Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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930 Part E Scattering Experiment

63.1.2 Deﬁnition of Cross Sections

The parameters which characterize collision processes

are the cross sections. Electron collision cross sections

depend on impact energy E

and scattering polar angles

θ and φ. The differential cross section, for a speciﬁc

well-deﬁned excitation process indicated by the index n

is deﬁned as

dσ

,Ω)

dΩ

| f

,Ω)|

, (63.1)

where Ω is the polar angle of detection, k

and k

are the initial and ﬁnal electron momenta, and f

the complex scattering amplitude (n = 0 refers to elas-

tic scattering). Integration over the energy-loss proﬁle

is assumed. If the energy-loss spectrum is broad, dif-

ferentiation with respect to energy loss also has to

be included. For certain processes it may be neces-

sary to deﬁne differential cross sections with respect

to angle and energy for both primary and secondary

particles.

Integration over all scattering angles yields the inte-

gral cross sections

) =

2π



dσ

,Ω)

dΩ

sin θ dθ dφ.

(63.2)

In the case of elastic scattering, the momentum transfer

cross section is deﬁned as

) =

2π



dσ

,Ω)

dΩ

(1 − cos θ) sin θ dθ dφ.

(63.3)

The total electron scattering cross section is obtained by

summing all integral cross sections:

tot

) =



) +



), (63.4)

where σ

are the cross sections for other possible

channels. Experimental cross sections typically rep-

resent averages over indistinguishable processes (e.g.,

magnetic sublevels, hyperﬁne states, rotational states

etc.). The cross section obtained this way corresponds

to an average over initial and sum over ﬁnal indis-

tinguishable states with equal weight given to the

initial states. (This may not always be true, as dis-

cussed later.) If the target molecules are randomly

oriented, the cross section averaged over these ori-

entations is independent of φ. In addition, there is

an averaging over the ﬁnite energy and angular res-

olution of the apparatus. It is important, therefore,

to specify clearly the nature of the measured cross

section, otherwise their utilization and comparison

with other experimental and theoretical cross sections

become meaningless. We denote the conventionally

measured differential and integral cross sections by

,θ) and Q

), with the various averagings

implied. Similarly, Q

) and Q

tot

) are the cor-

responding momentum transfer and total scattering cross

sections.

The collision strength for a process i → j,which

is the particle equivalent of the oscillator strength, is

deﬁned by

Ω

) = q

), (63.5)

where q

is the statistical weight of the initial state [q

(2L

+ 1)(2J

+ 1)], E

is in Rydbergs and σ

is in units

of πa

The rate for a speciﬁc collision process (e.g., excita-

tion) for electrons of energy E

is given as

) = NI(E

)σ

), (63.6)

where N is the density of the target particles



−3



and I(E

) is the electron ﬂux



−2

−1



; σ

is in m

yielding R

in units of m

−3

−1

. For nonmonoenergetic

electron beams, (63.6) must be integrated over E

to get

the average rate.

63.1.3 Scattering Measurements

Most scattering experiments, are carried out in a beam–

beam arrangement (Fig. 63.1). A beam of nearly

Energy

selector

Optics

Gun

Photons

Electrons

Ions

Detector

Signal

Energy analyzer

Optics

Fig. 63.1 A schematic diagram for electron scattering

measurements

Part E 63.1

Electron–Atom and Electron–Molecule Collisions 63.1 Basic Concepts 931

monoenergetic electrons is formed by extracting elec-

trons from a hot ﬁlament and selecting a narrow segment

of the thermal energy distribution. For the formation

and control of the electron beam, electrostatic lenses

are used and the energy selection is achieved with

electrostatic energy selectors. A magnetic ﬁeld may

also be applied to obtain a magnetically collimated

electron beam. The target beam is formed by let-

ting the sample gas effuse from an oriﬁce, tube or

capillary array with various degrees of collimation.

Target species which are in the condensed phase at

room temperature need to be placed in a crucible and

evaporated by heating. Extensive discussion of this

technique has been given by Scoles [63.4]. The elec-

tron beam intersects the target beam at a 90

◦

angle

and electrons scattered into a speciﬁc direction, over

a small solid angle



≈ 10

−3



, are detected. How-

ever the scattered electron is not necessarily the same

as the incoming electron. Exchange with the target elec-

trons may occur, and is required for spin-forbidden

transitions in light elements. The detector system con-

sists of electron lenses and energy analyzers similar

to those used in the electron gun. The actual detec-

tor is an electron multiplier which generates a pulse

for each electron. In the scattering process, secondary

species (electrons, photons, ions, neutral fragments)

may also be generated and can be detected individu-

ally or in various coincidence schemes. The experiments

are carried out in a vacuum chamber and it is im-

portant to minimize stray electric and magnetic ﬁelds.

More details about the apparatus and procedures can

be found in a review by Trajmar and Register [63.5].

The primary information gained in these experiments

is the energy and angular distribution of the scattered

electrons.

There are several methods used to carry out scat-

tering measurements. In the most commonly used

energy-loss mode, the impact energy and scattering an-

gle are ﬁxed, and the scattering signal as a function of

energy lost by the electron is measured by applying pulse

counting and multichannel scaling techniques. The re-

sult of such an experiment is an energy-loss spectrum.

The elastic scattering feature appears at zero energy loss;

the other features correspond to various excitation pro-

cesses and to ionization. Energy-loss spectra can also

be generated in the constant residual energy mode. In

this case, the detector is set to detect only electrons with

a speciﬁc residual energy E

= E

− ∆ E at a ﬁxed scat-

tering angle, and E

and ∆E are simultaneously varied.

Each feature in the energy-loss spectrum is obtained then

at the same energy above its own threshold. An exam-

250

200

150

100

20 21 22 23 24 25

c/s

Energy loss (eV)

90°

= 1.2 eV

n = 2

×8

Fig. 63.2 Energy-loss spectrum of He at a constant residual

energy of 1.2 eV and scattering angle of 90

◦

. The inelastic

features with the corresponding principal quantum numbers

are shown. IE is the ionization potential of He (24.58 eV).

No background is subtracted and true signal zero is in-

dicated by a dotted line under the expand portion of the

spectrum m. (Taken from Allen [63.6])

ple, taken from the work of Allan [63.6], ist shown in

Fig. 63.2.

The energy loss spectrum becomes equivalent to the

photoabsorption spectrum in the limit of small momen-

tum transfer K,whereK = k

− k

(i. e., high impact

energy, small scattering angle). The equivalence of elec-

trons and photons in this limit follows from the Born

approximation, and it can be used to obtain optical

absorption and ionization cross sections. The correspon-

dence is deﬁned through the Limit Theorem

lim

K→0

(K) = f

opt

, (63.7)

(K) =

∆E

dσ

dΩ

(k),

(63.8)

where f

is the generalized oscillator strength for

excitation process n,and f

opt

is the corresponding

optical f-value. Equation (63.7) was originally de-

rived by Bethe [63.7] from the Born cross section.

It was extended later by Lassettre et al. [63.8]to

cases where the Born approximation does not hold. In

this case, f

(K) is replaced by f

app

(K), the appar-

ent generalized oscillator strength. The Limit Theorem

implies that, in the limit of small K, optical selec-

tion rules apply to electron impact excitation. The

practical problem is that the limit is nonphysical and

the extrapolation to zero K involves some arbitrari-

ness. When K is signiﬁcantly different from zero,

Part E 63.1

932 Part E Scattering Experiment

0.6 0.8 20 0.2 0.4 0.6 0.8 21 0.2 0.4

Intensity (arb. units)

Energy loss (eV)

= 40.1 eV

125°

50°

5°

×50

×20 ×20

Fig. 63.3 Variation of energy-loss spectra (and DCSs) with

scattering angle for He at 40 eV impact energy. Spectra are

shown at 5

◦

,50

◦

and 125

◦

scattering angles

optical-type selection rules do not apply to electron-

impact excitation. As can be seen from Fig. 63.3,spin

and/or symmetry forbidden transitions then readily oc-

cur and can be an efﬁcient way of producing metastable

species.

Selection rules for electron impact excitations can

be derived from group theoretical arguments [63.9, 10].

For atoms, the selection rule S

↔ S

applies in gen-

eral and scattering to 0

◦

and 180

◦

is forbidden if

+ Π

+ L

+ Π

) is odd. Here, L

and L

are the

angular momenta and Π

and Π

are the parities. For

molecules, selection rules can be derived under two

special conditions: (a) rules concerning 0

◦

and 180

◦

scattering for arbitrary orientation of the molecule, and

(b) rules concerning scattering to any angle but for spe-

ciﬁc orientation of the molecule. An important example

of the ﬁrst case is the Σ

−

↔/Σ

selection rule for linear

molecules at 0

◦

and 180

◦

scattering angles.

19.2 19.3 19.4 19.5

(eV)

counts/ channel

θ = 90°

∆E = OeV

Fig. 63.4 The 19.37 eV He resonance observed in the elas-

ticchannelat90

◦

scattering angle

The energy dependence of cross sections is obtained

by ﬁxing the energy-loss value (scattering channel) and

studying the variation of scattering signal with impact

energy at a given angle or integrated over all scatter-

ing angles. In general, cross sections associated with

spin forbidden and optically allowed transitions peak

near and at several times the threshold impact energy

respectively, and they usually vary smoothly with im-

pact energy. However, resonances may appear at certain

speciﬁc impact energies. These sudden changes are asso-

ciated with temporary electron capture and are the result

of quantum mechanical interference between two indis-

tinguishable paths. An example is shown in Fig. 63.4 for

He in the elastic channel at 19.37 eV impact energy.

Integral cross sections can be obtained from ex-

trapolation of the measured D

,θ) to 0

◦

and 180

◦

scattering angles and integration over all angles. Re-

cently, incorporation of an “angle-changing” device has

enabled measurements to be extended over the whole

range of scatering angles [63.11, 12]. In certain cases it

is possible to measure integral cross sections directly by

detecting secondary products such as photons and ions.

These procedures and the resulting cross sections will

be discussed in some detail in Sect. 63.2.

Part E 63.1

Electron–Atom and Electron–Molecule Collisions 63.2 Collision Processes 933

63.2 Collision Processes

In addition to the basic elastic and inelastic processes

deﬁned in Sect. 63.1.2, we now also explicitly include

dissociation (to neutral and charged fragments) cross

sections Q

); and ionization cross sections Q

Each of these is now considered separately.

63.2.1 Total Scattering Cross Sections

Total electron scattering cross sections represent the sum

of all integral cross sections:

tot

) =



) + Q

(63.9)

tot

) are useful for checking the validity of scat-

tering theories, and the consistency of available data,

for normalization of integral and differential cross sec-

tions, and as input to the Boltzmann equation. At low

impact energies, elastic scattering dominates, while at

intermediate and high impact energies, electronic ex-

citations and ionization become major contributors to

tot

. Figure 63.5 shows the various cross sections for

electron–helium collisions. The data are from the rec-

ommended values of Trajmar and Kanik [63.13].

Two methods are commonly used for measuring

tot

): the transmission method and the target recoil

method (for details see [63.5,14]). Total scattering cross

sections measured by these techniques are, in general,

accurate to within a few percent. The extensive reviews

by Zecca and co-workers [63.15–17] should be noted.

63.2.2 Elastic Scattering Cross Sections

Elastic scattering cross sections Q

) are not as

readily available as Q

tot

). They are obtained from

differential scattering experiments over limited angular

ranges by extrapolation and integration of the measured

values. Typical error limits are 5 to 20%. For molecu-

lar species rotational excitation is usually not resolved

but is included in the D

,θ) and Q

) values.

In order to obtain the absolute D

,θ) directly from

the scattering signal, one has to know the electron ﬂux,

the number of scattering species, the scattering geom-

etry and the overall response function of the apparatus.

A direct measurement of these parameters can be made

at high energies (> 100 eV). However, at low electron

energies, this approach is not feasible. A number of

methods have been devised to derive relative D

,θ)

from the measured scattering intensities and then to nor-

0.1

0.01

0.001

1 10 100 1000

Cross section (10

–16

)

Energy (eV)

Helium

n =2

S+2

Fig. 63.5 Cross sections for various processes in the

electron–helium collision (see text for data sources)

malize the D

,θ) to the absolute scale. We brieﬂy

outline here only the most commonly used procedure.

The most practical and reliable method of obtain-

ing the absolute D

,θ)is the relative ﬂow technique

in which scattering signals for a known standard gas

and an unknown test gas are compared at each energy

and angle [63.5, 18–20]. The He elastic cross section is

the natural choice of standard since it is known accu-

rately over a wide energy and angular range, and He is

experimentally easy to handle. Only the relative elec-

tron beam ﬂux and molecular beam densities (and their

distributions) need be known in the two measurements.

The ﬂow rate of the test gas is adjusted so that the ﬂux

and density distribution patterns of the two gases are

identical, and all geometrical factors cancel in the scat-

tering intensity ratios. The absolute D

,θ) for the

sample gas is obtained from the measured scattering in-

tensity, target density, and electron beam intensity ratios

and the standard D

,θ)value. See [63.5, 20, 21]for

a detailed discussion of this technique.

63.2.3 Momentum Transfer Cross Sections

) can be obtained both from the elastic DCSs

and from swarm measurements. At low electron-impact

energies (from 0.05 to a few eV), where only a few

collision channels are open, the electron swarm tech-

nique is the most accurate (≈ 3%) way to determine

the momentum transfer cross sections. Beam–beam ex-

periments are mandatory at higher energies. A detailed

Part E 63.2

934 Part E Scattering Experiment

discussion of these techniques is given by Trajmar and

63.2.4 Excitation Cross Sections

,θ)and Q

) can be derived from energy-loss

spectra obtained in beam–beam scattering experi-

ments. The relative D

,θ) is usually normalized to

,θ) which in turn is normalized to the helium

,θ) by the relative ﬂow technique described in

Sect. 63.2.2. There are, however, complications and un-

certainties associated with this technique because of the

sensitivity of the instrument response function to the

residual energy of the scattered electrons. For more de-

tails, see Trajmar and McConkey [63.21]. Data obtained

by this procedure are rather limited, partly due to experi-

mental difﬁculties and partly due to the time required to

carry out such measurements.

For cases where an excited state j is formed which

can radiatively decay by means of a short-lived (dipole-

allowed) transition to a lower lying state i, the intensity

of the resultant radiation is directly related to the cross

section for production of the excited state in the origin-

al collision process. An optical emission cross section,

),isdeﬁnedby

) =

, (63.10)

where N

and n

are the densities in the excited and

ground states, respectively, Γ

is the branching ratio for

radiative decay from state j to state i, I is the electron

beam ﬂux, and τ

is the natural radiative lifetime of

state j. Since the excited state may be produced either

by direct electron impact or by cascade from higher-

lying states k, also formed in the collision process, we

may deﬁne the direct excitation cross section Q

)

) =



) −



). (63.11)

The last term subtracts the cascade contribution from

higher lying states. The quantity Q

) =



)

is known as the apparent excitation cross section

for level j. Clearly, to obtain Q

) from Q

the cascade contribution must be known.

In (63.10), N

/τ

gives the steady state number

of j → i photons per unit time per unit volume emitted

from the interaction region. Since observation is made

in a particular direction care must be taken to correct for

any anisotropy in the radiation pattern. Alternatively,

if observation is made at the so-called “magic” angle

(54

◦



) to the electron beam direction, the emission

intensity per unit solid angle is equal to the average inten-

sity per unit solid angle irrespective of the polarization of

the emitted radiation. However, even at this magic angle,

care must be taken to avoid problems with polarization

sensitivity of the detection equipment [63.22, 23].

The phenomenon of radiation trapping is often

a problem if the radiative decay channels of the excited

state include a dipole allowed channel to the ground

state. Repeated absorption and re-emission of the ra-

diation can occur and can lead to a diffuse emitting

region much larger than the original interaction region,

and the polarization of the emitted light can also be

altered. Often a study of the variation of the emitted

intensity or polarization with the target gas pressure is

sufﬁcient to reveal the presence of radiation trapping or

other secondary effects.

The emission cross sections of certain lines have

been measured with great care and now serve as

bench-marks for other work. Examples of these are the

measurements of van Zyl et al. [63.24]onthen

Slevels

of He in the visible spectral region or the measurements

of the cross section for production of Lyman α from H

the VUV region (see [63.25] for a full discussion of this

including many references). Use of secondary standards

is particularly important when crossed-beam measure-

ments are being carried out because of the cancellation

of geometrical and other effects which occur.

The Bethe–Born theory [63.26] provides a conve-

nient calibration of the detection system for optically

allowed transitions of known oscillator strength. At suf-

ﬁciently high energies, the excitation cross section, Q

of level n is given by

4πa

∞

opt

∆E

∞

ln(4c

∞

). (63.12)

Here ∆ E

is the excitation energy, and c

is a constant

dependent on the transition. A plot of Q

versus ln E

is a straight line with a slope proportional to f

opt

and the

intercept with the ln E

axis yields an experimental value

for c

independent of the normalization. For example,

the He n

P–1

S optical oscillator strengths are very

accurately known, as are cascade contributions. Thus

accurate normalization of the slope of the Bethe plot can

be made, yielding accurate excitation cross sections.

As mentioned above, the excitation cross sections

display characteristic shapes as a function of energy.

For optically allowed transitions, the cross section rises

relatively slowly from threshold to a broad maximum ap-

proximately ﬁve times the threshold energy. At higher

energies the (ln E

)/E

dependence of the cross section

Part E 63.2

Electron–Atom and Electron–Molecule Collisions 63.2 Collision Processes 935

predicted by (63.12) is observed. For exchange pro-

cesses, e.g., a triplet excited state from a singlet ground

state, the cross section peaks sharply close to threshold

and falls off at high energy as E

−3

. If the excitation is

spin allowed but optically forbidden, e.g., He n

D from

S , then the Bethe theory predicts an E

−1

dependence

of the cross section at high energies.

When excitation occurs to a long-lived (metastable

or Rydberg) state following electron impact, it is often

possible to detect the excited particle directly. Time-

of-ﬂight (T.O.F.) techniques are used to distinguish the

long lived species from other products, e.g., photons,

produced in the collisions.

63.2.5 Dissociation Cross Sections

Dissociation of a molecular target can result in frag-

ments which may be excited or ionized. Such processes

may be studied using the techniques discussed in the pre-

vious section or in the following section, where charged

particle detection is considered. Because a repulsive

state of the molecule is accessed, the fragments can

leave the interaction region with considerable kinetic

energy (several eV). If the fragment is in a long-lived

metastable or Rydberg state, T.O.F. techniques may be

used to distinguish the long-lived species from other

products such as photons, and also to measure the

energies of the excited fragments, and thus provide in-

formation on the repulsive states responsible for the

dissociation. For further discussion see the reviews by

Compton and Bardsley [63.27], Freund [63.28], and

Zipf [63.29]. If the detector can be made sensitive to

a particular excited species, its excitation can be iso-

lated and studied. Examples are the work of McConkey

and co-workers [63.30,31]onO(

S)andS(

) produc-

tion from various molecules. The detection of unexcited

neutral fragments is more challenging. One early method

was to trap selectively the dissociation products using

a getter and measure the resulting pressure decrease.

In a more sophisticated approach, Cosby [63.32]pro-

duced a fast (≈ 1 keV) target molecular beam by resonant

charge exchange and subjected it to electron impact dis-

sociation. The fast dissociation products were detected

by conventional particle detectors in a time correlated

measurement. Laser techniques, such as laser-induced

ﬂuorescence or multiphoton ionization, have also been

used recently to detect the dissociation products.

The Franck–Condon principle largely governs mo-

lecular dissociation. The principle states that if the

excitation takes place on a time scale which is short

compared with vibrational motion of the atomic nu-

clei the transition occurs vertically between potential

energy curves. Since dissociation rapidly follows a ver-

tical transition to the repulsive part of a potential energy

curve, compared with the period of molecular rotation,

the dissociation products tend to move in the direction of

vibrational motion. Since the excitation probability de-

pends on the relative orientation of the electron beam and

the molecule, dissociation products often demonstrate

pronounced anisotropic angular distributions. The an-

gular distributions have been analyzed by Dunn [63.33]

using symmetry considerations.

63.2.6 Ionization Cross Sections

Tate and Smith [63.34] some 60 years ago developed

the basic techniques for measuring total ionization

cross sections. These were later improved by Rapp and

Englander-Golden [63.35]. Full details of the experi-

mental methods are given in the reviews and books

already cited. Märk and Dunn[63.36] reviewed the situ-

ation as it existed in the mid 1980s. In the basic “parallel

plate” method, the electron beam is directed through

a beam or a static target gas between collector plates

which detect the resultant ions. Unstable species can be

studied by the “fast neutral beam” technique [63.37],

in which the neutral target species is formed by charge

neutralization from a fast ion beam, and is subsequently

ionized by a crossed electron beam. For the determi-

nation of partial ionization cross sections speciﬁc to

a given ion species in a given ionization stage, mass

spectroscopic (quadrupole mass spectrometer, electro-

static or magnetic charged particle analyzer or time of

ﬂight) methods are used. Fourier Transform Mass Spec-

trometry (FTMS) has also been used effectively to study

fragmentation with formation of both positive and neg-

ative ions. Reference [63.38] is a recent example of this.

Absolute total ionization cross sections have been meas-

ured for a large number of species with an accuracy

of better than 10%. Christophorou and colleagues have

presented helpful compilations of ionization and other

data of particular relevance to the plasma processing

industry, [63.39, and earlier references in this journal].

A large number of mechanisms can contribute to

the ionization of atoms and molecules by electron im-

pact. For targets with only a few atomic electrons,

the dominant process is single ionization of the outer

shell, with the resultant ion being left in its ground

state. The process is direct and is characterized by

large impact parameters b and small momentum trans-

fers. The cross section varies with incident electron

energy in a way very similar to the optically allowed

Part E 63.2

936 Part E Scattering Experiment

excitation processes discussed in Sect. 63.2.4. Processes

involving ionization of more than one outer shell elec-

tron become more important as the size of the target

increases. These events are associated with small b

and electron–electron correlations are usually strong.

Autoionization increases in signiﬁcance for heavier tar-

gets. Here also, collisions with small b dominate and

electron–electron correlations are strong. For heavier

targets, inner shell effects, such as Auger electron or

X-ray emission, become progressively more important.

For molecular targets, dissociative ionization (either di-

rectly or through a highly excited intermediate state) and

ion pair formation also play a signiﬁcant role.

In addition to measurement of gross ion production,

it is also possible to study the ionization process by

monitoring the electron(s) ejected or scattered inelas-

tically. Conventional electron spectroscopic techniques

are used for this purpose. The addition of coincidence

techniques (e–2e measurements) in which the momenta

of all the electrons involved are completely speciﬁed has

allowed many of the ﬁne details of the ionization process

to be extracted [63.40].

63.3 Coincidence and Superelastic Measurements

The cross section measurements described so far do not

yield complete information on electron scattering pro-

cesses. As mentioned in Sect. 63.1.2, these cross sections

do not distinguish for magnetic sublevels, electron spin

etc.and represent summation of cross sections over these

experimentally indistinguishable processes (summation

of the square moduli of the corresponding scattering

amplitudes). The quantum mechanical description of

a scattering process is given in terms of scattering am-

plitudes and under certain conditions requires summing

up amplitudes and squaring the sum. This leads to inter-

ference terms which arise from the coherent nature of

the scattering process. A complete characterization of

a scattering process, therefore, requires knowledge of

the complex scattering amplitudes.

Sophisticated experimental techniques have been

developed in recent years, which go beyond the conven-

tional scattering cross section measurements and yield

information on magnetic sublevel speciﬁc scattering am-

plitudes and the polarization (alignment and orientation)

of the excited atomic ensemble. The experimental tech-

niques fall into two main categories: a) electron–photon

coincidence measurements, and b) superelastic scatter-

ing measurements involving coherently excited species.

(We still consider unpolarized electron beams in the

description of these two techniques here and will ad-

dress the question of spin polarization in the following

section.)

In electron–photon coincidence measurements, the

radiation pattern emitted by the excited atom is deter-

mined for a given direction of the scattered electron.

A scattering plane is deﬁned by k

and k

, and hence

the symmetry is lowered from cylindrical (around the

incident beam direction) to planar, (Fig. 63.6).

It is now possible to determine, at least in principle,

both the atomic alignment (i. e., the shape of the excited

state charge cloud and its alignment in space) and its

orientation (i. e., the angular momentum transferred to

the atom during the course of the collision). Complete

sets of excitation amplitudes for the coherently excited

atomic states and their relative phases have been meas-

ured in some cases. A comparison with theory can then

be made at the most fundamental level. See [63.41]for

full discussion and analysis.

–

out

col

⊥

Fig. 63.6 Schematic illustration of a collisionally induced

charge cloud in a p-state atom. The scattering plane is ﬁxed

by the direction of incoming k

and outgoing k

out

momen-

tum vectors of the electrons. The atom is characterized by

the relative length (l), width (w), and height (h)ofthe

charge cloud, by its alignment angle γ , and by its inherent

angular momentum L

⊥

. The coordinate frame is the nat-

ural frame with the z-axis perpendicular to the scattering

plane and with the x-andy-axesdeﬁnedasshowninthe

ﬁgure relative to k

and k

out

Part E 63.3

Electron–Atom and Electron–Molecule Collisions 63.3 Coincidence and Superelastic Measurements 937

The electron–photon coincidence measurements can

be carried out in two ways: (1) measuring polarization

correlations, and (2) measuring angular correlations.

In (1), polarization analysis of the emitted photon in

a given direction occurs, while in (2), the angular

distribution of the emitted photons is determined with-

out polarization analysis. We will describe here only

method (1) in some detail.

Method (1) has the advantage that it measures di-

rectly the angular momentum (perpendicular to the

scattering plane) transferred in the collision. For P-state

excitation from a

ground state, four parameters

plus a cross section are needed to describe fully the

collisionally excited P-state. The natural parameters

introduced by Andersen et al. [63.41] are deﬁned as fol-

lows (Fig. 63.6): γ is the alignment angle of the excited

state charge cloud relative to the electron beam axis,



is the linear polarization in the scattering plane,

⊥

is the orbital angular momentum perpendicular to

the scattering plane that is transferred to the atom in

the collision, and ρ

is the relative height of the charge

cloud perpendicular to the scattering plane at the point

of origin. The + superscript indicates positive reﬂection

symmetry with respect to the scattering plane.

In polarization correlation experiments, one typic-

ally measures two linear (P

, P

) and one circular (P

)

polarization correlation parameters perpendicular to the

scattering plane. One additional linear polarization cor-

relation parameter P

is measured in the scattering plane.

Each parameter is the result of two intensity meas-

urements for different orientations of the polarization

analyzer:

I(0

◦

) − I(90

◦

)

I(0

◦

) + I(90

◦

)

−1

I(45

◦

) − I(135

◦

)

I(45

◦

) + I(135

◦

)

−1

− I

+ I

−1

I(0

◦

) − I(90

◦

)

I(0

◦

) + I(90

◦

)

−1

. (63.13)

Here I(α) denotes the photon intensity measured for

a polarizer orientation α with respect to the electron

beam axis, I

and I

refer to right- and left-handed

circularly polarized light and β denotes the polarization

sensitivity of the polarization analyzer. The relationships

between the experimentally determined polarization cor-

relation parameters and the natural parameters are given

γ =

tan

−1





+ P



1/2

⊥

=−P

(1 + P

)(1 − P

)

4 − (1 − P

)(1 − P

)

(63.14)

The total polarization P

tot

, which is deﬁned as

tot











⊥





1/2



+ P



1/2

, (63.15)

is a measure for the degree of coherence in the excitation

process. In the absence of atomic depolarizing effects

due to, for example, ﬁne and/or hyperﬁne interactions,

a value of P

tot

=+1 for the emitted radiation indicates

total coherence of the excitation process.

Much of the earlier work involved excitation of he-

lium n

P state. Here the situation is simpliﬁed as L–S

coupling applies strictly: P

= 1andρ

is zero. Excita-

tion of the 2

P state, for example, is fully coherent and

hence the excitation is completely speciﬁed by just two

parameters, γ and L

⊥



or P



since P





1 − L

⊥



1/2



More recently, the techniques have been applied to

heavier targets and more complicated excitation pro-

cesses [63.42–46].

The superelastic scattering experiments could be

looked at as time inverse electron–photon coincidence

experiments (although this is not exactly the case). In

these experiments, a laser beam is utilized to prepare

a coherently excited, polarized ensemble of target atoms

for the electron scattering measurement. The superelas-

tic scattering intensity is then measured as a function

of laser-beam polarization and/or angle with respect to

a reference direction. Linearly polarized laser light pro-

duces an aligned target (uneven population in magnetic

sublevels for quantum numbers of different |M

| value).

Circularly polarized laser light produces oriented targets

(uneven population in M

=+m and M

=−m mag-

netic sublevels). From these measurements the same

electron impact coherence parameters can be deduced

as from the coincidence experiments. This approach has

been applied to atomic species (mainly metal atoms)

which are conveniently excited with available lasers.

Detailed descriptions of the experimental techniques,

the underlying theoretical background, and the interpre-

tation of the experimental data are given in [63.47–56].

It should be noted that electron scattering by co-

herently excited atoms can be utilized not only for

obtaining electron impact coherence parameters for in-

elastic processes originating from ground state but for

Part E 63.3

938 Part E Scattering Experiment

elastic, inelastic, and superelastic transitions involving

excited states. These measurements yield information on

creation, destruction, and transfer of alignment and ori-

entation in electron collision processes which is needed,

e.g., in the application of plasma polarization spec-

troscopy [63.57].

63.4 Experiments with Polarized Electrons

So far we have considered the utilization of unpolarized

electron beams which yield spin averaged cross sec-

tions. Little information on spin dependent interactions

is gained from these experiments. However, these inter-

actions can be studied using polarized electron beam

techniques. Developments on both the production and

detection of spin-polarized electron beams have resulted

in a wide range of elegant experiments probing these

effects. The theory is also highly developed. For a de-

tailed discussion see the works of Kessler [63.58, 59],

Blum and Kleinpoppen [63.60], Hanne [63.61–63]and

Andersen et al. [63.44, 45] and the references therein.

Some basic concepts are presented here.

ThedegreeofpolarizationP of an electron beam is

given by

P =

N(↑) − N(↓)

N(↑) + N(↓)

(63.16)

where N(↑) and N(↓) are the numbers of electrons with

spins respectively parallel and antiparallel to a particular

quantization direction. Measurements of P both before

and after the collision enable one to probe directly for

speciﬁc spin dependent processes. For example, in elas-

tic scattering from heavy spinless atoms any changes in

the polarization of the electrons must be caused by spin–

orbit interactions alone since, in this instance, it is not

possible to alter the polarization of the electron beam

by electron exchange. Measurements have been carried

out for Hg and Xe and both direct (f) and spin-ﬂip (g)

scattering amplitudes, as well as their phase differences,

have been determined [63.59].

The spin–orbit interaction for the continuum elec-

tron caused by the target nucleus leads to different

scattering potentials and consequently to different cross

sections for spin-up and spin-down electrons (called

Mott scattering). Consequently, an initially unpolarized

electron beam can become spin polarized after scattering

by a speciﬁc angle according to



= S

(θ)

n , (63.17)

where nis the unit vector normal to the scattering plane,

(θ) is the polarization function and P



is the polariza-

tion of the scattered beam. For the same reasons, when

a spin-polarized electron beam is scattered by an an-

gle θ to the left and to the right, an asymmetry is found

in the scattering cross sections. Furthermore, an existing

polarization P



can be detected through the left–right

asymmetry A in the differential cross section, which is

given by

A ≡

(θ) − σ

(θ)

(θ) + σ

(θ)

= S

(θ)P



n , (63.18)

where σ

(θ) and σ

(θ) are the differential cross sec-

tions for scattering at an angle θ relative to the incident

beam axis to the left and to the right, respectively. For

elastic scattering, the polarization function S

,andthe

asymmetry function S

are identical and are called the

Sherman function.

When electron exchange is studied under conditions

where other explicitly spin-dependent forces can be ne-

glected, the cross sections for scattering of polarized

electrons from polarized atoms depend on the rela-

tive orientation of the polarization vectors. According

to [63.59]

σ(θ) = σ

(θ)[1 − A

(θ)P

· P

] , (63.19)

where P

and P

are the electron and atom polarization

vectors, and σ

(θ) is the cross section for unpolarized

electrons. Hence, an “exchange asymmetry” A

(θ) can

be deﬁned by

(θ) P

· P

↑↓

(θ) − σ

↑↑

(θ)

↑↓

(θ) + σ

↑↑

(θ)

(63.20)

where σ

↑↑

(θ) and σ

↑↓

(θ) denote the cross sections

for parallel and antiparallel polarization vectors respec-

tively. As Bartschat [63.64] points out, an asymmetry

can occur even if the scattering angle is not deﬁned. In

this case the function A

(θ) is averaged over all an-

gles. Differential and integral measurements of this kind

have been performed for elastic scattering, excitation

and ionization.

For heavy target systems it is necessary to consider

a combination of effects together with a description of

the target states in the intermediate or fully coupled

scheme. Consequently, the number of independent par-

ameters can become very large and the “complete”

experiments, which disentangle the various contribu-

tions to any observed asymmetry in the scattering, are

rarely possible.

Part E 63.4

Electron–Atom and Electron–Molecule Collisions 63.6 Electron Collisions in Traps 939

Even for very light target atoms, where conven-

tional Mott scattering is negligible, Hanne [63.61]has

shown that the “ﬁne structure” effect, in which elec-

tron scattering from individual ﬁne structure levels of

a multiplet occurs, can lead to polarization effects. In

fact it can be a dominating effect for inelastic collision

processes.

For full details of these various mechanisms and how

density matrix theory and other theoretical techniques

have been applied to scattering involving polarization

effects, the reader is referred to the review articles cited,

particularly Andersen et al. [63.44].

In certain cases, experiments involving spin po-

larized electron beams coupled with coincidence (or

superelastic) measurements allow one to extract the

maximum possible information for a given process, and

are termed as complete or perfect in the sense deﬁned

by Bederson [63.65–67].

63.5 Electron Collisions with Excited Species

There are many plasma systems where electron colli-

sions with excited atoms and molecules play a prominent

role, e.g., electron beam and discharge pumped lasers,

planetary and astrophysical plasmas. Especially im-

portant are electron collisions with metastable species

because of the long lifetime, large cross section and large

amount of excitation energy associated with them. Elec-

tron collision studies and cross section data in this area

are scarce mainly due to experimental difﬁculties associ-

ated with the production of target beams with sufﬁciently

high densities of excited species. With the application

of lasers for the preparation of the excited species this

problem can be overcome. However, this approach has

not yet been extensively exploited. Reviews of this ﬁeld

are given by Lin and Anderson [63.68], Trajmar and

Nickel [63.69]andChristophorou and Olthoff [63.70].

Since electron collisions with excited species nec-

essarily involve a method of preparation, they are two

step processes. Excitation and ionization in these cases

are frequently referred to as stepwise excitation and ion-

ization. The target preparation leads to mixed beams

containing both ground and excited atoms or molecules.

Preparation of excited atoms is achieved by electron im-

pact or photoabsorption. Fast metastable beams can be

produced by near-resonance charge exchange. For more

details see [63.69, 71, 72]. Electron impact excitation is

simple and effective but highly nonspeciﬁc, and charac-

terization of the composition of the mixture is difﬁcult.

Laser excitation is more involved but very well deﬁned.

Speciﬁc ﬁne and hyperﬁne levels of individual isotopes

can be excited. When laser excitation is used in con-

junction with superelastic electron scattering, an energy

resolution of 10

−8

eV is easily achieved, compared with

the 10

−2

eV resolution possible in conventional electron

scattering.

Depending on the method of preparation, the popu-

lation distribution in the magnetic sublevels of the target

atoms may be uneven and some degree of polarization

(alignment or orientation) may be present. The scattering

will then be φ-dependent. For polarized target atoms the

measured electron collision cross sections do not cor-

respond to the conventional cross sections (which are

summed over ﬁnal and averaged over initial experimen-

tally indistinguishable states, with equal populations in

the initial states). One, therefore, has to characterize pre-

cisely the state of the target beam in order to be able to

deduce a well deﬁned, meaningful cross section. Polar-

ization of atoms can be conveniently controlled, in the

case of excitation with laser light, through the control of

the laser light polarization (as discussed in Sect. 63.3).

Since the atomic ensemble is coherently excited in this

case, the scattering cross sections will depend on the

azimuthal scattering angle φ. These considerations also

come into play when one tries to relate measured inelas-

tic and corresponding inverse superelastic cross sections

by the principle of detailed balancing.

63.6 Electron Collisions in Traps

A technique which has recently begun to be exploited

involves collisions with trapped atoms. Pioneered by

Lin and colleagues [63.73, 74], using Rb targets the

technique has many advantages over more conventional

techniques, not least of which is the fact that the absolute

number density of the target need not be known. Cross

section data are obtained from measurements of trap loss

and electron beam current density. Because up to half of

the atoms in the trap can be in the excited state, it is pos-

sible to make measurements of cross sections involving

excited states as well [63.75]. Measurements involving

Cs targets have also been reported [63.76].

Part E 63.6