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DARESBURY SYNCHROTRON RADIATION LECTURE NOTE SERIES No.1
PHOTOIONIZATION
by
C. Bottcher
Science Research Council
DARESBURY LABORATORY Daresbury, Warrington, WA4 4AD DISCLAIMER
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Printed by McCorquodale Printers Ltd., NAwton-IA-Willow,;. M ersey side. DARESBURY SYNCHROTRON RADIATION LECTURE NOTE SERIES No.1
PHOTOIONIZATION
C. Bottcher Department of Theoretical Physics University of Manchester
Notes on a series of lectures given at Daresbury Laboratory, November 1973
Science Research Council DARESBURY LABORATORY 1974
iWASlER
FOREWORD
These lectures were given in November 1973 to experimental physicists using the Synchrotron Radiation Facility at Daresbury. The aim was a modest one, to survey the basic ideas of the subject and introduce some current theoretical developments.
I must thank Dr. Ian Munro for making all the arrangements connected with the lectures and members of the audience for providin~ stimulating feedback.
C. Bottcher
(i·ii) . v-·
<. 'c
THIS PAGE. WAS INTENTIONALLY LEFT BLANK CONTENTS Page Foreword ( ; ; ; )
1. BASIC PHOTOIONIZATION THEORY 1.1 Introduction 1.2 Expressions for the Cross Section 1.3 Oscillator Strengths and their Applications 5 1.4 Simple Examples 5 1.5 Ionization by Fast Projectiles 7.
2. CALCULATION OF NON-RESONANT PHOTOIONIZATION CROSS SECTIONS 8 2. 1 Introduction 8 2.2 Born Approximation 8 2.3 Hartree-Fock Approximation 9 2.4 The Method of Polarized Orbitals 10 2.5 Quantum Defect Theory 10 2.6 Close Coupling and Variational Methods 12 2.7 Recent Calculations 13
3. THEORY OF RESONANCES 23 3. 1 Introduction 23 3.2 Feshbach Projection Operator Theory 24 3.3 Fano Profiles 28 3.4 Calculations 31
4. PHOTOIONIZATION OF HEAVY ATOMS 33 4. 1 Introduction 33 4.2 Minima in Photoionization Cross Sections 33 4.3 Discrete Levels in the Far Continuum 36 .4.4 Other Topics 37
5. PHOTOIONIZATION OF MOLECULES 39 5.1 Non-resonant Photoionization of Molecules 39 5.2 Vibrational Resonances 42 5.3 Electronic Resonances 42
Appendix: SOME RESULTS IN SCATTERING THEORY 46
(v) BASIC PHOTOIONIZATION THEORY l. 1 Introduction Photoionization is the process + .. hv + A(p) ~ A (q) + e ( 1 ) where A(p) is an atom or molecule in state p; A, A+ are usually in their ground states but they can be in excited states. In photo-detachment the electron is removed from a negative ion
hv + A- ~ A + e . (2)
Any particle X moving sufficiently fast behaves like a photon; the cross section for X + A ~ A+ + e + X ( 3) is closely related to that for (1) and fast electrons, ions, etc. are increas ingly being used to complement traditional spectroscopy (see 1.5). Photo ionization is usually seen as a probe of atomic and molecular structure and it is increasingly used for this purpose in chemistry. However, these lectures ernphd~:;ise the relation between photoionization and collision processes.
The units used will usually be atomic units e = m = n = 1; atomic unit of length a0 ~ 5 x lQ-9 em, atomic unit of energy~ 27.21 eV. Otherwise CGS units 1 2 are used. Two recent review articles are highly recommended( ' ).
1.2 Expressions for the Cross Section We begin by considering a beam of photons of angular frequency w and energy spread ~E falling on an atom A. If the flux is N~E cm- 2 s-1 the rate of photoionization events is
,!» = a ( w) N~E s- 1 (4) where a is the cross section (cm2 ). The energy density in the photon beam is awN~E, a being the fine structure constant(~ 1/137, 1/c in atomic units). To calculate cr we must consider the Hamiltonian of an atom in an electro magnetic fie 1d; 1et~ be the unperturbed atomic Hamil toni an and dthe vector potential of the EM field. The perturbed Hamiltonian is
(5)
For a plane polarized wave
~ = ~ cos(wt- ~ • !), K = 2~ f wavelength.
0 If the wavelength ? 10 A, ~·r hardly varies through the extent of the atom so we can take
.st' = A cos wt . (6)
This is the dipole approximation. We are interested in a tran!ition between two unperturbed atom1 c states
Fermi's golden rule states that under a periodic perturbatinn Veiwt + V*e-iwt the rate of transitions 1 ~ 2 is
(7)
If we let K = K~z' A = A~x the energy density is
o(w) = awN (8) per unit energy interval. From (5) - (8)
2 !JP = (~ ~r p ( w) 1 < h 1 pX I
1 (9) It is easily shown that
whence
2 - 1) <
2 > > ( E E I I = i <
( 10}
2 If the radiation is unpolarized*
( 11)
We must look at ~ 1 and ~ 2 more closely. If 1 is the.bound state ~ 1 = ~B is normalized in the usual way,
< ~B I ~B > = 1
The final state 2 represents an electron being ejected with momentum ~' energy E = k2/2. In practice we observe electrons in a range of energies oc and solid angles bn. If the whole system is enclosed in a cubical box of side L the normalized states are asymptotically, -3/2 ik·r ~2 'V L e - -
The number of such states in the range bEbn is
2 n = [~,]3 k bkbn = [~.]3 k bEbn ( 12)
To get the rate of transitions into these states we multiply (11) by (12) and use (8),
(2TI) 2 2 bEbn n~ = (awN) kl < hlrl ~2 >1 3 [~,]' (21Tl2 2 = aw I< ~B lrl ~(~) > j NbEbn ( 13) 3 where f(~) is the ener·gy-normalized wavefunction
( 14) e ik·r - -
Comparing (13) with (4) we find that
* We are writing these formulae for a one-electron system. In a system with many electrons ! is replaced by N I r. j =1 -J
3 do(w} 41T 2 am = --3-- aw I< ~B 1~1~(~) > 12 ( 15)
The total cross section is obtained by integrating over all directions of ~'
o(w} ( 16)
Given the frequency w of the radiation, the wavenumber k of the ejected electron is fixed.
( 17)
The set of functions ~(~) have the desirable property that
<~(~)I'(~·)> = 6(E- E.) 6(n- n•) ( 18)
A commonly encountered case is the photoionization of the 1S ground state. This is only connected to the P wave of the continuum. ·
~(kp) = Fkp (r) Y1 m (e, ~), ( 19) sin(kr - ~ + 6R.} FkR. . (~]~ "" r
1 1 ~(kp} is energy normalized, < ~(kp)l~(k p) >=6(E-E ).
After some algebra (16) becomes
o(w} (20)
An important feature of (16) is that a involves the fine structure constant a and so photoionization cross sections are small ("-' lQ-18 cm2 as opposed to 10-16 cm2 for electron-atom cross sections).
We can use the principle of detailed balance to obtain the cross section for the inverse process, radiative recombination,
e +A+ -+ A+ hv . ( 21 )
If w8 is the statistical weight of the bound state,
2w 2 = cr(k ~ B) (22) c2
4 1.3 Oscillator Strengths and Their Applications The cross section (16) can be written in terms of the dimensionless oscil lator strength f(£), £ = k2/2, df o(w) a dE (23)
2 ~: = ~w j dn I < ~B Ir I '~' ( ~) > 1 (24) Denote the oscillator strengths for transitions from B to the discrete states s by fs and the energies of the discrete states by £s. The complete set of oscillator strengths, both discrete and continuous, satisfy certain sum rules. If we define the sums 00 fs d£ .df S(k) = z: k Cl£ (25) s (I+I£sl)k + f (I + £) 0
S(O) = N ( 26a)
2 N 1 ) < r.) 21 (26b) s ( = ~B I( Z: -J ~B > 3 j =1 S(2) = p (26c) where N is the number of electrons in the system and P the static polarizability. (26a) is the well-known Thomas-Kuhn sum rule. By combining the sum rules with experimental data, including photoionization cross sections, it is possible to build up consistent oscillator strength distributions for many systems. These distributions have a large number of applications, e.g. in calculating long range forces between atoms and molecules. More information is contained in 4 Dalgarno( 3 ) and Starkschall and Gordon( ).
1.4 Simple Examples The calculation of photoionization cross sections has been reduced to that of finding '!'(~) and evaluating (16). It is a comparatively simple matter in most cases to determine ~ 6 . Only one problem, the photoionization of hydro- genic atoms, is soluble in closed terms. In all other cases one must make approximations or use numerical methods.
5 [f we consider'the ls state of a hydrogenic system (nuclear charge Z) we can use (20). The matrix element is worked out in Bethe and Salpeter(s),
exp(-4v arc tan ~)v
(27) (v = Z/k)
We can see that
, 1 im df 1im df e:-+o ae = d£ =
It can be shown by calculating the discrete oscillator strengths that 1im n -+ n3f cl ~ n = It is a general result that the density of oscillator strength is continuous at a spectral· head. This is a useful check on experimental data. Some other properties of the hydrogenic system are perfectly general and worth noting. The binding energy I = Z2/2 so the dependence of f on I is
df I + e: d£ 'V I2
The larger I the smaller f. The matrix element (27) has poles at v =in, i.e. at£= -Z2j2n2, the bound states of the system.
An instructive example is provided by photo-detachment. Let the system A have binding energy K2/2. Neydlive systems are very diffuse so the wave function is well represented by its asymptotic form,
(28)
The·cont1nuum wavefunction is also represented by its asymptotic form,
(29)
Then the matrix element (20) is
(30)
6 The approximations leading to_ (30) should be valid if k >> K.. The matrix ele ment has a pole at the bound state £ = -K2/2. However, the cross section at short wavelengths has a different behaviour than in photoionization,
-3/2 a(£) '\, £
1.5 ·Ionization by Fast Projectiles
Consider a very fast particle of charge Z, velocity v0 , wavenumber k0 which collides with and ionises A. The cross section for producing ele_ctrons of 6 energy £ is (Matt and Massey( ))
a(£) + 4n df/d£ ( 31 ) 2 k0 I + £
This is only valid for small values of £. For E: » I the electrons in A behave as though they were free,
4n a ( E:) = (32) k~ £
Because of (31) photoionizntion cross section!; can be obtained fr-t:11il ~.,;ull"is1on experiments. However, the interpretation of these experiments may be rather complicated. For a given ko all values of E: are possible and so the ejected electrons must be energy resolved. Furthermore (31) is only valid at very high energies (1000 eV for electrons) where a is quite small. Thus experiments must sometimes be done at lower energies where the Bethe approximation is beginning to break down.
7 2 CALCULATION OF NON-RESONANT PHOTOIONIZATION CROSS SECTIONS
2~1 Introduction In this section we shall briefly look at the methods available for obtain ing continuum wavefunctions and then consider two simple case~, helium and the a 1ka 1i meta 1s, where comparison can be made between theory and experiment. Each method will be very briefly outlined; to arrive at numerical results an immense amount of detail must be mastered and this will be found in the references cited. We shall only consider atoms for the moment, since the photoionization cross sections of molecules cannot yet be calculated systemati cally. Some current research on molecules is described in section 5.
We recall that o{w) is proportional to the square of a matrix element,
N M(B < r.!l¥(k) > (33) ~ ~) = ~B I I -J - j =1 where ~B and 1¥{~) are eigenfunctions of the atomic Hamiltonian, 1 N N N N 1 (34) //J'rA = - 2 I - z I I I r .. j=l j=l i =1 j =i +1 1J
(35)
EA+ is the energy of the ionic core, 1 is the ionization potential, 8 w = IB + i k2.
2.2 Born A~~roximation Suppose the initial and fi na 1 wavefunctions have the form
~s(!'I !'N) = Fs(rd 4>A+(!'z !'N) ( 36) 1f'k ( !' 1 . . . !'N) = Fk (rd 4>A+(!'2 . . . !'N)
8 where the core wavefunction is unchanged and exchange with the core is neglected. Then ( 37)
as for a one electron system. The simplest approximation (plane wave Born) is to take the ejected electron wavefunctiori as a plane wave, unperturbed by A+, -3/2 1/2 . = (2n) k exp(l~·r) (38)
This is quite a good approximation 'for large k, say k2 ? 10 IB. It is instruc- tive to evaluate the Born approximation for hv + H(ls) and compare with the exact result (section 1). A much better approximation (Coulomb Born) is to take Fk as a coulomb wave. This is usually accurate to better than a factor of 2 for light systems, right down to threshold.
· 2.3 Hartree-Fock Approximation For the sake of illustration we consider the simplest case where there is more than one electron, hv + He(ls 2) ~ He+(ls) + e(kp). If we represent the final state by a product,
the continuum orbital F satisfies
[- .l2 v2- .l 2 k2 +'Yl d( ] + [! 0(() F(() The second term in (39) represents exchange.This approximation allows for short range interactions between the ejected electron and the He+ core but not for the polarization of the core. It is useful at lower energies than the Coulomb- Born approximation, especially in heavier systems. It is not generally appreciated that the Random Phase Approximation (RPA) which is usually discussed in the context of many-body perturbation theory can be obtained from a simple 7 generalization of (39) (Amusia et al( ); Jamieson(·s)). The initial state B may be calculated in a variety of approximations, though if ~k is not an exact 9 eigenstate of~ there is no reason to believe that a very sophisticated choice is better than a simple one, e.g. where $ is the ls orbital of He+. 2.4 The Method of Polarized Orbitals This is a generalization of the Hartree-Fock approximation in which the core orbital $(r 1 lr2 ) is corrected for the effect of a slowly moving electron at (40) where w(r) = o(r3) for small rand $(l) is a known function. The use of {40) leads to a long range polarization term in (39).The method is extremely useful and has an extensive literature (c.f. Drachman and Temkin( 9 )). 2.5 Quantum Defect Theory (QDT) Consider a series of Rydberg states converging on a continuum, e.g. He(lsnt) ~ He(lskt). The ionization potentials can be written I ( 41 ) nR. = where pnl is known as the quantum defect. The continuum orbitals have the asymptotic form 1 . -r s1n (42) field would In a pure coulomb (- Z/ r) o1 be zero. The fundamental result of QDT is 1im 1i m n ~ oo npnt = k ~ 0 ot(k), ( 43) so that the phase shift at k = 0 can be extrapolated from spectroscopic data. 10 To evaluate the photoionization matrix elements we write -r (44) where N is a normalization factor; Fi and Gi are regular and irregular coulomb wavefunctions. QDT has been generalized to the many-channel case when there . (10) are several thresholds close together e.g. 25ki, 2pkt' in Be . To understand where (43) comes from it is instructive to look at the coulomb approximation of Bates and Damgaard(ll) for bound-bound matrix elements. They considered that an electron in a Rydberg state was adequately r~presented by a one-electron wnvP.func:t.ion F.n)(, n satisfying. the equation - .l ~ + t(t + 1) - !:. + I ] F = 0 [ 2 · dr2 r2 r nt nt The effective potential is approximated by -Z/r but the ionization potential Int is fixed at its experimental value. Let zz 1 k2 I nt = I = 2(n - ~) 2 For large values of r the leading term in Fnt is (kr)n-~ exp(-kr) = exp [-kr + (n - ~) tnkrJ comparing with the continuum solution exp i [kr +I tn(2kr) +6] i z z and putting n - ~ = lk = r we find that 6 = (n - ~) .! 2 Thus n~ turns up as a phase shift when the bound wavefunction is continued into the scattering region. This is only a heuristic argument and does not give the correct relation between ~ and 6. 11 2.6 Close Coupling and Variational Methods We shall now discuss the methods used to study electron-atom and electron ion scattering at low energies and which involve large scale computing. Consider the scattering of electrons by an ion A+ whose eigenstates are ~{aSL); S, L are the total spin and orbital angular momentum and a includes.all other quantum numbers. We couple ~(aSL) to tne spin and orbital angular momentum of ·the scattered electron to form an eigenfunction of total spin and orbital angular momentum St and Lt' = ~ I < StMst ISMssms > < LtMLtl LML ~m~ > M t~smsm.t 5 (45) x ~ {aSLM5 ML) xsms Y~m~ The total wavefunction of the system e +A+ can be expanded as, (46) (r = aSLs~). The radial functions Fr(r) satisfy coupled integrodifferential ·equations which may be obtained by substituting (46) into the Schrodinger equation (35), F = L (47) [- i d~: + r r Vis a local (direct interaction) and Wa nonlocal (exchange) interaction. By solving these equations numerically one can determine the .elements of the T-matrix and hence the cross sections. _Those unfamiliar with scattering theory should consult the appendix, (A5). Photoionization matrix elements are calcu- lated from the Fr themselves. In practice (47) are only used for the lower partial waves while the Born approximation is adequate for the others. A full account of the derivation of (47) and of the finite difference techniques used to solve them is given in the book by Smith(lz). A recent development (e.g. 13 Burke and Webb( )) is to augment the eigenstates of A+, ~{a), by pseudostates, ~(~),which ~epres~nt the continuum of A+. As the number of pseudostates included in the calculation is increased the cross sections appear to converge to a 1imit. 14 The Kahn variational method( ) may be regarded as a way of solving (47). 12 Instead of integrating the equations numerically each Fr is expanded in terms of basis functions, Fr = sr + ).Cr + I a .U. ( 48) j fJ J where sin, cos (k r - ir 'If ) and u. is a square integrable (i.e. bound sr, cr"' r 2 J n·-1 -s ·r state type) function, e.g. r J e J ; Numerical close coupling techniques have been applied to most atoms (and many ions) up to argon. Except for the case of hydrogen the results are not yet quantitatively reliable. The variational method is much more difficult to implement in practice, but it has recently been applied with great success to e +He, Li, Na, K (Nesbet and Oberoi(ls)). The so-called "R-matrix inethod" probably offers most promise for the future. This expresses the T-matrix in terms of another matrix whose elements are given by = I ( 49) j where g . is independent of E. The outstanding advantages of this approach are CtJ that (i) one calculation suffices. for all energies and (ii) unlike-in the Kohn variational method no complicated i~tegrals need be evaluated. Fuller accounts are given in the references(lG,l 7 ). 2.7 Recent Calculations In this section we give an account of some problems which have been the subject of recent interest, and for which theory and experiment or different theories can be compared. Before coming to these we comment on figs.l - 4 which· illustrate general features of photoionization cross sections. · We con fine ourselves to non-resonant behaviour. Figure 1 compares the cross sections for CH 4 and Ne which are somewhat similar: both are closed shell systems with 10 electrons. One can see the L, • Kedges where inner electrons start to be ejected . 13 ILa.m IL, (Nol IK(C) '"(No) 10 l "'Q 1 b .01 .oo L_o ____JL.....J_...J....J....LJ..J..1J.L.oo_...___...... _.u..u1~0oo-:---'-·'·""~ 10,000 Fig. 1 Photoionization cross sections of neon andmethane. Z0 40 iO 10 100 1!0 .. lCD E!tV) Fig. 2 Photoionization cross section of neon theory and experiment. Figure 2 shows experimental and theoretical results on Ne. One can ex- press the dipole matrix element as either < ~ 8 lrl 1{1(~) ,. ur (I+ ~::}- 1 .: ~B 1~1 'i'(~)) • ( 1ength form) (velocity form) For exact wavefunctions these forms are identically equal, but if approximate wavefunctions are us~d they usually differ, as the figure shows. If they differ greatly the wavefunctions are certainly bad, but the converse is not true! 14 -6 -I -6 -4 -2 0 2 4 6 4 (o) E(eYI s 6 (b) 4 s 8 6 tO 8 \ Fig. 3 Oscillator .strength distribution in the discrete and part of the continuous spectra of (a) H (theory); (b) Li (experiments). The plot of Es vs s in the lower part of the figure relates to the construction of the histogram. 10.0 ... \ ' /AUTOIONIZING STATES. ~ . ' '\ ' ' 10 ~" ' .01 ... \, . ' ' '\:~. ' N' 0 ...... : ' .... E ...... ' u ...... !! .. ~ ...· ...·· 'Q . ',, ' b ----, -, '-...... -...... 0 ---, -...... ' t)...... '- ...... 001 ~- ...... -- -.....-._ ...... --...-..._ ' -, - -... , ...... o A ...... 0 100 E(e'l) 200 .300 Fig. 4 --- Born approximation;-·- hydrogenic approximation; -close-coupling. 15 Figure 3 shows the distribution of oscillator strength in He and Li, and in particular its continuity across the spectral head. In fig.4 theory and experiment are compared for He, the simplest system after H. At low energies, close coupling and at high energies, Born calcula tions agree perfectly with experiment. 0·6 0·5 0·4 I ;;:.. ££ 0·3 :01~ 0·2 0·1 0·0 0·5 1·0 1·5 E (Ry) Fig. 5 -- Norc:ross; -·-·- Dalgarno et al.; --· Barden. The photoionization of the metastable 23S and 21S states of He are of some interest to astrophysicists. They were originally calculated using close 19 coupling methods{Is). Some time later ·oalgarno, Doyle and Oppenheimer( ) calculated the triplet cross section using a Gre~n function technique, and obtained rather different results (fig.5). We shall not have space to describe the Green function method here, but it is· highly promising for the future. One calculates a Born matrix element with a scattering correction given by (~ 0 is a coulomb wave). GE is now replaced by a discrete finite sum 16 N I~· > < ~·I I J J j= 1 E - E. J where ~j are bound-type states obtained by diagonalizing the Hamiltonian in a discrete basis X.. Figure 5 also shows the results of a very simple model potential calcul~tion( 2 o) which agrees very well with the Green function results. 2 The cross sections have now been measured( l) and Table 1 ·shows that. they agree remarkably well with our model. TABLE Photoionization of He (2 35) o 10-22 m2 10-22 m2 >. nm 0 experiment model calculation 260 6.5 ± 1.0 7.2 240 5.5 ± 1.5 6.5 Our last example is the photoionization of the alkali metal atoms. We can see in fig.3 that the Li oscillator strength density has a minimum at n = 3 (i.e. at the 2s- 3p transition). In the heavier alkalis this minimum is found in the continuum just above the threshold. If we define matrix elements M1 = (where n0s is the ground state orbital) the cross section is proportional to 2 2 IM 1 1 + 2IM 31 . The minimum occurs because M1 and M3 change sign once as k 22 goes from 0 to oo, Seaton( ) pojnted out that because of the spin-orbit interaction M1 and M3 are not both zero at the same energy and so the cross section is never exactly zero. Fano( 23 ) further deduced that at a zero of : either M1 or M3 the photoelectrons ejected by circularly polarized light would have a definite spin, i.e. they would be polarized. This effect has now been used to produce low energy polarized electrons( 24 ). 17 A detailed theoretical study of near threshold effects has been made using a one electron model(zs)_ The alkali metal atom is regarded as a single elec tron moving in a potential V(r) adjusted to reproduce the ~bserved spectrum of Rydberg levels(zG). In this model one must correct the dipole operator in M by adding the dipole moment induced in the core by the valence electron, M = < 2S lr { 1 - ad w{r) }1 2P > . r3 . w is a cut off function such that as r ~ oo, w ~ 1 and as r ~ 0, w ~ r 4 . (Hameed 27 et al( ). The behaviour near the minimum is very sensitive to the core correction so Weisheit adjusted w until the position of the minimum agreed with experiment, fig.6. The model then predicts other aspects of photoionization; in particular the "perturbation function" which measures the polarization of the ejected electrons is in very good agree- ment with experiment and this is shown in fig.7a, b, c. In fig. 8a, b, c, d we summarize theoretical and experimental cross sections near the minimum. Table 2 presents all the available data on the position and magnitudes of the minimum cross sections. TABLE 2 Alkali-metal-atom ground-state photoionization-cross-section minima. The tabulated cross sections are in units lo-zo cm2 and the tabulated 0 wavelengths in A Present work Seaton Experi menta 1 a m1n. Am1n • a m1n. a m1n. Am1n • Na 0.0009 1940 0.001 < 0.3 1920 ± 30a 0.1 1950 ± 50b K 0.22 2720 0.03 0.4 ± 0.2 2725 ± 15a 0.2 ± 0.2 2675 ± 75c Rb 0.40 2500 0.4 0.8 ± 0.3 2480 ± 25a Cs 2.68 2685 3.0 6 ± 1 2650 ± 25a aMarr and Creek. bHudson and Carter. cHudson and Carter. 18 0 0 I-z 0.01 w ~ 0 ~ z Q I- (/) -0.01 z <( ·a: I- - 0. 0 3 L._J_l.--'-..L..J._J_..L...l-'-..L..J.__j_-L-J___j_-1---JL-L__J_J 0.06 0.10 ·0.14 0.18 0.22 0.26 PHOTOELECTRON MOMENTUM(o.u.l Fig.6 Potassium transition moments for ground-state photoionization. The moments M include the core-polarization correction,.while the moments R do not. Subsc~ipts (1) and (3) refer to transitions to degenerate· (j = !) and (j = 2 ) continuum levels, and the subscript (*) refers to moments computed by neglecting the spin-orbit perturbation of continuum p orbitals. When multiplied by (2/nk) 1/ 2 , these bound continuum moments join smoothly at the ionization threshold to bound bound moments. 12 10 w 8 ~ >< z 6 0 1- 4 u z ::::> u.. 2 z 0 0 X 1- Fig.7(a) Perturbation function x(E) for potassium. The filled circles are results obtained with rc = 4.220a 0 , the crosses are results obtained with rc = 4.635a0 , and the triangles are results obtained by neglecting the core-polarization correction to the dipole transition moment. The shaded area represents the width of 1 standard deviation in the • experimental determination. 19 RUBIDIUM UJ >C z 0 5 I= u z ::::> ~ z 0 ~ ~Ot-----~--¥.~.------1 ::::> ...... a:: UJ 0.. • • -5 • 4.0 50 6.0 PHOTON ENERGY E(eV) Fig.7(b) Perturbation function x(E) for rubidium. The filled circles indicate values computed with rc = 3.505a 0 , and the shaded area indicate.s the width of 1 standard deviation in the experimental determination of x. g >C 1.0 z 0 ~ uz 1\ ::::> u.. z 0.0 f------~------1 0 ~(I) a:: ...... ::::> ffi -1.0 0.. CESIUM - 2 .0 .____J,_ _,__ _.__..._____,~, _ __.____..x....J..._J 3.8 4.2 4.6 5.0 PHOTON ENERGY E(eV) Fig. 7(c) Perturbation function x(E) for cesium. The filled circles and triangles are results computed with rc = 4.834ao and rc ~ 6.100ao, respectively. The shaded area indicates the width of 1 standard deviation in the experimental determination; the full curve indicates the quadratic leas·t-squares fit by Kessler et al, to their experimental determination ofx(E). 20 N-; 20 ~u SODIUM • • 'Q • z 0 • f= (HUDSON) • u and 4.1 CARTER ~· Vl 12 Vl .. Vl 0a:: • u • z 8 0 • f= 6 <{ • N z 4 0 g 2 0r a.. 0 2400 2000 1600 1200 0 PHOTON WAVELENGTH (A) Fig.S(a) Sodium photoionization cross section. The curve indicates results obtained by including both the spin-orbit interaction and the core polarization correction to the dipole transition moment. · The filled circles are measurements reported by Hudson and Carter. N E 30 u ·2 POTASSIUM 'o 25 z 0 (HUDSON~ • f- 20 and u 4.1 CARTER (/) (/) 15 (/) 0 a:: u z 0 f- <{ 5 N z 0 6 0 f- 0 IIQC I 0 a.. PHOTON WAVELENGTH (A) Fig 8(b). Same as fig. 8(a) for potassium. The filled circles are measurements • reported by Hudson and Carter. 21 .c ~ 0.12 .---.---.,.---....,----.------.---.-----, RUBIDIUM I \ I \ \ I \ \ 5.0 6.0 7.0 PHOTON ENERGY (eV) Fig.S(c) Rubidium photoionization cross section (1 Mb = 10-18 cm2). The full curve and dashed curv~:> indir.atP., rP.!\pP.r.tively, rP.sults obtained with and without the core-polarization correction to the dipole transition moment. The bars indicate values measured by Marr and Creek, and the cross section minimum calculated by Seaton. ~ I ~ z I Q 0.20 ..... frl CESIUM (/) I (/) l (/) 0 16 a: u I z I 0 0,10 I ~ 0.00 ~- 4.0 5.0 6.0 7.0 8.0 PHOTON ENERGY (eV) Fig. S(d) Cesium photoionization cross section (1 Mb = 10-18 cm2). The full curve indicates results computed with rc = 4.834a 0 , the core-radius value obtained from the experimental determination of x(E). The triangles indicate results computed with rc = ~100a 0 , the core-radius value that yields agreement with the measured cross section at threshold. The bars indicate values measured by Marr and Creek, and the cross indicates the cross section minimum calculated by Seaton. 22 3 THEORY OF RESONANCES 3. 1 Introduction We now consider the influence of resonances on the photoionization cross sections. The prototype of all resonance systems is a damped oscillator .driven by a periodic force, II m(x + yX + wo2 x) = Fo e iwt The forced oscillations are given by iwt Fo eiwt Fo e X = "' -- m (wo2 - u/) + i ·yw 2mw 0 (wo w) + f while the power absorbed is proportional ·to The oscil.lator is out of phase with the driving force by an amount , ( W, e WQ )] ll This system illustrates many characteristics shared by all resonance situations: a natura 1 frequency w0 , a damping constant y, an externa 1 driving force a·nd a phase lag. The .earliest known resonances in atomic physics were the autoionizing states of helium seen in the 1930•s in absorption spectra, hv + He (ls2) .-+He (2s2p; 1p) -+He+ (ls) + e(kp) About 1959 resonances were found in electron scattering, e +He (ls2) -+,He- (ls2s2; 2S) -+ e + He (ls 2 ) ~ An entirely different line of development was the study of resonances as inter mediate states in physical and chemical reactions. Almost all chemical 23 reactions take place through the formation of an unstable 11 collision complex .. , A+B-+M*-+C+D I M* is a resonance in the scattering process, A on B. The most famous example of an intermediate state in atomic physics is probably that encountered in dis sociative recombination( 2 a), e + xv+ .. (XY)* .. x + v II The importance of resonances is that they lead to large cross ·sections for processes like I, II at very low (i.e. thermal) energies. 3.2 Feshbach Projection Operation Theory 29 Our theoretica·l development closely follows Feshbach( ). An alternative 30 discussion was given by Fano( ). We concentrate on the photoionization process hv + A -+ A* -+ A+ + e (kt) We can visualize this process as follows. The photon fie 1d .. drives 11 the 11 11 11 11 amplitudes of the states kt (the scattering background ) which in turn drive the amplitudes of the resonance states. Mathematically it is convenient to divide the states of the system A into two types P and Q. P contains states in wh1ch the core A+ is unexcited and Q the states in which A+ is excited. We also use P, Q to d~note the projection operators which select out each type of state. A projection operator is hermitian and satisfies the relation (i9empotency) p2 = p Q is defined as the complement of P, Q = 1 - P, and automatically satisfies Q2. = Q a 2s, 2p 2s 2p lsnl ls --He+ Levels He Levels ---- Fig.9 Resonance states of He. If we consider a two state system and represent the state vectors in the two states by column matrices / u = v = the projection operators which project onto ~~ y, are It should be noticed that the inverse. p-1 does not exist. In the simplest case, that of He, P selects states with a He+ (ls) core and Q selects states with excited cores, e.g. He+ (2s, 2p). Feshbach's theory hinges on the assumption that to a first approximation the Q states are a good description of the resonance states. The energy levels of He are illustrated in fig.9. The Q states 2s 2 , 2s2p lie below He+ (2s) but overlap the P continuum lski. The earliest theoretical work on resonances took .25 the view that say He (2s 2) is an autoionizing state which decays into the adjacent continuum because of the coulomb interaction between the electrons. From Fermi •s golden rule the rate of decay (or width of the resonance) is given by 2 1 2 21rl < 2s I - - I lsks > 1 r 12 . We shall see that the more elaborate theories do eventually link up with this simpler picture. The wave function satisfies Schr6'dinger•s equation (E - H) 'I' = 0 (50) and it can be split into Multiplying (50) by P, Q in turn we find the.pair of coupled equations (51 ) (E - HQ) 'I'Q = HQP '~'p (52) ~Hp = PHP, HQ = QHQ, HPQ = PHQ) 11 The resonances are con:tai ned in '~'Q which is dri ven .. by the term HQP '~'p.; energy is supplied through '~'p by the scattering experiment {whether photoionization or electron scattering) which also studies the re~ponse of '~'p to the 11 back coupling .. HPQ "'Q' The resonances are in a first approximation the bound states of HQ, (HQ - £~) ~~ = 0 (53) Having solved (53) it is tempting to solve (52) formally, < ~ > ~ I HQP I '~'p ~ E - £ ~ ~ This expression appears to have singularities for real values of E (in fact < ~~ I HQP I '~'p > has a pole at E = £~ which cancels the denominator) so we try another approach. 26 First we write down a solution of (51) with the boundary condition that the only ingoing wave is channel a. Such a solution can be obtained from (AlO) and (A15) of the Appendix, + H ~p = ~aE + GP PQ ~Q (54) Fr,om (Al5) the asymptotic form of ~P is the same as that of '~'aE' (A8), with SaS replaced by Returning to (52) we insert (54) on the RHS, = (56) where~is the complex and energy-dependent optical Hamiltonian 1JY = HQ + HQP Gp + HPQ (57) The solutions of (56) are <:§ ~Q = (E -Jf')-1 HQP ljlaE = HQP ljlaE (58) If the eigenstates of ~are given by (g'IJ -Jr) ~l.l = 0 (59) (58) becomes < ~ l.l I HQP I ljlaE > ~Q = I ~l.l (60) E - g' l.l l.l The poles of (60) occur at complex energiesW. We can evaluateW approxi- IJ IJ mately by assuming that ~ ==< ell in (59), and using (A12), . l.l l.l g'IJ = < elll.l I H I elll.l > + = e: + < ell I I ell > (61 ) IJ IJ HQP Gp HPQ l.l dE' = e: + I J 1 12 l.l a' E- E +in I vl.la'E' 27 where v~aE = < ~~ I HQP I ~aE > ~~ is usually written in terms of a width r~ and shift 6~, ~ = (e: + 6 ) - ~ r (62) ~ ~ ~ ~ ~ where, r = 21r I (63) ~ Thus the width = rate of decay into all available continua = inverse of the lifetime of the autoionizing state ~. The shift, given by dE' 2 6 P.V. 1 = E - E I V~a • E• 1 ~ f is less interesting. It is usually much less than r . ~ To simplify the results we shall consider only a single isolated resonance from now on, &r~ = Er- ir/2, and ~~ = ~. Then (60) becomes va ~ ljiQ = (64) Inserting (64) in (55), res = , .S aa I. 0 1 - 21Ti (65) a' { a a 3.3 Fano Profiles Cross sections are obtained by substituting (65) in (45). The simplest situation is elastic scattering with no excitations possible; the S-matrix has then just one element 2ia S = e where o is the elastic phase shift. The elastic cross section is a = 1T (66) 28 where the S-matrix is obtained from (65), 2io : - Er - i r /1 = e l (67) [ - E + r ir/2 Sres is also exp (2iores) where 'res = o- arctan [ /~1Er l (68) f (£,Q) -q 1/q ( Fig. 10 Fano Profile. This resembles the formula for the phase lag in the mechanical analogy of 3.1. From (66) and (67) we obtain the cross section (69) E - E r where £=--.;...,q= - cot a. Figure 10 shows the function r/2 f ( £ ,q) = 11 11 which is the famous Fano profile • Depending on the value of q the profile can range from a line (q = ~) to a window (q = 0). The integrated cross 29 section 0 dE = (70) J res The broader the resonance the more cross section it contains; this is worth bearing in mind when thinking about the role of resonances in physical processes: one generally wants to find broad resonances*. A Fano profile can be obtained for any matrix element, in particular the 2 photoionization cross section is proportional to 1 where IM a ( 71 ) and '¥1'1. = ljlp + ljiQ = ljlaE + ~E HQP ljlaE + G; Hp~o/HQp ljlaE Dropping the third term we have a very useful result, va ~ '¥a = ljlaE + (72) (E - Er + ir/2) The last step requires (54), (58) and (64). Substituting (72) into (71), V M(l) a = M( o) Ma a + (E - Er + ir/2) M(O) = M(l) (73) a < ~B ID I ~~ aE > ' - < ~B I Dl ~ > M(O) gives the nonresonance (background) photoionization. a The label a distinguishes several continua which in practice will be states of different angular momentum, e.g. in the photoionization of Ar we have s- and d-waves. The different a can be sorted out experimentally by looking at the angular distribution of the photoelectrons, a subject we shall not go into here, * Or one can add up a large number of narrow widths. This is the case with chemical reactions. 30 and what one effectively measures is (74) From (73) this becomes (e: + q where the shape parameters are given by (76) If there 1s just one continuum (15) in a standard Fano profile. At the present time it should be possible to calculate M(o) and M(l) quite reliably. Thus an a. experimental determination of the q would lead to values of V and r which are a. a. a much more severe test of theoretical predictions. (Remember that = 21T 2 ). r I Va. a. 3.4 Calculations To make the ideas of the last section more concrete we shall briefly explain how they can be used to calculate resonance scattering and photoioniza tion .. Th1s is the perturbation approach to calculating resonance behaviour in which P, Q spaces are initially regarded as noninteracting and HPQ is allowed for afterwards as a perturbation. It contrasts with say the close coupling methods in which the resonances come out of a large calculation. Though per- turbation theory is less accurate it exhibits the physics more clearly. Papers 31 34 which use this approach are included among the references( - ). Consider a two electron system A (e.g. He, H-, Ca, H2 ). If the (one electron) states of A+ are ¢P the wavefunctions of A can be expanded in the complete orthonormal set ~. pq = [ ¢~ 1) (77) (considering singlet states only). To obtain Q states we simply exclude all terms with either p or q = 0 (the ground state of A+) from (77.). Otherwise¢ • )..1 31 • and £ are calculated as though they were ordinary bound states. P space is ~ spanned by functions for which either p, q = 0. Thus a general P space wave- function is (78) ifF is a contiDuum orbital .it satisfies a single linear integrodifferential equation and 1jJ in the "static exchange" wavefunction for elastic e +A+ 6 scattering (c.~. Mott and Massey( )). Even for complicated systems F is quite easy to determine. Given 1jJ - ,,, and P - '~'aE 1jJ = $ = I cpq (jl q p ,q;z:Q pq it is a simple matter to evaluate V , M(o), M(l) etc. a a 32 4 PHOTOIONIZATION OF HEAVY ATOMS 4.1 Introduction This section is concerned with the results of two papers which repay close 35 36 study( ' ). In the first paper Cooper studied photoionization in a one electron model with a view to studying deviations from the hydrogenic approxi mation. 4.2 Minima in Photoionization Cross Sections The tota 1 p. I. cross section in an independent electron model is the sum of contributions from each subshell. Thus for the n, ~ subshell 2na 0 nt (£) = -3- (£ - £nt) (Ct-1 Rt-1 + cul Ri+l) (79) where (-£nt) is the ionization potential for removing an electron from the sub shell and Ct±l are constants arising from the angular integration. The radial integrals are = d r p n ( r) r p. n + 1 ( r) (80) n"' £,"'- and the bound and continuum rad1al wavefunctions are both solutions of d2 + 2£ I + v - t(t + 1)] p• = 0 ( 81 ) !dr 2 nt r2 The effective potential Vnt is derived from the Hartree-Fock bound state wave . HF function and energy PHFnt and · £nt , t(t + 1)] = - p~F [/: + 2£~~ - (82) nt r r2 Thus P~~ is automatically a solution of (81). Unfortunately ~he RHS of (82) has singularities at the nodes of P~~ which must be smoothed over. By the dipole selection rules s-shells photoionize only into the p-wave continuum; p-shells can go to s- or d-waves and d-shells to p- or f-waves. Where there are two possibilities the wave of larger angular momentum usually dominates. Cooper found that the radi.al integrals Rt±l showed two possible types of behaviour as functions of £. If the bound orbital is nodeless (ls, 33 2p, 3d, 4f, etc) the integrals have one sign for all e as in fig. ll(a) though they may have a maximum. But if the bound orbital has nodes (3p, 4p, ... , 4d ... , Sf .•. , etc) they generally change sign as in fig.ll(b). Ate= em (about 30 eV for 3p - ed in Ar) the cross section has a minimum (it is nonzero because of the other term in (79)). This is the famous Cooper minimum. A typical example is shown in fig.ll (c). The near threshold minima in the alkali metal atom cross sections discussed in. section 2 are really a special case of this effect (the transition is n0s ~ ep in each case). £ (a) ;:;-· 30 5 CD - 20 ·~ -0 10 1 2 J .. 5 6 £au (C) Fig. 11 Photoionizatjon matrix elements and cross sections in heavy atoms. (a) Behaviour of R1±l when n = 1+1. {b) Behaviour of R1±l when n > 1+1 e.g. 3p - ed in Ar. (c) Cross s~ction for.Ag+ (after Cooper). 34 To see how the Cooper minimum arises we consider the low energy d-waves and the 2p, 3p wavefunctions of Ne, Ar respectively. Obviously the sign of wave function is arbitrary but in fig.l2 we adopt the convention that the first r (a) ~-- 3p (Hydrogeni~) -.... -0.. r (b) Hg. 12 Radial wavefunctions (highly schematic). (a) Neon {b) Argon maximum is positive. At E = 0 most of the integral R(2p - Ed) comes from the only maximum of 2p and so is < 0. If E is large enough both integrals come from the innermost maximum of the bound state wavefunctions. Thus R(3p - Ed) must change sign at least once. The difference between a hydrogenic orbital and a realistic one (fig.l2{b)) arises from the stronger field near the nucleus in a realistic potential : the hydrogenic orbital is not so squashed up near the .• origin, so R(3p - Ed) is < 0 for E = 0 and the minimum is not found. 35 4.3 Discrete Levels in the Far Continuum The title of this section is taken from the title of the paper by Dehmer et 36 al( ), which deals with systematics of the spectra of elements from Sn (Z =50) and Lu (Z = 71) near the 4d threshold, i.e. in the photon energy range hv ~ 100 - 200 eV. Four types of behaviour can be distinguished in these elements. For Z < 57 a high broad peak is observed > 20 eV above threshold. From La (Z = 57) to Ho (Z = 67) there is a main peak 5-20 eV above the threshold with structure, decreasing in strength as Z increases, and numerous weak lines. From Z = 68 to Z = 70 only the lines persist and above Z ~ 71 no features can be seen. In these very heavy elements nd ~ Ef transitions dominate the photoioniza tion. ·For Z < 57 the centrifugal barrier keeps all f electrons outside the 4d orbit. At Z = 57 the nuclear attraction suffices to pull the 4f orbit inside the barrier and so the bound-bound transition 4d ~ 4f shou1d be strong (transi tions between states with the same principle quantum numbers are usually strong). The prominent feature in the continuum is nothing other than this 4d ~ 4f tran sition! But how does it get into the continuum? Consider the initial and final configurations written out in full, 37 The final state is a very complicated multiplet (c.f. Slater( )) about half the levels of which are raised into the continuum by the large exchange· interaction These levels can autoionize to and so they can appear diffuse or broadened. Figure 13(b) shows the variation of the transition matrix element I< 4d lrl Ef >1 2 and the exchange matrix element G1 (4d, Ef) with E for the case of cerium. Both are largest forE < 0 i.e. for Ef = 4f. Dehmer et al. suggest that similar effects should be seen in the transition matrix elements, e.g. the case of vanadium is illustrated in fig.l3(a). However photon energies in exces~ of 200 eV would probably be needed. 36 1/) 1/) 1·5 1·5 -c -c :::::> :::::> >- >- ...... 1·0 I""-, 1-0 m I I m...... :0-..... -:0..... <( 0·5 <( 0·5 ,.-- ...... -1 . o 1 2 -1 0 1 2 dA.U.) dA.U.) (a) (b) 36 Fig. 13 Transition and exchange matrix elements (after Dehmer et a.1( )). (a) Vanadium: -1< 3plr1Ed >1 2 . . . . GI ( 3p. Ed) (b) Cerium: -1< 4dlr1Ef >1 2 • • • • GI ( 4d. Ef) • 4.4 Other Topics The separate contributions of the continuum partial waves t 1 = t ± 1 may be studied by looking at the angular distributions of the photoelectrons. Several papers on this subject have appeared, notably that of Kennedy and 38 39 Manson( ). ·Connerade and Mansfield( ) studied the spectrum of Hg between 20 0 0 and 120 A. They found a broad transmission maximum around 80 A and a number 0 of identifiable discrete features .between 108 and 118 A. • 37 . ;r· < \. THIS PAGE. WAS INTENTIONALLY LEFT BLANK 5 PHOTOIONIZATION OF MOLECULES 5.1 Non-resonant Photoionization of Molecules The scattering of electrons by molecules and the photoionization of molecules are very much research fields. The complexity and wealth of phenomena observed in molecules as opposed to atoms arises partly from the extra degrees of freedom they possess (vibrational and rotational). Even if these degrees of freedom are ignored, molecules lack the spherical symmetry of atoms, so that calculations are an order of magnitude more complicated. We shall only discuss ' the simplest diatomic molecules here. I~ is worth beginning with a simple and instructive example. .Consider a symmetric diatomic whose orbitals have the form 1 ~ = -- [$(r + ~R) ± $ (r - ~R)J (82) ~ . -- -- We want the photoionization cross section out of ~ into a final state approxi• mated by a plane wave ~k = exp (i~.r) (83) If Do = < $ IZI ~k > and < $ 1~k > = 0 (true for exact wavefunctions) the dipole matrix element becomes D(B) = 1:2 D0 cos (~·B) (84) (84) should be averaged over all orientations of R. Then 52 = D20 [ 1 + sinkR kR) (85) which means that 52 is the same as for a single atom except for an interference factor which~ 1 as R ~ ~. (Remember there is only one electron in this case). The term sin kR/kR suggests that the cross section should have oscillations as a 40 41 function of energy. There is some evidence for such oscillations in N2 ( ' ). The only molecule for which the P.I. cross section can be calculated 42 exactly is the one electron system H~ ( ). The H~ calculation was later used to obtain a simple model of the photo- 43 ionization of H2 ( ) One imagines two charges ~e approximately at the 39 positions of the H2 nuclei. fig.l4(b). In describing the wavefunctions of the e e (a) (b) Fig.l4 Model of H2 • electron ejected from H2 it is important to allow correctly for the quadrupole + moment of the H2 core. This .was done in the model by placing the half-charges at a distance R1 (~ R) apart so that their quadrupole moment reproduces the known moment of H+ • The results agree quite well with experiments, though they are 20-30% too small. 44 The discrepancy has been removed in a new calculation( ). For the final state of the outgoing electron a Coulomb wave based on the midpoint of the nuclei was used and for the initial state a wavefunction containing about 40 variational parameters and describing the correlation between the two bound electrons quite accurately. This calculation does not allow for correlations between the two electrons in the final state and this must be the next advance in the theory. 44 Theoretical predictions of the ratio of production of H+ and Hi ions( ) can be compared with experiments by Fryar, Browning and Cunningham( 4 s). The 0 agreement is perfect down to A = 300 A, below which the theory predicts two few protons. This is either due to the opening of a new channel + hv + H2 ~ H2 (lo ) + e \.1 ~ H (ls) + H+ + e or to resonances (see section 5.3). 40 Fi 9.15 Vibrationally autoionizin9 states. -vibrational levels of H2i levels of H!. · H; FROM H (PARA, 78° K) 2 RES.-0.04 A z ; .o 16 15 14 13 12 It 10 9 8 -n I ~ ,--~ i :·,---r-- -~---- :~t ~ •; I • i! ,ill, _. ' ; : !. i: , : {\ ~ J •:0 ~ ~- YJ . ._.,_;._~._, __._·79·5.-A--- \J. ~ ~ --~ .______--.;,..__ PHOTON WAVELENGTH. A Fi 9. 16 Relative photoionization cross sectl'on (arbitrary units) of para-H2 • at 78°K. The resolution was 0.04 . 41 5.2 Vibrational Resonances Many types of resonance phenomena are encountered in molecules, but the simplest are probably vibrational resonances, such as those seen in the photo- o ionization of Hz just above threshold (810 A). These states arise because a vibrationally excited ion Ht (v > 0) can bind an electron in a Rydberg state {nim) and the 11 bound state 11 so formed can 1ie energetically above the ground state of the ion H~ (v = 0). It is a 11 bound state 11 in quotation marks because the motion of the nuclei, which breaks down the Born-Oppenheimer separation of electronic and nuclear co-ordinates, leads it to autoianize into the continuum H; (v 0) + e. The electronic transition involved is la nim + lagkim and the = 9 orbitals ni, ki would not overlap if the nuclei were fixed. Very little work 46 has been done on these states, apart from calculations by Bardsley( ). In fig.l5, the v = 3, 4 levels of Hz are autoionizing. 47 Figure 16 shows some of the resonances observed by Chupka and Berkowitz( ). 5.3 Electronic Resonances By electronic resonances we mean states like (lcruZ) 1E+g of Hz containing two excited electronic orbitals : even if the nuclei were fixed they would be autoionizing. Consider a resonance state of AB which supports bound states, fig.l7(a). The following theory applies equally to repulsive states, but the case shown is easier to visualise. The resonance r can decay into states of AB + denoted by a. Let ~r. be the electronic wavefunction of AB ** and ~a. (k) the electronic wavefunction of the system CAB+ + e(k)J; sr and sa. (e) are the associated vibrational wavefunctions. Then from 2(63) the width of r is obtained by summing the matrix elements ( 86) (squared) over all decay channels consistent with the conservation of energy, Er = £ + ~kz. HeR. is the electronic Hamiltonian, e is the vibrational energy of AB+ (or A + B+) and (87) The width becomes 42 2 . rr =. r 2• f dt kdk 6(Er- E- !k ) I< 'r l~.(k)l '· >12 (88) (88) shows that rigorously there is no such thing as a fixed-nucleus width Gr(R) because energy may be shared between the ion AB+ and outgoing electron in an As** r a Fig.l7 Electronic resonance states. infinite number of ways. However there is a useful 11 local 11 approximation in which ( 89) Cl ~kra(R) 2 = Wr(R) - Wa(R). At each value of R the molecule decays as though the nuclei were fixed. The validity of this approximation is uncertain. Much of the interest in molecular resonances arises from their role as 28 ~intermediate states in low energy reactions such as dissociative recombination( ). The cross section for 43 (90) has now been calculated from first principles( 4 a) and is in good agreement with experiment. The cross section is proportional to the transition rate from the initial state into the resonance, as given by a slightly modified form of Fermi 1 S golden rule, 2n < (v) >2 He z;;.1 ·r;:1r (k) I {, r 34 (Bottcher( )). z;;r is a vibratio~al wavefunction in the complex potential Wr - (i/2) Gr. The imaginary part of the potential allows for back ionization into the initial state. In the process (90), back ionization haS about a 30% probabi 1i ty and cannot be neglected. If one wants to predict the cross section for a specific final state, e.g. e + H+ (v) ~ H(ls) + H(2s) ( 91 ) 2 one must expand the total wavefunction of the system as 'I' = z;;. ~- (92) 1 1 49 (Bardsley and Mandl( )). Then the cross section for (91) is proportional to where z;;f c!>f is the final state wavefunction and~f = < ~f jPHetQI c!>r >. The same result can be obtained from 3(65). The inverse process to (91) is known as associative ionization. It 1s of sun1~ interest to spectro~copists, e.g. Na + Na* ~ Na+ + e 2 (93) has been studied by Marr and Wherrett(SO). Theoretical calculation on (93) are now quite feasible. The reason for mentioning (91) here was to lead into photoionization through resonances. Between 15 and 40 eV photon energy mos~ molecules have resonances of 1 L~ or lrru symmetry which produce large effects in photoioniza tion and photodissociation e.g. 44 ** + hv + H2 ~ H2 ~ H2 + e H + H+ + e (94) H + H* . + + * One can do experiments which detect e, H , H2 or H . The total wavefunctton including the resonances is given by (92) where i can stand for any of the 11 final 11 states in (94). The photoionization cross section is proportional to the square of M. 1 where < ~0 D >' < ~0 > and = I I·~.1 ID I ~r ~ 0 ~ 0 is the wavefunction of H2 in its ground state. Calculations based on (95) predict identifiable resonance peaks in all the final channels (94).* Experi mental evidence is available for the existence of resonance states near 24 and 29 eV in H2(sl). Thes~ states should be seen in photoelectron spectroscopy. The angular distribution of the photoelectrons, ·particularly if polarized photons are used, should identify the symmetry of the resonances involved. "* Note added in proof: detailed results are about to appear in a letter by the -· author, J. Phys. B: Atom. Molec. Phys. z, (1974). 45 APPENDIX SOME RESULTS IN SCATTERING THEORY Section 3 and to some extent Section 2 require a knowledge of multi- channel scattering theory. In this appendix we shall sketch the results used 2 52 53 in the text.· Fuller accounts are given in some of the references(&,·I , ' ), Consider the scattering of electrons by an atom or ion, whose eigenstates are ~(aSL). If we couple ~(aSL) to the orbital angular momentum and spin eigen- functions Ym xs~ of the scattered electron, the resulting functions 1 or X for short a are known as channel eigenstates. .The total .spin Stand orbital angular momen tum Lt are conserved. The only co-ordinate missing from Xa is the radial distance r of the scattered electron from the nucleus. Thus we can expand the total wavefunction in terms of x , a. (A 1) a (~t must be explicitly antisymmetrised, but we ignore this for- simplicity). When (Al) is substituted into the Schro.dinger equation we obtain a set of coupled equations (47). These equations can be solved subject to the boundary conditions, . k . lT -1 r+lR-- - H. -lTj s e z e 2. [ oae e - (A2) 2 2 provided the channel a is open, ka > 0. If a is closed, ka < 0, Fa should be exponentially decaying. , The physical interpretation of (A2) is that if an in- going wave of unit flux is sent into channel a, the flux in the resulting out 2 11 11 going _wave in channel f3 is 1Sa81 • The numbers Saf3 define the S-matrix • To get at the p~ysically meaningful cross sections we need solutions with the somewhat different boundary conditions, (A3) Then the cross sections are, 46 (A4) It turns out that Aae can be expressed in terms of Sae and by·going through a standard piece of algebra we find that lT/k 2 a ( 2S + 1 )( 2L + 1 ) ITSL (aS L s R. -+ + + a a a a (2S a 1)(2L a 1) (AS) where Tae = Sa.., a - oa ...a• The phases of the outgoing, ingoing waves in (A2) are appropriate for scattering by a neutral system. If the target is an ion of net charge Z, k r - R. .! a a 2 is replaced by e = k r - a a Most advanced developments in scattering theory make use of Green's functions. Suppose one has to solve the inhomogeneous equation (A6) knowing the solutions of the homogeneous system ( E - Hp ) ljJ aE = 0 (A7) The function ljlaE satisfies the boundary condition .that the only ingoing wave is in channel a, . r-v ne [ -i e +iee ljlaE I [2:er e e 0 ae. - e sae l (A8) r-+co 2ik r a 13 The normalization defined by (A8) is such that EI) < ljlaE I ljla'E' > = 0 aa' o(E - (A9) • We can formally solve (A6) by expanding both ljle and X in terms of ljlaE' e.g. 47 . ·.:~· . X dE • 1jJ a • E• < 1jJ a • E• X > = ~ 1 J I Then (AlO) where lji 1 S is any solution of (A7) and GE is the Green•s function, defined for complex values of energy by dE• (A 11) z - E1 The integral as written does not exist for real values of z. One must define GE as lim(z-+ E)Gz and the way one takes this limit determines the boundary con dition on ljl . Two possible Green•s functions are 8 G± - I f dE I (A 12) E a• E - E• ± in where n is a small positive number. G+ and G- lead to purely outgoing and ingoing waves respectively in GX. We shall now establish a lemma on the asymptotic form of G~X. From (Al2) and (A8) we have, dE• I X ) X ~. f E- E1 +in ,, xs -i a• I 6 e s - 5 :; e' a] [2:· ~r 2ik• r [ a'a a•s s 6 We must examine the integra 1s ±ik•r dE• F(E I) I± = !~ e (A 13) Ea E - E• + in where E1 = E + ~k· 2 and F is a slowly varying function.- Transforming to k1 a, the integrals are along the path OA in fig.l8. For large values of r, J = o(l), 08 r 48 so 2k'dk' ±ik'r f -(k_'_+_k_) (-k-' ---k-_-,-. n-) F (E I ) e BOA If k' = x + iy, e±ik'r = e±ixr e+yr, so for I+ the contour must be completed in the upper half-plane, enclosing the pole k' = k +in. For I the contour is completed in the lower half-plane enclosing no poles. Thus I rv 0 (Al4) Using (Al4) we see that < ~ 'E I X > L a S = outgoing part of ~aE with Sae replaced by - 2ni L sa'S < ~a'E I X > a' Complex k 1 plane B 0 + Fig. 18 Contour for I • 49 REFERENCES 1. A.L. Stewart, Adv. Atom. 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