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A UV Zeeman-effect polarizer 1985
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Title | A UV Zeeman-effect polarizer |
Creator |
Grant, Robert Wallace |
Publisher | University of British Columbia |
Date Created | 2010-05-13 |
Date Issued | 2010-05-13 |
Date | 1985 |
Description | It is shown that light emitted by mercury vapour at 253.7 nm can be polarized by passing the light through mercury absorption gas embedded in a magnetic field which is transverse to the direction of propagation of the light. The absorption lines of the mercury are split by the Zeeman effect, so that the absorber has an absorption coefficient which depends on both the polarization and wavelength of the transmitted light. A complete theory for the Hg²⁰² isotope is presented and the results are compared to measurements made with a natural mercury emitter and absorber. The observations are in qualitative agreement with the theory once isotope and hyperfine structure of the isotopes in natural mercury are included in the theory. Quantitative analysis was not possible because the emission line profiles could not be measured with the available equipment. |
Subject |
Zeeman Effect Polarization (Light) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-05-13 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0096090 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/24674 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0096090/source |
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A UV Zeeman-Effect Polarizer by Robert Wallace Grant B.Sc, The University of Manitoba, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULITY OF GRADUATE STUDIES (Department of Physics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1985 © Robert Wallace Grant, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of \hy$\c<, The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 i i ABSTRACT It i s shown that l i g h t emitted by mercury vapour at 253.7 nm can be polarized by passing the l i g h t through mercury absorption gas embedded i n a magnetic f i e l d which i s transverse to the direction of propagation of the l i g h t . The absorption lines of the mercury are s p l i t by the Zeeman effe c t , so that the absorber has an absorption coeff i c i e n t which depends on both the polarization and wavelength of the transmitted l i g h t . A complete theory for the Hg 2 0 2 isotope i s presented and the results are compared to measurements made with a natural mercury emitter and absorber. The observations are i n qual i t a t i v e agreement with the theory once isotope and hyperfine structure of the isotopes i n natural mercury are included i n the theory. Quantitative analysis was not possible because the emission l i n e p r o f i l e s could not be measured with the available equipment. i i i TABLE OF CONTENTS Abstract i i Table of Contents i i i L i s t of Figures v i Acknowledgements v i i i Chapter One Introduction 1 Chapter Two 3 I I . 1 The Zeeman Effect 4 11.2 Line Broadening 14 Natural Broadening 15 Resonance Broadening 17 Doppling Broadening 21 The Combined Line P r o f i l e 22 11.3 Theory 24 11.4 The Wave i n a Transverse and P a r a l l e l F i e l d 50 The Transverse F i e l d 51 P a r a l l e l F i e l d 57 i v II.5 The Polariz a t i o n of the Transmitted Wave 58 Completely Polarized Light 65 Unpolarized Light 67 P a r t i a l l y Polarized Light 68 Unpolarized Incident Wave 70 Simple Examples 72 Linearly Polarized Incident Wave 75 Chapter Three 111.1 The Experimental Design 89 The Light Source 92 The Gas C e l l 93 The Magnet 95 The Aperture Stops and Lenses 96 Polarizer and Quarter-Wave Plate 97 Monochromator 98 Detector 98 Effect of the Magnetic F i e l d on the Experimental Apparatus 99 111.2 The Experimental Procedure 100 The Unstable Light Source 104 111.3 Experimental Error 106 111.4 Experimental Results 110 V Chapter Four Comparing Theory and Experiment 120 IV. 1 The Effect of Using Natural Mercury i n the Absorption C e l l 121 Effect of Nuclear Spin 121 IV.2 The Shape of the Experimental Incident Line 124 IV.3 The Experimental Results 125 Chapter Five Concluding Discussion 129 References 132 Appendix 135 v i LIST OF FIGURES Figure Numbers Page r e l 1. Relative Intensities I of the Componet Lines for observations along the y-axis 12 2. Orientation of the Magnetic F i e l d 27 3. Orientation of |u>' Co-ordinate System 32 4. a) Polarization Measuring System b) Polari z a t i o n Orientation 62 5. Emission, Absorption and Transmission Line P r o f i l e 73 6. Geometry of Po l a r i z a t i o n E l l i p s e 78 CD 7. J as a function of Magnetic F i e l d Strength for Three Gas Temperatures, 300K, 350K and 375K D = 25 (10 - 3) cm - 1 J = J at zero f i e l d and i s independent of the xx yy strength of the f i e l d 86 CO 8. J as a function of Magnetic F i e l d Strength for Three Gas Temperature, 300K, 350K and 375K D = 40 (10 - 3cm - 1 ) J = J at zero f i e l d and i s Independent of the xx yy strength of the f i e l d 87 v i i GO 9. J as a function of Magnetic F i e l d Strength for Three Gas Temperatures, 300K, 350K and 375K D = 60(10- 3cm _ 1) J = J at zero f i e l d and i s independent of the xx yy r strength of the f i e l d 88 10. Experimental Apparatus 90 11. Temperature Regulator for the Water Bath 94 12. Im (J ) as a function of Magnetic F i e l d Strength 113 xy 13. J ' and J' as functions of magnetic f i e l d strength xx yy b for a natural mercury absorption gas at 22.6°C 117 14. J' and J' as function of magnetic f i e l d strength for xx yy e a natural mercury absorption gas at 30°C 118 15. Polarization P as a function of magnetic f i e l d strength for a natural mercury gas at 22.6° and 30.0°C 119 v i i i ACKNOWLEDGEMENTS Physics demands extended periods of isolated s o l i t a r y work. The reward for this e f f o r t comes from the time spent with others sharing ideas and learning. T would l i k e to thank my friends and colleagues i n the Physics Department for making my time there rewarding. I would p a r t i c u l a r l y l i k e to thank Dr. F.L. Curzon who suggested and supervised the work i n this thesis. Without his unflagging support this thesis would never have been completed. I would also l i k e to thank Mr. A. Cheuck for his assistance i n designing and constructing the experimental apparatus. The assistance of Dr. Bloom i n providing laboratory space and the electromagnet used i n the experiment i s also acknowledged. F i n a l l y , I would l i k e to thank Dr. F.L. Curzon, Susan Mah, Connie Zator, Young Yuen and Don Furseth for their help i n preparing this thesis. - 1 - CHAPTER I Introduction A number of spectroscopic techniques for analysing l i n e shapes and tr a n s i t i o n p r o b a b i l i t i e s require high quality linear polarizers. These polarizers currently exist for the v i s i b l e and near u l t r a - v i o l e t region, but because of the properties of the dispersive media used, they have low transmissivity i n the far u l t r a - v i o l e t region. The object of this study i s to examine the p o s s i b i l i t y of exploiting the Zeeman Effect to produce polarizers which have a high transmissivity i n the u l t r a - v i o l e t region. The proposed technique involves the charater- i s t i c s of the absorption spectra of a gas immersed In a magnetic f i e l d . The absorption spectra, when viewed i n a direction transverse to the magnetic f i e l d , s p l i t into a series of Zeeman Components which are l i n e a r l y polarized either transverse or p a r a l l e l to the magnetic f i e l d . By adjusting the degree of s p l i t t i n g and the shape of the absorption lines the gas can act as a linear polarizer for incident l i g h t at or near an absorption l i n e of the gas. The technique i s examined both theoret i c a l l y and experimentally. In Chapter I I a theoretical model i s developed for the propagation of l i g h t In a gas Immersed i n a magnetic f i e l d . The derived equations are more general than i s required for this study, but i t i s hoped that they may find application i n further studies. - 2 - The theoretical model i s used to calculate the effect of a gas composed of the Hg 2 0 2 isotope, on the state of polarization of an incident 253.7 nm l i n e of mercury. The 253.7 nm l i n e was selected because i t s properties are reasonably well known and therefore the calculation could be performed with a high degree of precision. The theoretical calculations indicate that the method can be used to produce highly l i n e a r l y polarized l i g h t with small transmission losses. The technique i s examined experimentally i n Chapter I I I . The state of polarization of the 253.7 nm l i n e of mercury i s measured after the i n i t i a l l y unpolarized l i g h t traverses a gas c e l l f i l l e d with natural mercury and immersed i n a magnetic f i e l d traverse to the direction of propagation of the l i g h t . While the use of natural mercury rather than an even isotope introduces a number of undesirable effects, the experiment supports the conclusion that the technique can be used to produce high quality polarizers. - 3 - CHAPTER I I In this chapter a theoretical model i s developed to describe the production of polarized l i g h t through the Z eeman Effec t . The purpose of this model Is to account for the observed polarization and i n so doing provide a framework for i t s study and assessment. The f i n a l result i s an equation for the transmission of l i g h t through a gas Immersed i n a magnetic f i e l d , where the frequency of the l i g h t i s at or near a reasonance frequency of the atoms of the gas. The equation indicates how to optimize the polarization and i n addition reveals the inherent limi t a t i o n s of this method of polarizing l i g h t . The central role of the Zeeman Effect i n this study necessitates a review of this phenomenon. However, since many comprehensive accounts of this effect already e x i s t , ( see Condon and Shortley 1964, Sobel'man 1972, and Kuhn 1969) only a cursory review i s presented. Brevity i s achieved i n part by omitting the mathematical derivations of most of the equations, and by presenting only those equations which are necessary for a clear under- standing of the e f f e c t , or which are required i n the subsequent theoretical development. - 4 - I I . 1 THE ZEEMAN EFFECT When atoms are placed i n an external magnetic f i e l d i t Is found that the f i e l d causes some of the spectral lines of these atoms to s p l i t into a series of component li n e s which are displaced about the frequency of the or i g i n a l l i n e . I f a spectral l i n e i s observed along an axis which i s transverse to the direction of the magnetic f i e l d , i t i s found that a given component l i n e i s l i n e a r l y polarized with i t s e l e c t r i c f i e l d vector either p a r a l l e l (n-component), or perpendicular (a-component), to the external magnetic f i e l d vector. I t i s these two re s u l t s , the s p l i t t i n g of the spectral lines and the polarization of the component l i n e s , which together comprise the Zeeman Ef f e c t . The Zeeman Effect can only be s a t i s f a c t o r i l y accounted for by using a quantum mechanical representation of the atom i n an external magnetic f i e l d as given by: H = H + H ( 2 - 1 ) o m v ' where H i s the Hamiltonian of the Isolated atom, and H describes the o m interaction between the external magnetic f i e l d \ and the magnetic moment of the atomu. H has the form m H M = - J . \ ( 2 - 2 ) The magnetic moment of the atom i s related to the t o t a l o r b i t a l angular momentum X, and the t o t a l spin angular momentum ̂ of the electrons i n the - 5 - t atom through the equation ; - - j p (t + 2 "I) (2-3) where ^ = | L - (2-4) e i s the Bohr Magneton; mg, i s the electron mass; and e i s the elementary charge. The Hamiltonian H should also contain a term H' which i s quadratic i n m ^ B. This term would account for the induced effect of atomic diamagnetism. However, i t can be shown (see Messiah 1962, page 541) that for the lower energy states of the atom: H' 1CT5 ZB (Tesla) (2-5) m where Z i s the atomic number of the atom. Thus, for the modest f i e l d s of less than 1 Tesla considered i n this study, the term H^ i s ne g l i g i b l y small and can be excluded from the Hamiltonian H. The Hamiltonian equation i s solved using perturbation techniques. The eigenstates of H q are given by | ajm > where j i s the quantum number for the t o t a l angular momentum J of the state, mh i s the eigenvalue of (the component of the t o t a l angular momentum operator along the z a x i s ) , and a labels a l l the remaining quantum numbers of the state. The eigenstates are tAt this point only atoms with a nuclear spin 1 = 0 are considered. - 6 - (2j + 1) - f o l d degenerate i n the quantum number m, i . e . states from the same l e v e l have the same energy.^ The degeneracy of the states dictates the use of a degenerate perturbation technique, which reduces the problem of f i n d i n g the f i r s t order change i n the energy A of each state to the diagonalization of a perturbation matrix, (see Merzbacher, 1970, page 425). In the representation | ajm > the perturbation matrix, which includes a l l the states of a given l e v e l , i s already diagonal and the equation for AE. reduces to: AE . = <a jml H la jm> jm J 1 m1 J r 1 = ~ - <ajm|(L + 2%) • B" | ajm> u = <ajm|ga j 5 • % | ajm> = a g . Bm (2-6) r o a j y where, i n the l a s t step, the z axis was chosen along 1$. § aj» known as the Lande g factor, i s a constant for each l e v e l , i . e . i t i s independent of m. tThere Is some ambiguity In the meanings of the terms ' l e v e l ' and 'energy l e v e l ' , (see Condon and Shortley pages 97 and 385). To avoid confusion the term 'energy l e v e l ' w i l l r e f e r e x c l u s i v e l y to the energy associated with a p a r t i c u l a r state, or when the state i s degenerate to the energy associated with the group of degenerate states. The term ' l e v e l ' w i l l r e fer to the group of states which d i f f e r only i n t h e i r m quantum number, even when these states are no longer degenerate. - 7 - Implicit i n this development i s the assumption that the change i n energy due to i s much less than the energy separation between the le v e l s ; or equivalently, the perturbation due to H i s much smaller than the m perturbation due to the spin-orbit interaction. Where this assumption i s no longer v a l i d the neighboring levels must also be included i n the perturbation matrix. For most of the levels of heavier atoms this i s necessary only for magnetic f i e l d s larger than 1 Tesla, which are not considered i n this study. The assumption i s easily s a t i s f i e d for the 253.7 nm l i n e of mercury (see the appendix). The effect of the magnetic f i e l d on the states can be deduced from equation (2-6). The degenerate l e v e l s p l i t s into (2j+l) different energy le v e l s , each of which corresponds to a unique value of m. These new energy levels are symmetrically displaced about the o r i g i n a l energy l e v e l since m = ±j, ±j -1, ... The magnitude of the s p l i t t i n g i s linear i n B. There i s no f i r s t order change i n the energy l e v e l of a state with j = 0 or m = 0. The effect of the magnetic f i e l d on the l i n e spectra of an atom can now be examined. The absorption or emission of a photon by an atom i s associated with the t r a n s i t i o n from one energy l e v e l to another. The angular frequency of a spectral l i n e resulting from such a t r a n s i t i o n i n the absence of a magnetic f i e l d i s given by OJ , where: OJ = - J - | E ° - E ° , , | o h 1 a j a ' j ' 1 (2-7) - 8 - and E°^ and are the energy levels of the unperturbed states |ajm> and ja'j'm^ respectively. In the presence of a magnetic f i e l d the energy levels E , become: a jm E . = E° . + AE . = E° . + g . a Bm (2-8) ajm a j jm a j 6 a j *o ^ U J and the l i n e s p l i t s into a series of component l i n e s ^ with angular frequencies u) l ., where: mm OJ mm or Umm' = W 0 + - ^ B ( 8 a f - g a ' j . m ' ) (2-9) This equation describes the observed s p l i t t i n g of the spectral lines by the magnetic f i e l d . However, the additional property of the Zeeman e f f e c t , namely the polarization of the component l i n e s , i s only i n d i r e c t l y related to the presence of the magnetic f i e l d . I t can only be properly accounted for by examining the radiative process which governs the t r a n s i t i o n between two states. Furthermore, while equation (2-9) describes a l l possible transitions between two l e v e l s , these transitions many occur with markedly different p r o b a b i l i t i e s . I t i s the radiative process that determines the probability of a pa r t i c u l a r t r a n s i t i o n or equivalently the strength of the associated component l i n e . tThe component li n e s are produced by the transitions between s p e c i f i c states of two given l e v e l s . - 9 - These observations indicate that a brief digression to the theory of radiative processes i s necessary for a complete understanding of the Zeeman Effect. For the purposes of t h i s study the only radiative processes of interest are those described by e l e c t r i c dipole t r a n s i t i o n s , since these produce the most intense l i n e s . The probability of an e l e c t r i c dipole t r a n s i t i o n between the states |ajm> and la'j'm^ re s u l t i n g i n the absorption or emission of a photon polarized i n the d i r e c t i o n of the unit vector ^ and t r a v e l l i n g i n the direction £ i s of the form: (Sobel'man page 305) W£ « |"R.<ajm |"p| a'j'm^ | 2 (2-10) where P i s the e l e c t r i c dipole operator Z "P = -e E "rr (2-11) i 1 and the sum i s taken over the position ^.of each electron with respect to the nucleus. Most of the interesting properties of the t r a n s i t i o n can be deduced from the matrix element <ajm | 1? j a ' j ' m ^ . In p a r t i c u l a r , by exploiting the Wigner-Eckart Theorem (see Messiah page 573), the dependence of the matrix element and hence W* on m and m' can be found as follows: - 10 - <ajm | $ | a'j«m'> = C(j'lm'q|jm) / ( 2 j +1) x <aj | | P | | a'j'> e* (2-12) where <aj || P || a'j'>^, the reduced matrix element, i s a scalar independent of m and m*, C ( j * l m'q|jm) i s a Clebsch-Gordon co e f f i c i e n t (these are tabulated i n Condon and Shortley page 76), q = m-m', e^ i s the complex conjugate of the unit vector e defined as f o l l o w s ^ : q e = - - ( i + i j ) ; e = k; e = ̂ ( i - i j ) (2-13) 1 /2 o -1 / T From the properties of the Clebsch-Gordon coefficients I t follows immediately that <<xjm | "P ̂ 'j'm'> = 0, and by equation (2-10) = 0 unless: j - j ' = 0, ± 1; j + j ' > 1 and; q = 0, ± 1 (2-14) Thus, only a r e s t r i c t e d class of states i s connected through e l e c t r i c dipole transitions and the selection rules contained i n equations (2-14) govern equation (2-9). t<aj || P||a'j'> Is related to the quantities <aj:P:a'j'> introduced i n Condon and Shortley. The relationships are tabulated i n Sobel'man (page 85). If these relations are substituted into equation 2-12 i t Is eas i l y shown that equation 2-12 i s equivalent to the equations given by Condon and Shortly at page 63. ttThe properties of these spherical basis vectors are presented i n Rose, 1957 page 103. - 11 - Equation (2-12) also accounts for the polarization and r e l a t i v e i n t e n s i t i e s of the Zeeman Components. This can be seen most readily by studying the special case of the emission or absorption of a photon t r a v e l l i n g along the y-axis, i . e . \ = j . Any direction of polarization 1t can now be resolved completely into components along the x and z axis, and no further r e s t r i c t i o n s are imposed by independently examining along each of these axes. Inserting equation (2-12) into equation (2-10), and l e t t i n g t = k w e fin d : Wk " 2jV 1 C(j'lm'q |jm)2 | <aj |1 P I! a'j'> | 2 when q = 0 (2-15) and W, = 0 when q = ± 1 k Now l e t t i n g I = i we f i n d : = 0 when q = 0 and W i " I 2l+T C ( J' l m , q 'jm)2 I <«J II p I U ' J '> I 2 when q = ± 1 (2-16) Thus photons polarized i n the k direction result exclusively from transitions where q = 0 and those polarized along the x-axis result exclusively from transitions where q = ± 1. - 12 - F i g . 1 : Relative Intensities l r e l of the Component Line, tor Observations along the y-axis Transition T R E L r . I 7 ,(m' = m) T R E L / . \ , (m' = m ± 1) - 13 - For t r a n s i t i o n s between two l e v e l s , a j •* a ' j ' the term 2J+1'' M II P!|a'j*> | 2 remains f i x e d . Therefore the r e l a t i v e p r o b a b i l i t i e s of t r a n s i t i o n s between the states of the two l e v e l s , or equivalently, the r e l a t i v e i n t e n s i t i e s of the component l i n e s I depend only on the square of the Clebsch-Gordon c o e f f i c i e n t s . These r e l a t i v e i n t e n s i t i e s for the s p e c i a l case of \ along the y-axis are tabulated i n F i g . 1. The o r i g i n of the observed p o l a r i z a t i o n produced by the Zeeman E f f e c t can now be understood. In the absence of an external perturbation, such as a magnetic f i e l d , the states ]a'j'm'> are degenerate i n m' and equally populated. Thus a l l the t r a n s i t i o n s between two le v e l s have the same energy and a l l the s p e c t r a l l i n e components have the same frequency. Each of the spectal l i n e components has a c h a r a c t e r i s t i c p o l a r i z a t i o n , but when these components are added together to produce the observed s p e c t r a l l i n e the c h a r a c t e r i s t i c p o l a r i z a t i o n s cancel exactly to give i s o t r o p i c unpolarized l i g h t . This r e s u l t i s e a s i l y demonstrated for the s p e c i a l case studied above by noting that f o r every s p e c t r a l l i n e the t o t a l Intensity p o l a r i z e d along the x axis i s equal to the t o t a l i n t e n s i t y polarized along the z-axis. (The t o t a l i n t e n s i t y polarized along each axis i s found by summing the r e l r e l r e l a t i v e i n t e n s i t i e s I and I , given i n F i g . 1, over a l l allowed values A . Li of m.) This i s the expected r e s u l t since It accords with the p r i n c i p l e that there i s no preferred d i r e c t i o n i n space for an i s o l a t e d atom. The s i t u a t i o n changes when a magnetic f i e l d i s applied. The component l i n e s no longer have the same frequency, making i t possible to observe each - 14 - component independently. Observations transverse to the magnetic f i e l d are equivalent to the special case examined e a r l i e r of photons t r a v e l l i n g along the y-axis, since the z-axis was chosen along % i n equation (2-6). The n-components can be immediately i d e n t i f i e d as the q = 0 transitions and the a-components as the q = ± 1 tr a n s i t i o n s . The r e l a t i v e i n t e n s i t i e s of the components w i l l s t i l l be given by Fi g . 1 provided the weak magnetic f i e l d approximation remains v a l i d , implying that the states do not deviate substantially from the unperturbed states | a jm>. The frequency of each component l i n e i s given by equation (2-9). These results provided a complete de s c r i p t i on of the main properties of the Zeeman E f f e c t . They are used i n a la t e r section, after the problem of l i n e broadening has been reviewed, to examine the f e a s i b i l i t y of polarizing spectral lines through the selective absorption of the component l i n e s . II.2 LINE BROADENING The results presented i n the previous section pertain to the idealized case of a perfectly isolated atom at rest. In this i d e a l i z a t i o n the atom can only absorb or emit photons with the discrete energy of a t r a n s i t i o n between two states. In practice an atom i s moving randomly through a gas and i s surrounded by perturbing particles which a l t e r i t s state. These deviations from the idea l have the effect of allowing an atom to absorb or - 15 - emit photons over a range of frequencies for each allowed t r a n s i t i o n . This effect i s known as Line Broadening and the function which expresses the dependence of the t r a n s i t i o n probability on the angular frequency, w, of the photon i s the Line P r o f i l e P(w)* In the case of emission lines the broadening processes are intimately related to the s p e c i f i c mechanism used to excite the atoms and therefore cannot be described without reference to the particular l i g h t source. Even when the excitation mechanism i s well understood the calculation of the resulting l i n e p r o f i l e i s generally very complex. In this study our chief interest i s i n the absorption of l i g h t by a single element gas through resonance t r a n s i t i o n , i . e . transitions connected to the ground state. In this case the problem assumes an unusually tractable form i n which the l i n e broadening Is dominated by only three well understood mechanisms. In t h i s section these three mechanisms, Natural, Resonance, and Doppler Broadening, are treated separately and then with some qu a l i f i c a t i o n s the results are combined to obtain the complete l i n e p r o f i l e . Natural Broadening An atom can never be isolated from the ubiquitous photon radiation f i e l d . The f i e l d interacts with the atom allowing energy to be transferred between the atom and the f i e l d . This transfer of energy i s , i n turn, t The Line P r o f i l e corresponds to the experimentally observed spectral l i n e shape only In the case of lines i n an o p t i c a l l y thin material. - 16 - accompanied by a t r a n s i t i o n from one atomic state to another. The existence of these allowed t r a n s i t i o n s between states implies that the states a j have uncertainty p r i n c i p l e AE . T , ~ h we f i n d that a f i n i t e l i f e t i m e implies an indeterminancy i n each energy l e v e l of the order AE ,. Thus the photon f i e l d leads to a f i n i t e l i f e t i m e and a f i n i t e l i f e t i m e means that a t r a n s i t i o n must occur at a range of energies rather than at a sing l e discrete energy. A complete analysis (see H e i t l e r 1954 page 181, or Messiah 1962, page 994) shows that a f i n i t e l i f e t i m e manifests i t s e l f i n the form of a Lorentz Line P r o f i l e L(u)). where y i s the f u l l width at hal f maximum. This l i n e p r o f i l e occurs whenever a perturbation has the e f f e c t of a l t e r i n g the l i f e t i m e of a state and Is therefore not unique to Natural Broadening. For Natural Broadening y = y^ and i s c a l l e d the natural l i n e width where a f i n i t e average or natural l i f e t i m e x a j . Applying the time-energy L(oj)du) = (2-17) 2rt (co- u 0 )2 + (L)2 -1 a j + x a ' j -1 , for the t r a n s i t i o n a j •*• a ' j ' (2-18) When an atom i s Immersed i n r a d i a t i o n which i t i s capable of absorbing, the l i f e t i m e of the ground state i s not s t r i c t l y i n f i n i t e . However, excepting the case of very intense r a d i a t i o n , the l i f e t i m e of the ground - 17 - state Is much greater than that of any excited state. Thus for resonance absorption y.~ % \ where a j i s the excited state. N a 3 Natural Broadening usually makes the least s i g n i f i c a n t contribution to the f i n a l l i n e p r o f i l e . Resonance Broadening Co l l i s i o n s among the atoms of a gas also lead to l i n e broadening. In a gas composed primarily of atoms of one element, the broadening of the resonance lines of this element Is dominated by c o l l i s i o n s among the atoms of this element. Consider two atoms d i f f e r i n g only i n that one i s i n the ground state ( a 0 j Q ) , while the other i s i n an excited state (aj) which has an allowed t r a n s i t i o n to the ground state. A c o l l i s i o n between these two atoms permits a resonance transfer of energy, i . e . the excited atom decays to the ground state while the ground state atom i s driven into the state a j . This resonance transfer e f f e c t i v e l y creates an additional mode of decay for the state a j and hence leads to a shortening of the l i f e t i m e of that state. This c o l l i s i o n induced shortening of the l i f e t i m e w i l l independently produce a Lorentz l i n e P r o f i l e characterized by a f u l l width at half-max. Yr, y R has been t h e o r e t i c a l l y calculated using a variety of techniques. The most complete treatments appear to be by A l l and Griem, 1965 and D'Yakonov and Perel 1965. A l i and Griem derive their result using the - 18 - Impact and c l a s s i c a l path approximations for a state a j which has an e l e c t r i c dipole t r a n s i t i o n to the ground state a Q j Q • They f i n d : e 2 f 2JQ + 1 „ * i n cgs units, or i n MKS units: where: H= 3 ' 8 4 * 2 j T T ~ N m ^ T - <2"19> e o e 2 f y = 3.84* N j (2-20) R 2j + 1 4n;enm w v ' 0 e o N i s the density of atoms i n the ground state, m i s the electron mass, e e Is the electron charge, f i s the absorption o s c i l l a t o r strength, and E Q i s the Per m i t t i v i t y constant D'Yakonov and Perel determine y using irreducible tensors to calculate the relaxation of the nondiagonal density matrix of an excited atom i n c o l l i s i o n with an atom In the ground state. They also use the c l a s s i c a l path and impact approximations and assume the existence of an e l e c t r i c dipole t r a n s i t i o n to the ground state. Their result does not exhibit the simple dependence on j and j of equation (2-20) and a numerical result i s provided only for the case where j = 1 and j =0. However, i f for this - 19 - s p e c i f i c case, their result i s put i n the same form as equation (2-20) they find that the numerical factor corresponding to 3.84 has a value of 2.55. The impact approximation, upon which these results are based, remains v a l i d provided the time between effective c o l l i s i o n s i s much larger than the duration of an effe c t i v e c o l l i s i o n Q_1 , and only frequencies within a range Aw from the l i n e centre are considered. These two v a l i d i t y c r i t e r i a are contained i n the equation from A l l and Griem: Max {y R, AOJ} « Q where, K T 0- (Jj-)3/1 (2-21) and, K R i s the Boltzmann constant, T i s the temperature i n K°, and M i s the mass of the atom. Y R has been experimentally determined by G. Stanzel 1974, for the 253.7 nm resonance l i n e of Mercury; he found: Y R = (5.3 ± 0.5) 10 - 9 N sec - 1 cm3 (2-22) for N ~ 5 x 10 1 8 cm - 3 and T ~ 553 K°. Using the characteristics of the 253.7 nm l i n e given i n the appendix, the theoretical values given by A l l and Grien and by D'Yakonov and Perel are respectively: and Y- = 6.03 x 10 - 9 N sec - 1 cm3 (2-23) K Y_ = 4.01 x 10"9 N sec - 1 cm3 (2-24) R - 20 - Thus both results are In f a i r agreement with experiment given that the v a l i d i t y c r i t e r i a are only weakly s a t i s f i e d at these temperatures and densities, i . e . from equation (2-21). Y R ~ 3 x 10 1 0 « 8X10 1 0 ~ 0 (2-25) The theoretical equations for y R were derived assuming that the atoms were not immersed i n a magnetic f i e l d . In the absence of a magnetic f i e l d a c o l l i s i o n between two l i k e atoms permits not only a resonance transfer of energy, but also an a l t e r a t i o n i n the exited state. The resulting excited atom can be i n a state with an m quantum number different from that of the excited atom before the c o l l i s i o n . When a magnetic f i e l d i s applied the states no longer have the same energy, therefore a change i n the m quantum number of the excited state must be accompanied by a change i n the k i n e t i c energy of the c o l l i d i n g atoms. This has the effect of decreasing the probability of a change i n the resulting exited state. The theoretical equations for y should remain v a l i d , however, provided the magnetic f i e l d induced l i n e s p l i t t i n g (u ), from equation (2-9) i s small compared to mm o the inverse of the duration of an effective c o l l i s i o n , i . e . u , « n « ~ w « Q (2-26) mm o (see D'Yakonov and Perel and Sobel'man page 464). For many lines t h i s inequality i s not s a t i s f i e d at even modest f i e l d strengths of less than 1 Tesla. At these f i e l d strengths the Zeeman - 21 - s p l i t t i n g of the l i n e becomes s u f f i c i e n t l y great to cause a s i g n i f i c a n t decrease i n the probability of a c o l l i s i o n induced t r a n s i t i o n to a state with a different m quantum number. This i n turn decreases the ov e r a l l probability of a resonance transfer of excitation energy since In effect the number of states able to participate i n the transfer has decreased. Thus the resonance l i n e width y R w i l l be smaller than predicted by equation (2-20) when the inequality (2-26) i s not s a t i s f i e d . Doppler Broadening Doppler Broadening of spectral lines i s the cumulative result of the random motion of the atoms. Ignoring a l l other broadening mechanisms, an atom w i l l only absorb or emit photons of frequency I O q as measured i n the atom's frame of reference. The frequency of these photons i n the reference frame of an external observer i s u, where according to the n o n - r e l a t i v i s t i c V Doppler Pri n c i p l e w = O J q + ~ U Q » a n (^ v I s t n e component of the atom's veloci t y along the dire c t i o n of observation. I f the d i s t r i b u t i o n function for V i s f(V) dV then the Doppler l i n e p r o f i l e , D(u)), produced by a unit volume of atoms i s : D(u>) du = f ( — (w-w )) — dw (2-27) o o For a gas In equilibrium f(V) dV i s just the Maxwellian v e l o c i t y d i s t r i b u t i o n and D(w) dto becomes: - 22 - 0\ D(w)dw = u _ 1 / 2 D-1 e x p [ - ( - g — ) 2 ] dco (2-28) Vo 2 K B T where D = U>q — i s known as the Doppler Width, and V = ( — y j ~ ) l / 2 (2-29) The Combined Line P r o f i l e The combined l i n e p r o f i l e P(io) dw i s a convolution of the character- i s t i c l i n e p r o f i l e s of each broadening process. I f two broadening processes are s t a t i s t i c a l l y independent and have p r o f i l e s Pĵ (w-to0) and P 2(o)-w 0), the combined l i n e p r o f i l e P'(U)-U)Q) i s given by: p'^> ° J°L pi<*> p2^-^) ^ (2-30) where P', P̂ and P 2 are expressed as functions of frequency differences, W-U)Q , rather than absolute frequencies u. Natural and Resonance Broadening are s t r i c t l y independent processes and the combined l i n e p r o f i l e L'(w) i s again a Lorentzian with y now given by: Y = Y N + Y R ( 2 - 3 1 ) - 23 - Doppler and Resonance Broadening are not s t a t i s t i c a l l y Independent, because a sin g l e c o l l i s i o n can produce both a resonance transfer of energy and a change i n v e l o c i t y . A complete analysis i n v o l v i n g the c a l c u l a t i o n of a c o r r e l a t i o n function (Sobel'man, page 401) shows that the c o r r e l a t i o n between these two processes can lead to a c o l l i s i o n a l narrowing of the Doppler width of the combined l i n e p r o f i l e . However, where: L « — (2-32) and L = — - — i s the mean free path and a i s the gas-kinetic cross section, /2a N ° o the two processes can be treated independently. The complete l i n e p r o f i l e P(w) i s therefore given by: P(u) = L ' ( 0 ) D ( u , - 0 ) d0 (2-33) or • ( f ) 2 P ( U ) - I _ n - l / 2 D-1 J " S l _ _ dx ( 2 _ 3 4 ) 2% ~ (a)-u)0-x)2 + (L)2 This type of p r o f i l e Is known as a Voigt P r o f i l e . In conclusion, where a l l the assumptions made i n t h i s development remain v a l i d , we expect a unit volume of atoms i n the gas to absorb photons over a range of frequencies with a p r o b a b i l i t y given by the Voigt P r o f i l e of equation (2-34). In the presence of a magnetic f i e l d the l i n e w i l l s p l i t - 24 - and each Zeeman Component w i l l have this same Voigt P r o f i l e , apart from a sl i g h t narrowing due to the effect of the magnetic f i e l d on the resonance broadening of each m-component. II.3 THEORY In this section a theoretical model i s developed for the transmission of l i g h t through a gas immersed In a magnetic f i e l d where the frequency of the l i g h t i s at or near a resonant frequency of the atoms of the gas. The theory i s based on a semi-classical approach similar to one used i n a related context by Corney et a l . , 1965 and Dodd and Series 1961. While more sophisticated and rigorous methods involving quantized f i e l d s are possible, l i t t l e additional information of importance to this study i s gained at the cost of much greater complexity. Therefore, while acknowledging the merits of r i g o r , I have opted for a simpler and I hope more readily comprehensible exposition of the problem. Conversely most of the results obtained through this development can be derived more simply by making perspicacious substitutions into the more familar equations describing the transmission of l i g h t through a gas. I t was f e l t however that only through the step by step development of a theory, i n which a l l the assumptions and approximations are given e x p l i c i t l y , can the a p p l i c a b i l i t y of the result be properly determined. Thus a complete and systematic development i s presented. - 25 - In this development the l i g h t Is treated c l a s s i c a l l y as an electromagnetic (em) wave governed by Maxwell's equations, while the atoms are treated quantum mechanically. The external ( s t a t i c ) magnetic f i e l d i s allowed to assume any orientation with respect to the dir e c t i o n of propagation of the l i g h t . While this provides a more general equation than i s necessary for t h i s study, and involves some additional complexity, the more general equation may be useful i n future research. For s i m p l i c i t y the theory i s developed using c.g.s. Gaussian units; however, the important resulting equations are also expressed i n MKSA units.t The e l e c t r i c f i e l d , 1j(r,t), of a c l a s s i c a l em wave t r a v e l l i n g through a non-magnetic, current-free medium obeys the equation: V 2 ^ ( r , t ) - V ( v " l ( r , t ) ) = — — CE(r,t) + 4n$ ) (2-35) c 2 a t 2 c where i s the dipole moment per unit volume. In the equation a l l higher e l e c t r i c moments have been neglected. This corresponds to the e a r l i e r approximation of considering only e l e c t r i c dipole t r a n s i t i o n s . To solve equation (2-35) for "E("r,t), "P must be determined for a gas i n a magnetic f i e l d . This i s accomplished by f i r s t finding the average dipole a moment P for a single atom immersed i n a magnetic f i e l d and interacting t J.D. Jackson's " C l a s s i c a l Electrodynamics" provides a table for converting any equation from one system to the other. - 26 - with an em wave. The average dipole moment corresponds to the expectation value of the quantum mechanical e l e c t r i c dipole operator (see equation (2-11)). Thus, V - < t | " * | t > (2-36) where |t> Is the state of the atom at time t . The problem has therefore been reduced to solving the time dependent Schrodinger equation: l h |_ | t> - H | t> (2-37) The Hamiltonian, H for this atom i s given by: u- H = H o + HD + TT 8 a j ̂ ' * + H I ( t ) ( 2 _ 3 8 ) where: H q i s the Hamiltonian for the unperturbed atom, and H q | ajm> = E .|ajm>. "o + + ^— ĝ .. J • B was introduced i n equation (2-6) and accounts for the effects of a weak magnetic f i e l d . With complete generality the magnetic f i e l d can be confined to the x-z plane with the angle between % and the z axis denoted by p. (see F i g . 2). Recall from equation (2-6) i f p=0, F i g . 2: Orientation of the Magnetic F i e l d - 28 - The operator i s introduced phenomenologically to provide for the exponential decay of the excited states e ' , where 1/y i s the l i f e t i m e of a given state. The f i n i t e l i f e t i m e can account for a l l spectral l i n e broadening mechanisms which produce a Lorentz p r o f i l e . Note that: H^.(t) describes the interaction between the atom and the electromagnetic wave. I f the wavelength of the em-wave i s much greater than the size of the atom (which i s true for o p t i c a l frequencies) then H (t) can be given by: which i s analogous to the c l a s s i c a l expression for the energy of a dipole i n an e l e c t r i c f i e l d . The interaction between the atom and the em-wave's magnetic f i e l d i s much weaker and has been neglected. To solve the Schrodinger equation the techniques of time-dependent perturbation theory are employed. The state |t> Is expressed as a superposition of states which are independent of time. H D | ajm> = -(1/2) i h T ( ( . | ajm> (2-39) l y t ) = 4 • "E(r,t) (2-40) - 29 - |t> = C g ( t ) I g> + I d C d ( t ) |d> (2-41) where: |g> represents the ground state which has been made non-degenerate: i . e . |g> = \t00>. While this r e s t r i c t i o n i s not essential i t greatly s i m p l i f i e s the derivation of |t>. The solutions for degenerate ground states are Introduced lat e r In an obvious, a l b e i t ad hoc, way for the special case of l i g h t t r a v e l l i n g transverse to the magnetic f i e l d . jd> i s simply a shorter notation for the excited states j ajm> where "d" represents a l l the quantum numbers of the state. C g ( t ) and C d ( t ) are the time dependent probability amplitudes for the states |g> and |d> respectively. Since a l l the states are orthonormal: C (t) = <g |t> and C (t) = <d |t> (2-42) The probability of an atom being i n a state other than the ground state i s very small, implying |C | ~ 1. This indicates that the state of the atom can be well described by just the f i r s t order terms i n the perturbation expansion. Defining the time-Independent operator H = H - Rj.U) (2-43) i t can be shown (see Messiah page 722) that to f i r s t order: - 30 - "|H(t-x) t " ^ H ( t - t 0 ) " i l ( t 0 - x ) |t> = e lg> + J L _ / d t o e H (t ) e |g> ( 2 - 4 4 ) , ih~ T ° where the o s c i l l a t i n g e l e c t r i c f i e l d producing H (t) was switched on at time t = t at which time the atom was i n the ground state. The terms of the form e - i - H | g> can be evaluated immediately since H q ]g> = E q |g> where E q i s the energy of the ground state. ]g> = 0, since H there i s no broadening of the ground state, and g^^ 5 • $ ]g> = 0 since |g> i s non-degenerate. Thus: - ( E f l ( t - r ) e |g> = |g> e ( 2 - 4 5 ) and equation ( 2 - 4 4 ) becomes - ^ E o(t-x) lt> = lg>e i h 1 t / dt e J x o l i c t - t o ) h E o ( t 0 - ^ ( 2 - 4 6 ) This equation for |t> although not d i r e c t l y solvable can be used with - 31 - equation (2-41) to obtain C. (t) and C ( t ) . To f i r s t order a g - i E Q ( t - 0 C d(t)= <d |t> ~ <d |g> e 1 t 4 « ( t " V "T E 0 ( t" T ) + i h ̂ d t o < d I e H l ( t o ) l g > e (2-47) The f i r s t term i s zero since the states are orthonormal. The second term can be solved i f R |d> can be found. Toward this end we consider the states |u>'̂ defined i n a new co-ordinate.system R'. R' i s chosen such that the z'-axis i s p a r a l l e l to the magnetic f i e l d 1i. Thus R' i s generated by a rotation about the y-axis through an angle {3 with respect to the old co-ordinate system; see F i g . 3. In the new co-ordinate system the operator H becomes: 5 = Ho + H D + ^ S a j B J z , (2-48) Therefore H |u>' - (E .- 1/2 i h y , + p, g .Bu | u>' = \ |u>' (2-49) ^ a j 'aj r o a3 1 u 1 v ' <d| i s related to '<u| through the equation <d| - < ajml - Y D j (0,p,0)'<aju I = Y D 3 < u l ' (2-50) 1 J 1 ^u mu ' K > J 1 L u mu 1 v where D^u(0,B,0) are elements of the rotation matrix, (see Rose, 1957). fTo simplify the notation the state |aju>' i s designated |u>* with the remaining quantum numbers implied. F i g . 3: Orientation of |u>' Co-ordinate System - 33 - Inserting these results into equation (2-47) gives: C d ( t ) = i h d t0 I Dmu 6 ' < u I HI< t0> b> e (2-51) u Now tranforming back to the o r i g i n a l co-ordinate system using the equation: '<u|-I ^*(0,p,0) <ajn | - J D̂ * < n | (2-52) we have nu 1 u nu n n . _ -~\ ( t - t ) 4 E (t-x) C d ( t , - ̂ d t o I 0 ^ . " « • <n ! B l ( , 0 , fe> . " • (2-53, un To evaluate the remaining matrix element <n | H^(t ) |g>, the time dependence of E(r,t) must be given e x p l i c i t y . Let ̂ ( r . t ) be given by the sinusoidal wave; Ifr.t) - V r ) ^ + e i w t (2-54) Where ̂ ( r ) may be complex, but as defined E(r,t) i s real as required for H (t) to be Hermitian. With t h i s d e f i n i t i o n <n | H T(t) |g> = (2-55) <n| - "p • V r ) |g> e'^ + <n | -1.1*(t)|elwt - - <n 1 J? lg> - 1l(r) e ' l w t - <n|"P|g> . l * ( r ) e i a ) t - 34 - The last step i s permitted because ^ ( r ) affects the states only through the operator The matrix element <n| 1»|g> i s just the familar e l e c t r i c dipole tr a n s i t i o n matrix, and equation (2-12) i s applied at a later stage to give the e x p l i c i t dependence on n. Invoking the e l e c t r i c dipole selection rules the element <n | "I |g> = 0 except for states |n> with j=l since |g> i s non- degenerate. Since <n 1> |g> • t l ( r ) i s independent of time the equation for C,(t) becomes a C,(t)-£E D ^ V " * e * V e ^ V < n | n > d i h u mu nu 1 B un 1 (X - E ) t n J T(.E(r) e 0 + E (r) e >̂J e d t Q (2-56) The integral can be eas i l y solved, with the result ¥ ( X U - E o - h W ) t o h t W ^ o B(r) [ f ]* + *Yr) i ] ' (2-57) 5- ( x u - E Q - to) F < V K o + h M > \ - F. Recall from equation (2-49) U , -° = T- (E .- E + a g _,Bu - (1/2) i hv . ) . h n a j o o a j a j The cases of interest occur when the frequency of the l i g h t i s at or near a E - E Q resonant frequency of the atom, i . e . OJ ~ a , . In this case the second h term i n equation (2-57) i s negligible compared to the f i r s t term and the - 35 - expression for C d ( t ) i s given f i n a l l y by: d Ih u n mu nu ^ 6 J e <n p |g> . Equations (2-41) and (2-44) can also be used to d e t u n e C ( t ) . To f i r s t order 8 % E ( t - r ) c g ( t ) = <g fe> e h ° + (2-59) - 36 - Using arguments similar to those used for C^Ct), i t i s apparent that the second term i s zero since <g|^|g> = 0. Thus, 4 E n < t - t > C (t) = e n (2-60) The expressions for C (t) and C,(t) can now be inserted into equation g d (2-41) yielding |t> to f i r s t order. With |t> known, <t f |t> can be found*. Expanding <t $ |t> and retaining only f i r s t order terms: <t|p |t>=<g|p ]g> c g * ( t ) c g ( t ) + £ <grp |d> C g*(t) C d ( t ) + ̂ <d|1?|g>Cg(t) C d*(t) (2-61) The f i r s t term i s zero since <gfp|g> = 0 The t h i r d term i s also just the complex conjugate of the second term; hence <t|p|t> can be written as: <t|P>> = 2 Re < g|1?|d> C* (t) C d ( t ) ) (2-62) where Re(Z) stands for the r e a l part of Z. tNote that <t|t> * 1, since the states |g> and |d> were made orthonormal; i. e . <d|d>=l. Thus, the proper expression i s ^ j ^ ^ . However, since Z C,(t) C, « 1 th i s small numerical correction can be ignored. , d d - 37 - Inserting the derived expressions for C (t) (equation 2-60) and C,(t) 8 d (equation 2-58) the expression i n brackets i n equation (2-62) becomes: Z <g|$|d> C*(t) C. (t) = Z <g|P|d> D(1)D*(1)<n|?|g>.E(r'5 . d , 1 1 mu nu 1 1 d dun -icot £ ( V V ^ -icox * ( " 6(X -E -ho) ' ) <2~63> u o The steady state solution, which i s the one of int e r e s t , occurs when the e l e c t r i c f i e l d has been Interacting with the atom long enough for a l l transient effects to have vanished. In the above equation this i s equivalent to taking the l i m i t of x •*• - ». Examining the function which i s dependent on x, i t i s clear that i t contains an o s c i l l a t o r y component and 1 / 2 W l / z \ ? the component e J . Since lim e J = 0 i t follows that the expression i n brackets i n equation (2-63) becomes: -iwt ((X ! E -ho)) <2-64> u o when x •*• - ». The summation over d i n equation (2-63) includes a l l excited states. However, i f the frequency of the l i g h t i s near a particular resonance l i n e , as we have assumed, the contribution from the states of neighbouring levels w i l l be minimal and the sum can be taken only over the states of the - 38 - relevant l e v e l . (Where the levels are closely spaced the sum should Include the states of the neighbouring levels.) With these modifications equation (2-63) becomes. E «l* >̂ ilKi1' <«l* ( o r ^ W ) <2-65, mun u o The matrix elements <g]~P ̂ n>, and <n|~P |g> can now be reduced using equation (2-12): <g]P \a> = C(llmq|00) <aj||P|| a'j'> e* (2-66) Since |g> i s non-degenerate, q = -m; i . e . each unit vector corresponds to a single allowed t r a n s i t i o n . From tables (see Condon and Shortley, page 76) the Clebsch-Gordon c o e f f i c i e n t s are found, to be: • 3 Thus C(llmq|00) = - (2-67) <g fy> = - <«j||p|| tt'j'> e* (2-68) /3 "™ * u From the property of the unit basis vectors, e = ( - l ) p e th i s becomes: u -u. <g|-p ^> = l L _ < a j | | p | | a'j'> era (2-69) /3 m - 39 - s i m i l a r l y <n $ |g> = ^L- <aj||p|| a \ 1 ' > * e* (2-70) /3 n Using these equations, (2-65) becomes: mun u 0 As an aid to further development, <t |~P |t> i s expressed i n terms of the complex p o l a r i z a b i l i t y tensor per atom, o f which i s defined by the equation: <t $ |t> - 2Re(a a • "E(r) e" l u 3 t) (2-72) It i s evident from equations (2-62) and (2-71) that a 3 Is most conveniently defined with respect to the unit vector basis e . Thus q a a \<ai\ H 1 « ' J ' > 1 2 D ( D p ( D * mn 3(\ - E -hu>) mu nu K > u u o and, •+ <t $ |t> • e = P a • e = 2Re(E a 3 E (r) e" i u ) t) (2-74) ' m c m ; n m n n ; v ' The dipole moment per unit volume "P , which Is required i n equati on (2-35), can be found by summing ^ a over the atoms i n a unit volume. It: is c " ~ s - 40 - convenient to define a complex p o l a r i z a b i l l t y tensor per unit volume a by the equation \ = 2Re(a • l ! ( ^ ) e " l u t ) (2-75) It follows from the d e f i n i t i o n of "p that a_ i s given by the sum of a 3 over a unit volume of atoms, or: a = X a * (2-76) mn 1=1 mn v ' At t h i s point we neglect the small number of atoms i n the excited s t a t e , which i s equivalent to ignoring induced emissions, and take the sum over atoms i n the ground s t a t e . By performing the usual macroscopic s p a t i a l averaging, (see Jackson 1975, page 226) the disc r e t e sum can be replaced by an i n t e g r a l over the v e l o c i t y d i s t r i b u t i o n of a unit volume of atoms. Assuming a Maxwellian v e l o c i t y d i s t r i b u t i o n and accounting for the floppier s h i f t of each atom's natural frequency as seen by an external observer, ( i . e . accounting for Doppler Broadening) the Integral i s given by: 1 I ^ 4 l l i » l l - . 4 . x l 2 <• N <aj||P||a'j'>| 2 S D V i'D mn 3h 1 J 1 1 1 1 J 1 mu nu „ w o u Vn%l,z £ d v (2-77) - * <oQ + -co0 + ^ g ^ B u "co - ( l / 2 ) i y a j - 41 - , 2 V ,1/2 r e c a l l VQ = ( - ± - ) N = density fo the atoms i n the ground state M = mass of the atom V = component of the v e l o c i t y along the d i r e c t i o n of observation w o mh <E„r E ) u n a j o This equation may be s i m p l i f i e d by introducing the complex err o r function W(a + ib) (see Abramowitz and Stegun 1972, page 295) W(a + ib) = I ( a Cihf.t (2-78) whence a becomes mn lNcu a 1/2 |<aj||p||a'j'>! 2 2 D ^ V a + ib) (2-79) mn 3h V Q cjg 1 J 1 mu nu u W h e r e a u = ( W " " 0 ' — g a j B U ) ( 2 " 8 0 ) ^ 0 - 42 - V0o)0 'aj (2-81) It Is conventional to introduce the diraensionless quantity f the absorption 3 o s c i l l a t o r strength for the t r a n s i t i o n a j a ' j ' , which i s defined by the r e l a t i o n (Sobel'man page 302): 2m < j j Q 1 f = _ e . |<aj1|Pl|a'j'>l2 3he 2 2j+l (2-82) With this d e f i n i t i o n a becomes, ( r e c a l l i=0): mn iNcf e 2 n 1 / 2 a = § £ D ( 1 ) D ( 1 ) W(a + ib) mn „„ 0 mu nu u 2 V 0 m e w o 2 u (2-83) To f a c i l i t a t e future development some elements of a are given e x p l i c i t l y . From Rose, 1957 the matrix: t> ( 1 )(0,p,0) = mu 4 (1 + cosp) — sinp Z /2 - sinp /2 2 ( l - c o s p ) cosp -^1-cosp) sinp • 2 /2 — sinB j (1 + cosp) (2-84) - 43 - Therefore: a l l = i K0 i j (l+cosp) 2W( a i+ib) + l / 2 s i n 2 p W(aQ+ib)+ + j (l-cosp^WCa^+ib)] (2-85) o c , ^ = IKQ [I (l-cos6) 2W( a i+ib) + i s i n 2 8 W(aQ+ib) + | (1+cosp) 2 W(a_1+ib)] (2-86) a - l l = a l - l = i K0 s i n 2 P [ j W ^ + l b ) --i-W(a0+ib) + | WU^+ib)] (2-87) Ncf e 2 u 1 / 2 where: KQ = - (2-88) 2 v 0 m e w 0 2 or Ncf e 2 * 1 ' 2 Kg = i n MKSA units (2-89) 8 l t e 0 V 0 m e w 0 2 For most l i n e s i n a gas of modest density l a m n l « l « For example, using the information i n the appendix: |a| ~ 10"5 for the 253.7 nm. l i n e of mercury at the vapour pressure of mercury at 20°C. Equation (2-75) for P c and equation (2-83) for a together contain the - 44 - relevant quantum mechanical r e s u l t s In a form suitable for further analysis using the c l a s s i c a l electromagnetic equation (2-35) given e a r l i e r . I n s ert- ing these quantities into equation (2-35) gives: V 2E(r , t ) - v ( v."E(r , t ) ) = — b— far,t)+4Ttt> ) c 2 a t 2 (ttf.t) + 4*(a • 1:(-r)e- l w t + a* • £ ' & ) e i w t ) ) (2-90) c 2 where E, , = ~E(r)e i u 3 t+"E * ( r ) e l w t i s the t o t a l e l e c t r i c f i e l d at each l o c a l atom. Equation (2-90) has the same form as the equation which describes the propagation of an em wave through an anisotropic c r y s t a l (see Born and Wolf 1975, page 665). In an a n i s o t r o p i c c r y s t a l the e l e c t r i c f i e l d need not be perpendicular to the phase v e l o c i t y ; i . e . there may be a component of the e l e c t r i c f i e l d i n the d i r e c t i o n of the phase v e l o c i t y . I f t h i s occurs the energy of the wave does not propagate i n the d i r e c t i o n of the phase v e l o c i t y . The component of the e l e c t r i c f i e l d along the d i r e c t i o n of the phase v e l o c i t y arises from the term v(V«^(r,t)) which i s zero for i s o t r o p i c materials. From Maxwell's equations: V*"! = -4itV • "p (2-91) c hence - 45 - V(V • 1) = -4nV [V .(a • "E'(r)e" l a ) t+a*, t'*(r)eia)t)] (2-92) Using t h i s equation i t can be shown that where U ^ l ^ l the termV(V«^) and the componet of E along the phase v e l o c i t y i s small compared to the component perpendicular to the phase v e l o c i t y . Thus, i n order to s i m p l i f y greatly the s o l u t i o n to equation (2-90), the term V(V»£) may be omitted, and the small difference between the d i r e c t i o n of the phase v e l o c i t y and energy flow may be ignored. In the equation f or ̂ c the e l e c t r i c f i e l d i n t e r a c t i n g with the atoms must be the t o t a l f i e l d at the atom, E l o c a l ' This f i e l d i s produced by both the f i e l d of the incident em wave "E(r,t), and the induced f i e l d of the neighbouring atoms. The r e l a t i o n between the incident f i e l d and the induced f i e l d i s extremely complex for high frequencies i n anisotropic materials. However where let I « 1 the induced f i e l d i s much smaller than the incident mn f i e l d and the approximation 1i(r,t) = ̂ , . ( r , t ) can be made. With these l o c a l approximations equation (2-90) becomes: V2(E Or) e - i w t + iV ) e 1 ^ ) = -1 [ (e . l(r) e" l u t c 2 5t2 ~ + e*. lYr) e 1 ^ ) ] (2-93) - 46 - where £ _ = ] _ + 4na_ (2-94) or 7^ E(r) e — [e • E(r) e J c 2 a t 2 ~ + V 2 S Y r ) e i w t - ( e \ l Y r ) e i w t ) = 0 (2-95) c 2 a t 2 ~ Since "E(r) and " l * ( r ) are l i n e a r l y independent and the d i f f e r e n t i a l operators are l i n e a r , t h i s equation can be true only i f : l*t(r) e " l w t - L_a_L_ ( . V r ) S 1**) = 0 (2-96) c 2 a t 2 and likewise for the complex conjugate term. Without loss of ge n e r a l i t y equation (2-96) can be solved for the case of a plane wave propagating along the z-axis. The solu t i o n i s of the form t - E 1 e 1 ( K z - w t ) e, e1**™^ where E + IE E + -+ ( X _ y) (2-98) _ i /2 and E and E are the amplitudes of the wave along the x and y-axis x y res p e c t i v e l y . - 47 - Substituting t h i s s o l u t i o n into equation (2-96) y i e l d s the two homogeneous equations: e 2 o — K 2E 1 - ( e u Ej + e 1 _ 1 E_ x ) = 0 (2-99) u 2 and c 2 o — K 2 E n - ( e _ u ̂ E_ L) = 0 (2-100) u 2 Solving these two simultaneous equations for K 2 gives: c 2 ( e _ i n + e u ) ± [(en " e . ^ ) 2 + 4 e2- l- 1]W2 — K 2 - (2-101) a)2 _ 2 where the fact that e - ^ - e i - i has been used. Substituting t h i s expression back into equations (2-99) gives E i T<en - e-i-i> ± [7; <en - e-i-i>2 + E i - i l 1 / 2 — ; ( 2 " 1 0 2 ) where the signs + and - are associated with the solutions K2^ and K 2 (see eq. 2-101) re s p e c t i v e l y . The form of the s o l u t i o n i s greatly s i m p l i f i e d by defining the complex function suggested by Corney, Kibble and Series, 1965: - 48 - 2 e n i tan 9 = (2-103) e l l - e - l -1 E1 cos 6/2 whence - — = s"ln" 9"fl f o r t h e s o l u t i o n K+ (2-104) E and — = ^ ^ Q ^ 2 f o r t h e s o l u t l o T l K l (2-105) The general s o l u t i o n for the d i f f e r e n t i a l equation (2-96) i s thus E = a+ (cos 9/2 + s i n 9/2 e ^ ) e i ( K + z _ a ) t ) + a_ (-sin 9/2 ^ + cos 9/2 e_j) e 1 ) (2-106) The factors a+ and a_ are determined by the boundary conditions. K+ and K_ are the roots of equation (2-101) which give an exponentially decaying s o l u t i o n . These expressions are examined i n d e t a i l i n a l a t e r s e c t i o n . For the purpose of t h i s study we can select an incident e-m wave propagating along the z axis and given by the equation: . i ( K z-wt) I = (E (to) i + E (to) j) e 1 (2-107) x y where, E x(w) = A x(to) e X , and (2-108) - 49 - A (u)) i s the amplitude, (a r e a l valued function of the frequency u) and <b x x i s the phase of the x component. E^(w) i s defined analogously. We choose the boundary of the gas at z = 0 normal to the z = axis, therefore the e l e c t r i c f i e l d i s continuous across the boundary and there i s no r e f r a c t i o n . Equation (2-107) can now be equated to equation (2-106) at z = 0 and solved for a+ and a_. Thus: — (E + IE ) e_, + — (-E + IE )e, • T x y 1 V2 X y 1 = ( a + cosG/2 - a_ s i n 9/2) e L + ( a + s i n 9/2 + a_ cos 9/2) e_ x (2-109) with the solutions E IE a+ = ( s i n 0/2 - cos 0/2) + — £ (cos 0/2 + s i n 0/2) (2-110) /2 /2" E IE a_ = (cos 0/2 + s i n 0/2) + — ^ (cos 0/2 - s i n 9/2) (2-111) • 2 /2 The r e f l e c t e d wave has been omitted because we are pri m a r i l y i n t e r e s t e d i n the change In the transmitted wave with the a p p l i c a t i o n of the magnetic f i e l d . It i s clear that the change In the r e f l e c t i v i t y of the gas with the ap p l i c a t i o n of the f i e l d i s small, hence there i s a neglegible e f f e c t on the transmitted wave. - 50 - The f i n a l s o l u t i o n for the em wave i n the gas can now be given: ^ E IE E = {[-^<sin 9/2 - cos 9/2)cos 6/2 + — £ (cos 9/2 + s i n 9/2)cos 9/2]e_, /2 /2 1 E IE + [ — ( s i n 9/2 - cos 9/2) s i n 9/2 + —2L (cos 9/2 + s i n 0/2)sin 9/2le_,} /2 /2 1 si(K+z-ut) + -E - i E { [ - ^ (cos 9/2 + s i n 9/2) s i n 9/2 + —-2. ( c o s 9/2-sin9/2) s i n 9/2]e •2 ' /2 1 E IE +[-JS(cos 9/2 + s i n 9/2)cos 9/2 + - ^ ( c o s 9 /2-sin9 /2)cos 9/2]e } e i ( K- Z" t i ) t ) • 2 /2 -1 (2-112) II.4 THE WAVE IN A TRANSVERSE AND PARALLEL FIELD The phy s i c a l e f f e c t s of the gas upon the incident l i g h t can now be ascertained from equations (2-112). While the equation i s quite complex, i t reduces considerably i n the case where the magnetic f i e l d i s eit h e r transverse or p a r a l l e l to the d i r e c t i o n of propagation. Since these two cases are of p a r t i c u l a r i n t e r e s t they are examined i n d e t a i l . - 51 - The Transverse F i e l d When the magnetic f i e l d i s p a r a l l e l to the x-axis, i t i s transverse to the d i r e c t i o n of propagation. From our e a r l i e r d e f i n i t i o n of the or i e n t a t i o n of the f i e l d t h i s corresponds to an angle 8 = n/2. Returning to equations (2-85), (2-86) and (2-87) and s u b s t i t u t i n g t h i s value for 6 we fi n d a l l = a - l - l = i K0 i j w < a l + l b> + 1 / 2 + i b ) + J w ( a - i + ib)) (2-113) a l - l = a - l l " i K0 i j W < a l + i b ) " \ + i b ) + J w ( a - i + ib)) (2-114) This increased symmetry s i m p l i f i e s equation (2-112). R e c a l l from equation (2-94): e =6 + 4-rua (2-115) ran mn mn v ' hence e l l = e - l - l a n d E - l l = e i - i Thus the complex function [equation (2-103)] 2 e l - i tan 0 = =o (REAL) (2-116) e l 1 " e -1 -1 and we have the simple r e s u l t 6 =it/2. Substituting t h i s value i n t o equation (2-112) y i e l d s : - 52 - IE (ei +e-) e i ( K + - w t ) /2 + — ( e n - e j e (2-117) /2 but from the d e f i n i t i o n of e^ t h i s i s simply ^ i(K_z - tot) _ i(K+z - tot) j E = E (to) e i + E (to) e (2-118) x y Thus the e l e c t r i c f i e l d of the incident wave divides n a t u r a l l y into two components along the x and y axis. The properties of these two components i n general w i l l be d i f f e r e n t as they are governed by the two d i f f e r e n t factors K_ and K+. The e f f e c t of the magnetic f i e l d i s contained i n the expressions for K_ and K+. Re c a l l equation (2-101): ,2 2 _ K ± ^ Invoking the unique symmetry of e m n for the transverse f i e l d we have: hence: J % ' h i ± e l - l < 2 " 1 1 9 > c 2 , — K2 = 1 + 4it ( a n ± a i n ) (2-120) to2 - 53 - The physical consequences of t h i s equation are most r e a d i l y demonstrated by introducing the conventional parameters n + , the index of r e f r a c t i o n and k +, the absorption c o e f f i c i e n t where^: J K ± - (n ± + i 1/2 £ k ± ) (2-121) It follows that: n±2 - I^-k| - <£- Re (K ±) (2-122) a)2 1 c 2 c . a)2 ^-n +k + = ^-Im (K +) (2-123) c 2 Solving these equations for n and k using the approximation la I « 1 we — -t mn f i n d n + ~ 1 + Zn Re ( a u ± al-l) (2-124) k + ~ 4* £ Im (au ± a i n) (2-125) Thus from equation (2-121): tThe factor 1/2 J J m u l t i p l y i n g the absorption c o e f f i c i e n t k + i s introduced to conform with the conventional d e f i n i t i o n of k as the c o e f f i c i e n t governing the exponential damping of the intensity,, I , which i s proportional to the square of the e l e c t r i c f i e l d , i . e . I = IQe~ z . - 54 - £ K ± =1 + 2% ( a n ± a i n ) (2-126) Substituting the complete expressions for and a^-^ from equations (2-113) and (2-114), we f i n d £-K+ = 1 + 2n i YQ [1/2 w(ax + ib) + 1/2 W(a_1 + lb)) (2-127) ^•K_ = l + 2 n i K 0 W(aQ + ib) (2-128) With these expressions inserted into equation (2-118) the equation for propagation transverse to the magnetic f i e l d assumes Its f i n a l form. Let us now examine the p h y s i c a l content of t h i s equation. The magnetic f i e l d manifests i t s e l f through the term a i n the functions W(a + i b ) . u u R e c a l l from equation (2-80) c ^0 a = (u) - ojn - T — g .Bu) u 0 h 6 a j vOuo The consequences of t h i s are c l e a r l y i l l u s t r a t e d by considering f i r s t the absorption c o e f f i c i e n t s k+ and k_ from equations (2-125), (2-113) and (2-114): k+=4it KQ ^ Re (1/2 W(ax + ib) + 1/2 W(a_1 + ib)) (2-129) k_ = to h ^ R e ( W ( a 0 + i b ) ) (2-130) - 55 - Examining k_, which determines the amplitude of the x component of the wave, we fi n d that the magnitude of the absorption varies over a frequency range determined by Re (W(aQ + ib)) which i s independent of the f i e l d . (Note that Re ( w(z)) i s proportional to a Voigt function.) In contrast the amplitude of the y component of the wave depends on k+ which i s a function of the magnetic f i e l d . As the f i e l d i s increased the peaks of the two functions Re(w(a x + ib)) and Re(w(a_ 1 + ib)) s h i f t to the frequencies u)Q + ^0 ^0 j j - g a j B and co0 - ^— r e s p e c t i v e l y . Hence the magnitude of the absorption w i l l decrease at the frequency wQ while increasing at the two frequencies displaced symmetrically about u 0 . Note that when 1$ = 0, k + = k_ = k where k i s the f a m i l i a r absorption c o e f f i c i e n t for a gas not immersed i n a magnetic f i e l d , (See Sobel'man, 1972 p. 381) The indices of r e f r a c t i o n n_ and n + determine the phase v e l o c i t y of the x and y components of the wave re s p e c t i v e l y . In the presence of a magnetic f i e l d n + * n_ and a wave propagating through the gas w i l l experience a r e l a t i v e phase s h i f t between i t s x and y components. Examining the equations for k + and k_ i n the l i g h t of our e a r l i e r discussions of the Zeeman e f f e c t (section II.1) leads to an obvious extension of the equations for K + and K_ to the more general instance of a degenerate ground s t a t e . In that discussion we noted that a _components could only absorb photons po l a r i z e d perpendicular to the magnetic f i e l d ( i . e . along the y - a x i s ) , while u components could only absorb photons p a r a l l e l to the f i e l d ( i . e . along the x-axis). I t i s therefore not sur p r i s i n g to observe that the expression for k i s simply proportional to - 56 - sum of the l i n e broadened a-components while k_ i s proportional to the sum of the l i n e broadened it-components for this non-degenerate case. Thus there emerges an intimate r e l a t i o n s h i p between the absorption c o e f f i c i e n t s and the Zeeman components. This r e l a t i o n s h i p can be demonstrated rigorously, (see Sobel'man 1972). Extrapolating t h i s r e l a t i o n s h i p to the more general degenerate case K + and K become - K - 1 + 2wiKn £ , C1 , W(a ,+ i b ) (2-131) u) + w mm mm mm ' where m - m1 = ± 1 £-K_ = 1 + 2*1^ E , C° W(a + ib) (2-132) co u mm mm mm where m - m' = 0 ^0 amm' = U " W 0 - — B K f " V j ' m , ) (2~133) The constants C° , and C1 , are the r e l a t i v e i n t e n s i t i e s of the Tt and a mm mm components r e s p e c t i v e l y , which have been normalized such that L - t c ° . - 1 and E , C1 , = 1 (2-134) mm mm mm' mm v where m'-m =0 where m'~m=±1 In the next section propagation p a r a l l e l to the magnetic f i e l d i s examined. - 57 - The P a r a l l e l F i e l d The magnetic f i e l d i s p a r a l l e l to the d i r e c t i o n of propagation when 8 = 0 . From equations (2-85), (2-86) and (2-87), a l - l = a - l l " 0 (2-135) a n = IKQ W( 3 l + ib) (2-136) a-!-! = IKQ W(a_1 + ib) (2-137) hence from equation (2-94) ^-l = 0 (2-138) and from equation (2-103) tanB = 0 (2-139) and therefore 0 = 0 Using this r e s u l t , equation (2-112) reduces to + "E + 1 E i(K+ - u t ) E + I E i ( K - u>t) E = 7 e e + y e e (2-140) / 2 1 / 2 "I Thus the wave divides n a t u r a l l y into two c i r c u l a r l y polarized components. From equation (2-101) c 2 2 — K2+ = e n and S- K2_ = e . ^ (2-141) C 0 Z ( 0 ^ - 58 - Again making the assumption |a| « 1 we f i n d ^ K + = 1 + 2n i K g W(ax + ib) (2-142) -̂ K_ = 1 + 2n i % WCa-! + ib) (2-143) These expressions together with equation (2-140) completely describe the propagation of l i g h t p a r a l l e l to the magnetic f i e l d . Note that each c i r c u l a r l y polarized component i s affected by a s p e c i f i c t r a n s i t i o n , u = ±1, of the atom. When a magnetic f i e l d i s applied these two components have d i f f e r e n t indices of r e f r a c t i o n and w i l l absorb l i g h t over a d i f f e r e n t range of frequencies. The d i f f e r i n g indices of r e f r a c t i o n lead to the f a m i l i a r e f f e c t of faraday r o t a t i o n of l i n e a r l y polarized incident l i g h t . These r e s u l t s can be extended to the s i t u a t i o n of a degenerate ground state by analogy with the case of a transverse f i e l d . When th i s i s done the f i n a l equation i s i n complete agreement with that obtained by Camm and Curzon, 1972, using a d i f f e r e n t approach. In the next section these r e s u l t s are used to examine the p o l a r i z i n g e f f e c t of a gas on an incident em - wave. II.5 THE POLARIZATION OF THE TRANSMITTED WAVE To t h i s point the theory has been developed for the sp e c i a l case of a monochromatic plane wave. R e c a l l equation (2-118) for the case of a transverse magnetic f i e l d . - 59 - i(K_z t i t ) i ( K + z -ut) E = E (u)e I + E (u>) e j (2-118) x y where K + and K_ are given by equations (2-127) and (2-128) and from equation I* i<j> (2-108): E (u>) = A (w)e X E (w) = A (to)e y x x y y where A and A are the amplitudes and <b and <b are the phases of the x y r y T x r incident em wave. A c h a r a c t e r i s t i c of monochromatic plane waves i s that three q u a n t i t i e s , examined at a given point z In space, remain constant for a l l time. They are the two amplitudes, - l / 2 k z - l / 2 k z A e - and A e + and the phase difference given by; These three independent q u a n t i t i e s are s u f f i c i e n t to determine completely the state of p o l a r i z a t i o n of the l i g h t represented by t h i s wave, ( K l e i n , 1970, p. 485). Since for a monochromatic plane wave these quantities are constant i n time, a monochromatic plane wave must have a fixed state of p o l a r i z a t i o n . A s t r i c t l y monochromatic plane wave i s , however, a mathematical i d e a l i z a t i o n which i s never r e a l i z e d i n p r a c t i c e . Consequently equation (2-118) must be modified s l i g h t l y to accord with r e a l i t y . This i s most re a d i l y accomplished by transforming equation (2-118) into the quasi-monochromatic plane wave: - 60 - i(K_z - tot) _ i ( K + z - tot)_ E = E (to,t)e 1 + E (to,t)e j (2-144) x. y where E (to,t) and E (io,t) are slowly varying functions of time compared to x y e ^ U t and are defined by the expressions: i<t> ( t ) E x(to,t) = A x(u,t)e X ( 2 - 1 4 5 ) E (w,t) = A (w,t)e y ( 2 - 1 4 6 ) where the modified amplitudes and phases are now stochastic functions of time. This new representation of the l i g h t wave requires a more sophisticated d e f i n i t i o n of p o l a r i z a t i o n than was necessary for s t r i c t l y nonchromatic plane waves since i t admits the p o s s i b i l i t y of unpolarized l i g h t . A number of equivalent schemes have been devised to represent the state of p o l a r i z a t i o n of l i g h t . The scheme used i n t h i s study i s a f t e r a method presented i n Born & Wolf ( 1 9 7 5 ) , p. 544 where the p o l a r i z a t i o n i s r e l a t e d to the degree of c o r r e l a t i o n between the i r r e g u l a r f l u c t u a t i o n s of A , A , <b , x y' T x and <)>y. An a t t r a c t i v e feature of this scheme i s that the t h e o r e t i c a l parameters are c l o s e l y related to the experimentally measured qua n t i t i e s because the t h e o r e t i c a l development c l o s e l y models the experimental technique used to analyse the l i g h t . This feature proves to be p a r t i c u l a r l y advantageous when we compare the re s u l t s of theory with experiment. - 61 - The theory i s developed by considering the following i d e a l i z e d experiment, (see F i g . 4). The Incident l i g h t , having traversed a length, X , of gas i n a transverse magnetic f i e l d , passes through a compensator (such as a quarter wave plate) which retards the phase of the y-component by an amount £ with respect to the phase of the x component. Using equation (2-144) the wave i s given by: -»• iK X i O . i ( - z - u t ) E = [E (oj,t)e " i + E (u,t)e e ^ j ] . e C (2-147) x y The wave next passes through a p o l a r i z e r with i t s pass d i r e c t i o n i n c l i n e d at an angle <\> with respect to the x ax i s . Whence the component passing through the p o l a r i z e r i s given by: i K A iK +Jt i ( — z - u)t) E (<KS,t) - [ E x ( u , t ) e cos cp + E y ( o j,t)e e ̂ s i n 4,] . e C (2-148) F i n a l l y , the i n t e n s i t y of the wave Is measured by a detector. The instantaneous i n t e n s i t y ^ , I', of the wave i s proportional to the square of the r e a l part of the complex e l e c t r i c f i e l d function, E(cj>,£,t), i . e . I* - Re2 (E(<|>,S,t)) (2-149) tTe c h n i c a l l y , t h i s quantity i s the fl u x density or irradiance and not the i n t e n s i t y , (see K l e i n , 1970 at pp. 122 and 508). However, the two are cl o s e l y related and many authors including Born and Wolf use the term i n t e n s i t y . Therefore, for convenience, the term i n t e n s i t y i s used here. a) | B / / LIGHT SOURCE GAS CELL IN TRANSVERSE MAGNETIC FIELD COMPENSATOR POLARIZER DETECTOR b) N3 F i g . 4: a) Po l a r i z a t i o n Measuring System b) P o l a r i z a t i o n Orientation - 63 - However, a r e a l detector, such as a photomultiplier or a photgraphic p l a t e , does not measure the instantaneous i n t e n s i t y ; It measures a time averaged i n t e n s i t y I. This experimentally measured i n t e n s i t y , I, i s t h e o r e t i c a l l y described by taking the time average of I 1 as described by the expression: I = < I' (t) > = ^/Jf I'(t) dt (2-150) where T i s the averaging time period. With the l i g h t source and detector used i n t h i s study, the functions E and Ey defined by equations (2-145) and (2-146) undergo a large number of random flu c t u a t i o n s within the response time T of the detector. I t i s therefore reasonable, for the sake of mathematical s i m p l i c i t y , to take the l i m i t T •*• °° rather than the actual response time of the detector i n ca l c u l a t i n g the time average of I'. That i s , I i s given by: I = <I'(t)> = lim jfj_l I ' ( t ) d t which i s proportional to lim j=rj I |Re ( E f o ; t ) |* dt (2-151) T-*» Using t h i s expression for the time averaging, i t i s possible to show that: 2<Re2 (U<\>,l;t)> = <m,Z;t) . E*(cK£;t)> (2-152) (see Born and Wolf 1975 at p. 498) Thus, the experimentally measured i n t e n s i t y I i s simply given by: I(<\>,l;u) = <E(4>,£,t) . E*((p,C;t)> (2-153) - 64 - where the unimportant constant of p r o p o r t i o n a l i t y has been set equal to 1. Inserting equation (2-148) i n t o the above, we have: * i ( K - K_*)A ^ i ( K - K*)A I(<K5;w) = <E xE x > e cos 2 4. + <E yE y > e s i n 2 4, (2-154) r * * i ( K _ - K i H * i ( K - K*H + [ <E E > e e x * + <E E > e e 1^] cos 4, s i n 4, x y y x Equation (2-154) can be written more su c c i n c t l y as I(4>,£;u>) = J ^ c o s 2 + J y y s i n 2 4, + (J + J e 1 5 ) cos 4. s i n 4, (2-155) xy yx where: J XX = <E E*> X X e l ( K _ - K _ ) * J yy = <E E*> y y el(K+-Kj)Jl j xy - <E E*> x y g i ( K _ - K j ) J l j yx = <E E*> y x el(K+-K*H The four terms J , J , J and J form the elements of a matrix c a l l e d xx yy xy yx the coherency matrix J. J J J = ( j X X j X y ) ( 2-156a) yx yy It i s evident from the d e f i n i t i o n s of J and J that: xy yx J = J * xy yx (2-156b) - 65 - Further, i t follows from equation (2-154) that the t o t a l i n t e n s i t y I T ( i . e . the i n t e n s i t y measured with the p o l a r i z e r removed) i s given by: I = J + J T xx yy (2-157) It i s clear from equation (2-155) that any two l i g h t beams with the same coherency matrix are i n d i s t i n g u i s h a b l e i n an experiment inv o l v i n g a p o l a r i z e r and a compensator.* Thus this matrix can be used to define uniquely the state of p o l a r i z a t i o n of the l i g h t . Let us b r i e f l y digress to i l l u s t r a t e this with some examples. For s i m p l i c i t y we use the coherency matrix of the incident l i g h t which from equations (2-107) and (2-144) i s simply: J = i <E V, > x x <E E *> y * <E E > x y <E E *> y y (2-158) Completely Polarized Light Completely po l a r i z e d l i g h t occurs when the random fl u c t u a t i o n s of the two amplitudes and two phases are correlated such that the r a t i o of the amplitudes and the d i f f e r e n c e i n the phases are time independent. This may be expressed as t i n interference experiments the waves may be distinguished by t h e i r c h a r a c t e r i s t i c coherence length and s p a t i a l coherence. However, since these are i n t r i n s i c properties of the l i g h t source which are only mildly a f f e c t e d by the absorbing gas, they are not considered here. - 66 - A (t) - 2 = q and x = 4> (t) - 0 (t) (2-159) A (t) x y x where q and x a r e constants. In t h i s case may be written; J P = <A2 > ( \ * e ± X) (2-160) i X q e ^ q 2 The superscript P has been added to denote completely polarized l i g h t . It i s e a s i l y shown that t h i s matrix i s i n d e n t i c a l to that of a s t r i c t l y monochromatic plane wave. Since two l i g h t beams with the same coherency matrix are i n the same state of p o l a r i z a t i o n i t follows that the matrix J' does indeed represent completely polarized l i g h t . From (2-160) note that a property of p o l a r i z e d l i g h t i s that Det J** = J J - J J = 0 (2-161) I xx yy xy yx v ' P By e x p l o i t i n g the r e l a t i o n s h i p s between and monochromatic plane p waves we can denote the various states of by the f a m i l i a r terms P applicable to monochromatic plane waves. Thus, i n general represents an e l l i p t i c a l l y polarized wave. If the wave i s l i n e a r l y polarized x = uni where m = 0, ±1, ±2 ... and the coherency matrix becomes <A > ( 1 q ) (2-162) / i \ m _2 x (-1) q q 2 The d i r e c t i o n of p o l a r i z a t i o n makes an angle a Q with the x-axis, where: <x0 = arctan [ (-l) mq] (2-163) - 67 - For c i r c u l a r l y p olarized l i g h t x = m n/2 and A = A and the matrix i s x y given by: < A x > (+1 l 1 ) (2-164) where the upper and lower signs represent righ t and l e f t c i r c u l a r l y polarized l i g h t r e s p e c t i v e l y . Unpolarized l i g h t We define unpolarized l i g h t as l i g h t whose i n t e n s i t y i s independent of the angle <J> of the p o l a r i z e r , and of the phase s h i f t i . e . I(4>,£,(jj) = constant for a given co. (2-165) It i s evident from equation (2-155) and (2-156b) that this i s true i f and only i f J = J and J = J = 0 xx yy yx xy An equivalent and perhaps more I n s i g h t f u l d e f i n i t i o n of unpolarized l i g h t i s a wave whose components and E y are completely incoherent so that <ExEy*> = 0 for any choice of x, y axes perpendicular to the d i r e c t i o n of propagation. I t follows that the coherency matrix for unpolarized l i g h t i s given by: - 68 - note: Det # 0 (2-168) P a r t i a l l y Polarized Light In general the state of a l i g h t beam l i e s between the two extremes of completely polarized and unpolarized l i g h t and th i s i s known as p a r t i a l l y polarized l i g h t . It can be shown that p a r t i a l l y polarized l i g h t can be uniquely expressed as the superposition of a completely polarized and an unpolarized wave, (Born and Wolf, 1975, p. 550). That i s U P J = J + J (2-169) where In accordance with matrices (2-160) and (2-167) we can define: and j P = ( D *?) ( 2 -^D U P The matrix elements of J and J are given i n terms of the elements of J by the following equations. A = 1/2 (J + J ) - 1/2 / ( J +~J )2 - 4 Det J xx yy xx yy (2-172) - 69 - B = J x x " A (2-173) C = J y y - A (2-174) D - J x y (2-175) D* = J y x (2-176) U P The elements of J and J thus provide a unique representation of the state of p o l a r i z a t i o n of the l i g h t . It i s useful to introduce one f i n a l parameter, the degree of p o l a r i z a t i o n P. P i s defined as the r a t i o of the Intensity of the pol a r i z e d portion I j_ to the t o t a l i n t e n s i t y of the l i g h t . That i s : T p o l B +C p = — = (2-17T) I_, J + J K A " L / / ; T xx yy note 0 < P < 1 (2-178) When P = 1 the l i g h t i s completely polarized and when P = 0 i t Is unpolarized. The degree of p o l a r i z a t i o n i s independent of the choice of x, y axes. We can now use these d e f i n i t i o n s to examine the p o l a r i z i n g e f f e c t of a gas i n a transverse magnetic f i e l d . From equation (2-155) the coherency matrix i s given by - 70 - <E E > e -k A i(K_-K*)A x x J = (2-179) <E E* > e y x I(K - K_*)A where equation (2-121) for k± has been used. If the state of p o l a r i z a t i o n of the incident wave i s known the above matrix w i l l describe completely the state of the wave a f t e r traversing the gas. Two cases are of p a r t i c u l a r i n t e r e s t : when the incident wave i s l i n e a r l y polarized and when the wave i s unpolarized. We examine these two cases i n order to e s t a b l i s h the conditions under which the gas may act as a p o l a r i z e r . where I 0(u)) i s here defined as the t o t a l Intensity of the incident wave. Thus, from matrix (2-179): Unpolarized Incident Wave When the incident wave i s unpolarized [from matrix (2-167)]: <E E*> = <E E*> - 1/2 I n ( u ) x x y y u * and <E E > = 0 x y -k J l 0 J = j IQ (O) ) e (2-180) 0 - 71 - One result i s immediately evident from this matrix. The gas affects the state of polarization only through the absorption coefficients k + and k_. Moreover, i t i s also clear how the gas i s able to polarize the l i g h t . Consider a s i t u a t i o n where at a given frequency w; k_Jl»l and k +A«l (2-181) then: J ~ 1/2 ^(o,) ( ° J) (2-182) Thus the l i g h t at frequency OJ becomes l i n e a r l y polarized along the y-axis. In fact only the condition k » k and not K 9 « 1 i s necessary for the gas to act as a po l a r i z e r . However i t i s highly desirable that the polarizer have as high a transmlttance as possible and i n this example t h i s i s achieved when k +Jl. •*• 0. I f the conditions of (2-181) can be arranged, two limi t a t i o n s to the technique are s t i l l apparent. The l i g h t can only be l i n e a r l y polarized and - k j t the l i g h t w i l l never be completely polarized since e i s never exactly zero. That i s , there w i l l be a strong l i n e a r l y polarized component along the y-axis and a weak unpolarized component. In the representation (2-169) this can be written as; J = I 0(u))/2 - k j l 0 - k J l + IQ (w)/2 0 0 0 ( e _ k + A -e" k _ J l ) (2-183) Clearly then the larger k j l the higher the degree of po l a r i z a t i o n . - 72 - Simple Examples To demonstrate that the conditions given by (2-181) can i n fact be r e a l i z e d we examine equations (2-129) and (2-130) which give k_ and k + for l i n e s with a non-degenerate ground state: k + = 4i Ko ^ Re (1/2 W(ax + ib) + l/2W(a_ 1 + ib) k_ = 4it Ko Re (w(ag + ib) Let us assume for the moment that a l i n e can be found such that at frequency u)Q , k_A » 1 for a gas of reasonable length and density. (It i s demonstrated l a t e r that t h i s i s indeed the case for the 253.7 nm l i n e of mercury.) In view of the e a r l i e r discussion of the e f f e c t of the magnetic f i e l d on k + i t i s c l e a r that i n p r i n c i p l e we can apply a magnetic f i e l d which i s s u f f i c i e n t l y large to cause k +A « 1 at the frequency O J q . Thus an unpolarized incident beam would become p a r t i a l l y polarized when passed through the gas. Our objective i s , however, much more ambitious. We want the gas to p o l a r i z e the l i g h t over i t s en t i r e frquency range and not just at the c e n t r a l frequency u>0 . That i s i f the Incident l i g h t has a l i n e shape with a c h a r a c t e r i s t i c width Av we require that the conditions k i » 1 and k A « 1, (see 2-181) both remain v a l i d over the frequency range A v . Examining the equation for k and assuming k Z » 1 at u>0 , i t i s evident that the f i r s t condition can be met over Av i f the absorption l i n e p r o f i l e W(aQ + ib) i s s u f f i c i e n t l y broad so as to remain nearly constant over Av, (see F i g . 5a). Of course i f k £ i s very large at the condition - 73 - FREQUENCY • F i g . 5: Emission, Absorption and Transmission Line Profine - 74 - k_A » 1 could be s a t i s f i e d even with a substantial change i n W(aQ + ib) over Av. While i t i s desirable to have a broad l i n e p r o f i l e i n attempting to s a t i s f y the f i r s t condition, just the opposite i s true of the second condition; k +£ « 1 over Av. For i f W(a2 + ib) and W(a_x + ib) i n the equation for k + became broader ( r e c a l l that W(ag + i b ) , W(a^ + ib) and -W(a_2 + ib) a l l have the same breadth), a larger magnetic f i e l d must be applied to s h i f t the centres of the two l i n e p r o f i l e s f ar enough from u)Q so that only t h e i r much smaller wings contribute to k + over the range Av. For many purposes It may also be important that k +£ remain reasonably constant over Av since changes i n k +£ over Av w i l l a l t e r the l i n e shape of the transmitted l i g h t . In conclusion any attempt to optimize the p o l a r i z i n g properties of the gas must reconcile the two c o n f l i c t i n g conditions k JL » 1 and k +A « 1 over Av. The foregoing comments would, of course, be less important i f we could not a l t e r the l i n e p r o f i l e width according to our design. However i n our e a r l i e r discussion of l i n e broadening we established that the l i n e width depended on the temperature and density of the gas. Moderate adjustments i n these two parameters "can lead to changes i n the l i n e width of an order of magnitude, which provides the l a t i t u d e necessary for optimizing the gas p o l a r i z e r ' s performance. There i s another s i t u a t i o n which can give r i s e to polarized l i g h t . Consider the case where k A » 1 at frequency w0, but the l i n e p r o f i l e of - 75 - the absorption l i n e Re(w(aQ + ib)) i s much narrower than the width Av of the Incident l i n e , (see F i g . 5b). For the x component, the l i g h t at the center of the l i n e i s almost completely absorbed, but the l i g h t i n the wings of the l i n e i s mostly transmitted. For the y component a weak magnetic f i e l d i s used to spread the l i n e p r o f i l e s Re(W(a1 + lb)) and Re(W(a_1 + ib)) over the frequency range Av rather than to s p l i t them away from Av as was done previously. (Note that t h i s e f f e c t can be enhanced by using an inhomogeneous, but highly i s o t r o p i c magnetic f i e l d . ) In t h i s way k+Jl » 1 i s maintained over Av and the transmitted l i g h t i s polarized along the x-axis. The obvious disadvantage of th i s technique i s that the absorbing gas, while p o l a r i z i n g the source l i n e , also d i s t o r t s i t s shape by almost completely absorbing the centre of the l i n e . Thus, t h i s technique for p o l a r i z i n g l i g h t Is only of use i n studies where i t i s acceptable for the source l i n e to be d i s t o r t e d , (although i n a predictable way) by the p o l a r i z e r . L i n e a r l y Polarized Incident Wave Consider now the e f f e c t of the gas on a l i n e a r l y p olarized incident l i g h t beam. This s i t u a t i o n can be completely i l l u s t r a t e d by considering the case where the incident l i g h t i s polarized at an angle of 45° with respect to the x-axis. That i s , from matrix (2-162) and equation (2-163): - 76 - The coherency matrix for the transmitted l i g h t i s then given by - k Jl i - (n_- n +)£ ~ 1 / 2 ( k + + k _ H (e c e e (2-184) , i ^ ( n + - n_)A - 1/2(k + + k_)A ) e where equation (2-121) has been used to define n + . Note that the transmitted l i g h t at frequency to remains completely polarized since Det J = 0. I f we again imagine a s i t u a t i o n where k I » 1 and k +A « 1 we obtain e s s e n t i a l l y the same r e s u l t as i n the case of an unpolarized incident wave, namely Thus the gas w i l l behave as a p o l a r i z e r much as i t did i n the case of an unpolarized incident beam. In p a r t i c u l a r , the e a r l i e r comments on how to optimize the p o l a r i z i n g properties of the gas remain v a l i d here. However the two cases are not exactly the same. The chief difference between the matrix (2-184). To demonstrate the e f f e c t of these terms consider again the s i t u a t i o n where k i » 1 and k^Jl « 1 over Av . For unpolarized incident l i g h t the f r a c t i o n of the l i g h t along the x axis which was not absorbed a f t e r traversing the gas appeared as an unpolarized component, [ r e c a l l J ~ ( IQ(W)/2) 0 0 0 1 (2-185) two Is the appearance of the indices of r e f r a c t i o n n + i n the coherency - 77 - (2-183)]. If the incident l i g h t i s polarized, however, t h i s f r a c t i o n does not appear as an unpolarized component since DET J = 0. To help i l l u s t r a t e t h i s s i t u a t i o n imagine f i r s t that n + = n_. In t h i s case the transmitted l i g h t remains l i n e a r l y polarized but i t i s now polarized along a d i r e c t i o n forming an angle a Q with the x-axis, where from equation (2-163) <z0 = arctan [ ( - l ) m ( j ) 1 ' 2 ] (2-186) Re(J ) a n d ( - 1 ) m = \Re(7 )) 1 xy 1 (See Born and Wolf, 1975, p. 27) Using the equations (2-173) and (2-174) for B and C: m 1 / 2 A ( k - k ) ct0 = arctan [ (-1) (e ' )] (2-187) If n + * n_, (which i s generally the case), the transmitted l i g h t i s e l l l p t i c a l l y p o larized and the shape of the e l l i p s e i s governed by the f a c t o r : 6 = (n_ - n +H (2-188) The e l l i p s e i s i n s c r i b e d into a rectangle whose sides are p a r a l l e l to the co-ordinate axes and whose lengths are 2/C and 2/B, (See F i g . 6). The e l l i p s e touches the sides at the points - 78 - F i g . 6: Geometry of P o l a r i z a t i o n E l l i p s e - 79 - (± /B, ± /C cos6) and (± /B cos6 , ± / F ) (2-189) The angle, <J>, which the major axis of the e l l i p s e makes with the x-axis i s given by the equation tan 24> = (tan 2aQ ) cos6 (2-190) Thus, i n general, when the incident l i g h t i s l i n e a r l y polarized the gas w i l l produce e l l i p t i c a l l y p olarized l i g h t , where the shape of the e l l i p s e i s a function of the frequency of the l i g h t . The discussion to th i s point has been f a c i l i t a t e d by using a representation which describes the state of p o l a r i z a t i o n of the l i g h t at a s p e c i f i c frequency w. This representation i s not, however, always the most convenient one. For instance i f the state of p o l a r i z a t i o n changes over the width of the incident l i n e (Av) i t i s d i f f i c u l t to assess the degree of p o l a r i z a t i o n of the en t i r e l i n e . Furthermore the t h e o r e t i c a l r e s u l t s are not compatible with the experimental r e s u l t s since an experiment n e c e s s a r i l y measures the Intensity over a range of frequencies and not at a d e f i n i t e frequency w. To overcome these d i f f i c u l t i e s we introduce a s l i g h t l y modified representation. Instead of considering the i n t e n s i t y at a s p e c i f i c frequency OJ as i n the de f i n i n g equation (2-154), we observe the i n t e n s i t y taken over the en t i r e frequency range of the l i n e . That i s , we are interested i n the i n t e g r a l of equation (2-154) over on. Therefore we define (2-191) - 80 - If only a part of the l i n e i s of i n t e r e s t the l i m i t s of the i n t e g r a l can be changed accordingly. In complete analogy with the e a r l i e r representation we can define a CO coherency matrix J by the equation and I(4>» x . ,) - J° cos 2 i b + f s i n 2 6 i xx ^ yy ^ + (J e xy + J e ) cos i b s i n i b yx ' T T (2-192) J = / CO * <E E > e do O X X f <E E > e ' o x y * (K_-K +)A dio * H , -k A / <E E >e dio f°° <E E*>e dco • ^ o y x ° y y (2-193) from which a l l the remaining parameters can be calculated In the same way as they were i n the e a r l i e r representation. Note that the t o t a l i n t e n s i t y of the x and y components of the incident l i n e are given by: 1° - / <E E*> do and 1° = f <E E*> dio o y '0 y y (2-194) and the t o t a l i n t e n s i t y of the incident l i n e , I , i s given by (2-194a) An important consequence of the fact that a r e a l detector measures the in t e n s i t y over a range of frequencies and not at a single frequency i s made + 1° x y - 81 - apparent i n this representation. For i f K + and K_ are not constant over the measured range of frequencies, then, (from matrix 2-193) even If the incident l i g h t i s completely p o l a r i z e d , the transmitted l i g h t i s only p a r t i a l l y polarized since Det £ 0. That i s , even though the l i g h t Is completely polarized at any given frequency u), the o v e r a l l l i n e i s found to be only p a r t i a l l y p olarized when i t i s measured experimentally. Consider the s i t u a t i o n where the gas acts as a p o l a r i z e r by absorbing the x component of the incident l i g h t . The o v e r a l l effectiveness of the gas as a p o l a r i z e r for the e n t i r e l i n e can be determined from the degree of p o l a r i z a t i o n P^ and the f r a c t i o n a l change i n the i n t e n s i t y of the polarized CO CO J J component — — . An i d e a l p o l a r i z e r would have P = 1 and —22. = 1. That i s 1° 1° y y the e n t i r e l i n e would be completely polarized and the transmitted y component would be unattenuated. In general i t can be assumed that the J y y smaller — — , the greater the d i s t o r t i o n of the l i n e shape of the incident i° y l i n e as a r e s u l t of traversing the gas. Therefore i t i s desirable to choose the condition of the gas such that J J P •> 1 and - 2 2 - , . 1, which implies - 2 2 + 1/2 1° 1° y To i l l u s t r a t e the ideas presented i n t h i s section J and J are yy xx calculated as functions of the magnetic f i e l d strength for the 253.7 nm l i n e - 82 - of mercury using the properties l i s t e d i n the appendix. The absorbing gas consists of a 3 cm length of H g 2 0 2 . The numerical c a l c u l a t i o n i s performed for a number of gas temperatures, with the gas density determined from the known vapour pressure. For s i m p l i c i t y the incident l i g h t i s unpolarized and has a Gaussian l i n e p r o f i l e . For convenience, the equations used i n the numerical c a l c u l a t i o n are co l l e c t e d below. From the matrix J° (2-193) J = J <E E > e ^do>, J = / <E E > ^ d e v here 1 = 3 cm. xx J o x x ' yy J o y x From equations (2-177), (2-173), (2-174) and (2-172); , f OO CD CO CO 4[ J J - J J 1 P - [ i - A x x y y — s i J 2 E i ] i / 2 ( 2 _ 1 9 5 ) (r + r )2 xx yy' But, since the Incident l i g h t i s chosen to be unpolarized, from matrix (2-180), J° = J° = 0 and xy yx CO CO I J - J p = I xx yy CO CO J + J xx yy (2-196) Furthermore, for an unpolarized incident beam, from equations (2-166) and (2-157) <E E*> = <E E*> = I n(to)/2. x x y y o v ' - 83 - The Gaussian l i n e p r o f i l e selected for the incident source l i n e i s given by: <V> - 2ff ( ¥ > 1 / 2 - r f " 1 " 2 h p V ] where D i s the half-width and the p r o f i l e has been normalized such that from equation (2-194a) the t o t a l incident i n t e n s i t y 1° i s equal to 1, i . e . 1° = /" 2 <E E*> dw = 1 (2-198) From equations (2-129) and (2-130) k+ = AitKg ^ Re [1/2 W ^ + i b ) + 1/2 W(a_x + ib)] , k_ = 4nKQ £ Re [w(a 0 + ib)] where, from equations (2-88) and (2-89) Ncf e 2 n 1 / 2 ^ = 1 i n MKSA units or 8ne0 V0mew2 Ncf e 2 i t 1 / 2 Kg = i n cgs units and 2V0mea)2 - 84 - N = the density of the atoms i n the ground state c = the speed of l i g h t f = the absorption o s c i l l a t o r strength (see equation (2-82)). e = the electron charge E Q = the P e r m i t t i v i t y of free space m = the electron mass e O)Q = the resonance frequency of the l i n e (see equation (2-7)) 2 K R T V 0 = [ ~ — ) i / 2 (see equation (2-29)) and = the Boltzmann constant T = the temperature In degrees Kelvin M = the atomic mass. W(a^ + ib) i s the complex error function defined by equation (2-78) and from equations (2-80) and (2-81) c ^0 b = ^ o ~ ( 1 / 2 ) ( 1 y*5> W h e r e U Q = the Bohr magneton (see equation (2-4)) g j a = the Lande g f a c t o r for the state a j , (see equation (2-6)) B = the magnetic f i e l d strength y . = the t o t a l Lorentz broadening width, (see equation (2-39)) From equation (2-31) Y a j = Y N + Y R - 85 - where y^ i s the natural l i n e width defined by equation (2-18) Y R i s the resonance broadening width. In the c a l c u l a t i o n the A l l and Griem expression (2-19) was used. CO In F i g . 7, J y y i s plotted as a function of magnetic f i e l d strength for three d i f f e r e n t gas temperatures, 300K, 350K and 375K. Note that J i s xx unaffected by the magnetric f i e l d and has a constant value equal to J at yy zero f i e l d . The h a l f width of the source l i n e , D, i s 25 10 - 3cm - 1. F i g s . 8 and 9 are the same as F i g . 7 except that the hal f width of the source l i n e has been increased to 40 10 - 3 cm - 1 and 60 10 - 3 cm - 1 r e s p e c t i v e l y . In the next chapter the p o l a r i z i n g properties of a mercury gas are examined experimentally. J ^ y as a function of Magnetic F i e l d Strength for Three Gas Temperatures, 300K, 350K and 375K D = 25 (ID" 3) cm-1 Jxx = J y y a t z e r o f i e l d and i s independent of th strength of the f i e l d 1— 1.5 - 1 — 2.0 — 1 — 2.5 MRGNET1C FIELD STRENGTH 3.0 IKGRU55) 1— 3.5 1.0 4.5 F i g . 8: j " as a function of Magnetic F i e l d Strength for Three Gas Temperatures, 300K, 350K and 375K o o 4.5 MAGNETIC FIELD STRENGTH IKGRU55) J^y as a function of Magnetic F i e l d Strength for Three Gas Temperatures, 300K, 350K and 375K 'yy -3cm_1) 0 0 J at zero f i e l d and i s independent of the strength of the f i e l d ~! 2.5 —|— 3.0 - 1 — 3.5 —T 4.0 -1 4.5 MAGNETIC FIELD STRENGTH (KGRUSS) - 89 - CHAPTER I I I I I I . I THE EXPERIMENTAL DESIGN A schematic diagram of the experimental arrangement i s given i n F i g . 10. The object of the experiment was to determine the state of p o l a r i z a t i o n , or more s p e c i f i c a l l y the coherency matrix J°, of l i g h t corresponding to the 253.7 nm l i n e of mercury a f t e r the l i g h t traversed a mercury gas c e l l immersed i n a magnetic f i e l d . As i s evident from F i g . 10, the experiment c l o s e l y p a r a l l e l e d the i d e a l experiment described i n the l a s t chapter, and the method for experimentally determining the coherency matrix i s indicated by equation (2-192) which defines J . R e c a l l : I(cb,£) = J cos24> + J sin2cb + (J e" 1^ + J e 1^) sintb coscb xx yy xy xy (where the superscript » has been omitted for convenience). Measuring the i n t e n s i t y I((b,£) for four independent combinations of the variables cb and £ leads to four independent equations which can be solved for J , J , J and J . While there are many acceptable choices of the xx yy yx xy four combinations of <\> and £ , the following set of measurements was used In the experiment: a. ) P o l a r i z e r along x-axis, thus I, (0,0) = J 1 xx b. ) P o l a r i z e r along y-axis, thus T^Cp/lyO) = (3 1) c. ) P o l a r i z e r set at 45°, thus I 3(u/4,0) = 1/2 (J + J ) + Re(j ) xx yy xy d. ) P o l a r i z e r set at 45° and quarter-wave plate inserted with fast axis along x-axis, thus ^ ( i t / 4 , n/2) = l / 2 ( J x x + J y y ) - Im ( j )• F i g . 10: Experimental Apparatus - 91 - These four equations are l i n e a r l y independent and e a s i l y solved f o r J , J , Re(Jxy) and Ira(Jxy), y i e l d i n g the complete coherence matrix, xx yy The experimental set of x-y axes was a r b i t r a r i l y selected, hence the experimental x-axis does not necess a r i l y l i e along the d i r e c t i o n of the magnetic f i e l d as i n the t h e o r e t i c a l d e s c r i p t i o n . This i s explained i n more d e t a i l i n a l a t e r s e c t i o n . B r i e f l y , the experimental method was as follows. A beam of l i g h t propagates through a c e l l f i l l e d with mercury vapour which i s immersed i n a magnetic f i e l d . The magnetic f i e l d i s transverse to the d i r e c t i o n of propagation. The l i g h t next traverses the quarter-wave plate and p o l a r i z e r , which are arranged i n one of the four preselected combinations discussed e a r l i e r i n (3-1). The l i g h t then passes through a monochromator which selects the l i n e of Interest. F i n a l l y the i n t e n s i t y of the en t i r e l i n e i s measured by a photomultiplier and an o s c i l l i s c o p e . - 92 - The Light Source It was e s s e n t i a l that the l i g h t source used i n the experiment produce a stable, intense, unpolarized and reasonably narrow 253.7 nm l i n e . The source most commonly used i n experiments of this kind i s a microwave excited e l e c t r o d l e s s discharge lamp, (see G. Stanzel, 1974). However, the extreme s e n s i t i v i t y of neighboring experiments to microwave r a d i a t i o n precluded i t s convenient use i n t h i s experiment. Two other l i g h t sources were considered; a high voltage mercury discharge tube and a standard low voltage laboratory mercury lamp f i l l e d with natural mercury and argon. The lamp was f i n a l l y selected a f t e r a spectroscopic analysis demonstrated that the lamp produced a narrower 253.7 nm l i n e than the tube at the required i n t e n s i t y . The s t a b i l i t y of the 253.7 nm l i n e produced by the lamp was examined by monitoring i t s i n t e n s i t y over a period of 30 hours. Following an I n i t i a l warm-up period of two hours, the i n t e n s i t y of the l i n e was found to d r i f t by as much as 30% over the 30 hour period. However, the d r i f t was not constant; there were long periods, t y p i c a l l y greater than four hours, during which no measurable change i n the i n t e n s i t y was observed. These long period of s t a b i l i t y were followed by r e l a t i v e l y b r i e f periods, during which changes i n the i n t e n s i t y of up to 20% were observed. These periods of i n s t a b i l i t y persisted despite e f f o r t s to locate the source of the i n s t a b i l i t y and eliminate i t . Ultimately, the experimental procedure was simply adapted to accommodate the I n s t a b i l i t y . This adaptation i s described i n a l a t e r section. The Gas C e l l The gas c e l l was a c y l i n d e r , 3.7 cm long and 2.5 cm i n diameter, containing a droplet of natural mercury and i t s vapour. Before the mercury was added, the c e l l was purged of impurities by heating i t as i t was evacuated by a high vacuum pump. The c e l l was equipped with high q u a l i t y quartz windows which, even under the s t r a i n of the p a r t i a l vacuum, had a n e g l i g i b l e e f f e c t on the state of p o l a r i z a t i o n of the l i g h t . Since the mercury vapour pressure (and hence the gas density) i s highly temperature s e n s i t i v e , the temperature of the gas c e l l had to be c a r e f u l l y c o n t r o l l e d . This was accomplished by immersing the c e l l i n the temperature regulated water bath sketched i n F i g . 11. Water was c i r c u l a t e d past the gas c e l l and into a large dewer vessel which contained a heating c o i l . An e l e c t r o n i c monitoring device was constructed which continuously measured the temperature of the water and automatically switched the heating c o i l on and off when the temperature reached c e r t a i n fixed settings above and below the desired temperature. By t h i s means the mercury temperature could be set at any value between 20°C and 30°C with an accuracy of ±1/2°C. Mercury condensation on the windows of the c e l l excluded temperatures greater than 30°C. HEAT SENSOR INPUT HEAT SENSOR- GAS CELL WATER BATH TEMPERATURE MONITOR TEMPERATURE READING TEMPERATURE CONTROL DEWAR VESSEL WATER HEATER COJL HEATER COIL OUTPUT 4 PUMP F i g . 11: Temperature Regulator for.the Water Bath - 95 - The Magnet The electro-magnet and i t s current supply were manufactured by Varian Associates for use i n Nuclear Magnetic Resonance studies and thus provided a very stable and homogeneous magnetic f i e l d . Measurements of the uniformity of the f i e l d , using an Incremental Gaussmeter H a l l Probe, showed that the f i e l d never varied by more than .3% over the length of the c e l l . The magnetic f i e l d was c a l i b r a t e d using both a Rotating C o i l Gaussmeter with a rated accuracy of .1%, (at the c a l i b r a t e d f i e l d strength of 25 KG) and, an Incremental Gaussmeter H a l l Probe. To ensure a consistent r e s u l t the f i e l d was adjusted, both during the c a l i b r a t i o n and the actual experiment, by f i r s t s e t t i n g the current at a large preselected value and then lowering the current to the large s t of the current settings to be used i n the c a l i b r a t i o n . The current was then lowered sequentially through the pre-determined current settings which were used both for the c a l i b r a t i o n and the experiment. In t h i s way v a r i a t i o n s due to hysteresis were reduced. The two sets of measurements were found to be consistent to about 1% over the complete range of experimental f i e l d strengths. When no current flowed i n the magnet a r e s i d u a l f i e l d of 90±1G was found to p e r s i s t . The Aperture Stops and Lenses The f i r s t aperture stop, (S L i n F i g . 10) encountered by the l i g h t served only to define a s u f f i c i e n t l y narrow l i g h t beam to prevent l i g h t from being r e f l e c t e d into the detector from the walls of the gas c e l l or the poles of the magnet. The second aperture stop (S 2) l i m i t e d the beam to a t o t a l divergence of les s than . 7 5 ° . The three lenses L^, and Lj (see F i g . 10) were a l l made of quartz. The purpose of L± was to concentrate the l i g h t emitted from the lamp into a narrow and intense beam, hence a lens with as small a f o c a l length as possible was selected. The lens collimated the beam before i t passed through the p o l a r i z e r and the quarter-wave plate and thus permitted the diameter of the second stop to be made as large or larger than the face of the p o l a r i z e r , which i n turn greatly increased the i n t e n s i t y of the l i g h t reaching the monochromator. The lens Lg was used to focus the beam on the entrance s l i t of the monochromator. The f o c a l length of L 3 was such that the f-number of the lens was only s l i g h t l y larger than the f-number of the monochromator. Thus e f f e c t i v e l y a l l of the monochrometer's grating was used and the dispersion of the instrument approached i t s rated value. P o l a r i z e r and Quarter-Wave Plate A. Glan-Thompson p o l a r i z e r , with a transmissivity of about 40% for a wavelength of 253.7 nm, was used i n the experiment. The p o l a r i z e r could be set at any angle i n the plane normal to the o p t i c a l axis with an accuracy of better than 1°. The quarter-wave plate was designed s p e c i f i c a l l y for use at the 253.7 nm wavelength. The plate was a true quarter-wave plate i n the sense that the r e l a t i v e phase s h i f t between the two components was n/2 and not it/2 + m 2% where m= 1, 2, 3..... . This property was necessary to ensure that the r e l a t i v e phase s h i f t never exceeded the f i n i t e coherence length of the l i g h t , which i n turn ensured that the coherence length did not manifest i t s e l f i n the experimental measurements. Most of the i n t e n s i t y measurements required only the p o l a r i z e r and not the quarter-wave p l a t e . Therefore the quarter-wave plate was fixed to a pivoting mount which enabled i t to be moved i n and out of the l i g h t beam without realignment. Measurements taken with and without the quarter-wave plate were made compatible by determining the attentuation of the l i g h t by the quarter-wave p l a t e . I t was found that the i n t e n s i t y of the l i g h t was diminished by a factor of 1/1.14, regardless of the angle of the quarter- wave plate's fast a x i s . Thus a l l measurements taken with the quarter-wave plate i n place were m u l t i p l i e d by the factor 1.14 before being combined with the measurements taken without the quarter-wave p l a t e . - 98 - Monochromator The monochromator was a Spex 1800, with a rated dispersion of 10A°/mm. i n the f i r s t order. The 253.7 nm l i n e was observed i n the 3rd order with the entrance s l i t set at 140 \im and the exit s l i t set at 500 p,m to maximize the i n t e n s i t y . This resulted i n a bandpass of about 1.7A° which was s u f f i c i e n t to exclude a l l other detectable l i n e s emitted by the source. Detector The i n t e n s i t y of the l i n e , i s o l a t e d by the monochromator, was measured by observing the o s c i l l i s c o p e trace of the output of a photomultiplier placed at the monochromator's ex i t s l i t . To reduce noise the l i g h t was chopped to give a ser i e s of 2 msec, pulses. Shorter pulse lengths were necessarily excluded by the long observed r i s e time of the o s c i l l i s c o p e trace. The l i n e a r i t y of the detector system was tested using a series of neutral density f i l t e r s to produce a range of i n t e n s i t i e s encompassing the range found i n the experiment. A least squares f i t to a s t r a i g h t l i n e was performed on the data with a r e s u l t i n g l i n e a r - c o r r e l a t i o n c o e f f i c i e n t of r = .998 for the 14 data points, (see Bevington, 1969). This high degree of l i n e a r i t y allowed the height of the o s c i l l i s c o p e trace to be taken as - 99 - d i r e c t l y proportional to the i n t e n s i t y over the i n t e n s i t y range of i n t e r e s t . E f f e c t of the Magnetic F i e l d on the Experimental Apparatus To minimize the e f f e c t of the magnetic f i e l d on the l i g h t source and the photomultiplier, each of these devices was surrounded by a u-metal s h i e l d . The magetic f i e l d strength inside the p.-metal s h i e l d of the l i g h t source was measured using an Incremental Gaussmeter H a l l Probe. I t was found that the f i e l d was as always less than .7% of the f i e l d at the centre of the magnet poles, which, even for the largest f i e l d s used i n the experiment, would lead to a n e g l i g i b l e Zeeman S p l i t t i n g of the source l i n e s . The e f f e c t of the f i e l d on the remainder of the system was tested by removing the absorption c e l l and monitoring the i n t e n s i t y as the magnetic f i e l d was increased. Even with f i e l d s up to 4 KGauss there was no detectable change i n the i n t e n s i t y . - 100 - III.2 THE EXPERIMENTAL PROCEDURE The He-Ne alignment laser was positioned to define an o p t i c a l axis p a r a l l e l to the surfaces of the poles of the magnet and through the centre of the gap between the poles. A l l of the o p t i c a l instruments, i n c l u d i n g the l i g h t source and the entrance s l i t to the monochrometer, were I n i t i a l l y centred on the o p t i c a l axis using the alignment l a s e r . The lens L x and the l i g h t source were placed on moveable mounts allowing the l i g h t source to be moved across the o p t i c a l axis and the lens L^ to be moved along the a x i s . These two devices were adjusted u n t i l the f o c a l point of was positioned on a region of excited gas i n the lamp which produced the most intense 253.7 nm. The lens Ig was placed on a moveable mount which enabled i t to be manoeuvered to the point i n the plane normal to the o p t i c a l axis which gave the maximum i n t e n s i t y . It was found that when the angle of the p o l a r i z e r was changed, the p o s i t i o n of Lj which gave the maximum i n t e n s i t y also changed due to a d e f l e c t i o n of the beam by the p o l a r i z e r . Although t h i s e f f e c t could be s i g n i f i c a n t l y reduced by c a r e f u l l y p o s i t i o n i n g the p o l a r i z e r i n Its mount, i t could not be completely eliminated. Thus whenever the po l a r i z e r was rotated the lens 1̂ was repositioned to give the maximum i n t e n s i t y . In general a monochromator polarizes the l i g h t which i t disperses. This property of the monochromator was studied because i n the course of the experiment i t was necessary to send l i g h t polarized at d i f f e r e n t angles into - 101 - the monochromator and the measured Intensity of t h i s l i g h t would be affe c t e d i f the monochromator acted as a p o l a r i z e r . To examine the e f f e c t , the absorption c e l l was removed so that only the unpolarized l i g h t from the source was incident on the p o l a r i z e r . The po l a r i z e r was sequentially rotated through 180° while the max. Intensity of the 253.7 nm l i n e was measured. I t was found that the max. i n t e n s i t y was unaffected by the angle of the p o l a r i z e r . This r e s u l t was greeted with some scepticism and the measurements were repeated for the 763.5 nm l i n e of argon. In t h i s case the Intensity varied by a factor of more than four over the 180° r o t a t i o n . Since the p o l a r i z i n g e f f e c t of a monochrometer i s known to be highly frequency dependent, with frequencies where there i s no p o l a r i z i n g e f f e c t , (see K. Rabinovitch et a l . , 1965 and G.W. Stroke, 1963), the r e s u l t for the 253.7 nm l i n e was regarded as one of those rare instances i n experimental physics where nature conspires to lessen the work of the experimenter. The x-y axes of the p o l a r i z e r were selected a r b i t r a r i l y rather than s e t t i n g the x-axls along the f i e l d as i n the t h e o r e t i c a l a n a l y s i s . This was done because the only method a v a i l a b l e for a l i g n i n g the axes was to f i n d the p o l a r i z e r angles which produced the maximum and minimum i n t e n s i t y for l i g h t p o l a r i z e d by the gas c e l l i n the magnetic f i e l d , and t h i s could only be done to an accuracy of a few degrees. It was f e l t that rather than introducing t h i s further source or error i t was simpler and more precise to c a l c u l a t e the maximum and minimum i n t e n s i t y from equation (2-192) using the J obtained from the a r b i t r a r y set of axes. - 102 - The experiment required the quarter-wave p l a t e , when i n place, to be oriented such that i t s fast axis was p a r a l l e l to the designated x-axis of the p o l a r i z e r . The most s e n s i t i v e method found for accomplishing t h i s was as follows. Another Glan-Thompson p o l a r i z e r (P 2) was placed along the o p t i c a l axis just past the second aperture stop. With the monochromator set for the intense l i n e of the He-Ne l a s e r , and the quarter-wave plate positioned out of the las e r beam, the p o l a r i z e r Pĵ i n F i g . 10 was rotated u n t i l the measured i n t e n s i t y was minimized, i . e . the p o l a r i z e r s were crossed. This p o s i t i o n of the p o l a r i z e r was henceforth designated as the x-axis. The quarter-wave plate was then swung into p o s i t i o n between the two p o l a r i z e r s , and rotated i n the plane normal to the o p t i c a l axis u n t i l the measured Intensity was again a minimum. This could only occur i f the fast and slow axes were aligned with the pass di r e c t i o n s of the two p o l a r i z e r s . It was found that using t h i s method the quarter-wave plate axes could be positioned along the x-y axes of the p o l a r i z e r Pĵ to within 1°. Of course t h i s method cannot d i s t i g u i s h between the s i t u a t i o n where the fast axis i s along the x-axis and the s i t u a t i o n where the slow axis i s along the x-axis. Returning to the equations (3-1) i t i s evident that the only e f f e c t of th i s uncertainty i s to leave the sign of l m J x y , or equivalently the handedness of the e l l i p t i c a l wave i n doubt. It was f e l t that t h i s was of l i t t l e importance and no attempt was made to d i s t i n g u i s h getween the fast and slow axes of the quarter-wave p l a t e . As a check on the experimental technique a mock experiment was performed using the Glan-Thompson p o l a r i z e r P~ which was known to produce - 103 - highly l i n e a r l y p o l a r i z e d l i g h t . With no current flowing i n the magnet, the p o l a r i z e r P 2 was placed between the poles of the magnet while the gas c e l l was placed between the second aperture stop S 2 and the lens L 2 . To be c e r t a i n that the r e s i d u a l f i e l d of the magnet would not a f f e c t the Hg gas, the c e l l was surrounded by a p.-metal s h i e l d . With t h i s arrangement the complete set of measurements (3-1) was taken. I t was found af t e r analyzing the data that P = 1.00 ± .03 and CO J = -.02 ± .13 xy (an explanation of the error estimates i s given l a t e r ) . Thus to within the accuracy of the experiment the l i g h t was found to be completely l i n e a r l y p o l a r i z e d . This r e s u l t indicated not only that the experimental system was performing as expected, but also that any depolarization of the l i g h t , by the gas or Intervening o p t i c s , was e n t i r e l y n e g l i g i b l e . The actual experiment was performed with the instruments positioned as i n F i g . 10. The i n i t a l l y unpolarized l i g h t from the source passed through the gas c e l l , which was held at a constant temperature and immersed i n a uniform magnetic f i e l d of known strength. With the p o l a r i z e r along the x-axis as i n (3-la) and the lens Lj positioned for maximum i n t e n s i t y , the height of the o s c i l l i s c o p e trace corresponding to the transmitted i n t e n s i t y of the 253.7 nm l i n e was recorded. The current i n the magnet was then lowered to the next s e t t i n g at which the magnetic f i e l d had previously been measured and the i n t e n s i t y was again measured. After a representative set of measurements had been taken covering f i e l d s from about 250 G to 3500 G, the p o l a r i z e r and the quarter-wave plate were set at the next combination - 104 - described by (3-1) and the process was repeated for the same magnetic f i e l d s e t t i n g s . This was done for the four combinations described by (3-1). The complete set of measurements provided a l l of the information necessary for determining J as a function of f i e l d strength at a given temperature. The problems created by the unstable l i g h t source are considered below. The Unstable Light Source The I n s t a b i l i t y of the l i g h t source affected the experimental procedure i n a number of ways. F i r s t , i n order to reduce the p r o b a b i l i t y of a period of i n s t a b i l i t y occuring during the taking of a given set of measurements, the number of measurements i n the set had to be li m i t e d to permit the complete set to be taken i n about 3 hours. Second, at the beginning, the end and p e r i o d i c a l l y during the course of the experiment, the i n t e n s i t y of the l i g h t at a f i x e d s e t t i n g was measured to ensure that the i n t e n s i t y of the source had remained constant throughout the set of measurements. When a measurable change i n t h i s i n t e n s i t y was observed the complete set of measurements was r e j e c t e d . F i n a l l y , i t might appear that because the magnetic f i e l d settings were not exactly reproducible, greater p r e c i s i o n would r e s u l t i f a l l four measurements (3-1) were taken before the f i e l d strength was changed. However, because a s i g n i f i c a n t period of time was required to reset the p o l a r i z e r and the quarter-wave plate and to r e a l i g n L j , the time necessary - 105 - to complete a set of measurements would be greatly increased i f t h i s approach were adopted and the i n s t a b i l i t y i n the i n t e n s i t y of the source would begin to Influence the r e s u l t s , thereby negating any gain i n p r e c i s i o n . In addition to the experiment using an unpolarized l i g h t source, an attempt was made to study the e f f e c t of the gas on a l i n e a r l y polarized incident l i g h t beam produced by i n s e r t i n g the p o l a r i z e r P 2 i n front of the gas c e l l . Unfortunately, t h i s attempt was unsuccessful because the po l a r i z e r P 2 had a low transraissivity and with the i n t e n s i t y of the l i g h t further diminished by the p o l a r i z i n g properties of the gas c e l l the r e s u l t i n g i n t e n s i t y was too weak to provide a meaningful i n t e n s i t y measurement. - 106 - III.3 EXPERIMENTAL ERROR The experimental i n t e n s i t y measurements could not, of course, be made with absolute p r e c i s i o n . Experimental error was introduced into the measurements p r i n c i p a l l y from three sources. F i r s t , there was error i n reading the height of the o s c i l l i s c o p e trace, i n part because of the inherent imprecision of such a reading, but primarily because the background noise of the photomultiplier caused the height of the trace to f l u c t u a t e . Second, the r e p o s i t i o n i n g of the equipment between each of the four s e r i e s of measurements described by equations (3-1) unavoidably perturbed the alignment of the equipment. F i n a l l y , because of the e f f e c t s of h y s t e r e s i s , the same electrogmagnet current settings may not have produced p r e c i s e l y the same magnetic f i e l d strengths for each of the four measurements (3-1). It i s d i f f i c u l t to determine d i r e c t l y the impact of each of these factors on the magnitude of the error i n the Intensity measurements. In general, the best estimate of the experimental error would be obtained by repeating the complete set of measurements u n t i l a s t a t i s t i c a l l y meaningful sample i s c o l l e c t e d , from which the standard deviation i n the i n t e n s i t y at each point could be c a l c u l a t e d . This approach was, however, unworkable i n th i s experiment because the time required to c o l l e c t a large sample would greatly exceed the time period during which the l i g h t source remained reasonably stable. I f the only consequency of the lamp's i n s t a b i l i t y was a d r i f t i n i n t e n s i t y i t would have been possible to renormalize the r e s u l t s - 107 - before comparing them. However, the observed change i n the lamp's i n t e n s i t y was accompanied by a change i n the l i n e shape of the source. This produced a v a r i a t i o n i n the measurements which was unrelated to the actual imprecision of the experimental technique. Thus, to achieve a better estimate of the actual error i n the experimental measurements an a l t e r n a t i v e approach was adopted. Instead of taking a large number of measurements at each magnetic f i e l d s e t t i n g , a large number of measurements was taken at only a small but representative group of f i e l d s e t t i n g s . Unlike the complete set of measurements, these measurements could be taken i n a s u f f i c i e n t l y short period of time to avoid the consequences of the changing i n t e n s i t y and l i n e shape of the source. Thus, the standard deviation In the i n t e n s i t y measurement at each of these representative settings can be taken as t r u l y i n d i c a t i v e of the magnitude of the error i n the experimental measurements. When the standard deviation i n the i n t e n s i t y measurements was calculated i t was found not to be simply proportional to the i n t e n s i t y of the l i g h t . I t was instead found to be v i r t u a l l y constant for measurements taken at the same o s c i l l i s c o p e voltage scale s e t t i n g , even when the height of the trace varied considerably. However, the standard deviation was found generally to increase when the voltage scale s e t t i n g was decreased ( i n order to observe less Intense s i g n a l s ) . This r e s u l t suggests that the p r i n c i p a l cause of error i n the measurements was the inherent imprecision i n determining the height of the o s c i l l i s c o p e trace against the background - 108 - noise, since the noise becomes more pronounced as progressively lower scale settings are used. Since the magnitude of the error remained reasonably constant for a given voltage scale s e t t i n g , i t was possible to provide a reasonable estimate of the error i n the i n t e n s i t y by as c r i b i n g the same absolute error a , to a l l measurements taken at the same scale s e t t i n g , (the subscript i i d e n t i f i e s the i n t e n s i t y measurement associated with the error a,). I t was th i s ascribed error which was used i n a l l subsequent ca l c u l a t i o n s i n v o l v i n g the experimentally measured i n t e n s i t i e s . The error i n the i n t e n s i t y measurements a,, leads to an error i n a l l of the q u a n t i t i e s , such as P, J , and J , that are calculated from the ' xx yy measured i n t e n s i t i e s . In general, each of these quantities i s a function of the four measured i n t e n s i t i e s given i n equations (3-1) and therefore may be represented by: f ( I L .Ij ,13 ,1̂ ) = f ( I ± ) (3-2) Since there i s no c o r r e l a t i o n i n the errors a. associated with each Intensity measurement I the error i n the experimentally determined quantity a, i s given by: 4 5 f ( I x ,L, ,13 ,1̂ ) °f 2 = i Z = l *l (3-3) - 109 - (See Bevlngton, 1969, p. 58.) The error i n each experimentally determined quantity presented i n the next section was found from t h i s equation. - 110 - III.4 EXPERIMENTAL RESULTS In t h i s section the experimental r e s u l t s are presented i n a form which permits a d i r e c t comparison with the t h e o r e t i c a l r e s u l t s . The experimental coherency matrix J, as i n i t i a l l y determined from the equations (3-1), can not be d i r e c t l y compared to the t h e o r e t i c a l coherency matrix because the experimental x-axis i s not along the d i r e c t i o n of the magnetic f i e l d . However, the two matrices can be made compatible by performing a transformation from the experimental (x,y) axes to a new set of axes (x',y'). The x' axis i s aligned with the d i r e c t i o n of the magnetic f i e l d , and forms an angle <)) with respect to the x-axis. The coherency matrix J transforms under t h i s r o t a t i o n into the matrix J ' which i s r e a d i l y expressed i n terms of the matrix elements of J and the angle <j>, (see Born and Wolf, 1975, P. 548). J C 2 + J S 2 + (J + J )CS xx yy xy yx (J - J )CS + J C 2 - J S 2 yy xx xy yx (3-4) (J - J )CS + J C 2 - J s 2 yy xx yx xy J S 2 + J C 2 - (J + J )CS xx yy xy yx where C = cos $ S = s i n <j> The elements of J ' are d i r e c t l y comparable to the t h e o r e t i c a l coherency matrix once the angle <j> i s determined. - I l l - The angle $ may be determined by noting that when the incident l i g h t i s unpolarized and the x'-axis i s aligned with the magnetic f i e l d the theory requires the coherency matrix of the transmitted l i g h t to be of the form (see equation 2-180) 0 J' xx J ' yy (3-5) That i s , the transmitted l i g h t may only consist of an unpolarized component and/or a l i n e a r l y p o l a r i z e d component with the d i r e c t i o n of p o l a r i z a t i o n along either the x' or y' a x i s . Thus, the angle $ may be found from the matrix (3-4) by s e t t i n g = J y x = 0 and solving the two equations f o r <J>. When th i s i s done i t i s found that: <)> = arctan {—) or <J> + n/2 = arctan (—J (3-6) where B and C are calculated from the experimental i n t e n s i t y measurements, (with unpolarized incident l i g h t ) using the equations (3-1), (2-173) and (2-174). A choice can be made between the two possible solutions by comparing them with the observed o r i e n t a t i o n of the experimental x-axis with respect to the magnetic f i e l d . - 112 - The foregoing method for determining <}>, (and hence J') i s of course only v a l i d i f the transmitted l i g h t i s found to consist e n t i r e l y of an unpolarized component and/or a l i n e a r l y polarized component as predicted by the t h e o r e t i c a l a n a l y s i s . For, i f the transmitted l i g h t i s found to contain an e l l i p t i c a l l y polarized component, that i s i f Im(J ) * 0, then from the xy matrix (3-4) J ' t 0 for a l l $ and the method i s i n a p p l i c a b l e . Thus, before xy the suggested technique for f i n d i n g $ can be used, the t h e o r e t i c a l p r e d i c t i o n that Im(J ) = 0, (for unpolarized incident l i g h t ) , must be xy experimentally v e r i f i e d . To t h i s end, Im(J ) was calculated at each xy magnetic f i e l d s e t t i n g using the experimental i n t e n s i t y measurements and the equations (3-1) and (3-3). The r e s u l t for a gas at 22.6°C i s plotted i n F i g . 12. It i s evident from F i g . 12 that Im ( J ^ ) Is zero at each magnetic f i e l d s e t t i n g to within experimental e r r o r . However, i t Is also evident that the average value of I m ( J x y ) i s not zero. In fact i f i s averaged over a l l f i e l d s e t t i ngs, i t i s found to be: -8.8 x 10 - 2 ± 1.7 x 10 - 2 Thus, i t would appear that there i s a small e l l i p t i c a l component i n the transmitted wave. This small component can, however, be completely accounted for by the systematic error which i s Introduced when, i n c a l c u l a t i n g Im(J ), the measurement I [see equations (3-1)] i s m u l t i p l i e d xy by the constant f a c t o r G = 1.14 In order to compensate for the attenuation of the l i g h t by the quarter-wave p l a t e . The error an i n determining G i s ± 0.04 and, using equation (3-3), t h i s produces a systematic error i n IM(JXY) IN RRBITRRRY UNITS -1.2 -1.0 -i 0.8 -0.6 J L _ -0.4 -0.2 _! I 0.0 I 0.2 _i Q.i 0.6 i 0.8 _ j 1.0 _i ft) CO c 3 o o 3 2 CL rr r» (0 3 OP - £IT - - 114 - IinCJ^) of about 0.12 for a t y p i c a l value of 1̂ . Thus, the apparent e l l i p t i c a l component can be a t t r i b u t e d e n t i r e l y to t h i s systematic error and the t h e o r e t i c a l p r e d i c t i o n that Im(J ) = 0 for a l l f i e l d strengths i s xy v e r i f i e d to within experimental e r r o r . This r e s u l t j u s t i f i e s using equations (3-6) to determine $ i n the experiment. The average value of $ was found to be 86.2°±0.8° For the purposes of comparing the experimental r e s u l t s with the t h e o r e t i c a l r e s u l t s the q u a n t i t i e s of primary i n t e r e s t , (when considering unpolarized incident l i g h t ) , are , J y y and the degree of p o l a r i z a t i o n P, (where again the superscript » has been omitted from P°°). These quantities were determined from the experimental measurements i n the following ways. The degree of p o l a r i z a t i o n P, i s independent of the choice of x,y axes and therefore was simply determined from the elements of the experimental coherency matrix J , which were i n turn determined from equations (3-1). From equation (2-195): 4(J J - J J ) 1/2 P - r i - xx yy xy yx y - i P - L l ( J — + J )2 J xx yy J ' and J ' can be calculated from the matrix (3-5) by simple xx yy s u b s t i t u t i o n , once the angle 4> and the matrix J have been determined. Instead, however, J ' and J ' were calculated using a set of equations i n ' ' xx yy n - 115 - which the values of J ' and J ' are not d i r e c t l y dependent upon the xx yy calculated value of ty. This set of equations i s again based on the f a c t that Im(J ) = 0 at a l l f i e l d strengths which implies that J ' = J' = 0. xy xy yx By further noting that the trace and the determinant of the matrix J ' are invariant under r o t a t i o n a l transformations the desired set of equations can be r e a d i l y derived from the matrix J ' . J' = (J + J ) - A and J ' = A xx xx yy yy when (3-7) C 1 = 0 and B' * 0 and that: J ' = A and J' = (J + J ) - A xx yy xx yy when B 1 = 0 and 0 where A i s given by equation (2-172) and B' and C are found from equations (1-173) and (2-174) and the matrix J ' which i s found from the ca l c u l a t e d values of ty and J . I t i s clear from these equations that the error i n determining ty only a f f e c t s the p r e c i s i o n with which B' and C are determined and these values are only used to choose between the two possible solutions for J ' and J' . xx yy Thus, when equations (3-7) were used to cal c u l a t e the experimental values of J' and J' , the uncertainty i n J ' and J ' was not affected by the xx yy xx yy J uncertainty i n determining ty . - 116 - The experimental values of J J ' and P for the two gas xx yy temperatures of 22.6°C and 30°C are plotted i n F i g . 13, 14, and 15. In the next chapter the t h e o r e t i c a l and experimental r e s u l t s are comared. o MAGNETIC FIELD STRENGTH (KGRUSS) - 120 - CHAPTER IV COMPARING THEORY AND EXPERIMENT When the experimental curves of and P, (given i n F i g s . 13, 14, 15) are compared with the t h e o r e t i c a l curves for J r o , and J 0 0 and xx yy (given i n F i g s . 7, 8, and 9) i t i s evident that there are s i g n i f i c a n t differences between the two. This r e s u l t i s , however, not unexpected since c e r t a i n c h a r a c t e r i s t i c s of the experiment do not accord with c e r t a i n assumptions made i n c a l c u l a t i n g the t h e o r e t i c a l curves. In p a r t i c u l a r , the t h e o r e t i c a l c a l c u l a t i o n s were c a r r i e d out using the properties of the s i n g l e , even isotope of mercury, H g 2 0 2 , (with nuclear spin I = 0), while the experiment was performed using natural mercury which consists of 7 isotopes, each of which has a unique spectrum. Furthermore, while the t h e o r e t i c a l curves were based on an incident source l i n e with a guassian l i n e p r o f i l e , the experimental source l i n e had a much more complex shape. In p r i n c i p a l i t i s possible to carry out the t h e o r e t i c a l c a l c u l a t i o n using the properties of n a t u r a l mercury and an Incident l i n e p r o f i l e matching the experimental source l i n e . However, i n practice t h i s complex c a l c u l a t i o n can only be performed with l i m i t e d p r e c i s i o n and no attempt was made i n t h i s study to perform such a c a l c u l a t i o n . In t h i s section the differences between the assumptions i n the t h e o r e t i c a l c a l c u l a t i o n and the c h a r a c t e r i s t i c s of the experiment are examined. It i s then suggested that when these differences are accounted for the theory and experiment are i n q u a l i t a t i v e agreement. - 121 - IV.1 The E f f e c t of Using Natural Mercury i n the Absorption C e l l When the spectrum of natural mercury i s examined the 253.7 nm l i n e i s found to consist of a se r i e s of very c l o s e l y spaced l i n e s , (Schweitzer, 1963). This complex l i n e spectrum, r e s u l t s from superposing the hyperfine structure df the various isotopes of mercury which are present i n natural mercury. Di f f e r e n t isotopes of the same element have d i f f e r e n t spectra because the energy l e v e l s of an atom are affected not only by the nuclear charge but also by other nuclear c h a r a c t e r i s t i c s , such as the nuclear angular momentum or spin, I, the nuclear mass and the nuclear charge d i s t r i b u t i o n . The e f f e c t of these nuclear c h a r a c t e r i s t i c s i s examined i n d e t a i l i n Kuhn, 1969 at page 329 and i n Sobel'man, 1972 at page 204. For the l i m i t e d purpose of explaining the r e s u l t s of the experiment i t i s s u f f i c i e n t simply to note that the chief e f f e c t of the differe n c e i n nuclear mass and charge d i s t r i b u t i o n between isotopes of the same element i s a small s h i f t i n the frequencies of the l i n e spectra. The e f f e c t of the nuclear spin I i s b r i e f l y considered below. E f f e c t of Nuclear Spin Isotopes with an odd mass number have a h a l f - i n t e g r a l nuclear spin, while isotopes with an even mass number have an i n t e g r a l nuclear spin. Isotopes with an even number of protons and neutons have zero nuclear spin. - 122 - The nuclear spin gives r i s e to a nuclear magnetic moment, p.̂ , which i s of the order of 1000 times smaller than the Bohr Magneton. The nuclear magnetic moment i n t e r a c t s with the magnetic f i e l d caused by the o r b i t a l motion of the electrons, and to a lesser extent with the magnetic f i e l d generated by the electron spin. This i n t e r a c t i o n leads to a s h i f t i n the energy l e v e l s , A E ^ , which i s given by: A E j = A/2 (F(F+1) - 1(1+1) - J(J+1)) (4-1) where A i s the magnetic dipole i n t e r a c t i o n constant, F i s the t o t a l angular momentum of the atom and i s produced by the coupling of I and J i n complete analogy with the coupling of S and L to produce J . The quantum values of F run from |j+l| to | j - l | . A quadrupole coupling which courses a smaller s h i f t i n the energy lev e l s has been ignored. It follows from equation (4-1) that isotopes where 1*0 w i l l e x h i b it a hyperfine s p l i t t i n g of the sp e c t r a l l i n e s while isotopes where 1 = 0 , w i l l e x h i b it no hyperfine s p l i t t i n g . With these r e s u l t s , the structure of the 253.7 nm l i n e depicted i n Schweritzer's paper can be understood. The even isotopes of mercury H g 2 0 4 , H g 2 0 2 , H g 2 0 0 and H g 1 9 8 produce 253.7 nm l i n e s at s l i g h t l y d i f f e r e n t frequencies because of the d i f f e r e n t nuclear shapes, but, since 1=0 for these isotopes, t h e i r l i n e s do not exhibit any hyperfine s p l i t t i n g . The odd isotopes, H g 1 9 9 and H f 2 0 1 also produces a frequency s h i f t e d 253.7 nm l i n e , - 123 - but, sine 1*0 for these isotopes, t h e i r l i n e s exhibit hyperfine s p l i t t i n g described by equation (4-1). In weak magnetics f i e l d s , (those producing l i n e s p l i t i n g which i s smaller than the hyperfine s p l i t t i n g ) , each hyperfine l i n e s p l i t s into Zeeman Components having a d e f i n i t e state of p o l a r i z a t i o n . The f i r s t order change i n the energy l e v e l s , AE^, i s given by: AF - F(F+1) + J(J+1) - 1(1+1) Z 2F(F+1) ( 8aj ^0 V ( 4 _ 2 ) where m has the values m = F, F - l , ... , -F When the magnetic f i e l d s p l i t t i n g exceeds the hyperfine s p l i t t i n g equation (4-2) i s no longer v a l i d , and the s p e c t r a l l i n e s p l i t s Into the u and o-components given by equation (2-9) and each of these components i s i n turn s p l i t into several equidistant hyperfine l i n e s as a r e s u l t of the nuclear spin. I t i s important to note that when 1=0, the Zeeman s p l i t t i n g of the l i n e s Is simply governed by equation (2-9) and not equation (4-2). It follows that when natural mercury i s immersed i n a weak magnetic f i e l d , the 253.7 nm l i n e s of the odd isotopes w i l l independently exhibit a more complex s p l i t t i n g described by equation (4-2). This complex l i n e s s p l i t t i n g makes i t d i f f i c u l t to describe the r e s u l t i n g absorption spectrum - 124 - with p r e c i s i o n , p a r t i c u l a r l y since equation (4-2) becomes i n v a l i d at r e l a t i v e l y weak f i e l d strengths of about 1 KGauss. The s p l i t t i n g of the l i n e s w i l l , however, be used i n section IV.3 to describe the shape of the experimental curves for J ' and J ' i n q u a l i t a t i v e terms. xx yy IV.2 The Shape of the Experimental Incident Line The l i g h t source used i n the experiment was a mercury lamp f i l l e d with natural mercury and argon. While the lamp produced an intense 253.7 nm l i n e , most of the l i g h t was produced at the centre of the lamp's gas bulb and had to traverse a volume of cold gas before being emitted from the lamp. As the cold gas would re-absorb some of the l i g h t the lamp would be expected to produce a highly s e l f - r e v e r s e d l i n e p r o f i l e . The 253.7 nm l i n e produced by the lamp was examined i n a spectrograph which i n the 5th order had a r e c i p r o c a l l i n e a r dispersion of 400 mA/mm. Unfortunately, t h i s did not provide s u f f i c i e n t l y high r e s o l u t i o n to observe the hyperfine structure or the d e t a i l e d shape of the l i n e . I t was, however, possible to estimate that the width of the l i n e was between 70 and 85 mA. Since Schweitzer's work shows that the width of the hyperfine structure i n natural mercury i s le s s than 50 mA i t may be assumed from the size of the measured l i n e width that each componet l i n e was s u f f i c i e n t l y broad to create a continuious l i n e p r o f i l e marked by narrow troughs produced by the re-absorption of the cold gas. In the next section th e f f e c t of t h i s l i n e p r o f i l e on the experimental curves for J ' and J ' i s examined, xx yy - 125 - IV.3 The Experimental Results The experimental and t h e o r e t i c a l r e s u l t s are i n broad agreement when the differences between the t h e o r e t i c a l assumptions and the experimental conditions are considered. In the t h e o r e t i c a l curves, F i g s . 7, 8 and 9 i s independent of magnetic f i e l d strength while i n the experimental curves, F i g s . 13 and 14, J x x varies with f i e l d strength. This difference i s , however accounted for by the presence of odd isotopes of mercury i n the experimental but not the t h e o r e t i c a l absorption gas. It was shown i n matrix (2-180) that the magnitude of J y y i s only affected by the strength and frequency of the a-components of the absorbing gas, while the magnitude of i s only affected by the strength and frequency of the n-components. The even isotopes of mercury have zero nuclear spin, (see the appendix) and a non-degenerates ground state and therefore, from equations (2-129) and (2-130), only the o-components of the 253.7 nm l i n e are f i e l d dependent. Thus, when the absorption gas consist e n t i r e l y of even isotopes of mercury, as i n the t h e o r e t i c a l c a l c u l a t i o n , only J ' should be f i e l d dependent for the 253.7 nm l i n e . yy In contrast, odd isotopes of mercury which were present i n the natural mercury used i n the experiment, produce both tt and a-components which are f i e l d dependent. This i s because odd isotopes of mercury have a mon-zero nuclear spin, I, (see the appendix) and t h i s produces a degenerate ground state with a non-zero angular momentum, F. The Zeeman s p l i t t i n g of the - 126 - 253.7 nm l i n e i s governed by equation (4-2). When equation (4-2) i s examined for a l l allowed t r a n s i t i o n s , i t i s found that some of the it-components are f i e l d dependent. Thus, when the absorption gas contains odd isotopes both J ' and J' are f i e l d dependent. xx yy While i t i s clear that the presence of odd isotopes i n the experimental absorption gas leads to the f i e l d dependence of the shape of the curves for i n F i g s . 13 and 14 can not be v e r i f i e d by d i r e c t c a l c u l a t i o n . This i s because the dependence of J ^ x on f i e l d strength depends c r i t i c a l l y upon the l i n e shape of the incident source l i n e , and i n the experiment t h i s l i n e shape could not be determined with p r e c i s i o n . The general shape of the J ' xx curves can, however, be explained. Referring to the emission spectrum of natural mercury (Schweitzer, 1963) and F i g . 13, the i n i t i a l decrease i n with increasing f i e l d strength i s caused by the s p l i t t i n g of the it-component absorption away from the centres of the absorption l i n e s of the odd isotopes, H g 2 0 1 and H g 1 9 9 . This s p l i t t i n g leads to an increased absorption of the n-component of the source l i n e and hence a decrease i n the magnitude of The increase i n J x x at a f i e l d strength of about 2.4 K Gauss which i s evident i n F i g . 13 i s l i k e l y due to three e f f e c t s . F i r s t , some of the it-components of the absorption l i n e s w i l l be s p l i t beyond the width of the source l i n e , reducing the t o t a l absorption of the it-component of the source l i n e . Second, some of the it-component absorption l i nes w i l l s h i f t to the same frequency, again reducing the t o t a l absorption of the it-component of the source l i n e . F i n a l l y , i n stronger f i e l d s the coupling between J and I breaks down and the Zeeman s p l i t t i n g i s no longer described by equation - 127 - (4-2). At these higher f i e l d strengths the f i e l d dependence of some of the TI-components i s reversed and there l i n e s again approach the centres of the incident l i n e s which reduces the t o t a l absorption of the n-component of the incident source l i n e . The most s t r i k i n g d i f f e r e n c e between the t h e o r e t i c a l and experimental curves for J y y i s the o s c i l l a t o r y appearance of the experimental curves. This feature can, however, be explained i f the e f f e c t of having a number of even isotopes i n the absorption gas i s examined. From F i g . 13, the magnitude of J y y peaks at f i e l d strengths of about 1.25 K Gauss and 2.5 K Gauss. Using equation (2-9) these f i e l d strengths lead to a s p l i t t i n g of the a-component absorption l i n e s of the even isotopes from the l i n e centre by 86.75 x 10 - 3cm - 1 and 173.5 x 10 - 3cm - 1 r e s p e c t i v e l y . By r e f e r r i n g to the emission spectrum of natural mercury (Schweitzer, 1963), i t can be seen that the emission l i n e s of the even isotopes are separated from each other as shown below a) between 202 and 204 174 x 10"3 cm - 1 b) between 200 and 202 177 x 10"3 cm - 1 c) between 198 and 200 160 x 10 - 3 cm - 1 Thus, a frequency s h i f t of about 86.75 x 10 - 3cm - 1 would place the o-component absorption l i n e s approximately half-way between the l i n e centres, while a s h i f t of about 173.5 x 10 - 3cm - 1 would place the o-component absorption l i n e s approximately at the centre of the adjacent absorption l i n e . I t therefore appears that the o s c i l l a t o r y appearance of J ' i s caused - 128 - by the crossing of the a-component absorption l i n e s which r e s u l t s i n a decrease i n the t o t a l absorption of the a-component of the incident source l i n e . In conclusion, the differences between the t h e o r e t i c a l assumptions and the experimental r e s u l t s preclude using the l a t t e r to v e r i f y d i r e c t l y a l l of the t h e o r e t i c a l p r e d i c t i o n s . However, when the t h e o r e t i c a l predictions could be tested d i r e c t l y as i n the case of 1^ ( J x y ) the theory and experi- ment were i n good agreement. Further, even when the theory could not be tested d i r e c t l y as i n the case of J 1 and J' , the theory was shown to be at xx yy least consistent with the experimental r e s u l t s . In the next Chapter the conclusion which may be drawn from t h i s study are examined i n more d e t a i l . - 129 - CHAPTER V Concluding Discussion The object of t h i s study was to examine the f e a s i b i l i t y of using the traverse Zeeman e f f e c t to produce a narrow band p o l a r i z e r i n the u l t r a - v i o l e t region. The t h e o r e t i c a l model indicated that an absorption gas consisting of the even isotope of mercury, H g 2 0 2 , would act as a high q u a l i t y p o l a r i z e r for narrow 253.7 nm emission l i n e s produced by an H g 2 0 2 source. The experimental r e s u l t s , while not inconsistent with the theory, were i n s u f f i c i e n t to v e r i f y the t h e o r e t i c a l predictions because of the complications Introduced by the hyperfine structure of natural mercury i n the absorption gas, and by the broad self-reversed source l i n e . The general t h e o r e t i c a l model which was developed for the transmission of l i g h t In a gas immersed i n a magnetic f i e l d i s applicable to a v a r i e t y of spectroscopic studies. Equation (2-112) provides a complete d e s c r i p t i o n of the propagation of an EM-wave i n a gas with a non-degenerate ground state where the magnetic f i e l d may assume any ori e n t a t i o n with respect to the d i r e c t i o n of propagation. A demonstration of how equation (2-112) could be extended to the case of a gas with a degenerate ground state was presented i n equations (2-131) to (2-134). A general d e s c r i p t i o n of the state of p o l a r i z a t i o n of an incident wave afte r traversing a gas i n a transverse magnetic f i e l d was provided by matrix - 130 - (2-179). The matrix i s v a l i d for incident waves with any i n i t i a l state of p o l a r i z a t i o n . The t h e o r e t i c a l c a l c u l a t i o n of the p o l a r i z i n g e f f e c t of an H g 2 0 2 absorption gas on H g 2 0 2 created 253.7 nm emission l i n e s with a v a r i e t y of l i n e widths indicated that the technique should produce high q u a l i t y narrow band p o l a r i z e r s i n the u l t r a - v i o l e t region. The experimental r e s u l t s supported some of the t h e o r e t i c a l p r e d i c t i o n s . For example, the t h e o r e t i c a l p r e d i c t i o n tht I (J ) = 0 for an unpolarized r m xy r incident source l i n e was v e r i f i e d to within experimental e r r o r . However, the hyperfine structure of the natural mercury used i n the experimental absorption gas, and the broad source l i n e generally precluded a d i r e c t comparison between theory and experiment. The width of the source l i n e l i m i t e d the v a r i a t i o n of J ' to a s i n g l e yy order of magnitude, while the t h e o r e t i c a l curves of J ' varied over several yy orders of magnitude. I t was f o r th i s reason that the t h e o r e t i c a l and experimental plots of J ' and J ' were not presented i n the same graph. yy x x The hyperfine structure caused JP^ to be f i e l d dependent contrary to the t h e o r e t i c a l model. Further, the isotope s h i f t of the 253.7 nm l i n e and the presence of a number of even isotopes i n natural mercury lead to l i n e crossing which i n turn caused the plots of J ' as a function of f i e l d strength to assume on o s c i l l a t o r y form. The precise shapes of the experimental curves of J ' and J ' could not xx yy - 131 - be t h e o r e t i c a l l y calculated because the source l i n e p r o f i l e , upon which the c a l c u l a t i o n c r i t i c a l l y depend, could not be measured with the l i m i t e d r e s o l u t i o n of the a v a i l a b l e spectrograph. Thus, while the experimental r e s u l t s confirm the theory i n some li m i t e d respects, the experiment could not provide a quantitative test of the theory. To test the t h e o r e t i c a l p r e d i c t i o n that the transverse Zeeman- e f f e c t can be used to produce a high q u a l i t y p o l a r i z e r a gas cons i s t i n g of a singl e isotope with zero nuclear spin should be used i n the experiment. Further the source l i n e should be reasonably narrow; i f i t i s broad, then i t s l i n e p r o f i l e should be known. - 132 - References 1. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover P u b l i c a t i o n s , New York, 1972. 2. A l l , A.W. and Griem, H.R., Physical Review, 140, A1044 (1965) and Erratum, P h y s i c a l Review, 144, 366 (1966). 3. Bevington, P.R., Data Reduction and Error Analysis for the P h y s i c a l Sciences, McGraw-Hill Book Co., New York, 1969. 4. Born, M. and Wolf, E., P r i n c i p l e s of Optics, Pergamon Press, Oxford, 1975. 5. Camm D.M. and Curzon, F.L., Canadian Journal of Physics, 50, 2866 (1972). 6. Condon, E.V., and Shortley, G.H., The Theory of Atomic Spectra, Cambridge U n i v e r s i t y Press, New York, 1964. 7. Corney, A., Kibble, B.P. and Series, G.W., Proc. R. S o c , A293, 70 (1966). - 133 _ 8. D'Yakonov, M.I. and P e r e l , V.I., Soviet Physics JETP, 21_, 227 (1965). 9. Dodd, J.N. and Series, G.W., Proc. R. S o c , A263, 353 (1961). 10. H e i t l e r , W., The Quantum Theory of Radiation, Oxford University Press, Oxford, 1954. 11. Jackson, W.D., C l a s s i c a l Electrodynamics, John Wiley & Sons, New York, 1975. 12. K l e i n , M.V., Optics, John Wiley & Sons, Inc., New York, 1970. 13. Kuhn, H.G., Atomic Spectra, Academic Press, New York, 1969. 14. Lurio, A., Physical Review, 140, A1505 (1965). 15. Merzbacher, E., Quantum Mechanics, John Wiley & Sons, Inc., New York, 1970. 16. Messiah, A., Quantum Mechanics Volumes I and I I , John Wiley & Sons, Inc., New York, 1962. 17. Rabinovitch, K., C a n f i e l d , L.R. and Madden, R.P., Applied Optics, 4, 1005 (1965). - 134 - 18. Rose, M.E., Elementary Theory of Angular Momentum, John Wiley & Sons, Inc., New York, 1957. 19. Schweitzer, W.G., Journal of the O p t i c a l Society of America, 53_» 1055 (1963). 20. Sobel'man, I . I . , Introduction to the Theory of Atomic Spectra, Pergamon Press, Oxford, 1972. 21. Stanzel, G., Z. Physik, 270, 361 (1974). 22. Stroke, G.W., Physics L e t t e r s , 5, 45 (1963). - 135 - APPENDIX Properties of the 253.7 nm Line A term diagram of the Atomic Spectra of Hg can be found i n Condon and Odishaw (ed.), Handbook of Physics, McGraw-Hill Book Company, New York, 196, at page 7-45. Lifetime of the state 3 Pĵ , from Lurio, 1965 T ,( 3P, ) = 1.14 x 10 - 7 sec. a j 1 which gives an absorption o s c i l l a t o r strength, f a , of fa = 2.54 x 10 - 2 Lande g ,( 3 P , ) , from L u r i o , 1965 a j ! g ( ^ ) = 1.486094(8) - 1 3 6 - d) Natural Isotopes of Mercury ISOTOPE % NATURAL ABUNDANCE NUCLEAR SPIN, I ATOMIC MASS Ha* 9 6 80 H 8 Hal 9 8 80 H 8 Hal 9 9 80 H S 80 1 ,Hg200 8 0 H § 80 201 H g202 80 Hg' 204 0.146 1 0 . 0 2 16.84 2 3 . 1 3 1 3 . 2 2 2 9 . 8 0 6 . 8 5 0 0 1/2 0 3/2 0 0 195.9658 197.9668 198.9683 199.9683 200.9703 201-9706 203-9735 (Taken from Handbook of Chemistry and Physics, The Chemical Rubber Co., Cleveland, 1969 on page B-491 to B-494) e) FREQUENCY AND WAVELENGTH OF THE 253.7 nm LINE X0 = 2536.519 A u 0 = 7.42613 x 1 0 1 5 Hz. (Taken from, Handbook of Chemistry and Physics Op. C i t . Page E-222) f) The Properties of the Hyperfine Structure of the 253.7 nm Line Structure of the 253.7 nm Line can be found i n Schweitzer, 1963.
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