Electron and Ion Mobilities

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 20

ELECTRON AND ION MOBILITIES GORDON R. FREEMAN Department of Chemistry University of Alberta Edmonton. Alberta, Canada

DAVID A . ARMSTRONG Department of Chemistry University of Calgary Calgary, Alberta, Canada

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Low-Density Gases (n/n, < 0.1). . . . . . . . . . . . . . . . B. Dense Gases and Low-Density Liquids (0.1 < n/n, < 2.0) . . . . 111. Ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Low-Density Gases (n/n, < 0.01) . . . . . . . . . . . . . . . B. Dense Gases and Low-Density Liquids (0.01 < n/n, < 2.0) . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

267 270 270 28 1 296 297 3 18 320

I. Introduction Empirical information about certain aspects of the transport of electrons and ions in fluids has thrust uncomfortably ahead of the related theory. The theories of electron and monatomic ion scattering by monatomic and diatomic gases at low number densities n are relatively advanced. Scattering by polyatomic molecules and in the multibody regime are now exciting challenges. Several examples illustrate the rich field of questions in urgent need of theoretical investigation. Two newly recognized phenomena have been interpreted only qualitatively or semiquantitatively. One is a molecular sphericity effect on electron scattering at energies
268

Gordon R. Freeman and David A. Armstrong

being equal (Gee and Freeman, 1980a, 1983a). The second is that thermal electrons in dense gases near the vapor- liquid coexistence curve can be quasi-localized by density fluctuations of suitable magnitude (Huang and Freeman, 1978a). Many nonpolar molecules with mean polarizabilities > 1.5 X l 0-30m3 exhibit a scattering cross section minimum for electrons at an energy < 1.O eV, the Ramsauer-Townsend (R-T) minimum (Massey et al., 1969).The R- T minimum results in a mobility (drift velocity/field strength) maximum as the electric field strength is increased and the electrons become heated. The R-T effect has a quantum mechanical interpretation. However, a similar mobility maximum occurs for simple ions in many gases as the field is increased (McDaniel and Mason, 1973). The interpretation is based on the energy-sensitive balance of the effects of the long range attraction and short range repulsion between the ion and molecule (Wannier, 1970; Bates, 1982a). It is curious that the electron and ion mobility maxima occur at similar drift velocities and energies (vd lo3 m/sec, E - 1 eV; Huang and Freeman, 1978a, see p. 1360; Takata, 1975; Thackston et al., 1980b). Furthermore the threshold drift velocity vp,above which the steady state mean energy of the charged particles is significantly greater than the thermal energy of the gas, is - lo2 m/sec for monatomic cations (Gatland et al., 1978) and anions (Thackston et al., 1980a) as well as electrons (Huang and Freeman, 1978a, 1981) in monatomic gases. The value of vF for both electrons and heavy ions is - 40% of the average speed of the monatomic gas molecules. For transport purposes the electron may be considered as a light anion. While that analogy is considered to be self-evident by some, it is thought by others to be misleading. At very low number densities, where the electron magnetic moments do not interact with each other, the analogy should be pursued to discover its limits. In particular, the phenomenological R-T scattering minimum is worthy of a new, more general examination by theoreticians. On another front, the mobilities of polyatomic ions in low-density gases do not conform to present treatments (Parent and Bowers, 1981 ). The clustering of neutral molecules about ions (Thomson et al., 1973; Sennhauser and Armstrong, 1978;Jbwko and Armstrong, 1982a)may be considered as a dynamic complication of the polyatomic ion problem. Quantitative interpretation of the mobilities of clustered ions has as a prerequisite an adequate model of the mobilities of polyatomic molecular ions. The important contribution of rotational transitions to the scattering of thermal energy ions by polar molecules has recently been emphasized (Takayanagi, 1980;Celli et d.,1980).The process is probably also significant for thermal ion scattering by nonspherical nonpolar molecules (Gee et al.,

-

ELECTRON AND ION MOBILITIES

269

1982).The importance of rotational transitions in low-energy electron scattering is illustrated by the depth and elegance of theoretical treatments ofthe process for diatomic molecules (Massey et al., 1969; Takayanagi and Itikawa, 1970; Golden et af., 1971; Golden, 1978; Burke, 1979). A different theoretical approach with different approximations might be required for more complex molecules. The molecular sphericity effect (Gee and Freeman, 1983a) might be a good point of initial attack. Excellent reviews of scattering by monatomic and diatomic molecules have been given previously in this series (Takayanagi and Itikawa, 1970; Golden, 1978; Burke, 1979) and in books by Huxley and Crompton ( 1974) and Mason and McDaniel ( 1985). Electron and ion transport in a selection of these gases at low densities is used in the present article as a comparative basis for that in polyatomic molecular gases. After increasing the number of atoms per molecule the next logical step is to increase the number of molecules per unit volume. Effects of increasing the fluid density through the gas, critical (n,) and into the liquid phase are described. There is a great need for deeper theoretical interpretations of electron and ion behavior at all densities. For example, the density-normalized mobility of electrons, going from the normal gas to the liquid, can either increase or decrease by several orders of magnitude depending on the molecular structure. Experimental techniques (McDaniel and Mason, 1973; Elford, 1972; Elford and Milloy, 1974a; Huxley and Crompton, 1974; Helm, 1978) and the conditions for steady state drift (Lin et af., 1977; Kumar, 1981) have been critically examined elsewhere. Although most measurements of mobilities have been made with drift tubes, the microwave conductivity method (Warman, 1982) and ion cyclotron resonance method (Huntress, 1971; Ridge and Beauchamp, 1976) are being increasingly used. An impressivebody of information on mobilities and momentum transfer integrals for the simpler ion-gas systems is summarized in Atomic Data and Nuclear Tables (Ellis et af., 1976b, 1978). Electron and ion mobilities in the “normal” liquid phase, which has a density more than double that of the critical fluid, > 2.0 n,, is beyond the scope of the present article. Suffice it to say that electron mobilities in dielectric liquids are strongly influenced by the molecular properties. If the molecules are spherelike, nonpolar, and do not capture the electrons, electrons tend to reside in a conduction band and have very high mobilities, up to 1 m2/V sec (Huang and Freeman, 1978a). If the molecules are distinctly nonspherical or highly polar, electrons tend to reside in localized states and have low mobilities, down to those of ions, lo-’ m2/V * sec (Schmidt, 1972). Mobilities may be found at all values between these extremes depending on the molecular properties and on the temperature and density of

-

-

-

270

Gordon R . Freeman and David A. Armstrong

the liquid (Dodelet et al., 1973). The study of electron transport in liquids serves as a probe of the liquid state and leads to the question of the nature of the electron itself.

11. Electrons A. LOW-DENSITY GASES(n/n, < 0.1)

1. Monatomic Molecules

The density-normalized mobilities np, of electrons, and their dependencies on the density-normalized applied electric field strength E/n, in helium (Pack and Phelps, 1961) and argon (Huang and Freeman, 1981) are compared in Fig. 1. These two examples were chosen because helium has a positive scattering length (6.3 X lo-" m; O'Malley, 1963) and is (of all molecules) the one that is most nearly a rigid sphere, while argon has a negative scattering length (-9.0 X lo-" m; O'Malley, 1963) and has a much cozier relationship with electrons. The polarizabilities of helium and m3;Landolt-Bornstein, 195 1). argon are respectively, 2.1 and 16.4 ( The former is not large enough for the charge-induced dipole attraction to dominate the short range repulsion even at the lowest energy and largest distance at which scattering has been detected. The latter polarizability provides a charge-induced dipole attraction that dominates the short range repulsion at low energies. The density of argon represented in Fig. I , 95 X 102smolecule/m3 or 35

1

10-3

10-2

10-1

1

Eln (Td)

FIG.1. Density-normalized mobilities ripe of electrons in low-density gaseous helium and molecule/m3 (Pack and argon, as functions of electric field strength. (0),He, 77 K, 4.0 X Phelps, I96 1 ); (O),Ar, 12 1 K, 95 X I 025molecule/m3 (Huangand Freeman, I98 I); Td = lo-*' V . m2/molecule.

ELECTRON AND ION MOBILITIES

27 1

Amagats, is at the upper limit of the low-density regime (Huang and Freeman, 1981). The mobility in helium is independent of field strength up to about 3 mTd (mTd = V m2/molecule), then decreases monotonically with increasingfield (Fig. 1). The decrease of mobility is due to heating ofthe electrons by fields 2 3 mTd, combined with the slight increase of the momentum transfer cross section with increasing energy (Fig. 2; Frost and Phelps, 1964).Electronsgain energy by acceleration in the field and lose it by collisions with molecules. At low field strengths the average amount of energy gained between collisionsis less than that normally exchanged during a collision, so the electrons remain near thermal equilibrium with the medium. At higher field strengths the electrons gain more energy between collisions than they normally exchange, so they increase in energy and velocity until a new steady state is reached..If the electron collision frequency changes as a result of the increased velocity, the mobility changes. It can either increase or decrease, depending on the velocity dependence of the momentum transfer cross section. In contrast with helium, the mobility in argon increases at fields > 1.4 mTd, passes through a maximum at 5 mTd, then decreases at higher fields (Fig. 1). This behavior reflects electron heating by fields 2 1.4 mTd, combined with the R-T minimum in the momentum transfer cross section at -0.23 eV (Fig. 2; Milloy et al., 1977; Haddad and OMalley, 1982). The mean energy of the electrons increases much more rapidly with field strength in argon than in helium (Warren and Parker, 1962). The shape of the electron energy distribution in argon at fields above the heating threshold is

-

1

v(l&m/sec)

5.9 i((11

0.01

I

'

18.8

8

I ""I

I

0.1

I

I

I

""r

59

1

E(W

FIG.2. Electron momentum transfer cross sections urn,= of low-density helium and argon and Phelps, 1964);Ar gases, as functions of electron energye and velocity 0.He (-)(Frost (---)(Frost and Phelps, 1964), (- - -) (Milloy el al., 1977), (-) (Huang and Freeman, 1981).

272

Gordon R. Freeman and David A. Armstrong 10

H

E

0 c \

0

c

0

001

01

E

10

(ev)

FIG.3 . Isotropic energy distribution function of electrons in argon gas at 293 K, at the indicated field strengths E/n in mTd V . m2/molecule)(Makabe and Mori, 1982). R - T indicates the location oftheRamsauer-Townsend minimum. Reproduced with min (---) the kind permission of the authors and publisher.

strongly influenced by the minimum in the scattering cross section at 0.23 eV (Fig. 3; Makabe and Mori, 1982).

2. Diatomic Nonpolar Molecules The diatomic molecules most frequently used in electron scattering studies are hydrogen and nitrogen (Pack and Phelps, 1961; Frost and Phelps, 1962; Warren and Parker, 1962; Lowke, 1963; Frommhold, 1968; Nelson and Davis, 1969; Robertson, 1971). Interest continues today (Taniguchi et al., 1978; Reid and Hunter, 1979; Braglia et al., 1981, 1982; Wada and Freeman, 198 1; Pitchford and Phelps, 1982). The value of np, for thermal electrons at 77 K in hydrogen is fourfold smaller than that in nitrogen (Pack and Phelps, 196 I), so at - 10-2 eV the m2; Englehardt momentum transfer cross section of hydrogen ( - 8 X m2; and Phelps, 1963) is fourfold larger than that of nitrogen ( 2 X Englehardt et al., 1964). The energy dependence of the cross section of hydrogen is much smaller than that of nitrogen at E < 0.1 eV. The comparison is somewhat analogous to that between the cross sections of helium and argon at a 10-fold higher energy (Fig. 2). The question is whether an R - T mz at 15 meV; minimum exists in the cross section of nitrogen (1.7 X Wada and Freeman, 198 1). The relative scattering behaviors of hydrogen and nitrogen imply that the cross section of hydrogen does not possess an R - T minimum, but that of nitrogen might. Among the molecules that have isotropic polarizabilities, the scattering length of helium is large and positive, whereas that of argon is large and negative (Table I; O'Malley, 1963). The scattering length of neon is small and positive (O'Malley and Crompton, 1980). Interpolation of the values of a and A in Table I indicates that if the isotropic polarizability were

-

273

ELECTRON AND ION MOBILITIES TABLE I PHYSICAL PROPERTIES He TJKY

5.19

E(10-3' m3)b a,,- ai( m3)b

2.1

n,(1OZ6 molecules/m3)" 104.3

8 , ( i O - 3 eV). a,( m2)d A( 10-l'm)p

0

0.0 5 6.3

Ne

Ar

Xe

H2

N2

CH4

CzH,

CzH,

190.6 305.5 282.4 44.4 150.9 289.7 33.3 126.2 60.8 40.6 46.7 50.5 92.7 67.1 144.4 80.5 40.1 8 18 26 45 40 4.0 16.4 0 0 0 3 7 0 8 21 0.0 0.0 0.0 7.5 0.25 0.65 0.08 0.08 0.3 6 -116 8 2 -20 -20 -11 1.1 -9.0 -34.4

Critical temperature and density; Reid ef al. (1977). Polarizability mean and anisotropy; Landolt-Bornstein (195 1); Bridge and Buckingham ( 1 966). Rotational constant; Herzberg (1950, 1967). Electron momentum transfer cross section at 0.01 eV; Frost and Phelps (1962, 1964);OMalley and Crompton (1980);Gee and Freeman (1981b). Scattering length; O'Malley (1963); O'Malley and Crompton (1980).

6X m3 the scattering length would be small and negative. When the polarizability is anisotropic a larger mean polarizability is required to produce an R - T minimum. For example, hydrogen has a mean polarizability of 8 X m3 (Bridge and Buckingham, 1966) but no R-T minimum, m3. The mean polarizability presumably due to the anisotropy of 3 X nitrogen is double that of hydrogen, but the electron momentum transfer cross section of the former at lo-* eV is only one-quarter that of the latter (Table I). Nitrogen either possesses an R - T minimum or is on the verge of doing so. The mean polarizability of nitrogen is larger than that of argon, but the anisotropy of the former would cause its R-T minimum to be much shallower and at a lower energy than that of the latter (Freeman ef al., 1979). The anisotropy effect is not understood beyond an unsatisfying qualitative rationalization. The detailed interpretation is evidently more complex than we would first suspect. Whether nitrogen possesses an R-T minimum can be resolved by refined measurements of np, at temperatures below 100 K and fieldsbelow 10 mTd (Wada and Freeman, 198I ). The values of ripe at 293 - 300 K agree from one laboratory to another within 3%(Fig. 4). However, at 78 _+ 1 K the values of Wada and Freeman (1981) are 10- 15% lower than those of Lowke (1963) and Pack and Phelps ( 1961) at fields < 10 mTd. The threshold drift velocities for electron heating seem unrealisticallylow in the earlier data for 78 k 1 K, but new measurements are required to settle the question. The spacing between rotational levels in nitrogen is sufficientlysmall (0 2 is 68, = 1.5 meV, Table I; Herzberg, 1950), and the anisotropy of polarizability is suffi-

-

274

Gordon R. Freeman and David A. Armstrong

a

FIG.4. Density-normalized mobilities n,ue of electrons in nitrogen as functions of electric field strength and temperature. Pack and Phelps (1961): (O),77 K; (+), 300 K. Lowke (1963): ( 0 ) 78 K; (o), 293 K. Wada and Freeman (1981): 79 K (A), 157 K; 295 K.

(o),

(o),

ciently large that inelastic energy exchanges between thermal electrons and the molecules are significant at 77 K (Englehardt et al., 1964). Inelastic scattering is the dominant process that removes from the electron the energy gained from the field, even at the lowest fields. The threshold drift velocity upfor electron heating in nitrogen should therefore be greater than the speed of sound c,, which is related to the average speed of the molecules (Gee and Freeman, 1979; Paranjape, 1980; Wada and Freeman, 1981). When the energy is removed from the electrons mainly by elastic collisions up is a few tenths of the average speed of the molecules and is referred to co by analogy with a similar phenomenon in semiconductors (Shockley, 195 1). The ratio up/coequals 0.5 in argon at 12 1 K, characteristic of energy loss by elastic collisions (Huang and Freeman, 1981). In nitrogen at 77 K the ratio should be significantly greater than one. The data of Pack and Phelps (1961) for nitrogen at 77 K correspond to u p / c o= 5 k 2; Wada and Freeman (1981) reported 10 & 2; Lowke's data (1963) correspond to a ratio < 2.1. The disagreement between these values reflects the difficulty of the measurements, but the true value at 77 K is probably 5 - 10. By contrast, the spacing between rotational levels in hydrogen is so large (0 + 2 is 6Bo = 45 meV, Table I ) that rotational excitation by thermal electron swarms is negligible at 77 K. At this temperature the value of up is 3 10 m/sec (Pack and Phelps, 196 1 ) and co = 669 m/sec, so u,"/co = 0.5 in agreement with the value for argon and energy loss by elastic collisions.

-

3. Polyatomic Nonpolar Molecules (Sphericity Eflect) The electron momentum transfer cross sections, om,,,of isotropically polarizable polyatomic molecules possess an R-T minimum at E = 0.25 k

ELECTRON AND ION MOBILITIES

275

0.05 eV, near that of argon. Examples are methane (CH,) (Ramsauer and Kollath, 1930;Cottrell and Walker, 1967; Pollock, 1968;Gee and Freeman, 1979; Sohn et al., 1983),silane (SiH,) (Cottrell and Walker, 1967; Pollock, 1968), and neopentane [C(CH3),] (McCorkle et al., 1978; Freeman et al., 1979). Upon replacing one of the H atoms of methane by a CH3group, making ethane, the molecule nearly doubles in volume but the value of nb, in the gas increases (Fig. 5). The average value a,,,,, = ( v ) / ( v/a,,,) of the larger molecule is smaller than that of the smaller molecule for thermal electrons at 200-300 K (Gee and Freeman, 1979, 1980a). The reason is that ethane possessesan R - T minimum at 0.12 eV, which is nearer the range of energies of the thermal electrons than is the R-T minimum for methane at 0.25 eV. This lowers the value of a,,,, for ethane at energies S O . 12 eV, covering the thermal region in question. The shift of energy of the R - T minimum, going from methane to ethane, is mainly the result of the change of the degree of sphericity of the molecule, rather than the change of molecular size. The spherelike methane (CH,) and neopentane [C(CH,),] are very different in size, but their R-T minima are at similar energies, 0.25 and 0.2 1 eV, respectively. By contrast, the distinctly nonspherelike ethane ( H3CCH3) and n-pentane (H3CCH2CH2CH2CH3) have R - T minima at 0.12 and 0.13 eV, respectively (Gee and Freeman, 1980a; Freeman et al., 1979). Results for many hydrocarbons are summarized in Table 11. The effect of molecular shape on am,,at E < 0.3 eV is illustrated by the cross sections of the three isomers of pentane, CSHIZ,in Fig. 6. There is a simple relationship between the external shape of the molecule and the energy dependence of its am.,,even for such complex molecules. This should

-

2 I00

200

300

T (K)

FIG. 5 . Density-normalized mobilities ripe of electrons in CH, (0)and C,H, (A) gas; n = 7 X loz5molecule/m3; Gee and Freeman (1979, 1980a).

TABLE I1

ELECTROF; SCATTERING PARAMETERS OF HYDROCARBOX GASES

Aknnes 0.0

0

0

I .0 7

90-160 294

-

0.13

23.2

0.14

1.2

15

325

0.0

52

0.21

1.7

5

300- 340

0.13

1.0

29

300

so.1

A

0.Y

15.7

0.05 0.132
Q

cn

0.23 0.12

0.19 0.14

0.084 N 4

4.0 3.4 8.2 12.4 21.9

1.1 1.2 1.0 I .3 0.9

0.0

0.13

0.14

15

293

3

276

-

-

0.0

5.8

0.10

0.2

-

-

-0.0

29.6

0.17

1.3

11

300

-0.0

28.4

0.15

1.3

14

300

-

Alkenes 0.0

4.3

0.10

0.5

24. I

0.16 0.12

0.9 1.2

11

38

0.0

12.5

0.08

0.5

-

0.3

33

0.13

1.6

20

291

0.50

15

0.16

1.7

14

297

0.12 -

-1.3 1.15 1.07

10 16 14

291 291 291

0.366 0.34

-15 19

227 297 297

Other CHjOCH, CH,OH HOH

1.30 1.69 1.84

220 340 590

-

C . m. esu * cm = 3.34 X ) , 300 K. Energy at which Ramsauer-Townsend minimum occurs. Assuming urncz E-Pbetween 0.03 and 0.15 eV; when R-T min is
1D = 1X

* uav= ( u ) / ( u/a,

278

Gordon R. Freeman and David A . Armstrong v(1o5 m/sec) 05'2

188

103

325

( u -

E

10 :

3 " 001

I

1

1

'

1

1

1

1

'

01

'

"

E (eW

FIG.6 . Electron momentum transfer cross sections of three isomers of pentane, C,H,*. (1, neopentane; (---), isopentane; (---), n-pentane; Freeman et al. (1979); (A), McCorkle el al. (1978). The relative degrees of sphericity of the molecules are indicated by photos of models.

relieve the anxiety of those who prefer to treat simple molecules and feel that even a diatomic molecule contains one atom too many. The R-T minimum occurs at a higher energy, and the cross sections at low energies are larger, for the more spherelike molecule. A similar relationship exists between the cross section curves for the butane isomers (see Fig. 9 in Gee and Freeman, 1983a). The sphericity effect is also apparent in electron scattering by the slightly polar isomeric butenes, which have dipole moments up to 0.5 D (Wada and Freeman, 1979). Molecular structure effects are a rich area for further investigation. The low-energy portions of curves such as those in Fig. 6 may be compared E - P over the energy by way of the value of p in the approximation range 0.03 < E < 0.15 eV. The values of p are near unity for most simple hydrocarbons (Table 11). Exceptions are molecules that have special properties. Molecules that have p << 1.0 tend to be reactive and possibly form transient anion states with the electrons, whereas those that have p >> 1.0 tend to be globular and rigid (Table 11; Gee and Freeman, 1983a). The ratio of the threshold drift velocity for electron heating to the speed of sound in low-density methane vapor is vF/c,, = 1.O k 0.1, at temperatures up to - 160 K and densities up to n/n, = 0.15 (Gee and Freeman, 1979). Thus elastic scattering is a major contributor to energy exchange between thermal electrons and methane molecules. The relatively large cross sections

ELECTRON AND ION MOBILITIES

279

at E < 0.1 eV, associated with the spherelike shape of the molecule, are due to an elastic process. The molecule is nonpolar and isotropically polarizable, so electron-collision-inducedrotational transitions have small cross sections at these energies in spite of the close spacing of the levels (Table I; Herzberg, 1967). The polarizability of ethane is anisotropic and rotational levels are closely spaced (Table I). Thus the inelastic mode is operative and the value of v,"/co is significantlygreater than unity, being 2.5 at 197 K and 7.9 at 326 K (Gee and Freeman, 1980a). 4. Polar Molecules

Electron scattering by point dipoles would have cross sections a a)2/ where a is the dipole moment (Altshuler, 1957). For real molecules with dipole moments in the range 0.3- 1.8 D (propene to water) and electrons with energies 0.02-0.2 eV, the scattering cross sections do vary approximately as However, the actual cross sections are about twice those estimated from the point dipole model (Christophorou and Christodoulides, 1969; Wada and Freeman, 1979; Gee and Freeman, 1982, 1983b). The momentum transfer cross sections in Fig. 6 and those from this laboratory in Fig. 2 were obtained by numerically fitting Eq. (1) to the measured mobility as a function of temperature. E,

w e=

-4ne

3m

[ v2 dfodv dv -0m.e

where e and in are the electron charge and mass, v is the electron velocity is the velocity-dependent electron relative to that of the molecules, ompe momentum transfer cross section of the molecule, and f-,is the isotropic term in the velocity distribution (Pack and Phelps, 1961; Huxley and Crompton, 1974). For a Maxwellian distribution fo = ( r n / 2 ~ k T ) ~ / ~ exp( - mv2/2kT) and we obtain

- 8n2e

( m [-expv3 (--rnv2)dv 2kT )5/2

npe-3m2nkT

um,e

Equation ( 1 ) involves the assumption that elastic collisions dominate the transport processes. It has been suggested that when inelastic processes make a major contribution to electron scattering,Eq. (3) is more appropriate than (2) for extracting as a function of v from p e , T data (Crawford et al., 1967; Garrett, 1972).

(

np, = 3& e -2kT )512/[ 8 m in

(- g) dv

v5am9,exp

(3)

280

Gordon R . Freeman and David A . Armstrong

Inelastic rotational processes are important in thermal electron collisions with polar molecules and anisotropically polarizable hydrocarbons at temperatures above about 100 K. This is verified by values of v ~ / c that o are much greater than unity (Table 11). The ratio equals - 13 for strongly polar gases at 297 K. Cross sections obtained from Eq. (3) are 10- 20% smaller than those from Eq. (2). The equations involve different averaging procedures and there is a 13% difference in the numerical factors. The averaging procedures give slightly different apparent energy dependences of om,e. For example, mobilities in methanol lead to the following cross sections from Eqs. (2) and (3), respectively (Gee and Freeman, 1983b): om,,

m2) = l . 2 O ~ - ~ . ~ ~0.02 ,

c 0.3 eV

(4)

Cm,e (

m2)= 1.12~-~.lO, 0.008 G E c 0.2 eV

(5)

GE

For water the corresponding cross sections are (Gee and Freeman, 1983b) am,,(lo-'' m2) = 2.6e-l.O7,

om,c(

m2) = 2.5e-l,O2,

0.3 eV

(6)

0.009 c E c 0.2 eV

(7)

0.02

GE G

The mobilities of the cations, most of which are (ROH),H+ with R = H or CH,, are two orders of magnitude lower than those of the electrons in the coexistence vapors. The factor p,/p+ is 130 in water and 400 in methanol. Cross sections om,+for ion momentum transfer were obtained by fitting the ion equivalent of Eq. (3) to sets ofp+,Tdata (Gee and Freeman, 1983b).For ions the m of Eq. (3) is replaced by the reduced mass M,, which for clustered ions colliding with gas molecules is approximately equal to the molecular mass M. The cross sections so obtained were m2) = 0 . 7 8 ~ - ' . ~ ~ ,0.008 G E G 0.2 eV

(8)

0.2 eV

(9)

methanol,

om,+(

water,

om,+(10-19 m2) = 1.46~-~.O',

0.008 c E

d

These values are fi smaller than those reported by Gee and Freeman (1983b), because they used M , = 0.5 M . The true values of M , probably lie in the range 0.9 -0.7 M . They would decrease with increasing temperature, which would decrease the energy dependence of om,+. Curiously, for water the apparent momentum transfer cross section of electrons (Eq. 7) is - 50% larger than that for cations at the same energy. For nonpolar molecules, the electron cross sections are usually 1-2 orders of magnitude smaller than those of ions at the same energy (Gee and Freeman,

28 1

ELECTRON AND ION MOBILITIES

198la, 1983a).' By contrast, polar gases usually have electron cross sections similar to but smaller than those of ions at the same energy (Pack et al., 1962; Cottrell et ul., 1968;Gee and Freeman, 1982;Giraud and Krebs, 1982).The large apparent cross section of water for electrons is mainly attributable to the large dipole moment (Table 11), but a factor of - 1.4 might be due to transient capture of electrons by van der Waals clusters of molecules.

+ (H,O), % (H20);

(10) where n = 1,2, . . . . The electrons spend less than half their time attached molecule/m3). The lifetime of to molecules in the low-density vapor ( sec/d, where 6 is the mean probability of the transient anion is attachment per collision of the electron, averaged over all species. r-

B. DENSEGASESAND LOW-DENSITY LIQUIDS (0.1 < n/n, < 2.0) In a low-density gas an electron interacts with only one molecule at a time. The dense gas regime begins when simultaneous interactions with two or more molecules become significant. This occurs at n/n, 2 0.1. In a liquid at or below its normal boiling point, the average distance between molecules (excluding the space occupied by the molecules themselves) is much less than a molecular diameter. A thermal electron inserted into a liquid interacts with many molecules, say 10, simultaneously. The densities of most liquids near their normal boiling points are 2.7 n,. As a liquid is heated under its vapor pressure its density decreases and the compressibility increases; the relative magnitudes of the density fluctuations increase. Transport properties indicate that deviation from normal liquid behavior occurs at n/n, 5 2.0, so it is convenient to classify the region 1.O < n/n, < 2.0 as a low-density liquid.

-

1. Monatomic Molecules

When the gas density is increased into the dense gas regime the value ofnp, may either decrease or increase, depending on the nature of the molecules. Both types of behavior are observed with monatomic molecules, the former I The electron cross sections for nonpolar molecules are usually an order of magnitude smaller than those predicted by the charge-induced dipole model (McDaniel, 1964; Wada and Freeman, 1979). The repulsive interaction between the low-energy projectile electron and the imperfectly polarizable molecular electrons cancels most of the attractive interaction between the charge and the induced dipole.

282

Gordon R. Freeman and David A. Armstrong

with helium and the latter with argon and heavier noble gases. Neon is borderline. In helium at 4.2 K the electron mobility plunges when the density exceeds about 0.1 n, (Fig. 7; Levine and Sanders, 1967). The interaction between electrons and helium atoms is repulsive and relatively strong (O’Malley, 1963).At 4.2 K the kinetic energy of the molecules is very low, - 0.36 meV, so they cannot push close to the electron. Balancing the strong repulsion, the low kinetic energy of the molecules and the zero point energy requirement of the electron results in a quasi-bubble around the electron in the dense gas. The quasi-bubble has to migrate with the electron, so the mobility is extremely sensitive to the fluid density. Increasing the gas temperature increases the kinetic energy of the helium molecules, thereby decreasing their distance of closest approach to the electron. The gas density required to localize the electron in a quasi-bubble therefore increases with temperature (Fig. 8; Hamson and Springett, 197la; Jahnke et al., 1975).This is a grossly oversimplified interpretation, but it outlines the observed behavior. molecule/m3, at 4.2 K, The beginning of the decrease in ye at - 5 X can be attributed to multibody scattering rather than localization. A multiple-scattering model of electrons in semiconductors (Foldy, 1945; Lax, 1951; Ioffe and Regel, 1960) has been adapted to electrons in dense gases. When the mean free path L of the electron is smaller than its de Broglie wavelength A, multiple scattering occurs. When the scattering length of the molecules is positive, multiple scattering decreases the value of npe (Legler, 1970; Atrazhev and Iakubov, 1977; Schwarz, 1980; O’Malley, 1980). The latter authors and others (Mott and Davis, 1979) use L / i = 27rL/A as the parameter. However, the threshold density for multiple scattering corresponds to L = I in dense polar gases (Giraud and Krebs, 1982; Gee and Freeman, 1983b). The multiple-scattering model of OMalley ( 1980) applies only to “hard gases,” in the temperature - pressure region where the compressibility factor 2 is 3 1.O. The model does not reproduce the steep plunge ofpe in Fig. 7 at n > 1OX molecule/m3, where 2 < 0.90 (McCarty, 1972). At T 3 20 K helium has 2 3 1.O at all densities, and the O’Malley (1980) model fits relatively well. The greater decrease of mobility in the “soft gas” with 2 < 1 .O, at T < 20 K, has been interpreted semiquantitatively by a two-state model (Eggarter and Cohen, 1971; Hernandez, 1972). As the density is increased the electrons spend an increasing fraction of their time in the localized (quasibubble) state. At 4.2 K and n/n, 3 0.2 the electrons spend 100%oftheir time in the localized state; in fact, the density-normalied mobility ripe = 2.6 X 1022molecule/mV/sec in the gas at n/n, = 0.20 (Levine and Sanders, 1967) is smaller than that, n,ue = 3.6 X loz2molecule/m V sec in the liquid at

- -

ELECTRON AND ION MOBILITIES

1

10

102

103

283

104

n (1025atom/m3) FIG.7. Effect of density on electron mobility in gaseous and liquid helium. (0),Levine and Sanders (1967); (+), Pack and Phelps (1961); (*), Griinberg (1968); (A), Ostermeier and Schwarz (1972).

n (1 oz7/m3)

FIG.8. Effect oftemperature on electron localization in dense helium gas. (---), Hamson and Springett (1971a); (O),Jahnke ef al. (1975).

284

Gordon R. Freemun und David A . Armslrong

n/n, = 1.8 at 4.2 K (Ostermeier and Schwarz, 1972). The minimum value of np, = 1.6 X loz2molecule/m * V sec occurs in the critical fluid. The increase of ripe with increasing density in the liquid phase is attributed to a decrease of the bubble radius that is relatively more rapid than the increase of

-

the liquid viscosity (Ostermeier and Schwarz, 1972). The difference between a soft and a hard gas is that density fluctuations are larger in the former. The larger fluctuations enhance the probability that an electron will encounter a multimolecular configuration that could localize the electron. The electron must drop into such a configuration through an inelastic interaction and can be ejected from the localized state by a superelastic interaction. In the gas at 4.2 K and - 13 X loz6molecule/m3 the mean residence time of an electron in the delocalized (quasi-free) state sec appears to be sec, compared to that in the localized state (Young, 1970;Schwarz and Prasad, 1975).These are surprisingly long times for configurations in a gas at n/n, = 0.13, but they might not be unreasonable at the very low temperature of 4.2 K. In hard dense argon or xenon gas electrons experience an increase of np,, rather than a decrease as in helium (Bartels, 1973; Huang and Freeman, 1978a, 1981). In the soft dense gas, near the vapor-liquid coexistence curve, thermal electrons form quasi-localized states that tend to decrease np, by a small factor (Fig. 9; Huang and Freeman, 1978a). At n/n, 3 0.1 the value of

1

1

,

,,,,,,,I

10

,

/

,

,

,

10‘

,

,

,

1-j

I , ,~, , ,

t

103

n (1025 mo1ecu1e/m3) “ C

FIG.9. Density dependence of the density-normalized mobility ripe in coexistence gas and liquid xenon (open points). (A), thermal electrons; (0),np,,,,, obtained from the peaks of curves analogous to that for argon in Fig. I . (O),T = 296 K = T, 6 K. Data from Huang and Freeman (1978a).

+

285

ELECTRON AND ION MOBILITIES

ripe for thermal electrons in xenon decreaseswith increasing n. Concurrently the temperature coefficient ofpe at constant n increases (Fig. lo), reaching the enormous value E,,e = 1 eV for a small temperature and density zone near the critical region. Heating the vapor at constant density reduces the depth of the minimum in the thermal np, curve in Fig. 9, and ultimately eliminates it (Freeman, I98 I). The dense vapor near the coexistence curve has a low compressibility factor, 2 = 0.6 at n/n, = 0.4 and 0.3 at n/n, = 0.9 (from data in Cook, 1961). Heating the vapor at constant density increases the value of 2, which equals 0.8 and 0.6 at the above two densities at TIT, = 1.3(from the law of corresponding states and argon data; Gosman ef af.,1969). The increase of 2 at a given density implies a decrease in the density fluctuations, and therefore fewer quasi-localization sites for the electrons. Heating the electrons with an elevated field also prevents quasi-localization. For example, the values of np,,,, obtained from the maxima in curves analogous to that for argon in Fig. 1, do not display a minimum. The value of increases at n/n, > 0.3 (Fig. 9), due to an incipient conduction band. The quasi-localizationprocess is represented by the following mechanism:

medium

+

site

(1 1)

eif site i=e , lo3

I

(12) I

I

L

13

4

5

6

1 OOO/T (K)

FIG.10. Arrhenius plots of thermal electron mobilities in xenon gas (Huang and Freeman, 1978a). (A), coexistence vapor, density increases with (0),constant density, numbers indiwere previously incorrectly cate n ( loz6rnolecule/m3). The temperatures for n = 25 X plotted.

Gordon R. Freeman and David A . Armstrong

286

where "site" represents a density fluctuation of sufficient amplitude and appropriate breadth, while eif and e; represent the quasi-free and quasilocalized electron, respectively. The number of molecules involved in a site appears to be lo2. The value of np, at any n and T is

-

(13)

= np8,nf

where p:,n is the quasi-free electron mobility at density n and

f= [eifl/([e;f1+ [ e ~ l ) = (1

+ [site]K,,)-'

(14)

where [site] is the concentration of sites and K I 2is the equilibrium constant of reaction 12. It follows that

(np:,Jnpe) - I

-

= exp(AS'/R)

exp(-AH'IRT)

(15)

where (AH' - TAS') = AG' is the standard Gibbs free energy change ofthe overall eif medium e; quasi-localization process, AS' = AS,", AS;2 = AS;, and AH' =AH,", AH,"2= AH,",(Gee and Freeman, 1979, 1980a). The threefold difference between np,,,, and np, at low densities(Fig. 9) is due to the field heating ofthe electronsinto the R-T scattering minimum. In the dense gas, quasi-localization of the thermal electrons increases the difference between np,,,, and np,. Both the R-T minimum and quasi-localization become negligible at n/n, = 2.0 in the liquid phase, so np, and np,,,, become the same (Fig. 9). The 15-folddifference between np,,,, and np, in the near-critical fluid at n, contains contributions of sevenfold due to quasilocalization and twofold due to the R-T effect. This corresponds to eV/K. A more AG' = -0.05 eV, so AH' - 1 eV and A S ' = - 3 X detailed treatment of the quasi-localization process is given in the section on polyatomic molecules. The main symptom of quasi-localization is the rapidly increasing value of E,,, at constant density with increasing density (Fig. 10). Nothing peculiar happens to np, in the critical fluid (Kimura and Freeman, 1974).The correlation lengths of the density fluctuations characteristic of the critical fluid, - lo-' m, are much greater than the electron - molecule m. The long-wavelength fluctuations are not interaction distances, effective in electron scattering (Freeman, 1980). In the dense liquid the low field value of np, increases so much that the field-induced hump in the np, curve is covered up (Fig. 11). The R-T minimum in scattering by a single molecule is related to the conduction band in the dense fluid. At sufficiently high densities the low energy wing of

+

+

i=

+

287

ELECTRON AND ION MOBILITIES 1 04

10

10-4

10-3

10-2

E/n (1 0-21 V . rn2/rnolecule)

lo-’

FIG. 1 I . Effect of electric field strength on electron mobility in liquid xenon at different densities (Huangand Freeman, 1978a). The numbers labelingthe curves are n ( loz6molecules/ m’), T W .

the om curve is completely obliterated by multibody interference. Furthermore, in the dense liquid the density fluctuations are too small for quasilocalization to occur.

2. Diatomic Molecules Very precise measurements detected a small percentage decrease of np, for thermal electrons in hydrogen and nitrogen at 77 K as the density was increased from lozs to loz6 molecule/m3 (Lowke, 1963; Crompton and Robertson, 197 1). The decrease was attributed to transient capture of the electrons by rotational resonance transitions of the molecules (Frommhold, 1968). At higher densities the decrease of np, becomes dramatic (Griinberg, 1968; Hamson and Springett, 1971b; Bartels, 1972; Wada and Freeman, 1980). The results in dense hydrogen are consistent with the quasi-bubble model that was developed for electrons in dense helium (Hamson and Springett, 197 1b). The mobilities in dense nitrogen seem to require a modification of the Frommhold model (Wada and Freeman, 1980), described below. In nitrogen at 127 K the value of np, for thermal electrons decreases rapidly as n is increased above 2 X lo2’ molecule/m3 (Fig. 12; note that T, = 126.2 K and n, = 6.7 X loz7molecule/m3). The value of np, for hot

288

Gordon R. Freemun and David A . Armstrong

n(l0*7 rnotecute/rn3)

FIG.12. Density dependence of r ~ p ~ / ( nforp thermal ~ ) ~ electrons in nitrogen gas. (O), I27 K; coexistence curve with temperatures 125- 129 K.( r ~ p , = ) ~7.8 X molecule/m * V * sec. Curves of Eqs. ( I 7), ( 1 8), and (19) represent theoretical results with m2, respectively. V ~= T 6.7 X m3/molecule, a = 4.7 X lo-" m and ( T . ~= 2.7 X DataofWadaandFreernan(1980).RecallthatT, = 126.2KandnC= 6.7 X 1027molecule/m3.

(a),near vapor-liquid

electrons, say at 3 Td, is independent of gas temperature and density up to the critical (cf. Figs. 4 and 13). The field dependence decreases and the threshold field for electron heating increases with increasing density, until at molecule/m3 the sign of the field dependence [d(np,)/d(E/n)] at 51 X E/n = 0.2 Td has changed from negative to positive (Fig. 13). Qualitatively similar behavior was observed in dense helium gas (Schwarz, 1980). At n G 48 X molecule/m3 the threshold drift velocity in the saturated vapor remains nearly constant at v h r = 1.3 km/sec, and the ratio vy/ c, = 7.3. Thus the electron energy is moderated mainly by inelastic collisions. At these energies only molecular rotations can be excited. At n > 48 X molecule/m3 the threshold drift velocity plunges, reaching 0.12 km/sec at 67 X 1020molecule/m3. The change in field effect is attributed to electron localization (Wada and Freeman, 1980). The localization process in dense nitrogen vapor is not of the same type as the quasi-localization that occurs in xenon. The temperature coefficient Ep,e= 0.0 eV at n/n, = 0.52, 0.75, and 1.00 in nitrogen (Fig. 14), whereas E,,, = I eV at similar densities in xenon (Fig. 10). The Frommhold mechanism of transient capture by rotational resonance states of the molecule is e-

+ N, 9 [N;] V'

(16)

289

ELECTRON AND ION MOBILITIES

FIG. 13. Field dependence of the density-normalized mobility of electrons in nitrogen vapor. along the coexistence curve or near the critical region. Densities and temperatures ( molecule/m3, K): (O), 0.96, 77; (A), 10, 106; (V), 30, 122; (0),42, 125; (O), 51, 126.3; (A), 59, 126.6; (0),61, 127.2; (T), 67, 127.0. Arrows indicate (Eln),. Data of Wada and Freeman ( I98 1).

I

'

'

"

'1""""

'

127K

it

-

"

I

'

I

r

I I

80

100 20

120

140

170

T (K) 22

200

240 24

3C J

log T

FIG. 14. Variation of the temperature dependence of the low field ripe with density in nitrogen vapor. n ( molecule/m3): (0),0.57; (V), 2.0;(O), 8.0; (A), 18; (+), 35; (X), 50; (O), 67 = n,: (0). coexistence vapor. Data of Wada and Freeman (1981).

290

Gordon R. Freeman and David A. Armstrong

The electron attachment rate is va = v:n and the autoionization mean lifetime ofthe anions is 7 .The mobility ofthe electrons is much greater than that of the anions, so reaction (16) leads to Eq. (17) (17) ( n p , ) o / ~ &= 1 + v:7n where (np,)o is the low-density limit of np,. molecule/m3, Equation (17) fits the results at 127 K up to n = 2 X m3/ with ( n , ~ ,=) ~7.8 X loz5molecule/m - V * sec and v:7 = 6.7 X molecule (Fig. 12).The value of v:z decreaseswith increasing thermal energy kT, being 7 X m3/molecule at 9 X eV and 1.7 X at 60 X eV ( Wada and Freeman, 1980). At n > 2 X molecule/m3 and 127 K the value of np,/(np,)o decreases much more rapidly than described by Eq. ( 17), so mechanism ( 16) is inadequate. Other possible explanations offered by Legler (1970) and Atrazhev and Iakubov (1977) are based on quantum mechanical corrections for multiple scattering when the mean free path of the electrons becomes similar to their thermal wavelengths. Legler's theory gives

np,/(np,)o =

1+ (x

b)-1/2x3/2e-xdx

(18)

where b = V(n)/kT,V(n)= h2an/2nm is the zero point energy of the electron in the gas, a is the positive scattering length of the molecules, and x is a variable of integration. In Iakubov's theory, np,/(np,)o is a linear function of n

np,/(np,)o = 1 - ha,n/(8mnkT)1/2

(19)

where a, = 4na2 is the average scattering cross section. The value of a, was taken as 0,

aau,e

= 2.40

= ( V ) / ( v/am.e)

x

107/~1/2( n ~ , ) ~

(20)

At 127 K we obtain a, = 2.7 X m2and a = 4.7 X lo-" m. With these parameter values Eqs. ( 18) and ( 19) are even less adequate than Eq. ( 17)

(Fig. 12). The steep decrease of np, at high densities is attributed to stabilization of the anion by collision or clustering (Wada and Freeman, 1980, 1981). One possibility is e-

+ N,

[NJ

& ( N*A-

...

(21)

with the mean lifetime T of the electron in the attached state increasing with

ELECTRON AND ION MOBILITIES 10

t

,

,,,

, , ,

,(,

29 1

I'"9

n (loz7rnolecule/m3) FIG.15. Apparent mean lifetime ST of temporary negative ions in nitrogen as a function of vapor density; S is the probability ofelectron attachment per collision; T = 125 - 129 K. Data of Wada and Freeman ( 1 980).

cluster size. The attachment rate is approximated by vLn = 6 ( v ) Dan. The value o f t from Eq. (17) is

where S is the probability of attachment per collision. The product ST is an apparent mean lifetime. The value of ST at 127 K is constant at 4 X 1O-I3 sec sec at molecule/m3, then increases rapidly to 6 X for n S 2 X 6.7 X 1OZ7 molecule/m3 (Fig. 15). sec at 127 K and low densities is two The apparent lifetime 4 X orders of magnitude greater than the duration of an ordinary elastic collision. The value of St decreases rapidly with increasing energy (Fig. 16). The anions are therefore least unstable at very low energies and may involve rotational resonance states as proposed by Frommhold ( 1968).They are not c

7

10 20 3( kT (1 O-3ev)

FIG. 16. Apparent mean lifetime ST of temporary negative ions in nitrogen gas at low densities, as a function of thermal energy kT. Data of Wada and Freeman (1980).

292

Gordon R. Freernan and David A . Armstrong

attributable to vibrational states, for which there is a resonance at 2.3 eV in low-density nitrogen (Schulz, 1964).

3. Polyatornic Nonpolar Molecules (Sphericity Efect) Quasi-localization of electrons in dense vapors is characterized by a large temperature coefficient ofp, at constant n (Fig. 10).There is sometimes, but not always (Huang and Freeman, 1978b, 198I), a small decrease of np, with increasing n in the coexistence vapor at n/n, = 0.1-0.5 (Fig. 9). When incipient conduction band formation causes (np,)o to increase faster than quasi-localization causes n , ~ ~ / ( ntop decrease, ~)~ no net decrease of np, is observed. Such is the case with argon (Huang and Freeman, I98 1). Quasi-localization also occurs in many hydrocarbon vapors. The density dependencies of np, in methane and ethane are compared to that in xenon in Fig. 17. The most spherelike of the three molecules, xenon, provides the

1000

-

100

0

al

. ? E

10

3

0

-

E

N v)

-

2

L-----l

0.1

0.01 0.1

10

100

n (lo2’ m o ~ e c u ~ e s / m ~ )

FIG.17. Variation of rip, with density for thermal electrons in the coexistence vapor and liquid of methane (0),ethane (B), and xenon (A). The arrows indicate the critical densities. Data of Gee and Freeman (1 979, I980a) and Huang and Freeman ( 1978a).

ELECTRON AND ION MOBILITIES

293

lowest value of np, in the low-density gas and the highest value in the dense liquid. Methane has tetrahedral symmetry and is the next most spherelike molecule; it provides the next lowest value of ripe in the normal gas and next highest value in the normal liquid. Ethane, H,CCH,, has a cylindrical shape and provides the highest gas phase and lowest liquid phase values of ripe. The large temperature coefficients E,,e at n/n, 3 0.5 near the vapor-liquid coexistence curve indicate that the degree of electron quasi-localization is similar in the three substances (Gee and Freeman, 1979, 1980a). The temperature coefficientsand the parameters from Eq. ( 15)for quasi-localization in methane and ethane are typical of those in many hydrocarbon vapors. The value of AH’ is approximately double that of the Arrhenius coefficient Ep,e (Table 111). The values of AH’ and AS‘ are large and negative, in agreement with the proposed quasi-condensation that is involved in the formation of the site, reaction ( 1 1). The values of AS’ correlate roughly with the structure factor S(0) = nkTXTin the dense gases away from the critical region;X, is the isothermal compressibility. The ratio AS’/S(O)in methane and ethane has eV/K (Table 111). values in the vicinity of - 20 X The values of AG’ are near zero (Table 111) in agreement with the relatively small extent of the quasi-localization process. The density dependencies of electron behavior in propane (C3H8;Gee and Freeman, 19800,1983a; Nishikawa and Holroyd, 1982)and C, to c6 alkanes (Gee and Freeman, 1983a; Gyorgy and Freeman, 1979; Huang and Freeman, 1978c) are largely governed by the degree of sphericity of the molecules. The more spherelike molecules generate a larger increase of n,ue on going from n/n, = 0.8 to 1.5, due to the formation of a less bumpy potential surface on the bottom of the conduction band. The behavior in benzene, cyclo-(CH),, is surprisingly similar to that in cyclohexane, cyclo-(CH,), (Huang and Freeman, 1978c, 1980b).The globular shapes dominate the difference in bonding. The density dependencies of Ep,eare remarkably similar in 18 substances ranging from xenon to alkanes, alkenes, and an ether that has a dipole moment of 1.3 D (Fig. 18). The quasi-localization process is only slightly sensitive to molecular structure. Molecular orientational disorder makes a much smaller contribution to the (shallow)traps in the dense vapor than to the (deeper) traps in the normal liquids of nonspherical molecules. The relative insensitivity to molecular shape in the dense vapor is due to the greater mean free volume and the greater rotational freedom than in the normal liquid. The duration of a quasi-localization event is estimated to be - lo-’* sec. Electrons in ethene (H,C=CH,; Gee and Freeman, 1981b) and cyclopropane [cyclo-(CH,),; Gee and Freeman, 1980b, 1983al display behavior intermediate between that in nitrogen (Fig. 12) and that in ethane (Fig. 17).

-

TABLE 111

PARAMETERS OF THERMAL ELECTRON TRANSPORT AND QUASI-LOCALIZATION IN VAPORS

Methaned 0.012 0.049 0.108 0.24 0.5 1 1.oo

117-297 180-295 153-208 170-187 185-196 192-196

0.006 0.012 0.0 16 0.039 0.094 0.42

22-34 27-34 29-33 37-39 50 84

0.03 0.07 0.23 1.o

0.3 0.4 1.2 5.1

+

0.02

0.0 1 0.00 -0.05

1.4 2.3 6 29

+0.03 0.00 -0.02

1.o 1.4 2.5 5.7 33

Ethane'

0.0 17 0.123 0.25 0.50 1.oo

197-326 254-325 276-309 298-308 307-310

0.013 0.025 0.040 0.17 0.63

31-43 37-43 45f 51 65

0.12 0.34 1.10

Equation ( 1 5). AG' = AH' - TAS'. S(0)= nkT.yT Gee and Freeman ( 1 979); n, = 6.08 X loz7molecule/m3. Gee and Freeman (1980a); n, = 4.06 X molecule/m3. f Value revised from 50, with consequent changes in AH', AS', and AG'

-

0.5 1.1 3.6

.o

1 1.1

20 20 20 18

20 20 I1

ELECTRON AND ION MOBILITIES

0

295

n I n,

FIG. 18. Arrhenius temperature coefficients of thermal electron mobilities in vapors at constant density, a few degrees from the coexistencecurve.(A): (0),CH,; (O),C,H,; (A), C3H,; (O), n-C,H,,; (0),trans-butene-2. B: (0),i-C,H,,; (O),n-C,H,,; (A),n-C6H,,; (A), propene; (O), cis-butene-2; (O), butene-I. C (0),neo-C,H,,; (O),cy~lo-C,H,~; (A),cyclo-C,HI,; (A), CH,OCH,; (0),i-butene; (0),i-C,HIo; (X), Xe. The (0)and (A)at n/n, = 2.0 and 2.3 correspond to behavior in the liquid under its vapor pressure, son varied with T. The constant density points would fall below these. The dashed lines are only guides (Geeand Freeman, I983a). Reproduced with permission of the Journal of Chemical Physics.

-

The two compounds evidently tend to form transient anions with electrons in the dense fluids, but less so than does nitrogen. 4. Polar Molecules

The values of np, decrease with increasing n in dense polar gases such as ammonia (Krebs et al., 1982;Krebs and Heintze, 1982;Christophorou et al., 1982),water (Giraud and Krebs, 1982;Gee and Freeman, 1983b), methanol (Gee and Freeman, 1983b), and dimethyl ether (Gee and Freeman, 1982). The onset of the decrease in each compound fits moderately well the multiple scattering criterion of Eq. (19). The strange thing is that the equation was derived for molecules that have a positive scattering length, which represents a repulsive interaction between the electron and molecule. An example is helium. Molecules that have a negative scattering length, representing an attractive interaction, display an increase of ripe with n. Examples are methane and xenon, and the appropriate equation is quite different from Eq. ( 19) (Atrazhevand Iakubov, 1977).Multiple scattering becomes important when the mean free path is reduced to the vicinity of the de Broglie wavelength of

296

Gordon R. Freeman und David A. Armstrong

the electron, and whether ripe increases or decreases depends on whether the interaction is attractive or repulsive. The interaction between an electron and a polar molecule is attractive. We might therefore have expected np, to increase with n in the dense gas. The observed decrease is attributed to quasi-localization by density fluctuations or van der Waals clusters. The latter are probably important at the low threshold densities observed for highly polar molecules. Equation ( 19) is based on a random distribution of molecules and does not consider density fluctuations. The correlation of Eq. ( 19) with threshold densities of quasi-localization must be based on a common underlying factor, not yet known (Gee and Freeman, 1983b). 5. Quasi-Localization and Multiple Scattering Threshold Densities

The section on electrons is ended with mention of a puzzle that is intriguing at the moment. The second term on the right side of Eq. ( 19)corresponds R the de Broglie “radius” of the electron averaged to 2@L, where Iz = A / ~ is over a Maxwellian velocity distribution, and L is the mean free path. According to this model, when 2A/L = 1 the electrons are completely localized. Threshold densities for the beginning of multiple scattering can be estimated from the ratio A/L = 1 (Ioffe and Regel, 1960). Equation ( 19)was intended only for “hard” gases of molecules that have a net repulsive interaction with thermal electrons. Its application, even in an approximate manner, is restricted to helium, hydrogen, and perhaps a few other molecules. However, threshold densities estimated from A/L = 1 are accurate within a factor of 1.5 for the onset of quasi-localization in a wide variety of “soft” gases, near the vapor-liquid coexistence region. The gases include highly polar and nonpolar molecules such as water, ammonia, methanol, and dimethyl ether, simple hydrocarbons such as methane, propene, and isobutene, and even xenon (Gee and Freeman, 1983b; Wada and Freeman, 1979). All of the polyatomic molecules are relatively rigid. Floppy chain hydrocarbons such as n-hexane have threshold densities an order of magnitude lower than those estimated from A/L = 1. The message has not been deciphered yet. Multibody effects are currently of great interest in many facets of physics and chemistry.

111. Ions An understanding ofthe motions of ionsdrifting in gases in the presence of electrical fields is of importance in aeronomy, astronomy, plasma science, and radiation physics and chemistry. When stationary conditions exist, the

ELECTRON AND ION MOBILITIES

297

drift velocity v d of a given ion is proportional to the applied field strength E: vd = p E . Even in the pioneering work of Langevin (1905) it was realised that the mobility p was determined by the cross section for momentum transfer, and contained information about the ion - neutral interaction potentials. Mobility studies have therefore a fundamental as well as a practical value, but it was not until the last decade that the determination of potential functions became truly feasible. We begin by emphasizing important developments in theory, which made that possible for atomic systems, and then proceed to treat successively more complex systems in an order chosen to provide insight into the various complicating factors introduced with polyatomic molecules. Since ion - molecule reactions do not specifically fall under the scope of this article, only a few examples are to be found. Cases where molecular or clustered ions are broken up or where secondary excitation or ionization occurred are omitted. Finally, we shall frequently refer to p,,, the mobility at the standard number density of n = 2.687 X 1019molecule/cm. A. LOW-DENSITY GASES(n/n,< 0.01) 1. Monatomic Ions and Gases

a. Theory. Accurate treatment of ion mobilities and diffusion coefficients requires solution of the Boltzman equation. When the comprehensive treatise of McDaniel and Mason (1973) was written the only satisfactory general solutions were those of Chapman and Enskog, and Kihara. These applied respectively to the regimes of very weak and low to intermediate electrical fields. The Chapman - Enskog theory is of such importance that we present its analytical form

The quantity q is the ion charge, whileM,and Tare, respectively,the neutral molecule - ion reduced mass and temperature. The momentum transfer collision integral R(lsl)(T)is the first of a family of integrals, which are familiar in the transport theory ofgases (McDaniel and Mason, 1973;Mason and McDaniel, 1985). It is evaluated from the energy-dependent momentum transfer (diffusion) cross section, which in turn is obtained from the ion-neutral interaction potential. It may be made to take into account quantum mechanical effects or charge transfer interactions (McDaniel and l)(T ) and higher Mason, 1973; Mason and McDaniel, 1985). Values of R(** collision integrals have been tabulated for several types of ion- neutral potential (Mason and Schamp, 1958; Viehland et al., 1975).

298

Gordon R. Freeman and David A . Armstrong

In 1973 there was no theory capable of covering the entire low field- high field range for all m/Mratios and types of interaction. However, a number of specific model cases had been treated successfully, for example, the cold gas model [ v d > ( 8 k T / 1 r M , ) ~ /the ~ ] , Maxwell model (constant collision frequency), the Lorentz model (m << M ) and the Rayleigh model (rn >> M). In addition, important studies by Wannier (195 1, 1952) had provided insight into higher field behavior. These were exploited later in the 1970s, when further attacks on the solution of the Boltzmann equation were made. In a valuable review, which also discusses the space-time development of swarms, representations of the collision integral and other important aspects of theory, Kumar et al. (1980) have classified and examined in detail approaches to the solution of the Boltzmann equation. “The method of most general applicability is that of moment equations,” and papers dealing with the required basis functions and other mathematical procedures have appeared (Kumar, 1980a,b). A lucid account of that method has also been given by Mason and McDaniel(l985). Viehland and Mason (1975a, 1978) devised a procedure for using basis functions, in which the ions of mass rn had a temperature Ti different from that of the neutral gas and defined as Ti = rn ( of ) /3k. According to this “two-temperature theory” the mobility and drift velocity is given at all E/N by

The term a, which is normally less than 0.1 and zero in first approximation, includes all of the higher order kinetic theory approximations. It depends in a complicated way on E/n, T, the ion and molecule masses m and M , and the is defined by ion -neutral molecule potential. The effective temperature Teff the relations

jkT,,

+M )

= +k(mT+M T i ) / ( m

(254

=+kT++Mv$(l +p)

(25b)

The term /3 is similar to cy and is also zero in first approximation. The effective temperature characterizes the mean ion eneigy in the center ofmass frame of the ions and neutrals and appears in the W.’)collision integral. Clearly when +Mvi << +kT, and a and p are negligible, T,, approaches T and Eq. (24) becomes the same as Eq. (23). As a further point, we should note from Eq. (24) that a choice of T,, and E/n fixes v d , and therefore also T [see Eq. (25b)l. Conversely, if T and T,,were chosen, then v d and hence E/n would be fixed. In effect T and E/n are replaced in the two-temperature theory by the single independent variable T e f .

ELECTRON AND ION MOBILITIES

299

Equation (24) has been shown to be accurate in first approximation (i.e., a = 0) to within 10%.A simple rearrangement then yields

This

equation was used by Viehland and Mason (1975a) to calculate Ter)from experimental mobility data for potassium ions in three inert gases at several field strengths but constant gas temperature (open circle in Fig. 19).The filled circles are values of T,,) derived with the aid of the Chapman Enskog equation [i.e., Eq. (23)] from zero field mobilities measured at several gas temperatures. The solid lines at high T,, were obtained by straightforward numerical integration of ion - neutral molecule potential functions derived from beam-scattering experiments. The success of the two-temperature theory, and the utility of the concept of T,, as an independent variable are now understood from the fact that the open circles merge smoothly with the beam-scattering data at high T,, and with the zero field mobility data at T,, below 700 K. The effective temperature range is very wide, which means that, in conjunction with this theory, mobility data can be used to probe potentials over a wide range of ion - neutral separations. The convergence of higher approximations for the two-temperature Q(lsl)(

0’ I02

L

I

I

II

I

Io4

I0’

Teff ( K )

FIG.19. Dependence on effective temperature of diffusion (momentum-transfer) collision integral found by analysis of mobility data: (0)calculated from mobility data as a function of held ‘strength at fixed temperature, using Eq. (24) for ud and Eq. (25b)for TeE;(0)obtained at high from zero field mobility measurements as a function of temperature; (-) polarization asymp temperatures-calculated from results of ion beam scattering; (---) totes. From Viehland and Mason (1975a) by kind permission of the authors and publisher.

300

Gordon R. Freeman and David A . Armstrong

theory has been shown to be satisfactory (Viehland and Mason, 1975a, 1978), and it has been compared to accurate calculations based on exact theories for the special models (see above) or the Monte Carlo results of Skullerud (1973). For the third approximation agreement with these was within a few percent, except for the case of m/M > 1 at high field strengths. Thus the Viehland- Mason theory is the first rigorous kinetic theory, which is valid at arbitrary field strengths and has no restriction on the form of the ion -neutral potential function or on the m/Mratio, with the above exception.

b. Comparison of’ theory and experiment and the determination o j ion neutral potentiul ,fimctions ,from mobilities. During the past 7 years the two-temperature theory has been used extensively to test potential functions by comparing calculated mobilities with experimental results (see, e.g., Gatland et al., 1977a,b; Lamm et al., 1981; Gatland, 1981; Viehland et al., 1981b). Also the inverse process of finding potentials from mobilities over a range of field strengths has been undertaken (see, e.g., Gatland et al., 1977b; Gatland, 1981 ;Viehland et al., 1981b). Both procedures are nontrivial computational exercises, and careful tests have been made to establish an acceptable level of accuracy (Viehland et al., 1976; Viehland, 1983).Given that the best gas phase mobility data are now accurate to k 290,Viehland et a/. ( 1976) suggest that it should be possible to determine ion- neutral potentials to within about 5% over a wide range of separation distances. The effect of small changes in potential function on mobility is aptly illustrated in Fig. 20a,b, where some data reported by Gatland ( 1981) for Li+ ions in argon are reproduced. The circles in Fig. 20a are the experimental mobilities of Akridge et al. (1975) for standard argon density (2.687 X 1019 molecule/cm3) and temperature T = 300 K, while the curves are mobilities calculated from the five potential functions shown in Fig. 20b with the aid of the Viehland- Mason theory. The Gordon- Waldman (GW), Kim Gordon (KG),self-consistentfield (SCF), and configuration interaction (CI) potential functions are seen from Fig. 20a to give reasonable and progressively better fits to the experimental mobilities, but there is still a significant difference. The best fit is given by the solid line, which was calculated from a potential determined directly from the mobility data. The fact that it differs significantly from the CI potential only from 6 to 8 a.u. internuclear separation indicates the sensitivity of the mobility to the form of the potential and demonstrates the utility of mobility data for the refinement of such functions. As T,, and the kinetic energy of relative motion of the ion - neutral pairs change, different regions of the potential energy function are sampled. This causes the mobility to exhibit a certain form of dependence on T,, (or E/n). For example, at low T,, the long-range r4 polarization attraction potential

30 1

ELECTRON AND ION MOBILITIES

20

50

200

I00

E/n (Td)

0'01

-

2.5

k"

I

~

'

~

1

'

~

"

b Li Ar

7.5

5.0

I 0

R(a.u.)

FIG.20. (a)Reduced mobilitiesofLi+ionsinargongasat300 Kderivedfrom the potentials in (b) and experimental data. (b) The Kim-Gordon (KG), Gordon-Waldman (GW) and Olson and Lin SCF and CI theoretical potentials for Li+-Ar, together with the potential determined directly (DD) from the experimental data. Both R and V are in atomic units (0.5292 A and 27.2 1 eV, respectively). The crossing point 0 and minimum v, in the DD curve are indicated. From Gatland (1981) by kind permission of the author and publisher.

dominates and the mobility tends toward pWl,

0

PWl=

n m r

where the constant 0 is independent of T and E/n and a is the molecular polarizability (McDaniel and Mason, 1973; Mason and McDaniel, 1985).

302

Gordon R. Freeman and David A . Armstrong

At high TeB,short-range repulsion dominates. Generalized mobility curves, such as the idealized example in Fig. 21, take the form of p/ppl plotted against T,*,= kTe& where E is the depth of the potential well (Gatland et al., 1977a;Takebe, 1983).The maximum in Fig. 2 1 can be shown to occur in the region where the short-range repulsive forces most effectively cancel the longer range attractive forces. This usually corresponds to ion - neutral interaction distances in the range of 0 to rmin Fig. 20b. Its height aboveppl is a measure of the extent to which this cancellation occurs. For example a “softer repulsion” yields a higher and broader maximum, while addition of shorter range (e.g., 1O-a) attractive forces augments the polarization forces and reduces the maximum. Mobility ratiosp/pWlfor the alkali ions Li+ through Rb+in He, Ne, and Ar at 300 K are shown as a function of E/n in Fig. 22. These data from Gatland et al. ( 1977a)illustrate the types of mobility dependence on E/n (or TeB)seen in real systems. The circles are ratios ofexperimental mobilities topwl,while the curves are for calculated mobilities based on Waldman- Gordon potentials. Considering their relative simplicity, the latter give quite good overall agreement with experiment. The values of E in the required potential functions have been shown to vary over quite a wide range (Takebe, 1983).If the E / n scale in Fig. 2 1 were converted to T,*,, the curves for different ion neutral combinations would therefore correspond to different parts of the generalized curve in Fig. 21. Thus for Li+-Ar, with E = 0.55 eV (Takebe, 1983), the data cover roughly the same T:ffrange as the generalized curve, while for K+- He with E = 0.023 eV they are pushed to the right-hand part of the generalized curve. This effect of E and the factors discussed above can

I

10.‘

I

10.8

10

kT,fi/E

FIG.2 I . Generalized curve of ion mobility as a function of effective temperature. The arrows indicate regions dominated by the potential minimum rm and crossing point u. The curves in Fig. 22 correspond to fragments of the generalized curve starting from different T,, as indicated by the short vertical lines (where well depth e2 < c, ). From Gatland et al. ( 1 977a) by kind permission of the authors and publisher.

303

ELECTRON AND ION MOBILITIES '

10'

10'

10'

10'

10'

10'

10'

10'

I

I

(Td)

FIG.22. Calculated and experimental ion mobilities at 300 K as a function of the ratio of the electric field strength to the gas number density. The circles represent experimental measurements, their diameters the experimental uncertainty. Solid curves are calculated from Waldman-Gordon electron-gas model potentials. From Gatland ef al. (1977a) by kind permission of the authors and publisher.

account qualitatively for the different forms of curve seen in Fig. 22. The fall-off in p/pWlat high E/n for all systems is due to the dominance of the repulsive interaction. As a means of improving the accuracy of the theory for m/M > 1 at high fields, Lin et al. ( 1979c)built anisotropy into the ion temperature used in the basis functions for the moment equations, defining temperatures perpendicular to, T L ,and parallel to, TI,,the field direction by the equations

The theory, thus derived, is now called the "three-temperature theory." The numerical results were rapidly convergent and gave much better agreement (0.5%at third approximation) for m/Mas large as 4 in the high E/n region. At the same time, the agreement achieved in the two-temperature theory for lower m/M was unspoiled. However, a heavy price in increased computational complexity is paid in the three-temperature theory and it will probably only be used where absolutely necessary.

c. Special effects. This section is concerned with special types of interaction or potential. i. Systems with valence interaction: H+-He and others. The hydrogen ion - helium gas system exhibits a number of special effects and warrants

304

Gordon R. Freeman and David A. Armstrong

separate consideration. The solid line and points showing the temperature dependence of the low field mobility of H+ in He in Fig. 23 were both calculated from the accurate Kolos-Peek potential for that system with the aid of the Chapman-Enskog theory. The line was obtained by Lin et al. ( 1979a) using a classical cross-section formulation, while the points represent Dickinson and Lee’s quantum mechanical calculation ( 1978). The quanta1deviations seen below 50 K will be discussed later. Here we may note that above that temperature the two sets of calculations agree within their uncertainty limits with each other and with the experimental measurement of Orient (1 97 1) at 300 K, which is shown by the lower triangle. This is particularly gratifying, for the accuracy of the Kolos- Peek potential is considered to be such that in principle the mobility is now calculable to a greater accuracy than that to which it can be measured (Lin et al., 1979a). Other measurements of H+ and D+ mobility in He have been compared by Howorka et al. (1979). There is general agreement within the limits of the experimental uncertainties. The upper triangle in Fig. 23 is the mass-scaled mobility of D+ from Orient ( 1972). The molecular ion HHe+ is a homolog of H, with a binding energy of 2 eV and rm = 0.8 A.Indeed at sufficiently high pressures and low E/n it is stable in helium and its mobility has been determined (Snuggs et al., 1978).Thus to observe the true mobility of H+ in He Orient worked at helium pressures of <4 T o n for 100 psec drift times. At intermediate pressures, H+ would be expected to spend part of its time as HHe+ and the correct mobility would

FIG.23. Zero field mobility of H+ in He as a function oftemperature. Classical calculations ) of Lin, Gatland, and Mason (1979a); quantum mechanical calculations ( 0 )of Dickinson and h e ( 1978);lower triangle, measurement of Orient ( 197I ); upper triangle, scaled measurement for D+ in He (Orient, 1972). From Lin et al. (1979a) by kind permission of the authors and publisher.

(-

305

ELECTRON AND ION MOBILITIES

not have been observed. This is phenomenon, against which precautions must always be taken (McDaniel and Mason, 1973, pp. 39 -44). The interaction between H+ and He is properly called a valence interaction. Such interactions may be anticipated with other open-shell ions, particularly in gases with compatible orbital energies. However, their characteristics may be quite diverse, relatively few having as large a bond energy as that dislayed by HHe+. Here the values of E and rm (2 eV and 0.8 A) are significantly different from those of the systemsdiscussed above, for which E = 0.3 eV and r , = 2 A would by typical. The net effect of the strong interaction is to cause the curve ofp versus Te,for H+ in He to fall below that which would otherwise be expected (Mason and McDaniel, 1985). ii. Quantum and charge transfer effects. The quantity TeB)has to be calculated from the transport cross section, which is formulated differently in quantum and classical mechanics (Mason and McDaniel, 1985). Quantum effects are expected to be most important in light atom systems at low temperature. One effect arises from the discrete nature of angular momentum, which can cause sharp fluctuations in the transport cross section as a function of energy. These fluctuations are associated with the phenomenon of orbiting resonances. As we can see from Fig. 23 this effect is small, since the quantum mechanical deviations do not exceed the numerical uncertainties in the quantal and classical calculations for H+ in He until the temperature drops below 50 K, and even at 10 K the difference in mobility is only about 3%. Calculations by Gatland et al. (1977b) for the “slightly heavier” system Li+ in He have shown that the quantal and classical values of R(’x’)(TeR)agree down to 5 K. For other heavier systems at the temperatures ordinarily used quantum mechanical effects on cross sections should therefore be negligible. A further effect of orbiting resonances has been postulated to account for a small pressure dependence of the zero field mobilities in more massive systems at 293 K. Careful work by Elford and Milloy ( 1974a)demonstrated reproducible reductions in the mobility of K+ ions over the pressure range from 1 to 100 Torr. The authors showed that the fall-off, which amounted to 1.3% in argon and much less than that in helium and hydrogen, could be explained by the formation and break up of transitory complexes,

a(’*’)(

K+

K+M*

+ M & K+M*

(29d

+ M -.LK+ + M + M

(29b)

kd

The results required that these have lifetimes for spontaneous decay (i.e., k,’ ) of 1 to 2 nsec. Using reasonably reliable potentials, Watts (1 974) calculated lifetimes of K+Ar* complexes formed by quantum mechanical tunnel-

306

Gordon R. Freeman and David A. Armstrong

ing through the centrifugal potential barrier. He found several virtual states of accessible energy with lifetimes in the region required and concluded that their existence could indeed provide conditions for a small pressure dependence of mobility at room temperature. This interesting hypothesis certainly deserves further attention, since the effect, though small, might be more common than quanta1 perturbations in Ter). The last quantum mechanical effect to be considered here occurs when the cores of the ions and the gas are identical. The resulting symmetry then causes resonance attraction and repulsion, and resonant charge transfer (McDaniel, 1964). The effect of resonant charge transfer on mobility is usually profound, since momentum of the ion in the drift direction is effectively lost when the charge is transferred to a randomly moving neutral. Lin and Mason ( 1979) have proposed a modified moment method for dealing with mobility in this case. In the following paragraphs we examine briefly a few experimental studies carried out in this area since 1973. Helm (1975) and Helm and Elford (1977b) have measured the mobilities of the ground state ions of the inert gases, and have been able to resolve the and metastable (2P,12)ions in Ne, Kr, and Xe as mobilities of the (2P312) parent gases. Since the cross section for charge transfer falls only slowly with relative energy or velocity, the term in Eq. (24) means that mobilities dominated by a charge transfer interaction will rapidly fall below pyl as Tee or the mean energy ( eCm)in the center of mass system rises. This is illustrated by curves Bj, Bf, and C which show the ( E , ) dependence of Ne+(2P3,,)and Ne+(2P,l,) in neon as determined experimentally by Helm and Elford (1977b). A similar trend is predicted by the calculated curves At and Aj of Cohen and Schneider. The agreement between theory and experiment is only fair, but the dependences on energy are similar. Only one mobility peak could be detected for Ar+ ions in argon and it appears that there the two ion species must have mobilities within & 1.59/0of the observed mobility. In xenon and krypton the relative magnitudes of the ion mobilities were opposite to those in neon, the (2P,12) species exhibiting 5 and 3% larger mobilities in xenon and krypton, respectively. In further studies (Helm, 1977;Hegerberg et al., 1982)the mobilities were extended to higher field strengths, and used in conjunction with equations of Skullerud ( 1973)to derive cross sections for resonant charge transfer. These were found to agree reasonably well with recent theoretical results. The old rule that the momentum transfer cross section is twice the charge transfer cross section ( McDaniel, 1964) remains a good approximation at large field strengths. The effect of resonant charge transfer in more complex systems was discussed by McDaniel and Mason (1 973). iii. Runaway. When the average momentum acquired from the field in unit time by an ion is not balanced by the average loss of momentum per unit

307

ELECTRON AND ION MOBILITIES

- 1 ’

LO

loo

Loo

(Ecm) (mew

FIG.24. Variation of Ne+ (2P,,z)and Ne+ (2P3,z)mobilities in neon with (c,) , the mean center of mass energy-curves Bf, Bj,respectively,and C. The curvesAf and At are theoretical values calculated by Cohen and Schneider for the two ions, respectively. Curves B and C were derived from experimental data at 77, 78, and 294 K, respectively, using (em) = fkTctrand Eq. (25b). From Helm and Elford (1977b) by kind permission of the authors and publisher.

time, the ion will accelerate and a steady drift velocity will not be acquired, Under these conditions the concept of a mobility is meaningless, but an apparent mobility may still be found from the ion transit time in a drift tube under particular conditions. Such apparent mobilities may be expected to depend separately on E and n, rather than on the ratio E/n. Also they will vary with the drift distance L, which will usually couple with n as a single variable nL (Waldman and Mason, 1981b). Lin et al. (1979a) showed that an approximate condition for runaway would be [(m M)/m]1/2(E/n) > ~ E Q ( ~ ) ( E )where , Q(’)(E) is the momentum transfer cross section and E the mean kinetic energy of relative motion of the ion and gas molecule. They also made the first prediction of a runaway effect for a specific systemH+(and D+) in He. Figure 25 presents their plot of EQ(I)(E)vs E for that system. The curve never rises above the first maximum, where the value is 1.1 X lo-” eV cm2. This is equivalent to the ion in a field strength of 1 10 Td. Thus runaway should be seen when the mass-scaled field strength, [(m M)/m]l/Z(E/n), significantly exceeds that. However, these authors pointed out that even below 1 10 Td a significant high-energy tail should develop on the ion distribution. Experimental verification of a runaway effect for H+, and indeed also D+, in He was first provided by the results of Howorka et al. (1979). Contrary to the usual finding of a narrowing in the amval time histogram with increasing field, they observed a broadening at E/n = 70 Td and an “early toe.” The latter effect is attributable to a high-energy tail as discussed above, while the

+

-

+

308

Gordon R. Freemun und David A. Armstrong

10

Id

1

cleVl

FIG.25. Collisional momentum loss as a function of relative collision energy, showing the ); calculated for the conditions for runaway; calculated for the actual H+-He potential ((8-4) potential model (---). The energy limits for orbiting collisions are marked for both potentials. From Lin ef a/. (1979a) by kind permission of the authors and publisher.

broadening is an indication of the effect of runaway on the longitudinal diffusion coefficient D,, (see below). Their mass-scaled mobilities for D+ and H+ have been plotted against the mass-scaled field strength in Fig. 26 as filled and open triangles. The upturn in mobility at 80 Td is most unusual for ions in helium and is thought to be a further indication of runaway. This figure was actually taken from the paper of Lin and co-workers (1979a) and shows also their curves for the first, fourth, and fifth orders of approximation for mobilities calculated from the Kolos- Peek potential by means of the Viehland-Mason theory. In the region below 40 Td, where steady drift occurs, convergence of the fourth and fifth approximations is seen. However, above 60 Td convergence ceases. The increasing divergence there was attributed to runaway, since the higher order moments solutions are more sensitive to the high-energy tail. Finally Moruzzi and Kondo (1980) have shown that the apparent mobility depends separately on E and n, and there appears to be no doubt ofthe runaway phenomenon for H f a n d D+ in He. A corresponding phenomenon for electrons in gasses has of course been known for some time. Acceleration need not continue indefinitely, since it will usually be terminated by inelastic collisions.

309

ELECTRON AND ION MOBILITIES

-% F

1L

>

N \

-k P 9 -f +

13

E

-E

r.

12

0

50

Km + M)/mI” (E/N (Td)

100

FIG.26. Mass-scaled mobilities of H+ and D+ in He at 300 K as a function of mass-scaled field strength, showing the onset of runaway. The curves are calculations of Lin et a/. (1979a): H+ (-), D+ (---), the numbers refer to the order of approximation. Smoothed values (&A) from the measurements of Howorka ef al. (1979). From Lin ef a/. (1979a) by kind permission of the authors and publisher.

d. Ion difiuion -generalized Einstein relations. The fact that ions drifting in a gas under the influence of a field may also undergo diffusion is of practical as well as fundamental importance, since the arrival time histogram is thereby broadened and must be carefully deconvoluted to extract the drift time and mobility (McDaniel and Mason, 1973).Diffusion occurs both parallel and transverse to the field direction with diffusion coefficientswhich are, respectively, D,,and D, . At low fields, where the energy distribution of the ions is Maxwellian, these two are equal and related to the mobility by Eq. (30), -qD - kT P

Weinert and Mason ( 1980) have pointed out that although it is most commonly referred to as the Einstein Relation, Eq. (30) was in fact derived independently and prior to Einstein’s work on Brownian motion by both Nernst and Townsend. Strictly speaking it is therefore more properly called the Nernst - Townsend - Einstein relation (Mason and McDaniel, 1985). In principle both D,,and D, can be determined experimentally, but the measurements are more difficult and the present accuracy ( 2 10%)consid-

-

310

Gordon R. Freeman and David A. Armstrong

erably less than for mobility. For this season there is great practical value in having relations similar to Eq. (30), which are valid at all field strengths. The first such generalized Einstein relation (GER) was derived by Wannier ( 1952) on the basis of a simplified model,

1

--k7',,[1+ 4DIl d dln(E/n) lnK P This form was later confirmed by Robson (1972) using nonequilibrium thermodynamics. Since those publications extensive theoretical work has been done in this area (Viehland et al., 1974; Viehland and Mason, 1978; Skullerud 1976; Robson, 1976; Waldman and Mason, 198la; Waldman et al., 1982), largely because the derivation of GER provides insight into the parallel and transverse components of the ion energy. Furthermore, where D,, and D, have been determined experimentally their comparison to values calculated from experimental mobility with the aid of GER provides an important test of the reliability of theory and experiment. Experimental values ofp and D, in several systems have been used in tests with GER based on Eq. (31) and ion temperatures calculated from the two-temperature theory (see, e.g., Viehland and Mason, 1975b; Thackston et al., 1980a; Holleman ef al., 1982, and references therein). The results in general agree within the uncertainties ofthe determinations ofD,. The most rigorous theoretical treatment is that of Waldman and Mason (198 la), which employs the formalism of the three-temperature theory to develop GER from the Boltzman equation without restrictions of m/Mor E/n. It has been tested against the calculations of Viehland and Mason and Skullerud, and experimental D,,and D , for the alkali ion-inert gas systems. The agreement for the three theoretical approaches was comparable. However, in principle the three-temperature theory is more adaptable as well as rigorous. Finally we may note that GER have been modified to take into account the case of resonance charge transfer (Waldman et al., 1982). 2. Polyatomic Ions and Gases

Polyatomic systems introduce three new major complications: (1) The ion - neutral interactions may no longer be spherically symmetric, (2) energy contained in internal degrees of freedom ofthe ion or molecule must now be included in the energy balance equations, and (3) inelastic collision processes must be included in the kinetic theory treatment. The Wang Chang, Uhlenbeck and de Boer equation is a generalization of the Boltzmann equation for

ELECTRON AND ION MOBILITIES

31 1

particles with internal degrees of freedom and can be made to take into account the inelastic collisions. With this as a starting point and some modifications of the techniques used earlier for the Bolzmann equation, Lin et al. (1 979b) and Viehland et al. (1 98 la) have, respectively, derived solutions for mobilities of structurelss and polyatomic ions in polyatomic gases. The form of the master equation for mobility in first approximation is the same as Eq. (24) with a! = 0, except that Te5is replaced by a new quantity Tk,&. The latter can be found from #kTk,eE= [ t k T + fMv:1[ + (M/m)t(Tk,e5)l-' = jk(mT MTk)/(m 4- M )

+

(32a)

(32b) Here Tk is the ion translational temperature, while Tk,&is the effective translational temperature in the center-of-massframe of the ion and neutral molecule. The quantity t(Tk,,5), which does not appear in the corresponding equations in Section III,A, 1,a for atomic systems, is a dimensionless ratio characterizing the energy loss due to inelastic collisional exchanges. It is defined as (33) Where p is the momentum in the ion-molecule center-of-masssystem and A indicates the change in a collision. c(Tk,eE)

= ( APz ) / ( 2P

AP )

a. Polyatomic and clustered ions in monatomic gases. Detailed balance shows that when steady state conditions are reached in these systems the internal temperature of the ions, T,, must be equal to Tk,,(Viehland ef al., 1981a). Also ((Tk,e5) is zero so that Eq. (32a) reduces to Eq. (34a), an expression of the same form as that for atomic systems, [#kT+ 3Mv$1 (344 With these similarities in the kinetic theory expressions, the major differences from purely atomic systems lie in the possible asymmetry of the interaction potentials and the fact that they are now averaged over the rotational and vibrational levels of the ion. The effects of these factors appear to be unspectacular, because the field dependences of mobilities of molecular ions, for example, COfand N20+ in He, Ne, and Ar (Ellis et al., 1976b, 1978),exhibit similar trends to those seen in Fig. 22 for K+or Rb+. However, it remains to establish experimentally whether changes in Tk,,Eproducedby altering E/n are equivalent to those induced by changing the gas temperature T, and experimental data (above and Dotan et al., 1976)warrant further analysis. Most of the studies of larger ions, like S F f , SF;, and S02Fywere contkTk,e5=

312

Gordon R. Freeman and David A . Arrnstrong

ducted only at low field (Patterson, 1972). However, it was analysis of this kind of data which emphasized the need to consider more realistic potentials than the hard sphere repulsive polarization attraction and other simple models (Patterson, 1972; Mason et al., 1972; McDaniel and Mason, 1973). Usually the mobilities can be fitted with potential functions of the form

with n ranging from 8 to 12. This also appears to hold for clustered monatomic ions like C1-*(H,O), in argon, krypton, and xenon (Jbwko and Armstrong, 1982a). A core model, consisting of a (12-4) central potential displaced from the origin, can also be used to reproduce the mobilities of polyatomic ions (Mason et al., 1972). Both this and the n-6-4 potential may drastically alter the dependence of mobility on Tea,completely suppressing the maximum seen in Fig. 2 1. This resembles what is seen experimentally when temperature is varied, but n - 6 -4 potentials do not give realistic agreement (Parent and Bowers, 1981). A special type of interaction can occur in the case ofdiatomic inert gas ions in their parent gases (Helm and Elford, 1978). The process involves the transfer of the monatomic ion from one neutral atom to another, e.g., He:+ He

-

He

+ He:

(35)

It has a relatively high cross section and causes the mobilities of the dimer ions to be lower than for ions of similar mass with normal interaction potentials (Mason and McDaniel, 1985).Helm and Elford point out that the phenomenon of fragment ion exchange may apply to more complex systems as well.

b. Monatoinic ions in nonpolar polyatomic gases. The introduction of internal degrees of freedom into the buffer gas molecules has no obvious effect on the field dependencies of mobilities measured at a given temperature, which frequently resemble those in monatomic gases (see Ellis et al., 1976b, 1978). Also we find that p approaches pp, at low field. However, experiments by Viehland and Fahey (1983) have provided evidence for the , ~ ~occurs ) , in Eq. (32a). The open squares in Fig. 27, parameter c ~ T ~which which is taken from their paper, represent zero field mobilities of C1- measured in nitrogen gas at several temperatures and plotted against these temperatures as the values of Tk,eaon the abscissa scale. The open circles and triangles show mobilities all measured at 300 K, but with different E/n. They are plotted against values of Tk,,,calculated on the assumption that <(Tk,ea) is negligible [i.e., using Eq. (34a)l. The authors attribute the lack of coinciand dence of the two sets of data to the existence of finite values of <(Tk,eff)

313

ELECTRON AND ION MOBILITIES

26~""""'"""""~ 0 loo0 Zoo0

Ma)

4 m

Tk,eff (K)

FIG.2 7 . Reduced mobility p o as a function of Tk,,,for CI- ions in N,. (0),high-pressure drift tube mass spectrometer results measured at low field strengths and the temperature shown as Tk,e,;(A,O), data measured in a flow-drift tube at 300 K and 0.200 and 0.120 Tom, respectively, at different field strengths with Tk,FB calculated from Eq. (34a). From Viehland and Fahey (1983) by kind permission of the authors and publisher.

neglect any possible contribution from a specific dependence ofG('") on the internal temperature of the molecules, T g , which clearly differs for the two types of experiment. Since the T temperature dependence for the squares in Fig. 27 is fairly similar to that for C1- in argon (Ellis et al., 1978, p. 194), the assumption that R('.')does not depend strongly on T; in fact seems reasonable. The reason the two sets of data points coincide at 300 K is as follows. Detailed balance requires that when the ions and gas are in thermal equilibrium energy transferred from the ions to internal modes of the molecules is + 0 as Tk,eff + T (Viehoffset by the reverse process, and therefore t(Tk,eff) land et al., 1981a). This condition applies to all of the squares and their point ofjunction with the other data. As E/n is increased with Theld at 300 K the translational temperature of the ions departs more and more from the rotational and vibrational temperature of the molecules and (( T&&)becomes larger. Since the ions are in a trace quantity only, the average gas temperature is unaffected by the energy transfer. rose Viehland and Fahey concluded from analysis of the data that t(Tk,cff) from zero at the 300 K reference temperature to 2.0 at 1665 K. Polyatomic ions gave similar results. It is evident that this type of experiment could provide a new way of measuring rates of translational + vibrational and can be related to cross sections in a rotational energy transfer, if <( Tk,eff) meaningful way.

c. Polyatornic and cliistered ions in nonpolar gases. The practical importance of mobility measurements has led to a continual increase in interest in complex systems (Ellis et al., 1978). In several cases temperature dependences have been determined for mass-analyzed ions. One or two, such as NO:, NO;, and CO; in nitrogen, exhibit mobilities with rather strong and as yet unexplained dependencies on gas temperature (Eisele et al., 1980).

314

Gordon R. Freeman and David A . Armstrong

Rather interestingly, some polyatomic systems, e.g., COT and COTin 0 2 , appear to exhibit a near equivalent effect for changes in Tk,eastemming from alteration of either E/n or T (Perkins et al., 1981, Fig. 2), but data are at present limited. Attempts to relate mobilities of polyatomic systems to potential functions have invariably led to the conclusion that the simple hard sphere repulsive polarization attractive potential is inadequate (e.g., Patterson, 1972; Ridge and Beauchamp, 1976; Huang and Freeman, 1979). In the case of CH:, C,Hf, and C,H;and a few other ions in nitrogen and hydrogen (Ridge and Beauchamp, 1976) the mobilities at 298 K could be fitted with either a 12-6-4 potential or an accentric potential. For CHfin CH4 both forms of potential yield rm - 0.4 nm and E = 0.15 eV. Both curves fall significantly below the polarization potential from r = rm to r even as large as 1 .O nm, which explains why p is - 8090less thanp,, at 298 K. However, an examination of the temperature dependence of the mobilities of polyatomic and monatomic ions in polyatomic gases again reveals inadequacy in the n - 6 -4 form of potential function (Parent and Bowers, 1981). An evaluation of potentials in the SF, system has recently been published (Brand and Jungblut, 1983). As experiments have been pushed to higher and higher pressures the interest in clustered ions has become more evident. For example, the past decade has seen new determinations of the equilibrium constants for reactions (36) and (37) as well as the measurement of the mobilities of the ion species present

F=H:+

(36) (37) (Elford and Millory, 1974b; Milloy, 1975). In the case of reaction (37) there is good agreement between the equilibrium constants derived from mobility measurements and from high-pressure mass spectrometry, but serious discrepancies between the results from the two methods remain for the hydrogen system. The mobility data have been checked recently by Elford (1983), and the earlier results were confirmed. Rates of clustering and declustering, and free energies are now known for several systems (Kebarle, 1977; Meot-Ner, 1979). The former are generally sufficiently rapid that above 0.1 atm, an equilibrium distribution between the members of a given ion family, e.g., m+.M,, H f + 2H2

H,

o:+ 2 0 2 === o:+ 0,

m+.M,

+2

M e m+-M,+,

+M

(38) should be maintained during drift. Under these circumstances all ions ofthe same family have the same apparent mobility, which is given by Eq. (39),

315

ELECTRON AND ION MOBILITIES

where p iand zi are the mobility and lifetime of the ith member (Smirnov, 1967).

/=-

c i

(39) Ti

i

The validity of this rule has been established experimentally for the ion families CO;.(CO,),, CO;-H,O.(CO,), with x up to eight and for a positive ion family in C02at n = 4.0, 1 1.8, and 24.7 X lo1*molecule/cm3 (Ellis et al., 1976a). It is self-evident that when ion clusters become very large the identity of the ion core will have very little influence on mobility. This effect is seen in CO, where the two negative ion families and the positive ion family noted above all have reduced mobilities in the range 0.99 - 1.09 cmZ/V* sec (Ellis et al., 1976a;Jbwko and Armstrong, 1982b).However, free energies of clustering depend sensitively on core geometry and size, and it is not surprising that the foregoing effectis not universal for all systems in the 0.2-2 atm pressure region. Thus the positive and negative ions in N,O have different mobilities even though the molecule is slightly polar and should form clusters at least as readily as C 0 2(Jbwko and Armstrong, 1982b). However, in strongly polar gases, where clustering is always strong and ion-dipole forces of long range dominate the interaction potential anyway (see below), ion cores generally have little effect (Sennhauser and Armstrong, 1980). Equation (39) does not actually take care of the dynamics of momentum or energy exchange in cluster collisions. In place of the elastic collisions assumed in evaluating for each ion species, viz.

M + m+.M.x(T,,e,) + M + m + * M x ( T k , e E ) , these may take the form of reaction sequences,

(41a)

M + rn+.M,(T,,eff)+ rn’.M,+,(Tf) rn+*M,+,(T’)

+

rn+.Mx+,(T’) M rn+.M,+,(T’)

+M

--

(40)

+

m+.M,(T”) M rn+.M,+,(T,,,,) + M

(4 1b)

m+.M,+,(T”’)

(41d)

m + . M x + l ( T f f ‘+ ) rn+-M,(T,,,)

+M

+M

(4 1 4 (4 1e)

Here the internal temperatures T’ and T”’ exceed the equilibrium tempera(see above), while T” may be larger, smaller, or equal to it. First, if ture Tk,eff

316

Gordon R. Freeman and David A . Armstrong

their lifetimes k;,'b are significant relative to T ~ the , species m+.M,+, ( T ' ) should be included in Eq. (39). Second, the overall transfer of momentum resulting from the sequence (4 la)- (4 1b) may be different from that occurring in the elastic collision (40). The same may apply to the sequences (41a)-(41c) and (41a)-(41d)-(41e). When k41b is fairly large the sequence (41a)-(41b) is equivalent to the fragment ion exchanges, which have a profound effect on the mobilities of the rare gas dimer ions (see Section III,A,2,a). For the x = 1 molecular systems H{in H, and CHfin CH,, Ridge and Beauchamp ( 1976) concluded that symmetrical proton transfer made little contribution to the momentum transfer cross section. However, this does not agree with earlier discussions ofthe H:-H2 system (McDaniel and Mason, 1973). More recently Gee and co-workers (198 1, 1982) have postulated sticky collisions, which are formally equivalent to (41a)-(4 1b), to explain low mobilities in a number of systems. Thus a general theoretical valuation of the effects of cluster collisions and molecular exchanges on the momentum transfer cross section is badly needed. d. Ions in polar gases. Early interest in these systems stemmed from the desire to explain the rates of ion - polar molecule reactions (Moran and Hamil, 1963),but theory has been developed to the point where the motion of ions in polar gases in the low field region can also be treated quantitatively. The relationship between the two phenomena can be expressed by rewriting the Chapman - Enskog equation in the form

kcolis the thermally averaged collision rate constant (8k7'/nMr)1/2 where Q(',I)( T ) .Actually the rate constant of interest in reaction rate theory is not k,,,, but the rate constant for orbiting collisions b.The latter neglects grazing collisions and is smaller -about 10% smaller for a pure polarization potential (Chesnavich et af., 1980). A different procedure is therefore required in the calculation ofthe omega integral fork,. However, in either case the major problem lies in evaluating the contribution ofthe ion-dipole force to the interaction potential. The usual procedure, which is justified at low relative energies of motion, has been to neglect the short-range repulsive and attractive forces and assume that the potential is of the form of Eq. (43), where pDis the dipole moment of the neutral,

a q 2 - -COS 9pD V(r,0)= - 2r4 r2

0

(43)

r the distance of the center of mass of the neutral from the ion, and 0 the angle between the dipole and r. Methods of obtaining thermally average

ELECTRON AND ION MOBILITIES

317

values of k, have been reviewed and discussed in several recent papers (Ridge, 1979;Turulski and ForyS, 1979; Chesnavich et al., 1980;Celli et al., 1980; Bates, 1982a,b)and agreement with experiment for exothermic reactions is good. Recently Celli et al. (1980) have developed a thermodynamic model for the evaluation of kcol.The rate constant and mobility depend only on the reduced polarity parameter P = pD/(cwkT)lI2.The theory yields the ratio p/pPolin terms of Eq. (44).The parameter a varies from unity at small P

to 0.578 at large P, where p/pu,,+ 1.6 1P-l or p/ppo!= 1.6akT'I2/pD. A comparison of experimental mobilities with the prediction based on Eq.(44) is given in Fig. 28. The closed triangles are from low-pressure ICR experiments and the squares from high-pressure drift tube mass spectrometry. In these cases the ion identities were all established by mass analysis. This was not the case for the open triangles and two lines, which were taken from drift studies and for which there is some uncertainty as to the number of

FIG. 28. Dependence of p / p p o lon the reduced polarity parameter P = pD/(cykT)1/2 for ) of Celli et al. ( 1 980). several ion - polar molecule systems. Theoretical calculation (Points are from: low-pressure ICR studies-(A), Ridge and Beauchamp (1976) for Na+ in C,H,O isomers; (V),Buttrill (1973) for CH:, CH2F+, H,S+ in CH,, CHIF, and H2S at two different T in order of increasing P; high-pressure pulsed-source mass spectrometry-(.), Polley et al. (1980) for CH:in CH,, and NHZand NH:.NH, in NH, (coincident points); and drift expriments with clustered ions-(V) Sennhauser and Armstrong (1980) for ions in NO, HBr, HCI, and NH,; (0)JBwko and Armstrong (1982a) for CI-.4H20 in H20, and Gee and Freeman (1983b) for ions in CHIOH (---)and H,O (-).

318

Gordon R. Freeman and David A . Armstrong

neutrals clustered around the ion cores. However, as mobility depends only on Mil2the uncertainty from this source is not large. Clearly the dependence on P predicted by theory is followed by both the lighter ions and the heavier clustered ones (see the figure caption for ion identities). This is attributable to the long-range nature of the ion-dipole interaction. However, it is an apattractive comproximation to neglect entirely short-range repulsive and r 6 ponents, and this may contribute to the scatter ofthe points above and below the line. Inclusion of these forces has been shown to markedly improve the agreement -particularly at low values of P (Ridge, 1979). The thick line represents the mobility ratio p/pPl for H,O+.n H 2 0 in water vapor over the temperature range 300 to 500 K with n assumed equal to 6 in the calculation of,uP,. Since n probably decreased with temperature, which would cause pPl to rise slightly, the agreement between the experimental temperature dependence and the line shown by theory should be somewhat better if correct ion masses were used in the evaluation ofpu,,. A similar argument applies to the data for positive ions in methanol, wherep(,, was calculated for CH30Hf.3CH,0H. However, here there is a much stronger temperature dependence, which suggests that the bulkier methanol molecules may have introduced other factors into the interaction. Parent and Bowers (198 1) also found that the temperature dependence p/pPl agreed only semiquantitatively with theory. In addition they point out that may somecare should be taken in comparisons of ICR results, where Tk,ea times exceed T by up to 25%. In summary, the theory has provided a framework for understanding the gross effectsof polarity, but the finer details of combining ion -dipole interactions with repulsive and short range attractive forces and systematic experimental studies of mobilities for systems of different ion sizes require attention.

B. DENSEGASESAND LOW-DENSITY LIQUIDS (0.01 < n/n, < 2.0) Systematic study of the mobility of ions in a fluid as density is vaned provides insight into the mechanisms oftransport and the changes in properties which occur as the fluid density rises from that of dilute gas to liquid. Figure 29 from the work of Gee and co-workers (1982) shows the densitynormalized cation mobility np in argon, xenon, nitrogen, and methane (open symbols) plotted against the ratio of fluid density n to the density under critical conditions n,. With the exception of xenon the curves are flat in the region n/n, = 0.01 to 0.3. As n/n, rises above this, all four substances exhibit a rise in np. Beyond the maximum, which occurs near n/n, = 1.5 to

319

FIG.29. Density dependenceofdensity-normalizedtransport coefficientsof cations np and neutral molecules n (De/kT)in coexistence vapor and liquid. Densitiesare normalizedto those of the critical fluids, n, ;argon (A,A), xenon (O),. N2(0),CH, (0,W). Open symbolsare cations and filled symbols are neutral molecules. From Gee et al. (1982) by kind permission of the authors and publisher.

2.5, np falls abruptly. Similar effects have been seen for ions in dimethyl ether and a variety of hydrocarbons (Gee and Freeman, 198la, 1982;Huang and Freeman, 1979, 1980a). For comparison, the filled symbols in Fig. 29 represent the normalized self-diffusion coefficients of the neutral molecules nDe/kT, where the factor e/kTmakes them equivalent to np. In contrast with the latter quantity these exhibit no maximum, but above n/n, = 1 they fall off, as do the ion mobilities. The rise in np as n/n, increases from 0.5 to 1.5 has been attributed to “destructive interference of attractive electrostatic interactions, when the ion is near to more than one molecule at a time” (Huang and Freeman, 1979).There is no equivalent ofthis effect for neutrals. The fall-offin np and nDe/kT above n/n, = 1.5 can be attributed to the obstruction of translational motion as the repulsive cores become more closely spaced. Application of Stokes’ law to the liquid phase mobility and diffusion processes yields for the neutral molecules effective radii, which are close to the “rigid sphere radii” calculated from viscosity and other gas phase data, viz. 0.20, 0.18, 0.21, and 0.18 nm for methane, nitrogen, xenon, and argon, respectively (Gee et al., 1982). The ion radii tend to be larger: 0.37, 0.40, 0.35, and 0.20 nm for the gases in the same order. This effect is attributable to electrostriction. Other studies (Gee and Freeman, 1980c)showed that the extent of electrostriction increased with compressibility of the fluid and appeared to decrease as the molar volume rose. The temperature coefficient of mobility has been measured at different densities in several vapors. Near the coexistence curve at constant n it be-

320

Gordon R. Freeman and David A . Armstrong

comes very large. Typical trends in the activation energy of mobility are E,, 30,20, and 7 kJ/mol at n/n, = 1,0.5, and 0.3 in simple hydrocarbons (Gee el al., 1982).These large activation energies exist for only a few degrees above the coexistence curve, and are attributed to the evaporation of large clusters as the temperature rises. However, at densities near to and above n, the distance between molecules is no longer greater than the cluster diameter. The concept of a cluster then ceases to be useful and “the continuum concept ofelectrostriction is more appropriate.” Roughly this is the region in which the destructive interference of electrostatic interactions referred to above comes in. Many attempts have been made to devise a formula that would describe the transport of a particle as a function of increasing density from the free molecule to the continuum regime (Annis et al., 1972). Attempts to apply such formulas to the cation mobility results of the Edmonton group thus far have failed (Gee et a[., 1982). Possibly more success could be achieved if experiments were limited to variation of n at T,. However, the present formulae do not appear to take into account the effects of electrostriction, and it is doubtful if progress can be made for ionic transport until that has been done. Other difficulties have been discussed by Gee and co-workers ( 1982). L-

ACKNOWLEDGMENTS The authors wish to thank Drs. Elford, Fahey, Mason, McDaniel, and Viehland for making preprints oftheir articles available prior to publication. They are also indebted to the publishers of the Journal ofChemical Physics.the Journal ofhysics,Annals ofPhysics, and the Canadian Journal ofchemistry for permission to reproduce the figures thus designated in the captions, and for which these publishers hold copyright.

REFERENCES Akridge. G. R., Ellis, H. W., Pai, R. Y., and McDaniel, E. W. (1975). J . Chern. Phys. 62,4578. Atlshuler, S. (1957). Phys. Rev. 107, 114. Annis, B. K., Malinauskas, A. P.. and Mason, E. A. (1972). Aerosol Sci.3, 55. Atrazhev, V. M., and Iakubov, I. T. (1977). J. Phys. D 10,2155. Bartels, A. ( 1972). Phys. Rev. Lett. 28, 2 13. Bartels. A. (1973). Ph.w Left.44A, 403. Bates, D. R. (1982a). J. Phys. B 15, L259. Bates, D. (1982b). Proc. R. Soc. London Ser. A 384, 289. Braglia, G. L., Bruzzese, R., Solimeno, S., Martellucci, S., and Quartieri, J. (198 I ) . Liltt. Nuovo C i m 30, 459. Braglia. G. L., Romano, L.. and Diligenti. M. (1982). Phys. Rev. A 26, 3689.

ELECTRON AND ION MOBILITIES

32 1

Brand, K. P., and Jungblut, H. (1983). J. Chem. Phys. 78, 1999. Bridge, N. J., and Buckingham, A. D. (1966). Proc. R. SOC.London Ser. A 295, 334. Burke. P. G . (1979). Adv. .4t. Mol. Pliys. 15, 471. Buttrill, S. E., Jr. (1973). J. C‘hrm. Phys. 58,656. Celli, F., Weddle, G., and Ridge, D. P. ( 1980). J. Chem. Phys. 73, 80 I. Chesnavich, W. J., Su, T., and Bowers, T. (1980). J. Chem. Phys. 72, 2641. Christophorou, L. G., and Christodoulides, A. A. (1969). J. Phys. B 2,7 I. Christophorou, L. G., Carter, J. G., and Maxey, D. V. (1982). J. Chem. Phys. 76,2653. Cook, G. A. (1961). “Argon, Helium and the Rare Gases.” Wiley (Interscience), New York. Cottrell, T. L., and Walker, I. C. (1967). Trans. Faraduy Soc. 63, 549. Cottrell, T. L., Pollock, W. J., and Walker, I. C. (1968). Trans. Furaday Soc. 64, 2260. Crawford, 0. H., Dalgarno, M., and Hays, P. B. (1967). Mol. Phys. 13, 181. Crompton, R. W., and Robertson, A. G. (1971). Aust. J. Phys. 24, 543. Dickinson, A. S., and Lee, M. S. (1978). J. Phys. B 11, L377. Dodelet, J.-P., Shinsaka, K., and Freeman, G. R. (1973). J. Chem. Phys. 59, 1293. Dotan, 1.. Lindinger, W., and Albritton, D. L. (1976). J. Chem. Phys. 64,4544. Eggarter, T. P., and Cohen, M. H. (1971). Phys. Rev. Lett. 27, 129. Eisele, F. L., Perkins, M. D., and McDaniel, E. W. ( 1980). J. Chem. Phys. 73, 25 17. Elford, M. T. (1972). In “Case Studies in Atomic Collision Physics” (E. W. McDanieland M. R. C. McDowell, eds.), Vol. I, Chap. 2. North-Holland Publ., Amsterdam. Elford, M. T. (1983). J. Chem. Phys.. 79, 5951. Elford. M. T., and Milloy, H. B. (1974a). Aust. J. Phys. 27, 21 I. Elford, M. T., and Milloy, H. B. (1974b). Airst. J. Phys. 27, 795. Ellis, H. W., Pai, R. Y., Gatland, 1. R., McDaniel, E. W., Wernhund, R., and Cohen, M. J. (1976a). J. Cliem. Phys. 64, 3935. Ellis, H. W., Pai, R. Y., McDaniel. E. W., Mason, E. A,, and Viehland, L. A. (1976b). At. Data A’irc~l.Tables 17, 177. Ellis, H. W., McDaniel, E. W., Albritton, D. L., Viehland, L. A., Lin, S., and Mason, E. A. (1978). At. Data Niicl. Tables 22, 179. Englehardt, A. G., and Phelps, A. V. (1963). Phys. Rev. 131,2115. Englehardt, A. G., Phelps, A. V., and Risk, C. G. (1964). Phys. Rev. A 135, 1566. Foldy. L. L. (1945). P h j x Rev. 67, 107. Freeman, G . R. (1980). J. Phys Sor. Jpn. 48,683. Freeman. G. R. ( I 98 I). In “Electron and Ion Swarms” (L. G. Christophorou, ed.), pp. 93- 102. Pergarnon. Oxford. Freeman, G. R., Gyorgy, I., and Huang, S. S . 4 . (1979). Can. J. Chem. 57, 2626. Frommhold. L. (1968). Phys. Rev. 172, 118. Frost, L. S.. and Phelps, A. V. ( 1962). Phys. Rev. 127, I62 1. Frost, L. S.. and Phelps, A. V. (1964). Phys. Rev. A 136, 1538. Garrett, W. R. (1972). Mol. Ptiys. 24, 465. Gatland, I. R. (198 I ) . J. Chem. Phys. 75,4162. Gatland, I. R., Viehland, L. A.. and Mason, E. A. (1977a). J. Chem. Phys. 66, 537. Gatland, 1. R., Morrison, W. F., Ellis, H. W., Thackston, M. G., McDaniel, E. W.. Alexander, M. H., Viehland, L. A,, and Mason, E. A. ( 1977b). J. Chem. Phys. 66,5 12 I . Gatland, I. R., Thackston, M. G., Pope, W. M., Eisele, F. L., Ellis, H. W., and McDaniel. E. W. (1978). J. Chern. Phjs. 68, 2775. Gee. N., and Freeman, G. R. (1979). P/J.W.Rev. ‘4 20, 1152. Gee, N., and Freeman, G. R. (1980a). Phys. Rev. A 22, 301. Gee. N.. and Freeman, G. R. (1980b). Radiat. Phjs. Chem. 15,267. Gee, N.. and Freeman, G. R. (1980~).Can. J. Chcm. 58, 1490.

322

Gordon R. Freeman and David A . Armstrong

Gee, N., and Freeman, G. R. (l981a). Can. J. Chem. 59,2988. Gee, N., and Freeman, G. R. ( I 98 I b). Phys. Rev. A 23, 1390. Gee, N.. and Freeman, G. R. (1982). Can. J. C‘hem. 60, 1034. Gee, N., and Freeman, G. R. (1983a). J. Chem. Phys. 78, 1951. Gee, N., and Freeman, G. R. (1983b). Can. J. Chem. 61, 1664. Gee, N., Huang, S. S.-S., Wada, T., and Freeman, G. R. (1982). J. Chem. Phys. 77, 141 1. Giraud, V., and Krebs, P. (1982). Chem. Phys. Lett. 86, 85. Golden. D. E. (1978). Adv. At. Mol. Phys. 14, I . Golden, D. E., Neal, F. L., Temkin, A., and Gejuoy, E. (1971). Rev. Mol. Phys. 43, 642. Gosman, A. L., McCarty, R. D., and Hurst, J. G. (1969). “Thermodynamic Properties of Argon.” NSRDS-NBS 27. Washington, D.C. Grunberg, R. (1968). Z. Naturforsch. 23a, 1994. Gyorgy, I., and Freeman, G. R. (1979). J. Chem. Phys. 70,4769. Haddad, G. N., and OMalley, T. F. (1982). Aust. J. Phys. 35, 35. Hamson, H. R., and Springett, B. E. (1971a). Phys. Lett. 35A, 73. Harrison, H. R., and Springett, B. E. ( I97 1 b). Chem. Phys. Lett. 10,4 18. Hegerberg, R., Elford, M. T., and Skullerud, H. R. (1982). J. Phys. B 15, 797. Helm, H. (1975). Chem. Phys. Lett. 36, 97. Helm, H. (1977). J. Phys. B 10, 3683. Helm, H.(1978).J. Phjr. D 11, 1049. Helm, H., and Elford, M. T. (l977a). J. Phys. E 18, 3849. Helm, H.. and Elford, M. T. (1977b). J. Phys. B 18,983. Helm, H., and Elford, M. T. (1978). J. Phys. B 11, 3939. Hernandez, J. P. (1972). Phys. Rev. A 5,635. Herzberg, G. ( 1950). “Spectra of Diatomic Molecules.” Van Nostrand-Reinhold, Princeton. New Jersey. Herzberg, G. ( 1967). “Electronic Spectra of Polyatomic Molecules.” Van Nostrand-Reinhold, Princeton, New Jersey. Holleman, F. B., Pope, W. M., Eisele, F. L., Twist, J. R., Neeley, G. W., Thackston, M. G., Chelf, R. D., and McDaniel. E. W. (1982). J. Chem. Phys. 76,2106. Howorka, F., Fehsenfeld, F. C., and Albritton, D. L. (1979). J. Phys. B 12,4189. Huang, S. S.-S., and Freeman, G. R. (1978a). J. Chem. Phys. 68, 1355. Huang, S. S.-S.,and Freeman, G. R. (1978b). J. Chem. Phys. 69, 1585. Huang, S. S.-S.,and Freeman, G. R. (1978~).Can. J. Chem. 56, 2388. Huang, S. S.-S., and Freeman, G. R. (1979). J. Chem. Phys. 70, 1538. Huang, S. S.-S., and Freeman, G. R. (1980a). J. Chem. Phys. 72, 1989. Huang, S. S.-S.,and Freeman, G. R. (1980b). J. Chem. Phys. 72,2849. Huang, S. S.-S., and Freeman, G. R. (1981). Phys. Rev. A 24, 714. Huntress, W. T. (1971). J. Chem. Phys. 55, 2146. Huxley, L. G. H., and Crompton, R. W. (1974). “The Diffusion and Drift of Electrons in Gases.” Wiley, New York. Ioffe, A. F.. and Regel, A. R. (1960). Prog. Semicond. 4,237. Jahnke, J. A., Silver, M., and Hernandez, J. P. (1975). Phys. Rev. B 12, 3420. Jbwko, A., and Armstrong, D. A. (1982a). J. Chem. Phys. 76,6120. Jbwko, A,, and Armstrong, D. A. (1982b). Radiat. Phys. Chem. 19,449. Kebarle, P. (1977). Annu. Rev. Phys. Chem. 28, 445. Kimura, T., and Freeman, G. R. (1974). J. Chem. Phys. 60,4081. Krebs, P., and Heintze, M. (1982). J. Chem. Phys. 76, 5484. Krebs, P., Bukowski, K., Giraud, V., and Heintze, M. (1982). Ber. Eimsenges. Phys. Chem. 86, 879.

ELECTRON AND ION MOBILITIES

323

Kumar, K. ( 1 980a). Aust. J. Phys. 33,469. Kumar, K. (1980b). Ausf. J. Phys. 33,449. Kumar, K. (1981). J. Phys. D 14, 2199. Kumar, K., Skullerud, H. R., and Robson, R. E. ( I 980). Aust. J. Phys. 33, 343. Lamm, D. R., Thackston, M. G., Eisele, F. L., Ellis, H. W., Twist, J. R., Pope, W. M., Gatland, 1. R., and McDaniel, E. W. (1981). J. Chem. Phys. 74,3042. Landolt-Bdmstein (195 1). “Zahlenwerte und Funktionen,” 6th Ed., Vol. 1, Part 3. SpringerVerlag, Berlin and New York. Langevin, P. (1905). Ann. Chim. Phys. 5,245. An English translation is given in Appendix I1 of McDaniel(l964). Lax, M. ( I95 1). Rev. Mod. Phys. 23,287. Legler, W. (1970). PhyJ. Lett. A 31, 129. Levine, J. L., and Sanders, T. M. (1967). Phys. Rev. 154, 138. Lin, S. L., and Mason, E. A. (1979). J. Phys. B 12,783. Lin, S. L., Viehland, L. A., Mason, E. A., Whealton, J. H., and Bardsley, J. N. (1977). J. Phys. B 10, 3567. Lin, S. L., Gatland, I. R., and Mason, E. A. (1979a). J. Phys. B 12,4179. Lin, S. L., Robson, R. E., and Mason, E. A. (1979b). J. Chem. Phys. 71,3483. Lin, S . L., Viehland, L. A., and Mason, E. A. (1979~).Chern. Phys. 37,411. Lowke, J. J. (1963). Ausf. J. Phys. 16, 115. McCarty, R. D. ( 1972). “Thermophysical Properties of “He from 2 to 1500 K with Pressures to 1000 atm.” NBS Tech. Note 631. NTIS, Washington, D.C. McCorkle, D. L., Christophorou, L. G., Maxey, D. V.,and Carter, J. G. (1978). J. Phys. B 11, 3067. McDaniel. E. W. (1964). “Collision Phenomena in Ionized Gases.” Wiley, New York. McDaniel, E. W., and Mason, E. A. (1973). “The Mobility and Diffusion of Ions in Gases.” Wiley, New York. Makabe, T., and Mon, T. (1982). J. Phys. D 15, 1395. Mason, E. A,, and McDaniel, E, W. (1985). “Transport Properties of Ions in Gases.” Wiley (Interscience), New York. Mason, E. A., and Schamp, Jr., H. W. (1958). Ann. P h p . 4,233. Massey, H. S . W., Burhop, E. H. S., and Gilbody, H.B. (1969). “Electronic and Ionic Impact Phenomena,” 2nd ed., Vol. I. Clarendon, Oxford. Mason, E. A,, O’Hara, H., and Smith, F. J. (1972). J. Phys. B 5, 169. Meot-Ner, M. (1979). In “Gas Phase Ion Chemistry” (M. T. Bowers, ed.), Vol. I, p. 197. Academic Press, New York. Milloy, H. B. (1975). Ausf. J. Phys. 28, 307. Milloy, H. B., Crompton, R. W., Rees, J. A., and Robertson, A. G. (1977). Ausf. J. Phys. 30,61. Moran, T. F., and Hamil, W. H. (1963). J. Chem. Phys. 39, 1413. Moruzzi, J. L., and Kondo, Y. (1980). Jpn. J. Appl. Phys. 19, 141 1. Mott, N. F., and Davis, E. A. (1979). “Electronic Processes in Noncrystalline Materials,” 2nd ed. Clarendon, Oxford. Nelson, D. R., and Davis, F. J. ( 1 969). J. Chem. Phys. 51, 2322. Nishikawa, M., and Holroyd, R. A. (1982). J. Chem. Phys. 77,4678. OMalley, T. F. (1963). Phys. Rev. 130, 1020. O’Malley, T. F. (1980). J. Phys. B 13, 1491. OMalley, T. F., and Crompton, R. W. (1980). J. Phys. B 13, 3451. Orient, 0. J. (1971).J. Phys. B4, 1257. Orient, 0. J. (1972). J. Phys. B 5, 1056. Ostermeier, R. M., and Schwarz, K. W. ( 1 972). Phys. Rev. A 5,25 10.

324

Gordon R . Freeman and David A . Armstrong

Pack, J. L., and Phelps, A. V. (1961). Phys. Rev. 121, 798. Pack, J. L., Voshall, R. E., and Phelps, A. V. (1962). P h j Rev. ~ 127,2084. Paranjape, B. V. (1980). Phjs. Rev. A 21, 405. Parent. D. C., and Bowers, M. T. (1981). Cliem. Ptiys. 60, 257. Patterson, D. L. ( I 972). J. C h o n . P/JJS.56,3943. Perkins, M. D., Eisele, F. L., and McDaniel, E. W. (1981). J. Chem. PIiys. 74, 4206. Pitchford, L. C.. and Phelps, A. V. (1982). Phys. Rev. A 25, 540. Polley, C. W., Jr., Illies, A. J., and Meisels, G. G . (1980). Anal. Chem. 52, 1797. Pollock, W. J. (1968). Trans. Faraduy Sor. 64, 2919. Ramsauer, C., and Kollath. R. (1930). Ann. Pliys. (Lerpzig) 4,9 1. Reid, 1. D., and Hunter, S. R. (1979). Airst. J. Phys. 32, 255. Reid, R. C., Prausnitz, J. M., and Sherwood, T. K. ( I 977). “The Properties of Gases and Liquids,” 3rd Ed. McGraw-Hill, New York. Ridge, D. P. (1979). I n “Kinetics of Ion Molecule Reactions” (P. Ausloos, ed.). Plenum, New York. Ridge, D. P., and Beauchamp, J. L. (1976). J . C ’ l i t w . Phys.64, 2735. Robertson, A. G. (1971). Ausr. J. Phy.7. 24,445. Robson, R. E. (1972). Airst. J. Ptiys. 25, 685. Robson, R. E. (1976). J. Pliys. B 9, L337. Schmidt, K. H. (1972). Int. J. Radiat. Phys. Clrem. 4,439. Schulz, G. J. (1964). P h j ~R. e v A 135, 988. Schwarz, K. W. (1980). Phys. Rev. B 21, 5 125. Schwarz, K. W., and Prasad, B. (1975). Pkys. Rev. Li>tt. 36, 878. Sennhauser, E. S., and Armstrong, D. A. (1978). Can. J. Chem. 56, 2337. Sennhauser, E. S., and Armstrong, D. A. (1980). Can. J. Chem. 58,231. Shockley, W. (1951). Bell Sy.st. Tech. J. 30,990, see pp. 1018- 1024. Skullerud, H. R. (1973). J. Phys. B6, 728. Skullerud, H. R. (1976). J . Pliys. B9, 535. Smirnov, B. M. (1967). Sov. Phyx Usp. 10, 313. Snugs, R. M., Voltz, D. J., Schummers, J. H., Martin, D. W., and McDaniel, E. W. (1978). J. Chem. Ptiys. 68, 3950. Sohn, W., Jung, K., and Ehrhardt, H. (1983). J. Phjx B 16, 891. Takata, N. (1975). J . Phys. B 8, 2390. Takayanagi, K. (1980). Comments At. Mol. P h p . 9, 143. Takayanagi, K., and Itikawa, Y. (1970). Adv. A / . Ma/. Phys. 6, 105. Takebe. M. (1983). J. Chem. P k ~ ~ .78, s . 7223. Taniguchi, T., Tagashira, H., and Sakai, Y. (1978). J. Phys. D 1 1 , 1757. Thackston, M. G . ,Sanchez, M. S., Neeley, G .W., Pope, W. M., Eisele, F. L., Gatland. 1. R., and McDaniel, E. W. (1980a). J . Chem. Pliys. 73, 2012. Thackston, M. G., Eisele, F. L., Pope, W. M., Ellis, H. W., McDaniel, E. W., and Gatland, 1. R. (1980b). J . C h m . Phys. 73, 3183. Thomson, G. M., Schummers, J. H.. James, D. R., Graham, E., Gatland, 1. R., Flannery, M. R., and McDaniel, E. W. (1973). J. Chem. Phvs. 58, 2402. Turulski, J., and ForyS. M. (1979). J. Phys. Chcm. 83, 2813. Viehland, L. A. (1985). Chem. Phys. 78,279. Viehland, L. A., and Fahey, D. W. (1983). J. Chern. Plrys. 78, 435. Viehland, L. A,, and Mason, E. A. (1975a).Ann. Phvs. 91,499. Viehland, L. A., and Mason, E. A. (1975b). J. Chem. Phys. 63, 2913. Viehland, L. A., and Mason, E. A. (1978). Ann. Phys. 100,287. Viehland, L. A,, Mason, E. A., and Whealton, J. H. (1974). J. Phys. B 7, 2433.

ELECTRON AND ION MOBILITIES

325

Viehland, L. A,, Mason, E. A., Momson, W. F., and Flannery, M. R. (1975). At. Data N u d . Tables 16, 495. Viehland, L. A,, Hamngton, M. H., and Mason, E. A. (1976). Chem. Phys. 17,433. Viehland, L. A., Lin, S . L., and Mason, E. A. (1981a). Chem. Phys. 54,341. Viehland, L. A., Mason, E. A., and Lin, S . L. (1981b). Phys. Rev. A 24,3004. Wada. T., and Freeman, G. R. (1979). Can. J. Chem. 57,2116. Wada, T., and Freeman, G. R. ( I 980). J. Chem. Phys. 72,6726. Wada, T., and Freeman, G. R. (1981). Phys. Rev. A 24, 1066. Waldman, M., and Mason, E. A. ( 1 98 la). Chem. Phys. 58, 12 I . Waldman, M., and Mason, E. A. ( I98 I b). Chem. Phys. Lett. 83,369. Waldman, M., Mason, E. A., and Viehland, L. A. (1982). Chem. Phys. 66, 339. Wannier, G. H. (1951). Phys. Rev. 83,281. Wannier, G. H. (1952). Phys. Rev. 87, 795. Wannier, G. H. (1970). Bell Syst. Tech. J. 49, 343. Warman, J. M. ( 1982).In “The Study ofFast Processes and Transient Species by Electron Pulse Radiolysis” (J. H. Baxendale and F. Busi, eds.), pp. 129 and 433. Reidel, Dordrecht. Warren, R. W., and Parker, J. H. (1962). Phys. Rev. 128,2661. Watts, R. 0. (1974).Aust. J. Phys. 27, 227. Weinert, U., and Mason, E. A. (1980). Phys. Rev. A 21,681. Young, R. A. (1970). Phys. Rev. A 2, 1983.