This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 169.230.243.252

This content was downloaded on 27/08/2014 at 12:55

Please note that terms and conditions apply.

Quenching of excited atoms by collisions with stable molecules

View the table of contents for this issue, or go to the journal homepage for more

1977 J. Phys. B: At. Mol. Phys. 10 2853

(http://iopscience.iop.org/0022-3700/10/14/019)

Home Search Collections Journals About Contact us My IOPscience

J. Phys. B: Atom. Molec. Phys., Vol. 10, No. 14, 1977. Printed in Great Britain. @ 1977

Quenching of excited atoms by collisions with stable molecules

C Bottchert and C V Sukumar Daresbury Laboratory, Science Research Council, Warrington WA4 4AD, England

Received 15 February 1977, in final form 27 April 1977

Abstract. We are concerned with relatively simple semi-empirical models for the quenching of excited atoms by stable molecules, i.e. the transfer of electronic energy into vibrational and translational degrees of freedom. The most notable example is the quenching of Na(3’Pj to the ground state by N,. The cross sections are first related to those for free-electron-molecule scattering in a ‘first-order’ theory and it is shown that this simple theory leads to numerical values which are far too small. Thus we invoke an ionic collision complex formed by partial charge transfer and succeed in obtaining reasonable total cross sections and final-state branching ratios from a hybrid theory. Calculations are carried out for excited Li, Na and K colliding with H,, Nz and 0,. We show that a unified quantal theory can be developed which gives a better account of branching ratios (so far as they can be compared with experiments) than the hybrid theory. The unified theory is used to make quantitative predictions on the quenching of Na(32Pj by Hz and N I , and qualitative remarks on the quenching by 02, C O and CO,. Other possible applications of partial charge-transfer complexes (e.g. to catalysis) are discussed briefly.

1. Introduction

The quenching of excited atoms in collisions with molecules is of great practical importance in gas discharges, lasers and aeronomy. We stress that the molecules are stable to exclude reactive collisions and electronic excitation of the molecule. Nor shall we be concerned with those cases in which the electronic energy lost by the atom nearly coincides with a strong multipole transition (rotational, vibrational or electronic) in the molecule. Such resonant transitions are usually available if the energy transferred is less than 1 eV or greater than 10 eV, and first-order perturbation theory using long-range interactions gives a reasonable account of the cross sections. Where no resonant transition with a large Franck-Condon factor exists, experiments show that the process is still efficient, so that we must look for another means of enhancing the cross sections, e.g. curve crossings with an ionic surface. The most famous quenching process is

(1) which has been studied extensively by classical gas-cell methods (Lijnse and Elsinaar 1972) and more recently in crossed molecular beams (Hertel et aE 1976, 1977). The total cross section at thermal energies is about 40A2 (there is little variation with

Na(2P) + N2(u’’) -+ Na(*S) + N 2 ( 4

t Also: Department of Theoretical Physics, University of Manchester, Manchester M13 9PL, England.

2853

2854 C Botrcher and C VSukumar

temperature) and if the initial vibrational state is c“ = 0, the branching ratios in the final state U‘ appear to favour U’ = 3 or 4 rather than U’ = 6 for which the energy transfer is almost resonant. Hertel er al (1977) have also measured the distribution of energy transfer in collisions of Na(’P) with Oz, Hz, D,, CO, COz, N20 and CzH4. We shall compare our predictions with these results in due course.

The only serious theoretical attacks on the problem (Bauer et al 1969, Bjerre and Nikitin 1967) have invoked curve crossings with an ionic state Na’N; involving the short-lived N; state observed in electron scattering experiments. Bauer et al (1969) used a Landau-Zener approximation; to obtain reasonable cross sections they were forced to make ad hoc assumptions about the polarisability of N2(c) which they admitted had no physical justification. Bjerre and Nikitin (1969) assumed that the colliding species follow classical trajectories except for surface hoppings from the initial to the intermediate (ionic) surface. and from the intermediate to the final surface. The details of their calculation can be criticised severely (see below, $5) but their physical picture is unquestionably correct and similar to the one that we shall adopt. Our viewpoint is that the cross sections for (1) must be closely related to the cross sections for the electron scattering process

e + N2(u”)-+ e + N,(u’) (2) which has a prominent resonance (a short-lived state of N;) at an initial electron energy of about 2 eV. We shall explore physical models. of necessity fairly simple ones, whlch relate the processes (1) and (2). The ideas developed will be seen to have a wide application in atomic physics and chemistry, and ought to provide an incentive to the experimental study of electron scattering by complex molecules.

A complementary approach is to calculate accurate potential surfaces and matrix elements for the different electronic states of NaN, involved in (1) and then to carry out a large scattering (in fact, close-coupling) calculation from which the correct physics will emerge. This method has been successful in studying elastic scattering and rotational excitation where no change of the electronic or vibrational state takes place (Bottcher 1975, Amaee and Bottcher 1977). The latter work has now been extended to electronic-vibrational transfer. Preliminary results indicate that the best hope for progress lies in a more accurate numerical version of the resonance model described in $5 below.

In most of this paper we shall ignore the rotational structure of the molecule. If calculations on electronic-vibrational processes are performed with interactions averaged over all molecular orientations the cross sections so obtained should be a good approximation to cross sections summed over all final rotational states. Most experiments in the near future will not, in any case, resolve the final rotational state. It is not difficult to make some simple statements about the rotationally resolved cross sections, which we shall do in $5.

The plan followed in this paper is as follows. We develop a ‘first-order’ theory of (1) which in its simplest form is inadequate to explain the measured cross sections ($2). Thus we introduce a ‘complex-formation’ model in which the atom and molecule partially transfer electronic charge, and become bound by the resulting Coulomb attraction (53). Detailed results are presented for a hybrid model, combining the first-order and complex-formation theories ($4). As a final step a unified theory is developed, using quantal resonance theory, which provides a justification for the hybrid theory. With some further assumptions the unified theory leads to a very simple expression for the branching ratio into final vibrational states which appears

Queizching of excited atoms 2855

Table 1. Data

Vibrational cg - (2p-2s)f Molecule spacing w, (eV) we

Hz 0,545 N2 0,293 0 2 0.196

3.5 6.5 9.7

Position Width Resonance (eV) (ev)

H; 3.2 2.0 N; 1.8 0.45 0; 1 0 0.05

t Energy difference ('P-'S) in Li, Na and K z 1.9 eV

remarkably successful ( $ 5 ) . In conclusion we mention some other possible applications of partial charge-transfer complexes.

Atomic units are used throughout, unless otherwise specified. It is helpful to have available the numerical data in table 1.

2. The first-order theory

We consider the collision of an atom, which may be electronically excited, with a molecule in its electronic ground state,

A(M) + M(6,) -+ A(P) + M(u~) . (3) For simplicity the atom will be supposed to have a single valence electron. In the impact-parameter formalism the cross section for (3) is given by

where P is the probability of a transition for collisions at impact parameter b ; the electronic states of the atom before and after the collision are denoted by 3 and P and the vibrational states of the molecule by U, and E , . The internal energy of the total system is given by

E,, = EA(!-4 + E,(c,) ( 5 ) where p = a, ,8 and EA and E, refer to the atom and molecule respectively. The quenching probability is related to the transition matrix element q,] by

P(b) = lT,p12 (6) where to first order

2856 C Bottcher and C V Sukumar

C O , ~ = E , - E,, and l&(t) is the matrix element of the interaction between the valence electron of the atom and the molecule at any time t on the classical trajectory.

We denote the distance between the atom A and the centre of mass of the molecule M by R, the position of the valence electron relative to the nucleus by Y and the distance from the valence electron to the centre of mass of M by R‘ = R - Y. If the initial and final valence-electron wavefunctions are $ x and $, the matrix element in (7) becomes

where U,, is the matrix element of the interaction between the initial and final vibrational states of M. If we now replace $, and by Fourier integrals over the momentum wavefunctions, (8) is transformed into an integral over two momenta q , and q2, involving the integral

exp[-i(q, - q 2 ) . R ’ ] U,,(R‘)dR’. 271

The last expression is just the Born amplitude fB(ql, q 2 ) for scattering of an electron by M when the wavevector changes from q1 to q 2 and the vibrational state from ux to zip. An improved approximation (equivalent to including certain interactions to all orders) is obtained on replacing f;, by the exact amplitude f ip for the same process, and inserting the resulting matrix element in (7). If we transform the momen- tum wavefunctions back to coordinate space, the modified form of the interaction (8) is given by

T/,,(R) = Jexp[i(q, - q 2 ) . R - iq,.x + iq2.y]

x $,(4*$pb)fi,j(41, q 2 ) d q 1 dq2dx- dy. (9)

= q 2 - 91 9 = & I 2 + 41) (10)

It is more instructive to use relative and centre-of-mass momenta,

so that (9) becomes

To obtain a tractable approximation from (ll), the dimensionality of the integral must be reduced to not more than three. This can be achieved if f i , is a function only of K or only of q . The former case leads to the well known ‘impulse approxima- tion’, valid when the interaction time is very short. We are concerned with the oppo- site limit, the ‘slow-encounter’ approximation, in which the projectile remains close to the target for a long time. For such small values of q the scattering amplitude will not depend strongly on K (i.e. only low partial waves will be significant). Integrat- ing over K, (1 1) becomes

Quenching of excited atoms 2857

where

is an effective interaction: in the Born approximation G,, is again proportional to UX/](2(). Another way of writing (12) is to introduce a 'form factor' containing all the dependence on atomic wavefunctions,

E/ ,h R ) = exP(- 2iq . 5 ) $ x ( R + 5)*$/i(R - 5 ) d5. (14) i Inserting (12) in the expression (7) for the transition matrix element we have

The projectile is supposed to move with uniform velocity U so that the trajectory is given by R = h + ut . The cross section follows from (4) and (6).

The scattering of low-energy electrons by most molecules (except for some with large dipole moments, and certainly including those in which we are interested) is dominated by shape resonances. We shall limit ourselves to the case in which the scattering amplitude is dominated by a single term of the Breit-Wigner form

where the resonance is in the (2 ,A) partial wave at energy E,, r, is the partial width into the P, vibrational channel and r is the total width; q l , q 2 are referred to a frame fixed in the molecule. If we transform q1 and q 2 to a frame fixed in space and average over momentum transfer, it is easily seen that the product of spherical harmonics in (16) is replaced by 1 Yfj,(n)12 where n is the orientation of the nuclear axis; on averaging over n this is in turn replaced by 1/4n. The effective interaction (13) now simplifies to

where qc is a complex wavenumber such that

44: = E , - 4ir. (18) We are interested mostly in the quenching of the first excited (2P) state of an

alkali atom to the ground state (2S). Hence the valence-electron wavefunctions have the forms

$&9 = $&)Yo*

0.8 = A,, + B,p cos2@

$ p ( 4 = $,(4YlO which, when used to evaluate (15), lead to a cross section of the form

(19) 0 being the angle between the quantisation axis of the atom and the direction of the incident molecule, i.e. 0 specifies the orientation of the atom. It is worth retaining 0 as a parameter in the theory in anticipation of experiments with polarised atoms.

2858 C Bottcher and C V Sukuinar

For unpolarised atoms the measured cross section is A + $3. The coefficients A and B in (19) can be expressed in terms of the integrals

f“ 7

J2(b ) = d Z ?sin(@ J(b , Z) O R

I , = 3n(&’j2 1; J,(b) b db (1. = 1, 2) V I 4 c I

where R , = /R & (1, t = R.</R( and IC = W , ~ / V . Thus

A,, = I1 B,, = 212 - I I . (23)

Thus the simple first-order theory gives the cross sections in terms of four-dimensional integrals involving only the atomic valence-electron wavefunctions and the parameters of the shape resonance describing the scattering of free electrons by the quenching molecule. For transitions involving highly excited states of the atom, the first-order theory can be simplified further (Fermi 1934). In appendix 1, the simplification of the first-order theory in the limit of highly excited states is outlined.

Having established the formalism we set out to obtain numerical results for Li, N a and K in collisions with H2, N2 and 02. For the radial wavefunctions $s and $, we used LCAO representations obtained by diagonalising model-potential Hamil- tonians (Bottcher 1971). A most important role in the quenching process is played by the recoil of the quenched atom, represented in the first-order theory, e.g. in (15), by the factor exp(io,oZ/c). This factor pulls the cross section down, the transfer of momentum from an electron to a heavy particle being an inefficient process. It is important not to overestimate the effect by using the initial (usually thermal) vel- ocity ci; we always use the geometric mean of the initial and recoil velocities ( U ~ V ~ ) ’ : ~ .

a i ) in each case; more details are contained in $3. There are two reasons why a single-pass binary-encounter model should predict that the process is highly inefficient. We have already referred to the recoil effect : nearly-resonant transfers with small recoil involve large changes in the vibrational quantum number which are discouraged by the par- tial widths (see (16)), in effect by Franck-Condon factors. The second difficulty is that the initial- and final-state momenta of the valence electron are only well matched for energy transfers in a very small region of configuration space, an effect described by the form factor in (15). Clearly we should look for some mechanism which increases the time during which the molecule remains in the region where the form factor is large, the most obvious such mechanism being the formation of a long-lived ‘intermediate complex’. We develop a ‘complex-formation’ model in 43 and further argue that the branching ratios into final states are still given approximately by the first-order theory. Thus we shall return to the results of the first-order theory in due course.

Our calculations led to extremely small quenching cross sections (around

Quenching of excited atoms 2859

3. Complex-formation model

We are considering atom-molecule collisions in which the atom has a single weakly bound valence electron and the molecule has some degree of electron affinity. It is likely that the electron will spend part of its time attached to the molecule and that this partial charge transfer will give rise to a weak Coulombic attraction between the atom and molecule. The concept of a partially ionic bond is familiar in quantum chemistry; if, as in the present case, the system must separate into neutral components, the Coulomb attraction can only be effective over a finite range and does not extend to arbitrarily large distances. In the course of a collision the Coulombic potential energy can become equal to the relative kinetic energy, in which case the molecule will be captured into a bound orbit, forming a long-lived collision complex (figure 1). The lifetime of the complex must be finite since the molecular negative ion is unstable and will eventually decay into a free electron (to be picked up by the atomic ion) and a neutral molecule. Before the complex decays the two colliding systems will remain close to each other at a separation favourable to energy transfer for a much longer time than in a direct encounter. Indeed we shall assume that whenever a complex is formed quenching must take place, this being the process most likely to destroy the complex. A further point to notice is that the final state produced is much less correlated to the initial state than in a direct process, offering the possibi- lity of transferring several vibrational quanta at once, as observed in a number of the experiments on quenching. To avoid confusion we should probably remark that the orbital capture we are talking about here is quite distinct from the ‘orbiting’ which occurs in attractive R-” potentials for n > 2, but not in a Coulomb potential.

We shall thus consider a model in which the molecule moves through the charge cloud of the atom, picking up a small amount of negative charge. The largest impact parameter for which capture into a Coulomb orbit is possible is the effective radius for quenching. Since the molecule is interacting with a diffuse atomic state whose spatial extent is much larger than the molecular dimensions, it is possible to treat the bound electrons as a free-electron gas whose density and kinetic energy depend on the position of the molecule. A more detailed justification of this approximation is discussed in appendix 2. If c is the net charge transferred to the molecule, the net rate of flow of charge into the negative molecular-ion state is given by

Figure 1 . Schematic illustration of ‘capture’ into a bound orbit in a slow collision process. The sodium-ion core, the N2 molecule and the 3p-electron orbit are shown in realistic relative dimensions (from Hertel et al 1977).

2860 C Bottcher and C V Szikumar

The first term on the right of (24) represents the rate at which electrons flow into the resonant state of the molecule; 11 is the local number density of atomic electrons, v the local velocity of atomic electrons and G~~~ the cross section for resonant scatter- ing of free electrons by the molecule. The density of electrons at a distance R from the atomic nucleus is determined by the wavefunction $i( of the valence electron in the Rydberg state ci

‘2 = l$,iR)I2 (25)

+v2 + V(R) = E ,

while the local velocity (also the local momentum k in atomic units) is given by

(26) where Vis the effective potential of the valence electron in the atomic field, to suffi- cient accuracy -l /R, and -E, is the ionisation potential. The second term in (24) gives the rate at which charge leaks out of the resonant state, being the width (decay rate) of the resonance. The width also enters G~~~ which is assumed to have the simple Breit-Wigner form (17); on averaging over the molecular orientation,

71 r2 k Z ( E - E,,)’ + $r2 Gres = -

where E, is the position of the resonance. We can rewrite (24) in the form

where

The relative velocity in thermal collisions is typically less than lop3 while r > so that the solution of (28) is close to its quasi-equilibrium value i o ( R ) at each point of the trajectory. It is convenient to average lo over the orientation of the atom, so that $,(R) is replaced by the radial wavefunction q,,(R). Capture occurs when the total relative energy of the reactants is zero, i.e. when

E + AE = io(R)2/R (30) E being the relative kinetic energy and AE the increase in relative kinetic energy due to acceleration in the Coulomb potential,

Energy is apparently not conserved since the Hamiltonian of the heavy-particle system is time dependent: the missing energy has simply become the potential energy of the electron in the field of the molecule which our model does not describe in detail. The capture radius R, is determined by (29)-(31), and the total quenching cross section is

G~ = nR;. (32)

An apparent difficulty with (29) is that io can become greater than unity for some impact parameters. In practice this does not happen for b > R, so the calculation

Quenching of excited atoms 2861

Table 2. Capture radii (in ao) (see (27)-(32)).

Molecule

Atom Hz NZ 0 2

Li 2.70 5.32 9.22 N a 1.75 4.98 2.12 K 1.19 6.31 2.93

of C T ~ is not affected. It is worth pointing out that this problem almost always appears when one attempts to apply rate equations to electron-transfer problems. At thermal energies R , is an insensitive function of energy, since capture tends to take place when the local electronic kinetic energy is comparable to the resonance energy. For the same reason the calculated cross section is not affected much by our drastic averaging over the molecular and atomic orientations.

Capture radii for Li, Na and K in their first excited 'P states impinging on H2, N2 and O2 are shown in table 2. The resulting cross sections are of approximately the right size, though perhaps a little on the small side, e.g. the accepted experimental cross sections for N2 and H2 on Na are 40 and 10A2 respectively (Massey 1971).

4. The hybrid theory: results

At the end of the preceding section we calculated total quenching cross sections for various pairs of collision partners. Our model for the total cross sections assumes the formation of a bound complex which allows the reactants to interact repeatedly until quenching takes place. If each interaction during the evolution of the complex is approximately the same as if only a single pass had taken place, the number of systems in each final vibrational channel will be given approximately by the first- order theory derived in $2. This assumption leads to the hybrid model in which the total cross section is taken from the complex-formation theory. In effect the first-order cross sections are multiplied by a factor equal to the number of interactions during the lifetime of the complex. The hybrid theory will break down if the quench- ing probability approaches 50%, since in this limit all the branching ratios must become equal. These ideas will be placed on a more rigorous basis in $ 5 .

First-order cross sections are obtained from (15t(23). A knowledge is required of the positions and widths of the negative-ion resonances (table 1) and of the partial

Table 3. Branching ratios for vibrational excitation of H, and N 2 by electron impact.

U

0 1 2 3 4 5 6

f ( a ) (H2)t: 0.96 0.34 ( - 1) 0'27 ( - 2) 0.27 ( - 3) f'(v)(N,)$ 0.78 0.16 0'48(-1) O ' l l ( - l ) 0.18(-2) 0.17(-2) 0.5(-5)

+ f ( a ) = cr(0 --t u)/C, .o(O + ti'), o(0 + c) summed over all resonances. H 2 data from Golden et id (1966). Linder and Schmidt (1971) and Ehrhardt et nl (1968). N2 data from Birtwistle and Herzenberg (1971).

~l The numbers in parentheses indicate the power of 10 by which the previous number should be multiplied.

2862 C Bottcher and C V Sukumar

Table 4. Results on A* + H,, A = Li, Na and K (see (19H23) for A and E )

A + fB(10- '4cmZ) Energy

Atom (ev) c = o 1 2 3

L1 0.1 0.51 ( - 3)t 0.86 ( - 5) 0.15 ( - 5 ) 0.11 (--5)

N a 0.05 0.43 (-3) 0.38 (-4) 0.12 (-4) 0.21 (-4)

0.5 0.88 (-3) 0.15 (-3) 0.55 (-4) 0.81 (-3) 0.1 0.43 ( - 3) 0.42 (-4) 0.20 (-4) 0.66 (-4)

K 0.1 0.62 (-3) 0.41 (-4) 0.19 (-3) -

B cm')

t ' = o 1 2 3

Li 0.1 -0'40(-3) -0.38 (-5) -0.99 (-6) 0.23 (-5)

N a 0.05 0.22 ( - 4) 0.41 (-4) 0.16 (-4) 0.27 (-4) 0.1 0.38 (-3) 0.62 (-4) 0.29 (-4) 0.14 (-3) 0.5 0.11 (-2) 0 2 5 (-3) 0 8 5 (-4) 0.13 (-2)

K 0.1 0.47 (-3) 0.72 ( - 4) 0.17(-3) -

t The numbers in parentheses indicate the power of 10 by which the previous number should be multiplied.

widths for different final channels; the latter are obtained from experiment and the values we adopted are given in table 3, with references. Only for Nz and H 2 is the partial-width data good enough to make branching-ratio calculations worthwhile.

We present results for Hz and N z in tables 4 and 5 respectively. Collisions with Li, Na and K at 0.1 eV and with Na at 0.05 and 0.5 eV have been considered. The behaviour of the branching ratio is determined by competition between the partial widths which decrease as the vibrational quantum number increases, and the recoil effect discussed in 92 which goes the other way. The results show that in all cases the recoil loses out, favouring the lowest vibrational state. In some cases, e.g. Hz = Na*, Nz + K*, the branching ratio turns up slightly at the highest vibrational state, an effect explained simply by a very near coincidence of energies in these systems. The Nz + Na* ratios (table 5) do not show the peak at v = 3,4 observed by Hertel et a1 (1976) but there is some indication of a flattening in the distribution between v = 1 , 4 which becomes more pronounced as the energy increases. Regarding H2 + Na", the results in table 4 are consistent with the peak at U = 3 observed by Hertel et a2 (1977).

5. A unified model

In the preceding section we derived cross sections from a hybrid theory, supported by reasonably convincing arguments. Nonetheless it would be better to find a unified theory which would provide a deeper justification. Furthermore we would like to have a quantal theory of the foundation of the collision complex, since a semiclassical

Quenchiizg of excited atoms 2863

Table 5. Results on A* + NZ, A = Li, N a and K (see (19)-(23) for A and B)

A + iB(lO-I4 cm') Energy

Atom (eV) u = o 1 2 3 4 5 6

Li 0 1 0.44(-3)7 0.13(-3) 0.60(-4) 0.58(-5) 0.17(-5) 0.23(-6) 0.28(-7)

N a 0.05 0'18(-3) 0.39(-4) 0'13(-4) 0.58(-5) 0.19(-5) 0.17(-5) 0'18(-4) 0.1 0.48 ( - 3) 0.28 ( - 3) 0.74 (-4) 0.76 (- 5) 0.18 ( - 5) 0.47 ( - 6) 0.48 ( - 6) 0.5 0.12(-2) 0'97(-4) 0.44(-4) 0.87(-5) 0.24(-5) 0.38(-6) 0.38(-6)

K 0 1 062(-3) 0.39(-4) 0.90(-4) 0.30(-4) 0.62(-5) 0.7(-1) -

B (10- l 4 cm') Energy

Atom (eV) c . = o 1 2 3 4 5 6

Li 0.48 ( - 7)

N a 0.05 0.14(-3) 0.38(-4) 0.15(-4) 0.36(-5) 0.25(-5) 030(-5) 036(-4) 0.1 0.23 ( - 3) 0.16 (-3) 0.15 ( - 3) 0.15 ( - 5) 0.16 ( - 5) ,0.41(- 6) 0.60 ( - 6) 0.5 0.39 ( - 3) 0.57 (-4) 0.19 (-4) 0.47 ( - 7 ) -0.18 ( - 5) 0.17 (-6) 0.64 ( - 6)

0.1 -0.25 (-3) -0.76 ( - 4) -0.47 ( - 5) -0.38 ( - 5) -0.18 ( - 5) -0.18 ( - 6)

K 0.1 0.11(-2) 0.52(-4) 0.25(-5) 0.42(-8) 029(-5) 0.11 -

t The numbers in parentheses indicate the power of 10 by which the previous number should be multiplied.

theory of resonances does not exist. Both objects may be achieved at once by applying the standard quantal theory of resonances (Mott and Massey 1965).

Figure 2 shows the potential-energy surfaces W(R) involved in the process

M + A*-+M- + A + + M + A (33) where M - is a resonant state of negative electron affinity. We use the subscripts p, r and r to refer to the three states in (33) in the order written, the labels including the internal vibrational state of M (or M-) as well as the electronic state of A (or A'). The crossings of W, with W,, W, are denoted by R,, R, and that with the MA' surface by R,,. We can easily see that the intermediate state is an electronic resonance in the system e + MA of the type invoked to explain dissociative attach- ment,

(34) e + AB+AB-+A + B-.

R

Figure 2. Schematic illustration of the potential-energy surfaces involved in the process M + A* -+ M- + A' -+ M + A. The labels are described in the text.

2864 C Bottclzer and C V Sukumar

Thus, suppose that the separation R of M and A is held fixed. For R > R,, M- + A’ can decay into M + A with nuclear recoil; for R > R o , M - + A + can actually autoionise into M + A + + e without nuclear recoil. These decay processes could be described by a width depending on R in a non-local way: a local approximation would not suffice here because nuclear recoil is important. However it is not necessary to introduce complex potentials explicitly as the resonances of interest are all vibra- tionally bound in the potential Wr, and each vibrational state has its own associated width. To a good approximation Wr is purely Coulombic so that its vibrational levels are hydrogenic eigenstates, with the reduced mass that is appropriate to M + A.

It should be clearly understood that even for R < Ro, M - A + is not a stable molecular state. Because of crossings such as R , the ionic state can internally convert into a normal molecular state MA without nuclear recoil (providing the latter is energetically accessible). If M - A t were stable the adiabatic surfaces would have to be drastically redrawn, e.g. NaN, would be bound by about 2eV. For the ionic states which are familiar in quantum chemistry, e.g. Li’ F-, the ionic potential usually lies lowest in energy and so represents a stable state.

For clarity the term ‘vibrational’ will henceforth refer to the internal state of M, and ‘nuclear’ to the relative motion of M and A. Let the adiabatic wavefunctions describing the electronic and internal vibrational state of MA at fixed R in the three states be $z, $ p and q5r. The associated nuclear wavefunctions will be denoted by c,, cg and 5,. To calculate the coupling between r and a, p we neglect the nuclear kinetic energy and retain only H e [ , the electronic-vibrational Hamiltonian of MA, i.e. we assume that the nuclear and electronic-vibrational wavefunctions can be separ- ated in the Born-Oppenheimer fashion. Then the couplings are ( p stands for a or

from here on)

while the partial widths for decay into different channels are

The nuclear wavefunction [,,(E) is calculated for the total energy E, and energy norma- lised; E , is the total energy associated with &.. If the resonances do not overlap the cross section for (33) is the sum of single Breit-Wigner terms. On averaging over an energy interval AE which contains a large number of resonances, the cross section becomes

where the sum is over resonances such that E < Er < E + AE and the total width is

A particular simplification results if one of the partial widths rrp is much larger than all the others, (37) then reducing to,

Queizching of excited atoms 2865

Our next step is to calculate Vr,, from (35) which requires some knowledge of &.. An attempt at calculating this matrix element for N2 + Na* was made by Bjerre and Nikitin (1967) but we believe that their results are invalidated by several mistakes. They assumed incorrectly that the extra electron is bound to an excited state of N2 and that its wavefunction is a simple decaying exponential; the coupling was calculated in a one-electron approximation, forgetting about the supposedly excited core?. These assumptions led to a high probability of crossing on one pass, contrary to our conclusions. The low-energy shape resonances in N2 and other molecules are quasi-bound states in the polarisation potentials of the molecular ground state. The physical concept of a resonance demands that the wavefunction be mostly con- fined to a limited region of space; what happens outside that region is not physically interesting and depends on the particular mathematical formulation chosen to de- scribe the resonance. Thus the important contributions to (35) must come from the overlap of the tail of the A* wavefunction with the interior of the M - wavefunction rather than the other way about as assumed by the earlier authors.

To proceed we make the further approximation of replacing H e , by Hk,, the Hamil- tonian of M - alone, neglecting its interaction with A'. The wavefunction of A or A* near M can be approximated by a plane wave

$p - $Im exp(i k,(R) 4

yrrp = $,(RI (4rIHL~Ikp~ P )

(40) where k,,(R) is the local wavevector described in $3 and appendix 2. Then (35) becomes

(41) where lk, p) is the product of an unnormalised plane wave and a vibrational wave- function. To evaluate the matrix element in (41) we note that the partial width for the decay of the shape resonance into a scattering state with wavevector k is given by

This formula is derived in appendix 3. The superscript 'el' distinguishes widths in the M - system from those in MA-. Usually the resonance couples predominantly to one partial wave LA such that the final wavefunction of the detached electron is

(43) in a frame of reference fixed in the molecule. Then the width (42) arranged over all orientations of the molecule becomes

(rO4lkLA) = (2k/7~)'/~ j,(kr) YL,,(O, 4 )

or simply I-;;, the observed partial width for the decay of M-(r) into e(k) + M(p). From (41) and (44) we can express VrU in terms of T $ / k . The width is only observed at the resonance energy 3k;; its variation with k is usually not known, but it should be smooth and such that r - 0 as k + 0 . Thus we take T $ / k as a constant

G r p r:i,(krLA)/kr (45)

t They also quote a width of 0.15 eV rather than the well established 0.45 eV, but this may be a misprint since we know of no possible source for the former value.

2866 C Bottcher and C V Sukumar

and then we have

vrP = ( 2 4 3 : 2 ~ , : / 2 $ p ( ~ ) . (46) The final step in calculating the widths rr, is to evaluate the matrix element

in (36) using JWKB wavefunctions and the method of steepest descent (Landau and Lifshitz 1965). We assume that the potentials W, and W, are respectively constant and purely Coulombic respectively, the curves crossing at R,. The widths depend on the total angular momentum J , or equivalently the impact parameter b,

b = J/Mxii (47) where M is the reduced mass and xi, the asymptotic velocity in the channel p. The wavefunction $ , (R) is replaced by its radial part qt,(R)/R on averaging over all orien- tations of A. The semiclassical matrix element contains, as usual, the squared cosine of a rapidly varying phase which we replace by i. At last we find that

where BE, is the mean spacing of the resonance energy levels and x i the radial velocity at the crossing point,

(49) 2 R2 1 , 2 x: = ;cp(l - b / J . To obtain the cross sections we must look more closely at (48). The widths should

satisfy the Wigner bound

27tr,, G AE, (50)

so that if (48) violates this bound one cannot calculate the widths in first-order perturbation theory; however in such a case the upper bound itself can be used as an approximation. In the initial state p, xa is a thermal velocity so that Tra >> r,,, implying that (39) rather than (37) can be used. The largest of the widths rra is in most cases, and certainly in N;, that for which the vibrational state of M-(vr) is the same as M(va). Then the initial state p determines the set of resonances r ( P ) which contribute most to (39). On summing over all values of the total angular momentum, we have

which is of course independent of AE,. The factor Z,, must be less than unity so that nR?, the effective area of the curve-crossing region, gives an upper bound on the cross section.

We shall now apply (52) to the quenching of Na(3’P) by N2 and H2. The mol- ecules are assumed to be initially in their ground vibrational states which are coupled

Quenching of excited atoms 2867

Table 6. Branching ratios f and p for e + M and Na* + M (electron scattering, quench- ing) processes respectively.

Molecule ~

0 1 2 3 4 5 6

NZ f(4 (%I+ 16.0 31.0 28.0 16.0 6.3 1.7 0.33 P ( U ) (%I 13.0 28.0 28.0 19.0 8.0 3.0 1.0

HZ f (4 (%I 002 0.29 1.88 6.98 P ( U ) (%I 0.08 1.58 13.23 85.11

DZ f(4 (%I 0.00 0,004 005 0.32 1.3 P ( 4 (%I 0.01 0.13 1.94 14.32 83.60

t Birtwistle and Herzenberg (1971). Note that these values are different from those in table 2 which are summed over all resonances.

most strongiy to specific resonance states. In N; the favoured resonance is or = 0 and in H; it is the dissociating state whose turning point lies in the Franck-Condon region of the ground state of Hz. We write the partial widths in the form

I-;; = f*(v,) r;’ (53 )

where r:‘ is the total width (table 1) and the branching ratios are given in table 6, with references. Using the same wavefunctions as in 93 for the N a 3p and 3s orbitals we find that?

2 n <rr , /AEr) = x(MK(vp)/~p (54) where X(N2) ‘v l o v 3 and X(H,) = 2 x For an initial energy 0.04eV, X, rr Thus the bound (50) is violated in the upper but not in the lower state, and the branching ratios Z,, are proportional to rr,. From

and 1, 1 8 x

(54)

Z,j, = constant x J(v) 1 - - i :YZ ( 5 5 )

where co is the value of U for which energy transfer is resonant (see table 1). The sum of Z,, over final states is close to unity so that the total quenching cross section is simply nR; , oQ(Nz) = I T ~ i ( 6 * 7 ) ~ , oQ(HZ) = l - I~i(5.4)~. The former is close to the experimental value while the latter is larger by a factor of two (Massey 1971).

The results of ( 5 5 ) are given in table 6 for N2, Hz and Dz. The branching ratio normalised to a percentage is denoted by p(v). In the case of N2 this remarkably simple prescription is successful in predicting a distribution which peaks around v = 1-3. The observed distribution peaks at v = 3, 4 but for the present we are satisfied if we can only say whether the peak is at v = 0, vo or in between. For H z and Dz we agree with Hertel et a1 (1977) in predicting a strong peak at U = 3, 4 respectively. It should be noted that the e + H2, Dz branching ratios are calculated from the theoretical complex potentials of Bardsley et a1 (1966) and not from experi- ments on vibrational excitation (e.g. Golden et a1 1966, Linder and Schmidt 1971,

t By chance the coefficients are almost the same in the upper and lower states

A V I ’ l I 3 I I O I 4 t i

2868 C Bottcher aiid C V Sukumar

Ehrhardt et al 1968). The peaks observed just above the thresholds for excitation in these experiments have long been identified with the 'X: shape resonance. The embarrassing fact that the branching ratios totally disagree with calculations which account perfectly well for dissociative attachment has been glossed over. We have strong reason to believe that the experiments have been interpreted wrongly, and that the measured peaks are simply direct scattering in the forward direction. The excited sodium experiments confirm this interpretation, if our theory proves to be reliable. We can make qualitative remarks on the quenching by O,, CO and CO2. In 02, vo = 9.7 so for small U denominator of (55) does not vary rapidly, and the branching ratio must peak at z: = 0, following the electron scattering ratios (Spence and Schulz 1970). Another peak at higher c is possible, but the electron scattering data here is not good enough to reach a conclusion. In many respects CO should be similar to N,, except that the shape resonance is much broader (at least 1 eV). Thus the partial widths fall off less rapidly and a peak at u0 is likely. It appears to be a trend that broad resonances (greater than 0.5eV) lead to a peak at U = co, and narrow resonances to a peak at U = 0, with N, an intermediate case. Finally, in CO,, c0 = 8'f but inspection of electron scattering data shows that the partial widths are increasing up to this point so that a distribution peaked at u0 is inevitable. The measured branching ratios for 02, CO and CO2 agree qualitatively with these predictions. Thus all the experimental results with which our theory can be compared are explained qualitatively.

To discuss the effect of a quenching collision on the rotational state of the mol- ecule we must allow for the orientation-dependent parts of the atom-molecule interac- tions W For the calculated NaN, surfaces (Bottcher 1975) the orientation dependence is relatively small so that only pairs of states for which the change in rotational quantum number A j = & 2 are connected appreciably. In (37), Arz and rlii both satisfy this selection rule so that overall IAjI 6 4. Hertel et al (1977) find evidence for an even greater spread of A j which might be explained by the undoubtedly larger anis- tropy of the ionic Na'"N; surfaces.

Finally we can show that (53) is approximately equivalent to the first-order theory of $2 in predicting the branching ratios. The impact-parameter formulation is an approximation to a fully quantal treatment in which the cross section is proportional to the square of

Since I)~ and q5r have a large overlap we can put Vz,) 2. Yrs. The matrix element between t, and tp does not have a point of stationary phase, but on any reasonable model of the behaviour of Yz,, e.g.

it is easily shown that

the constant being independent of c(. Thus the stated result is established.

t In the symmetric-stretch mode which appears to be highly favoured in resonant excitation.

Queiichiizg of excited atoms 2869

6. Conclusion

The results from the theories outlined in this paper agree well with available experi- mental results. However more experimental results would be necessary to test the predictions of our models. The model of 45 can be improved in two ways: (i) by calculating more accurately the dependence of W, and VrP on the internuclear separ- ation, and (ii) by deriving T r p from a coupled-equations approach which would auto- matically satisfy (50). This approach is now being pursued.

The concept of partial charge transfer invoked in the discussion of the complex- formation model might have wider applications. For example one of the major factors in catalysis is the availability of special reaction paths. The availability of different states of different nominal charge close together in energy in the relevant range of energy for a reaction enhances the reaction rate by providing alternative reaction paths (Anderson 1976). Such telescoping of valence states of adsorbed atoms on a surface might possibly be understood from the point of view of partial charge transfer. Another major factor in catalysed reactions is the energy required for a reaction to take place. The weak Coulombic attraction arising from partial charge transfer would provide Coulomb energy towards satisfying the energy needs for the reaction.

Acknowledgments

We thank Professor I Hertel and Dr H Hofmann for useful communications.

Appendix 1. First-order theory in the limit of highly excited states (Fermi approxima- tion)

We wish to evaluate (14) when c( and p are highly excited states. This means that we can use the ‘plane-wave’ approximation (appendix 2) to and $o in evaluating 9. Thus if

$2(R - 5 ) = I$,(R)lcosCKm * 5 - o,(R)I (Al.l)

where 4.; is the local kinetic energy, 9 involves terms like

* 6 [ q - 4(Kz -t K/i)li$~$/dexp Ci(ox + O p ) ] .

If K’, << I,I is the first excitation potential of M, we can replace the 6 function 6(q) simply. For a Rydberg state n,l,, IC, = 1/n2. Then (15) becomes

(A 4

where we recall that R = h + Vt. -The amplitude for scattering with a change in momentum (z is given by

db exp(i(z * h ) rqi (h )

2870 C Bottcher and C VSukumar

(A1.3)

since Qz 2~ W , ~ / V . In line-broadening theory the important quantity is the forward elastic amplitude

A,,(O) = %z(O) (A1.4)

which is the Fermi approximation.

Appendix 2. The plane-wave approximation for highly excited states

We frequently have to evaluate integrals which depend on the behaviour of the wave- function of a Rydberg state in a small region, i.e. small compared with the extent of the state. It would thus be useful to have an approximation valid over such a region, and we can find an expression based on the intuition that the electron should behave as if it were free, with an appropriate local momentum. Let us write the wavefunction, in the neighbourhood of R, as

(A2.1)

where 1 $ 1 is slowly varying so that it can be replaced by l$(R)l, a constant. Then

$(R + I*) = 1 $ 1 cos d

o(R + r ) 2~ a(R) + r . Vo(R)

so that

Y ( R + r ) N IY!(R)l cos[k(R).r + a(R)]

where k(R) = Va(R)

(A2.2)

(A2.3)

is the local momentum. If the energy level is E ( < O ) we must have

)k(R)’ = E - V(R) (A2.4)

Vbeing the potential seen by the electron (usually -l/R). The direction of k can be determined partly if the total angular momentum 1 is known. Since 15 R x k,

(A2.5) (k. R)’ N (kR)z; - (1 + i)’. If R is chosen as the Z axis, m, = 0.

Appendix 3. Integral expression for the width of a shape resonance

For simplicity consider s-wave potential scattering. The scattering states have the form

(A3.1)

outside r = a. An R-matrix state (Burhop 1961) is defined by

$ = A sin (kr + 6)

ka + 6 = (m + ))n

loa 4’dr = 1 (A3.2)

Quenching of excited atoms 2871

where m is an integer. If q5 is identified with a shape resonance, the width is given by

r = kA2. (A3.3)

If H = T + I/ is the Hamiltonian, then

joa q5(H - E ) x dr = &kA sin(6' - 6)

where

x = sin(kr + So)

is a solution of (T - E)x = 0. If we choose ka = nx, 6' = 0, (A3.4) becomes 7 r a

A = ; ! q5(H - E ) ~ d r . 0

(A3.4)

(A3.5)

(A3.6)

The value of r should not be sensitive to the exact choice of a, so we can replace q5 by a bound-state approximation which tends to 0 as r-+ x and let a-+ x in (A3.6). With some rearrangement we find

r = 2zi(q5lvjk)l2 (A3.7)

where l k ) is an energy-normalised plane wave. This formula is valid for any superposi- tion of partial waves in the final state.

References

Amaee B and Bottcher C 1977 to be submitted Anderson P W 1976 Caoendish Laboratory Preprint, Cambridge to be published Bardsley J N, Herzenberg A and Mandl F 1966 Proc. Phys. Soc. 89 305-19 Bauer E. Fisher E R and Gilmore F R 1969 J . Chem. Phys. 51 4173-81 Birtwistle D T and Herzenberg A 1971 J. Phys. B: Atom. Molec. Phys. 4 53-70 Bjerre A and Nikitin E E 1967 Chem. Phys. Lett. 1 179-81 Bottcher C 1971 J . Phys. B: Atom. Molec. Phys. 4 1140-9 -1975 Chem. Pliys. Lett. 35 367-71 Burhop E H S 1961 Quantum Theory vol 1, ed D R Bates (London: Academic Press) p 414 Ehrhardt H, Langhans L, Linder F and Taylor H S 1968 Phys. Rev. 173 222-30 Fermi E 1934 Nuooo. Cim. 11 157-62 Golden D E, Bandel H W and Salerno J A 1966 Phys. Ret.. 146 40-2 Hertel I V, Hoffman H and Rost K A 1976 Phys. Ret.. Lett. 36 861-5 -1977 Chem. Plzys. Lett. to be published Landau L D and Lifshitz E M 1965 Quantum Mechanics 2nd edn (Oxford: Pergamon) pp 325-7 Lijnse P L and Elsinaar R J 1972 J . Quant. Spectrosc. Radiat. Transfer 12 1115-28 Linder F and Schmidt H 1971 2. Naturf . 26a 1603-14 Massey H S W 1971 Electronic and Ionic Impact Phenomena vol 3 (Oxford: Oxford University Press)

Mott N F and Massey H S W 1965 The Theory of Atomic Collisions 3rd edn (Oxford: Oxford University

Spence D and Schulz G J 1970 Phys. Ret.. A 2 1802-1 1

p 1693

Press) pp 437-44