University of Southern California Dissertations and Theses

A Theoretical Analysis Of The Rare Gas Autoionization Between The Doubletp(3/2) And Doublet P(1/2) Series Limits, With Applications To Argon

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A Theoretical Analysis Of The Rare Gas Autoionization Between The Doubletp(3/2) And Doublet P(1/2) Series Limits, With Applications To Argon

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Content This dissertation has been microfilmed exactly as received 6 8 -1 2 ,0 4 9 MENDEZ, Antonio Juan, 1938- A THEORETICAL ANALYSIS OF THE RARE GAS 2 AUTOIONIZATION BETWEEN THE P , AND 2 3/2 P SERIES LIMITS, WITH APPLICATIONS TO Ar. 1/2 U niversity of Southern California, Ph.D ., 1968 P h ysics, spectroscopy University Microfilms, Inc., Ann Arbor, Michigan A THEORETICAL ANALYSIS OP THE RARE GAS AUTOIONIZATION BETWEEN THE 2P3/2 AND 2p SERIES LIMITS, WITH APPLICATIONS TO Ar by Antonio Juan Mendez A Dissertation Presented to the FACULTY OP THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Physics) January 1968 UNIVERSITY O F S O U T H E R N CALIFORNIA T H E G RA D U A TE SC H O O L U N IV ER SIT Y PARK LO S A N G E L E S. C A L IFO R N IA 9 0 0 0 7 This dissertation, written by under the direction of /sis.—. Dissertation Com mittee, and approved by all its members, has been presented to and accepted by the Graduate School, in partial fulfillment of requirements for the degree of Antonio Juan Mendez D O C T O R O F P H I L O S O P H Y Dean Date Japuary,_1968 DISSERTATION COMMITTEE Chairman TABLE OF CONTENTS Page LIST OF TABLES................................... . iii LIST OF F I GURES.................................... iv Chapter I. INTRODUCTION . 1 II. THE ABSORPTION CROSS SECTIONS OF THE RARE GASES BETWEEN THEIR 2P3/2 AND 2Pt SERIES L I M I T S .................. 8 III. THE NOTION OF FANO'S "BACKGROUND" CROSS SECTION FOR AUTOIONIZATION ............. 24 IV. THE ANGULAR DISTRIBUTION OF PHOTOELECTRONS ........................... 36 V. THE FORMULAE OF AUTOIONIZATION........... 45 VI. APPLICATIONS TO A R G O N .................... 63 APPENDIX............................................. 98 LIST OF REFERENCES.................................... 12£ ii LIST OF TABLES Table 1. The Observed and Rydberg Formula 5 2 ' Calculated Ar 3p ( P^)nd Levels . . . • 2. Comparison of Experimental and Theoretical Absorption Cross Section D a t a ........................... .. 3. The Ar P. (H-F Ground State, Due to 3p Hartree-Hartree) . • ................ 4. The Ar PEd (STF, E = 0.001 Ry, from Scaled K+ T-F Potential) . . . . . . . . 5. The Ar P9d (STF, from Scaled K+ T-F Potential) ................................ 6. The Ar PEg (STF, E = 0.001 Ry, from Scaled K+ T-F Potential .................. 7. The Ar P-^g (STF, from Scaled K+ T-F Potential) LIST OF FIGURES i ‘ Figure Page ! I 1. Densitometer Traces of High Resolution Argon Absorption Spectra Taken by j : 1 : i Ogawa. ........... .. ..... . H 2. Krypton Absorption Coefficient 12 j 1 | 3. Xenon Absorption Coefficient ••••••• 13 j 4. Argon Absorption Cross Section Near the p ' ^2/2 Series Limit. . . . . . . . . . . 14 | /V N 5. j-j and j-1 Coupling X ^ Purities as j I Functions of RX/R2 29 1 j * i | J 6. a and c. as Functions of X Purity • 31 a b E 7. The Angular Distribution of Photoelectrons ........................... 40 8. The Ratio ( a + a.) I a D e = 0°/C a + (S.) a D 0=90° 44 9. The Autoionization Line Shape, Including ' the Influence of the Neighbour Levels . . 1 66 j | j 10. The Autoionization Line Shape, Including [ - t ! o ! the Influence of the Neighbour Levels . .2 67 ! 11. The Autoionization Line Shape, Including 1 i the Influence of the Neighbour Levels . .3 68 j iv i Figure P&9e j 12* The Autoionization Line Shape, Including ! the Influence of the Neighbour Levels • . . 4 69 13, The Autoionization Line Shape, Including the Influence of the Neighbour Levels . . . 5 70 14. The STF Discrete (9d) and Continuum (kd, E = 0.001 Ry) Radial Wave Functions .... 77 15. The STF Discrete (11s) and Continuum (ks, E = 0.001 Ry) Radial Wave Functions • • • • 78 16. The H-F Ground State and Excited State 3p Radial Wave Functions ......... ••••• 79 5 17. The Observed and Calculated 3p 9d Multiplet. 84 5 18. The Observed and Calculated 3p 11s Multiplet............... 85 t 19. The Theoretical 11s Line Shape. T = 4.6 cm-1, q = 4.4, o = 0.3Mb. . . . 89 S t I 20. The Theoretical Autoionized 9d Line Shape. a = 6.6 Mb, a. = 3.5 Mb, a 7 b T = 67 cm”1, q = 1.65 . . ........... 90 » 21. The Superposition of the Theoretical 9d and * 11s Autoionized Levels with the Observed' Level Separation . . . . • • • o . . . . . 92 CHAPTER I | | INTRODUCTION I The independent particle model is a convenient, land rather good, device for characterizing the energy [spectrum of an atomic system. Invariably, a single set of j principal and angular momentum quantum numbers (consistent jwith the Pauli principle), a "configuration," incorporated linto a product wave function is sufficient to describe thej observed level structure of an atomic system. This is i amply demonstrated in the extensive use and usefulness of ; the spectroscopic notation based on the independent ;particle model. j But there are exceptions where a proper descrip- I !tion of the observed spectrum requires more than one con- I figuration. A particular type of exception occurs at j energies above the first ionization potential. At these j energies, electronic configurations can be devised which j alternately describe the atomic system as having been ! 2 |constructed by adding an electron to the ground state of j |the (parent) ion or adding an electron to an excited j state of the (parent) ion. If both of these configura- |tions have the same symmetries, then at these energies the I j added (emission) electron will in practice have some i jsemblance of a bound and some semblance of an unbound | i i !electron. These dichotomous states of an atomic system ! I I i ; ; are said to give rise to "autoionization,” and they mani- j ;fest themselves particularly strongly in absorption J spectra (1-4) and in scattering cross sections. (5-7) ! I | !Similar states are responsible (11) for the Auger effect and for cascading events following an inner electron ! excitation in an atomic system. j i , i ! Autoionizing states are found also in molecular j systems (where they are sometimes ascribed (8) to npre- j ionization”), but for the sake of argument only the atomic! | type will be discussed. j Where more than one configuration can describe an ; atomic system in a given energy range, the method of con- j figuration mixing or configuration interaction (9) can be I used to find an improved description of the atomic system,! i j keeping in this way some of the conceptual advantages of j | the independent particle model. According to this method,! in energy ranges where autoionization is possible, a good | wave-mechanical description can be had by taking an ad mixture of the bound electron configuration (the Indiscrete” state) and a band of unbound electron con- i ;figurations (the ''continuum” states). The bound- and i I unboundedness is referred to a particular ion. The I appropriate admixture results from considering any terms j j ;in the Hamiltonian that have been neglected in arriving j I at the independent particle description. I ; i i Autoionization can be described as a radiationlessj : | transition: the discrete state relaxes by emitting an electron. (11) Then it is the discrete state which is to |be called the ”autoionizing state.” i ! There is a formal apparatus available for decoup- ! t : i .ling the discrete and continuum type solutions of a given j j i system and its complete Hamiltonian. This technique uses j i the Peshbach projection operator, (45-52) an operator I ■ i 1 which defines the discrete states as those solutions | which vanish at large distances from the center of the system. So far a representation of this operator is known; i only for two electron systems. (47, 52) But the Peshbachi ; i projection operator formalism does lead to a more or less t j rigorous manner of describing the autoionizing (discrete) states and the continuum states to which these relax or 1 I 1 I | decay. (52) j The Feshbach projection operator formalism gener- ; ates Hamiltonians for the discrete and continuum functions j as well as a coupling Hamiltonian. Since these modified ! Hamiltonians are not generally of the form used in the traditional study of the quantum theory of atomic struc- I |ture, the solutions (usually obtained by a variational ;principle) (51) are not necessarily recognizable as "con- j I figurations i Relying on the independent particle model, the attitude taken in the configuration mixing approach to the! : j jproblem is that an understanding of the atomic structure, I as represented by the normal bound state energy levels, j can give a very good initial solution. The autoionizationj is ascribed to a degeneracy involving a configuration I i j identified as one giving rise to bound states and a con- ! ;figuration giving rise to unbound states (the configura- ' I i tions have different parent ions). The derivations | iinvolved in this approach emphasize the diagonalization ofj the Hamiltonian, with the solution formed from an admix- j ture of the two types of configurations. i j Prom this viewpoint it can be said that the spectral evidence of autoionization is a property of the admixture, and thus the admixture can be referred to as j the "autoionized state" or "autoionized level." j The configuration mixing theory applied to auto- ; i j ionization (10) leads to certain indices useful in describing the autoionizing states. One of these indices measures a shift in the energy associated with the "center" of the autoionizing state, the shift being measured from the position of the independent particle |discrete state. This index is the "level shift.” Another i j index measures the line width (more properly, a transition probability) of the autoionizing state. ! | | | | A third index is related to the shape of the | j | I transition probability from an initial state to the auto- ! ionized level under some operator as a function of energy. ;A fourth index appears in the theory if more than one | : t ■ i icontinuum state type is included in the configuration 'mixing: the interaction with the discrete state parti- I i itions the continuum types into interacting and noninter- | acting superpositions. i In this research a complex atomic system known to j i I have autoionizing states in a certain part of its energy spectrum will be theoretically analyzed. Fano's con- j figuration mixing formalism (10) will be used (for an j equivalent treatment, but a different viewpoint, see Shore's papers on the parametrization of autoionized levels). (11, 12) Numerical estimates will then be made i for the autoionization indices. In order to maximize the i insight into the physics of the situation and minimize the! i i ^numerical trappings, highly accurate wave functions will I not be derived. Reasonable wave functions (Scaled Thomas-j j Fermi) (13) bolstered by a study of the total spectro- 'scopic picture of the atomic system will be used to obtain the numerical estimates. This Gestalt method, it will be seen, is quite enough to characterize completely 6 j ;the observed autoionization phenomena in the systems being |analyzed, the heavy rare gases. A search of the literature has revealed that most |of the theoretical effort has been expended in high accur- | ^ |acy numerical studies of the first two indices mentioned i above (see, for example, the review article of Smith) (14)^ for relatively simple systems. Not much work has gone j I : !into determining the third and fourth indices (but see Pano) (10) (Fano and Cooper) (15). Thus, there is a |scarcity of work in which the positions of the autoioniz- j ; ’ I |ing states in the energy spectrum as well as the influencel of these states on, say, absorption cross sections is |given. The close coupling calculations (32) have been | very valuable in this respect. j : ‘ 1 In the next chapter, the problem of the autoioniz-; 2 2 ing states between the ^3/2 and series limits in the rare gases will be defined in detail. Data will be shown i which exhibit the influence of the autoionization on the photon absorption cross section in this energy region. Inj ! | subsequent chapters the analysis of the autoionization by , the method of configuration mixing will be carried out formally, and then the analysis will be applied to the j rare gases. Although the analysis presented here is directed towards understanding the phenomenon of autoionization as | found in the absorption spectroscopy of the rare gases, it: I 7 i I |should at least be mentioned that under special conditions i jit has been possible to observe the reverse transition, !that is, radiative transitions from "autoionized levels" I in the rare gases Kr and Xe. (42) This situation is of !practical interest in astrophysics. (43) j CHAPTER II THE ABSORPTION CROSS SECTIONS OP THE RARE GASES BETWEEN THEIR 2P3/2 AND 2P^ SERIES LIMITS The spin-orbit coupling in the rare gas ionic 5 I core, p , is very strong, so j-j coupling is applicable and the ionic configuration splits up into the two levels : 5 2 5 2 ■ p ( P3//2) and P ' * The level lies above the j j j=3/2 level by the amount of the spin-orbit splitting. j The normal levels of the neutral atom can be obtained by j coupling an emission electron to either form of the parent! 5 2 ion: the multiplets are derived from p ( P ^ ^ ^ 1 and 5 2 ’ ! p ( P^)nl , where the principal quantum number n and angu lar momentum 1 label the emission electron. The observed j 5 2 ^ spectroscopic series terminate at the p ( P3/2) or ^rie i 5 2 p (Pi ) series limits. For the neutral atom, a j-j (9) or "2 j-1 (16) coupling scheme is suitable. In doing a Slater-Racah analysis of the rare gas multiplets below the first ionization potential, it is customary (9) to neglect any coupling between levels in a i I 2 ; configuration which belong to different parents ( P3/2 ; 2 P^). The reason for this is that the spin-orbit split ting of the ion is much greater than the energy correc- 1 I itions (like the electron-electron interaction) due to the j emission electron. It is quite sufficient (9) to par- j Itition the multiplets of a configuration into the "upper j ! 2 I |levels," belonging to P^, and the "lower levels," belong-! 2 i ing to ^3/2* and to Perform a perturbation calculation j ionly within each set Cupper or lower) of levels* The j i 5 i |p nd configurations, for example, (9) can be understood fully in this way. j i ' ! j j ! if these classifications are continued beyond the ; I :first ionization potential, say, to the energy region | 5 2 5 2 i between the p ( and p ^ series limits, then the ; 5 2 configurations in that energy range would be p ( 5 2 * i i.e., continuum states, and p ( P, )nl , discrete states. ' ^2 : i On a photographic plate, the absorption from the ground j 5 2 istate to the p ( (J=l) continuum would be expectedi to be a smooth function of the photon energy, and the ; 5 2 i absorption of the p ( P-. )nl (J=l) states would be expected; ! ^ I to be very narrow Lorentzian lines, with the level separa-j tions predictable from the series terms on the longer wave; length side of the first ionization potential. But actual; jphoton absorption experiments (photographic data, photo- j ionization and total cross section data) indicate that the' ' ' ..... . ......... .”... lb. i ■ !cross sections do not have the simple properties mentioned. i iThis is seen in Figures 1, 2, and 3. Figure 1 is a set of densitometer traces due to Ogawa (17) of high resolution I (0.9A/mm or 150 cm*"1 /mm plate factor) Argon absorption 2 2 . spectra in the region including the P3/2 an<^ series limits. Figure 2 is the measured absorption cross section 2 2 of Kr in the P3/2 ~ region, and Figure 3 is the equivalent for Xe. Figures 2 and 3 are due to Huffman, ; et al. (1) Figure 4 gives the initial portion of the |absorption cross section of Ar, at a resolution of 0.04A; these data were taken by Hudson. (18) There is some j photographic evidence available (19) indicating that Ne exhibits the same form of absorption in its corresponding j i ! I energy region. j The points of interest are that, in the energy I 2 2 region between the ^3/2 an<^ series limits of the rare gases, the absorption cross sections show strong i oscillations of the same generic form for all these gases,! i the absorption cross sections have nonzero minima, the j ; i oscillations can be grouped into broad asymmetric peaks ; accompanied by a ”spike” on the shorter wave length side | | 1 of the maximum of the asymmetric peak, and Kr and Xe have ! i r\ at the P3/2 a small feature unlike other peaks (an i ”odd peak,” in the words of Huffman), (1) while Ar does j i not have this. j These characteristics of the rare gas absorption FIGURE 1 DENSITOMETER TRACES OF HIGH RESOLUTION ARGON ABSORPTION SPECTRA TAKEN BY OGAWA.| __ __ ___ _ _ _ ____ a.. n m a m nnriPHTtnPn OGAWA. TAKEN BY SPECTRA ABSORPTION ARGON RESOLUTION TRACES HIGH OF DENSITOMETER THE THREE TRACES CORRESPOND TO DIFFERENT PRESSURES ABSORPTION COEFFICIENT cm 12 4p - 4p ns 3000— 2000— 1000— 890 870 880 860 WAVELENGTH (A) 850 840 FIGURE 2 KRYPTON ABSORPTION COEFFICIENT ABSORPTION COEFFICIENT i ns 5000 H XENON E u 4000 H 3 /2 3000 H 2000 H iooo H 920 930 940 950 960 970 980 990 1000 1010 1020 WAVELENGTH ( f t j* j XENON ABSORPTION COEFFICIENT W ABSORPTION CROSS SECTION, Mb I4T 72 ° 783.8 784.3 784.9 785.4 0 785.9 786.4 786.9 WAVELENGTH (A) FIGURE 4 ARGON ABSORPTION CROSS SECTION NEAR THE 2P3/2 SERIES LIMIT. THE POINTS ARE CALCULATED VALUES TO BE DISCUSSED IN THE TEXT. jcross sections have been known since the work of Beutler, I (20) and are sometimes referred to as Beutler lines. 1 The series of spikes in the cross section follows |very closely (21) the Rydberg formula fitting the long i 5 2 * series p ( Pi)ns , and thus is identified with the ! 5 2 ' Ip ( P . , )ns series beyond the first ionization potential. I I ^ i I The series of broad asymmetric peaks is ascribed to the j i 5 2 ' ! series p ( P^)nd in the autoionizing region. These I identifications will find support in the theoretical i |estimates of the autoionization indices given in this i research. j The shape that the rare gas cross sections take in i ! p 2 their P- - Pt spectral region is complicated by the Of d . fact that the level width is of the order of the level | spacing. The spectral shapes of isolated levels (level j 'width much smaller than the level spacing) can be under- j stood (10) by considering alone the interaction of the j local discrete state with the continuum states in the same! 2 2 i energy region. But in the P3/2 “ re9ion second | order effects of autoionization are at least as important | as the first order ones: the interaction between members | j j of the series of discrete states, coupled through the j continuum, becomes a most important factor in determining ; the line shapes. To show this the formal apparatus of ; i i configuration mixing must be developed. This will be done! ! I I i in a later chapter. j 16 Before going on to the formal theory, it seems appropriate to list the states that partake in this in stance of autoionization, and to give the relevant inter actions, Since the analysis proposes to explain the ex perimental evidence available from photon absorption experiments, it is sufficient to specialize the discussion to the odd purity total angular momentum J = 1 states, the C t initial state in the absorption being a p Sq, ' 5 2 1 The odd purity J = 1 states are the p ( P^ins, I p ^ P3/2^nSV P * P5^nd3/2’ P * P3/2^nd3/2 and P * P3/2* j ndf>/2* As exPla^-ned earlier, in the region between the j 2 2 i 3/2 and h series limits, the terms belonging to j 2 ' the P3/2 Parent are actually continuum states, and ; 2 t those belonging to Pa are discrete (of course, in the ; I i independent particle model, and without so far considering: all of the interactions present). j For brevity of notation, the following symbols | I i will be used to label the interacting states: j (1) (2) P ^ P3^nd3/2 ? R 2 ' p ( eh )n s H (3) p5(2p3/2)Ed3/2 = (4) P (2p3/2)Bd5/2 = (5) p5(2p3/2)Esa5 “ For definiteness j-j coupling is used initially. Later oni the purity (9) of the autoionizing and non-autoionizing j . _ ' ” . - . jL7" continuum superpositions is discussed in terms of j-j and j-1 coupling. The Hamiltonian can be written as (6) » = »o + He-e> where Hq includes all the single electron operators and H is the dominant correction (in this case), the e—e electron-electron interaction. For the numerical work, H is tfsmeared out," and H is simulated by a Scaled e—e 1 J Thomas-Fermi Hamiltonian. (13) \ ; The autoionization is due to the mixing of con figurations by Hg_e. The matrix of the interaction for ij-j coupled states, derived by standard methods, (22) will jnow be exhibited. The matrix will show all of the jpossible couplings between the states and ^ . j n I J jThen plausible arguments will be given to reduce the Icomplexity of the matrix. The entries are in terms of the : K V | Slater integrals (22) R(i,j ; r,t), k * ( 7 ) R M R n i(r,)R n , ta) r - > * * 0 0 t t | x — r 2 r2 d r . d r z ! r k + l 1 z 1 Z f > 1 !and the R -i.(rm) 3-s the radial wave function of the mti> I I A A ! electron, having principal and angular momentum quantum numbers and 1^ , respectively. The matrix is shown on page 18. Only the matrix | |elements in the first and fourth rows (or columns) play a J ; i role in the autoionization; so only these contributions 4 (n) 1 5F°(p,nd) + 2/45 G^Ptnd) -63/245 G3(p,nd) 2/25V5 R2(p,nd;p,Ed) -4/45V5 R^pjiidjEd.p) -1/25V5 R2(p,nd;p,Ed) +4/15V5 R1(p,nd;Ed,p) -2/9V2 R1(p,nd;n's,p) 4/9 R^(p,nd;Es,p) p -1/5 R (p,nd;p,Es) 2/25V/ 5 R2(p,nd;p,E'd) 5R°(p,Ed;p,E'd) -2/25 R2(p,Ed;p,E'd) 2/75V10 R1(p,E,d;n's,p) -8/225VS"R1(p,Es;E'd,p) LU -4/45V5 R1Cp,nd;E,d,p) -1/25 R2(p,Ed;p,E’d) -14/45 R1(p,Ed;E'd,p) -63/245 R3(p,Ed;E'd,p) -4/15 R3(p,Ed;E d,p) +1/25VI0 R2(p,E'd;p,n's) +8/250VT R2(p,Es;p,E*d) -1/25V5-R2(p,nd;p,E"d) -2/25 R2(p,Ed;p,EMd) 5R°(p,Ed;p,E"d) -2/15V10 R1(p,E"d;n’s,p) 4/I5V5 R1(p,Es;E, 'd,p) ^2E" +4/15V5' R1(p,nd;E"d,p) -4/15 R1(p,Ed;E"d,p) -4/25 R2Cp,Ed;p,E"d) +2/5 R1(p,Ed;E"d,p) -63/245 R3(p,Ed;E"d,p> + 2/25V10 R2 :(p,E"d;p,n's) -1/25V5 R2(p,Es;p,E"d) (n‘ ) *2 -2/9V5-R1(p,nd;n s,p) 2/75V10 R3(p,Ed;n's,p) +1/25V10 R2(p,Ed;p,n's) -2/15V10 R1(p,Ed;n's,p) +2/25VlO R2(p,Ed;p,n's) 5F“(p,n s) -7/9 G1(p,n's) -2/9V2"R1(p,Es;n's,p) V" 1 4/9 R (p,nd;E s,p) ____ T It 1 -8/225VT R (p,Ed;E s,p) 4/15VfT R1 (p,Ed;e"' s,p) ____ 1 I M l -2/9VTR (p,n s;E s,p) I II 5R,(p,Es;p,E s) ^ I I I -1/5 R (p,nd;p,E s) ____ ^ I I I +8/250VT R^Cp.Edjp^ s) _ O H I -1/25V^ R (p,Ed;p,E s) 1 " • -5/9 R (p,Es;E s,p) i ' ■ ” ' .... '. 19 need be considered. Also, the diagonal terms are measured with respect to the average of the multiplet; besides giving this relative position, they are not important. In a later chapter, Scaled Thomas-Fermi radial Iwave functions for and YjE are derived. One of the jsalient features of the excited s-state radial wave func- ! itions, in the energy region of interest, is the large 1 jnumber of lobes within the "size” of the ionic core, i.e., | lup to radii where the ionic core's p-state radial wave jfunctions have their exponentially decreasing tails. Over ithese dimensions the p- and d-state radial wave functions have few lobes (two for the Ar 3p and one for the Ar 9d, i !for example). Thus it can be expected that there will be ! I jstrong cancellation in the Slater integrals mixing the s— | and d-states. This will be particularly true for the I exchange type integral R (p, -d; -s, p), since the ionic core's p-state radial wave function essentially truncates :the range of integration to the size of the ionic core. For this reason the mixing of the s- and d-states will be i Ineglected. This approximation reduces the problem to the i autoionization of the s—states, indicated by the matrix ishown on page 20, 20 (n) 0 2 0 . 3E 0 (n) 5F°(p,ns) -7/9 G (p,ns) -2/9V2” R1(p,ns;E s,p) 0 3E -2/9V2 R1(p,ns;Es,p) 5R°(p,E s;p,Es) 1 1 ■5/9 R (p,E s;Es,p) and the autoionization of the d-states, indicated by the matrix on page 21. In j-1 coupling, the He_e matrix for the d-states takes the form (J=l) shown on page 22. The terms omitted along the diagonal do not influence the following dis cussion. < t > (n) IE ib » 2E * (n)5F°(p,nd) + 2/45 G1(p,nd) 1 3 1 -63/245 G (p,nd) 2/25V5 R2(p,nd;p,E'd) -4/45V5 R1(p,nd;E,d,p) -1/25V§" R2(p,nd;p,E"d) +4/15vS" R1(p,nd;E"d,p) 2/25V5 R (p,nd;p,Ed) 1E -4/45VT R1(p,nd;Ed,p) 5R°(p,E d;p,Ed) -1/25 R2(p,E'd;p,Ed) -14/45 R1(p,E,d;Ed,p) -63/245 R3(p,E'd;Ed,p) -2/25 R2(p,E"d;p,Ed) -4/15 R1(p,E"d;Ed,p) -1/25V5 R (p,nd;p,Ed) 2E +4/15V5 R1(p,nd;Ed,p) -2/25 R2(p,E'd;p,Ed.) 5R°(p,E d;p,Ed) -4/15 R1(p,E,d;Ed>p) -4/25 R2(p,E"d;p,Ed) +2/5 Rl(p1EUd;Ed,p) -63/245 R^CpjE d;Ed,p) i \ j H i ] a < - x « » | rH a x> ; i —i # » » x> (0 ! i _i to • * ! XJ * # w XJ to XJ to ! ^ C # ■ * ! eg * > a a S —* ! rH | 04 H PS C M PS ! Ol ■ m Ch N ; a N - s t 1 1 / —1 PN a I a ! p H x j •» XJ : I — l to x j E ! A* to CO a • * N ! i_ j • •* ■o X) ! 10 x> c CO ; CO c « * c * •* a a e g a v ^ " w * i \ rH rH - 00 r g PS os : 04 OS ■ eg c r > CO s - * in s N in \ ’ S t ! °* rH 1 1 / —s j a /> 1 XJ a rH m x j i 1 —1 CO m 13 1 A* r » CO I pH a • • s “to . l —l • • * XJ • w ! Xl XJ a XJ i C c w * c ; pn # * a # * ■ X a S ^ x * a 04 r _ j eg eg os - PS PS m o \ a in s 0> s • S t * N p H 1 >st I rH l— l p H A™ 1 — 1 rH p H V M * 4 \ 1 — 1 1 — t 1 __1 A " XJ XJ rH — E 1 __1 CO to X) c eg eg \ s A ~ 00 co a a a C M eg eg * —> LO in in a a a it n ii c C O H l - s -H 1 - ^ I ' ............... ' ." 23 | It was mentioned in the Introduction that radi ative transitions from autoionized levels have been iobserved in Kr and Xe. These transitions have been ! i jidentified as originating in the even parity autoionizing jpVp^np and p^(2P^)nf states of Kr and the p5(2P^)nf j states of Xe. (42) The final states are the "normal" i i ' j jlower lying d-states in Kr and the normal lower lying j ] is- and d-states in Xe. CHAPTER III 1 | THE NOTION OF FANO'S "BACKGROUND" CROSS SECTION FOR AUTOIONIZATION i The interaction between the discrete and the con- t j itinuum configurations generates a partition of the con- tinuum states into two linear combinations: (8) = A 1 *IK + A 2 I p Y 2E ;and (9) x ™ = Bl * 1E + B2 2E, 'with (10) H / X (b)> 1 e-e E = 0 and (11) ( x <a> \x ) - 0 • E E This partition is of interest because the minima in the oscillatory part of the absorption cross section correspond (10) strictly to transitions to X ^ , in the case of a single or isolated discrete state. Fano (10) has defined a correlation coefficient for autoionization 24 ...... ~~... '.. 25 whose square measures the ratio of the background cross section (transitions to X ^ ) to the total cross section | (transitions to xjj^ and Xg^ ). | But in the case of two continuum types it is easy lb) i(and more meaningful) to explore the existence of Xg - fa) for information concerning the purity (44) of Xy and | x » • Then statements can be made about the spectro- : fa) lb) : scopic strength of Xg and Xg , and about the angular distribution of the photoelectrons as a function |of photon energy. j | From Eqn• (10), ! (n) (b) ; (12) ( 4> |He_e | x > 1 e E Cn) (n) - Bl ( *1 lHe-el ' V + B2( *1 I He-eI I (n) (n) = B1D1E + B2D2E = o, ii.e (13) B2 = -B1 PIE (n) E__ (nT“ °2E This means quite generally that the background cross section is a property of the continua and the given discrete state, and the Bi should reflect this: (14) B± = Bi 26 (n) (Note that in arriving at Eqn. (14) it has been tacitly assumed that either no mixing between continuum states of different energies occurs, because it has been i • (accounted for in a "previous diagonalization," or that the! mixing is negligible. In any case, the general expres- I isions can be easily synthesized from the results presented] i I (here.) : In spite of this, constant (or slowly varying) j lean define sufficiently well the background cross section j » I for an autoionizing Rydberg series and two continua pro vided the ratio is slowly varying with principal I quantum number n. M (b) The orthonormalxty of X g and X ^ together (with Eqn. (13) can be used to give * r = c [ o o M O ' r ' 1 , a s , p i < » . A r = ^ ;us> A t ' - A t j f i / £ - C [ O f i f + O f i f f * If the ratio g is denoted by J/g , then Eqns. (8), (9), and (15) to (18) give 27 (19) X U) - °(n) C(D(n))2 + 13"** < f r E E E IE + [(D(n))2 + 1]“^ ^ E 2E and (20) X - C(D(rl))2 + 1 r H t E E IE - DU) C(D(n))2 + I]"4* i , j E E 2E. j ! If a normalized function can be expanded as a I sum of orthonormal functions 4^ , and 4^ , ^ = (^1^) ir (&1 then the number J ( ' J >|4^)|*' is said to be the purity of ^ in terms of 4^ • Obviously, o p r w T t i r ^measures the purity of X.^ , in terms of • Thus the interac- | tions that give rise to the autoionization also partition : (as described) the continua and determine the purity of ; j ithe partitions, I ; j Now, from the interaction matrix, it follows that j (21) 1 i 2/5 - 4/9 R <P»nd;Ed,p) j D^n^= ___________ R2(p,nd;p,Ed) for j-j couplingj -1/5 + 4/3 R^(p%nd;Ed,p) R2(p,nd;p,Ed) and 4/9 R. 1(P^n. d.»E.d ,£,)_ R2(p,nd;p,Ed) % fQr j_1 coupling. 4/9 R1(p1ndi,Ed1£)_ R2(p,nd;p,Ed) (The approximation is sometimes made for terms of j a Rydberg series that derived parameters of the type j R1(p,nd;p,Ed) and R1(p,nd;Ed,p) can be approximated by a ! reduced parameter R1(p,d) and a normalization constant, (15) | (23) R1 (p,nd;p,Ed) SS N(n)R1(p,d). | This approximation is equivalent to the statement ; 4 ! that most of the contribution to R (p,nd;p,Ed) comes from regions within the ionic core. With this approximation , given in j Eqns, (21) and (22), are roughly independent of n. Thus a t general background cross section can be discussed.) The purity of Xg1?^ does not depend on the values ; of the Slater integrals involved, but only on their ratios. Figure 5 shows the X£ purity functions, \ If r . i o — ( k ) . O IE ^ XE ** and ^2E I XE » f°r and coupling, in the physically meaningful range 0.0 < R1(pd)/R2(pd) < 1.0. It is interesting to note that pure j-1 coupling obtains not only at R^(pd) = 0, Racahfsi case, (16) but also at R " * " (pd)/R2 (pd) = 0.45. i Quite a bit can be said about the background (22) 1/5 - 5(n> PURITY O F X 29 100.0 Ar CALCULATED Xe MEASURED 9 0 .0 - Kr MEASURED 8 0 .0 - 7 0 .0 - 60.0 - A r MEASURED 5 0 .0 - % OF d 1 1 /21 IN 40.0 - % OF d „ IN o 3 0 .0 - 20.0 10,0 - 0.5 ,2 (b) FIGURE 5 j-j AND j-1 COUPLING X PURITIES AS FUNCTIONS OF RVR ->u / T- \ | A cross section O^oC J ( X£ |r j $Q) | , where < J > Q is the ground state) appearing in the absorption spectroscopy of the rare gases without actually having to calculate any matrix elements. This is so because the background cross section is very sensitive to the purity of Xgb^ . As far (b) as radiative transitions are concerned, XE is composed j of Edg^ and Ed5/2* and the sPectrosc°Pic strengths of I transitions to <*3/2 and 65/2 from the ground state are in ! the ratio of 1:9. I C I d ) 2 I Figure 6 shows the function ( XE |r | ‘ i 5 ) , in I tYl\ iarbitrary units, plotted as a function of the X f E* ipurity. The total cross section CT = <Ta + (T^» where *Ea>H V | 2- The oscillatory behaviour of the actual cross ! section is introduced by a form factor that modifies • cl In the range 0.15 £ R^(pd)/R2(pd) s 0.9, the (b) dependence of XE on the purity, in terms of ^5/2’ 1S given by (the ionic core being understood) i (24) XEb)= - VP <Ed5/2) + (Ed3/2), ! and, therefore, j (25) <rb «c |C x^b)|r | 4.0)|2 = |( -VP (Ed5/2) + Vl^P (Ed3/2)|r | < f > Q) | 2 ! ©c (2Vp + 2/3Vl-*P)2. 0.8- 0.6- 100 AS FUNCTIONS OF PURITY FIGURE 6 G AND co Similarly, V E = v^^5/2' - * ■ rr v^3/2 (26) X ^ - VT^P (Ed,/0) + V P (Ed-.-), so that (27) «Ta oC |(4>|rM0 )|2 oC ( 2V1-P - 2 /3 Vp ) 2 • | j ! The proportionality in (27) indicates that at a certain purity (P = 0.45, corresponding to (pd)/R2(pd) = j 0.225) the cross section <51 will be zero. This has c l implications for the phenomenon of autoionization which j will be discussed after the formalism is established. j i This chapter will be terminated with an estimate j /g\ I of the purity of X' 1 , based on experimental evidence j E and on the numerical work yet to be described. j It has been shown (10) that the cross section for j ! j a transition involving interacting discrete and continuum j states as the final state can be written as i I (28) CT(E)= (energy dependent form factor) j x ^ ♦ ®i>, j j i ! and furthermore, that the positive, energy dependent form! | factor goes to zero at certain energies. At these zeros 1 : of the form factor, obviously, <5‘(E) = These zeros j ■ i ; are the minima seen in Figures 1 to 4. Then, on the longj ! J 2 . wave length side of the Pa series limit, where the ' ’ i . j 33 ' instrumental resolving power is too weak to define the ! oscillations in the form factor, the measured cross sec tion corresponds to <Ta + ^b* The sura ®“ a + 0"b ^-s | not expected to vary much over the range of energies of \ interest, since the range is of the order of the spin- j orbit splitting in the ion, and this splitting is much \ smaller than the ionization potential. j Thus both (at each Rydberg term) and the sum j ; CT_ + <J\ can be read from the absorption cross section j o L D |data. I j | If <T. is taken at the first minimum (where the I : b i ; resolving power should be good enough to give a fair indi-j i i i cation) in the data of Cook, et al.„ (23) Huffman, et al. j ; i (1) and Hudson, (24) and if the sum < 5* + OTv, is taken J 1 7 a b | i p immediately before the P, series limit in these same j ^ | data, then the following estimates of the Ar, Kr and Xe lh) ! f purities for X result: s I E 0.715 corresponding to 0.476 corresponding to 0.610 corresponding to 0.405 corresponding to 0.305 corresponding to 0.19 corresponding to 0.206 corresponding to 40% d5/2» 60% d3/2 (Cook) 18% d5/2* 82% d3/2 (Hudson) 30% d5/2’ 70% d3/2 (Huffman) 13% d5/2’ 87% d3/2 (Cook) 7% d5/2 * 93% d3/2 (Huffman) 2.2% d5/2’ 97.8% d3/2 (Cook) 2.8% d5/2 * 97.2% d3/2 (Huffman) u> 35 The data on Ar show quite a scatter. Judging from i i Ithe detail appearing in the data plots, Hudson's figures jare more reliable. The profiles in Hudson's photoelectric data resemble those found by Ogawa in high resolution spectroscopy of Ar, whereas Cook's and Huffman's profiles (k ) are somewhat washed out. Thus the 18% ^5/2 for x ^ / based on Hudson's work will be taken as an upper limit. /t_ \ It is interesting to note that X E in Ar comes very close to being almost pure j-1 coupling, but Kr and i Xe tend very definitely towards j-j coupling, and Xe is i !almost pure j—j coupled. The numerical work derives directly the ratio i 1 2 |R (pd)/R (pd) for Ar; the calculated value is 0.53, which, I /-u\ iaccording to Figure 5, corresponds to X£ being 9% d^/2 and 91% ^3/2* I The indication is that <^5/2 tends to participate in autoionization, and ^en<^s participate in the "conventional” photoionization, loosely speaking. In the above arguments, no consideration has been given to the influence that the s-continuum might have on ! the observed values of the cross section. This is because the s-continuum states contribute very little to the absorption cross section. (25) With the STF wave func tions calculated in the work, the d-state absorption cross |section was found to be larger than the s-state cross i ! ! section by a factor of 30. CHAPTER IV | » THE ANGULAR DISTRIBUTION OP PHOTOELECTRONS (b) ! The purity of X is mirrored, not only in the E i ratio of the background cross section to the total cross ; section, a + ^b^’ kut also in the angular distri bution of the photoelectrons associated with the ionizing (b) transition to X • This aspect of the background cross E section will now be discussed. With sufficient instrumental resolution, this ; i angular distribution can be monitored at the minima of the! observed cross section, where <r(E) = <rb. The angular j i distribution has a significance if it is referred to an incident polarized radiation. Then, assuming that the ! polarization vector coincides with the z—axis of the j coordinate system in which the atom is described, the | angular distribution is implicit in the symmetries of the | i unbound orbital and the specification that AM =0. 6 * Since the initial state is p SQ, the angular — : ■ --- ~. 37 distribution (following the absorption of ionizing polar- |ized radiation) is~ a property of the J = 1, M = 0 sub- r ^ C O | states of p ( ^3/2^ ^*d3/2 and ^ ^ ^*3/2^ ^d5/2* ^ -^1 I 2 the total angular momentum of the P3/2 core and j2 the total angular momentum of the unbound orbital Ed.-, then J2 ; the linear combinations of forms (j^* ; J2* m2^ such |that J = 1 and M = 0 with j2 = 3/2 or j2 = 5/2 are given by (29) P5(2p3/2)d3/2 = 3/10V5(3/2, 3/2; 3/2, - 3/2) , - l/10V§(3/2, hi 3/2, - h) - l/10V5(3/2, - Jg; 3/2, h) i + 3/10V5(3/2, - 3/2; 3/2, 3/2) I and I(30) P5(2p3/2)d5/2 = -Vl75(3/2, 3/2; 5/2, - 3/2) +V3/iO(3/2, %; 5/2, - h) -V37l0(3/2, - 3* 5/2, h) +V1/5(3/2, - 3/2; 5/2, 3/2). Then, looking at the unbound orbital behaviour of the probability density, it is found that the angular depend- :ence of the unbound orbital is given by ;(31) jd3/2 I^ an9Ular dependence = (3/10V§)2 |d3^2 _ 3/2 12 ang. dep. + (1/10V5-)2 | d3/2 h |2 an9* dep. + (l/10V5)2|d3/2^ 2 ang. dep. + (3/10V5)2 |d3/2j 3/2 |2 an9• dep. I and / | 5/2 cLRy • uepi (VT75)2 1 ^5/2 - 3 / 2 l2 an9» d eP + (VJ/iO)2 | d5/2t_3^| 2 ang.'. dep. + (V3710)2 | d5/2 2 ang. dep. + (VI75)2 |d5/2 3/2 |2 ang. dep. Now, the d spin-orbitals are constituted in the following manner: d3/2 3/2 = -'f i L75(1+> + 2 VI75(2"), ang. dep. = 1/5 [Y^j + 4/5 j Y2 j 2 j d3/2,-3/2 = -2V^/5<-2+) +^75C-1-), ang. dep. = 4/5 |y”2|2 + 1/5 j Y ^ 2; d^/0 «. = -£75(0+) +V575(i“), ang. dep. = 2/5 | Y®|2 + 3/51 Y21 ^ 5 d3/2,-^ = J&75(-1+ ) + v§75f(D, ang. dep. = 3/5 j Y ^|2- * - 2/5 | Y^| ^5 d5/2, 3/2 = 2V^75 <1+) + VI75 (2”), ang. dep. = 4/5 | Y * | 2 + 1/5 |Y2|2; d5/2 _3/2 = VI75(-r2+) + 2VT75 (-1"), ang. dep. = 1/5 |Y22|2 + 4/5 j Y^j2; d5/2, ^ = (0 + ) + (1”}» ang. dep. = 3/5 |y°|2+ 2/5 j Y ^ j2 ; d,- . = V275(_l+) + V375(0“), ang. dep. = 2/5 Iy'1^ 3/5 |y^ | 2 . Thus the angular dependences turn out to be 39 | (33) |d3/2 |2 ang* dep# = 45/100(4/5|y22|2 + 1/5 I y ^ 2 ) i + 5/100(3/5|Y21|2+ 2/5 | Y° |2 ) + 5/100C2/5|Y^ |2+ 3/5 |Y\ |2 ) ; + 45/100(1/5|Y2 |2+ 4/5 |y2 |2 ) I S 1/5(7 - 6 cos2 6) , ! ! and j (34) |d5/2 |2 ang* dep* = l/5(l/5|y“2 |2 + 4/5 Jy”1 |2 ) + 3/10(2/5 | Y-l| 2 + 3/5 | y£|2 ) + 3/10(3/5 |y°|2 + 2/5 |y*]2 ) + 1/5(4/5|y||2 + 1/5 |y||2 ) = 1/5 ((6) cos2 © + 3). ;These distributions are shown in Figure 7. I (b)l2 The angular dependence of | can be con structed in the following manner: for each core (j^, m^) j I there will be an unbound orbital component of the type - V T dc/o m + Vl-P do/o 5 / 2 , — 3/2> —m^ y 1 2 in the appropriate range of R (pd)/R (pd) ratios. Thus i <35> x<e = [VPVT75 d5/2)_3/2 + 3/10V5 Vl^P d3/2 j - 3/2^ + v57io d5/2)_,s - i/iovs vr=p d3/2)_%] + [ V p V§7io d5/2j ^ - 1/10V5 V l - P d 3 / 2 ^ i + Vi/s d 5 / 2 ^ 3/2 + 3 / 1 0 i / 5 V I — P d 3 / 2 3/2*^° 40 340 330 350 330 340 350 320 i ; 320 3/2 310 : 310 300 [ POLARIZATION VECTOR 300 290 ; 290 280 280 270 270 260 100' 100 260‘ 250 250 120 ; 240' 240 : 130 ; 230' 130 220": j : 140 ! 2201 200 1 6 0 ' 170 190* 190 170‘ 150' 210 200 FIGURE 7 THE ANGULAR DISTRIBUTION OF PHOTOELECTRONS Since the core components are orthogonal, inte- i lb),2 grating the probability density |X^, [ over all the coordinates but those of the unbound orbital results in the expression (36) | Xjj, | ? unbound orbital ang. dep. = P|^5/2| an9* dep. + (1-P) |d3/2| ang. dep. +2|3/10VP <1-51 d*/2^_3/2 d3/2)_3/2 ang. dep + l/lOV3/21fe (1-P) d5^2 d3/2 ang. dep. * -1/10V3/2VP (1-P) d5^2 ^ d3/2 % ang. dep. -3/10Vfc (1-P) d*/2) 3/2 d3/2? 3/2 ang. dep.) j = 1/5 [cos2 e (12 P + 9 VP (1-P) -6) j +(7 -3VP (1-P) —4P)] j The parameter P is the purity of X£ in terms of d ^ 2. j It should be noted that even with the interfer- ! I 2 ence effect introduced by VP (1-P), a pure cos 6 angular; i dependence never obtains; an isotropic distribution occurs i at P = 0.2, which corresponds to R1(pd)/R2(pd) = 0.45 and; to <T. / (cr+<T,)=0.5. ; b a b | ! If the X ^ purity is taken from Hudson's measure-! E i ments for Ar (P = 0.18) and from the average of Cook’s and Huffman's measurements for Kr (P = 0.10) and Xe j (P = 0.025), then the following angular dependences are j calculated from Eqn. (36), 0 being measured from the i polarization vector (as in Figure 7), I '.'. 42 p Ar ang. dep. oC [5.2 -0.6 cos 0] — 2 Kr ang. dep. oC [5.7 -2.1 cos 0] 2 Xe ang. dep. oc, [6.43-4.3 cos 0] These numbers are applicable at the first minimum of the observed absorption cross section. The angular character of X ^ also is quite well defined. Complementing can be written as E E (37) x‘* > P5(2P3/2) Ed5/2 +VPrp5(2P3/2) Ed3/2) where it is understood that the J = 1, M = 0 substate is being specified. Then, using the same procedure as before, 1 3 i \ 2 (38) |X j f unbound orbital ang. dep. 2 ^ | = (l-P)|d^^2 |2 ang. dep. + p |d3/2 f ang. dep.j +2VP(1-P)j -3/10 ^5/2}_3/2 d3/2,-3/2 -1/101/372 dz/2,-% ! + 1/101/372 d5/2^ ^ dd/2, H * . +3/10 d5/2^3/2 ^3/2,3/2J = 1/5 [cos2 0 (6-9 VP (1-P) -12 P) + (3 + 3/P (1-P) + 4 P)]. Here, again, an isotropic distribution is possible (at 2 P = 0.2), but a pure cos 0 is not. The angular distribution of photoelectrons due to absorption into both X ^ and X ^ can be evaluated, E E essentially by using cra and 0“ ^ as spectroscopic |weighting functions• This is written j I |(39) ang. dep. of photoelectrons i (b) . 2, | (a) .2 oc [ J X | ang. dep. + 6* a | X £ | ang. dep • ] | oCl I cos2 0 (12P + 9Vp (1-P) - 6) - (4 P + 3 YP (1-P) ) + | 7<rb + 3 ^a I I I ;making use of (36) and (38) : i The ratio of this expression evaluated at 0 = 0 O S to that at 0 = 90 is given by I (40) (y(Tb) 0o ^ <j^(8P + 6 V P ( l - P ) + 1) - 0 * a (8P + 6 a /p ( T p ) - 9) ! ^‘ b(7-4P-3VP(l— P)) -t + ) I This relation is plotted in Figure 8; it is experimentally useful only if the oscillations in the actual cross sec- 2 tion cannot be resolved (for example, near the P, series 4 limit). In practice, under good resolution, <r is seen to a be modified by an oscillatory form factor. Thus, Eqn (40) must go over to 1(41) (<T(E))0o _ <jy8P + 6YP(1-P) +1) — f(E)g^(8P-f 6VP(1~P) — 9) 1 (<T(E))90« , 0^(7- 4P 31/P( l-p)) -f-f(E) <Ta(3 + 4P +31/P( 1-P) ) In the next chapter the study of the nature of the form factor f(E) is taken up. CO o 4^ o P ) b* CD I I o e P » C D I I VO O • PP o CO PHOTOELECTRONS EMITTED A L O N G 6 = 0 PHOTOELECTRONS EMITTED A L O N G 6 = 90° o g n 8 s- O M O' + < _ r ? _ O VJ. O cr 00 o TOTAL CHAPTER V ! THE FORMULAE OF AUTOIONIZATION Because of Eqn. (10) the interaction H will not; d>M v W mix the discrete state 7^ with /L^ , so it is suffici- j ent, according to the theory of configuration mixing, to j i consider the admixture ; <42> x 6 w = 1 + At' \ for the d-states, and, for the s-states, (43) W ^ + ? / v ^ ' ) f 3£- J*' i These both have the form of a single level interacting J 1 with a single continuum. The problem in either case involves finding expansion coefficients a(E), b(E) and j b(E,E#) such that i r 1 45 46 i i (44) (45) ( X ' M j X ' V ) ) - S ( £ - f % given that ( * * ' l « / ¥ ” ) -- £ w , , ./til Y '£ f t / #/ f ' ) - v ? , and ! ( % . \ » l % ) = e s ( £ '-£>- ; \b Y ^ ) (It is understood here stands for » 3£ or A~£ ? I depending on whether the s- or d-states are being dis- rk(h) ^ cussed. Similarly, T stands for or Tj •) I ; i ! The derivation of the coefficients has been given j i t ; in detail by Pano. (10) The important results will be described here, and then a derivation will be given which ; incorporates the changes and improvements required by the J : experimental evidence. j After some lengthy but straightforward algebra, it turns out that „ N v f c h > i (47) <x(£) r --- ( e- £c">- F^Csj) + i n i v n <48> 1(e) = «. (e) V (n) 1 v* » (49) L(£,£') = a.(E ) V (? — — e E - E ' In these expressions M 47 \/(») (50) V\ = O W W t ) - * W « > E 1 ( x ? i H t - J € ) t n (n)/^ < 5 1 > F % = /.„/ s k u = ? [ J l k J l 4£' (52) £^) ( ^ h / H / ^ ^ -for s-sUiesy ($[ * I H/ ^ n>) {**<* «Us;M«s. The admixture of discrete and continuum functions | i has a(E) as a common factor, and fa(E)l ^ has a Lorent^ian j ( n) (n) ' shape as a function of energy, with center at E + F , 1 and with a half-width of fT /ifI? (53) 11/ C* ’ 5 I a | a ( E ) | l= |Ve 1 ( e ~ e m ~ f w ) * jt1 | v J T : '.■ ■ ■ ■ ■ “.....'..'.........48 1 I This is indicative of a resonance between the dis- |crete state and a band of continuum states centered about E Cn). The properties listed so far belong to the atomic |system and the Hamiltonian assumed. Now transitions iinvolving %£ can be investigated: under the operator T, j :transitions from some state ^ to X. involve the matrix j ° E \ element ! : ^ 4 ) ( x e i t i % ) = < * < « ) f (h> + & ) % + | = L * (.e)(%ITI%) \ J E-£' | At this point an equivalent matrix element can be | I defined to simulate, in terms of ( t l r l t ), the contribu- ! <hC«> 0, j i tion from T and g* transition, j :<55, ( f » + d E ,,Tj f } J £ - E ' I I ! | ‘ i I Note that ^ depends on the operator T. The main inter- ; est here is on radiative transitions. ! i i With this equivalent matrix element Eqn. (54) i ;becomes < 5 6 ) ( y . . m v = )$ /T n > Since CT(E)oc/(Xf / "P| ^0)|^ and Or<< /ftg | T | 4^)1 » it follows that (57) <r(E) r ------- § + 0 w ) 4 1 r|V«|* +1 / * T t |v ' T e*+ 1 The last equality is written in an energy scale with units of * IV /. This energy unit, labeled is :related to the width of the "line" seen in autoionization <*( n ; 1 (see Eqn. (53)) and to the probability that T will relax j by the emission of an electron, (11) Prom an experiment-! (n) p(H) al viewpoint, and I are the important parameters, ;because of the role they play in the line shape. The coefficient of <y in Eqn. (57) is the form a factor f(E) required in Eqn. (41). i i The form factors of the observed isolated auto- j |ionized levels can be understood fully in terms of the j p 2 family of (e + q) /(e + 1) curves. The curve fitting | then yields the indices q and P . (10) However, it is : well known that these curves will not fit the rare gas j 50 2 2 | data taken between the P3/2 anc* series limits unless !(26) q and F are taken to be noticeably energy dependent ;over a range as small as a level spacing. I It will now be shown that, in fact, the observed ;deviations from Eqn. (57), for constant q and F , are due 1 to strong second order effects, and the theory will be |presented which, introducing the second order effects, I gives a new family of curves which fit the observed data I ■ i j and which in one limit go over to the usual Fano shapes, j | I | The basic assumption required is that the devia- j ! i 1 • ! tions from the usual Fano profile are due to second order j effects, and that these second order effects are primarily due to the influence of the "nearest neighbour” levels. Then instead of starting with something like Eqn. (42), the configuration mixing technique requires the inclusion j I of the nearest neighbours: o) /_N («»),.s d>“"Q 1 (58) I with lL satisfying (59) tfll£)= and t “ “.... ' ...." 51 I (60) (Xe.IX £) = 6(e - e'), j and the 4 ^ and ^g satisfying Eqns. (46). A word of caution should be interjected here. ) j The formulae that result are applicable to the lowest I ! autoionized level (in which case the formulae are cor rected for the influence of the next two higher autoion- I izing levels) or to the ntb level (in which case the j formulae are corrected for the effect of the next lower j i and the next higher neighbours). In any case, the j | formulae fit one level at a time, and not the whole ener- | ; gy range in one neat package. | Fano (10) has, in his formal theory of a Rydberg ! series' autoionization, implicitly given the solution to ] I • i the problem as presented here. But the formal operations; i J i required are in practice very inconvenient. The analysis j given here is a simplification which is meant to reveal j : i explicitly the main source of deviations from the usual 1 Fano profiles, and to derive a useful formula which has j broader applications than the usual Fano formula. In the! ; process it will be seen that, with the insight offered by i I . ! the new formula, the qualitative features of the observed j level spacing can be used to give estimates of q and T • The 4 will be taken to be eigenfunctions of [ ! H with H smeared out (i.e., eigenfunctions of the samej | Hamiltonian with different energy eigenvalues) but with ! 2 ^ j the ionic core, and the to be eigenfunctions of a ! 2 similar Hamiltonian but with the **3/2 ^ -on^ - c core. In ; this context the set of $ and the set of ^ can be | considered to be (10, 27) "previously diagonalized." The Eqns. (58) to (60) then enter the formal analysis in the following way: (61) ) = £( J E) = £ El and r*-1 + P f U e,£') V e. p <62) (rt j H i x £ )* e ( r s. i x £) = e[l(e)l(£-f)i- p f (,$£') SCE'-rW'] = Z a f y E ) (v'?/+i(E)ES(£-£") j j8"'1 + P f SCe‘ -E-)dE'. These can be rearranged to give (63) &•-£*>) a Ifc) (v^) 53 £) t P f iL*&Ve. M £' and m (64) (£-£"•) LtZ,*") ~ E ) • n-1 E ' V e Hence ^ <«> J f e O - E >*n-i £ V e and (66) /E - E( V % ) = ift)VeW + P d£' \ ^JSlh‘1 E-eV Taking a hint from (65) and the single level results, j b(E) can be assumed to be of the form <67) ■ i &) * z ( f ) j: fc) (VgY. ! i Inserting this statement in Eqn. (66) gives rise j (i) ' to a homogeneous matrix equation with the a (E) as un knowns. As usual, in order not to have only trivial j (i) * solutions for the a (E), the determinant of the matrix I (i) multiplying the a vector must be zero. This condition; 1 / • \ jdetermines 2 (E). Then the ratios of the a 1 (E) can be jknown and, finally, the values of the a 1 (E), by invoking ;the normalization of Xg* Eqn. (66) includes terms in the "generalized level shift," 4£* • Where experimental evi- jdence is available, for example where a Rydberg formula ihas been developed for a series which has its higher mem bers autoionizing, (28) the indication is that the level ishift is an extremely small correction (of the order of a cm”' * ' ) as shown in Table 1. It can be said that the level ishift, besides being difficult to pinpoint experimentally, j is not too informative. Notice, however, that there is a dispersion relation derived by applying a form of the Feshbach projection operator to the transition operator (for scattering in two electron systems). This dispersion relation connects the analytical properties of T , the llevel width, to those of the level shift. (53) Dropping the level shift terms in Eqn. (66) and linserting the identity in Eqn. (67), <6 8 ( V ^ T V f . The coefficients a^^(E) can be eliminated by multiplying both sides of Eqn. (68) by (Sm£^) 1 ( v f r and summing over the index i: TABLE 1.— The observed and Rydberg formula calculated Ar 3p^(^P^)nd levels n Vobs (N.B.S.) vobs (°gawa) vcalc <°9awa> 3 115366.90 cm"1 115367.2 cm-1 115367.2 cm"’1 4 121011.979 121011.8 121012.1 5 123815.53 123815.3 123814.9 6 125286.28 125286.5 125286.5 7 126167.4 126168.6 8 126740.6 126739.4 ionization limit = 127109.9 9 127130 127122.3 127122.6 10 127410 127397.9 127397.9 11 127610 127599.8 127599.7 12 127760 127752.3 127752.5 13 127880 127872.8 127872.9 “..... '..' .' .... 56 <« »> lf>u> ( O ' - [ C * " * ) ( f ) ' , $ **»“ ! so that (70) [xffUsj; W 3 1 & - E & With this value of z(E), the various equations in (68) / * \ give that the a are related by <71> _ e-E M ^ (e) y(») ! Utilizing the orthonormality of the and the I normalization condition on *X> leads to £ <72> t ( £ - e n ) = 2 ( S ' c r f S t f r - f f " ) + l*(F)l(£';E)+i(£")C(£',£") + 3>fc%e')U£*, £')<&'■ A straightforward application of Eqn. (68) to Eqn. (72) shows that the first, third and fourth terms in the last equality of (72) cancel out. This leaves 57 (73) $(£-£") =• t>%) i Is ") StF-E") + p f b * ( E , S . But M\“ „ £) <74, & : ) %■ ** ■(?-*')"(£ "l the Fourier analysis (10, 29) of ^jP-E1) leads to (75) (e-£•)'*(£"-E'f1 a (f- E)'1 [ (r- £')'1- (B f')'J] + Wl S (£ *-£) S(£ ' f (e1- £V ‘ The partial fractions give level shift—type contributions* Consistent with the approximations being used, these will be dropped. Then Eqn. (73) becomes i / P (76) S ( C - E " ) = S ( e ~ £ ' ! ) + TTZ (*.* (£$ J 4 = S (E-e'J i t£)UE") 7 tj1 2*(E)z (£.") making use of (67). Hence z (£) (77) L(E) = x ( E)i. U7r ’ ! i I iand „ (i) (78) = F - f * 0 Z(£)+<TT i Turning now to transitions from a state to the I state under the influence of an operator T, the following is obtained: <79, cxei ti C £ ) f )+ ? f u v t f e> d E 'lT i i ) ] - A. , ; i i Again, an equivalent transition moment for the j i I bracketed contributions from and can be defined, i I ! ;Since / this equivalency properly is assumed to be of the form <80, ( f * ' * p f ( y * T f Je'lTl t ) J Then Eqn. (79) becomes, with this definition, ( . ! » f t | T | « , = [ “ • V T ] j •it, 'T| Now, the formula being sought should apply well to the ntt term, and not necessarily so well to the (n±l)& term. Also, in a series it is expected that (15) the q will be slowly varying with index i. Hence it will be taken that I ^ 2(n*1\ ;And then I (82) a j T f t f ) - \ z . ( E ) - * t r ± E - £ U) In this way the cross section a(E), proportional to can be related to ( Ta , which is propor tional to with the same constant of propor tionality. Thus k f r ^ (83) o-(F; y - p z z + I v f 'l But it can also be expected (15) that, in a series r <»*fi iw (**n > 7. (..wii r fr>> — = l V * I - l V e I = I i T ’ 60 Z so that (83) takes the form « , *■->*>= o * A The cross section has now been written with the I superscript n to indicate that the formula is derived for i | ;the nfl» level, j With q^n^and £* as variables, this formula i gives the new family of curves, correcting the usual Fano | I formula for the effect of the nearest neighbour levels, iTo the approximation of the derivation., the most general formula, applicable to the case where two or more -continu- i jum types are interacting with the discrete level of inter est and its nearest neighbours, is given by (85) 9 M (E)= ' ° 1 jk-1 0- -t- or 1 ♦ p^) In the limit where |e(i1 ~ 1)-E(n)| » -^— , Eqn. (85) Z reverts to the familiar isolated level formula, in the sense that for . . J ( | \ £ £ « - D + £ ( » ) E b r '*t > z Eqn. (85) and 61 -V <rb 1+(£< "> )2(e-em )“z give essentially the same result. The practical boundary ! for one regime or another seems to be STor^ib E^|. ! iThis will be shown with Ar as an example. i i In Eqn. (85), the "undisturbed" cross section <j*a appears. It was remarked in the study of <T, that, at a certain purity level of /(.-,<!" can be zero. This E ^ jtneans that the definition of q, a convenience for normal izing the transition moment of (XelTl t) to that of i lT l t ) , ceases to be a sensitive parameter. Since = 0, it can be expected that for E, O^'/T/ fBho< so that r | - f c w y d £ . „ 0 . J t-F' * ! then i |c86) ( X f/T/<£)= (a(£) - (V(£)+ T>fUE,V)fe '4t'\T\%) ~ &*(£)( Tl ( p 0). This means that (87, (<r Ce)-<*i) °c This proportionality shows that the line shape of the transition from (p^ to under the operator T will 62 be Lorentzian (or of the Breit-Wigner form (31)), because of Eqn. (53), and the cross section will be proportional to the oscillator strength of the discrete transition from < ? > 0 to 4 > (n) . The form of the cross section given by (87) should I also be a good approximation under the following very | similar conditions: if the cross section due to the continuum component alone ( <r ) is relatively small (of cL the order of one megabarn) and if the oscillator strength j of the discrete component is relatively large. These con-! t I ditions hold in the outer p shell excitation of the alkalij atoms, and the observed line shape (30) of the lowest j i j autoionized level seems to be a very strong, Lorentzian j line. ! CHAPTER VI I APPLICATIONS TO ARGON In the previous chapter, a formula was derived which shows the effects of autoionization on the cross jsection for a transition (induced by the operator T), taking into account the influence of other energetically close autoionizing states of the same symmetries. The formula as derived is applicable separately to each level of an autoionized Rydberg series. This formula, Eqn. (85), rather than the usual fano formula, &qn. (57), has to be used if (for the nib jlevel) r (n) « (level spacing)/50. To show this, Ar is ! taken as an example. But this example includes at least |the other heavy rare gases (Ne to Xe), because of the ■similarities discussed in Chapter II. According to the NBS Table of Atomic Energy Levels, (33) just above the first ionization potential (127109.9 cm”1 above the ground state) of Ar occur the J = 1 levels p5 (2Pi) 9d and p5(2P*) 11s at 127130 cm”1, 63 p5(2P^) lOd and p5(2P^) 12s at 127410 cm-1, and p5(2Pjg) lid and p^(2P4i 13s at 127610 cm"1. Prom the information presented in Chapter II showing the complex form of the "lines” or "levels,” it should be understood that the energies ascribed to the configurations by the NBS Tables are not as exact as they could be, in that they do not |differentiate between the broad asymmetric component and 1 ;the superimposed spike. Ogawa (34) has obtained a classi fication giving the position of the spike (at the peak) j |and the position of the steep slope of the broader, asym- I metric peak. This information is valuable for reconstruc-' [ting the profiles from the presumably additive s and d Icomponents. The energies given by the NBS Tables, though,! i i |are good enough for the sake of the arguments to follow. |(Note that the level shift is being neglected.) j If-the ntb level is to be studied, then it is con-; / n \ venient to center the energy scale at E = E . Then j Eqn. (85) takes the form ! (8 8 ) <rfn)(E)-<r = I b 1 ' - 1 ‘ 200) 1 ( 1 + q(n)p(n)|(£+280) + E + ( E - (l+ (I (lV [ (E 280)"1 E-1 (E 200)"1] 2 ) )2 -1 The energies are in terms of cm , and the formu-j la is written for the p^(2P^ )10d, with p^(2P, )9d and ! 'i. j 5 2 5 2 ■ p ( P^)lld as neighbours, or for the p ( P^,)12s, with the ; 5 2 5 2 ip ( P^)lls and p ( P^)13s levels as neighbours. i ' ~.......................' ............... ~ .........................65 i But the formula also is applicable to the 1 ilevels; then the next two higher levels are included as icorrections. The energy range covered is exactly that |shown in Figure 4. i { Figures 9 through 13 show plots of the coefficient! of a in Eqn. (88), for the set of parameters I i I j r ( n ) , q ( n ) ) | 5 ,1. 0 ; 5,1.5; 5,2.0; 50,2.0; I 50,1.5; 50,1.0; 100,1.0; 100,1.5; 100, 2.0 ) . j i • I ; J Also shown on these plots, in open circles and triangles, I | . j are the usual Fano profiles. The data shown in Figures 1 | jthrough 4 evidently can be parametrically represented by j j | |the new formula. For example, on Figure 4 are shown data | —I' jpoints calculated by using Eqn. (88), assuming T= 50 cm ; | _ i ;and q = 1.5 or 2.0, and adjusting the calculated curve to | I i : i fit the experimental curve at the maximum. This procedure! I gives a to fall in the range 10.3 Mb to 15.8 Mb. The ! usual Fano formula is a good approximation to the new j ’formula only if 50r^n^ lE^n±1^ — E^n^ | , over the range 1 A E < ( E < n > - E ^ n _ 1 b / 2 . | Hence for any experimental data which seem to be j (n) ! ;of the Fano profile, an upper limit on T can be ! j i obtained just by noting the level spacing. A point of information can be brought out here: I i subtracting the background cross section the observed: 1 oscillating cross section is the product of a form factor ; 3 .0 - FANO FORMULA, r = 5 cm NEW FORMULA FANO FORMULA, r = 5 cm NEW FORMULA 2.0- .0- 0.0 100 -100 0 200 -300 -200 E(cm ') FIGURE 9 THE AUTOIONIZATION LINE SHAPE, INCLUDING THE INFLUENCE OF THE NEIGHBOUR LEVELS 67 ! ] • FANO FORMULA, r = 5 cm" 1 ' / NEW FORMULA 3 .0 - I i i i i | I I [ i 100 200 300 -400 -300 -200 -100 0 m FIGURE 10 THE AUTOIONIZATION LINE SHAPE, INCLUDING THE INFLUENCE OF THE NEIGHBOUR LEVELS • FANOFORMULA, r = 50 cm , q = 1.0 NEW FORMULA A FANO FORMULA, r = 50 cm-1 , q = 1.5 NEW FORMULA 3 .0 - 2.0 - -300 -100 100 -200 E(cm S FIGURE 11 THE AUT0I0NI2ATI0N LINE SHAPE, INCLUDING THE INFLUENCE OF THE NEIGHBOUR LEVELS • FANO FORMULA, r = 100 cm , q NEW FORMULA 4 FANO FORMULA, T = 100 c m '1, q NEW FORMULA 3.0- 2.0- .0- 0.0 -300 -200 -100 100 E(cm FIGURE 12 THE AUTOIONIZATION LINE SHAPE, INCLUDING THE INFLUENCE OF THE NEIGHBOUR LEVELS FANO FORM ULA, r = 50 cm", q = 2.0 NEW FORM ULA FANO FORMULA, r = 100 cm'1, q = 2.( NEW FORM ULA 6.0- 5 .0 - r - 100 cm 4 .0 - r - 50 cm q = 2.0 3 .0 - 2. 0- 0.0 -300 -200 -TOO 200 100 E(cm V FIGURE 13 THE AUTOIONIZATION LINE SHAPE, INCLUDING THE INFLUENCE OF ^ THE NEIGHBOUR LEVELS o ' .................. " 71 j (which does not distinguish between the energy regions j f above and below the first ionization potential) and °"a» j i i roughly speaking a step function, with the step at the j jfirst ionization potential. The step can occur at some : ! |energy on the shallow slope of the asymmetric broad peak | ;of the form factor or on the steep slope. ! If it occurs on the shallow slope, then the onset jof the absorption continuum shows an incomplete broad i t | I jpeak, an "odd peak" as in Kr and Xe. And if it occurs on j 1 i : I |the steep slope, then roughly a whole asymmetric peak is seen at the onset of the absorption continuum, as in Ar. \ Thus the appearance of an odd peak in the absorption cross! i !sections of Kr and Xe, and its absence in Ar, is really of no consequence and is not a serious distinction. Con- | sidering, for example, in Figure 13 the T = 50 cm , !q = 2.0 curve, if on the energy scale shown the step in i -1 —1 i ! the function cr occurred between —200 cm and -50 cm , cL then the Kr - Xe case would result, and if the step were j j to occur between -50 crn-^ " and -20 cm""'1 ', then the Ar case j 1 i would be reproduced. By comparing Figure 4 with Figures 9 through 13, | i | it can be estimated that T, for the broad peaks shown, is I i of the order of 50 cm”^, and q seems to fall in the range I . i 1.5 to 2.0. Taking for the spike to be the full width I !at half maximum, it can be estimated that this component j ;has r <5.5 cm-' * ' , and, therefore, it can be expected that ' j ' 72 I the usual Fano formula can be used to fit it. However, J |superimposed as it is over a grosser peak, nothing posi- i tive can be said about the q of the spike. Now the analysis will turn to the autoionization parameters of Ar, derived from Scaled Thomas-Fermi (STF) jwave functions. (13) This part of the research will be ! ; ! i I used to identify and classify the origins of the two | i I ! I |approximately additive components in the structure seen 1 o - ;in the absorption cross section between the ^3/2 anc^ I iseries limits, and to show that, indeed, to a remarkable j j | degree the observed cross section can be reproduced by twol i additive components of very different indices T and q and i I very different amplitudes <ra. ! In order to use Eqn. (85), except for curve fit ting, the theory must supply the parameters , q^n^, ; j tr and 0" , • The definitions of these in terms of a b I matrix elements have been given in Chapters III and V. i According to the separation discussed in Chapter II, the j parameters for the s—states and those for the d—states j i can be considered independently. Using the matrix elements of the electrostatic ! ! :interaction (Chapter II) and the formulae of Chapter V, j the parameters to be calculated are the following: | 73 M I.. i iU \l 2. r “ , » . v W x A i K J W s AIT l(pr(1fji)hsjHe.e |pS(iP^)Es)| = 2 n l - ^ R lCpinsi^,p)l) specifically for n = 11 and E 0.001/Ry ^ (90) r W(d) = a* |V > ) |* = X 7 T K f * l U9. J X ?)/1 * nrlfp^^B d/zt^gJ + A 'l'^ ( l^1 )£ ^ )) » H [ * r M r M ] U - £ ♦ f r ( $ ) % specifically for n = 9 and E *£ 0.001/Ry• In (90) the (n) (n) lA± and A2 of Eqns. (17) and (18) were used. The n ^ bracketed polynomial in R /R appearing in Eqn. (90) 1 2 ranges from a minimum of 0.02 to 0.2, for O.O^R /R - 0.9, 1 2 and with the minimum value corresponding to R /R = 0.225. For radiative transitions from the ground state the q*s are given by o » I T and (n) (n) r i (a) , (92) q (d) = (<l>1 + P/ba(E,E ) Xe* dE j rj <ftQ) Tn5 (a) (a) *<^1 lHe- J X E )( X E I r| ‘ J V (n) <$! U l j ) __________________• ~ Tn5 Ta) Tal I He-e I X e ’ < X B U l « J > 0 ) - (p ( P^)nd3^2|r| < p Q ) ^[(p ( p^^nd3/2lHe-elp * P3/2*Ed3/2^p * P3/2^Ed3/2^ r l^o^ + (p5(2POnd3/2|He__e[p ( P3/2)Ed5/2) (p ( P3/2^Ed5/2 I r I Now, it can be shown (22) that (for< | ^ 0 = (np)6 and for the J=l, M=0 substates of •the upper state) oo an (p5(2P3/2)Es|r| fo) - - 2/3 / P PES rdr, (p5 (^ n ' s |r 1^1 = ^sVS/p P^rdr, r=o r=o CO 00 (P5 (2 p3 / 2 > E d5 /2lr l^o> ‘ - 2VI75/ Pnp PE d rdr> (P5 <2 p3 / 2 )Ed3 / 2 M V = 2 /3 '/I7 ' 5 / PnpPE drdr cdT s o r = r O (p5(2 p3 s)n'd3 /2| r |^ ) = - 2 /3 /P np P n* d rdr, S r=o 75 | where, if R ^ is the radial wave function, then Pnl = r Rnl* Also» for Ar» = (3p)^, for Kr, (4p)^, 6 ! and for Xe, (5p) • With these dipole length integrals representing I the radiative transition moment and interaction matrix (n) | element taken from Chapter II, the q (1) become I i (93) ,(n 9 1 q (s)~--- —------ ; ------- 4ir R (p,n s;Es,p) • 00 I P P ' rdr o np .n s r 00 Jq P~e rdr np £s and R2(p,n’d;p,Ed) (l/5-8/9R?- (- P- »- ^,d^■ Ed^ )) R (p,n d;p,Ed) •00 np n d L p— P-'- rdr •/O r GO Jo P P_ , rdr np Ed In order to evaluate the integrals corresponding to the first asymmetric peak and its spike, STF radial wave functions P^i Plls* PEd* PEs* E ~ 0*001 RY» were derived, and a Hartree-Fock ground state P ^ was obtained from the literature. (35) Having developed the wave functions, it was noted that, within the ionic core size (the effective extent of the P3p> about 5 Bohr radii)j 76 the discrete (Pgd and Plls) a**d continuum (PEd and PEs) wave functions have very closely the same shape, except for normalization: N(9d) PQ . ~ -------- P_, 9d N(Ed) Ed* _ N(lls) EllS ~ N(Es) Pes> where the N(nl) are the appropriate normalization factors, and the P , are normalized STP radial wave functions, nl The un-normalized wave functions are shown in Figure 14 and Figure 15 to illustrate this point. Figure 16 shows H-F ground state 3p of Ar, (35) as well as the H-F 3p of Ar for the excited configuration 3p^4s. (36) The close proportionality of the Pn^ and P ^ in the range of r of major importance can be used to reduce the problem of evaluating the ratio of the dipole length integrals to that of evaluating the ratio of the normali zation constants. However, the dipole length integral for the transition 3p — Ed or Es must still be evaluated in order to obtain < r (S) and tr (d). a a Thus r0 0 r0 0 N(lls) (95) I*3 p Pl l s rdr _ I P3P i F ( £ 3 PE . r d j _ NUls)^ ” 3 o r co N ( E s ) f % PEs rdr / P3 p P E s “ X r r \ r d r 77 9d, -0 .0 1 2 5 Ry (T -f) kd , 0.001 Ry (T-F) To norm alize 9d, m ultiply this graph by N= 0.0348 0 . 5 - 10.0 12.0 0 6.0 r(BOHR RADII) 0.0 - 0 . 5 - i .0- ! 1 ! -2.0 FIGURE 14 THE STF DISCRETE (9d) AND CONTINUUM (Jed, _________E = O.OOOl Ry) RADIAL WAVE FUNCTIONS______ > • ARGON CONTINUUM WAVE FUNCTION E - 0.001 R v ♦ ARGON 11s WAVE FUNCTION 0 .6 - 0.4- 0.0 - 0 . 2 - -0.4- On the scale shown on this graph, the normalization constant for the Ar 11s is N = 0.0452 -0.6- -0.8 - r (BOHR RADII) FIGURE 15 THE STF DISCRETE (11s) AND CONTINUUM (Jcs E « 0*001 Ry) RADIAL WAVE FUNCTIONS 0.40 -I—* - ------------------------------------------------- 3 ------------------------------------------------- : ------- r ; 0 . 3 0 - • A 4 • 0 . 2 0 - t A 0 . 1 0 - # r(BOHR RADII) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.00 1 --------------------1 --------------------1 --------------------1 -----------------4 — ------- A- I A...« A " I +------------------- (•----------------- f*---------^-------< » • A . A • * . *• ‘ I* - 0 . 2 - A • * - 0 . 3 - A * * • • HARTREE & HARTREE Ar 3s23p6 - 0 . 4 “ • ^ Ar 3p A A R. S. KNOX Ar3p 4s - 0 . 5 - ^ 1 A* -0 .6 - A • • A - 0 . 7 - A # • A A* 1 - ° - 8 ~ f #A r A A : - 0 . 9 - ■! z l . P r .. — :------———----.........------ — — • — ---- . . . . ■ ............- ^ ..............-. FIGURE 16 THE H-F GROUND STATE AND EXCITED STATE 3p RADIAL WAVE FUNCTIONS 3 4 • t A # r(BOHR RADII) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 A 1 1 1 1 1 $ L a ...§ i H i ---------»A f»---------------- 1#-------- A- A • A A A • A A . * A • A* A * • m UADTDCC i UADTPPC A, • q Ar 3p A • HARTREE & HARTREE Ar 3s 3p A R. S. KNOX Ar 3p^4s A* A • • A A • • A » t • #A “ ' “ ' 80 , a V° ■ Wfrl*) p ? ri)*x-o T> ®o = J h v ' i O® ...... x "....... ~.81 ( 9 9 ) M i f , U i > e r f ) = f f rn V ° r> g v •" f 14 ^ A/flU) Jf w ( w ; The G^(3p,nl) are the usual exchange Slater inte- j ! ? ^ grals and F (3p,9d) is the direct Slater integral. ! The approximations in (97) and (98) are particu- j | larly good, because the P^p wave function modulates the ] * P9d* Pns» PEd and PEs ’ and the P3p starts to decrease | ! exponentially at about 2 Bohr radii. For the exchange j ■ type Slater integrals, the range of interaction is effec- : tively truncated to the region where the and PE1 are ! ;most similar. : The approximation in (99) will be good provided j } i I ;• the major contribution to the integral comes from the I ; region r < 10 Bohr radii. The integration involved in 10 Bohr radii, and the zeros of the and | PE{^ ate progressively displaced. These two effects : probably lead to significant cancellation beyond = 10 Bohr radii. These approximations lead to 82 (100) r(s) = 16- tt 81 N (Es) N(lls) ■ ] 1 G1(3p,lls)12, (101) (102) q ( s ) = N(lls) 4* 7 r 1 N(Es) 2 / _ 2 , T(d) = 2ir r N(Ed) f~ /F^(3p,9d)j N(9d) I G (3p,11s) 2 (103) 1/25 —8/45 G (3p,9d) + 32/81/(G (3p«9d) \^1 F2(3p,9d) F2(3p,9d) ' q(d) = /7r f N(Ed) I2 F2(3p,9d) N(9d)^ (8/9 GX(3p.9d) _ 1/5" 1 F2(3p,9d) ■ o : The energies GX(3p,9d), GX(3p,lls) and F2(3p,9d) :can be obtained without calculations, by fitting the !observed lower levels of the configurations 3p^ 9d and 5 3p 11s. The upper levels of these configurations are i p above the P3/2 series limit, so they are subject to !autoionization; the J = 1 levels are found in absorption, |but no radiative transitions from any of the upper levels I are known. Condon and Shortley (37) have given the (j-j j ........... “...." " "". ' " ."..'.. ' 83 5 ; coupling) levels of the rare gas configurations p nd as ! v v ! functions of the parameters F , G and the spin-orbit parameters p and d. Using the Condon and Shortley I t | formulae, a good fit of the observed levels was obtained j with |P° (3p,9d) = -278 cm-1, F2(3p,9d) = 252 cm"1, j | G1(3p,9d) = 135 cm"1, G3 (3p,9d) = 106 cm"1, I = 954.3cm"1, = 0.4 cm"1, ; and the part of the Hamiltonian not depending on the ! emission electron or the spin-orbit coupling of the 3p i ! electrons was found to be | H(p,p) » 127587.0 cm"1. The fit obtained with these parameters is shown in Figure ; 17. 5 The same technique was applied to the 3p 11s multiplet. Xt was found then that P° (3p,lls) = -275 cm"1, G ^ p ^ l s ) = 45 cm"1; ■ the fit is shown in Figure 18. The observed levels in Figures 17 and 18 were j taken from the NBS Tables. (33) (Incidentally, from Eqns. (98) and (99), j^Qp.gdiEd.Sp) _ G1 (3p.9d) R2(3p,9d;3p,Ed) F2(3p,9d) 1 2 Then, using the values of G and F obtained by fitting the observed levels, it is seen that 84 125700 i 125600 i ' la 1 ■3a' ■ 2a' ■3b' ■2b' ■ 4a ■lb’ ■Ob ARGON OBSERVED la 3a 2a 3b 2b 1b p i r„,„ )9d CALCULATED 127100 - 127000 “ ARGON Id \ \ > ) 9d I OBSERVED CALCULATED _____________I ________ Id •3c 2c '2d 5 FIGURE 17 THE OBSERVED AND CALCULATED 3p 9d MULTIPLET OBSERVED CALCULATED 127200- 1 127132.4 0 127122.7 1 127130 12 7 1 0 0 - 127000- ; 1 2 5 8 0 0 - 1 125715.50 2 125709.45 1 125712.7 2 125691.3 125700- : 125600 125500 FIGURE 18 THE OBSERVED AND CALCULATED 3P* 1 11s MULTIPLET " ' 86 R1/R2 = ■ ■ ■ ■ --- = 0.535. This is the num- 252 cm her entered in Figure 5, the purity curve as a function of rV r2.) The burden on the derived STF radial wave func tions is then to supply the normalization factors N(Ed), N(9d), N(Es), N(lls), and the dipole length integrals JP2p PEd r<^r an<* fP3p PEs rdr* T^e stalls are given in the Appendix, and the relevant results quoted here. On the scale shown in Figures 14 and 15, it was found that N(9d) = 0.035 and N(lls) = 0.045, where the discrete wave functions were normalized in the usual way. Normalizing the continuum functions per unit energy range by a technique described by Cooper, (25) it was found that N(Es) = (1.12)-1 and N(Ed) = (1.64)-1. The dipole moment integrals were found to be 00 J p 3 PEd rdr = - 2.1 atomic units, o and 00 Jp3 PEs rdr = +0.71 atomic units. O ^ These numbers correspond to <r(d, total) = 10.11 Mb, <r(d3/2) = 1.01 Mb, (r(d5/2) = 9,1 105» <r(s) = 0.3Mb. The observed cross section is roughly double the value <r(d, total) + cr(s) = 10.4 Mb. On i 87 i ! 2 j the short wave length side of the P^ series limit, the | calculated cross section is 15.65 Mb (note that this in- ! 1 2 i eludes contributions from continuum states having the P^ | parent ion, as well as those with the ^3/2 Parent ion). | The experimentally determined cross section on the short 2 ; wave length side of the P^ series limit is 32 Mb. Putting the normalization factors and the fitted value of G^Opjlls) into Eqns. (100) and (101), it is j ! found that i i ! (104) r(s) = 167T / 81 (1.12)(0.045) f ( ■ 45 (13.6)(8000) (105) q(s) = (13.6)(8000) cm”1 = 4.6 cm"1, (1.12)2 (0.045)2 (13.6) (8000) = 4.4. 4*7r 45 And, similarly, (106) r(d) = 2vr (1.64)(0.035) (13.6)(8000) 2 5 2 x 1/25 - 8/45 (135/252) + 32/81 (135/252) 21 = 67.0 cm - 1, (107) q(d) = [ IT- 252 (1.64)2(0.035)2 (13.6)(8000) (8/9 135/252 - 1/5)] -1 = 1.65. ' ' " ' “ " 88 (Note: to be entered into the formulae for q(l) I V ! and r ( l ) , the G and F must be expressed in Rydbergs. | | This is the source of the additional numerical factors I i in Eqns. (104) through (107).) | | Figure 19 shows the theoretical cross section, with and without the correction due to the nearest j I neighbours. Since o-^ = 0 for the s-states, Eqn. (84) I was used for the cross section formula, with r(s) = ; 4.6 cm-1, q (s) = 4.4, and <r (s) = 0.3Mb. Obviously the t i i Fano formula would be good enough for this component. i i i i These results show that the spike in the observed cross I section can be associated with the s-states. j Figure 20 shows the theoretical cross section f C | | around the p 9d (J = 1) level, with and without the I | influence of the nearest neighbour. Figure 20 has also | taken into account the calculated purity of the states j j | X ^ and X ^ ? j_n showing the background cross I E E i ; section: since <r(d, total) = 10.11 Mb and R^/R^ = 0.535, i > o\ = 3.5 Mb and <r = 6.6 Mb. The theoretical curve b a 89 7 .0 6.0- 5 .0 - 4 .0 - s t ) 3 .0 - o. 2.0- 0.0 -30 -10 10 -40 -20 20 30 0 E cm FIGURE 19 THE THEORETICAL 11s' LINE SHAPE* r = 4 * 6 cm q = 4*4, o = 0 * 3 M b • a THE NEAREST NEIGHBOUR LEVEL CORRECTION IS NEGLIGIBLE 90 i A FANO FORMULA • CORRECTED FOR INFLUENCE OF NEIGHBOUR LEVELS 25- 200 100 0 -100 E cm FIGURE 20 THE THEORETICAL AUTOIONIZED 9d'LINE SHAPE. a = 6.6 Mb, o r = 3.5 Mb, T = 67 cm”1, q = 1.65 t i ishows that the broad, asymmetric curve of the observed i | jcross section can be associated with the d-states. Figure i j20 shows also that most of the line shape of the broad ;peaks seen in the rare gas absorption cross sections is i i J |in fact due to the influence of the nearest neighbour j ! . ! levels. The departure from the Fano formula j <r (E) - o- = (q + e )2 <ra 1 + e2 |commonly used to fit the data is very noticeable. In order to reconstruct the total cross section by ithe superposition of the curves in Figures 19 and 20, the ;origins of the respective abscissae must be displaced withj I | respect to each other. There are experimental guidelines j i I ;for doing this• ! ; i Ogawa (38) has located the midpoint of the steep slope of each of the broad peaks and the midpoint of each | ;of the spikes. Analytically, the midpoint of the steep j —1 ' ! slope corresponds to E s 0.0 cm . For the 9d peak he j ' 1 I gives the midpoint at 127122.3 cm” , and for the 11s* spike, 127143.2. If the calculated data of Figures 19 ; i and 20 are superimposed additively with the measured j 1 i :separation, the Figure 21 results. The first ionization j jpotential, located at 127109.9 cm“\ according to the j |NBS Tables, falls between the minimum and the midpoint of I the calculated steep slope. Thus, if 0" is approximated a- t ; t :by a step function, multiplying the curve of Figure 21 by j 92 i — IONIZATION POTENTIAL 127109.9 cm i- 3p5(2pi/2) p 3 p 5 ( 2P ./2 ) U s' 127143.2 cm 127122.3 cm 40.0 3 0 .0 - 2 5 .0 - b 1 5 .0 - 10. 0 - 5 .0 0.0 250 0 200 50 150 50 100' -100 -150 | F -i I E cm FIGURE 21 THE SUPERPOSITION OF THE THEORETICAL 9d*AND llS* AUTOIONIZED LEVELS WITH THE OBSERVED LEVEL SEPARATION 93 <r results in an absorption cross section which starts immediately with the broad asymmetric peak, as seen experimentally. The findings of this research can be summarized in Table 2. TABLE 2.— Comparison of experimental and theoretical absorption cross section data Data Experimental Theoretical 1. (d) a . 2. cb(d) 3• X purity E 4. o(total), short wave length side of 2„ 10.3 to 15.8 Mb (deduced by taking into account the form factor) 12.8 Mb (monitored at first minimum of absorption cross section, Hudson's data) <18% (from the purity curve and he ratio of < J , a1 b minimum to the sum 32 Mb the ratio of at the first °a + % ) 6.9 Mb (obtained from STF and HF wave functions, and calculated purity of X^ ) E 3.5 Mb (obtained as described in entry for ”a(d)) 9% (from the calculated and deduced value of R1/!*2) 15.65 Mb 5. o (s) Cl 0.3 Mb 6. q(s) 7 . r ( s ) <5.5 cm spike) -1 (from f.w.h.m. of 4.4 4.6 cm -1 TABLE 2— Continued Data Experimental ; 8. t ( s ) =fi/r(s) = life time against autoionization i 9. Good fit of spike with Fano formula? 10. q(d) 11. T(d) 12. t ( d) =tt/r(d) 13. Good fit of asymmetric peak with Fano formula? I 14. Shape of absorption cross section I at onset yes 1.5 - 2.0 (comparing data to family of new curves derived in text) 50 - 100 cm no -1 same as #10 no "odd peak" Theoretical 1.14 x 10”12 sec. yes 1.65 (from STF wave func tions and deduction of energy parameters) 67.0 cm-’ * ’ same as #10 7.5 x 10“14 sec. no step of < r a occurs on steep slope of calculated peak; hence no "odd peak" TABLE 2— Continued Data Experimental Theoretical 15, Identification of spike and broad asymmetric peak spikes fit Rydberg formula derived for p^ns; steep slopes of broad peaks fit Rydberg formula derived for p^nd the form factor— spike and broad peak— can be repro duced by assuming s- and d-state contributions are additive; calculations give spike for s component and broad peak for d; calcula tions show steep slope corresponds to "center” of autoionizing state, i.e., position of p^nd. APPENDIX APPENDIX In this appendix the derivation of the radial wave functions Pn s> PEd* and PEs wil1 be described. The wave functions were derived from a Thomas- Fermi potential. (41) It is known (13) that by treating the "siz^' of the T-F ionic radius as a slightly adjust able parameter, the T-F excited state wave functions can be altered so as to match a calculated number (a cross section, for example) to an experimental one. This tech nique can give rather good results with relatively ele mentary numerical operations. The wave functions developed in this way are called Scaled Thomas-Fermi (STF) wave functions. The Ar wave functions needed in this research were those of the 9d, 11s, Ed, and Es (E= 0.001 Ry) emission electrons in the field of the (T-F) Ar+ ionic potential. The potential which was used was an accurate K+ T-F potential, (40) scaled down so that the boundary conditions of Ar+ at r = o and at the ionic radius were 98 i 99 i I met. In this respect the solutions can be called STP type. A further variation of the Ar+ T-F ionic radius was considered (this could have been done to double the calcu lated absorption cross section, to match the measured | cross section), but, really, the initial results were good |enough to explain all of the features of the observed i jphotoionization cross section to the experimentalist, so no further variation was undertaken, j The boundary conditions on the T-P function 0 ( x ) , !which is a solution of i <A x) = x " ^ ( 0 ( x ) ) 3 / 2 , where space, and x = r fi 71 r = radial variable in "real” fi = 0.8853z , z = 18 for Ar, are that 0(0) = 1 0 (xQ)= 0, xQ = "size" of ionic core, x o ^ (x0^= " fractional degree of ionization = - 1/18, for Ar+. Stewart and Rotenberg (13) have given a high accuracy polynomial fit for obtaining xQ as a function of the fractional degree of ionization. But it was found that their expansion gave definitely wrong results for the Ar4 - range (this was confirmed by an Errata (54) I ' ...." '. ...... .. ... . ' 100 i i i published after the calculations reported here were com- jpleted). So the xQ for Ar+ was obtained by interpolating I the high accuracy alkali results of Brudner. (40) Then the K+ T-F potential was scaled down in size by the factor x (Ar+)/x (K+). The x (Ar+) was taken as 15.25. o o o The radial wave equation for an electron of : i quantum numbers n and 1 in a T-F potential is (z is the atomic number) Pnl<r> * Enl " 1(1 ^2 ^ Pnl(r> “ where r is in Bohr radii and E is in Rydbergs. The func tion is given by ; #(x) =*A(-) = <Kx) + f- ’ x s xo ! V 18 xo 1 , x i x , for Ar+, ~ 3.8 * ~ ~ o' Note that ^ is such that near the nucleus the emission electron ''sees" the full nuclear charge, then, up to x = xQ, the T-F function gives some charge shield- 1ing, then, beyond xQ, the field of the Ar+ ion is essentially hydrogenic. The discrete P , and continuum P_n are zero at thei I n± El origin. The integration of the wave equation then is radially outward. For the Pn-p was taken from the NBS 2 Tables, (33) measured from the P, series limit. For the 2 PE]L, E was taken to be 0.001 Ry, measured from the P3/2 101 series limit. The energy E as taken corresponds roughly jto the center of the first observed broad asymmetric peak, j To bring the zeros of the Pnl closer to a uniform jspacing, it is convenient to change the independent vari- j \ jable from r to r^ (- y). But this introduces first j derivatives into the wave equation; these can be removed by the transformation Pnl(r) = p(y) = y^ q(y). |Then the wave equation goes from *2 p" (r ) = + V(r) - E] P (r) = K(r) P(r), r ;where ito V(r) = - - 2~ - ", . r 16 1(1 +1 ) + 3 . 2-. # * q (y) = L--------5-------- + (V(y )-E) 4y Jq(y) 4y = K(y)q(y). Both of these equations can be solved by Numerov's method (55) of numerical integration. The form in q(y) seems more complicated, but it yields more information with a given uniform integration grid. According to Numerov's method, the equation i n — j I q (y) = K(y)q(y) ! i ! lean be step-wise integrated by means of the recursion j 102 relation 2 + (5/6 s2)K(y ) <5 n+i ---------- —---- ^ n 1 - (s /12)K(yn+1> - 1 - <s2/12)K(yn-i) qn_i 1 - (s2/12)K(yn+1) starting with qQ = q(o) = 0 , and progressing in units s. The numerical integration was carried out in this fashion to several nodes of the wave function outside of the T-P ionic core size* The T-F core size is about 5 Bohr radii; at larger distances than this the potential is hydrogenic. j Then, for the discrete functions, after matching jboundary conditions at one of the external nodes of the i i T-F wave function, the function was continued by means of !the WKB approximation. The discrete wave function was jthen normalized in the usual manner: / % n l > 2 ar . 1. The unnormalized wave functions P_, and P,.„ and yd 11s their normalization constants are shown in Figures 14 and 15. The continuum solutions were normalized per unit i ! ..... ' ... " “ " 103 ' i | energy range by a method described by Coopero (25) Essentially, the method is a WKB continuation inwards ]from the standing wave, normalized asymptotic form of the | function. Using two points of the solution obtained by inte-i j I grating outwards, and matching these to the WKB form, the j j l |phase of the asymptotic function and the amplitude of the j I i T-P function can be obtained. 1 i • i | Cooper's technique is the following: The wave j ! equation asymptotically is j PE1(r) + A(r) PEi^r^ = °» I | iwhere j | A(r) = — + E — - U l + Li, i = o,2. I r r I(Note that this holds outside the T-F ionic core.) Then i !the solution is assumed of the form ! | i PE1 = C( 1) ( i r x) 2 s in©(r), x = pp. i I j Thus j i 2 ./ \ -H d2 ( I 1 x = A(r) + x z — ? (x z). i dr j ! i i % I Asymptotically, A(r) —- E, or x — E^. ;At two points r1 and 104 and Also al^rl^ = x^ r].) PEl^rl^ = ^ C(1) sin9(r1) a2^r2^ = PEl^r2^ ~ ^ ^ C(l) sinGCr^)* r_ / a = I xdr = “ ©Cxr^> • i ri | 5 The function PE1 implied above is unnormalized. !Thus x must be known to abstract C(l). To a good I approximation, (25) j x2 . A . A' 3/2 f * |y U',2 A" 5/2. i i | In this research, r^ was taken to be a node of !P^i(r), and r^ was taken to be an adjacent extremum. Then ! ! a = ir/2. Trigonometric identities can be used to give C(l) a i r * * [(a^ + a 2 - 2aia2 cos2^ /sin20C]^. Picking r1 and r2 as described, C(l) = ir a2 = PEl^r2^* H ~ . 105 ! For the s-electron, r2 was found to be 28 Bohr radii; then x^ = 0.519, and (for the scale shown in j Figure 15), C_ = 1.116. s For the d—electron, r2 was found to be 12.8 Bohr U radii, so that x z = 0.588, and, to be used to normalize I Figure 14, = 1.64. j i To obtain the absorption cross section, the > i formula (25) 1 I | j t . | < t(e ) = (0.855) C-~ -) | ( f | r | i) | 2 Mb j was used. The number E is the energy of the photoelect- j ron, and I is the ionization potential. Both of these |energies are to be expressed in Rydbergs. For Ar, |l = 1.17 Ry; and E s; 0.001 Ry, for the asymmetric peak in 2 ! these calculations. The spacing between the P3/2 anc* ^P^ series limits is about 0.013 Ry. The final state f is one of ^E or ^3E* u s i n 9 the nomenclature of j 6 ' i Chapter II, and i is the p SQ ground state. 1 1 The dipole length formula was used to calculate j 1 ; the radial contribution to (f | r| i). The Hartree and | . i i j | Hartree (3 5) ground state Ar 3p was used as the lower | ! i state and the normalized STF of this work as the j El | ; upper state. The integration 106 was done graphically. (The integration giving the normalization constants of the discrete Pn^ also was done graphically.) In atomic units, the dipole length inte- i | gral gave +0.71 for the 3p-Es transition and -2.1 for the I ! 3p-Ed transition. These numbers correspond to a total i ^3/2 anc* ^5/2^ d-cross section of 10.11 Mb, and 0.28 Mb _1 2 | for the s-cross section, at ~ 100 cm above the p ^/2 I i series limit. j The following tables give the Hartree and Hartree | ground state Ar 3p, and the STP Pn s» PEd* PEs J t derived for the work presented here. TABLE 3.— The Ar P3p (H-F ground state, due to Hartree-Hartree) r P3u(r) r P3p<r) r P~ (r) _.3p _ 0.0 0.00 0.28 0.347 2.8 -0.355 0.005 0.001 0.30 30.327 3.0 -0.302 0.010 0.0045 0.35 0.261 3.2 -0.256 0.015 0.010 0.40 0.175 3.4 -0.216 0.020 . 0.017 0.45 0.078 3.6 -0.182 0.03 0.035 0.50 -0.023 3.8 -0.153 0.04 0.057 0.55 -0.124 4.0 -0.128 0.05 0.082 0.60 -0.222 4.5 -0.082 0.06 0.108 0.7 -0.399 5o0 -0.0515 0.07 0.135 0.8 ; -0.545 5.5 -0.032 0.08 0.162 0.9 -0.656 6.0 -0.020 0.09 0.188 1.0 -0.735 6.5 -0.0125 0.10 0.214 1.1 -0.786 7.0 -0.0075 0.12 0.260 1.2 -0.813 7.5 -0.0045 0.14 0.298 1.4 -0.814 8.0 -0.003 0.16 0.329 1.6 -0.771 9.0 -0.001 0.18 0.351 1.8 -0.705 10.0 -0.0005 0.20 0.365 2.0 -0.629 0.22 0.371 2.2 -0.553 0.24 0.369 2.4 -0.481 0.26 0.361 2.6 -0.414 TABLE 4.— The Ar P_. (STF, E = 0,001 Ry, from scaled K+ T-F potential) 2 y r = y KE d ^ PEd^r^’ unnormalized 0.00 0.00 0.00 0.01 0.0001 247346.2 1.0 x 10"11 0.02 0.0004 61731.2 6.6 x 10"10 0.03 0.0009 27346.4 7.52 x 10~9 0.04 0.0016 15325.45 4.21 x 10"8 0.05 0.0025 9757.7 1.60 x 10’7 0.06 0.0036 6733.4 4.74 x 10-7 0.07 0.0049 4910.6 1.19 x 10“6 0.08 0.0064 3727.49 2.62 x 10"6 0.09 0.0081 2916.94 5.25 x 10"6 0.10 0.01 2337.8 0.977 x 10“! 0.15 0.0225 975.37 1.03 x 10”4 0.20 0.04 493.88 5.29 x 10“4 0.25 0.0625 280.1 1.79 x 10-3 TABLE 4— Continued y N I I U *Ed(*} ; 0.30 0.09 167.86 0.35 0.1225 103.35 : 0.40 0.16 64.94 0.45 0.2025 39.91 0.50 0.25 25.0 o l £ > o O 0.36 9.39 0.70 0.49 3.15 0o80 0.64 0.976 o . O 0.81 -0.1559 1.00 1.00 +0.4466 1.1 1.21 +0.4236 1.2 1.44 +0.949 1.3 1.69 +0.6359 1.4 1.96 +0.5043 1.5 2.25 +0.2996 1.6 2.56 -0.0731 PEd(r)» unnormalized 4.65 X 10-3 9.99 X 10-3 1.87 X 10“2 3.16 X 10“2 4.90 X 10"2 0.84 X 10”1 1.32 X 10"1 1.89 X 10”1 2.54 X 10"1 3.24 X 10”1 4.00 X 10”1 4.82 X 10"1 5.74 X 10-1 6.74 X 10_1 7.82 X 10”1 8.97 X 10”1 109 TABLE 4— Continued y r = KEd(y> PEd(r), unnormalized 1.7 2.89 -0.5429 1.02 1.75 3.0625 -0.5708 1.075 1.8 3.24 -0.0538 1.134 1.9 3.61 -1.2007 1.246 2.0 4.00 -2.457 1.345 2.25 5.0625 -3.27 1.421 2.50 6.25 -4.055 1.192 2.75 7.5625 -4.745 0.622 3.00 9.00 -5.272 -0.187 3.25 10.5625 -5.682 -1.000 3.5 12.25 -6.009 -1.514 3.75 14.0625 -6.274 -1.489 4.00 16.00 -6.492 4.25 18.0625 -6.673 4.50 20.25 -6.827 4.75 22.5625 -6.957 TABLE 5.— The Ar P9d (STF, from scaled K+ T-F potential) y r = y2 K$d(Y) P9d(r), unnormalized 0.00 0.00 0.00 : o . o i 0.0001 247346.2 1.0 x 10”11 0.02 0.0004 61731.2 6.6 x 10"10 j 0.03 0.0009 27346.4 7.52 x 10~9 0.04 0.0016 15325.45 4.21 x 10”8 0.05 0.0025 9757.7 1.60 x 10”7 0.06 0.0036 6733.4 4.74 x 10”7 0.07 0.0049 4910.6 1.19 x 10”6 0.08 0.0064 3727.49 2.62 x 10”6 0.09 0.0081 2916.94 5.25 x 10”6 0.10 0.01 2337.8 0.977 x 10”5 0.15 0.0225 975.37 1.03 x 10“4 ! 0.20 0.04 493.88 5.29 x 10”4 Ill TABL&--5-— (Continued) y r - y 2 K9d(Y> P^Cr), unnormalized 0.25 0.0625 280.1 1.79 x 10“3 0.30 0.09 167.86 4.65 x 10“3 0.35 0.1225 103.35 9.99 x 10“3 0.40 0.16 64.94 1.87 x 10“2 0.45 0.2025 39.91 3.16 x 10“2 0.50 0.25 25.025 4.90 x 10~2 0.60 0.36 9.409 0.84 x 10”1 0.70 0.49 3.174 1.317 x n r 1 o . 00 o 0.64 1.01 1.895 x 10"1 0.90 0.81 -0.114 2.542 x 10”1 1.00 1.00 +0.499 3.241 x 10”1 1.1 1.21 +0.487 3.999 x 10"1 1.2 1.44 +1.025 4.830 x 10”1 1.3 1.69 +0.725 5.756 x 10”1 1*4 lo96 +0.607 6.801 x 10”1 ZTT TABLE 5— (Continued) Y r = Y2 K9d(y> P9d^r^* unnormalized 1.5 2.25 +0.418 7.945 x 10-1 1.6 2.56 +0.061 9.161 x 10"1 1.7 2.89 -0.392 1.046 1.75 3.0625 -0.410 1.109 1.8 3.24 -0.684 1.174 1.9 3.61 -1.012 1.299 2.0 4.00 -2.248 1.414 2.25 5.0625 -3.006 1.381 2.50 6.25 -3.728 1.065 2.75 7.5625 -4.349 4.551 x 10"1 3.00 9.00 -4.800 -3.363 x 10"1 3.25 10.5625 -5.129 -1.091 3.50 12.25 -5.366 -1.548 3.75 14.0625 -5.537 -1.505 4.00 I 16.00 113 TABLE 6.— The Ar PEs (STF, E = 0.001 Ry, from scaled K+ T-F potential) 2 _ y r = y KE s ^ PEs^r^* unnormalized 0.00 0.00 0.00 0.01 0.0001 7356.2 1.0 x 10~4 0*02 0.0004 1431.2 3.74 x 10"4 0.03 0.0009 689.4 9.60 x 10~3 0.04 0.0016 325.7 2.236 x 10”2 0.05 0.0025 157.3 3.838 x 10“2 0 o06 0.0036 66.5 5.739 x 10”2 0.07 0.0049 12.4 7.893 x 10“2 0.08 0.0064 -22.61 1.026 x 10"1 0.09 0.0081 -46.113 1.280 x 10- "1 0.10 0.01 -62.216 1.543 x 10”1 0.15 0.0225 -98.950 2.833 x 10"1 0.20 0.04 -106.712 3.582 x 10"1 114 TABLE 6— (Continued) y r = y2 *Es(Y> PEg(r), unnormalized 0.25 0.0625 -103.90 3.312 x 10-1 0.30 0.09 -98.81 1.945 x 10-1 0.35 0.1225 -92.57 -0.2287 x 10”1 0.40 0.16 -85.06 -2.687 x 10”1 0.45 0.2025 -78.60 -4.854 x 10"1 0.50 0.25 -70.99 -6.238 x 10-1 0.60 0.36 -57.28 -5.821 x 10"1 0.70 0.49 -45.83 -1.696 x 10"1 0,80 0.64 -36.53 +3.989 x 10"1 0.90 0.81 -29.79 8.951 x 10"1 1.00 1.00 -23.55 1.171 1.1 i j j i | 1.21 -19.41 1.180 115 TABLE 6— (Continued) 2 ~ y r = y KEs^^ PEs^r^’ unnormalized 1.2 1.44 -15.72 9.475 x 10_1 1.3 1.69 -13.57 5.337 x 10_1 1.4 1.96 -11.74 8.145 x 10”3 1.5 2.25 -10.37 -5.590 x 10”1 1.6 2.56 -9.45 -1.105 1.7 2.89 -8.85 -1.579 1.75 3.0625 -8.40 -1.752 1.8 3.24 -8.27 -1.931 1.9 3.61 -7.85 -2.137 2.0 4.00 -8.46 -2.177 2.25 5.0625 -8.01 -1.468 2.50 6.25 -7.90 +9.206 x 10~2 116 TABLE 6— (Continued) Y r - y2 KEs(y) PES(r), unnormalized 2.75 7.5625 -7.92 1.767 3.00 9.00 -7.94 2.718 3.25 10.5625 -7.95 2.386 3.50 12.25 -7.97 8.367 x 10-1 3.75 14.0625 -7.98 -1.243 4.00 16.00 -7.99 -2.849 4.25 18.0625 -8.00 -3.141 4.50 20.25 -8.01 -1.896 4.75 22.5625 -8.02 +3.597 x 10-1 5.00 25.00 -8.03 +2.560 5.25 27.5625 -8.04 3.598 5.50 30.25 i I 117 TABLE 7.— The Ar Plls (STF, from scaled K+ T-F potential) r = y Klls(y) Pllg(r), unnormalized 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.00 0.0001 0.0004 0.0009 0.0016 0.0025 0.0036 0.0049 0.0081 0.01 0.0225 0.04 7356.2 1431.2 689,4 325.7 157.3 66.5 12.4 -22.61 -46.11 -62.22 -98.95 -106.71 0.00 1,0 x 10 -4 3.74 x 10 9.60 x 10 -4 -3 2.236 x 10 3,838«x 10 5.739 x 10 7.893 x 10 1.026 x 10 1.280 x 10 1.543 x 10 2.833 x 10 3.582 x 10 -2 -2 -2 -2 -1 -1 -1 -1 -1 118 TABLE 7— (Continued) Y r = y2 Klls(y) P ^ g(r), unnormalized 0.25 0.0625 -103.9 3.312 x 10”1 0.30 0.09 -98.81 1.945 x 10”1 0.35 0.1225 -92.56 -0.2287 x 10"1 ; 0.40 0.16 -85.05 -2.687 x 10”1 0.45 0.2025 -78.59 -4.854 x 10-1 0.50 0.25 -70.98 -6.238 x 10”1 0.60 0.36 -57.26 -5.821 x 10’1 0.70 0.49 -45.80 -1.696 x 10’1 o 00 • o 0.64 -36.50 +3.989 x 10-1 0.90 0.81 -29.74 8.951 x 10"1 1.00 1.00 -23.50 1.171 1 1.1 1.21 -19.35 1.180 TABLE 7— (Continued) 2 y r = y Kl l s ^ Plls^r^* unnormalized 1.2 1.44 -15.64 9.475 x 10”1 1.3 1.69 -13.48 5.337 x 10-1 1.4 1.96 -11.64 8.145 x 10“3 1.5 2.25 -10.25 -5.583 x 10”1 1.6 2.56 -9.31 -1.103 1.7 2.89 -8.70 -1.579 1.75 3.0625 -8.24 -1.754 1.8 3.24 -8.10 -1.935 1.9 3.61 -7.66 -2.148 2.0 4.00 -8.25 -2.199 2.25 5.0625 -7.75 -1.528 2.50 6.25 -7.57 -1.302 x 10“3 TABLE 7— (Continued) y r = y2 “l l s ^ Plls(r), unnormalized 2.75 7.5625 -7.52 +1.667 3.00 9.00 -7.47 2.718 3.25 10.5625 -7.40 2.568 3.50 12.25 -7.33 1.209 3.75 14.0625 -7.24 -0.807 4.00 16.00 -7.15 -2.599 4.25 18.0625 -7.06 -3.349 4.50 20.25 -6.95 -2.670 4.75 22.5625 -6.84 -7.914 x 10_1 5.00 25.00 -6.72 +1.530 5.25 27.5625 -6.65 i5.50 30.25 LIST OF REFERENCES LIST OF REFERENCES 1. 2. 3. 4. 5. 6 . 7. 8. 9. 10. 11. 12. 13. 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XXIX Pt. 2 (The Institute of Physics and the Physical Society, London, England, 1966), p. 373. Fano, U., and Cooper, J. W. Phys. Rev. 138 A 400 (1965). Racah, G. Phys. Rev. 6JL 537 (1942). Ogawa, M. Private communication, 1966. j Hudson, R. D. Private communication, 1967. I Madden, R. P., Codling, K., and Ederer, D. L. ! Phys. Rev. 155 #1 26 (1967). Beutler, H. Z. Physik £3 177 (1935). Ogawa, M. Private communication, 1966. Slater, J. C. Quantum Theory of Atomic Structure (McGraw-Hill Book Company, Inc., New York, 1960) Vol. II. t Cook, G. A., Metzger, P. H., and Ogawa, M. Private communication, 1965. i i i Hudson, R. P. Private communication, 1967. ! i i Cooper, J. W. Phys. Rev. 128 681 (1962). j I Comes,-F. J., and Salzer, H. G. Phys. Rev. 152 i 29 (1966). 1 Fano, U. and Prats, F. Atomic Collision Processes, edited by M.R.C. McDowell (North-Holiand Publishing, Co., Amsterdam, 1964), p. 600. Ogawa, M. Private communication, 1966. Lighthill, M. J. 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Creator Mendez, Antonio Juan (author)

Core Title A Theoretical Analysis Of The Rare Gas Autoionization Between The Doubletp(3/2) And Doublet P(1/2) Series Limits, With Applications To Argon

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Degree Doctor of Philosophy

Degree Program physics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

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Language English

Advisor Stephens, Phillip J. (committee chair), Kyner, Walter T. (committee member), Ogawa, Masaru (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-618847

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