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Charged-Particle–Matter Interactions 91.4 Energy Loss Straggling 1383

fraction of the particles do not lose any energy in the

absorber.

91.4.2 Analytic Methods for Calculating

Energy Loss Straggling Function

Three principal methods are convolutions, Laplace

transforms and the use of moments [91.5]. They are prac-

tical for different thicknesses of absorbers, as indicated.

Convolutions

For this method, a complete collision cross section

must be available [91.47,50], and numerical calculations

are required. The probability density function for par-

ticles having suffered n collisions is given by the n-fold

convolution of the single collision spectrum [91.5, 95]:

∗n

(∆) =

∆



σ(E)σ

∗(n−1)

(∆ −E) dE , (91.39)

with

∗0

(∆) = δ(∆) , σ

∗1

(∆) = σ(∆) . (91.40)

Assuming that successive collisions are statistically in-

dependent, the number of collisions is described by the

Poisson distribution

−n

, (91.41)

where P

gives the fraction of particles suffering n

collisions, and n

= xM

is the average number of colli-

sions for all particles. The complete straggling function

is then

F(x,∆)=

∞



n=0

∗n

(∆) . (91.42)

Usually, this method is only used for absorbers thin

enough that the single collision cross section changes

negligibly during the passage through the absorber.

This approach would become quite complex for

large n

. The doubling method [91.96] uses the follow-

ing procedure: the function for an absorber of thickness

2x is calculated by convoluting F(x,∆)with itself, viz.

F(2x,∆)=

∆



F(x,∆−g)F(x, g) dg . (91.43)

An initial distribution is calculated for an extremely

thin absorber of thickness dx from

F( dx,∆)= δ(∆)(1 −M

dx) +σ(∆) dx , (91.44)

where δ(∆) = F(0,∆)and M

dx ≈ 0.001.

For Si, the convolution method has been tested and

compared with experiments for as many as 10

colli-

sions [91.5]. The limitations of the algorithm are ≈10

collisions. Since P

= e

−n

, there will be particles that

pass through an absorber without making a collision.

Laplace Transforms

The Laplace transform method is mathematically

equivalent to the convolution method. The method, as

implemented by Landau [91.6], is based on σ

(E).

The implicit number of collisions is therefore too large,

and the Landau function is too narrow, as shown in

Figs. 91.1 and 91.6. A reﬁnement proposed in [91.7]

gives straggling functions which are too wide (Fig. 91.1).

Use of Moments

If the skewness γ

is not small, a Gaussian function is

not a good approximation to a straggling function. This

can be remedied by using a modiﬁed Gaussian of the

form [91.55, 56]

f(y) =

√

2π



×exp



−



γ +

δ ln









(91.45)

where y = (∆ −∆)/σ and

γ and

δ are related to

and M

[91.55, 56] such that the second and third

moments are reproduced exactly.

The moments can be calculated as a function of

particle speed, and thus the method is useful for mean

energy losses up to about 0.7T

. At these losses, some

particles reach their full range and thus disappear from

the beam. The moments are then distorted and other

procedures are needed [91.11, 97].

Thick Absorbers, Ranges and Range Straggling

In radiation therapy with charged particles, a parallel

beam is directed at the body. Usually, the energy T

the particles is adjusted to penetrate a given distance into

the body. A primary measure of this distance is the mean

range R calculated with the continuous slowing down

approximation (CSDA)

R(T

) =



S(T)

(91.46)

Because of energy-loss straggling, there also is strag-

gling in range amounting to a few percent of R [91.84].

Part G 91.4

1384 Part G Applications

In the practical application, a depth-dose curve is

needed. At a given depth, the dose D is given

by (91.6). For the ﬁrst 50% of the range, the

spread in energy of the beam is relatively small,

and the dose is approximated closely by the stop-

ping power S(T ) for the mean energy of the beam.

For larger thicknesses, a detailed transport calcula-

tion must be made in order to ﬁnd φ(T , x) [91.11].

Range straggling functions can then also be ob-

tained. Close agreement between measurement and

calculation has been achieved [91.97].Similarly, the

mean energy loss T

in a thick absorber is given

implicitly by

x =



S(T)

(91.47)

Usually, values are obtained from a range energy table.

Such tables may be found in [91.25, 63–65, 79, 82–87].

91.4.3 Particle identiﬁcation (PID)

In particle physics the determination of particle

types produced in the collisions of fast particles is

needed [91.59]. Since the cross sections in Sect. 91.2

only depend on particle speed β and charge Z

e,energy

deposition measurements combined with the measure-

ment of particle momentum p can be used to determine

particle masses. This process is called particle identiﬁ-

cation. A large amount of empirical information about

PID has been accumulated, but little effort has been

made to correlate this information with the theory de-

scribed here. A major problem is the dependence of

∆

/x,Table91.2,onx. In addition, ∆

/x is less than

dE/x and its dependence on particle speed, Fig. 91.5,

differs from that of dE/dx,Fig.91.5. Comparisons of

calculated and measured functions F(∆) can be used to

diagnose problems of the performance of the particle

detector.

91.5 Multiple Scattering and Nuclear Reactions

Coulomb scattering of particles heavier than electrons

by nuclei and electrons usually produces many, but very

small angular deﬂections [91.4, 98].

The Molière theory [91.99] giving the distribu-

tion f(θ) of angular deﬂections θ agrees well with

experimental data [91.100]. As a ﬁrst approximation,

a Gaussian distribution f(θ) = e

−y

,wherey = θ/θ

can be used. The value θ

is given by θ

=θ

,where

= 0.157

+1)

xγ





radians (91.48)

and B

has values of the order of 10. A more detailed

description plus several tables is given in [91.84].

For a broad beam of particles, multiple scatter-

ing produces only a minor correction in the range.

In a narrow “pencil” beam, it causes a broaden-

ing, e.g., a 320 MeV proton beam spreads to a width

of about 2 cm after traveling through 40 cm of

water [91.101].

It may be necessary to take into account the inﬂuence

of nuclear interactions. As a ﬁrst approximation, the total

cross section is roughly [91.25]

tot

= π



1.3×10

−13

1/3

+λ



, (91.49)

where λ = λ

/2π.

91.6 Monte Carlo Calculations

In Monte Carlo calculations, the interactions occurring

during the passage of the particles through matter can

be simulated one at a time, collision by collision, and

including those for the δ-rays [91.102, 103]. Particles

travel random distances x

between successive colli-

sions, calculated by selecting a random number r

and

determining the distance to the next collision from

=−λ

mfp

ln r

. (91.50)

The energy loss E

is selected with a second random

number from the integrated collision spectrum

Q(E) =



σ(E)

dE . (91.51)

This process is repeated until



exceeds the absorber

thickness. The total energy loss ∆ of the particle is

∆ =



Part G 91.6

Charged-Particle–Matter Interactions References 1385

91.7 Detector Conversion Factors

For all detectors, a study must be made of the pro-

cess in which energy lost by the particles is converted

into the observable signal, see Chapt. 92. The primary

products of collisions are excited states, electron–

ion and electron–hole pairs, and secondary electrons

also called δ-rays. Secondary products are chemical

species, ﬂuorescent photons and Auger electrons. An

important problem is the export of energy from the vol-

ume under observation by secondary radiations (δ-rays,

mesons, X-rays, neutrons). Prominent among the ef-

fects [91.104] are ionization currents measured under

applied electrical ﬁelds (in gaseous and solid state ion-

ization chambers), photo effects by ﬂuorescent radiation

(scintillators), measurements of chemical yields (fer-

rous sulphate dosimeters, photographic emulsions, etc.),

and the release of light by stored crystalline defects in

thermoluminescent measurements.

The effect J can be related to the energy deposited

D by a conversion factor W as J = D/W,whereW rep-

resents the amount of energy needed to produce a unit of

observable effect (e.g., an electron–ion pair). W depends

on the absorber, particle type, and speed. For ioniza-

tion, typical values are W ≈ 25 eV gases, W ≈ 3.5eV

semiconductors [91.84, 105].

For chemical effects [91.106], J is sometimes writ-

ten as J = YD. Usually Y is given as the “yield per

100 eV”. A typical value for a Fricke dosimeter is that,

for fast electrons, 15 ferrous ions are converted into

ferric ions per 100 eV of deposited energy. Usually, Y

depends on time after the passage of the particles, and

the full values are only reached after say 1 µs. For slow

ions, Y is much smaller. Light emission (scintillations)

has been observed from gases (typically, W ≈ 20 eV),

liquids and solids (typically, W ≈ 1000 eV) [91.104].

The subsequent measurements of these radiation

effects are done with current or charge ampliﬁers, pro-

portional avalanche ampliﬁcation, photo multipliers,

channel plates, etc. [91.104].

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1386 Part G Applications

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Part G 91

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Part G 91

1389

Radiation Phy

92. Radiation Physics

Radiation physics entails studies of the in-

teractions of ionizing radiation with matter.

The term ionizing radiation refers to any en-

ergetic particles, either charged or uncharged,

that can ionize atoms or molecules in matter.

These particles include photons in the ultra-

violet, X-ray, or γ-ray spectral region; electrons

and positrons; mesons; protons and deuterons;

α-particles; heavier ions including molecular

ions; and neutrons. The matter under con-

sideration includes substances in every phase

of atomic aggregation (i. e., gas, liquid, solid,

or plasma). Strictly speaking, the term “ion-

ization” signiﬁes an event in which at least

one electron leaves an atom or molecule and

eventually becomes free. This notion applies

to a dilute gas, but not to condensed mat-

ter, for which it would be more precise to

use the term “electronic activation” encom-

passing all modes of excitation, as sketched

in Sect. 92.3.5.

92.1 General Overview ................................1389

92.2 Radiation Absorption

and its Consequences .......................... 1390

92.2.1 Two Classes of Problems

of Radiation Physics .................. 1390

92.2.2 Photons ...................................1391

92.2.3 Charged Particles ...................... 1391

92.2.4 Neutrons ..................................1391

92.3 Electron Transport and Degradation ......1392

92.3.1 The Dominant Role of Electrons...1392

92.3.2 Degradation Spectra

and Yields of Products ............... 1392

92.3.3 Quantities Expressing the Yields

of Products...............................1394

92.3.4 Track Structures.........................1395

92.3.5 Condensed Matter Effects ........... 1396

92.4 Connections with Related Fields

of Research .........................................1397

92.4.1 Astrophysics and Space Physics ...1397

92.4.2 Material Science........................1397

92.5 Supplement ........................................1397

References .................................................. 1398

92.1 General Overview

The motivation for studying the interactions of radiation

with matter and their consequences is manifold. First,

nuclear and particle physics experiments often require

radiation detectors – devices that detect, score, or ana-

lyze energetic particles. The working principles of these

detectors rest on knowledge of elementary processes of

radiation interactions with matter [92.1]. Second, mea-

surements of radiation ﬁelds, called dosimetry [92.2],

are important to many purposes, including the protec-

tion of workers in industry and medicine, as well as of

the general public; risk estimates for exposures to ra-

diation from natural sources (such as cosmic rays and

terrestrial radioactivity) or from artiﬁcial sources (such

as accelerators, man-made radioisotopes, and nuclear-

power facilities); and good performance of radiation

diagnosis and therapy. The principles of dosimetry are

largely based on radiation physics. Finally, ionizing ra-

diation is useful for processing materials such as plastics

and semiconductors, to endow desirable properties. To

optimize methods of radiation processing, knowledge of

radiation physics is highly valuable.

Atomic, molecular, and optical physics provides

important underpinnings for radiation physics in two

general contexts: instrumentation for the measurement

of radiation and its effects on matter on the one hand,

and elucidation of the atomic and molecular mechanisms

leading to the effects on the other. The term instrumen-

tation here means all devices that enable one to detect,

identify, and quantify consequences of radiation inter-

actions, such as particle kinetic energies and ionization,

luminescence, and spectral analyses of atomic and mo-

lecular species produced. Conversely, needs in radiation

physics have led to new atomic and molecular tech-

niques such as resonance ionization spectroscopy [92.3],

which was originally developed to determine the yield

of excited atoms in gases exposed to radiation, but is

Part G 92

1390 Part G Applications

now regarded as a branch of multiphoton spectroscopy

valuable for studies of excited states of atoms and

molecules (Chapt. 74). Thus, the instrumentation de-

velopment constitutes a mutually beneﬁcial interface

between atomic, molecular, and optical physics and

radiation physics.

The elucidation of microscopic mechanisms of radi-

ation effects requires a grand synthesis of knowledge

from various areas ranging from atomic, molecu-

lar, and optical physics to condensed matter physics,

statistical physics, chemical kinetics, and molecular bi-

ology [92.4, 5]. Yet, the role of atomic, molecular, and

optical physics is most fundamental [92.6, 7]. Interac-

tions of radiation with matter lead to numerous collisions

of energetic particles with atoms and molecules, and re-

sult in the production of various excited and ionized

states, as well as electrons and other secondary par-

ticles. These have a wide range of kinetic energies, and

in turn collide with atoms and molecules to generate fur-

ther products. A detailed analysis of the chain of these

early events demands knowledge in three topical areas:

ﬁrst, spectroscopic data (i. e., energy values and quantum

numbers) of excited and ionized states; second, modes

of decay and relaxation of these states, such as lumines-

cence, molecular dissociation, and internal conversion

and the branching ratios of each mode; and ﬁnally, the

cross sections for all of the processes involved. All such

knowledge is, in a broad sense, the fruit of work in

atomic, molecular, and optical physics.

The following sections will concern key topics in

basic radiation physics selected from a vast range of

possibilities. The treatment will concentrate on basics

and principles, and examples are meant to be illustrative

rather than exhaustive; yet, the author hopes that the dis-

cussion will convey the charm and challenges of current

research.

92.2 Radiation Absorption and its Consequences

92.2.1 Two Classes of Problems

of Radiation Physics

It is useful to distinguish between two classes of

problems in radiation physics [92.6], as illustrated in

Fig. 92.1. Problems of Class I concern the fate of radia-

tion after interactions with matter. Problems of Class

II concern the fate of matter after interactions with

radiation.

Problems of Class I are exempliﬁed by the deter-

mination of the energy losses of particles penetrating

matter, and the attenuation of a beam of photons in

matter. Experimental studies on these problems are

straightforward in principle; one only needs to analyze

the kinetic energies of transmitted particles in the ﬁrst

problem and to count the number of transmitted photons

in the second problem. Problems of Class I are also often

simple to treat theoretically; for instance, in the Bethe

theory of stopping power, discussed in Chapt. 91,the

use of sum rules enables one to bypass detailed knowl-

edge about excited and ionized states. As a consequence,

many of the problems of Class I have been solved. Fur-

thermore, many of the applications of radiation physics

are based on the established knowledge of solutions of

problems of Class I. For instance, diagnostic uses of

X-ray photons in medicine and industry rest on the ﬁrm

knowledge of the attenuation of photons in matter (more

precisely, on its dependence on the atomic number).

In sharp contrast, problems of Class II are funda-

mentally difﬁcult; indeed, none of them has been solved

completely. Reasons for the difﬁculty are manifold.

First, any piece of irradiated matter is a new material that

needs to be fully characterized. Second, irradiated mat-

ter is in general in a nonequilibrium state that changes

physically and chemically with time. Finally, existing

tools and techniques, both experimental and theoretical,

for material characterization are limited.

To illustrate the nature of Class II problems, it is

useful to consider a prototype problem that has been

treated reasonably well, but certainly not completely.

Suppose a single electron with energy 10 keV enters

a dilute hydrogen gas of a sufﬁciently large volume.

The incident electron will collide with a hydrogen mol-

ecule. Possible outcomes include excitation of molecular

states, ionization leading to H

in various vibrational

and rotational states, and ionization leading to dissoci-

ation into H

+ H (in various excited states or in the

ground state). An electron resulting from an ionizing

event may have enough kinetic energy to cause further

excitation and ionization. Eventually, all the liberated

electrons will lose enough energy to become subexcita-

tion electrons unable to excite even the lowest electronic

level of H

at 9.5eV. Table 92.1 summarizes the vari-

ous atomic and molecular species produced. To produce

this answer, it was necessary to survey all the spec-

troscopic and electron collision data on H

,andto

Part G 92.2

Radiation Physics 92.2 Radiation Absorption and its Consequences 1391

Class-I Problems:

Fate of radiation

Class-II Problems:

Fate of matter

Matter

Incident

radiation

Fig. 92.1 Schematic diagram showing the deﬁnition of

Class I and Class II problems. Incident radiation comes

from the lower left, interacts with matter, and trav-

els toward the upper right, having been scattered. The

crosses symbolize interaction events taking place within

matter

carry out an analysis of electron transport. Problems

of this kind will be further discussed in Sect. 92.3.2 and

Sect. 92.3.3.

92.2.2 Photons

High-energy photons (i. e. X-rays, γ -rays, or a major

part of synchrotron radiation), are important as the ion-

izing radiation most frequently used in applications.

Their initial interactions with matter result in a spec-

trum of energetic electrons. Detailed discussion is given

in Chapts. 61 and 62.

92.2.3 Charged Particles

Energetic charged particles penetrating matter cause

ionization and electronic excitation of atoms and

molecules as they lose kinetic energy (Chapt. 65). The

most fundamental index to characterize the energy

loss process is the total cross section for inelastic

collisions, which is the mean number of inelastic col-

lisions per unit path length in a material of unit

molecular density. The next fundamental index is the

stopping power, which is the mean energy loss per

unit path length. For many purposes, a more de-

tailed description of particle penetration is necessary.

An example is energy loss straggling, fully discussed

in Chapt. 91.

Ionization is generally accompanied by the pro-

duction of secondary electrons, which in general have

a broad energy spectrum. This topic is reviewed by Rudd

and coworkers [92.9–11].

92.2.4 Neutrons

Neutrons with kinetic energies in the MeV domain

cause nuclear reactions, generally leading to the pro-

duction of a number of charged particles. Neutrons

of any kinetic energies collide with nuclei in mat-

ter elastically (viz., without causing nuclear reactions)

and transfer substantial kinetic energy to nuclei. Con-

sequently, neutrons may be regarded as a source of

charged particles of various kinds, most importantly

protons and light nuclei, all having broad energy spec-

tra [92.12, 13].

Table 92.1 The mean number N

of initial species pro-

duced in molecular hydrogen upon complete degradation

of an incident electron at 10 keV, and the energy absorbed

abs

(in percent).

The left column of this table indicates

kinds of species tersely. For instance, “Lyman” here means

the B

state, which emits the Lyman band. The term

“Werner” means the C

state, which emits the Werner

band. The designation “H(2p)” means the production of hy-

drogen atoms in the 2p state. The designation “Slow H(2s)”

means the production of hydrogen atoms in the 2s state with

no appreciable kinetic energy. The designation “Fast H(2s)”

means the production of hydrogen atoms with kinetic en-

ergies of several eV. The designation “Triplet” means all

the triplet states combined. The designation “Subexcitation

electrons” means all the electrons that are not energetic

enough to cause further electronic excitation

Initial species j N

abs

) (Lyman) 96.8 11.7

) (Werner) 112.1 14.0

H(2p) 5.6 0.8

Slow H(2s) 29.6 4.4

Fast H(2s) 5.3 1.7

H(n = 3) 3.2 0.5

Higher excited states 12.2 1.7

Ions 295.6 45.9

Triplets 102.6 10.0

Subexcitation electrons 295.6 9.4

Adapted from Douthat [92.8]

Part G 92.2

1392 Part G Applications

92.3 Electron Transport and Degradation

92.3.1 The Dominant Role of Electrons

In many Class II problems (i. e., radiation actions on

molecular substances such as gases, molecular liquids

or solids, and biological cells), energetic electrons gen-

erated by ionizing radiation play a central role because

they are numerous, and deliver to atoms and molecules

a large fraction of the total energy of radiation. What

follows is a resume of our current understanding.

92.3.2 Degradation Spectra

and Yields of Products

The prototype H

problem discussed in Sect. 92.2.1 may

be generalized as follows. Consider the total number N

of a particular molecular species j produced as a result

of complete slowing down of electrons in a chemically

pure medium consisting of n molecules per unit volume.

The product species j could be ions or excited states

speciﬁed by a set of quantum numbers. Let σ

(T ) be

the cross section for the production of j in a collision

of an electron of kinetic energy T with a molecule.

During its passage over an inﬁnitesimal path length dx,

an electron of kinetic energy T contributes the amount

= nσ

(T ) dx to N

. Therefore, one may write

= n



(T ) dx . (92.1)

To carry out this integration, one must have a relation

between T and x. To the extent that one regards x and T

as related through an analytic function, one may write

dx = ( dT / dx)

−1

dT , (92.2)

where dT / dx is the mean energy loss per unit path

length traversed, or the stopping power as deﬁned

in Chapt. 91. This treatment, called the continuous-

slowing-down approximation (CSDA), is justiﬁed if T

greatly exceeds the majority of energy losses upon

individual collisions.

However, the CSDA is inadequate for electrons in

general because the energy loss of an electron upon an in-

dividual collision is not always small compared with the

current kinetic energy, and because the CSDA does not

account for secondary electrons, which are abundantly

produced. An appropriate formulation, introduced by

Spencer and Fano [92.14], is obtained by writing

dx = y(T ) dT

(92.3)

in place of (92.2) and determining the function y(T )

through full analysis of electron transport, production,

and slowing down. This function, called the degradation

spectrum, has been an object of extensive studies [92.6,

15]. Once y(T ) is determined, the yield N

of product

species j is evaluated as

= n

max



1, j

(T )y(T )dT , (92.4)

where E

1, j

is the threshold energy for the production

of j,andT

max

is the highest kinetic energy of electrons

in the medium.

Three basic properties of y(T ) are as fol-

lows [92.15]. First, y(T ) is proportional to the electron

energy distribution (as treated by the Boltzmann equa-

tion) multiplied by electron speed v,or(2T /m)

1/2

the nonrelativistic case.

Second, y(T ) obeys an integral equation called the

Spencer–Fano equation. This equation expresses the

balance of the number of electrons arriving at kinetic

energy T with the number of electrons departing from

it. One may write the cross section for a collision pro-

cess in which an electron of kinetic energy T collides

with an atom or molecule and an electron of kinetic

energy T



emerges as σ(T → T



). Then, the expres-

sion



σ(T



→ T )y(T



)dT



−



σ(T → T



)dT



y(T )

represents the change in the number of electrons at

kinetic energy T due to all collisions in a medium

of unit density. For convenience, one may write the

above expression as K

y(T ) using a linear operator K

called the collision operator. In the simple case where

a source steadily generates u(T ) dT electrons having

kinetic energies between T and T + dT in a medium

consisting of n atoms or molecules of a single species,

the Spencer–Fano equation takes the form

y(T ) + u(T ) = 0 . (92.5)

Third, in the domain E

< T < T

max

,whereE

the lowest electronic-excitation threshold energy of the

medium, y(T ) is invariably bimodal, as exempliﬁed

by Fig. 92.2. The behavior at high T is largely under-

standable from the CSDA (92.2), which approximately

holds at T  E

; then, y(T ) is the reciprocal of the

stopping power, which is given by the Bethe theory

and corrections as explained in Chapt. 91.Forexam-

ple, for electrons with T between 1 keV and 10 keV,

the stopping power of molecular hydrogen is a mono-

tonically decreasing function of T , and therefore y(T )

is a monotonically increasing function of T , as seen

in Fig. 92.2. In contrast, the steep rise of y(T ) at low

Part G 92.3

Radiation Physics 92.3 Electron Transport and Degradation 1393

T occurs because most of the secondary electrons pro-

duced in an ionizing collision of any energetic charged

particle (including an energetic electron) have low ki-

netic energies, corresponding to a few multiples of the

ionization threshold energy, as fully documented by

Rudd et al. [92.9–11].

As T varies, the principal contribution to the yield

expression (92.4) comes from ranges where both σ

(T )

and y(T ) are large. A large y(T ) means a long path

length of electrons at that T , and hence a modest energy

loss per unit path length. From (92.2), the time interval

dt during which electrons stay in the energy interval dT

dt = ( dt/ dx)

−1

dx = v

−1

y(T )dT . (92.6)

The quantity v

−1

y(T )dT represents the sojourn time

of electrons at energy T . Equation (92.4) expresses

how N

is determined by the competition between

electron degradation, and the cross section σ

(T ) for

production of j upon individual collisions. The product

(T )y(T ), called the yield spectrum for production

of j, plays a central role in the theory.

Since T ranges from a few eV up to keV or MeV, it is

convenient to rescale the yield spectrum to ln T . In fact,

ln T is more meaningful as a variable because it rep-

resents, in effect, the mean number of elastic collisions

required to reduce T by a given amount. (In the neutron

slowing down theory [92.16], ln T is called lethargy by

Fermi.) Equation (92.4) then becomes

= n



T σ

(T )y(T )d(ln T ), (92.7)

–2

–3

–4

–5

20 50 100 200 500

1000

2000

5000

10 000

20 000

345678910

ln T(eV)

y(T

, T) (cm/ eV) T(eV)

Fig. 92.2 Electron degradation spectrum in molecular hy-

drogen. Source electrons of kinetic energy T

max

= 10 keV

steadily enter the hydrogen gas at 0

◦

C and 1 atmosphere

at the rate of 1 electron

−1

−3

. Data are taken from

Douthat [92.8]

with T σ

(T )y(T ) regarded as a function of ln T .

The product T σ

(T ) is proportional to the collision

strength in the theory of atomic collisions, as treated

in Chapt. 47. It is also a key quantity in the Bethe

theory [92.17].

Figure 92.3 is an example of a yield spectrum for ion-

ization in molecular hydrogen. The area under the curve

over any given interval of ln T represents the number

of ions produced in collisions of electrons in that inter-

val. One sees a sizable contribution from high energies

(between 1 keV and 10 keV). Below 1 keV, where the

spectrum is roughly constant, each decade of T con-

tributes roughly the same amount to the total ionization

yield. This observation applies to any product j that

results from dipole-allowed transitions.

In contrast, the yield spectrum for a product j

resulting from forbidden transitions is dominated by

values at low T ,whereσ

(T ) is appreciable, as

Fig. 92.4 illustrates. An example is the production of two

ground-state hydrogen atoms from a hydrogen molecule,

which occurs solely from the lowest triplet repulsive

state.

Finally, once electrons fall below E

, they moder-

ate much less rapidly than they do above E

.These

subexcitation electrons [92.18], lose energy through mo-

mentum transfer upon elastic collisions with molecules,

rotational excitation, and vibrational excitation, as dis-

cussedinChapt.47. The behavior of the subexcitation

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

20 50 100 200 500

1000

2000

5000

10000

20000

ln T(eV)

34 5678910

T)Tσ

(T)/T

(eV

–1

) T(eV)

Fig. 92.3 The yield spectrum for ionization in molecular

hydrogen. Source electrons are the same as for Fig. 92.2.

The vertical axis represents ny(T

max

, T )T σ

(T )/T

max

where n is the number density of molecules, T

max

= 10 keV

is the source-electron energy, and σ

(T ) is the ionization

cross section of H

for electrons of energy T .Dataare

taken from Douthat [92.8]

Part G 92.3