Fourier Transform Spectroscopy................. 422.1 The Mathematics of Fourier Transform Spectroscopy ......................................... 433 The Theory of Untuned Resonators ............... 533.1 The Theory of Homogeneous Cavity Sample Cellsfor Gaseous Absorbers ............................ 533.2 Solids in Homogeneous Cavities ............... 633.3 Absorption of Parallel Sided Sheets .......... 673.3.1 Calculation of Absorption .................. 673.3.2 Interferometer Transmission ................ 713.3.3 Reflection from Vacuum Cavity Window....... 743.3.4 Absorption of Parallel Sided Sheet within anUntuned Resonator ................................ 773.4 Inhomogeneity ............................... 794 Cation Vibrations within Zeolites ............... 805 Experimental Techniques and Equipment.......... 865.1 Characterisation of Zeolites ................. 865.1.1 Supplied Zeolites .......................... 865.1.2 Preparation of Ion Exchange Samples ........ 875.1.2.1 Zeolite L ................................ 875.1.2.2 Zeolite X ................................ 875.1.2.3 Zeolite A ................................ 885.1.2.4 Silicalite-1 ............................. 885.1.2.5 Mordenites ............................... 885.1.3 Hydration .................................. 885.1.4 Dehydration ................................ 885.1.5 Polythene Pellets .......................... 895.1.6 Analysis of Zeolites ....................... 90
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5.1.6.1 Wet Chemistry :- Analysis for Silicon andAluminium ........................................ 905.1.6.2 Electron Spectroscopy for Chemical Analysis(E.S.C.A.) ....................................... 905.1.6.3 Thermogravimetric Analysis ............... 915.1.6.4 Mid-Infrared Analysis .................... 915.2 Far-Infrared Spectroscopy .................... 925.2.1 Instrumental ............................... 925.2.2 Computation ................................ 955.3 Untuned Resonator Far-Infrared Work .......... 975.3.1 Common Experimental Technique .............. 975.3.2 The Original Cavity System................. 995.3.3 The Novel Cavity System .................... 1016 The Far-Infrared Spectra of Zeolites ........... 1056.1 The Effect of Hydration on Cations withinZeolite X ........................................ 1056.1.1 Sodium Zeolite X ........................... 1086.1.2 Potassium Zeolite X ........................ 1096.1.3 Ammonium Zeolite X ......................... 1136.1.4 Lithium Zeolite X .......................... 1146.1.5 Common Themes From Zeolite X ............... 1156.2 Hydration of Zeolite A ....................... 1166.2.1 Sodium Zeolite A ........................... 1186.2.2 Divalent Cation Zeolite A .................. 1206.2.3 Themes From Zeolite A ...................... 1246.3 Cations in Zeolite L ........................ 1256.3.1 Cation Site Identification in Zeolite L .... 1276.3.2 Effect of Hydration On Zeolite L ........... 1346.3.3 Varying Degree of Exchange in Zeolite L .... 1376.3.4 Themes From Zeolite L ...................... 1416.4 Natural Clinoptilolite ...................... 1426.5 Zeolon 700 - Natural Ferrierite .............. 1466.6 Mordenites ................................... 1496.6.1 Themes From Mordenite ...................... 1587 Computer Simulation of Untuned Resonators ....... 1607.1 Objective ................................... 1607.2 Background ................................... 1607.2.1 Simplifications ............................ 1617.3 The Components of the System................. 1637.3.1 The Source ................................. 1637.3.2 The Cavity ................................. 1647.3.3 Absorber ................................... 1647.3.4 Detector ................................... 1657.3.5 Radiation .................................. 1657.4 Mathematics of the Simulation ................ 1657.4.1 Ray Bouncing Program....................... 1657.4.2 Absorber Interaction Program ............... 1757.4.3 Cavity Radiation Program................... 1757.4.4 Cavity Hole Calibration .................... 1788 Results of Computer Simulations of An UntunedResonator ............ 1798.1 Background ................................... 1798.2 Results ...................................... 1828.2.1 The Effect of Varying the Plug Size ........ 182
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8.2.2 The Effect of Varying the Entrance and ExitHole Sizes ....................................... 1858.2.3 The Effect of Varying the Exit Hole Size .... 1888.2.4 The Effect of Varying Exit Height .......... 1918.3 Design Considerations for an Untuned Resonator 1948.4 Lessons Learnt ............................... 1959 Untuned Resonator Results ...................... 1979.1 Determination of the Cavity Q ................ 1979.2 Polymers ..................................... 1999.2.1 Synthetic Polymers ......................... 1999.2.2 Natural Polymers ........................... 2019.2.2.1 Cellulose................................. 2019.2.2.2 Lignin.................................... 2059.3 The State of Water within Zeolitic Cavities ... 2069.4 Determination of Effective Pathlength In theNovel Cavity ..................................... 2119.5 Uniformity of Radiation Field within the NovelCavity ........................................... 2159.6 The Effect of Hydration on the VeryFar-Infrared Spectrum of Silicalite .............. 21710 Conclusions ............................... 22210.1 Far-Infrared Spectroscopy of Zeolites ....... 22210.2 Untuned Resonator Studies ................... 223
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FIGURES1.1a Building Units of Zeolite L Structure ....... 201.1b Atomic Structure of Zeolite L perpendicular tothe C-Axis ....................................... 201.1c Bond Network of Zeolite L perpendicular to theC-Axis ........................................... 201.Id Atomic Structure of Zeolite L parallel to theC-Axis ........................................... 21l.le Bond Network of Zeolite L parallel to theC-Axis ........................................... 211.2a Structure of Sodalite Cage in Zeolite X ..... 221.2b bond Network of Zeolite X ................... 221.3a Atomic Structure of Zeolite A ............... 231.3b Bond Network of Zeolite A ................... 231.4a Atomic Structure of Potassium Mordenite ..... 241.4b Bond Network of Potassium Mordenite ......... 241.4c Atomic Structure of Calcium Mordenite ....... 251.4d Bond Network of Calcium Mordenite ........... 251.5a Secondary Building Unit of Silicalite ...... 251.5b Atomic Structure of Silicalite-1 ............ 261.5c Bond Network of Silicalite-1 ................ 262.1 Fourier Spectrometer ......................... 432.2 Interferogram of Single Frequency Source ..... 462.3 Interferogram of Broad Band Source .......... 493.1 Untuned Resonator ............................ 533.2 Photons Through Cavity Hole .................. 583.3 Screening Effect ............................. 663.4 Parallel Sided Sheet Geometry ................ 683.5 Interferometer Interference .................. 723.6 Transmission of 6 Micron Beamsplitter ........ 733.7 Transmission of 35 Micron Beamsplitter ....... 733.8 Transmission of 100 Micron Beamsplitter ...... 733.9 Transmission of 135 Micron Beamsplitter ...... 733.10 Variation of Transmission of llOum CavityWindow with Wavenumber ........................... 753.11 Variation of Window Transmission withAbsorption Coefficient ........................... 763.12 Variation of Window Transmission withRefractive Index ................................. 763.13 Variation of Window Transmission withThickness ........................................ 763.14 Variation of Window Transmission withWavenumber ....................................... 763.15 Variation of Absorption Integral withAbsorption Coefficient ........................... 773.16 Variation of Absorption Integral withRefractive Index ................................. 773.17 Variation of Absorption Integral withThickness ........................................ 783.18 Variation of Absorption Integral Transmissionwith Wavenumber .................................. 785.1 RIIC FS-720 Fourier Spectrophotometer ........ 935.2 Construction of a Golay Cell ................. 945.3 Original Cavity Set-Up ....................... 1005.4 Original Cavity Signal Flow .................. 1015.5 Novel Cavity Set-Up .......................... 103
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5.6 Novel Cavity Signal F l o w .................... 1046.1a Cation Site I' in Zeolite X ................. 1066.1b Cation Site I in Zeolite X .................. 1066.1c Cation Site II' in Zeolite X ................ 1076 . Id Cation Site II in Zeolite X ................ 1076.2 Far-Infrared Spectra of Sodium Zeolite X ..... 1086.3 Far-Infrared Spectra of Potassium Zeolite X ... 1106.4 Far-Infrared Spectrum of Ammonium Zeolite X. .. 1136.5 Far-Infrared Spectrum of Lithium Zeolite X .... 1146 .6a Cation Site I in Zeolite A .................. 1166 .6b Cation Site II in Zeolite A ................. 1176 .6c Cation Site III in Zeolite A ................ 1176 . 8 Far-Infrared Spectrum of Sodium Zeolite A .... 1196 . 8 Far-Infrared Spectrum of Magnesium Zeolite A .. 1216.9 Far-Infrared Spectrum of Calcium Zeolite A .... 1226.11 Far-Infrared Spectrum of Magnesium CalciumZeolite A ........................................ 1236.11a Cation Site A in Zeolite L ............. 1256.11b Cation Site A in Zeolite L ............. 1266.11c Cation Site C in Zeolite L ................. 1266 . lid Cation Site D in Zeolite L ................. 1266.12 Far-Infrared Spectra of Dehydrated PotassiumZeolite L ........................................ 1296.13 Far-Infrared Spectra of Dehydrated CaesiumZeolite L ........................................ 1296.14 Far-Infrared Spectra of Dehydrated RubidiumZeolite L ........................................ 1316.15 Far-Infrared Spectra of Dehydrated AmmoniumZeolite L ........................................ 1316.16 Far-Infrared Spectra of Dehydrated SodiumZeolite L ........................................ 1336.17 Far-Infrared Spectra of Dehydrated LithiumZeolite L ........................................ 1336.18 Far-Infrared Spectra of Hydrated PotassiumZeolite L ........................................ 1356.19 Far-Infrared Spectra of Hydrated SodiumZeolite L ........................................ 1356.20 Far-Infrared Spectra of Hydrated LithiumZeolite L ........................................ 1376.21 Far-Infrared Spectra of Hydrated RubidiumZeolite L ........................................ 1376.22 Far-Infrared Spectra of Potassium/AmmoniumZeolite L at 108K ................................. 1386.23 Far-Infrared Spectra of Potassium/AmmoniumZeolite L at 303K ................................. 1386.24 Far-Infrared Spectra of Potassium/SodiumZeolite L at 108K ................................. 1396.25 Far-Infrared Spectra of Potassium/SodiumZeolite L at 303K ................................. 1396.26 Far-Infrared Spectra of Potassium/LithiumZeolite L at 108K ................................. 1406.27 Far-Infrared Spectra of Potassium/LithiumZeolite L at 303K ................................. 1406.28 Far-Infrared Spectra of Potassium Zeolite L at108K ............................................. 1416.29 Far-Infrared Spectra of Potassium Zeolite L at303K ............................................. 141
6.30b Bond Network of Clinoptilolite ............. 1436.30a Atomic Structure of Clinoptilolite ......... 1436.31a Clinoptilolite Cation Site A ............... 1436.31b Clinoptilolite Cation Site B ............... 1446.32 Far-Infrared Spectra of Natural Clinoptilolite.................................................. 1456.33a Atomic Structure of Ferrierite ............. 1476.33b Bond Network of Ferrierite ................. 1476.33c Magnesium Cation Site in Ferrierite ........ 1476.33d Sodium Cation Site in Ferrierite ........... 1486.34 Far-Infrared Spectra of Natural Ferrierite -Zeolon 700 ....................................... 1496.35a Potassium in Mordenite Cation Site II ..... 1506.35b Potassium in Mordenite Cation Site I V ..... 1506.35c Potassium in Mordenite Cation Site VI ..... 1516.36 Far-Infrared Spectrum of Dehydrated HydrogenMordenite ........................................ 1526.37 Far-Infrared Spectrum of Dehydrated SodiumMordenite ........................................ 1526.38 Far-Infrared Spectrum of Dehydrated LithiumMordenite ........................................ 1536.39 Far-Infrared Spectrum of Dehydrated PotassiumMordenite ........................................ 1536.40 Far-Infrared Spectrum of Dehydrated RubidiumMordenite ........................................ 1546.41 Far-Infrared Spectrum of Dehydrated CaesiumMordenite ........................................ 1546.42a Calcium in Mordenite Cation Site I .......... 1546.42b Calcium in Mordenite Cation Site III ....... 1556.42c Calcium in Mordenite Cation Site I V ......... 1556.42d Calcium in Mordenite Cation Site VI ........ 1556.43 Far-Infrared Spectrum of Dehydrated MagnesiumMordenite ........................................ 1566.44 Far-Infrared Spectrum of Dehydrated BariumMordenite ........................................ 1566 .45 Far-Infrared Spectrum of Dehydrated CobaltMordenite ........................................ 1576.46 Far-Infrared Spectrum of Dehydrated NickelMordenite ........................................ 1576.47 Far-Infrared Spectrum of Dehydrated CopperMordenite ........................................ 1577.1 Two Dimensional Cavity Replicas .............. 1617.2 Plan Views of Cavity ......................... 1627.3 Orientation of Axes .......................... 1667.4 2-D Horizontal Reflection .................... 1677.5 Off-Axis 2-D Horizontal Reflection ........... 1697.6 Infinite Cavity .............................. 1707.7 Reflections of Plug .......................... 1728.1 Source Rays .................................. 1798.2 Radial Positions ............................. 1808.3 Vertical Positions ........................... 1818.4 Fraction of Rays Detected vs. Plug Size ...... 1838.5 Detected Energy vs. Plug Size ................ 1838.7 Inhomogeneity vs. Plug Size .................. 1848 . 6 Detected Pathlength vs. Plug Size ............ 1848 . 8 Q vs. Plug Size .............................. 1858.10 Detected Energy vs. Hole Size ............... 186
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8.9 Fraction of Rays Detected vs. Hole Size ..... 1868.12 Inhomogeneity vs. Hole Size ................. 1878.11 Detected Pathlength vs. Hole Size ........... 1878.13 Q vs. Hole Size ............................. 1888.15 Detected Energy vs. Exit Hole Size .......... 1898.14 Fraction of Rays Detected vs. Exit Hole Size . 1898.17 Inhomogeneity vs. Exit Hole Size ............ 1908.16 Detected Pathlength vs. Exit Hole Size ...... 1908.18 Q vs. Exit Hole Size ........................ 1918.20 Detected Energy vs. Exit Height ............. 1928.19 Fraction of Rays Detected vs. Exit Height .... 1928.21 Detected Pathlength vs. Exit Height ......... 1938.22 Inhomogeneity vs. Exit Height ............... 1938.23 Q vs. Exit Height ........................... 1949.1 Far-Infrared Spectrum of Cavity with Calibration Hole ........................................ 1989.2 Far-Infrared Variation of Cavity Q ........... 1999.3 Far-Infrared Absorption Coefficient of PVC .... 2009.4 Far-Infrared Absorption Coefficient of PVDF ... 2009.5 Far-Infrared Absorption Coefficient of PTFE ... 2019.6 Structure of Cellobiose ...................... 2029.7 Structure of Cellulose ....................... 2029 . 8 Far-Infrared Spectrum of Qualitative FilterPaper ............................................ 2039.9 Far-Infrared Spectrum of Hardened Filter Paper 2039.10 Far-Infrared Spectrum of Wiper Paper ........ 2049.11 Far-Infrared Spectrum of Paper Computer Tape . 2049.12 Far-Infrared Spectrum of Cartridge Paper .... 2059.13 Far-Infrared Spectrum of Lignin ............. 2069.14 Far-Infrared Absorption Coefficient of SodiumZeolite A ........................................ 2099.15 Far-Infrared Absorption Coefficient of SodiumZeolite X ........................................ 2109.16 Far-Infrared Spectrum of Interferometer Systemwith Water Vapour ................................ 2129.17 Far-Infrared Spectrum of Cavity ContainingWater Vapour ..................................... 2129.18 Far-Infrared Dependence of Cavity Pathlength . 2149.19 Vertical Cavity Homogeneity Tests .......... 2169.20 Horizontal Cavity Homogeneity Tests ......... 2179.21 Far-Infrared Absorption Coefficient of DriedSilicalite ....................................... 2199.22 Far-Infrared Absorption Coefficient of WetSilicalite ....................................... 220
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TABLES1.2 Cation Vibrations for Zeolites X and Y afterButler et al ..................................... 331.2 Cation Vibrations for Zeolites X, Y and E afterBrodskii and Zhdanov ............................. 351.3 Cation Vibrations for Zeolites X, Y and US-EXafter Peuker and Kunath .......................... 361.4 Cation Vibrations for Zeolite A after Kosslicket al ............................................ 375.1 Golay Cell Parameters ........................ 986.1 Calculated Vibrational Band Assignments forCations in Zeolite L Based on Bands Observed in Potassium Zeolite L .............................. 1306.2 Vibrational Band Assignments for Cations inZeolite L ........................................ 134
Corrigenda
Pages 8 6 , 130-141 : The formula used for zeolite L is based on eight cations per unit cell in the original material. It is more normal to state the formula in terms of 36 tetrahedral atoms per unit cell. The original zeolite L formula stated in this form isNa0.2k 7.6 C (a 1 0 2 ) 8 (s i -02) 28 3 • (H2° )24 -
Page 122 : The excess form used in figure 6.9 is a non-washed calcium exchanged zeolite A containing the calcium exchange salt, calcium chloride.
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1 Introduction
1.1 Zeolites
Zeolites are tectosilicates; crystalline, three dimensional frameworks of silica tetrahedra in which some of the silicons are replaced by aluminium^/2 ,3, The replacement of silicon by aluminium gives rise to a framework negative charge which is balanced by cations located at energetically favourable sites about the framework. Zeolites have the general formula :
M V [ { A l 0 2)x{ Si 02)y] . z H 20
where M is a cation of charge n+ and x, y and z are numbers of atoms.
The three dimensional framework is an open structure in which large void volumes, cavities and channels, are present. In these cavities and channels it is possible to adsorb gases or liquids or build-up metal clusters^.These pores allow the cations present to be exchanged under mild conditions. Sites on the framework or the exchangeable cations or metal clusters present can act as catalytic centres for reactant species which enter the channel structure^. in normal atmospheric conditions most zeolites absorb water vapour from the atmosphere to fill their cavities.
1.2 Major Uses of Zeolites
The uses of zeolites are many and varied, ranging from catalysis to meat production. The main uses and the unique characteristics which command their use over other materials are outlined below.
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1.2.1 Sorption
One of the first commercial uses of zeolites was as a desiccant; several zeolites both natural and synthetic have a very high capacity for water uptake and gases dried through zeolites have a very low residual water content^. Sodium zeolite A produced by Union Carbide? as Linde Molecular Sieve 4A has a water uptake of 29% by weight and pellets can be produced with a capacity of 22%. Zeolites are used as moisture scavengers in polyurethane formulations to stop break-up of the pore system causing blistering and as hydrogen absorbers in high zinc paints8 . Some paper coatings now incorporate zeolites which scavenge water to stop adhesion failure between the coating and the paper, roughen the surface and give heat and light resistance^.
A p a r t f r o m t h e i r a b i l i t y t o s e l e c t i v e l y r e m o v e w a t e r
d u e t o t h e i r p o l a r n a t u r e , t h e y a r e a b l e t o r e m o v e a w i d e
v a r i e t y o f p o l a r m o l e c u l e s f r o m g a s e s a n d l i q u i d s ^ .
Z e o l i t e s a r e u s e d f o r t h e r e m o v a l o f w a t e r , c a r b o n
d i o x i d e a n d s u l p h u r c o m p o u n d s f r o m o i l r e l a t e d g a s e s ;
p o l l u t i o n c o n t r o l i n e x h a u s t g a s ; f o r t h e r e m o v a l o f
m e r c u r y v a p o u r a n d o x i d e s o f n i t r o g e n f r o m f a c t o r y w a s t e
g a s e s ^ .
1.2.2 Molecular Sieving
Z e o l i t e s h a v e l o n g b e e n u s e d a s m o l e c u l a r s i e v e s , i n
t h a t t h e y s e l e c t i v e l y a b s o r b m o l e c u l e s o n t h e b a s i s o f
t h e s i z e o f t h e molecules^. Z e o l i t e s , b e i n g c r y s t a l l i n e , h a v e a p r e c i s e l y d e f i n e d p o r e d i a m e t e r i n t o t h e
t h r e e d i m e n s i o n a l c h a n n e l a n d c a v i t y s t r u c t u r e .
M o l e c u l e s w h i c h p o s s e s s a V a n d e r W a a l ' s d i a m e t e r g r e a t e r
t h a n t h i s c r i t i c a l p o r e d i a m e t e r a r e u n a b l e t o e n t e r t h e
c h a n n e l s t r u c t u r e a n d a r e t h u s n o t a d s o r b e d ; w h e r e a s
m o l e c u l e s w h i c h p o s s e s s V a n d e r W a a l ' s d i a m e t e r l e s s t h v n
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the critical pore diameter can enter and be adsorbed. Linde Molecular Sieve 5A, calcium zeolite A is used for the separation of n-type paraffins from petroleum fractions. It works by admitting the n-type paraffins and excluding iso-type paraffins and aromatics due to the size of its pores^ (0.5nm).
1.2.3 Ion Exchange
Many zeolites behave as ion exchangers, in that ions from the zeolite exchange with ions in a solution in which the zeolite is placed. The ultimate degree of exchange is controlled by the thermodynamics of the exchange and the relative size of the ions and the controlling apertures to the cation sites. Some zeolites are selective for specific cations. The naturally occurring zeolite clinoptilolite has been used for waste water purification to selectively remove ammonium ions^. A major development in the use of zeolites for ion exchange is the gradual replacement of phosphate water softeners by sodium zeolite A in detergents, due largely to the serious ecological problems caused by phosphate ions. Sodium zeolite A selectively exchanges sodium ions for the calcium and magnesium ions^5/ which cause hard water.
1.2.4 Catalysis
The primary advantage of zeolites for use as catalysts in various petrochemical processes is their shape selectivity. This is due to the fixed size of the cavities near catalytically active siteŝ -G. The primary catalytic sites within zeolites are the Bronsted and Lewis acid sites, similar to those in amorphous aluminosilicates, but due to the molecular shape selectivity the product distribution obtained is far more favourable and can be controlled by the choice of zeolite
17
catalyst. The acid sites, present within the zeolites, catalyse cracking and isomerization reactions of alkanes and alkylation of aromatic hydrocarbons1?. Hydrocracking can be accomplished by the incorporation of metals normally used for homogeneous type catalysis, i.e. lanthanides and platinum^.
The largest step change in zeolite technology occurred with the discovery of Mobil's zeolite ZSM-5 (Zeolite Socony M o b i l Z S M - 5 is able to catalyse the conversion of methanol to high octane petrol at low temperatures and pressures. The largest use for ZSM-5 is in the isomerization of xylenes, in which it produces large excess quantities of para-xylene over the other isomer because of the diffusion controlled reaction.ZSM-5 has very low contents of both cations and aluminium. It also possesses a pore structure devoid of cavities and therefore its catalytic life is extended due to lack of coke build-up which needs cavities to occurrapidly^O.
1.2.5 Curious Uses
Some of the most unconventional uses of zeolites have been in agriculture and horticulture. Zeolite addition to soil in Japan has been carried out throughout recorded history^l. The scientific basis for its use is now understood to be the inhibition of ammonium ion leaching, as well as the concentration and retention of heavy metal ions, stopping their introduction into food crops22. The use of zeolites as feed supplements for pigs23 and chickens24 has been shown to give rise to weight rises and increased egg yields.
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In the future the most spectacular advances are likely to be made in the field of zeolite catalysis following in the footsteps of ZSM-5 rather than in novel agricultural uses. Work is already being undertaken to fully understand the total nature of shape selectivity in catalysis within the complex three dimensional cavities of zeolites2 .̂ Detailed crystallographic work to precisely locate positions of c a t i o n s 2 **/27 has been going on for some time and now this is being complemented by computer graphics2** which allow true visualisation of the processes involved. This increased understanding, given by energy calculation programs2**, should allow predictions to be made as to the most advantageous geometries and compositions for the optimal catalyst.
1 . 2 . 6 F u tu r e U se s
1.3 Zeolite Structure
Zeolites are hydrated aluminosilicates with uniquely defined crystal structures. They possess a pore structure within which are sited cations. The cation sites are uniquely defined forming a set of sites of differing energies, in which the cations are distributed according to their type and statistics. It is possible to say only that a particular cation site has a fractional occupancy, meaning that only a particular fraction of the total number of sites of that type are occupied by cations. The occupancy fraction of a particular set of sites will vary with the type of cation due to size and charge differences, as well as with differences in aluminium to silicon ratio and degree of hydration. Certain cations are not able to fit into certain sites and only half the number of divalent cations compared to monovalent cations are needed to balance a given framework charge.
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The framework is composed of silicon surrounded by oxygen in a tetrahedral arrangement. The central silicon atom can be replaced by an aluminium atom giving a tetrahedron with a negative charge. The various tetrahedra are then joined, so that each oxygen is shared by two tetrahedra. The linking of groups of tetrahedra give rise to the so-called secondary building units of zeolite structure. The eight secondary building units are capable of building all the zeolitic frameworks. The similarities between the different zeolite structures can be used to divide them into families according to the secondary building units used.
Hydrophillic zeolites, both natural and synthetic, contain water adsorbed within the channels and cavities of the crystals. The volume of water adsorbed by a zeolite is used as a measure of the total intracrystalline pore volume. This assumes that the water molecule is small enough to penetrate all intracrystalline pores and that the density of water within zeolites is the same as normal water. The second assumption is a good approximation, but is flawed as the density will be affected by both the cation charge and framework silicon to aluminium ratio.
1.3.1 Zeolite L
Barrer and Villager used X-ray crystallography^ to obtain the correct crystal structure of zeolite L. Zeolite L is based on the double six ring (hexagonal prism) and the eighteen tetrahedra unit (cancrinite cage). These are shown in figure 1.1a.
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Figure 1.la
Cago Prism
The double six ring joins the eighteen tetrahedra units so that they are symmetrically arranged (see figures 1 .1b and c).
Figure 1.lb Figure 1.lc
® S i l i c o n Q> S o d i u m
The columns of eighteen tetrahedra units are joined by single oxygen bridges giving rise to twelve membered rings which produce wide channels parallel to the C-axis, as shown in figures 1 .Id and e.
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Figure l.d Figure 1.le
® Oxygen • Silicon
0 Potassium O Sodium
The cation environments in zeolite L are described in section 6.3.
1.3.2 Zeolite X
Zeolite X is isostructural with zeolite Y having the structure of the natural zeolite Faujasite. Zeolites X and Y are both synthetic with zeolite Y (48-76 Al atoms) having a lower aluminium content than that of zeolite X (77-96 Al atoms). It is constructed from sodalite cages (see figure 1 .2a) which are joined via oxygen bridges through four of the eight six faces, forming hexagonal prisms (see figure 1 .1a), in a tetrahedral array^l. in this way the sodalite cages are arranged tetrahedrally with respect to one another, giving rise to four, six and twelve faces (see figure 1.2b). The supercage formed between the sodalite units controls the channel system with a free diameter of 0.78nm.
22
Figure 1.2a Figure 1.2b
The cation environments in zeolite X are described in section 6 .1 .
1.3.3 Zeolite A
Zeolite A is constructed from sodalite cages (see figure 1 .2a) which are in a cubic array and joined by oxygen bridges between the six four faces of the cages32. The restricting aperture in the channel system of Zeolite A is the eight ring which has an aperture of 0.5nm. The aperture of the cavity is further restricted by the type of cation which is present as one cation site is within the eight ring window. Figure 1.3a shows a single unit cell of zeolite A, while figure 1.3b shows four unit cells joined.
23
Figure 1.3a Figure 1.3b
O x y g e n @ S o d i u m
0 S i l i c o n
The cation environments in zeolite A are described in section 6 .2 .
1.3.4 Mordenite
Mordenite is a complex structure to visualise. It is made up of linked five ring units in a series of chains which are joined in such a way as to give two channels, one an eight ring and the other a twelve ring33f as shown in figures 1.4a and b. Eight water and cation sites can be identified, the major cation sites being in the centre of the eight rings of one channel, close by the five rings lining the large channel and coordinated to the four oxygen atoms in adjacent rings.
Figures 1.4a and b show the structure of potassiummordenite34 #
24
Figure 1.4a Figure 1.4b
The structure of calcium mordenite^S shows the occupation of different cation sites from potassium mordenite.
25
Figure 1.4c Figure 1.4d
® Silicon Site I ^ Calcium
The cation positions in Mordenite are described in section 6.4.
1.3.5 Silicalite-1
Silicalite-l36 is isostructural with ZSM-5. They are members of the pentasil series of zeolites, with silicalite- 1 being the very low aluminium end member oft h i s s e r i e s ^ O ,
The ZSM-5 structure is made by the stacking of the units shown in figure 1.5a.
Figure 1.5a
26
The structure of silicalite is shown in figure 1.5b and c.
Figure 1.5b Figure 1.5c
° Silicon
Intergrowths of other pentasil members are reported to be a common occurrence caused by stacking faults giving rise to small amounts of ZSM-11 type structure. The primary difference between the pentasil zeolites and most other zeolites is their lack of cavities, having only a constant diameter pore system, only slightly enlarged at intersections. This is an important advantage for catalysis as the tendency for coke formation is much decreased.
1.4 Infrared Spectroscopy for the Study of Zeolite Structure
Initially all structural work on zeolites was carried out using X-ray diffraction techniques. Neutron and X-ray diffraction are the only techniques readily available for entire structure elucidation using single
27
crystal studies, but once the structure has been elucidated it is not as well suited to small changes in structure as are infrared studies.
Vibrational bands within the infrared spectrum of zeolites are specifically related to certain structural features. Study of these bands allows a crude picture of an unknown zeolite to be built up, or alternatively and more appropriately, variations with different treatments of a known zeolite can be used to understand slight changes.
Flanigen et al^7 systematically studied a wide range of different types of zeolites. From this study some principal correlations emerged:
i) each zeolite structural unit has a typical infrared pattern.ii) there are strong similarities between zeolites of the same structural type and in the same group.
They also found they could classify the vibrations between 1300cm”1 and 200cm” 1 into 2 classes:
i) internal vibrations of TO4 tetrahedra, not distinguishable between Si04 and AIO4 . These are insensitive to change in the framework.ii) vibrations related to external linkages between tetrahedra, which are sensitive to changes in the framework. These vibrations show up the presence of secondary building units and building block polyhe- dra.
The bands over which these vibrations occur and the structural features to which they are assigned are listed below^?:
28
TO4 Vibrations.
Asymmetric Stretch Symmetric Stretch T-0 Bend
1250-950cm-1720-650cm-1500-420cm“l
External Linkage Vibrations.
Asymmetric Stretch Symmetric Stretch Double Rings Pore Opening
1 1 5 0 -1 0 5 0 c m - 1
8 2 0 - 7 5 0 c n r 1
650-500CIH-14 2 0 -3 0 0 cm - 1
Several other general observations can be made on the factors effecting the infrared spectrum:
the frequency of any band shifts with changes in the silicon to aluminium ratio;the position of bands is also related to the type and class of zeolite e.g. for the Faujasite zeolite group the double six-ring vibration is always symmetric and between 540-585cm~l;zeolites devoid of double-rings or larger polyhedra show only weak bands in the 540-630cm“l range; "breathing" motion of isolated rings forming pore openings in zeolites between 300-420cm”l is seen as a distinct band;the cation composition affects both the frequency and intensity of the infrared bands38/39.
The use of infrared spectroscopy for checking structural changes caused by a variety of treatments is thus highly effective.
29
1.5 Magic Angle-Spinning Solid State Nuclear Magnetic Resonance
Nuclear magnetic resonance spectroscopy studies atomic nuclei which have magnetic moments arising from nuclear spin, in the presence of an applied magnetic field. The different chemical environments of atomic nuclei give rise to different frequency chemical shifts, which are expressed as parts per million (ppm):
c h e m i c a l s h i f t ( p p m ) =C h e m i c a l S h i f t { H z ) x 106
O b s e r v a t i o n F r e q u e n c y (// z )
this is then independent of the field strength of the magnetic field.
Magic angle-spinning solid state nuclear magnetic resonance (MAS-NMR) has enabled high resolution spectra to be obtained. The sample is spun at the "magic angle" of 54°44' (with respect to the axis of the magnetic field) to remove the line broadening shift of the anisotropy caused by the solid state of the sample giving different nuclear orientations. This is combined with high power decoupling and cross-polarisation in order to enhance the signal from nuclei present at low concentrations.
The use of silicon and aluminium MAS-NMR has given increased information about the environment experienced by cations and adsorbed molecules within zeolites. The exact arrangement of silicon and aluminium tetrahedra relates precisely to the electrostatic field experienced by cations within the structure. The most fundamental result obtained from MAS-NMR is the confirmation of the validity of Lowenstein's rule, this is that two aluminium atoms will not occupy adjacent tetrahedral sites.
30
The work of Lippmaa et al^, as repeated and extended by Thomas et al^l, has been neatly correlated considering geometrical as well as chemical considerations by Smith and Blackwell^. They have shown a semi-quantitative relationship between mean interatomic distances and angles determined by X-ray diffraction and the NMR chemical shift for silicons bonded to silicon tetrahedra only.
MAS-NMR allows one to determine the ratio of the various different types of silicon or aluminium tetrahedra,i.e. the number of aluminium atoms surrounding a silicon tetrahedron or vice versa. The maximum shift seen with different atomic environments is 40ppm as compared to a maximum geometrical shift of 12ppm. The large size of the geometrical shift as compared to the more fundamental chemical shift means that any interpretations of MAS-NMR spectra must be based, at least in part, on comparisons with spectra of similar zeolites with different silicon to aluminium ratios to show a trend.
1.6 Far-Infrared Spectroscopy
The far-infrared is considered to be from 0.05mm to 1mm wavelength or 2 0 0cm“l to 1 0cm“*l, above this is the infrared region and below this is the sub-millimetre region. Over the entire region very few effective laboratory radiation sources exist. For this reason the development of spectrometers was many years behind that of the infrared, where energetic sources are plentiful. The only effective laboratory sources in the far-infrared are blackbody sources and these need to be filtered in order to protect sensitive detectors from the shorter wavelength radiation. It was not until effective room temperature detectors were invented that commercial spectrometers were made possible.
31
The study of the far-infrared was begun by Rubens and his co-workers between 1892-1922^3, it was continued after 1922 by others using grating instruments^. it was in the late 1960's and early 70's that prolific growth occurred in the use of far-infrared spectroscopy^. This is well demonstrated by the growth in publishing on the subject. Only 100 papers on far-infrared spectroscopy were published in 1964 but by 1968 this had grown to nearly 200 per year. Even though the rate of growth appeared to have slowed, in 1985 in excess of 700 papers were published. Several factors have combined to produce this effect:
i) the development of interferometric spectrometers, providing signal to noise levels previously unobtainable with the low power radiation sources available.ii) the increased interest in liquid and solid state chemistry and physics about which the far-infrared is an abundant source of information.
1.7 Far-Infrared Spectroscopy of Zeolites
Brodskii, Zhdanov and Stanevich^^ were the first people to apply far-infrared spectroscopy to the study of zeolites. They carried out a study of zeolite X substituted with monovalent cations and found a systematic variation of the vibrational bands with the cation mass. They thus assigned the observed vibrational bands to the vibrations of cations with respect to the framework. They found that the vibrational frequency of cations in particular sites was proportional to the inverse of the square root of the product of the mass and the cube of the cation radius. This would be the type of relationship suggested by a simple harmonic oscillation in which the framework is considered to have infinite mass and the cation is able to vibrate at a frequency determined by its mass and the force constant. They were able by selective ion exchange to assign the cation
32
dependent bands to particular cation localisation sites within the zeolite framework. They also showed how the vibrational bands were affected by the dehydration of the zeolite causing the cation to be more firmly bound to the framework.
Brodskii, Zhdanov and Stanevich^ consolidated their previous work by studying a range of both cation compositions and silicon to aluminium ratio zeolites with the Faujasite structure(i.e. the synthetic zeolites X and Y). They observed the effect of decreasing aluminium content on the occupancy of various cation sites and hence the disappearance of certain bands. They saw the technique as being a way of understanding the energy inequivalence between various sites within the zeolite framework.
Dyrikheev, Kiselev, Lygin and Tul'chinskii^ studied the effect of dehydration on the far-infrared spectrum of calcium X. They showed how the cation is highly solvated in the hydrated state as is shown by the lack of well resolved bands. On dehydration, localisation of the cation occurs and well resolved bands are then visible in the far-infrared spectrum.
In 1977 Brodskii, Zhdanov, Krasaavttseve and S a m u l e v i c h 4 9 published a paper on a synthetic chabazite zeolite exchanged with monovalent and divalent cations. This zeolite has only two crystallographic cation positions, but several bands can be found in the far-infrared spectra which vary depending on the cationic form of the zeolite. They suggest that this shows a greater inequivalence in cation site energies than suggested by crystallographic studies. They were also able to observe a large change in the spectra on prolonged extreme heat treatment, showing the effect of cation migration.
33
An extensive study of dehydrated and solvated, mono- and divalent cationic zeolites was carried out by Butler, Angell, McAllister and Risen^O. They partially repeated the work of Brodskii, Zhdanov and Samulevich but without reaching all the same conclusions. They extended the previous work by studying the effects of a variety of different solvents on the far-infrared spectra. They were able to explain the spectral effects in terms of the geometry that could be expected for the solvated ions. They tried to interpret the vibrational frequencies in terms of cationic conduction by several possible mechanisms. They reached the conclusion that cationic conduction was likely to be a co-operative phenomena and therefore did not lend itself well to modelling by simple models of the single jump kind.
Table 1.1 summaries the assignments made by Butler etal.
Table 1.1 Cation Vibrations (cm-1) for Dehydrated Zeolites X and Y after Butler et al.
Site I II IIICation X Y X Y X YLi+ 380Na+ 160 167 190 180 67K+ 107 156 133 58Rb+ 108 48Cs+ 86 62 39 30Ag+ 50 82Ca2+ 287 256 273 227Sr2+ 189 150Ba2+ 137 107
34
In 1979, Peuker and K u n a t h ^ l studied a variety of previously unstudied zeolites. Ultra-stable zeolite X was found to have a weak band which could be associated with sodium cations in zeolites X and Y at Site I, presumably due to residual cations in this site (110cm~l). They also showed the non-conforming nature of lithium cationic vibration bands due to the high charge to radius ratio of the lithium cation producing a distortion of the simple harmonic model. The study of chloromethane adsorbed on lithium X showed the direct effect of the adsorbate on the cations within the super-cage.
At the Fifth International Zeolite Conference in Naples in 1980 Brodskii and Zhdanov52 presented a paper which reviewed both their own work and that of others in the field of far-infrared spectroscopy of zeolites. They presented improved spectra showing:
i) dependence of vibrational bands on the mass and charge of a cation;ii) change in cation vibration frequencies with changes in silicon to aluminium ratios;iii) changes in cation vibrational bands with adsorption into the zeolites.
Brodskii and Zhdanov summarised their own work on cation vibrations of faujasite (X and Y) and chabazite(E) zeolites, this is reproduced in the table below:
35
Table 1.2 Cation Vibrations (cm”*) for DehydratedZeolites X, Y and E after Brodskii and Zhdanov.
Table 1.2 when compared to Table 1.1 shows a high degree of agreement over the assignment of cations in sites II and III, but no agreement over the assignment of cations in site I.
Peuker and Kunath^3 presented results of their further studies on monovalent cation containing zeolite X, showing the effect of different adsorbates and the deviation of lithium from the mass dependence relationship, which they explained in terms of a reduction of the force constant. They published the definitive assignment of cation bands to monovalent cations.
36
Table 1.3 Cation Vibrations (cm“l) for DehydratedZeolites X, Y and US-EX after Peuker and Kunath.
Site I I r II IIICation X Y US X Y X Y XLi+ 153 119 202Na+ 156 156 151 110 113 189 185 67K+ 106 73 156 56
A paper presented at a workshop in Hungary in 1982 by Kosslick, Walther, Roethe and Roethe54 developed a relationship for the vibrational frequency based on the mass of the cation and the cation-oxygen distance. This effectively takes into account the force constant problem the necessity for which was widely realised. They obtained very good agreement between vibrational frequencies predicted from their model and experimentally observed vibrational frequencies. This model appears to accurately predict the vibrational frequency of lithium cations, for which no previous model had been able to account.
Also in this paper, Kosslick et al presented, for the first time, assignments for a variety of alkali and alkaline earth metal cations in zeolite A. These are shown in table 1.4.
37
Table 1.4 Cation Vibrations (cm"l) for DehydratedZeolite A after Kosslick et a l .
Cation I II IIILi+ 320 170Na+ 210 100 87K+ 130 75 55Rb+ 55-60 46 20Mg2+ 315Ca2+ 270Sr2+ 160Ba2+ 106
Loeffler, Peuker and Kunath55 at the same workshop presented observations on the interaction between cations and adsorbed benzene molecules. These showed the modification of certain cation bands by their association with benzene. Peuker, Moeller and Kunath^G also presented a paper specifically on the interactions of a series of zeolite Xs.
Pechar57/58,59f between 1981 and the present time, has been publishing papers on the study of a wide range of natural zeolites' far-infrared spectra. He has been able to assign vibrational bands to cations by correspondence with similar sites in previously studied zeolites.
A review by F r i p i a t ^ O in 1982 provides an excellent comparison of the similarities of cationic environments in clays and zeolites, by reviewing some of the work done on the far-infrared spectra of both.
38
Tabourier, Carru and Wacrenier^l studied both the dielectric and far-infrared properties of cations in X type zeolites. They used the existence of independent cationic vibration modes to suggest that cation-cation repulsion effects, as well as cation-framework repulsion effects, should be responsible for the dielectric properties. The dielectric response measured suggested that the cation-cation repulsions are very weak compared to cation-framework repulsions.
Ozin, Baker and Parins62 used far-infrared spectroscopy to study the auto-reduction and clustering of silver cations in Y type zeolites induced by heat treatment. They were able to make tentative assignments of vibrational bands to modes of silver ions, atoms and clusters. This is the first study where bands due to uncharged non-molecular species have been postulated.The extension of this technique to the study of iron containing zeolites may well be relevant to the study of Fischer-Tropsch type catalysts.
Stock, Dombrowski, Fruwert and Ratajczak^3 studied ammonium and hydrogen forms of zeolites X and Y. In order to correctly interpret their spectra they re-examined the data presented in tables 1.1, 1.2 and1.3. They concluded that four regions in the spectrum could be assigned to sodium cation bands in faujasite type zeolites, site II : 182-196cm“l, site I : 156-161cm"l, site I' : 110cm“ ̂and site III : 63-66cm”l. They discovered bands due to ammonium cations but no bands due to hydrogen ions or hydrogen bridged hydroxyl groups, only the disappearance of sodium cation vibrations.
Ozin et al^4,65,66,67,68 have used far-infrared spectroscopy as their primary tool in further studies of silver clustering in zeolites A and Y. They have
39
extended their work to consider assignments of bands and their intensities by GF-Matrix methods on simplified cation sites with increased symmetry to make the problem tractable. Their work studying Faujasite type zeolites largely agrees with the assignments of Peuker and KunathS^ (c.f. table 1.3), with the addition of identifying some previously unassigned bands, broadly in agreement with Stock et al^3.
No et al69 have carried out similar calculation work to Ozin et al but on zeolite A structure, using assumed net atomic charges and force constants in order to be able to carry out a normal mode analysis for sodium zeolite A.
1.8 The Untuned Resonator Technique
The commonest long pathlength cell is the white cell. This type of cell is only suitable for gases, as scattering by solid samples will destroy the desired effect.
An untuned resonator is a vessel in which a sample and radiation interact. It has three apertures, one for radiation entrance, one for radiation detection and one providing a known loss to calibrate the cavity. The cavity is designed in such a way as to have multiple reflection paths, minimum reflection losses and resonant modes widely spaced to give an isotropic radiation field within the cavity.
The true origin of this technique is from the measurement of acoustic properties of r o o m s ’7^ 71 which was extended and used as an analogy by Lamb72 to develop the theory for electromagnetic radiation. In 1946, Becker and A u t l e r 7 3 built a 2.5m cube spectrometer to
40
study a water vapour rotation line at 0.833cm"^ which interfered with radar. The technique was not revived for another 30 years until at the Appleton Laboratories^4,75 a smaller resonator was constructed to study anomalous absorption of water vapour in the sub-millimetre region which was affecting remote sensing and sub-millimetre wave astronomy. These untuned resonators were also used to study polytetraf luoroethylene solid samples?*^??.
The untuned resonator technique has several advantages for study in the sub-millimetre region:
a) radiation is collected over large solid angles, hence weak radiation sources can be used.b) no diffraction problems exist, as no images are formed.c) the long photon path in the cavity is either fully interacting with gaseous samples or partially with solid samples, but non-interacting photons are recirculated.d) the isotropic radiation field allows solid samples in any form to be used and complex dielectric properties to be determined?9,80.
The technique has been used to study polymers^l, biological materials*^ and materials with potential electronic applications^^/84.
1.9 Objectives of the Work
The primary aim of this work was to investigate the state of various extra-framework species within zeolites using far-infrared spectroscopy.
One major area of interest was the state of water within zeolites. The state of water within zeolites is of interest as many of the commercially useful processes
41
involve hydrated zeolites. The state of water will affect the ion exchange of radioactive cations into the cavities. It also presents a unique environment in which to study the properties of water itself. In order to enable this study to be conducted, it was necessary to further develop the untuned resonator technique. The further development of untuned resonators was seen to be necessary to allow the study of zeolite powders below lOOcm-l wavenumbers. This further development was also seen to open up hew possibilities for the examination of previously inaccessible systems, e.g. cement and biological materials.
The other area of interest was the study of the environment of cations within zeolites. Multiple cation exchanges of zeolites could be carried out and the resulting cation forms investigated by the use of far-infrared spectroscopy. It was intended to use conventional transmission spectroscopy but a flexible method of sample preparation was necessary in order to allow many samples to be studied.
42
2 Fourier Transform Spectroscopy
Fourier Transform Spectrometers are based on the interference between two coherent light beams of differing pathlength and first became widely used in the far-infrared where the inherent advantage of improved signal to noise ratio was necessary to overcome low energy output of available radiation sources. This improved signal to noise ratio comes from two advantageous effects, which are described below.
The main advantage of Fourier Transform Spectroscopy results from the fact that every element of the interferogram contains information about all the elements in the final spectrum: this is the Fellgett advantage^. If one considers a spectrum containing N elements then in a Fourier Transform Spectrometer each element is observed for the total observation time of T. Whereas in a monochromating spectrometer each element is only observed for T/N. If one observes a spectrum with both instruments for the same length of time the sig- nal-to-noise ratio of the Fourier Transform Spectrometer is square root of N times that of the monochromated instrument. This is a multiplexing advantage which is gained by astronomers using grating spectrometers which spread the entire spectrum over a photographic plate thereby getting the same advantage as by using a Fourier transform instrument. The computer used by Fourier Transform Spectrometry and the photographic plate used in monochromating spectrometry allow one to multiplex.
The second major advantage of a Fourier Transform Spectrometer is the throughput advantage first noted by Jacquinot86. The circular symmetry of a Fourier Transform Spectrometer means that compared to a monochromating instrument of the same resolving power the Fourier Transform Spectrometer will always have higher energy throughput.
43
2.1 The Mathematics of Fourier Transform Spectroscopy
This section outlines the mathematics of Fourier Transform Spectroscopy and is drawn from many sources, the main ones being Chamberlain's^7 and chantry's** 8 excellent books.
A diagrammatic Fourier spectrometer is shown in figure 2.1.
Figure 2.1 F o u r i e r S p e c t r o m e t e r .
S o u r c e
Radiation from the lamp is split at the beam splitter, half is transmitted to the moving mirror and half is reflected to the fixed mirror. The radiation is reflected from the two mirrors and recombines with interference at the beam splitter. The radiation is again split with half going to the detector and half returning to the source. The type of interference which occurs on recombination at the beam splitter will depend on the distance of the two mirrors from the beam
44
The difference between the distance travelled by the radiation from the two mirrors to the beam splitter is called the path difference. When both mirrors are the same distance from the beam splitter then constructive interference will occur no matter what the wavelength of the radiation, this is called zero path difference. If one considers a single frequency source, then the intensity of the signal detected will vary as a function of the path difference. When the path difference, p, is an integral number of wavelengths then constructive interference will result in maximum intensity.
s p l i t t e r . I f t h e m ovin g m ir r o r i s sc a n n e d to w a r d s andaway from t h e beam s p l i t t e r th e n t h e i n t e r f e r e n c e w h ic ho c c u r s on r e c o m b in a t io n c o n t i n u a l l y c h a n g e s .
In expressing the mathematical analysis of Fourier spectroscopy it is convenient to use wavenumber, defined by:
1
whereA., is the wavelength andv 0 is the frequency in wavenumbers.
( 2 .2 . 1)
The interference maxima and minima will occur at path differences, p, given by
p = n \ 0 /t = 0 ,+ 1 ........oo for maximum,
n = 0, + 1 ........ oo for minimum.
45
The actual intensity of radiation detected if the moving mirror is scanned across a range of pathlengths is given by:
where IQ is the intensity of the single frequency source.
For a hypothetical source emitting N single frequency lines the intensity would vary as the sum of the individual contributions:
where 1^ is the intensity of the kth line of wavenumber vfc.
In the case of a single frequency line it would be possible to calculate the frequency of the line from the interferogram by simple mathematics and knowing the separation between the maxima (figure 2.2). For more than two lines this simple approach becomes impossible.
/(p) = 2/0(l +cos(27rv0p)} ( 2 .2 .2 )
(2.2.3)
46
F i g u r e 2 . 2
I n t e r f o r o g r a m o f S i n g l e F r e q u e n c y S o u r c e
W a v e l c n g t h / 2
The method of extracting the spectral information from the interferogram of a source comprising a band of frequencies becomes one of applying a Fourier transformation. The Fourier transform is outlined below.
If one considers the single frequency source and only the oscillatory part of the interferogram:
Now, multiplying this by a function due to a single frequency which varies between 0 and infinity, the integrated intensity is given by:
U p ) = / ( p ) - 2 / 0
( 2 . 2 . 4 )
= 2/ ol cos (2 /ri'0p ) c o s ( 2 / r v p ) d p ( 2 . 2 . 5 )
47
Now:/(v) = 0 i f v £ v 0
* 0 i f v = v 0
/(v) is the Fourier transformation of I(p) (and a representation of the spectrum that would be obtained with a dispersive spectrometer).
In the case of N single frequencies, v* :
r + »7 ( v ) = J 7 ( p ) c o s ( 2 7 r v p ) d p ( 2 . 2 . 6 )
¥= I k i f v = v^then/c = 1 , N
= 0 i f v ¥= v k t hen k = 1 , N
In practise the interferogram is not obtained from minus infinity to plus infinity and hence can not be carried out over this range, but only over a finite range (”Pmax
48
a r e t h e i n s t r u m e n t a l l i n e s h a p e f u n c t i o n s c a u s e d b y
t h e t r u n c a t i o n o f t h e i n t e r f e r o g r a m , r e s u l t i n g i n a
f i n i t e h a l f - w i d t h a t t h e c e n t r a l m a x i m u m , w h i c h l i m i t s
r e s o l u t i o n . T h e o t h e r e f f e c t s o f t h e i n s t r u m e n t a l l i n e
s h a p e f u n c t i o n , t h e s i d e - l o b e s , c a n b e m i n i m i s e d a t t h e
e x p e n s e o f d e c r e a s e d r e s o l u t i o n , b y c a u s i n g t h e
i n t e r f e r o g r a m t o a p p r o a c h z e r o s m o o t h l y r a t h e r t h a n
a b r u p t l y b y i m m e d i a t e t r u n c a t i o n . T h e m i n i m i s a t i o n i s
a c h i e v e d b y m u l t i p l y i n g t h e i n t e r f e r o g r a m b y a f u n c t i o n
w h i c h i s u n i t y a t z e r o p a t h d i f f e r e n c e a n d d e c r e a s e s
t o w a r d s z e r o a t t h e t r u n c a t i o n . S e v e r a l f u n c t i o n s c a n b e
u s e d f o r t h i s p r o c e s s w h i c h r e d u c e s t h e i n t e n s i t y o f t h e
s e c o n d a r y m a x i m a , b u t i t a l s o d e c r e a s e s t h e r e s o l u t i o n -
i n t h e c a s e o f t r i a n g u l a r a p o d i s a t i o n b y a p p r o x i m a t e l y a
f a c t o r o f t w o .
T h e r e s o l v i n g p o w e r i s o f t h e f o r m :
C o n s i d e r i n g t h e d i a g r a m m a t i c s p e c t r o m e t e r , t h e s o u r c e
e m i t s a c o n t i n u o u s r a d i a t i o n s p e c t r u m . T h i s w i l l g i v e
r i s e t o a n i n t e r f e r o g r a m c o m p o s e d o f c o n t r i b u t i o n s f r o m
e a c h i n f i n i t e s i m a l f r e q u e n c y e l e m e n t , t h u s :
^ V P max ( 2 .2 .8 )
( 2 . 2 . 9 )
49
F i g u r e 2 . 3
Intcrferogram of Broad Band Source
2
Ji
50
7(p) = / ( p ) - ^ / ( 0)
- / ( p ) - / ( ~ )
- 2 o / (v) cos ( 2 n v p ) d p ( 2 . 2 . 1 2 )
The choice of apodisation function determines the spectral window of the spectrometer as in practice the interferogram, I(p) is multiplied by the apodisation function, A(p) which relates to the line shape function A(v) hence:
I ca IcO)* P max
A ( p ) I (p) cos (2/rv p ) d pP min
I ( v ) { A ( v - v ' ) + A ( v + v ' ) } d v ( 2 . 2 . 1 3 )
7colcO ) =+ 00
f ( v ' ) { A ( v - v ' } d v 'CO
( 2 . 2 . 1 4 )
wherev* is the dummy variable.
Thus the spectrum which is obtained is a convolution of the spectrum with the line shape function. This is equivalent to the convolution of a dispersion spectrometers spectral window(slit size) with the spectrum.
The calculation of the interferogram is normally carried out using a digital computer, the use of which is not discussed here. It is however necessary to consider the effects on the resulting spectrum of taking a finite number of sam p l e s .
51
When an interferogram is recorded for computationa finite number of points on it are digitised at a set interval over the travel of the mirror. It is hence possible that an actual sample was not taken at the exact zero path difference, but a point near to it. This will result in the whole interferogram being shifted, thus a constant error. This error in zero path difference will manifest itself as a frequency dependent phase error in the spectrum
52
^ c C l ' W U L e C ' O l M / ^ C v ) ) 2 (2.2.18)
It is not always convenient or sensible to record an entire double sided interferogram, and hence a single sided interferogram is recorded, from slightly before zero path difference to pmax* In this case the distortions introduced by the cosine transform are entirely intolerable. Since the phase error can be considered to be a slowly varying function it can be obtained with sufficient accuracy from a small double sided portion of the interferogram around zero path difference, p=0, i.e.:
These few values of the phase error obtained can either be used as points for further interpolation, or the function can be convoluted with the full single sided inter ferogram from 0
53
3 The Theory of Untuned Resonators
3.1 The Theory of Homogeneous Cavity Sample Cells for Gaseous Absorbers
The derivation of useful quantities from cavity measurements will be described in terms of the ideal cavity shown in figure 3.1.
Figure 3.1Untuned Resonator
Detector
The rotating mode scrambler is necessary when either the radiation source is monochromatic or high resolution spectra are required. The mode scrambler ensures uniform energy density and that all the normal modes of the cavity are excited.
The simplest method of derivation is that first used by Lamb72r the rate theory explanation of photon absorption.
In the evacuated cavity shown, the rate of photon increase is given by:
54
d N k
dt M *(0 CLkS N K (3.1.1)
whereN]̂ is the number of photons of type k in the cavity.S is the surface area of the cavity walls. M]̂ (t) is the rate of creation of photons. a k is the constant of proportionality for absorption.
The equation 3.1.1 is purely for one single excited mode, k, within the cavity. Once equilibrium has been reached within the cavity,
d N k dt = 0 (3.1.2)
i.e. rate of creation = rate of absorption.
On the introduction of an absorbing gas into the cavity equation 3.1.1 is changed to:
d N k
d tM k( t ) - a k S N k - p k V N k (3.1.3)
whereV is the volume of the cavity and hence of the gas.p k is the constant of proportionality for the gaseous absorption.
Once again at equilibrium, d N kd t
55
The process can then be considered in terms of the Q of the cavity and its individual processes, where Q is defined by:
1 rate of loss of photons .— = ----------- ;------------ ( o . 1 . ■Q (jo x no of photons
and to is the resonant angular frequency.
The total Q of the cavity is made up of the Q's of the individual absorption processes i.e.
_J_______ 1 ( 1Q W Qi/(A:) + QC(A:) (3.1.5)
whereQw is the Q due to the walls.Q q is the Q due to the gas.
from the definition of Q (equation 3.1.4):
a, 5 =Qur(k)
(3.1.6)
PkV QgW
where to* is the resonant frequency of mode k.
Now representing 3.1.3 in terms of Q:
d N k t a ) kN k | a) kN k d t +Qt/(A:) + QC(A:) ( 3 . 1 . 7 )
56
or in terms of the total Q of the cavity:
d N k | aj kN k cit + Q(fc) M * ( 0 ( 3 . 1 . 8 )
The average, over a number, of pulses of equation3.1.8 gives the average number of pho t o n s :
N = Q(fc)CUjfc < M fc(0>„u (3.1.9)
In order to generalise the expression to all excited modes, the average is taken over all excited modes:
n k- 1
_ _1 y' Q(fc)7i7A; n k.\\ w k
(3.1.10)
where n is the number of excited modes, and the time average ĵ v for the kth mode is ~Mk.
If all modes are equally excited and photons in all modes are equally likely to be absorbed 3.1.10 becomes:
W = ( 3 .1 .1(JO
Now, if one once again considers an evacuated cavity i.e. with no absorbing gas, then Q = Qg and the number of photons is given by:
A7, QvMCO
( 3 . 1 . 1 2 )
57
If the cavity is refilled with enough gas to halve the number of photons then:
yv2 60 2 1 ( 3 . 1 . 1 3 )
From this Q = Q„/2 and hence QU = QC (3.1.14)
The attenuation coefficient of the radiation in the gas can then be determined from Q. If all other losses in the system were negligible, then the radiation would decay as exp-cut/Q. As in t seconds a photon travels x = ct, this becomes exp-cux/cQc. Thus the mean free path in the absorbing medium is expressed by cQc/a)r so that the absorption coefficient is given by:
“ = ̂ (3.1.15)
It is not possible to derive a simple expression for Qw as the cavity is not ideal as described above.
Qw can be obtained from experiment by introducing a hole into the wall of the cavity for which the loss of radiation can be calculated, as Qjj. The total Q of the evacuated cavity with a hole in it is given by:
I = _ L 4. J _Q Q w Qh (3.1.16)
If the hole is of sufficient size so as to halve the number of photons in the cavity then as before, QH = Qw . Qpĵ for a given hole size, is calculated giving QWt Then Q q can be determined by a further experiment with the cavity filled with gas.
58
An expression for Q jj can be obtained relatively simply from the kinetic theory of ideal gases. Figure 3.2 shows a hole in a cavity wall.
Figure 3.2 Photons Through Cavity Hole
td>
Wall
The number of photons which will hit the area A of the hole in a short time will be:
AccosOdt (3.1.17)
where c is the speed of light.
Assuming a uniform density and angular distribution of velocities throughout the cavity, then the number of photons per unit volume moving between 0 and e+de is:
^ £ s i n 0 d 0 ( 3 . 1 . 1 8 )4/r V
where V is the volume of the cavity.
Thus combining 3.1.17 and 3.1.18 the number of photons escaping through an area A of the wall per second i s :
N Acn2
cosf ls in OdQ2V 0( 3 . 1 . 1 9 )
59
N A c 4 1 /
H en ce from 3 . 1 . 1 1
_ o ) NQ " ~ N A c
= 8 l t V XA
( 3 . 1 . 2 0 )
It is then possible to calculate the absorption coefficient of the gas by measuring the photon density of the evacuated cavity, the evacuated cavity with a hole in the wall and the cavity containing an absorbing gas.
The above derivation makes the following assumptions about the radiation within the cavity:
i) the radiation density within the cavity is homogeneous throughout the entire volume;ii) absorption of radiation by the source and detector are negligible, whe n compared to the walls of the cavity;iii) all cavity modes are equally excited;iv) detector responsivity is linear.Condition i) is not true near to the calibration
hole, source or detector, but the effect of the inhomogeneity is minimal as the cavity is filled by the absorbing gas. The mode stirrer ensures that all modes are nearly equally excited.
A n y radiation detection devices must have a linear relationship between energy measured and output. In order to avoid having to calculate this relationship ratios of detector outputs can be used in calculations.
60
IfDw is the detector output from the evacuated cavity. Dfj is the detector output from the evacuated cavity with a hole in it.Dq is the detector output from the cavity filled with absorbing gas.
Then:
D h = Q H D \/ Q h + Q w
( 3 . 1 . 2 1 )
R GWD c _ Q c D w Qc + Q w
( 3 . 1 . 2 2 )
Now as Q jj can be calculated as shown above, Qw can be derived in terms of Q jj and R h w *
Q w ~ Q h(1 - R hv)
R„ w
From 3.1.22 Qq can be derived:
( 3 . 1 . 2 3 )
QcQ w R o w 1 ~ R gw
( 3 . 1 . 2 4 )
Then substituting for from 3.1.23 into 3.1.24:
QcR hw (1 - Rc w)
( 3 . 1 . 2 5 )
Then to obtain the absorption coefficient from equation 3.1.15, substitution of Qq gives:
61
(3.1 .26)- ̂ ̂ }lw ( ̂ ^ci/) 1c Row (1 “ R h\ / )Qh
The alternative method of carrying out the experiment is to introduce a gas of known absorption coefficient instead of a hole and calibrate in this manner.
If D c is the detector output from the cavity containing the calibration gas,then:
RcwD c _ Q cD \/ Qc + Qw
( 3 . 1 . 2 7 )
Then by similar derivation as in the hole calibration:
_ to Rct/{ 1 Rev) 1 c R c i / ( 1 ~ R ci/)Qc
( 3 . 1 . 2 8 )
Then substituting for the known absorption coefficient of the calibration gas:
otc Rcurjl R g / ) R c v ( l - R c v ) ac
( 3 . 1 . 2 9 )
One thus has two alternative procedures for obtaining the absorption coefficient of a gas.
The single frequency derivation above can be used for any frequency of source. However, the advantage of using a cavity is limited in frequency range, as shown below.
62
In an evacuated cavity Q is related solely to the losses in the cavity walls, i.e. absorption and transmission. The frequency dependence of the reflectivity of a metal was derived empirically by Hagen and Rubens89 and can be expressed as:
R~ 1 - 22 go e 0
a( 3 . 1 . 30 )
whereco is the angular frequency of the radiation, e0 is the permittivity of a vacuum and a is the d.c. conductivity of the metal.
Now considering the losses in the cavity wall, the cavity wall absorption coefficient is given by:
y « ( l - / 0
oc(Joe0
a( 3 . 1 . 3 1 )
If the radiation is over a relatively narrow band, thus allowing conductivity to be considered as constant, then:
( 3 . 1 . 3 2 )
From this it can be seen that the Q of the evacuated cavity is related to the frequency of the radiation by:
( 3 . 1 . 33 )
63
The Q of the cavity relates to the average time spent by a photon in the cavity and hence the average photon path length i.e.
( 3 . 1 . 3 4 )
Hence the effective pathlength of radiation within the cavity decreases with increasing frequency.
The decreasing reflectivity of the metal results in two effects:
i) losses in the cavity walls become greater.ii) sensitivity of the cavity becomes less, as it has less effective pathlength.
These two factors result in an upper limit on the frequency at which a cavity is a useful cell for studying gaseous a b s o r b e r s .
The lower frequency limit is defined by the need to be able to obtain a homogeneous radiation field of equally stimulated modes. The normal criteria used for this is that the minimum dimension of the cavity should be at least one hundred times greater than the wavelength of the minimum frequency being studied76.
3.2 Solids in Homogeneous Cavities
When solids are placed within an evacuated cavity the effect is considerably different from that produced by gases. The solid, in whatever form, has now introduced a new set of boundaries into the cavity. These boundaries are where the refractive index changes and hence reflection can occur.
64
The effect of reflection is not the same as in plane wave transmission spectroscopy where the energy will be reflected to the source and is effectively lost. Inside the cavity any reflected radiation is recycled. The effect is more subtle as now one has to consider the results of interference of both reflected and transmitted r a d i a t i o n .
An effect seen for solid samples, which should be avoided if possible, is the shielding of the interior of the samples when they have a high absorbance and/or a large volume. A fundamental relationship of the cavity as derived earlier showed an inverse linear relationship between the Q of a sample and its linear absorption coefficient and volume (equation 3.1.15).
The loss of detected signal when a solid is placed inside the cavity is proportional to the absorption cross-section of the sample i.e. its equivalent Black Body area. The absorption cross-section can be calculated by ignoring interference effects as:
wherea, is the linear absorption coefficient, d is a measure of the thickness of the sample and 6s is an element of the surface.
When a,d«i then this simplifies as now, exp(-a,d) ~ i-a,d hence:
(3.2.1)
(3.2.2)
65
where V is the volume of the sample.
The method for determining the linear absorption coefficient once the Q of the sample, Q s , has been calculated from equation 3.1.25 is outlined below based on the derivation of Llewllyn-Jones et al76.
Q s can be related to the Black Body area, A, by equation 3.1.20:
Qs8 i t V c
A.A(3.2.3)
whereV c is the volume of the cavity.
Now integrating equation 3.2.2 in isotropic radiation and neglecting effects of reflection due to differences in refractive index between the sample and the cavity volume:
o-0 = 4a,I/s (3.2.4)
whereV s is the volume of the sample.
In order to correct for the effects of differences in refractive index Llewllyn-Jones et al?6, calculated a correction factor by a ray tracing method. So that the equivalent Black Body area is 1.6 times the value calculated in equation 3.2.4. Hence substituting the corrected equation 3.2.4 into equation 3.2.3 one obtains the linear absorption coefficient of the sample:
66
67
i) Llewllyn-Jones et who considered theeffect of the real coefficient of refraction on the imaginary and derived a shape correction factor by a "ray-tracing" approach. They found the shape correction factor was constant over a large range of shapes and sizes of solid samp l e s .ii) Kremer et a l 7 9 , 8 0 f who worked out a method for the solution of both the real and imaginary parts of the refractive index using a series of very high accuracy single frequency measurements.
The major approach in this work has been to use samples of known optical properties and of the same geometry and volume as the experimental specimens to calibrate the cavity, allowing similar samples of materials of unknown optical properties to be studied. This approach is most effective when the materials have similar optical properties, e.g. silica and silicates.
3.3 Absorption of Parallel Sided Sheets
There have been several instances in the course of this work where it has been necessary to know the absorption properties of parallel sided sheets. These have arisen for:
a) beam splitter transmission of interferometers;b) calibration hole transmission from evacuatedcavit i e s ;c) absorption of samples within a cavity.
3.3.1 Calculation of Absorption
A Pascal computer program was written which allows one to obtain information on the absorption of samples in the above three situations. Specifically, the program can calculate the following quantities:
6 8
a) reflection, transmission and absorption versus angle of incidence.b) fixed angle of incidence reflection, transmission and absorption versus wavenumber.c) integrated angle of incidence reflection, transmission and absorption versus wavenumber.d) integrated angle of incidence reflection, transmission and absorption versus thickness.e) integrated angle of incidence reflection, transmission and absorption versus imaginary part of the complex refractive index.f) integrated angle of incidence reflection, transmission and absorption versus real part of the complex refractive index.g) iteration of experimentally determined cavity absorption to determine the range of possible complex refractive index, both real and imaginary.
Figure 3.4
whereI is the incident radiation power.R is the fraction of radiation reflected. T is the fraction of radiation reflected.
69
d is the thickness of the sheet.n,k are the real and imaginary parts of the complex refractive index.
The method of calculation of the coefficients is outlined below.
Hadley and Dennison^l solved the Maxwell equation boundary-value problem for a parallel sided sheet of absorbing material in an infinite lossless dielectric medium. Kremer et a l 7^ reduced these equations to a simpler form where the dielectric was air or vacuum. The interaction of electromagnetic radiation with parallel sided samples is determined by the real (n) and imaginary (k) complex refractive indices, the wavelength (A.), the thickness of the sheet (d) and the angle of incidence (0).
The formula for the reflection (R), transmission (T) and absorption (A) coefficients of a parallel sided sheet as shown in figure 3.1 are elucidated below, the formula are highly algebraic and very little physical significance exists in any of the intermediate values. The subscript "s" signifies the sample and the subscript "o" signifies the medium surrounding the sample.
For radiation polarised perpendicular to the plane of incidence:(Values specific to perpendicular polarisation are signified by use of the subscript a . )
f coshy- tcosa ~Da
(3.3.1.1)
T a
70
/!„= 1 - T „ - R „
D a = ^ c o s h y + x s i n l v y + ^ c o s a + c o s i n a
y =4- i tqsad
k
a = 4 / r p sgdk
q sa = ^ \ ] J ^ f [ { n 2- k 2- s i n 2 0)2 + 4 n 2fc2] - ( r i 2- k 2~ s i n 2 9)}
p sa “ ( V U ^ 2- ̂ 2" sin20)2 + 4rt2A;2] + (ft2 - A:2 - sin20)j
_ 2 _ 2 , _ 2PsO P SO P SO
p 0 = n Q cos 0
£-(p L + 2^2 4 p L p 2o
P 4o
T =(p L - P » ) 2 + 4 c?2 2 so P o
p 40
0 = (P2s o
2^2 4 p L p ;Po
x = 4 p sg(pLPo
p = 4 q 2saPo-{Pso2
P 40
71
4
72
F ig u r e 3 .5
Interferometer Radiation Intensities
RIT
TI
Source
2RT1
Detector
Each beam encounters one reflection from and one transmission through the beam splitter. The detected portion of the incoming radiation is given by the expression below:
where R a and T a are the polarisation averaged reflection and transmission coefficients of the beamsplitter at 45°.
The computer program was used to calculate the transmission of the interferometer over a range of wavenumbers for the thicknesses of beam splitters used in the present work (figures 3.6, 3.7, 3.8 and 3.9). The beam splitter was made of Melinex polyester film. The values of the complex index of refraction of biaxially orientated polyester film have been taken from the work of Igoshin et al ^ 2 .
(3.3.2.1)
Frac
tion
al C
oeff
icie
nts
Frac
tion
al C
oeff
icie
nts
73
F i g u r e 3 . 6 F i g u r e 3 . 7
6u Me l in ex B e a m s p l i t t e r n 1.69, k 0.06
35u Melinex Beamsplitter n 1.69 k 0.06
Figure 3 . 8 Figure 3.9lOOu M el in ex B e a m s p l i t t e r
n 1.69, k 0.06135u Melinex Beamsplitter
n 1.69 k 0.06
74
3.3.3 Reflection from Vacuum Cavity Window
Most of the present work was carried out under vacuum in order to avoid interference from the spectrum of water vapour. In order to keep the vacuum it was necessary to use a Melinex (polyester) window over the calibration hole. The hole was therefore not acting as a true black body due to reflection from the polyester film.
The radiation passing through the cavity window is incident from all angles. The correction to the derivation of Lamb for a hole in an untuned resonator involves the integration of the coefficient of transmission over all angles of incidence from 0 to n / 2 .
The transmission coefficient of the window is :
T w=s 2 ) T a(
75
z r - ° -—77 f 3( T a ( 8 , n , k , c l , A . ) + A a ( 6 , n , k , d , \ . ) } (3.3.3.3)Q u 4 Tt V J o
The corrected value Q of the calibration hole was used in the present work.
The variation of the integrated sum of absorption and transmission is shown below for llOum polyester film used as window material for calibration.
Figure 3.10A p p a r e n t W i n d o w T r a n s m i s s i o n
k 0 .0 6 , n l . 6 9 , l l O u m
The variation of transmission with various parameters, as calculated by the computer program, are shown below:
Fra
ctio
nal
Coe
ffic
ient
s F
ract
iona
l C
oeff
icie
nts
76
Variation of Absorption w ith k
F i g u r e 3 . 1 1
n l .6 9 , l lO u m , 1 0 0 c m - l
k
Figure 3.13Variat io n of A b sorp t i on
w i t h T h i c k n e s s n l . 6 9 , k0.06, 1 0 0 c m - l
Thickness(mm)
Varia t ion of Absorp t i on w i t h n
F i g u r e 3 . 1 2
k0.06, l lOum, 1 0 0 cm - l
Figure 3.14Varia t ion of Absorp t i on
w i t h W avenum ber n l . 6 9 , k0.06, l lOum
77
3.3.4 Absorption of Parallel Sided Sheet within an Untuned Resonator
A parallel sided sample placed within the cavity is equivalent to a two sided hole as modelled by Lamb and outlined in section 2.3. Photons will strike the sample with equal probability from either side and at any angle, the number of photons will be twice the value calculated from Lamb's derivation, as 0 now varies from 0 to 4/r not 2n as in the case of the calibration hole. The Q of the parallel sided sample is thus given by:
1 K a r 2— = - — — A a( 0 , n , k , d , A.)sin0cos0d0 ( 3 . 3 . 4 . 1 )Q s ZJT V Jo
Graphs are shown of the variation of the absorption integral with wavenumber, thickness, imaginary and real parts of the complex refractive index.
Figure 3.15Var ia t io n of A b sorp t i on w i t h k
n3.88, 1mm, 1 0 0 c m - l
k
Figure 3.16Variat ion of Absorp t i on w i t h n
k0 .0226, 1mm, 1 0 0 c m - l
Fra
ctio
na
l C
oef
fici
ents
78
V a r ia t io n of A b s o r p t i o n
F i g u r e 3 . 1 7
w i t h T h i c k n e s s k 0 .0 2 2 6 , n 3 .8 8 , 1 0 0 c m - l
Var ia t io n of Absorpt io n
F i
