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

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10 Charged Particle and Photon Interactions with Matter

focus on vacuum ultraviolet light is that the maximum value of the photoabsorption cross section is

usually

given in this range.

proceed by using a semiclassical model in which light is treated classically while the atom or

molecule is described by quantum mechanics. However, we often use the term “photon” for conve-

nience. The semiclassical model is valid when the light intensity is so high that the number of photons

in a single mode can be treated as a continuous variable. Consequently, the absorption and stimulated

emission of light are usually described well by the model, while the spontaneous emission of light is

not, because only one photon is concerned. The electric dipole approximation (Bransden and Joachain,

2003: 183–236) is valid in the present range of incident light wavelength. The rate of transition from

state |i> with energy E

to state |k> with energy E

by the photoabsorption of light of angular frequency

in the electric dipole approximation, W

, is expressed by (Bransden and Joachain, 2003: 183–236)

( )

⋅

πε



ki ki

( )ω

(2.1)

r r

ki s

k i=

∑

(2.2)

k i

E E

−( )



, (2.3)

where

is the velocity of light in vacuum

Planck’s constant divided by 2π

e is the elementary charge

is the permittivity of free space

I(ω

) is the energy ux density per unit range of the angular frequency of light

the unit polarization vector

is the position vector of the electron in an atom or molecule

ket-vector |j> is normalized as

j j

′′ ′

″ ′

= δ ,

(2.4)

where

′′

′j j

is Kronecker’s delta. Equation 2.1 is derived for the case in which the absorption line is

so narrow that only the light of the angular frequency ω

is absorbed. It is convenient to introduce

a dimensionless quantity, f

, called the oscillator strength for the transition from state |i> to state

|k>.

It is dened by

m E E

k i

−

∑

( )



(2.5)

where

m is the mass of the electron. Equation 2.5 is rewritten in a more useful form as

E E

k i

−

∑

( )

(2.6)

where

is the Rydberg energy

is the Bohr radius

Oscillator Strength Distribution of Molecules in the Gas Phase 11

We can readily understand from Equation 2.6 that f

is dimensionless. In fact the initial energy

level, E

, and the nal energy level, E

, are often degenerate, and we dene the oscillator strength

for the transition from energy level E

to energy level E

by averaging the right-hand side of Equation

2.6

for initial states |i, a> and summing for the nal states |k, b> as

E E

R g

k b i a

k i

−

∑

∑∑

( )

, ,

(2.7)

where

is the degree of degeneracy of energy level E

a and b specify each member of the degenerate states

We assume that the population of each initial state |i, a> is uniform. A ket-vector │j, c> is normal-

ized

j c j c

j j c c

″ ″ ′ ′

″ ′ ″ ′

, , = δ δ . (2.8)

The initial energy level, E

, is usually the ground energy level, E

, and so, let us use f

instead of f

set of E

− E

and f

characterizes a discrete spectrum of photoabsorption.

To discuss a transition from the ground energy level, E

, to continuous energy, E, which is due to

photoionization, we dene the density of the oscillator strength per unit range of energy E, simply

called

oscillator strength distribution, by

d .

E a













∑

∫

∑∑

ξ ς

, , ,

(2.9)

The subscript c in Equation 2.9 is different from the symbol c in Equation 2.8. The nal state |E, ξ, ζ>

is specied by the continuous energy, E, and sets of other quantum numbers, ξ and ζ. Note that ξ

is the set of continuous quantum numbers, for example, the direction of the ionized electron, while

ζ is the set of discrete quantum numbers. The origin of energy is taken at the ground energy level,

, in Equation 2.9, which we follow from now on. The initial state |0, a> is normalized following

Equation

2.8, while the nal state |E, ξ, ζ> is normalized as

′′ ′′ ′′ ′ ′ ′

′′

−

′ ′′

−

′

′′ ′

E E E E, , , ,ξ ς ξ ς δ δ ξ ξ δ

ς ς

( ) ( ) , (2.10)

where δ is the Dirac delta function. The quantity (df/dE)

has the dimension of the reciprocal of energy.

Let us then dene the oscillator strength distribution also for the discrete transitions from the

ground energy level, E

, to {E

}, a set of discrete energy levels other than the ground energy level, as

( )

f E E

k k













= −

∑

δ .

(2.11)

The quantity (df/dE)

has also the dimension of the reciprocal of energy. The oscillator strength

distribution

for all the transitions from the ground energy level, E

, is dened as

c d

























(2.12)

12 Charged Particle and Photon Interactions with Matter

The energy of an atom or molecule, E, ranges from zero to innity. The quantity df/dE has the

dimension of the reciprocal of energy. The oscillator strength distribution for all the transitions,

df/dE, which is a function of E, is often called the oscillator strength distribution for photoabsorp-

tion. It represents the magnitude of the interaction of a photon of energy E with an atom or molecule.

The

most remarkable character of the oscillator strength distribution for photoabsorption is

E Z













∫

, (2.13)

where Z is the number of electrons in the atom or molecule of interest. Equation 2.13 is well known

the Thomas–Kuhn–Reiche sum rule.

supposed from Equation 2.1, the cross section, σ, for the absorption of a photon of energy E,

that is, the photoabsorption cross section, is proportional to the oscillator strength distribution for

photoabsorption,

df/dE, dened as

σ π α( )

,E a

E R

= 4

( / )

(2.14)

where α is the ne structure constant. Note that the photon energy is equivalent to the energy of an

atom

or molecule of interest. Equation 2.14 is more conveniently written as

σ( )

= ×













−

1 098 10

. (2.15)

where

σ(E) is expressed in cm

and df/dE in eV

−1

The

partial oscillator strength distribution of a channel q, (df/dE)

, is also dened as

( )

























η (2.16)

where η

(E) is the branching ratio of the channel q following the excitation of an atom or molecule by

the absorption of a photon of energy E within the electric dipole approximation. Accordingly, the partial

cross section of channel q originating from the absorption of a photon of energy E, σ

(E), is dened by

σ η σ

q q

E E E( ) ( ) ( ).= (2.17)

Equation 2.14 holds for σ

(E) and (df/dE)

, and thus the oscillator strength distribution, df/dE

or(df/dE)

, has a corresponding cross section, σ(E) or σ

(E), respectively. The summations of

(df/dE)

and σ

(E) over all channels give (df/dE) and σ(E), respectively:













∑

(2.18)

σ σ( ) ( ),E E

∑

(2.19)

since

( ) = 1

∑

Oscillator Strength Distribution of Molecules in the Gas Phase 13

In this chapter, we discuss the formation and decay of singly inner-valence excited and mul-

tiply excited states as superexcited states produced in the interaction of light in the vacuum

ultraviolet range with an isolated molecule, that is, a molecule in the gas phase. Their dynamics

is probed by the partial oscillator strength distribution dened by Equation 2.16 or the partial

cross section dened by Equation 2.17, as described in detail in Section 2.3. The oscillator

strength distribution for photoabsorption, that is, total oscillator strength distribution, has pre-

viously been discussed by Kouchi and Hatano (Kouchi and Hatano, 2004: 105–120), and thus

this chapter is a continuation of it. In Section 2.2, an overview of the superexcited states of

molecules is given.

2.2 superexCited states oF moleCules

Superexcited states of atoms and molecules are electronically excited states of energies higher

than the rst ionization potentials (Hatano, 1999: 109). They are hence degenerate with ionization

continua or, in other words, embedded in ionization continua. The concept of superexcited states

was rst introduced by Platzman to understand the primary interaction of ionizing radiation with

matter (Platzman, 1962a: 372; Platzman, 1962b: 419). There are two distinct kinds of superexcited

states depending on how they have energy higher than the rst ionization potential (Nakamura,

1991: 123). In Figure 2.1, we take a diatomic molecule AB. The solid curves are potential energy

curves of neutral electronic states of AB, and the curves with shadows represent the continua of

potential energy curves of AB

+ e

−

, where e

−

is a free electron and, thus, it has continuous energy.

As seen in Figure 2.1a, the potential energy curve, that is, the solid line, is embedded in the

(b)

(a)

Internuclear distance

EnergyEnergy

Internuclear distance

–

AB**

Superexcited state of the first kind

Superexcited state of

the second kind

Figure 2.1 Potential energy curves of the diatomic molecule AB for the superexcited states of the rst

(a) and second (b) kinds (solid curves) and those of AB

+ e

−

(solid curves with shadows). Horizontal lines

indicate the vibrational–rotational levels, and the downward arrows connecting them indicate the autoion-

ization. The length of the arrow shows the kinetic energy of the emitted electron with respect to the nuclei-

xed frame. (Based on Figure 19 of Nakamura, H., Int. Rev. Phys. Chem., 10, 123, 1991. With permission.)

14 Charged Particle and Photon Interactions with Matter

electronic continuum. Such states are called superexcited states of the rst kind. Their electronic

energy exceeds that of AB

+ e

−

with zero kinetic energy of e

−

at least in a certain range of the

internuclear distance. On the other hand, the potential energy curves, that is, the solid lines, in

Figure 2.1b are below the electronic continuum. Hence, some readers may think that such states

are not superexcited states. However, the vibrational−rotational level can be higher than the zero

point energy associated with the potential energy of AB

+ e

−

with zero kinetic energy of e

−

, that

is, the lower edge of the continuum. Such states are called superexcited states of the second kind.

Their energies exceed the lowest energy of AB

+ e

−

by vibrational or rotational excitation, and

thus they are embedded in the ionization continuum. The superexcited state of the rst kind is a

superexcited state when we take only the degree of freedom of electronic motion into account.

The superexcited state of the second kind, on the other hand, is a superexcited state when we take

account of the degree of freedom of nuclear motion as well as that of electronic motion.

As shown in Figure 2.1, the superexcited molecule can spontaneously ionize since it degener-

ates with the ionization continuum. This process is called autoionization, which is indicated by the

downward arrow connecting upper and lower vibrational–rotational levels. The length of the arrow

shows the kinetic energy of the emitted electron with respect to the nuclei-xed frame. The auto-

ionization of superexcited states of the rst kind is attributed to the breakdown of the independent

electron model, that is, the electron correlation. In other words, a mixing of discrete and continuous

energy levels of the electronic system causes the autoionization. Such autoionization is called elec-

tronic autoionization. The superexcited state of the rst kind is described by local complex potential

energy,

U(R) (Nakamura, 1991: 123):

U R W R

d d

( ) ( ) ( ),= −

Γ (2.20)

W R d H d

d el

( ) ,= (2.21)

Γ π

R d( ) 2=

∑

cont H

cont

, (2.22)

where

|d> is the electronic state of the superexcited AB molecule

|cont>

is the continuous electronic state of AB

+ e

−

is the electronic Hamiltonian dened at internuclear distance R in the framework of the

Born–Oppenheimer

approximation

The

summation in Equation 2.22 is carried out for continuous electronic states, |cont>, that are

degenerate with the superexcited state, |d>, at R. Both W

(R) and Γ

(R) have the dimension of energy.

The real part of U(R), W

(R), is the potential energy of the superexcited state |d> and shown by the

solid curve in Figure 2.1a. The imaginary part, −(1/2)Γ

(R), expresses decay of the superexcited

molecule in the |d> state through autoionization. The lifetime of the autoionization from the super-

excited

state |d>, τ

(R), is given by

( ) .=



Γ ( )

(2.23)

Oscillator Strength Distribution of Molecules in the Gas Phase 15

The quantity Γ

(R) is hence referred to as the autoionization state width or resonance width of

the superexcited state |d>. It is difcult to calculate the autoionization state width as a function of

internuclear distance except for the lower-lying doubly excited states of hydrogen molecules. Figure

2.2 shows their results. The state width of 100 meV gives a lifetime of 6.6 × 10

−15

s according to

Equation

2.23.

The

autoionization mechanism of the superexcited state of the second kind is much different

from that of the superexcited state of the rst kind. It originates from the breakdown of the Born–

Oppenheimer approximation. In other words, energy transfer from the degree of freedom of nuclear

motion, that is, vibration and rotation, to that of electronic motion brings about this autoionization.

This kind of autoionization is thus called vibrational or rotational autoionization. The state widths

for vibrational autoionization in H

are shown as a function of a vibrational quantum number, v

, of

the superexcited (1sσ

)(npσ

)

states in Figure 2.3. The state width of 10cm

−1

gives the lifetime

of 5.3 × 10

−13

s according to Equation 2.23. Vibrational autoionization is, in general, much slower

than electronic autoionization, as seen in Figures 2.2 and 2.3, which is reasonable since vibrational

autoionization requires energy transfer from the degree of freedom of the vibration to that of the

electronic motion, while electronic autoionization only requires energy transfer within the elec-

tronic system. The typical value of the state width for rotational autoionization in H

is 2.3 cm

−1

which gives a lifetime of 2.3 × 10

−12

s (Lefebvre-Brion and Field, 2004: 578). Rotational autoioniza-

tion is, in general, slower than vibrational autoionization, which is reasonable since the period of

rotation

is much longer than the period of vibration.

Superexcited

molecules decay through not only autoionization but also neutral dissociation.

In other words, neutral dissociation competes with autoionization. The lifetime of neutral dis-

sociation is considered to be in the same order of magnitude of the vibrational period, which

is typically ≈10

−14

s. Hence, we can easily understand that the competition takes place based

on the estimation of the autoionization lifetimes mentioned above. The most remarkable fea-

ture of the dynamics of superexcited molecules is this competition. In general, the lifetime

of the emission of uorescence is longer than nanoseconds, and thus it competes with neither

0.30.20.10

(1)

Internuclear distance (nm)

Franck–Condon region

Autoionization state width (meV)

Figure 2.2 The autoionization state widths of the doubly excited states of H

as a function of internuclear dis-

tance calculated by Martín’s group. (From Sánchez, I. and Martín, F., J. Chem. Phys., 106, 7720, 1997; Sánchez,I.

and Martín, F., J. Chem. Phys., 110, 6702, 1999; Fernández, J. and Martín, F., J. Phys. B, 34, 4141, 2001.)

16 Charged Particle and Photon Interactions with Matter

autoionization nor neutral dissociation. The formation and decay of a superexcited molecule is

schematically shown below (Kouchi and Hatano, 2004: 105):

AB AB e direct ionization+ → +

+ −

hv : (2.24)

** :

→

AB superexcitation (2.25)

→ +

+ −

AB e autoionization:

(2.26)

→ +

A B neutral dissociation: (2.27)

, ,

→

Other minor processes for example ion-pa i r formation (2.28)

The interference effect between direct ionization (Equation 2.24) and autoionization (Equation

2.26) following superexcitation (Equation 2.25) is expected. In fact the effect results in the appear-

ance of the asymmetric peak shape, called the Fano prole or the Beutler−Fano prole, in the

ionization cross-section curves against incident photon energy (Fano, 1961: 1866). We stress that

superexcited molecules play an important role as reaction intermediates in radiation elds, process-

ing plasma, upper atmosphere, and so forth, where electrons, ions, and excited atoms and molecules

exist

(Nakamura, 1991: 123; Hatano, 1999: 109; Ukai and Hatano, 2004: 121–157).

this chapter, we take singly inner-valence excited and multiply excited states as superexcited

states of the rst kind, since they are not amenable to the independent electron model and cannot be

eigenstates of the electronic Hamiltonian, H

, dened at a xed internuclear distance because

of the degeneracy with the electronic continuum, as shown in Figure 2.1a (Nakamura, 1991: 123).

1 2 3

Initial vibrational quantum number, v

Autoionization width, Γ (cm

–1

)

4 5 6 7

Figure 2.3 Vibrational autoionization state widths as a function of the initial vibrational quantum number,

, of H

(1sσ

)(npσ

)

states, the superexcited states of the second kind. The numbers attached to the lines

indicate the principal quantum number, n. (From Nakamura, H., Int. Rev. Phys. Chem., 10, 123, 1991. With

permission.)

Oscillator Strength Distribution of Molecules in the Gas Phase 17

It is a subject of current interest to reveal the formation and decay dynamics of the singly inner-

valence excited and multiply excited states of molecules. The coupling between the superexcited

state of the rst kind, |d>, and the continuous electronic state, |cont>, through H

or the electron

correlation

is a remarkable feature, and is expressed as

V R cont H d

cont d el,

( ) .= (2.29)

cont,d

(R) is called electronic coupling and is related to the autoionization state width, Γ

(R), as

shown

in Equation 2.22.

2.3 observing the singly inner-valenCe exCited

and

ultiply

xCited

tates

The usual way to observe the excited states of a molecule is measuring the photoabsorption cross

sections, that is, the oscillator strength distribution for photoabsorption, df/dE, in Equation 2.12, as

a function of incident photon energy, where they appear as peaks. This is certainly the case in the

energy range below the rst ionization potential of the molecule, for example, as shown in Figure

2.4 for methane (Kameta et al., 2002: 225). The vertical ionization potentials of CH

are indicated

in this gure by short vertical bars, which were measured by means of photoelectron spectros-

copy (Bieri and Åsbrink, 1980: 149). Photoabsorption cross sections well above the rst ionization

potential are dominated by the transition to continuous electronic states of CH

, that is, the direct

ionization in Equation 2.24, and thus direct ionization prevents the superexcited states from being

observed, as seen in Figure 2.4. The cross sections just decrease with the increasing incident photon

energy except for the small undulation in the range of 20–23eV. The key to observing superexcited

states of molecules, in particular, the singly inner-valence excited and multiply excited states, is

measuring cross sections free from ionization as a function of incident photon energy. As shown

in Equations 2.24 through 2.28, neutral dissociation is an ionization-free process. It is more dif-

cult to detect neutral fragments directly than charged species. However, they are often electroni-

cally excited, and thus emit uorescence. Hence, we can observe singly inner-valence excited and

Photon energy (eV)

Cross sections (10

–18

)

10 15 20 25

×10

(1t

)

–1

(2a

)

–1

Figure 2.4 Photoabsorption (σ), photoionization (σ

), and neutral-dissociation (σ

) cross sections of CH

as a function of incident photon energy measured with a wavelength resolution of 0.1nm, which corresponds

to the energy resolution of 32meV at an incident photon energy of 20eV. The relation among σ, σ

, and σ

= σ− σ

. The vertical ionization potentials are indicated by the vertical bars (Bieri and Åsbrink, 1980: 149).

(From

Kameta, K. et al., J. Electron Spectrosc. Relat. Phenom., 123, 225, 2002. With permission.)

18 Charged Particle and Photon Interactions with Matter

multiply excited states in the cross sections for the emission of uorescence from neutral fragments,

that is, the oscillator strength distribution for the emission of uorescence, as a function of incident

photon energy. This sort of oscillator strength distribution is of course partial (see Equation 2.16

as well). There is another way to measure ionization-free cross sections as a function of incident

photon energy: the photoabsorption cross sections and photoionization cross sections are measured

as a function of incident photon energy by means of the double ionization-chamber method or the

fast electron impact dipole-simulation method, and the difference of both cross sections gives the

neutral-dissociation cross sections, as shown in Figure 2.4 (Kouchi and Hatano, 2004: 105–120).

However, this method is not suitable for observing the singly inner-valence excited and multiply

excited states since the difference between photoabsorption cross sections and photoionization cross

sections is smaller in the energy range of superexcited states of the rst kind than in the energy

range

of those of the second kind (see Figure 2.4).

2.4 osCillator strength distribution For the emission

F Fluores

CenCe

rom

eutral

Fragments

In this section, the cross sections or oscillator strength distributions for the emission of uorescence

in the visible and ultraviolet range from neutral fragments in the photoexcitation of CH

(Kato et al.,

2002: 4383), NH

(Kato et al., 2003: 3541), and H

O (Kato et al., 2004: 3127) as a function of inci-

dent photon energy are discussed. All of them have 10 electrons and are different from each other

in terms of symmetry: CH

has T

symmetry in the ground electronic state, NH

has C

symmetry,

and

O has C

symmetry. The ground electronic state of CH

CH :X A (1a ) (2a ) (1t ) ,

1 1



(2.30)

where

the

orbital is an inner-shell orbital

the

orbital is an inner-valence orbital

the

orbital is an outer-valence orbital (Herzberg, 1991: 348)

The

ground electronic state of NH

NH :X A (1a ) (2a ) (1e) (3a ) ,

1 1

2 4



(2.31)

where

the

orbital is an inner-shell orbital

the

orbital is an inner-valence orbital

the

1e and 3a

orbitals are outer-valence orbitals (Herzberg, 1991: 609)

The

ground electronic state of H

O is then

H O:X A (1a ) (2a ) (1b ) (3a ) (1b ) ,

1 1



(2.32)

where

the

orbital is an inner-shell orbital

the

orbital is an inner-valence orbital

the

, 3a

, and 1b

orbitals are outer-valence orbitals (Herzberg, 1991: 585)

An outline of the apparatus is shown in Figure 2.5. The synchrotron radiation monochromatized

by a 3 m normal incidence monochromator (BL-20A, Photon Factory, KEK, Tsukuba, Japan)

Oscillator Strength Distribution of Molecules in the Gas Phase 19

(Ito et al., 1995: 2119) was introduced into the gas cell lled with sample gas. Fluorescence due to

the photoexcitation of the sample molecule was collected with a pair of lenses and focused onto the

entrance slit of the 30 cm monochromator for ultraviolet and visible light. The 30cm monochroma-

tor was set such that the longitudinal direction of the entrance slit was parallel to the incident photon

beam to collect more uorescence photons, and thus uorescence was dispersed vertically. The dis-

persed uorescence was detected by a liquid-nitrogen-cooled and back-illuminated charge-coupled

device (CCD), which consists of 1340 × 400 pixels. The counts read out from the pixels aligning

perpendicular to the dispersion direction were summed, and a wavelength-linear one-dimensional

image covering the range of 280 nm width was obtained, which is a uorescence spectrum. An

example of uorescence spectra is shown in Figure 2.6, which were measured for the photoexcita-

tion of H

O (Kato et al., 2004: 3127). Such uorescence spectra were recorded at a wide range of

the

incident photon energies.

The

pressure in the gas cell was carefully chosen such that the uorescence count is proportional

to the sample gas pressure even in the range of the incident photon energy where photoelectrons

have enough energy to produce uorescence through colliding with sample molecules. Such a sec-

ondary effect was seen for CO (Ehresmann et al., 1997: 1907). The intensities of the uorescence

spectra obtained were normalized for the ux of incident photons, the sample gas pressure, and the

accumulation time, and then the normalized intensities were converted into absolute values of the

uorescence cross sections. Details of this procedure were described by Kato et al. (2002: 4383). In

brief, the spectra of N

v′ → X

v″) uorescence in the photoionization of N

were mea-

sured in the range of the uorescence wavelength of 357–522 nm as a reference to obtain conversion

factors from normalized intensities to cross sections as a function of the uorescence wavelength,

since the cross sections for this uorescence have been known (Samson etal., 1977: 1749; Woodruff

and Marr, 1977: 87). We note that the conversion factors were measured online. These were mea-

sured in the range of the uorescence wavelength of 357–522nm, as mentioned above, so that the

cross sections for the emission of uorescence out of this range were not placed on an absolute scale,

but

on a relative scale.

The

clearly observed uorescences are as follows:

1. CH

a. Balmer-α, -β, -γ, -δ, and -ε

b. CH(A

Δ → X

Π) and CH(B

−

→ X

Π)

Gas cell

Au-mesh

PMT

CCD

Entrance slit

Lenses

Picoammeter

Monochromatized

synchrotron radiation

Picoammeter

30 cm monochromator

Glass window coated with

sodium salicylate

Figure 2.5 The outline of the apparatus used to measure the oscillator strength distributions for the emis-

sion of the uorescence in the visible and ultraviolet range from neutral fragments in the photoexcitation of

molecules.

(From Kato, M. et al., J. Phys. B, 35, 4383, 2002. With permission.)