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UBC Theses and Dissertations

Zero field level crossing in the helium atom Bardsley, Robert Eckardt 1970

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ZERO FIELD LEVEL CROSSING IN.THE HELIUM ATOM by ROBERT ECKARDT BARDSLEY B.Sc, University of British Columbia, 1968 .A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS -We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f PHYSICS  The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date September , 1970 ABSTRACT Zero-field level crossing techniques have been used to measure some upper state lifetimes of the helium atom. The upper states of the atom were excited in a discharge between two capacitor plates. A radio frequency voltage was applied to these plates by a 450 MHz source. The emitted light was modulated by rotating a quarter-wave plate in front of a polaroid in the beam of emitted light. The polarization, which is directly related to this modulation, was measured by a phase sensitive lock-in amplifier. -The halfwidths of curves obtained by plotting the polarization against the magnetic field strength for the n^ D - 2^ P transitions 8 1 8 yielded lifetimes of; 2.71 x 10" sec. for the 3 D state, 3.97 x 10" 1 -8 1 sec. for the 4 D state and 4.48 x 10 sec. for the 5 D state. - i i i ' " -TABLE OF CONTENTS Chapter Page I INTRODUCTION 1 II " THEORY 2.1 Introduction . . . . . . . 4 2.2 Classical Mechanical Theory of the Hanle Effect 4 2.3 Quantum-Mechanical Description of the Hanle Effect 7 2.4 The Excitation Matrix Elements . . . . . . 11 2.5 The Helium Atom 12 III EXPERIMENTAL DETAILS 3.1 Experimental Arrangement 14 3.2 The Discharge. 16 3.3 The Optical System 19 3.4 Lock-In Amplifier 20 3.5 Rotating Quarter-Wave Plate and Reference Signal . . . . . . 21 3.6 Vacuum System. 23 3.7 R.F. Supplies and Coupling . 25 3.8 Helmholtz Coils . . . . . 27 3.9 Photomultiplier 31 .3.10 X.Y. Recorder 31 ? >• \f I 3.11 Data Processing . . . ...... . . . 31 ' v/ 3.12 Special Optical System 32 IV RESULTS 4.1 Lifetimes - . 34 4.2 Cross-section . . . . . . . . . . . . . . . 34 iv -TABLE OF CONTENTS (cont'd.) Chapter Page 4.3 Stability and Polarization of the Discharge . . . 44 4.4 Experimental Errors a) Magnetic Field 49 b) Pressure in the Discharge 49 c) Temperature in the Discharge 49 V CONCLUSIONS . . . . . . . . . . 51 APPENDIX I ROTATING QUARTER-WAVE PLATE 53 APPENDIX II CROSS-SECTIONS . . . 56 LIST OF REFERENCES 58 - v -LIST OF TABLES Table Page I " Properties of Electron Motion in an R.F. Electric and D.C. Magnetic Field 18 II. Lifetime of the 51D State of Helium 40 III Lifetime of the 41D State of Helium . .41 IV Lifetime of the 31D State of Helium 42 V .Cross-Sections 44 - vi -ILLUSTRATIONS AND FIGURES Figure Page 1 Splitting of Zeeman Sublevels 2 2 Classical Radiating Dipole . . . . . . . . 5 3 Typical Polarization Curve . . . . . . 7 4 -Four Level System 8 5 Energy Levels of Helium Atom 13 6 Apparatus 15 7 Cross-Section of Discharge Cell and Capacitor Plates 17 8 Quarter-Wave Plate Rotator 22 9 Vacuum System 24 10 Matched Resonant Circuit 26 11 -Helmholtz Coil Power Supply . . . 28 12 Pre-Amplif ier and R. C. Circuit 29 13 Photomultiplier Wiring System . . . . . . . . . . . . 31 14 Special Optical System . 33 15 Experimental Level-Cross Curve for the 4^ D Line . . . 35 16 Least Squares Fitted Curve for the 4*D Line 36 17 Plot of Hn as a Function of Pressure for the 1 h 5 D Line . . . . . . . . . . . . . . . . . . . 37 18 Plot of H, as a Function of Pressure for the I2 4 D Line 38 19 Plot of H, as a Function of Pressure for the 1 2 3 D Line 39 2 20 Plot of the Cross-Sections as a Function of n . . . 43 - v i i -ILLUSTRATIONS AND FIGURES (cont* d.) Figure Page 21 Plot of Intensity as a Function of Magnetic Field . . .45 22 Polarization Plotted as a Function of Magnetic Field . . . . . . A6 23 Fitted Curves of Runs at Low Pressures 48 24 Optical Schematic 53 25 A Sample Cross-Section Plot „ 56 - v i i i ACKNOWLEDGEMENTS I wish to express my sincere appreciation to Professor F. W. Dalby for his encouragement and his guidance. I am particularly thankful for his ceaseless enthusiasm for this project. I am indebted to Dr. Jacob Vanderlinde for designing the apparatus and for patiently describing its operation. The efficient typing of the thesis by Mrs. Sherri MacRaild is greatly appreciated. Also, I thank my wife and my fellow graduate students for their worthy assistance. CHAPTER I INTRODUCTION 1.1 Introduction The lifetimes of states in atoms and molecules, or the transition probability between these states have been, for many years, a difficult property to measure. In the past, elaborate methods have been used which involve interconnected systems which have properties which are difficult to measure accurately. In my opinion, the difficulty in measur-ing these properties or lack of knowledge of these properties produces unreliable results. For example, the Hook method involves an accurate measurement of the partial pressure of the absorbing gas. In the following experiment the lifetimes have been measured by the Hanle effect. This is a straight-forward method that involves measuring only the polarization of the emitted light as a function of the magnetic field strength. Basically, the Hanle effect measures the natural width of a spectral line with a resolution of the order of 10 "* A. This will seem incredible when one considers today's spectrographs and monochromators -2 • which have a resolution of about 10 A. Consider two Zeeman levels split by a magnetic field as shown in Fig. 1. Notice that they are completely resolved at a large magnetic field, (case 1). As the magnetic field decreases, the levels come closer together and the two levels can interfere with each other other, (case 2). By measuring the degree of interference as a function of the magnetic field, the width of the energy levels can be calculated. (This. interference will later be shown to be related to the polarization! of emitted light.) The interference effects, mentioned above, are similar to the interference effects of a double s l i t diffraction pattern. Im the double s l i t experiment one cannot distinguish which s l i t a certaxe photon came through. Similarly in our experiment, when the two states of the atom overlap, one cannot distinguish what state the atom was in. The Hanle effect has been used in the past to measure lifetimes of atoms. But, its use was limited mainly to transitions excited by absorption of resonance radiation. In the present experiment, electron excitation is used. Now, one would expect that the selection of miiDle-cular states or atomic states that one can measure, is almost unlimited. I I — : > 0 M A G N E T I C F I E L D * . G A U S S 4 0 FIG. 1—Splitting of Zeeman Sublevels - 3 -S p e c i f i c a l l y , t he l i f e t i m e s o f t he 3^D, 4 *D , and 5^D e n e r g y l e v e l s o f t he h e l i u m atom have been m e a s u r e d . These l i f e t i m e s a r e known r e l a t i v e l y a c c u r a t e l y , t h u s , t h e v a l i d i t y o f t he u se o f e l e c t r o n e x c i t a -~ t i o r i t o o b s e r v e H a n l e e f f e c t s i g n a l s i s s u b s t a n t i a t e d . CHAPTER II THEORY .2.1 Introduction There is a classical and a quantum mechanical treatment of the theory. The classical treatment, I feel, is beneficial to the experi--nnenter in visualizing what is taking place at the time of the experiment. The quantum treatment can be used to obtain a theoretical value for -polarization. During the discussion of the quantum theory a description of the excitation mechanism will be given. Following this there will be a short description of the mode of excitation of the helium atom. 2.2 Classical Mechanical Theory of the Hanle Effect In order to discuss the Hanle effect classically one uses, as a model for the atom, an oscillating electric dipole which possesses a magnetic dipole momenty(£ , perpendicular to the dipole axis. If this dipole is set into oscillation along the x-axis with a frequency If i t will emit radiation in a direction perpendicular to the dipole axis. The electric vector of the emitted radiation will be parallel to the dipole axis (see Fig. 2). If one observes this :radiation-along the z-axis, the signal will appear polarized 1007o in the x direction. A Now, one places a magnetic field H = H K along the z-axis. If we •^ assume yu. to be of the form /u, = Y L, where L is the angular momentum, ithe magnetic field puts a torque on the dipole, and causes i t to precess about the z-axis with an angular frequency LJ = Now, the angle to - 5 -< E L E C T R O N B E A M : X O S C 1 L L A T 1 N G M A G N E T I C D I P O L E - M A G N E T I C F I E L D : ; D E T E C T O R rcjdiation with to X - ax i s FIG. 2--Classical Radiating Dipole the z-axis of the E vector of the emitted radiation will be: © (t) = LJ t = A H t (1) The amplitude of the emitted radiation is radiation damped, with a time constant 2 % , where Y represents the mean li f e of the upper state involved. Thus the observer, looking along the z-axis, will see time dependent electric field components of the form: - i ( v t + 5 ) _ t / 2 (t) = A 0 cos(u/t)e Ay (t) = A 0 s i n ( u / t ) e - i ( V t + n e'"^ or the intensity of radiation with electric vectors along the x- and y-axis is: (2) - 6 -I* ( t ) = A 0 cos2(uy t) e" t y i c 2 -t/r ( 3 ) I Y (t) = A 0 sin (u/t)e -Usually one observes, over a long period of time, T, in comparison to 2" . Now the intensities become: T oo Ix = ^ I x ( t ) d t = ^ I (t)dt 0 0 T oo I y = ^ I y(t)dt = ^ I (t)dt (4) The polarization "P" is defined to be: Ix - Iy •' P = _* £ ( 5 ) I*. + Iy Substituting Equation 4 into Equation 5: ^ (cos^U/t - sin^u/t) e t^ T dt . C -tl-c ^ J e dt v • 0 performing the integration and substituting Cc/= /. H in Equation 6 we find P = 1 (7) 1 + (2/6lIt ) 2 If-one plots "P" as a function of H, a curve of the form shown in Fig. 3 is obtained. -The lifetime, TJ" , can be found by noticing that when p o y 1 P(0) i s given by: - 7 -P O L A R I Z A T I O N • -- M A G N E T I C F I E L D FIG. 3--Typical Polarization Curve In the case of an atom, one must evaluate the magnitude of the .magnetic moment .ju. = gju , where ^ is the Bohr magniton: In the preceding description i t is implied that the only depolar-ization can also be caused by a collision of one atom with its neighbour. Another constraint on this theory is the time of the excitation -process. That is, i t must be short in comparison to f . Our excitation -14 ^process by electron collision is of the order of 10 seconds. 2.3 Quantum-Mechanical Description of the Hanle Effect -In 1924 Hanle observed the depolarization of polarized light by a (3) magnetic field. In 1933 Breit derived a quantum mechanical expression -awhich described the Hanle effect. Then in 1959 Franken^"^ rederived - 8 -i t for level crossing experiments (zero field level crossing is called the Hanle effect). Breit was puzzled with the lack of -completeness in the mathematical description for the following situation: One considers a four-level system, with ground states |a^ and excited states |b^  , |c^ , and |d^ . (See Fig. 4.) Let states | b/ and |c^ be Zeeman sublevels of one-state, then the separation between these states will be a function of the magnetic field. Now there are three cases which have to be explained; i) completely separate states | and 1 c^ i i ) completely overlapping states |b^ and |c^ "- " i i i ) an intermediate situation between case (i) and " ( i i ) . FIG. 4--Four Level System Before 1933, cases (i) and (ii) were completely explained mathe-matically. But, there was no expression for case ( i i i ) . Breit explained the third case. Below, I will give the expression for a l l three cases. F(fg) is the radiation emitted when light of polarization f_ excites state |a/ to states |b) and |c) and then decays down to state by emitting light of polarization j*. 2 2 case (i) R(f g) = | f a b gbd | +| f a c g^] (9) case (ii) R(fg) = j f g + f g | 2 (10) 1 ab bd ac cd' -case (iii) R(fg) = c £ f i a £ a i §id §di (11) i j 1 - i T(Ei - Ej) -B The expression for case ( i i i ) is called the Breit formula, where j and i refer to the Zeeman sublevels of the excited state. E^ is the energy th of the i excited substate. f±a and gi<j are dipole transition-matrix elements between the states in question, given by: f. = <oC, J, i l £ * . r |cC', J - l , a> l a g i d = <cC> J> M £ ' £ J~l> d> where r is defined to be a vector along the dipole axis. The above expressions for f^ a and g^a are for dipole transitions which stipulate thait .A J = In this experiment electron excitation was used to excite the helium atom from a ' s state ( J = 0) to a 'D state ( J = 2). "Thus the expression for .f^ cannot be used and for the time being, will be replaced with Q a^ which will be dealt with later. For a large magnetic field, E^ - E c ^ ^ 1, the expression for R(fg) :reduces to the same expression given for case (i) earlier. For a zero -magnetic field E^ - E £ = 0 and R(fg) reduces to case (ii) as is expected. Now, from the expression for R(fg) we can see that for a moderate magnetic field, the factor pL - i T(E^ - Ej)/fQ ^remains in some of the terms. - 10 -Hence R(fg) is dependent on both t and the magnetic field. Therefore, by controlling the magnetic field one is capable of solving for T . In order to adapt the Breit formula for this experiment, i t is broken up into two parts; one dependent and the other independent of the magnetic field. The independent part is segregated by noticing that in terms when i = j; E^ - E^ = 0, the respective terms are independent of the magnetic field. These terms are represented by R Q; R = T Q. 2 g. 2 (12) o ia °id i Rewriting Equation (11) in the two parts as stated above and setting: Q i a «aj hd hi ' A ( a l j d ) ( 1 3 )  R < « - E o + 5 . 1 -Ai%}f. - <14> 1 f J 1 J R ( f g ) " R o + . 1 - l t ( E . A* . ^  ^ -(E. - E.)+i 1 + i t ( E . - E.) -h i> J i J i J , i T(E. - E.) -, R(fg) = R + E 9 / (A+A*) + 1 ^— (A-A*) I (15) ° i > j 1 + T 2 (E. - E.) 2 L • fc J It can be shown that A =Q unless |m^  -mjj = 2, and that A has the form A = A Q exp(2i(j)) where (J) is the angle between the x and y axes of the atom's coordinate system and the § and 5 axes of the observer's coordinate system. Assuming that the levels exhibit a linear Zeeman effect in a magnetic field, then: •hgeH/U, m. E. = E, + i J 2mc •hgeH/A hence; E. - E = (16) i j mc Substituting Equation (16) into Equation (15) and simplifying, 2 A R(fg<t» = R + Z 2 — 2 C c o s 2(t>- • S £^ £ s i n 2<bl (17) The polarization, with respect to the axis is defined to be: P = R(<t>) - R«t> + 7T/2) R«D) + R(d> + Substituting Equation (17) into Equation (18) and simplifying, one gets: P * " Z^T? ' f 2 *, («»s 2* - s i n 2+ ) (19) 1 • ° »/, P P = 2 o * (cos 2<*> - g e H / r sin 2<*>) (20) mc i + ( M I ) mc 2 -where P = — Z A (aijd) o R o J o . . . 1 > J Comparing Equation (12) with the expression for the polarization derived classically (Section 2.2), one can see by setting 4> = O , (this was -done automatically in the classical explanation in order to simplify it) the two expressions are the same, except for a difference in the defini-tion of P . 2.4 The Excitation Matrix Elements In order to apply Equation (12), we must find an expression for the perturbation matrix elements Q. . Even though laws governing 13. electron atom collisions at non-relativistic energies are completely known, the calculation of low energy scattering cross-sections is extremely complex and has only been attempted for the most simple atomic cases. Referring to Equation (12), i t should be noted that as long as 0 and is independent of the magnetic field i t will have no effect -12-on the shape, of the curve. Since i t was experimentally observed that P S&O there was no need to calculate Q. . o *r* x i a 2.5 The Helium Atom A diagram of the pertinent levels of the helium atom is given in 3 Fig. 5. The S state normally has a relatively large lifetime because dipole transition from a triplet to a singlet state in helium is forbidden. Thus after a period of time this state becomes populated. This could make the mode of excitation as is shown by the dotted lines in Fig. 5. But there are the following complications which make the previous mode unlikely. Firstly, since the atom in the triplet state frequently collides with the walls of the discharge cell, its lifetime is shortened. And secondly, since the electron density is small, there is a relatively long period of time between collisions of atoms and electrons. Thus 3 the population of the S state is considerably smaller than expected. From the above argument, the transition ''"S - (full line in Fig. 5) would be more probable. A suitable explanation would say that a percentage of the atoms are excited to the '"D states directly from the ground state, 3 and a percentage is excited via the metastable S state. The proportion depends on the density of electrons with energies of 20 ev. in the discharge cell. FIG. 5--Energy Levels of Helium Atom - 14 -CHAPTER III EXPERIMENTAL DETAILS 3.1 Experimental Arrangement^^ The experimental system required to detect the Hanle effect described earlier, is composed of four main parts; 1) a cell containing a radio frequency excited discharge, 2) a homogeneous variable magnetic field in the discharge region, 3) an optical f i l t e r which passes only the spectral line of interest, and 4) a device for detecting and measuring polarization. A schematic diagram of the apparatus is shown in Fig. 6. The discharge cell is placed between a pair of capacitor plates which lie parallel to the y-z plane. These plates are connected to a power supply which produces a radio-frequency electric field of the form E = (E QCOSU/ tYx, thus causing free electrons to oscillate in the X^ direction. A homogeneous magnetic field is produced by two sets of Helmholtz coils. One is used to cancel the earth's magnetic field in the y direction, and the other is used to produce a variable magnetic field A H = Hk in the z direction. The light travelling in the z direction is passed through a monochromator which is set to transmit the wavelength of interest. This radiation is detected by using a photomultiplier, the output of which is fed into the signal channel of a lock-in amplifier. To detect the polarization of the signal, a rotating quarter-wave plate was used. Before entering the monochromator, the signal passes through a rotating quarter-wave plate, and a fixed polaroid with its X - Y R E C O R D E R FIG. 6--Apparatus - 16 -axis in the y direction. The polarized component of the light is thus modulated at four times the rotational frequency of the quarter-wave plate. This rotational frequency provides a reference signal for the lock-in amplifier (see Section 3.5). The output of the phase-sensitive detector now shows the degree of polarization. The output of the lock-in amplifier is connected to the y-channel of the x-y recorder. The x channel is controlled by the voltage across the A 'Helmholtz coils producing a field H = Hk. These-coils are driven by a .power supply and an amplifier whose output is slowly swept to produce fields of the form H = H + 0 t where H Z - 15 gauss and H Z + 15 o o max gauss. .3.2 The Discharge The discharge apparatus consisted of a cylindrical pyrex discharge cell 5 cm in diameter and 2.5 - 3 cm in length. The cell was connected to the vacuum system by a 1 cm O.D. pyrex tube as shown in Fig. 9. The ends of the discharge cell are slightly convex in shape to provide, mechanical strength.against air pressure. A circular brass plate of 7 cm in diameter is placed at either end so that i t extends about 1 cm beyond the edges of the cell. The plates were parallel to the z-y plane. The output of a radio frequency transmitter is coupled to these plates in a manner described in Section 3.7. The cell and plates are shown f u l l scale in Fig. 7. For convenience, ;the coordinate axes used throughout the discussion are also shown. ;The discharge is assumed to consist of a dilute gas of neutral atoms and a much smaller number of free electrons and ions. The motion of the electrons in a radio-frequency electric field, assuming a complete - 17 -FIG. 7--Cross-section of Discharge Cell and Capacitor Plates absence of collisions and a zero magnetic field, is governed by: mx = eE cos u/1 o eE X ~ — COS ti/t mu/ and the kinetic energy K.E. is given by 1 - 2 1 , Qx2 . 2 K.E. = ^ mx = — (rzjn sin u/t The application of a small magnetic field H = Hk yields: mx = eE cos u/t - eHy o J my - eHx Solving these equations: e Eo ' x(u>t) = - (— -) cos(wt) m U / 2 . y(<^t) = ^- x(u/t + -7T/2) - 18 -where ^ = — m This describes an electron moving in an ellipse whose ratio of major to minor diameter is ~ . For our case — (see Table I). In order for the above description to meet the conditions of the discharge cell, three conditions must be satisfied; 1) The mean free path of the electron should exceed the amplitude of its motion. 2) Collisions of electrons with the walls of the discharge cell must also be rare events. This merely means that any dimension of the discharge cell ought to exceed considerably, the amplitude of the electron's motion. 3) Electrons must be sufficiently energetic to sustain the discharge; that is, they must have sufficient energy to ionize the occasional molecule in order to make up for electrons lost by recombination. Table I lists some of the properties of the motion of the electrons in an oscillating electric field. TABLE I.--PROPERTIES OF ELECTRON MOTION IN AN R.F. ELECTRIC D.C. MAGNETIC FIELD _ „ . Minor Axis to E Required . .. . „ , - ... „ 2 , Amplitude of Major Axis Ratio Frequency ^ To P r°duce Electron Motion in 10 Gauss (*10 Hz) 'max '_ (Major Diameter) Magnetic Field 200 200 V/cm 2 mm .035 500 500 V/cm 0.8 mm .014 In order to see how well the f i r s t condition is satisfied we com-pute the mean free path, L, between molecule of cross-section , dis-- 19 -tributed with a density p ; L = ^  p. . Then at a density of 6.68 10 /cm and assuming a cross-section of 10A; L — 1 cm. From these figures we see that the condition is fairly well satisfied. By consulting Table I, one can see that condition (2) is easily satisfied for the discharge cell that was used. Condition 3 is satisfied when the pressure in the cell is greater than .060 mmHg. (This value was found experimentally.) 3.3 The Optical System The light from the discharge passes through a pair of plano-convex lenses , one of which is placed at its focal length away from the discharge, the other is placed at its focal length away from the entrance s l i t of the monochromator. A rotating quarter-wave plate is placed in the parallel beam of light between the two lenses (see Fig. 6). The aperture of the quarter-wave plate is large enough not to constitute a "stop" in the optical system (see Fig. 9). In front of the s l i t of the monochromator a piece of polaroid was placed perpendicular to the axis • of observation. Observations were performed at wave lengths of 4387A, 4922A, and 6678A. The monochromator, an F/8 Spex, one metre instrument had, when used in second order, a dispersion of 6A/mm, and a resolution of better than 0.IA. Since there were no spectral lines within 30A of the above mentioned lines, i t was possible to use slitwidths of 0.3 mm yielding a resolution of Oz2A. The rotating quarter-wave plate followed by a polaroid with its axis of acceptance in the y direction, was used to detect the polarization of the light, instead of what seems more straightforward, a rotating polaroid. The former method was used because the efficiency of the monochromator was found to depend upon the polar-- 20 -ization of the emitted light. By placing the polaroid in front of the entrance s l i t of the monochromator, the effect of the monochromator was cancelled. A detailed calculation in Appendix (1) will show that when a signal is emitted from the discharge, which is polarized at an angle © to the x-axis, the system explained above produces a signal of the form: where is the frequency of the rotating quarter-wave plate. 3.4 Lock-In Amplifier The lock-in amplifier consists of a tuned pre-amplifier with a Q of about 10 and a phase sensitive detector. The lock-in amplifier used in this experiment is a Princeton Applied research Model 120, which has 4 + a linearity of 1°L and a gain of 10 . The output is D.C. -10V at fu l l scale. In the mode i t was used, the reference signal triggers an internal sinusoidal reference signal. In this experiment a three-second time constant was used throughout. the output signal of the lock-in amplifier is proportional to the polarization, P, of the emitted light referred to the x-axis. With an input signal V(t) an output signal, S, is of the form: 2 With the chopped input signal from the photomultiplier, 2 7T U/ T T/27TU/ 2(n + 1)7T n = 1 2n where: d> is the selected phase angle V is the tuned frequency T is the time constant - 21 -So long as T is sufficiently large, incoherent signals will average to zero. Substituting in the expression for V(t), determined earlier as the signal from the photomultipler (see Section 3.3): .S oC 0 2 or 1 + cos4 wt J cos 2© +-| sin4 wt sin 2& - l^cos(-vt - <j>)d(vt) where UJ is the frequency of rotation of the quarter-wave plate. From the explanation in Section 3.5 i t can be seen that V = 4^ , (i.e. four cycles are gone through for each revolution of the quarter-wave plate). By setting <p - 0 27T 2 „ _ x , , cos 4 cut cos 20 , cos4u/t cos 20 S cC \-{-cos4ut + + \ -J  >\7> ,-^ sin4 u>t cos4"-'t sin 2© , , + )• d(4cu»t) 2 2 S oC cos 2 © = cos & - sin & = P Thus the output signal, S, of the phase sensitive detector is proportion to the polarization, P, of the emitted light. -.3.5 Rotating Quarter-Wave Plate and Reference Signal The rotator is shown in Fig. 8. A quarter-wave plate is glued to a two-inch I.D. brass pipe which is fitted tightly inside large ball-bearings. A sewing machine belt is laid over the pipe and the motor pulley rotates the quarter-wave plate. The motor, model CH3GRH, Universal Electric Company, runs at 1050 r.p.m. and delivers .05 H.P. The motor pulley has a 1.2 inch diameter. Thus the quarter-wave plate is rotated at approximately 10 cycles per second. The revolving quarter-wave plate also furnishes the reference FIG. 8--Quarter-Wave Plate Rotator - 23 -signal; painted sections were symmetrically placed around the perimeter of the quarter-wave plate in order to interrupt a light signal falling on a photodiode at a frequency 4W (see Fig. 8). 3.6 Vacuum System The vacuum system is shown in Fig. 9. It is a conventional glass system pumped by a "Cenco-Hyvac 14" mechanical pump which is capable -4 of reducing the pressure in the system to less than 5 x 10 mm Hg when a liquid nitrogen cold trap was used to prevent backstrearning of pump -.oil. Helium gas was leaked into the system through a needle valve. The leakage rate established the equilibrium pressure of the system. Starting from the reduction valve on the pressure bottle, helium at — 2 pounds per square inch above atmospheric pressure was passed through a flexible rubber hose to a cold trap f i l l e d half-way with activated charcoal. This cold trap was utilized to purify the commercial ^helium that was used. From there, the gas proceeds through some more flexible rubber hose to the needle valve and from there, at low pressure, to the discharge cell. About halfway between the discharge cell and the -needle valve, a Pirani gauge was attached. Just above the discharge cell a McLeod gauge was attached. Both of these were connected to the system via glass stopcocks. Proceeding downstream from the discharge cell another cold trap was encountered before the gas was pumped out by the mechanical pump. Al l connecting glass tubing has an inside diameter of 7 mm. Some flexibility in the position of the discharge cell was pro-vided by the ground glass swivel joints, by which i t was connected to the vacuum system. ..When data was being taken, the pressure was measured with the N E E D L E V A L V E M C L E O D G A U G E T O C H A R C O A L T R A P P I R A N I G A U G E \ r D I S C H A R G E C E L L T O P U M P V J C O L D T R A P FIG. 9--Vacuum System McLeod gauge before and after each run. i t was noticed that, for pressures less than .075 mm Hg, when the stopcock to the McLeod gauge was opened the intensity of the discharge changed in appearance to the naked eye. Thus the stopcock was opened only while pressure readings were taken, then i t was closed and we waited about five minutes until the discharge returned to a steady state before the run was taken. The Pirani gauge was used as a check for pressure fluctuations. Runs were taken for various pressures between .300 and .050 mm Hg. The pressure readings were accurate to ^7%. The discharge was periodically observed with a small hand spectro-graph in order to check the purity of the spectrum. 3.7 R.F. Supplies and Coupling The source was a 450 Mz transmitter.. It consisted of an R. C.A. M.I. -17436-1 transmitter used as a "driver" and a Canadian Marconi, Model . 163-107 high frequency power amplifier. With an output impedance of 50jtl. an output power of approximately 50w was transmitted. The R.F. voltage available is 50 V. Since we needed a field of order, 400 V/cm. between the discharge capacitor plates, and the plates were spaced at 3 cm, then 1200 V is required. Voltages of this order were obtained by the " P matched" resonant circuit shown in Fig. 10. A length of RG 8-U cable carried the power to this circuit. By adjustment of the contact point A, (see Fig. 10), an approximate impedance match between the resonant circuit and the transmission line may be obtained. The circuit is tuned to resonance by moving the cross-bars closer together or further apart, while keeping them equidistant from the capacitor plates. The advantage'of using such a high frequency source is that i t has a negligible effect on other parts of the apparatus such as the phase sensitive - 26 -R G 8 - U T T R A N S M I S S I O N L I N E f " ^ U . H . F C O N N E C T O R C A P A C I T O R P L A T E ' S r 5 C M TT (9 S L I D I N G C R O S S B A R FIG. 10— - Matched Resonant C i r c u i t - 27 -detector or the x-y recorder. 3.8 Helmholtz Coils The applied magnetic field in the x-direction was designed to be homogeneous in the area of the discharge cell to about 1 part per 4,000. In general, inhomogeneity near the centre of the Helmholtz coils are r 4 •of the.order (—) ; where "r" is the radius of the test area (discharge R .cell), and "R" is the radius of the coils. Therefore, Helmholtz coils with a diameter of 37 cm and a spacing of 18.5 cm were used in series. These coils each had 100 turns of #18 wire. The current delivered to these Helmholtz coils is supplied by a pair of "EICO 1064" power supplies and regulated by a D.C. power ^amplifier, whose schematic is shown in Fig. 11. The amplifier delivers up to 7A of either polarity into a 1J1 load with reasonable linearity; has a voltage gain of approximately unity and an input impedance of -approximately 10K.fi. . This enables the coils to give a magnetic field of -anywhere between T15 gauss. The sweep of the magnetic field is controlled by an R.C. circuit whose output is fed into a preamplifier before going into the circuit, as shown in Fig. 12. The time constant of the R.C. circuit is 20 minutes, thus by varying the gain of its preamplifier, the sweep time can be selected between 3 and 20 minutes. Most runs were done with a sweep time of 7 minutes. The calibration of these coils was done with a "Bell 240" Hall r-gaussmeter. The accuracy of this gaussmeter was tested by using i t to measure a magnetic field of — 1.5K gauss that was known to be accurate to 0.57o. The "Bell 240" Hall gaussmeter measured the strength of this VFIL.TR. 15-ft-—fA/W 2 M 1 1 3 2 ALL RESISTORS f WATT CAPACITORS 10 VOLT ro oo FIG. ll--Helmholtz Coil Power Supply 37K 10K, MFD -15V +4v A A A 5K 1M J O O K T V V V + 15 V FIG. 12--Pre-Amplifier and R.C. Circuit - J U -magnetic field to within 0 . 5 7 » of the known value. Since the specifica-tions for the gaussmeter stipulated that its scale could be changed with an error of less than 1 . 5 7 o , this gaussmeter was believed to be accurate to 2 7 » . In order to save calibrating the x-channel of the x-y recorder, the position of the pen was measured as a function of the field. With the recorder on the 2 V/in. scale, (which was always used in its calibration mode), the field in the discharge region was found to be 3 . 9 0 1" . 0 8 gauss per inch of the pen movement. The direction of observation which was also the direction of the applied magnetic field was align with the horizontal component of the earth's magnetic field. With this orientation the earth's magnetic field was cancelled by the applied magnetic field. The vertical component of the earth's magnetic field is cancelled by a set of smaller Helmholtz coils which were placed inside the pre-viously mentioned ones. These coils can be made smaller with the same absolute homogeneity in magnetic fields because one is dealing with a smaller magnetic field. These coils are 1 9 . 5 cm in diameter and 9 . 7 cm apart. Each coil has 5 0 turns of # 2 4 copper wire. The current for the above coils is provided by a small power supply delivering up to 1 V and 1 5 0 ma. Thus, when the 2 coils are placed in series the field produced at the centre can reach approximately .7 guass. Since the object of these coils was to make the vertical component, (y-axis), of the magnetic field zero, the calibration, of sorts, was done with a dip needle. That is, the dip needle was placed in the position of the discharge cell, and the current was adjusted until the dip needle read zero magnetic field. Fields down to approximately . 0 1 gauss could - 31 -be measured this way. 3.9 Photomultiplier The photomultiplier is an E.M.I. 9558QB with an S-20, (NaKSbCs), surface and a quantum efficiency of —20%. It was operated with a cathode to anode potential of -1280 V. The dynode chain resistors were a l l 33K_fiL . The anode is connected to ground by a 100KXL resistor. The abbreviated circuit is shown below in Fig. 13. D, HAM/v 33 K D, D, anode 100K A A A / V V A A A A A ^ V - y v ^ A / V v V v 33K 33K 33K 6 V=-1280 FIG. 13--Photomultiplier Wiring System 3.10 X-Y Recorder The x-y recorder used was a Varian model F100 having a linearity of 17» and input impendance of 100K SL into each channel. 3.11 Data Processing When running the experiment the phase angle c£ w a s difficult to set accurately to zero. Also, the polarization signal was not noise-free. - 32 -Thus, to obtain a more accurate value of the halfwidth, H, , the graphs plotted by the x-y recorder were subject to numerical processing. Relative values of the polarization were read from the graphs at 0.2 inch intervals to provide 33 data points. The points were then fitted by a function of the form: P = C, + —T~ o [cos 29 - (H-H )Sin 2e| 1 1 + - L . ( H- H ) I \ o J where C^ , H^ , Ho, and Q are parameters fitted by a computer "least" squares" fitting routine, (U.B.C. L.Q.F.). From these fitted curves, the H^  for about 7 different pressures between 0.050 and 0.300 mm Hg were obtained for each transition. These values of H, were plotted as a function of pressure and extrapolated to zero pressure. From these extrapolated values the radiative lifetimes of the 3^ D, 4^ D, and 5^ D levels were obtained. 3.12 Special Optical System For the 3*D - 2^ P spectral line at 6678A, i t was found that, by scanning the spectrum, there were no neighbouring lines between 6100A and 7010A. Thus, i t was possible to replace the monochromator by an Optics Technology Inc. interference f i l t e r ; set no. 3099, f i l t e r no. 666 (see Fig. 14). The specifications on this f i l t e r indicated a pass-band of 6700A - 200A. A line which showed up at 7010A had no appreciable effect when viewed through the f i l t e r . Since this f i l t e r had no polar-izing effects, i t was possible to measure the absolute amount of polar-ization, and the steadiness of the overall signal as the magnetic field was swept. To measure the percentage of polarization as a function of f i l t e r polaroid ^ l ^ i RM. FIG. 14--Special Optical System. the magnetic field, the rotating quarter-wave plate was removed. The light was transmitted, in turn, through a polaroid, the f i l t e r , and then into the photomultiplier. The signal from the photomultiplier was then fed directly into the x-y recorder, bypassing the phase sensitive detector. This procedure was repeated, except with the polaroid removed, to measure how the entire signal varied with the magnetic field strength. This method was used only to measure the polarization and stability of the light emitted from the discharge. The previous optical system was used to measure the lifetime of the 3^ D state of helium. - 34 -CHAPTER IV RESULTS 4.1 L i f e t i m e s In t h i s experiment the l i f e t i m e s of three energy l e v e l s were measured. As mentioned p r e v i o u s l y they were the 3D, 4D, and 5D s i E g l e r s t a t e s of the helium atom. I n F i g . 15 i s shown a t y p i c a l exaerimeniT.gl p l o t of the 4^D p o l a r i z a t i o n curve. I n F i g . 16 i s i t s " l e a s t square:s" f i t t e d curve. The h a l f w i d t h , H j , was measured f o r Hanle s i g n a l s a t s e v e r a l pressures. Graphs of Hj p l o t t e d as a f u n c t i o n of pressure a r e showm i n F i g s . 17, 18 and 19 f o r the 5 1D, 4*D and 3^"D l e v e l s r e s p e c t i v e l y - T i e h a l f w i d t h of a l i n e a t zero pressure was obtained by e x t r a p o l a t i o n - Tb= upper s t a t e l i f e t i m e s , , were computed from the f o l l o w i n g e q u a t i o n : The l i f e t i m e s obtained f o r these l e v e l s are l i s t e d i n Table LI, III, P.T\? IV along w i t h t h e i r r e s p e c t i v e l i f e t i m e s obtained by v a r i o u s other- "workers. I l l Besides the three s t a t e s mentioned above, the 6 D, 7 8 D s u a 9*D s t a t e s were attempted through the t r a n s i t i o n s n^D - 2^?. But tnha s i g n a l to noise r a t i o was so small f o r these t r a n s i t i o n s , tha.t no w c r i a -w h i l e data was recorded. 4.2 C r o s s - s e c t i o n s The l i f e t i m e of a c e r t a i n l e v e l i s shortened as a resmtt of csIIiL-s i ons of an atom w i t h i t s neighbour. The number of c o l l i s i o n s p e r secrad. Pola r i za t ion • Magnet ic field (gauss) 1 1 1 1 ; 1 1 1 1 i -15 -12 - 9 , -6. • -3 0 3 6 9 12 15 FIG. 15--Experimental Level-Crossing Curve for the 4^ D Line j M A G N E T I C FIELD -10.0 - 8 0 -6.0 -4.0 -2.0 0.0 20 4.0 6 -0 8.0 10.0 GAUSS FIG. 16--Leaat Squares F i t t e d Curve for the 4 D L ine halfwid th (gauss) 5 0 100 150 2 0 0 2 5 0 300 p ressu re (x10 mm. Hg.) FIG. 17--Plot of Hx as a Function of Pressure of 5^ D State Hal fwid th j (gauss) FIG. 18--Plot of H. as a Function of Pressure of 4iD State FIG. 19--Plot of H, as a Function of Pressure of the 3D State 4 U -TABLE II.--LIFETIME OF THE 51D STATE OF HELIUM - 8 (xlO sec) Author Technique Sate 6.6 ±.4 I. M a r t i s o n e t a l . ^ " ^ Beam f o i l 1969 4.91+.2 (14) Descomps et a l . Magnetic Resonance 1960 4.9 +.5 (13) J. P. Descoubes Level Crossing 1967 7.9 +.6 Pendleton & Hughes Di r e c t observation of decay 1965 4.6 +.3 (9) Kindleman & Bennett Delayed coincidence 1963 6.3 +.9 B r i d g e t t & K i n g ^ 1 0 ^ D i r e c t observation of decay 1967 4.6 +.3 Osherovich & Verolainien^'''''"^ Delayed coincidence 1968 7.16 (12) Wiese et al. T h e o r e t i c a l 1965 4.48+.4 Ours Level Crossing - 41 -TABLE III.--LIFETIME OF THE 4 D STATE OF HELIUM - 8 (xlO sec) Author Technique Date 3.8 ±.3 I. Martison et a l . ^ ~ ^ Beam f o i l 1969 3.91±.2 (14) Descomps et al. Magnetic Resonance 1960 4.1 ±.5 (13) J. P. Descoubes Level Crossing 1967 4.7 3.5 ( Q\ Pendleton & Hughes Direct observation of decay 1965 3.0 +5 (9) Kindleman & Bennett Delayed coincidence 1963 3.5 ±.4 Fowler et al. ' Direct observation of decay 1964 3.9 ±.5 Bridgett & King^ 1 0^ Direct observation of decay 1967 3.8 +.5 Allen et a l / 1 7 * Direct observation of decay 1969 3.66 (12) Wiese et al. Theoretical 1965 3.97+.4 Ours , Level Crossing - 42 -TABLE IV--LIFETIME OF THE 3 D STATE OF HELIUM - 8 (xlO sec )Author Technique Date 1.2 +.3. (13) J. P. Descoubes Level Crossing 1967 1.6 +.2 Pendleton & Hughes Direct observation of decay 1965 1.6 +.4 (9) Kindleman & Bennett Delayed coincidence 1963 1.8 +.5 Fowler et a l / 1 6 ^ Direct observation of decay 1964 1.55+.5 Allen et a l / 1 7 ^ Direct observation of decay 1969 1.6 +.1 Osherovich & Verolaimen^^ Delayed coincidence 1968 1.5 (12) Wiese et al. Theoretical 1965 2.71+.3 Ours Level Crossing -44- ' i s proportional to the pressure. Therefore, from the slope of the H, versus pressure graphs i n F i g s . 17, 18 and 19, one can c a l c u l a t e reasonable values f o r the cross-sections of the helium atom i n various energy l e v e l s . From Appendix 2 the cross-section, , i s : 1 -12 2 CT = T (slope of H, curve) (2.59 x 10 )cm where / V i s the mean v e l o c i t y of the atoms (assume that -Jc i s independent of CT ) and i s the density of the atoms at a pressure, F^, which i s the pressure i t would take to d i v i d e the l i f e t i m e i n h a l f . The cross-sections of the helium atom i n the 3^D, 4^D, and 5^D states are l i s t e d i n Table V and p l o t t e d as a function of the square of the quantum number n i n F i g . 20. TABLE V.--CROSS-SECTIONS Atomic State n Cross-section 266 A 366 A 440 k 4.3 S t a b i l i t y and P o l a r i z a t i o n of the Discharge I t was important to know i f the amplitude of the s i g n a l from the d i s -charge varied with the magnetic f i e l d . This was tested by looking at the 3^D - 2^P t r a n s i t i o n as discussed i n Section 3.12 ( i . e . using an interference f i l t e r instead of the monochrometer). Below i s shown a trace from the x-y recorder, of the amplitude of the emitted l i g h t as a function of the magnetic f i e l d ( F i g . 21). As we assumed i n the theory, i t was unaffected by the magnetic f i e l d to w i t h i n 17D. Another i n t e r e s t i n g feature was investigated. This was the percentage of p o l a r i z a t i o n of the s i g n a l . Again, t h i s was done by observing 3 ^ 5 XD 15 M A G N E T I C F I E L D 15 z —\ m z: in H -< i FIG. 21—Plot of Intensity as a Function of Magnetic Field 5 G <— MAGNETIC FIELD z —i O x 1 + 15 G. — > pressu re =0.150 m m FIG. 22--Polarizaticm Plotted as a Function of Magnetic Field the 3 D - 2 P transition line through an interference f i l t e r . As one can see from Fig. 2 2 , at a pressure of .095 mm Hg, the polarization is about 37o, using the phase sensitive detection system we were able to obtain a large signal to noise ratio for pressures between .060 mm Hg and .250 mm Hg. . The minimum pressure of helium at which the discharge would con-tinually run was .060 mm Hg. At .050 mm Hg the discharge could be started with a Tesla coil, but the intensity would decrease to zero within 30 seconds. I assumed that the electron source was helium atoms. Thus at lower pressures the discharge did not function because of too low a density of electrons. The above mentioned use of the Tesla coil strengthens this argument. At .060 mm Hg the discharge would run proficiently only when the stopcock to the McLeod gauge was left open. (It was assumed that mercury vapour was present.) Within about 3 minutes after this stopcock was closed, the amplitude of the intensity and polarization of the-light -signal decreased by a factor of 5. Also, the width of the curve became broader. Below, in Fig. 2 3 , are shewn two polarization curves; one with •the stopcock opened and mercury vapour present and the other with no mercury present. Bothhave a total pressure of .060 mm Hg. This change in intensity can be explained by a change in the electron density. Since the ionization energy of mercury is less than helium, a discharge •of a mixture of mercury-and helium at .060 mm Hg will have a higher density of free electrons than a discharge of pure helium. The discrepancy of H, in Fig. 23 could have been caused by an unstable discharge. At .060 mm Hg of pure helium the discharge is more or less at its excitation threshold (i.e. unstable). Thus the magnetic 4 A 5 Q - 2 P P R E S S U R E : Q 0 6 0 m m H g ( M e r c u r y vapor present) p o l a r i z a t i o n 10 rr 5 D'- 2 P . PRESSURE-*0 .060mm Hg. ( P u r e hel ium) polarization magnetictield (gauss) 1 - i r 1 1 1 1 1 p -4 6,. .A magnet ic fielcr(gauss) FIG. 23--Fitted Curves of Runs at Low Pressures v - 49 -field could have had a large effect on the intensity of the emitted light which would have an effect on H, . k 4 . 4 Experimental Errors ' a) Magnetic Field The calibration of the magnetic field was done with a gaussmeter that was capable of measuring the magnetic field to an absolute accuracy of 0 .2 gauss. The linearity of the magnetic field calibration could be effected by; ohmic heating of the coils, hysteresis effects in nearby ferro-magnetic materials, and residual earth magnetic fields. To investigate this, the output of the gaussmeter was fed into the y-channel of the x-y recorder and the magnetic field, which was fed into the x-channel, was swept from - 1 5 gauss to +15 gauss. From this recording i t was observed that the magnetic field was linear to within 17». b) Pressure in the Discharge Due to the scale on the McLeod gauge, the accuracy to which the pressure could be measured was 77o. From Fig. 10, i t is noted that the pressure in the discharge cell could be different from that in the McLeod gauge. But these pressures are s t i l l proportional to each other. Therefore the extrapolated values of H, are correct, but the slope of the versus pressures curves could be out by 207o. This would effect only the calculation of the cross-sections. c) Temperature in the Discharge The temperature of the gas in the discharge affects only the cal-culation of the absolute value of the cross-sections. Since the relative values are unaffected, this error has no effect on the straight line for - -2 the plot of the cross-sections as a function of n in Fig. 20. The temperature of the discharge was assumed to be 300°K, the room temper-3~ o ature. The computed cross-section varies as T2, thus a "i"25 K error in the assumed temperature of the discharge will affect the results by only 47o. d) R.F. Broadening The radio frequency electric field, E, of the discharge is about 300 - 400 volts/cm. In general this will broaden the polarization curve. In a weak magnetic field (R.F. frequency -v»<«A - a Stark term is taken into account and the polarization curve then has the form: P P = T+ T z [c zE^ + (u/± -When one compares this expression for P with the one that was derived earlier for P, i t becomes apparent that the presence of the Stark polar-2 4 izing term, c E , will tend to broaden the polarization curve. In a / 1 Q \ paper by Lombardi this broadening was reported to be as much as 307o for low energy transitions. A shielding of the radio frequency electric field, E_, by free electrons in the discharge could possibly be a reason to disregard broadening due to Stark polarization in this experiment. As mentioned in Section 4.3, there was a broadening of H, at. low pressures (and hence low electron densities), which would be consistent with the free electron shielding hypothesis. Also the values of H, in this -experiment are on the whole, narrower than values measured previously by other workers. CHAPTER V CONCLUSIONS The accuracy of this method, as in most experiments, depends on the signal to noise ratio. Before the signal reaches the photomultiplier i t must pass through a quarter-wave plate and a monochromator which reduces the entire signal by about 50%. The signal which is being measured, is proportional to the percentage of polarization of the emitted light. As was shown in Section 4 . 3 this is only about 37o. Therefore, a criterion for performing this experiment is a strong light source. An improvement which would simplify the experimental procedure and increase the signal to noise ratio would be to replace the monochromator with an interference f i l t e r than has no polarizing effects. Excitation of an atom or molecule by an electron beam permits Hanle effect measurements of most states of atoms which have strong dipole transition to a lower state and possess Zeeman sublevels. An upper limit on the lifetimes that can be measured (how narrow Hx is in •2 F i g . A), depends on the linearity of the varying magnetic fields and the accuracy of the control of the earth's magnetic field. In this case -the upper limit is 10 7 sec. The lower limit .of lifetimes is affected by the size of the product of g % . If g t is large the polarization curve that fits the equation P P - — - 2 J w i l l be extremely broad. To measure broad curves a high signal to noise ratio is required. Thus with the light source used in this experiment a lower limit of 10 ^ sec is set for the product of gT. From the graphs in Figs. 17, 18 and 19, one can see that the experimental values of H, plotted as a function of pressure form a 12 convincing straight line. The straight line agreement of the graph in Fig. 20, which is a plot of the cross-sections as a function of the square of the quantum number, agrees with the classical theory of the atom. There is good agreement between the values we measured for the lifetimes of the 4^ D and 5^ D states and the values previously found, as can.be seen in Tables II and III. There is a noticeable discrepancy between our results and previous measurements of the lifetime of the 3^ D level. We used the same technique -to measure T , for a l l three levels, and the extrapolation shown in Fig. 19 seems reasonable. We are unable to suggest why our results and those of previous measurements for the 3^ D level are in such bad agree-ment. - 53 -APPENDIX I ROTATING QUARTER-WAVE PLATE The optical system described in Section 3.3 is schematically shown below in Fig. 24. FIG. 24--Optical Schematic A signal, polarized at an angle to x-axis, is transmitted from the discharge cell in the z-direction. The x and y components of the signal are: E x = A Q C O S © exp r i(kx-vt)J Ey = A Qsin© exp £ i(kx-vt)J (1) The polarization is defined to be: P = 2 2 E - E . . • 0 x y 2^ .2 — x 2 = C O S ^ - s i n © E - E x y - 54 This signal was then passed through the rotating quarter-wave plate. To express the outcome, I set up a rotating coordinate system. It was fixed to the quarter-wave plate, such that E|j was parallel to the slow axis and Ej_ was perpendicular to the slow axis. This coordinate system was rotating at a frequency u> . The E,, and Ej_ coordinates, expressed in terms of E and E , are: x y En = E cosfu/t +<$) + E sin(c*/t + <J>) " x y Ej_ = EyCOs(u/t +<p) - E x s i n ( c J t + where dp is an arbitrary phase angle. Setting ($> = 0 and expressing the signal after i t had passed through the quarter-wave plate in terms of E || and E_L . i E J ( = E ,| exp(i 6) i E_j_ = Ej_exp i(6" - 7r/4) = - i E_,_exp(i<S) Transforming into the stationary x-y coordinate system: i i i E x = cos(u;t)E|1 - s i n ^ l ^ E ^ • i i E y = sin ( L^t)E - cos^^E^ At this point the signal passes through a polaroid with its axis in the y direction. Now the above signal becomes: II . E = 0 x II i E = E y y . The polaroid was align with its axis of polarization parallel to the axis of polarization of the monochrometer. Therefore the signal which passed through the monochrometer and reached the photomultiplier was the same as the input signal to the monochrometer. The signal recorded by the photomultiplier is proportional to: |E J2 = |sin(u/t)E/, - i cos(u/t)EX (2) By substituting the above values for E ,, and Ej_ into Equation (2) and performing some algebraic operations, one gets: 2 e „ I 2 y | 2 = - f - [- 1 + [ C° S 4 2 + 1 ] cos 2* - \ s i i f u,$sin 2 e } (3) As is shown in Section 3.4, when this signal is analyzed by the phase sensitive detector the output signal L is: 2 Ao « T 2 • 2*1' L cC —j-g— cos © - sin©| Using the definition of polarization given earlier, i t is seen that the output of the phase sensitive detector is proportional to the polarization. APPENDIX II CROSS-SECTIONS The following assumptions were made about the discharge: i) The number of free electrons in the discharge cell is small compared to the number of neutral atoms, i i ) The number of excited atoms is small compared to the number of ground state atoms. From these assumptions, the lifetime of an excited state of an atom is shortened by collisions with atoms only in the ground level. If one considers a plot of H i . as a function pressure and remember that H a oC then the following properties are noted (see Fig. 25). FIG. 25--Sample Cross-Section Plot When the pressure is zero no transitions are caused by collisions. At pressure P = V^, T is \ of its value when P = 0. The number of * 1 transitions per sec, N = by definition. Therefore, the number of transitions, N, at P = P„ is twice the number of transitions, N* a t P = 0. Half of N is a result of the collision and half is a result of spontaneous radiation. Therefore, at a pressure P = T^y the number of * 1 _ collisions is M = N = -^r- but M =/vpo~. Therefore, the following expression can be derived for cr : <y = 1 = (Slope of H, curve) • (2.59 x 10_12cm2) where O" is the cross-section, p is the density at a pressure of P = P„, and / v i s the mean velocity of atoms. 1-- 58 -LIST OF REFERENCES 1 A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms, pp. 93-153 (London: Cambridge University Press, 1961). .2 R. L. DeZafra and W. Kirk, Amer. Journal Phys. 35, 573 (1967). 3 W. Hanle Zeits, f. Physik 30, 93 (1924). 4 G. H. Breit, Revs. Modern Phys. 5, 91 (1933). 5 F. D. Colgrove, P. A. Franken, R. R. Lewis and R. H. Sands, Phys. Rev. Letters 3, 420 (1959). . 6 J. Vanderlinde, unpublished thesis, University of British Columbia. 7 J. Vanderlinde, unpublished thesis, University of British Columbia. 8 W. R. Pendleton and F. H. Hughes, Phys. Rev. 138A, 683 (1965). 9 P. T. Kindleman and W. R. Bennett, Bull. Amer. Phys. Soc. 8, 87 (1963). 10 K. A. Bridgett and T. A. King, Proc. Phys. Soc. A92, 75 (London, 1967). 11 Oshervovich and Verolaimen, Opt. Spec. _24, 81 (1968). 12 W. L. Wiese, M. W. Smith and B. M. Glennon, Atomic Transition Probabilities, Vol. NSRDS-NBS4 (Washington, D.C: U. S. Government Printing Office, 1966). 13 J. P. Descoubes, Physics of the One- and Two-Electron Atom, loc. cit., p. 341. .14 B. Descomps, J. C. Pebay-Peyroula and J. Brossel, Compt. Rend. Acad. , Sci. 251, 941 (1961). 15 I. Martison, W. S. Bickel, J. Bromander, H. G. Berry, L. Lundin, R. Buchta, and I. Bergstrom, J. Opt. Soc. Amer. 60, 352 (1970). 16 R. G. Fowler, T. M. Holzberlein, C. H. Jacobson and S. J. B. Corrigan, Proc. Phys. Soc. A84, 539 (London, 1964). ; - 59 -LIST OF REFERENCES (cont'd.) 17 L. Allen, D. G. C. Jones, D. G. Schofield, Jour. Opt. Soc. Amer. _59, 842 (1969). 18 M. Lombardi, Compt. Rend. Acad. Science 266, 60 (Paris, 1968). 


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