Z. Phys. D 31, 235-244 (1994)
ZEITSCHRIFT
FORPHYSlKD
© Springer-Verlag 1994
Comparison of rate coefficients for Rydberg electron
and free electron attachment
D. Klar, B. Mirbach, H.J. Korsch, M.-W. Ruf, H. Hotop
Fachbereich Physik der Universit~it, Erwin-Schrrdinger-Strasse, D-67653 Kaiserslautern, Germany
Received: 7 April 1994/ Final version: 16 May t 994
Abstract. New experimental results on attachment reactions involving free electrons at sub-meV resolution allow for the first time a conclusive comparison of measured rate coefficients for Rydberg electron attachment
with those calculated from the measured free electron
cross sections on the basis of the quasi-free electron model for Rydberg electron collisions. Using classical velocity
distributions for the high n Rydberg electrons and our
measured free electron attachment cross sections, we calculate Rydberg electron attachment rate coefficients k,•
for the two c a s e s S F 6 and HI for Rydberg binding energies IE, l of 0.1-40 meV. We find a significant increase in
k,,: towards lower binding energies, especially for HI,
which is due to the deviation of the free electron cross
section from the limiting s-wave behaviour ao~E -~/2.
The increase at ]Enl ~<2 meV is in qualitative agreement
with our high n Rydberg data (n > 80) if (-mixing due to
residual electric fields is taken into account. For low (,
Rydberg rate coefficients k,t(IE,I) are significantly larger
than free electron rate coefficients ke(E=lEnt), while for
circular orbits ({= n - 1 ) they agree. On average, attachment reactions of Rydberg electrons in low ( orbits proceed with an effective collision energy substantially
smaller than the binding energy [En[.
the energy range 0-150 meV at typical resolutions of
6-8 meV [4-6]. More recently, we realized a laser photoelectron source based on resonant two-photon ionization of metastable Ar*(4s 3/'2) atoms in a collimated beam
and measured the negative ion yield due to electron attachment to several molecules at sub-meV resolution in
the energy range 0-170 meV [7, 8]. In these experiments
the useful electron currents were of order 10-12 A.
Two other approaches exist which also provide important information on electron collision processes at very
low energies. The electron swarm method [9, 10], which
has been recently substantially refined and improved [ 11 ],
is capable to provide accurate rate coefficients ( k e ) =
(aev), which are the velocity-averaged product of the
collision cross section of interest ae (v) times the electron
velocity v. Going back to ideas of Fermi [ 12], electrons
in Rydberg orbits have been used as a source of low
energy electrons, see e.g. [13, 14]. Within the concept of
the quasi-free electron model for the description of Rydberg electron collisions [12, 15-17] the rate coefficient
kn e for a particular process to occur with Rydberg electrons is related to the cross section a~(v) for the same
process involving free electrons by [14, 16, 17]
oo
PACS: 31.50.+w; 34.60.+z; 34.70.+e; 34.80.-i
1. Introduction
Several techniques are now available to study the collisional behaviour of very slow electrons (energies
E~< 0.1 eV) under controlled conditions. The application
of photoelectron sources involving VUV photoionization
of ground state rare gas atoms has allowed the measurement of total, elastic and rovibrationally inelastic scattering cross sections for molecules at energies as low as
10meV with a projectile energy width around 4meV
[1-3] and studies of electron attachment processes in
kne= ~ ae(V)vf~e(v)dv,
(1)
0
where £e(v) represents the three-dimensional velocity
distribution function of the highly excited n e-electron in
the Rydberg atom A**(ng). For a proper test of the
quasi-free electron model accurately measured rate coefficients k. e should be compared with calculated values,
obtained from (1) on the basis of accurately measured
cross sections ae(v ).
For electron attachment processes a detailed comparison has not been possible so far, mainly because of
the lack of accurate, highly resolved free electron attachment data, but also due to perturbing effects associated
with the presence of the A + ion core in Rydberg electron
collisions at intermediate or low principal quantum numbers (postattachment effects [14, 17-20]). In this paper
236
we present such a comparison for the process of (dissociative) electron attachment by comparing measured rate
coefficients for Rydberg electron attachment (n = 80-200)
to X Y : SF6, HI with those calculated from cross sections
for free electron attachment measured with sub-meV resolution. The investigated processes are
cated and the average according to (1) extends over a
broad v-range with an emphasis on v < v~s. This was
accounted for more recently by Ling et al. [22] by introducing the median velocity vm as a more appropriate average velocity in relating free electron cross sections to
measured k,,e:
A **(n 0 + SF6--~A + + SF 6
(2a)
¢Te(E= Em)=kne/Vrn,
e - (E) -Jr SF6--*SF6
(2b)
ae(v)vf~e(v)dv= S ae(V)vf, e(v)dv=kne/2.
**(he) + HI--,A + + I - + H
e - (E) + H I - - I - + H.
(3a)
(3b)
The reactions (2a, b) and (3a, b) are thought to occur
by s-wave attachment [4-8, 14, 18, 21-23], for which
theory predicts the following limiting ( E ~ 0 ) behaviour
of the attachment cross section [24, 25]:
a~ (E, ;t = O) = a o (Eo/E)'/2 = ao (Vo/V)
(4)
while attachment through a partial electron wave with
orbital angular momentum Z > 0
is given by
a~(E,A),,,E ~-1/2 [24,25]. If the s-wave behaviour (4)
were valid over a substantial velocity range (enclosing the
major part of the velocity distribution of the Rydberg n gelectron for n > no) the rate coefficient for Rydberg electron attachment would be independent of n and e and
take the constant value
~.~ =
where vm is defined by the prescription
vm
and
A
(8)
a0 v0,
(5)
where we assume the velocity distribution f,,e(V) to be
normalized to one, i.e.
f , e ( v ) d v = 1.
(6)
o
Essentially constant values of the measured rate coefficients k,t have in fact been observed at values n>40,
notably by Dunning and colleagues [ 14, 22], who recently
extended their measurements for XY = SF6, CC14 up to
n = 4 0 0 [22]. These results were interpreted to reflect a
1/v (i.e. s-wave type) behaviour of the free electron attachment cross section [ 14, 22] at the corresponding electron velocities. Until recently, measured Rydberg rate
coefficients k,t and free electron cross sections cr~ were
often related by the simple relation
°'~ (I E, I) =k,,e/Vrm~(n),
(7)
where V~s(n) is the root-mean-square velocity of the
Rydberg electron which is related to the Rydberg binding
energy E, by [E,[ = m v ~ J 2 = R / n *z (m = electron
mass, R = Rydberg energy= 13.6 eV, n * = effective principal quantum number). Formula (7) can, however, be
expected to be a good approximation only for circular
Rydberg orbits g = e~= = n -- 1, for which the velocity distribution f,, e=,-l(V) is narrow and centered at V~m~.
For orbits with g,~ n, as typically encountered in Rydberg collision experiments, the situation is more compli-
0
(9)
Vm
Note that the median velocity vm and the associated median energy Em=mV~/2 depend on both n and L This
is true also if ae (v). v = constant in the relevant range of
velocities. For K**(nd) Rydberg states with n ~ 100 the
data of Ling et al. [22] infer that the median energy E m
is about 8 times smaller than the Rydberg binding energy
IE. 1. The fact that for orbits with g~n E m is substantially smaller than IE,[ is qualitatively seen by calculating the local kinetic energy Eki n of the Rydberg electron
at the average distance (r)n. e.~n~ (3) n*2 ao ' for which one
obtains Eki n = I En I/3.
Recently, we have measured the free electron attachment cross sections for several molecules including SF 6
[7, 8] and HI over the energy range 0-170 meV with submeV resolution and at the same time obtained rate coefficients for Rydberg electron attachment at high n > 80.
The free electron cross sections were found to decrease
more rapidly with rising energy than the E-1/z behaviour
for s-wave attachment, especially for HI, where the dependence is close to a 1/E behaviour for energies above
a few meV. Correspondingly, one would expect on the
basis of (1) an increase of k,e with rising n, and our data
exhibit such an increase for n ~>80. In this paper we present these experimental results and compare them with
values k, e calculated from (1) on the basis of the measured free electron cross section ae and classical velocity
distributions, thereby providing a detailed test of the
quasi-free electron model for Rydberg electron attachment in the range of principal quantum numbers, where
postattachment interactions between the ion core A + and
the negative ion can be neglected. The following paper is
organized as follows: in Sect. 2, we discuss the classical
velocity distribution of the Rydberg electrons for arbitrary values of n and g =<n - 1 ; in Sect. 3 we present the
experimental results for the reactions (2) and (3) and
analytical expressions for the absolute free electron attachment cross sections ae (E); in Sect. 4, we present calculated rate coefficients k, e ( [En [ ) in the binding energy
range 0.1 to 40 meV for different values of g (including
an average over 2) and we compare them, for IE.I
< 2 meV, with measured values and discuss the effects
of electric fields on the measured rates.
2. Velocity distributions of Rydberg electrons
The quantum probability distribution function f , e (v) for
the electron velocities in a hydrogen-like atom is known
237
to be given in terms of Gegenbauer functions C~Z~_~
[26]. Gegenbauer polynomials require a rather high computational accuracy for large values of n and f, as needed
in the present work, and tested computer programs for
this range (n>~ 100) do not exist to our knowledge. Moreover, it will be sufficiently accurate at high principal
quantum numbers to use the corresponding classical distribution functions which were given, e.g., by Gryzinski
[27]. Before we became aware of this work, we derived
the results independently in a somewhat different way.
In the following we summarize aspects of general interest
and give formulae relevant in the context of the present
paper.
Let L be the classical angular momentum (i.e.
L = ( f + 1/2)h semiclassically) and E < 0 the energy for
the bound electron, moving in the Coulomb potential
V(r) = - c/r(c > 0). The maximum classically allowed
value of L at fixed energy is L m a x = l / / - m c 2 / 2 E for a
circular orbit.
The distribution wL(r) of the radial coordinate is obtained by parametrising the Kepler ellipse by time, leading to
wL (r)dr = 2 dt=-Tv2(r ) dr,
(10)
With r e = ] / / ~ / m
distribution
one obtains the required velocity
4v3v
J~(v) (0 5+
2
v2 -
,
I L•l
2
(° 2 +
(17)
2
or, in terms of the dimensionless velocity u= v/v E, the
universal function
8u
A(u)
/r (U2+ t) 2 ]//4u z -
),2(uz4-
1) 2
(18)
with 7 = L v e / c = L / L m ~ = V-I ~--~-, which depends only
on the eccentricity e of the ellipse [27]. Equation (18) can
be rewritten as
8u
J L ( U ) = n y (U2+ 1) 2 ]//~-t~2 -- b/2)( u 2 - u 2 - )
(19)
The classical velocity distribution diverges at the maximum and minimum radial velocities
1
u± = - - (I + e ) .
Y
Some further remarks may be useful:
(20)
- The distributions are normalized:
where vL (r) is the radial velocity and
U+
j" f L ( u ) d u = 1.
T=nc
(21)
(11)
3
- The mean squared velocity is unity
is the period of motion on the ellipse. The factor of two
in (10) appears since each r-value occurs twice on the
ellipse. The distribution of r can be transformed into a
distribution of any variable which is a one-to-one function of r.
The distribution of the velocity v = I vl is given by
fL(V)=WL(r)]~[
1~
.
(12)
The r-dependence of the velocity v as well as the radial
velocity vL is determined by conservation of the total
energy
m 2 c_rn
L 2
E=~ v -r-2
v~ q 2mr2
c
(13)
r
yielding
u+
u2fL(u)du= 1
<U2> = I
(22)
u-
as already known from the virial theorem, which states
that v2s = (vZ> = v2e.
- In the limit of a circular orbit, e = 0, the classically
allowed regime u = < u < u + shrinks to the point u = I,
i.e. v = ve, as illustrated in Fig. 1.
- In the opposite limit of a free fall orbit (e = 1) the
distribution simplifies to
4
fo(u) - rt (u 2 + 1) 2 "
(23)
The usefulness of this distribution in a description of
atomic scattering processes has been demonstrated by
Gryzinski [27]. In this limit the average velocity is given
by
co
L=
r(v) =
E
2mr 2+r
o
m v2
2
,
'
(14)
(u> = j" ufo(u)du =2~ .
Tt
0
(15)
- In some cases, in which the angular momentum L of
the electron is not specified, the average over all angular
momenta is of interest:
E
and
Zmax
f(u)=
dr ] _
mvc
2•
(16)
(24)
Zmax
I fL(u)2LdL/
I 2LdL
0
0
2
-- 2
Lmax
Lm~x
~ fL(u)LdL.
o
(25)
238
10-2
I t=116
correspondence identity between classical and quantum
mechanics, where an entire set of probability distributions
coincides. The deeper origin of this observation is still
unclear.
In other situations, e.g. for g-mixing in homogeneous
electric fields (projection m e conserved, see Sect. 4), it is
appropriate to omit the weight factor 2 L, and one then
arrives at a microcanonical distribution, which corresponds to a 2 D-average in (26):
10-3
i g=100
10.4
lff s
t=50
I
io-6
~
10 -7
~
10 8
_o
4u
jr(u) - (u 2 + 1) 3 .
10.9
0
0_J
~ 104°
It should be noted that there is a systematic progression
in the 1D, 2D, and 3D distributions (23), (28), and (27),
respectively, which can be conveniently expressed as
\"%'
',,\
,,,,
Ud- 1
0<1041
i.e.
\
10-13
(A>
E = - 1meV
O
103
-
104
",
(B)
1044
l
f(d)(u)=Nd
k
~ 0 1 0 -12
l
S
~
105
VELOCITY v [m/s]
(28)
10s
107
Fig. 1. Classical velocity distributions of Rydberg electrons for
several values of the angular momentum at the fixed binding energy
E = - 1 meV (vE=l.88-10am/s). The dashed curve (A) and the
dashed-dottedcurve(B) represent the L-averaged distribution (27)
and (28), respectively
(U2+ 1)a+ ~ ( d = 1, 2, 3),
(29)
f(i)= fo, f(2)= f,f(3)= f.
In Fig. 1 we present the classical velocity distribution
(17) for several values of the angular momentum at the
fixed binding energy E = - t meV. All these distributions
have the same root-mean-squared velocity v~, = r e =
l / / - ~ / r n = 1.88- 104 rn/s. For low angular momentum
(i.e. e close to unity) the distribution is quite extended
and close to the free fall distribution (23), and one recognizes the power-law decay as v -4 for large velocities.
As the angular momentum increases the velocity bound-
The weight factor 2 L arises from averaging over all
orientations of the vector L, corresponding to a statistical
mixture of all e and m e states in quantum mechanics.
This integral can be evaluated analytically [27]. A more
direct approach is, however, possible:
f(v)~
~(V-Vo)~(E-
Z
_o
v~-C ~ d3rod3vo
to~
I--I-.a
ro /
,~02r4(u),'~(~v2E)4
i) 2
(26)
O
O
_d
iii
>
_Ol
O3
with r(v) taken from (15).
Written in terms of u = v/vE the normalized distribution
is
32 u 2
1) 4 '
f(u) - rr (u 2 +
5
o 1
(27)
the so-called microcanonical distribution, in agreement
with the result obtained by Gryzinsky [27]. Most interestingly, this distribution precisely agrees with the corresponding quantum result [28] at the bound state energies
E , of the hydrogenic atom. This is a unique case of a
1
103
104
10s
VELOCITY v [m/s]
106
Fig. 2. Classical velocity distributions of Rydberg electrons for a
fixed value of the angular momentum (¢ = 2) and various binding
energies E
239
aries v± = u± ve (compare eq. (20)) approach each other
and coincide in the case of a circular orbit (e = 0) at
v = V~ms. Also shown are the L-averaged microcanonical
distributions (27) (dashed curve (A)) and (28) (dasheddotted curve (B)). For large velocities (u>> 1) the distribution (27) decreases as u -6 and (28) as u -5. Figure 2
illustrates the velocity distributions for a fixed value of
the gmgular momentum (e = 2 as relevant for our experiment) and various binding energies E.
SF s- /
SF s
103
)
_..I
LU
>=
3. Experimental results
The experimental data for Rydberg electron and free electron attachment were obtained with a laser photoelectron
attachment method described elsewhere in detail [7, 8].
Briefly, a collimated beam of metastable Ar* (3p 54s 3P2)
atoms is excited to Rydberg states (ns, nd) or to
the ionization continuum by two lasers via the
At* (3pS4p 3/)3) intermediate level. A cw single mode dye
laser (Al=811.75nm) transversely excites the closed
transition Ar* (4s 3P2 - 4 p 3D3), and a small fraction of
the Ar* (4p 3D3) population is excited further by an intraeavity ew multimode dye laser (mode spacing about
40 MHz, bandwidth for the present measurements 12 GHz
50 ~teV FWHM). Both lasers are linearly polarized with
electric vectors parallel to each other and to the direction
of ion extraction. Tuning the second laser over the
wavelength range 462.30-461.10nm around the fieldfree ionization limit Ar + (2P3/2) (vacuum wavelength
2;=461.957 nm) accesses bound Rydberg levels n~>80
(E < 0) and yields free electron energies (E > 0) up to
5 meV, i.e. the energy range for which we present data
here. The experiment is pulsed at a repetition rate of
140 kHz to allow essentially field-free conditions during
electron production and attachment (phase I, 2.7 txs) and
application of a delayed pulsed electric field (23 V/cm) for
negative ion extraction (phase II) [8]. During phase II
the infrared laser and thereby the electron production are
switched off. For details of the experiment, the reader is
referred to [8]. The obtained data represent relative rate
coefficients for Rydberg electron attachment (E < 0) and
relative cross sections for free electron attachment. The
latter are placed on an absolute scale through normalization to thermal energy rate coefficients ( T = 3 0 0 K),
determined with electron swarm techniques [10, 29].
Previously we have reported absolute cross sections
for the attachment of free electrons to SF 6 molecules
( T = 300 K) in the electron energy range 0.3-170 meV. At
energies below the first vibrationatly inelastic threshold
( E = 9 5 . 4 meV, excitation of one quantum of the symmetric stretch vibration in SF6) the cross section was
found to be accurately represented by a simple analytical
formula [8], whose functional form was suggested by
Klots [30],
tre (E) = ( t r , / E ) [ 1 - exp ( -- fl E'/2)]
(30)
with E = electron energy in naeV, tr~ = 7130.10- J6 cm 2
( ± 5%) and B = 0.405 ± 10%. In the limit E ~ 0 , this for-
102
-2
,,,
, i . i ,1,
I
I
'
I
Il
I
I
I
12
1I
I
I
'
3I
I
I
'
I
4I
I
t
I
5
ELECTRON ENERGY E [meV]
Fig. 3. Yield of SF~- formation in collisions of high n Ar** Rydberg
atoms (E < 0) and of free electrons (E > 0) with SF6 molecules in
the threshold region. During this experiment the residual electric
field was in the range (0.03-0.1) V/re. The smooth curve for E > 0
represents the result of a calculation for the free electron attachment
yield on the basis of the cross section in (30) with fl =0.405 (see
text), which takes into account the influence of an electric stray
field F, = 0.03 V/m on the electron trajectories (for details, see [31 ]).
The vertical dashed line (E= 0) marks the position of the (strayfield reduced) Ar + (2/03/2) threshold
mula yields the s-wave threshold behaviour [8]
o e( E ~ 0 ) =
(2888+ 400). 10-16 cmZ/E 1/2.
(31)
The rate coefficient k~ ( E ) = tr~ (E)v for free electron attachment drops significantly towards higher energies from
the limiting s-wave value k ~ ( E ~ 0 ) = 5 . 4 8 . 1 0 - 7 c m 3 / s
( _ 14%) [8], as discussed in Sect. 4.
In Fig. 3 we present experimental attachment yield
data (circles) obtained with an optical resolution of
50 I~eV ( F W H M bandwidth of ionizing laser) and a residual electric field in the range (30-100) ixV/mm [8, 31]
over the energy range - 1 . 8 to + 5 meV. In this range
the (energy-averaged) photoexeitation ( E < 0) and the
photoionization (E > 0) cross sections are nearly constant (see Fig. 4 in [8]); in the evaluation of the attachment yield we have taken the weak energy dependence
of the excitation/ionization cross section into account.
The rate for Rydberg electron attachment is (almost) constant in agreement with previous results [14, 22]. Upon
closer inspection, the attachment yield reveals, however,
a weak increase towards E ~ 0 ; when simply linearized,
this increase amounts to about 4% per meV in the binding
energy range 1.5 to 0.3 meV. For E > 0 the smooth curve
through the data corresponds to the functional form (30)
with the parameters listed above; the influence of an electric stray field (Fs = 0.03 V / m ) and of the laser bandwidth
were taken into account (see [31] for details).
In Fig. 4 we present experimental attachment yields
for the molecule HI; only I - ions were observed in the
240
4. Rate coefficients for free electron
and Rydberg electron attachment
r/HI
104
¢-I
q
Ul
>-
10 3
it,~|t=,,
-2
-t
,,,,lllvtlllt,|,,l,l,*t
0
1
2
3
4
5
ELECTRON ENERGY E [meV]
Fig. 4. Yield of I - production in collisions of high n Ar** Rydberg
atoms (E < 0) and of free electrons (E > 0) with HI molecules in
the threshold region. During this experiment the residual electric
field was about 0.5 V/m. The smooth curve for E > 0 represents
the result of a calculation on the basis of the cross section in (30)
with # = 1.6, taking into account the influence of an electric stray
field F~=0.55 V/m. The vertical dashed line ( E = 0 ) marks the position of the (stray-field reduced) Ar + (2P3/2) threshold
Using the classical Rydberg electron velocity distributions described in Sect. 2 and the analytical expressions
for the free electron attachment cross section presented
in Sect. 3, we have calculated rate coefficients for Rydberg electron attachment on the basis of the quasi-free
electron model (1) and for free electron attachment
(ke=a~(v)v). In Fig. 5 we present the results for SF 6 in
the energy range 0-40 meV. For Rydberg electron attachment the energy corresponds to the binding energy
[ E, [. The rate for free electron attachment drops significantly with rising energy from the limiting value
ke(E--*O ) = 5.48- 10 - 7 cm3/s to about 1.95- 10 - 7 cm3/s
at E = 40 meV. Correspondingly, the deviation from the
s-wave behaviour k~ = const is substantial. As expected,
the calculated rates for Rydberg electron attachment involving circular orbits (e= em~x = n - 1) agree with those
for free electron attachment. In contrast, the rates for
Rydberg electron transfer from highly eccentric orbits
( t = 0, 2) are significantly larger than those for free electrons and circular Rydberg electrons, and they decrease
5.2
energy range of the experiment. For E > 0 results were
obtained up to 170 meV, as presented and discussed in
comparison with previous work [5, 23] in a forthcoming
paper [32]. Analysis of the free electron data showed that
over the range 0.8-170meV the cross section is well described by the functional from (30) with parameters
a l = 7 5 0 0 . 1 0 - 1 6 c m 2 ( + 6 % ) and f l = 1 . 6 + 0 . 3 . The
relative yield data were normalized to the thermal
attachment rate coefficient k e (T = 300 K) = (3.0 + 0.9)
• 10 - 7 c m 3 / s due to Smith et al. [29]. The listed uncertainties for a~ and fl are estimates with reference to combinations of a~ and fl, which yield essentially the same
averaged rate coefficient ke = 3.10-7 cm3/s. As indicated
by the much higher value of B, the cross section for I formation from HI has a much steeper decline towards
higher energies than the one for SF 6 formation from SF6,
and it exhibits essentially a G e ' E -1 dependence for
E > 10meV. Correspondingly, one expects a strong energy dependence of k e (E) (see Sect. 4) and also a nonconstant rate coefficient k,e for Rydberg electron attachment. In fact, the measured Rydberg attachment yield
(Fig. 4) clearly shows a rise towards E = 0, which - when
linearized - amounts to about 10% per meV. We note
that during the HI measurements the residual electric field
was not as low as in the SF 6 experiments. Model calculations of the attachment yield function incorporating
various values of the residual electric field F~ indicated
F,~550 l~eV/mm for the data of Fig. 4. As a result gmixing will occur and influence the measured rates (see
Sect. 4).
SF 6- / SF 6
4.8:
#
o
y:
3.E
",, "~
kl.
kL
(.3
tU
2.!
k E)
0
10
~
,
~
20
ENERGY E [meV]
30
40
Fig. 5. Rate coefficients k,e for Rydberg electron attachment to S F 6
in the range of binding energies 0. I to 40 meV, as calculated on the
basis of the quasi-free electron model Eq. (1) with the classical
velocity distributions Eq. (17) and the free electron cross section
Eq. (30) (see text). The diagram also includes rates for the L-averaged velocity distributions Eq. (27) (dashed curve (A)) and Eq. (28)
(dashed-dotted curve (B)) and the rate ke for attachment of free
electrons (dots), which agrees with the Rydberg attachment rate
coefficient for circular orbits (e = em~)
241
less strongly to values around 3 . 2 . 1 0 - T c m 3 / s at
IE, I = 40 meV.
Experimental rate coefficients k, e for negative ion formation can only be directly compared with the calculated
values if postattachment interactions between the positive
ion core A ÷ and the negative ion (here SF 6 ) can be
neglected [14]. Towards low principal quantum numbers
(n < 30) an increasing fraction of the ion pairs, created
in the primary electron attachment process, is formed at
internuclear distances such as to prevent direct escape
from the attractive Coulomb potential [14, 18]. As a result, the measured rate coefficients stay below those calculated with formula (1) [14]. For the conditions of the
experiments carried out so far [14, 18-20, 22, 23, 33],
postattachment effects play a minor or negligible role for
principal quantum numbers larger than about 30 (binding
energies <~15 meV). For this range of n, absolute measurements of rate coefficients were reported by Dunning
et al. [14,22] for low f Rydberg states (e<3) and n up
to 400 and found to be essentially constant with an average value of (4 + 1). 10-7 cm3/s. Unfortunately, the scatter in the individual data points (as expressed by the given
uncertainty) is too large to reveal a variation ofk, e, which
our calculations predict to be a 18% decrease from
[ E , [ = l m e V to IE, l = 1 0 m e V - Our Rydberg data
(Fig. 3) indicate a decrease of about 4% in the range
0.5 < [E, I < 1.5 meV in qualitative agreement with the
calculated results.
For comparison with the rates for circular and highly
eccentric Rydberg electrons we have included in Fig. 5
rates calculated with e-averaged velocity distributions, as
given in (27) and (28). The former average corresponds
to a statistical ensemble with complete g and me-mixing,
while the latter case represents e-mixing in a homogeneous electric field (e completely mixed, me conserved).
As expected, both averages yield rate coefficients between
those for highly eccentric and circular orbits, and the
result for the (2 f + 1)-weighted average (dashed curve
(A)) is closer to the one for the circular orbit.
In Fig. 6 we present rate coefficients for I - formation
from HI, obtained in an analogous way as for SF 6 formarion in Fig. 5. The decrease of the rates is much
stronger than for SF6; over the range 1-40 meV the free
electron rate k e falls off by a factor of 5 and the Rydberg
rate for e = 0 by a factor of 2.3. The e-averaged rate
coefficients fall in between these extremes as for SF 6. In
Fig. 7 we show the range 0.1 to 2 meV on an expanded
scale. The circles correspond to our measured Rydberg
data (Fig. 4), which are normalized to the calculated rate
coefficients for e = 2 at E = [En[ = 1.8 meV. One observes
that the measured values stay below the calculated rates
towards lower energies. As pointed out in Sect. 3, the
I'/HI
20f
18
=2
I/HI
15
14
6
'~.,,
%
8
I--
,,z, ~o
(A) ........
w
O
o
,. "
e=0
W
10
ke(E)
0.0
f = Lfrnax
ke(E)
0
10
20
ENERGY E [rneV]
30
40
Fig. 6. Rate coefficients k,e for Rydberg electron attachment to HI
in the range of binding energies 0.1 to 40 meV, as calculated on the
basis of the quasi-free electron model (compare caption for Fig. 5).
The diagram also includes rates for the L-averaged velocity distributions Eq. (27) (dashed curve (A)) and Eq. (28) (dashed-dotted curve
(B)) and the rate k, for free electron attachment
0.5
1.0
ENERGY E [meV]
1.5
2.0
Fig. 7. Rate coefficients k,, e for Rydberg electron attachment to HI
in the range of binding energies 0.1 to 2 meV, as calculated with
the quasi-free electron model (see Fig. 6 and text). The diagram
also includes the rates for the L-averaged velocity distributions
Eq. (27) (dashed curve (A)) and Eq. (28) (dashed-dotted curve (B))
and the rate coefficient ke for free electron attachment. The open
circles represent the measured rates for Rydberg electron attachment, which are influenced by an electric stray field of about
Fs = 0.5 V/m, which leads to g-mixing and consequently to a weaker
rise towards lower energies than the one predicted in the calculation
for e=2
242
experimental results for HI were obtained with a residual
electric field F s of about 0.5 V / m ; correspondingly one
expects that e-mixing affects the measured rates at lower
quantum numbers n than for SF6, where the residual field
was in the range (0.03-0.1)V/m. Strong g-mixing occurs
above the critical value ne,~ 176 (Fs[V/m]) -1/5, i.e. for
F~ = 0.5 V/m at n >200 or energies IE , 1~0.4 meV [31 ].
Therefore we attribute the deviation of the measured rates
from the values calculated for low e orbits to the effect
of g-mixing. We note that the field-induced redistribution
of the oscillator strength for Rydberg excitation should
have negligible effects on the efficiency for high Rydberg
production (n > 80) under our experimental conditions
(broadband laser 50 tieV F W H M with narrow modespacing); field-dependent studies of Rydberg electron attachment to SF 6 have shown this in a direct way [31].
As shown in Figs. 5 and 6, rate coefficients for Rydberg electron attachment involving eccentric orbits will,
in general, be larger than those for circular orbits or for
free electron attachment due to the combined effects of
10
,
For the comparison of rates for free electron attachment
ke (E), determined at a certain free electron energy E,
with Rydberg rates kne involving eccentric orbits which
are normally produced by laser excitation, it is of interest
to establish the value of the Rydberg binding energy for
which the Rydberg rate coefficient k, e (g~ n) is found to
agree with a given k~ (E).
Figures 5 and 6 show that these values of [E. I are
larger than the corresponding free electron energy E by
a scaling factor s, which amounts to values of 3 to 9 in
the energy range covered in Figs. 5 and 6. In Fig. 8 we
illustrate the energy dependence of this scaling factor s (E)
in the range (0.5-9) meV for several values of the parameter fl -including the two relevant cases SF 6 (/~ = 0.405)
and HI (p = 1.6) - and fixed angular momentum g= 2.
For SF6, the scaling factor varies slowly between 3 and
4.6. For HI, s (E) rises from 4.9 to a maximum value of
nearly 8.8 around 7 meV. Even for the small value p = 0.1
the scaling factor is much larger than unity (the value
reached in the limit ,6--+0). We conclude that Rydberg
data at low e represent the energy dependence of the free
electron rate on a strongly expanded energy scale and
previous interpretations, regarding IE , ] as equivalent
with the free electron energy E (e.g. [14, 23]), were not
correct to an amount which depends on the system.
At this point we should mention that the influence of
the heavy particle motion (relative velocity Vr.1 around
600m/s) on the effective collision velocity of the
(quasi)free Rydberg electron has been neglected in all the
calculations of the Rydberg rate coefficients presented in
this paper. For very low Rydberg electron velocities
v(v~ V~el) the effective collision velocity v~r= Iv - Vre][
will be essentially given by the value of V~¢~.Correspondingly, our calculated rate coefficients will be somewhat
,
'
,
,
I
,
'"~.........
r/HI
9
[3=1.6
~=2
/
9=1.0
7
O
/
,,<6
J"
~e/O
4
/
oj~
/
,~
,.-~
i) a free electron attachment cross section, which decreases more rapidly than E -]/2 with rising energy,
ii) the fact that eccentric orbits possess a substantial fraction of velocity components smaller than the root-meansquare velocity Vr~ = (2 t E, ]/m) 1/2 (i.e. the velocity on
the circular orbit).
, ",
i
i
i
~
/
o'~
13:o.1
i
2
FREE
i
i
I
i
1
t
4
ELECTRON
I
6
ENERGY
,
1
l
I
8
,
l
l
10
E [meV]
Fig. 8. Energy dependent scaling factor s (E) = IE, t/E, relating the
Rydberg binding energy IE. 1 with the value of the free electron
energy E, for which the respective electron attachment rate coefficients k, e- 2( IE, I) and k, (E) agree. On average, attachment processes involving Rydberg electrons in low e orbits proceed with an
effectiveelectron kinetic energy, which is substantially smaller than
the Rydberg binding energy IE, I- The parameter fl is a measure
of the deviation from the s-wave threshold behaviour, see (30)
too low towards low binding energies and especially for
low L The influence of this effect can be judged by an
inspection of that fraction of k,e which arises from low
Rydberg velocities. We have therefore calculated integrals
/).
kale(v*) = I ae(v)vfL(v)dv
(32)
O_
which represent the fraction of the total rate coefficient
k. e, which is accumulated from the lower boundary v = v_
to a variable upper bound v = v , .
In Fig. 9 we present ratios kacc (E,)/kne (e= 2), as calculated for Rydberg electron attachment to HI over the
energy range E,=rnv~/2 from l0 -4 to 2meV at seven
Rydberg binding energies IE,] from 0.1 to 10meV. To
a first approximation the effect of a finite relative velocity
Vre] can be simulated by using V~eI as a lower boundary
in the integral (32). This corresponds to a value E,=
1 0 - 3 meV in Fig. 9. One observes that the corresponding
decrease in the total rate coefficients k,,e=2 ranges from
about 15% to 3% for binding energies between 0.1 to
2 meV.
Figure 9 also contains information on the values of
the median velocity v,, and the median energy Era=
mv~/2, as introduced in (9). For the case of electron
243
1.0
are based on classical Rydberg velocity distributions and
on the validity of the quasi-free electron model (1). F o r
a further test of these ideas it appears desirable to experimentally determine n-dependent rate coefficients at
high n with sufficiently low uncertainties as to reveal
variations of k, e with n and g.
0.9
0.8
This work has been supported by the Deutsche Forschungsgemeinschaft (Ho 427/17-1, Ko 686/4-4).
0.7
0.6
"~" 0.5
References
111
~0A
0.3
0.2
0.1
10-4
10-3
10-2
104
ENERGY E, [meV]
100
Fig. 9. Fraction k ~ ( E , ) / k , l = z of the rate coefficient for Rydberg
electron attachment to HI, as accumulated in the velocity interval
v = v _ to v = v , ( v . = l / / 2 E . / m ) , see (1, 32) and text. The dots and
arrows indicate the values of the median energies E,,, at which the
accumulated fractions reach one half of the respective total rate
coefficients (E,,=15.5, 28.5, 62, 108, 184, 357, 5821xeV for
IE.I =0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0 meV, respectively)
attachment to HI, presented in Fig. 9, the median energies
E m are found to be 6.5 to 17 times smaller than the corresponding Rydberg binding energies I E. I, ranging from
0.1 to 10 meV, i.e. the ratios IE.I/E,. are about 1.5 to
2 times larger than the scaling factors s (E) displayed in
Fig. 8. As is true for s (E), the ratio I E , I ~Era depends
on the parameter B. In the limit B ~ 0 , i.e. for the case
cre (v) v = constant, the ratio I E , t/E,,, is nearly constant;
for e = 2 and n >>e we find I E , I/E,,, = 5.1 (this value is
somewhat lower than the factor 8 to be inferred from the
data in [22]).
So far we have measured free electron attachment cross
sections at sub-meV resolution for five molecules (SF6,
HI, CC14, CFC13, 111-C2C13F3); in all cases these cross
sections were found to be well described by (30) for
energies below the first vibrationally-inetastic threshold
[32]. The parameter B, which is a measure of the deviation from the s-wave E - 1/2 behaviour of the attachment
cross section, is smallest for SF 6 and largest for HI. Therefore, the scaling factor for the other three molecules takes
values intermediate between those for SF 6 and HI, as will
be discussed in a future publication [32]. There we shall
also present calculated rate coefficients for free electron
attachment involving a Maxwellian electron ensemble as
a function of electron temperature.
In conclusion, we emphasize that the presented calculated rate coefficients for Rydberg electron attachment
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Note added in proof. After submission of the manuscript we became
aware of several references relevant to Sect. 2 of the present paper.
In particular the distribution (27) has been presented in [34-37],
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