Parameterization of Absorption-Line Profiles * Scientific Report # 23 Bruce W. Shore Harvard College Observatory November 1967 ABS TRACT The parameterization for attenuation cross-sections is discussed, with attention to the following details: prescriptions for "exact" calcula- tion of profile parameters, in which the effects of "distor- tions" are separated from the effects of multiple "scatterings": the validity of assuming independent (non-interfering) reso- nances: the specific case of autoionizing lines: connections with alternative parameterizations; the prohibition on negative cross-sections assured by unitarity: behavior at threshold: and the applicability of this parameterization to emission lines. n -1- I. INTRODUCTION Observations of photon (or neutron) attenuation typically disclose cross-sections with the energy dependence K Applied to photon projectiles, incident with energy w on an atom in state I, this attenuation cross-section reads a(1,u.l) = C(w) + (+rK) B K + (W - w IK) AK . (W - w + (trKla IKl2 K (Here, and throughout this paper, I use atomic units, e = h = m = 1; c = l/a ZE 137). The rapid variation of ~(I.,u)) with photon energy near the resonance energies u) = E - E IK K I traces the profile of an absorption line. Nuclear physicists have, for many years, used such param- eterizations, although the validity of formula (1.1) is by no means restricted to nuclear collisions. Until the recent revival of interest in ultraviolet spectroscopy, atomic spectroscopists had little need for such elaborate parameterization: for non- autoionizing lines, the parameter A vanishes, B is equal to K K 27ra times the oscillator strength, and the observed width TK -2- reflects conditions in the absorbing medium rather than the natural radiative width. For autoionizing lines, the profile parameters A B and E each have empirical and physical K’ K’ ‘K’ K utility2, just as do the more familiar parameters of quantum defect and oscillator strength. Equation (1.1) is only one of several mathematically equiva- lent representations (parameterizations) of Q(E). Other expres- sions have also been suggested. Burke3, Smith4, and Peterkop and Veldre6 have recently reviewed the theories for explaining resonance structure in collision cross-sections for electron and photon scattering; Burke3 and Smith4 summarize the presently available values for profile parameters. To date, there has been little effort to determine, either theoretically or exper- imentally, profile parameters other than resonance width and resonance position. I hope the present article will stimulate experimental tests of formula (1.2) for the description of complicated photoionization cross-sections and will encourage computation of profile parameters. Before we judge the usefulness of formula (1.1) for fitting and predicting cross-sections, several points deserve attention. Are the resonances really independent, or is there interference between resonances? How does formula (1.1) compare with other commonly used parameterizations? Is formula (1.1) consistent -3- with the unitary property of the scattering matrix or will it give erroneous negative cross-sections? Do autoionizing lines seen in emission have the same profiles as absorption lines? The present paper addresses these questions, and provides a more refined prescription for the computation of profile pa- rameters than the formulas given in reference l. -4- 11. BASIC DEFINITIONS' For a system comprising projectile and target, whose composite quantum numbers are a and whose combined energy is E, the attenuation cross-section o(a, E) is wa where F denotes the projectile flux corresponding to the choice a of normalization for the incident wave Jr , T =- <$cITIJra) is a ca the transition amplitude linking initial state JI with final a state Jr and denotes a generalized sum over final states (a C' 5 C sum over discrete labels and an integration over continuous labels). Making use of the unitary property of the scattering matrix S = 6 - 21-1 &(Ea - Eb) Tba, one can write (2.1) in ba ba the alternative form 2 O(a,E) = - - Im Taa . Fa The optical theorem, Eq. (2.2), expresses mathematically what experimenters long ago recognized: in studying neutral projectiles it is simpler to measure beam attenuation in the forward direction than to collect the scattered flux from all -5- directions. It should be clear that a(a,E) has the form Q(E) of Eq. (1.1) if T can be written aa K + D - iC. (2.3) Fa aa K Thus if the scattering amplitude can be expressed as the sum of independent resonance contributions, attenuation cross-sections will display the energy dependence Q(E) of equation (l.l), with no interference between resonances. -6- 111. THE DEFINITION OF INDEPENDENT OVERLAPPING RESONANCES Methods for breaking the operator 1 T-V+V- V E V + VGV E+-H into resonant (T ) and non-resonant (T ) parts, T = T + T Q P P Q’ have been discussed by Fonda and Newton7, Feshbach*, and Zhivopistsevg amongst others. The resonance structure can be brought out most readily by the use of projection operators, l=P+Q PP = P QQ = Q PQ = QP = 0 (3.2) such that Q projects resonance states, and P projects possible initial and final states”. The introduction of an operator1 t = V + VP [E+ - Ho - WP1-l PV (3.3) then permits one to write T = V + V [E’ - Ho - VI-’ V = t + tQ [E - Ho - QtQ1-l Qt (3.4) -7- and so to identify the non-resonant part (elastic scattering and direct processes) T =t (3.5) P and the resonant part T = tQ [E - Ho - QtQI'l Qt (3.6) Q We can now introduce', at least formally, a set of resonance states 4 K' which satisfy the equation [Ho + QtQ - EK1 djK = 0 (3.8) with complex eigenvalue (3.9) -8- Since t is not Hermitian, the familiar orthogonality theorem for eigenstates having different eigenvalues applies to <$:I <$:I qL> rather than to the usual (3,. I qL) , where @' is the adjoint of % . By using the fact that Q[H"+t]Q = Q[H"+@VP E - 1PHP PV - iir VPb(E-II)PV]Q Q[H" + VG V - i7r V I V]Q (3.10) P P (8 denotes principal value) where H" and QWQ are real, one can * show that1" @:= GK. The bi-orthogonal expansion is therefore (3.11) L, K Just as with conventional calculations of atomic structure, states of different energy are orthogonal, but degenerate states need not be. Equation (3.11) requires that we determine our degener- ate states to diacpnalize the interaction QtQwithin a manifold of degenerate states. This requirement can be met with conven- tional approaches employing angular-momentum coupling and/or the diagonalization of comparatively small matrices. The reso- nance part of the scattering amplitude can now be written (3.12) K -9- We thereby obtain, as desired, -.&e resonance scattering amp1 tude as a sum over independent (though not necessarily well- separated) resonance terms. To determine the resonance parameters, it proves convenient to write n t=V (GOV - irg V) (3.13) ) P P L n=o where P 6(E - Ho)P 9P GoP E @[P E - Ho 1 Go @[Q E - HO (3.14) Q We can then write a perturbation-theory solution of Eq. (3.8) as $K = [l +GoV (G'V + GoV - i-rrg V) (3.15) Q Q P P n=o is a combination of degenerate where, as in reference 1, cpK eigenatates of Ho chosen to diagonalize V within a degenerate -10- manifold. We may also write Eq. (3.15) as (3.16) The preceding prescription yields an unnormalized state 4K which satisfies the condition since {cp Icp } = 1. (3.17) KK Similarly, the complex value & can be written K (3.18) The preceding expressions, extensions of previous results13, do not separate explicitly the real and imaginary parts of K' For that purpose, it is useful to introduce (real) states '4' K' (GOV + GoV)n qK 2 A'pK. (3.19) Y=K iQP L n=o This expression for Y taken with the requirement that degen- K' erate Y have diagonal elements of V, K if E = E and K # L, (3.20) K L -11- is a description of the states obtained in conventional calcu- lations of bound states: all integrations over continuum states require principal values. These Y contain mixtures of config- K urations, including continuum functions, whose energy differs from E Thus unlike the states they do not belong exclu- K' K sively to the Q set: QYK + PYK = but PTK # 0. K (3.21) BY rearranging sums, one can then obtain the formulas (3.22a) rK = 2~ (YK/vgpvpK) + 0(v4) (3.23a) (3.24a) B =-{2 KF I<'Klvl la)la - T2 I < PKIVgp vAI$a)12 + 0(v6), a 1 (3.25a) n where O(V ) indicates that further terms involve n products of V. such corrections are of the form -12- and thus they describe the effects of multiple scatterings between equal-energy states. In contrast, the operators VG V, VGOV, and A describe "distortions" which mix configura- P Q tions of different energy. Formulas (3.22) - (3.25) therefore give a prescription for profile parameters which separates the effects of distortion and of multiple scattering. In obtaining the preceding formulas I have neglected correc- tions to the normalization of 4 K: 2 1. (3.26) Here e is the eigenvalue of an unperturbed state: HoqK = eKqK. As we shall note in section VI, this approximation is consistent with unitarity restrictions. It is also useful to introduce (real) distorted "continuumii the counterparts of the "bound" states Y states Ya , K'- Y= V + Go V)" fa (3.27) a P A$,. L n=o -13- The Ya states are the Y1 states of Lippmann and Schwinger14; the wavefunctions are standing waves.