Hatano Y., Katsumura Y., Mozumder A. (Eds.) Charged Particle and Photon Interactions with Matter - Recent Advances, Applications, and Interfaces

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220 Charged Particle and Photon Interactions with Matter

sets g(r) = 0 at a small value of r. Thus, in neopentane at 25°C, one gets r

min

= 3.0Å in Figure 9.4.

However, there is some exibility in determining r

max

, depending on the method of integration and

pre-symmetrization of the data for g(r) (Mikolaj and Pings, 1967). We have consistently used such a

value of r

max

where the normalized value of g(r) returns to one for the rst time beyond the princi-

pal peak (Hug and Mozumder, 2008). This procedure automatically gives nearest-neighbor separa-

tion. For liquid methane, it correctly predicts Z = 12, a well-known value for this liquid (Petz, 1965;

Karnicky and Pings, 1976). Methane has the highest value of Z that is theoretically possible, while the

situation is expected to be similar for TMS and its analogues. The so-determined K = Z − 1 and the

nearest-neighbor separation (r

) are given in Table 9.1 for different hydrocarbon liquids. Also given are

the critical values of (W/2V)

for the given connectivity, K, of the liquid, obtained from Equation 9.8.

In the Anderson theory of localization, it is not the diagonal energy per se, but the width of its

uctuation that is the determining factor (Anderson, 1958). Of the various possibilities of the source of

this uctuation, the involvement of the electron-bond (micro) dipole interaction has been discounted

by Funabashi (1974), on the ground that it is too small, despite the possibility of internal rotation.

It has also been discounted by Kimura etal. (1977) because it would predict less stability with the

availability of CH

groups, contrary to experimental evidence on spin-echo data. Freeman and his

4.0 6.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

= 0.00519 molecules Å

–3

g (r)

r (Å)

Figure 9.5 Intermolecular pair (density) distribution in liquid isopentane, normalized to the average

number density. g(r) was calculated from the structure factor, S(k), in Stephens (1986) using the analysis of

Narten (1979).

table 9.1

Connectivity

(K) and n

earest-neighbor

eparation

) in hydrocarbon liquids

molecule/T (k) K r

(Å)

(W/2V)

≡ σ

Ethane (105) 8.55 5.5 17.37

Propane

(300) 7.1 6.0 12.95

n-Pentane (300) 5.0 5.0 6.86

Isopentane (300) 5.1 6.22 7.14

Neopentane (256) 7.6 7.4 14.56

Neopentane (298) 7.7 7.7 14.83

Neopentane

(423) 6.1 7.9 9.93

Electron Localization and Trapping in Hydrocarbon Liquids 221

associates (Dodelet and Freeman, 1977; Gyorgy and Freeman, 1979) have given experimental evi-

dence of the involvement, among many other things, of the anisotropy of molecular polarizability.

This has been supported theoretically by Funabashi (1974). The anisotropy of molecular polarizability

can be measured by depolarized Rayleigh scattering, or by the Kerr effect (Gelbart, 1974), of which

the rst gives the difference between the parallel and perpendicular components (α

∥

− α

⊥

), while the

latter can, in principle, give all three components. For the diagonal energy uctuation, one needs only

the difference between the maximum and minimum polarizability components (vide infra), as the

two perpendicular components are equal for alkanes. For methane, neopentane, tetramethylsilane,

etc., all three components are equal giving no anisotropy. Consequently, one does not expect much

localization in these liquids. Table 9.2 gives the values of the mean polarizability and the difference

between the maximum and minimum components in the n-alkane series, along with certain other

parameters explained later. Both of these increase with the carbon number, the rst because of the

size effect and the second indicating that the molecule is increasingly becoming “cigar like.” Cases

of isopentane and neopentane have been included for intercomparison within the isomeric series.

The polarizability components of isopentane are not readily available in the literature. These were

obtained from the mean polarizability

α α α= +









⊥

1 3 2/ ( )



, usingthe Lorentz–Lorenz equation and

a measured value of the Rayleigh depolarization ratio, Δ = 0.053 (Stuart, 1962). The latter is related to

δ α α α

2 2 2

10 6 7 2 9≡ − = −

⊥

∆ ∆/ ( ) ( / )( ) / [( ) ]



. For isopentane, this yields

α = 10 1

. Å

and ∣α

∥

− α

⊥

∣=

6.57Å

. Therefore, α

∥

= 14.48Å

and α

⊥

= 7.91Å

. The polarizability ellipsoid for isopentane is shown

in Figure 9.6 and compares it with those of n-pentane and neopentane. The progressive sphericity, or

loss of anisotropy, is apparent as one goes from n- to iso- to neopentane. However, notice that it is due

mainly to the contraction of the major axis, rather than by the expansion of the minor axes.

We may now prescribe the width of uctuation of the electron–molecule polarizability interaction

in the liquid as

W Z













α − α

⊥||

(9.14)

table 9.2

polarizability,

iagonal

isorder,

and t

ransfer

nergy

in h

ydrocarbon

iquids

molecule

(Å

) |α

∥

− α

⊥

|(Å

)

W (ev) V

(ev) V (ev)

max

(Å)

% initially delocalized states

Ethane 4.47 1.51 1.20 0.2 0.74 55 97.6

Propane 6.29 1.92 1.38 −0.1 0.18 96 83.9

n-Pentane 9.95 3.13 3.46 0.0 95 0.0

Isopentane 10.68 6.57 1.63 −0.2 0.25 112 73.7

Neopentane 9.8 0.0 0.0 −0.4 100

14.48 Å

7.91 Å

Isopentane

(axes ratio 1.83)

n-Pentane

(axes ratio 2.35)

17.32 Å

7.365 Å

9.8 Å

Neopentane

(axes ratio 1.0)

Figure 9.6 Relative shapes of polarizability spheroids for pentane isomers.

222 Charged Particle and Photon Interactions with Matter

where e is the electronic charge. Using Z = K + 1 from Table 9.1 and ∣α

∥

− α

⊥

∣ from Table

9.2, third column, we calculate W, which is given in the fourth column of Table 9.2. With the

increase of the carbon number, molecular anisotropy increases, but Z decreases up to pentane.

These two effects partially cancel each other, so that one can expect the variation in r

reect in W. Beyond pentane, both Z and r

remain sensibly constant, causing W to be roughly

proportional to |α

∥

− α

⊥

|, that is, to the size of the molecule. For highly symmetric molecules,

such as methane and neopentane, we may use W ∼ 0, due only to the polarizability interaction.

A nonzero contribution may be expected due to electron–microdipole interaction, which is cur-

rently neglected.

At present, our knowledge of the electronic site energy is not sufciently accurate to give a reli-

able estimate of the transfer energy, V. However, Funabashi (1974) hypothesized a value of ∼0.1eV

for a typical hydrocarbon liquid. Model calculations by Hug and Mozumder (2008) for the percent-

age of initially delocalized states as a function of V in the alkane series show progressive delocaliza-

tion with an increase of V for a given carbon number and progressive localization with the carbon

number

for a given V. Both are expected effects (cf. Equation 9.12).

Taking

a cue from the mobility edge trajectory, we can set E

= V in Equation 9.11. Further, for

a given liquid with specic disorder σ ≡ (W/2V), we must have E

= V

, since all states would be

delocalized above this energy. Since σ(E

) = σ, an implicit equation for V from Equation 9.11 is

obtained,

that is,

V V V

c c

1 2 1 2

1 1

= − −

























= − −

























ln ln .

/ /

σ σ

(9.15)

Given σ

and W from Tables 9.1 and 9.2, respectively, V can be calculated from Equation 9.15

when V

is known experimentally (Allen, 1976; Nishikawa, 1991). Some values of V obtained

in this manner are shown in Table 9.2, but no such V could be obtained beyond n-butane, since

in these liquids already σ > σ

, implying that all states are initially localized in the Anderson

sense. If we formally apply Equation 9.15 when W is very small (e.g., in methane), we would

obtain

VW V

= =

2 4 56 eV

. ( )

, by inserting appropriate values for methane from Table 9.1 and

Nishikawa (1991). This means V >> 4.56eV since W is small. Therefore, localized states would not

result

in such cases.

The

fraction of initially delocalized (doorway) states at low energy (near thermal or epithermal

energy) may now be computed with values of σ and σ

from Tables 9.2 and 9.1, respectively, using

Equation 9.12. The fraction of initially delocalized states falls precipitously from propane, due to

both the increase of σ and the decrease of σ

. It is zero from n-pentane onward. In the case of iso-

meric pentanes, the effect of molecular symmetry is apparent, contributed both by the isotropicity

of polarizability and the connectivity. For ethane and propane, the initial states are highly delocal-

ized; only trapping and thermal equilibrium produce eventual localization, as is evident from stud-

ies

of mobility and its activation energy.

the three pentane isomers, isopentane occupies an intermediate position in respect of polar-

izability, V

, and electron mobility. Its computed connectivity (K) is close to that of n-pentane,

nearest-neighbor distance (r

) is intermediate, while the percentage of initially delocalized states is

to that of neopentane (Mozumder and Hug, 2009).

Disorder-induced

Anderson localization gives electron wave functions with random amplitude

and phase, having an exponentially decaying envelope, ∼ exp(−r/ξ) (vide supra). Denoting the maxi-

mum and minimum energies of the localized electron by E

max

and E

min

, respectively, equating E

with

(vide supra), and employing Equation 9.9, we get

Electron Localization and Trapping in Hydrocarbon Liquids 223

max

min

max

min

( )

−













−

V E

(9.16)

where, according to Chang etal. (1990), ν = 1. This, however, is not well established. Further,

Equation 9.16 is strictly not applicable to equilibrated trapped electrons, since trapping involves

lattice relaxation, which is not considered in the Anderson localization. Nevertheless, Hug and

Mozumder (2008) obtained some idea of the spatial extent of the wave function envelope by set-

ting V

− E

min

= ΔE

act

, the activation energy of mobility. For V

− E

max

, an intermolecular vibra-

tional quantum, hν, of 0.01 eV is used following Mozumder and Magee (1967) and Rassolov and

Mozumder (2001). Any higher value will make the electron delocalized (ξ → ∞). Thus, roughly,

max

/ξ

min

= ΔE

act

/hν. Values of ξ

min

= r

are taken from Table 9.1 and the corresponding values of ξ

max

may be computed using the experimental activation energies (Allen, 1976; Nishikawa, 1991). The

last two columns of Table 9.2 give ξ

max

and the percentage of initially delocalized states. Another

estimate of the extent of ground-state wave function is obtainable from the eld dependence of

mobility, using the quasi-ballistic theory of Mozumder (2006). Generally, these are ∼1.5–2.5 times

larger than the r

values listed. Currently, it can only be surmised that ξ

max

, in most liquid hydro-

carbons, will be bracketed between ∼50 to a couple of hundred Å. The number of molecules, n

interacting with the ground state of the electron may be taken as Z = K + 1, from Table 9.1. Then the

number n

interacting with the quasi-free electron will be given by n

= ξ

max

/ξ

min

= ΔE

act

/hν.

Using the data in Table 9.1, one obtains n

= 130, 114, and 120, respectively, for propane, pentane,

and hexane. These may be compared with an estimate by Berlin and Schiller (1987) in the range

of ∼60–120

by a completely different line of reasoning.

9.3 trapping

9.3.1 general featureS

In the literature, there is considerable confusion as to what constitutes localization and trapping.

Often these terms are used synonymously, including that used in a recent benchmark paper on elec-

tron solvation in ammonia cages (Lee etal., 2008) and in another seminal paper on the stability cri-

terion of an excess electron in nonpolar uids (Springett etal., 1968). As we have remarked before,

Anderson localization refers to no medium rearrangement, whereas trapping always involves some

rearrangement of medium molecules, the extent of which may be debated.

Generally, a distinction is made between preexisting and self-dug traps. Electrons in liquid

hydrocarbons and in LRGs may be divided into three classes. The rst class is where the electron

is highly delocalized at all times and where its mobility is large (≥ several hundred cm

−1

)

without any evidence of positive activation energy. In such cases, as in LAr, LKr, LXe, liquid

methane, and TMS, the drift mobility and the Hall mobility are about the same (Munoz, 1991).

The second class is where the electron is initially delocalized, but trapping is evidenced by posi-

tive activation energy, resulting in a signicant difference between drift and Hall mobilities. Such

cases occur in neopentane, isooctane, and possibly in ethane and propane (Hug and Mozumder,

2008). Electron mobility in these liquids may be small or large, but seldom exceeding 100 cm

−1

. The third class is where the electron is initially localized, the electron motion is activated,

and the room temperature mobility is small (≤10 cm

−1

). Such cases occur in the liquid alkane

series starting from butane, and in many other liquids. The Hall signal is generally unobservable in

these liquids because the drift mobility is small. Nevertheless, at least in the two-state theories

of transport, a quasi-free state with a high mobility has been postulated (Holroyd and Schmidt,

1989; Munoz, 1991). Liquids in the rst class are essentially trap-free. It stands to reason that

traps are self-dug in the second class. In the third class, traps may be either preexisting or self-dug,

but probably preexisting in the appropriate cases of liquid hydrocarbons.

224 Charged Particle and Photon Interactions with Matter

9.3.2 ModelS of preexiSting trapS

There are several models of trapping, preexisting and self-dug. Space does not permit us to go into

details. Here, we discuss two cases briey. The theory of Springett etal. (1968) obtains a criterion for

the stability of an electron in a bubble embedded in a nonpolar uid. Without any positive electron

afnity, the existence of the bubble depends on the electron–medium interaction, which is mainly

repulsive. The authors distinguish between three classes: (1) a stable quasi-free state, (2) a metastable

bubble, and (3) an energetically stable bubble, conditioned by E

< V

, where E

is the ground-state

energy of the electron bubble and V

is the lowest energy of the electron in the quasi-free state. After

a complicated set of reasoning, the stability criterion is given as follows. A quasi-free state will be

stable when

F V≡ ≥

4 0 087

πβγ

/ .

, where γ is the liquid surface tension, β = ħ

/2m, in which, ħ is

the reduced Planck’s constant and m is the electron mass. A metastable state exists when 0.047 ≤ F ≤

0.087, and a stable bubble state requires F ≤ 0.047. Examples occur for delocalized quasi-free states in

LAr, LKr, and

LXe with negative V

values. With positive V

, stable bubble states are found in liquid He,

, and D

, while metastable states occur in LNe.

Both the theories of Kestner and Jortner (1973) and of Schiller etal. (1973) relate the observed

electron mobility in a liquid hydrocarbon with its V

value and the allowed fraction C of the

liquid volume. In the allowed region, the electron energy E ≥ V, where V is the local potential.

Transport is forbidden when E < V. However, there are some fundamental differences between

these theories. In the Kestner–Jortner model, the medium is treated as inhomogeneous with

rotational uctuation resulting in high- and low-mobility regions, characterized respectively by

∼ 100 and μ

∼ 0.1 cm

−1

. Making some drastic assumptions, and adjusting the parameters

of the Gaussian, the authors calculate the allowed fraction, C, of volume available for transport.

A critical value, C *, separates open and closed channels for transport, as in percolation models.

For C > 0.4, the mobility is simply given by μ = μ

(3C/2 − 1/2). For C < 0.4, the mobility drops

sharply, ultimately reaching μ

when C ≈ 0. In the application of this model to liquid hydrocar-

bons, Kestner and Jortner make parametric adjustments to obtain the best agreement with experi-

ments. The resultant mobility, μ, as a function of the quasi-free energy, V

, shows a precipitous

fall around V

= −0.2 eV. This model is partially successful in explaining the variation of electron

mobility in liquid hydrocarbons with the quasi-free energy, but a serious objection has been

raised with respect to the Hall mobility (Munoz, 1991).

In the model of Schiller etal. (1973), the medium is treated as homogeneous. Electron energy,

, in the bubble is the sum of the quantum mechanical electron energy, E

, a surface bubble

energy, and a pressure–volume work. Together, they are the medium reorganization energy, of

which the pressure–volume work is neglected being too small, in the same manner as in the

Kestner–Jortner theory. To these, a term −(e

/R)(ε − 1)/2ε is added representing the reversible

work needed to bring the electron from vacuum into the bubble. This is estimated to be ∼−0.80 eV.

The localization (actually trapping) criterion is taken as E

< V

. The probability p(E)dE that the

electron in the bubble would have an energy between E and E + dE is assumed to be Gaussian,

given by

p E

E E

( ) ( ) exp

( )

= −

−













−

2 1 2

πσ

The

total trapping probability is now expressed as follows:

P p E dE

−∞

∫

( ) .

Electron Localization and Trapping in Hydrocarbon Liquids 225

The observed mobility is written as μ = μ

(1 − P), where μ

is the quasi-free mobility. The

Gaussian parameter, σ, of the probability distribution is not adjustable, but is given for a canoni-

cal ensemble with constant bubble radius by σ

= k

, where k

is the Boltzmann constant,

T is the absolute temperature, and C

is the specic heat at constant volume. At T = 300 K, the

authors estimate σ= 0.12 eV, a constant value in all liquids, which does not appear to be very

likely. Comparison with experiment requires the parametric values of μ

and the total energy E

of the electron in the bubble. The former is taken to be equal to the measured mobility in TMS,

that is, 90cm

−1

, and the latter is adjusted to −0.26 eV to be in agreement with the measured

mobility in n-hexane. Therefore, in this model, the only other non-adjusted quantity is the quasi-

free energy. Taking V

from experiment gives only a moderate agreement with mobility measure-

ments. Notice that the factor 1 − P is not quite comparable to the allowed volume fraction, C, of

the Kestner–Jortner model (1973). Even in the high-mobility regime, the electron mobility is not

given simply by the product of C and the quasi-free mobility in the Kestner–Jortner model, but in

the model of Schiller etal. (1973) the mobility is always given by μ

(1 − P) (vide supra). Among

other reasons, the use of the percolation model and an effective medium theory are responsible

for the difference.

9.3.3 Structure of trapS in nonpolar Media

Davis etal. (1973) proposed a possible model for the electron trap in nonpolar hydrocarbons. It is

assumed that the electron is trapped in a cavity (preexisting) and the interaction potential arises

from both the electron-bond dipoles of the molecule and the electron-induced dipole (polarizability)

the entire molecule as follows:

V r

N e

ND e

r R

V r V r R

( ) ,

−

− <

= >

4 2

for

(9.17)

where,

is the coordination number of the electron

is the mean polarizability of the molecule (spherically averaged)

is the separation of the center of the cavity from the center of the nearest-neighbor molecules

= r

− ã, in which ã is the rationalized molecular radius

The quantity D

represents the total dipole moment of the molecule, arising mainly from CH

groups

that do not add up to zero because only a part of the molecule is involved. The potential of Equation

9.17 is of the square-well type in three dimensions, for which the binding energy, the difference

between the ground-state energy, and V

are rationalized with the activation energy of electron

mobility in n-hexane (4.06kcal mol

−1

) for N = 6 and

α = 11 8

. Å

(the value for n-hexane). This

requires D

to be 0.78 Debye, for which the potential is V(r) = −2.4eV (r < R). Considering the bond

dipole

moment of CH to be 0.4 Debye, the authors considered the model to be plausible.

Davis

etal. (1973) criticized Dodelet and Freeman’s (1972, 1977) conjecture relating the decrease

of electron mobility with the anisotropy of molecular polarizability. According to these authors, the cor-

relation is weak and uncertain. It has recently been shown by Hug and Mozumder (2008) that the

anisotropy of molecular polarizability, together with lattice connectivity, is central for determining

the initial electron localization. This, however, does not preclude the nal trapping being due to the

isotropic part of the polarizability with or without the interaction of the electron and bond dipoles.

On the other hand, the suggestion of Davis etal. (1973) that only the interaction of the electron with

the CH

group of the molecule is important should not be taken literally, as it would tend to produce

226 Charged Particle and Photon Interactions with Matter

similar activation energy in very different liquid hydrocarbons, while ignoring other correlations.

Also, the coordination number of six is not always supported by x-ray determination (Hug and

Mozumder,

2008).

The

measurement of the Hall mobility in a hydrocarbon liquid can give some useful information

about electron trapping. In the presence of mutually perpendicular electric (E

⃗

) and magnetic (B

⃗

)

elds,

the average electron velocity



is given by



  

ν µ µµ= + ×E E B

(9.18)

where

is the ordinary drift mobility

is the so-dened Hall mobility

Equation 9.18 derives from the Lorentz force on a (nearly) free electron. It, therefore, only applies to

the quasi-free electrons. The Hall mobility should approach the drift mobility in liquids where the

transport is mostly in the quasi-free state, although some difference is expected anyhow, because

these mobilities represent different moments of the momentum relaxation time, τ (Holroyd and

Schmidt,

1989; Munoz, 1991).

is clear from Equation 9.18 that the Hall signal is proportional to the drift mobility. Therefore, as

yet, signicant measurements are limited to liquids in which the drift mobility is greater than ∼10cm

−1

. In the case of TMS, the two mobilities are virtually the same over the entire temperature range

studied (Holroyd and Schmidt, 1989). This liquid is then considered to be trap-free for electrons. In

neopentane, the Hall mobility is about the same as the drift mobility up to ∼140°C, but systematically

higher. Since there is a small activation energy of mobility in this liquid, it could be interpreted in the

two-state model as due to a small population of electrons residing in shallow traps. Beyond 140°C,

the Hall ratio in neopentane rises signicantly above unity, being ∼5 at the critical temperature. This

has been attributed to localization due to density uctuation (Holroyd and Schmidt, 1989).

9.3.4 tiMeScale of trapping and trap Size

Since trapping requires some medium rearrangement (vide supra), it is evident that trapping from

either a localized or delocalized state would need a nite time. Closely associated with it is the ques-

tion of the trap size. At present, neither of these is available from fundamental theory, but consider-

able information is available from experiments and from phenomenological theory. Due to space

limitation,

we will not elaborate on this aspect.

9.4 mobility

9.4.1 baSic conSiderationS

It should be remembered that the primary reason for the investigation of electron localization and

trapping is their relationships with mobility. Certain other considerations related to electron mobility,

such as its variation with density, pressure, and external electric and magnetic elds (Hall effect),

are

clearly outside the purview of this chapter.

When

excess electrons are generated in an isotropic liquid in the presence of an external electric

eld, the steady-state electron velocity (v) will be parallel to the eld (E) and, for not too high an

external eld, will be proportional to it. The constant of proportionality, μ = v/E, is dened to be drift

mobility. The importance of mobility derives from the fact that the electron is a very sensitive probe

of interaction with molecules and also with the liquid structure. Therefore, it can vary greatly from

one liquid to another within the same class of hydrocarbons having very similar densities. Mobility

Electron Localization and Trapping in Hydrocarbon Liquids 227

study can be utilized to extract information about the state of the electron in the liquid (Holroyd and

Schmidt, 1989; Munoz, 1991). When the mobility is low and increases with temperature, the activa-

tion

energy, ΔE

act

, may be dened by the relation μ(T) = A exp(−ΔE

act

T), where μ(T) is the elec-

tron mobility at absolute temperature T, A is a certain constant, and k

is the Boltzmann constant.

Usually, the activation energy is extracted by measuring the mobility over a range of temperature

of ∼30°C or so. Different methods for measuring electron mobility in hydrocarbon liquids have been

discussed

by Munoz (1991) with their relative advantages and disadvantages.

Extensive

tabulations exist for electron mobilities and their activation energies in many liquid

hydrocarbons, as well as in other liquids including LRGs (Allen, 1976; Munoz, 1991; Nishikawa,

1991). From these, it is apparent that electron mobility in liquid hydrocarbons can vary by three

orders of magnitude or more under similar conditions of temperature and density (Allen, 1976;

Nishikawa, 1991). Activation energy can also vary, but not so much; typical values range from ∼5 to

20kJ mol

−1

(Nishikawa, 1991). Typical mobility values in straight-chain hydrocarbons are ∼0.1−0.2cm

−1

at room temperature, with activation energies ∼15–20kJ mol

−1

. Highly branched hydrocar-

bons exhibit much higher mobilities, often with negative activation energies (i.e., mobility decreasing

with temperature). For example, electron mobility in neopentane and in TMS at room temperature

has been reported to be 70 and 100cm

−1

, respectively (Nishikawa, 1991). Some intermediate

cases are also known. Generally speaking, smaller mobility and higher activation energy are indica-

tive of considerable electron trapping, while the reverse is true for electrons existing mainly in the

quasi-free state. In either situation, some electrons may be in the trapped state while the remaining

electrons

are in the quasi-free state, with an established equilibrium between them.

9.4.2 eMpirical relationShipS

Although the experimentally measured mobility in a hydrocarbon liquid decreases with V

, a satis-

factory theoretical explanation is hard to come by. The inhomogeneous percolation theory (Kestner

and Jortner, 1973) and the homogeneous uctuation theory (Schiller etal., 1973; Berlin and Schiller,

1987) (vide supra, Section 9.3.2) aim to do just that, but the quantitative agreement is not good,

except

in the high- or low-mobility regimes (Holroyd and Schmidt, 1989).

Notwithstanding

the theoretical situation, an empirical relationship between V

and mobility has

been

advanced by Wada etal. (1977) as follows:

µ =

125

1 360 15

exp( )

(9.19)

In Equation 9.19, the mobility, μ, is expressed in cm

−1

and the quasi-free electron energy,

, in eV. This equation encompasses a surprisingly large amount of experimental data, although,

understandably, not all the data are equally well tted by it. As yet, no theoretical derivation has

been

proposed for this equation.

There

is yet another correlation between electron mobility and free-ion yield, G



, proposed by

Jay-Gerin etal. (1993). The free-ion yield is the number of electrons escaping geminate recombina-

tion and homogeneously collected by a vanishingly small electric eld, normalized to 100eV of the

deposited energy. Experimentally, it has been known for a long time that the free-ion yield usually

increases with electron mobility. In the theory of Onsager (1938), the probability of escaping gemi-

nate recombination depends on the initial separation, interpreted to be the thermalization distance.

In this theory, it is independent of the electron’s diffusion coefcient, and therefore of the mobility.

The experimental observation then must be rationalized on the dependence of the thermalization

distance with the mobility. In any case, the importance of the nding of Jay-Gerin etal. lies in

establishing G



∝ μ

0.31

(albeit approximately, and with some deviations), when the mobility ranges

from ∼0.1 to ∼100cm

−1

, but the same remains essentially constant at ∼0.1 for very low electron

228 Charged Particle and Photon Interactions with Matter

mobility (<0.1cm

−1

). This total correlation cannot be explained by the commonly accepted

theories of electron transport, namely, the hopping model or the two-state model. The quasi-ballistic

model of electron transport, advanced by Mozumder (1993, 1995, 2002), can give a reasonable

explanation of the phenomena in terms of the elastic and inelastic mean free paths of epithermal

electrons. These determine both the thermalization distance distribution and the thermal quasi-free

electron mobility. The former gives the free-ion yield, and the latter can be converted to an effec-

tive

mobility using the electron-trap concentration and the binding energy. Thus, a relationship can

beobtained between free-ion yield and mobility. When the electron mobility is very low, transport

is governed by random trapping and detrapping only, being independent of the quasi-free mobility.

Then

the mobility and the free-ion yield become essentially decoupled.

Over

many years, Freeman and his associates have strived to correlate the variation of electron

mobility with some intramolecular property of the liquid (Freeman, 1963a,b, 1986; Tewari and

Freeman, 1968a,b; Robinson etal., 1971; Dodelet and Freeman, 1972, 1977; Fuochi and Freeman,

1972; Dodelet etal., 1973, 1975, 1976; Freeman and Huang, 1979; Gyorgy and Freeman, 1979; Gee

and Freeman, 1983, 1987; Gee etal., 1988). These correlations are suggestive, however, without pro-

viding a theoretical basis. The early investigations (Freeman, 1963a,b; Tewari and Freeman, 1968a,b)

correlated the conductance (current overshoot) with molecular sphericity. The more spherical a mol-

ecule is, the greater would be the free-ion yield. The free-ion yield of LMe was measured to be 0.8,

which is much greater than that of other hydrocarbons (Robinson etal., 1971). It was inferred that

electrons penetrate further in LMe, and this was attributed to its sphere-like structure. The mobility

in LMe was measured to be 300cm

−1

; it is much less in other hydrocarbons with decreasing

sphericity (Fuochi and Freeman, 1972). A similar correlation was found in both mobility and free-ion

yield in C

− C

alkane liquids (Dodelet and Freeman, 1972, 1977). The presence of a small molecular

dipole moment (≤0.5 Debye) did not seem to affect electron mobility (Dodelet etal., 1973).

Dodelet etal. (1975) sought and found a correlation between the experimental activation energy of

mobility and V

in various hydrocarbons and in TMS (Dodelet etal., 1975). Generally, the activation

energy increased with V

, negating a theoretical expectation by Kestner and Jortner (1973), based on

percolation theory, that predicted a maximum at V

∼ −0.15eV and a minimum at V

∼ −0.27eV. The

correlation between molecular structure on one hand, and mobility and electron range on the other

hand, was sought in various liquid hydrocarbons by Dodelet etal. (1976). The penetration range, b

was found from the free-ion yield by tting to a modied Gaussian distribution with a power tail. The

results were interpreted by suggesting that less rigid molecules would allow longer penetration range,

thus partially modifying the earlier correlation with sphericity. It was further suggested that the epi-

thermal electron interaction with a molecule is essentially conned to two C–C bonds in series. The

authors analyze their ndings in terms of a model, which depends, in part, on Schiller etal.’s theory

(1973), which in turn is based on the equilibrium energy uctuation in the liquid (vide supra).

A particularly important study is the intercomparison of electron mobilities in liquid pentane

isomers: normal pentane, isopentane (2-methyl butane), and neopentane (Gyorgy and Freeman,

1979). The mobilities in the liquid phase increase by orders of magnitude going from less to more

spherical molecules. The values (in cm

−1

) range from ∼0.1 in n-pentane, to ∼1.0 in isopen-

tane, to ∼50–70 in neopentane. These values clearly indicate the effect of the molecular shape. The

authors offer alternative reasons: (1) the Ramsauer–Townsend effect, implying minimum scattering

cross section occurring at lower velocity for less spherical molecules, and (2) a low-lying transient

negative-ion state. None of these could be veried, especially because a Ramsauer–Townsend mini-

mum has never been found in the liquid state. However, the authors succeeded in demonstrating that

electron transport, in all these cases, is activated and that the process is similar in n-pentane and

isopentane,

but rather different in neopentane.

similar correlation between electron mobility and molecular shape has also been seen by

Bakale and Schmidt (1973), Holroyd and Cipollini (1978), and Schmidt and Allen (1970). However,

Stephens (1986) pointed out that the mobility in isomeric pentanes reverses its order in the gas phase

Electron Localization and Trapping in Hydrocarbon Liquids 229

and that the density-normalized mobility curves cross over near the critical density. Therefore,

intramolecular symmetry, or some other property, cannot be the sole determinant. To our knowl-

edge, he rst suggested the involvement of the liquid structure factor, although it is well known for

electron mobility in LRGs. Stephens (1986) stressed the peak of the liquid structure factor as deter-

mined by low-angle x-ray scattering. Freeman (1986) acknowledged this as a useful procedure, but

correctly pointed out that detailed correlation depended, in part, on the differences in the structure

factors that are likely to be within experimental uncertainty. In a recent article, we have shown

that the entire liquid structure needs to be taken into account, not just the peak position (Hug and

Mozumder, 2008). Actually, this only gives the initial localization in the Anderson sense. Further

considerations

are needed to connect trapping with localization.

Several

other correlations have been proposed to connect molecular shape with mobility, such

as the anisotropy of molecular polarizability (Dodelet and Freeman, 1972, 1977; Funabashi, 1974;

Gyorgy and Freeman, 1979), which is an important factor. Together with connectivity, it determines

the extent of Anderson localization probability (Hug and Mozumder, 2008). However, it is the abso-

lute difference between the maximum and minimum values of the polarizability that determines the

width of diagonal energy uctuation, which is important, rather than the mean polarizability, as was

earlier

thought to provide for the trapping potential.

the face of many correlations and conjectures regarding free-ion yield, mobility, molecular

shape, etc., a few things stand out. These are the anisotropy of molecular polarizability, the liquid

structure, and the scattering of low-energy electrons. At a deeper level, these may be interrelated,

but, as yet, not much is known about this.

9.4.3 theory

Since electron mobility in the condensed phase varies over several orders of magnitude, and since

it depends sensitively on the interaction of the electron with both the molecule and the medium

structure, it is not reasonable to expect that all experimental results will be explained by the same

theoretical model. We shall now focus on two theoretical models specic to liquid hydrocarbons.

These

are the hopping model and the two-state models.

9.4.3.1

hopping

odels

In this model, the electron is assumed to exist in a preexisting trap. Transport occurs by trap-

to-trap hopping, sometimes with phonon assistance (Holstein, 1959; Funabashi and Rao, 1976;

Schmidt, 1977). Random hopping generates a Markov process with the resultant mobility given by

the

Nernst–Einstein equation as follows:

µ = eL w k T

/ ,

(9.20)

where

is the mean distance between nearest-neighbor traps

is the corresponding mean transition frequency

Classically,

this transition probability at low electric elds is given by w

= v exp(−ε

T), where

v is the attempt frequency and ε

is the energy barrier to hopping. A quantum mechanical version

of the transition probability is also available (Funabashi, 1974), which however is not applicable

to hydrocarbon liquids. Schmidt (1977) has reviewed the mobility in several liquid hydrocarbons

where electron mobility is low (<1cm

−1

). By tting into Equation 9.20, over the temperature

interval of 100–300K, it was found that for ethane, propane, butane, and cyclopentane, good agree-

ment with experiments could be found for reasonable energy barriers (∼0.1–0.16eV) and also for

reasonable attempt frequencies (1.4 × 10

–2.0 × 10

−1

). However, the resultant jump lengths were