NBSIR 77-1283

Spectra From Energy

Degradation of Fast Electrons

in Water

Martin J. Berger

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

August 1977

Interim Report for Period October 1, 1976 to July 15, 1977

Prepared for Office of Naval Research

Arlington, Virginia 22217

and

Energy Research and Development Administration

Washington, D.C. 20545

NBSIR 77-1283

SPECTRA FROM ENERGY

DEGRADATION OF FAST ELECTRONS

IN WATER

Martin J. Berger

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

August 1977

Interim Report for Period October 1 , 1 976 to July 1 5, 1 977

Prepared for Office of Naval Research

Arlington, Virginia 22217

and

Energy Research and Development Administration

Washington, D.C. 20545

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Or. Sidney Harman, Under Secretary

Jordan J . Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Acting Director
i 'S

' '

!

' •***

• :

•

‘ , n . _

-'V
SPECTRA FROM ENERGY DEGRADATION OF FAST ELECTRONS IN WATER*

1

Martin J. Berger

ABSTRACT

This report gives data for energy degradation spectra in

water vapor. Calculations utilized the Monte Carlo method and

included Auger electrons. Results include 1) a new energy scaling principle at low energies, 2) values at different source energies for W, the mean energy loss per ion pair formed, 3)

data on the electron range calculated by continuous-slowing-down

approximation, and 4) data on stopping powers and restricted

stopping powers as compared with results from the Bethe theory.

Work supported by Office of Naval Research, Arlington, Va. and

U.S. Energy Research and Development Admin., Washington, D.C.

+ This is an interim report and the work is continuing. The report

is to be superseded by a future publication which will receive

general distribution and should be cited as a reference. Please

consult the NBS Office of Technical Publications to obtain the

proper citation.

INTRODUCTION

Fast electrons lose energy to the surrounding medium by many kinds of

interactions, including a) knock-on processes with individual electrons

which follow approximately the Miller cross section, and b) glancing

processes for which the electric pulse generated when a fast electron

passes an atom can stimulate both excitation and ionization.

The cumulative result of these interactions is the progressive loss of

energy, often called energy "degradation," by the fast electrons.

A source of electrons can thus give rise to electrons of all lower

energies; and these electrons include not only the "primaries" whose

energies have been degraded, but also "secondaries" produced in ionizing

interactions and defined to be the free electron of lower energy of

the two which emerge from such interactions. The resulting spectrum

is commonly called the "degradation" spectrum; and it has importance in

many fields dealing with radiation effects on matter.

Since the early exploratory calculations of Spencer and Fano (1954),

calculations of these spectra for a variety of materials have

been made; although the requirement for a complete set of inelastic

cross sections at all energies below the source energy T makes these

calculations difficult. In this brief interim report we give many of our results on degradation spectra for water vapor. We have not yet

completed comparisons with other calculations, including those for

comparable materials; these will be included in the final publication

on this subject.

SUMMARY DESCRIPTION OF THE CALCULATIONS

The Monte Carlo method has been applied to the calculation of energy

degradation spectra in water vapor. Calculations have been made at a large

number of energies between 100 keV and a few eV. The successive inelastic

events which an electron makes in the course of slowing down in an extended

medium have been simulated in direct analogy to the physical processes, and

all generations of secondary electrons were followed. The number of electrons

whose energy histories were simulated varied depending on the initial energy

T

q

of the primary electron; for example:

T , eV

o No. of secondaries followed

No. of primaries followed per primary, on the average

100,000

10,000

1,000

100

500

2,000

10,000

3270

331

33

2.8

As far as possible, the various inelastic events were sampled according to cross sections derived from experiments. The sources of cross section data were as follows:

Excitation

Dissociative excitation

Dissociative attachment

Ionization (total)

Ionization (K-shell)

Spectrum of Auger electrons

Ionization (differential)..

Olivero, Stagat and Green (1972);

Kutcher and Green (1976)

Beenakker, de Heer, Krop and Mohlmann

(1974)

Compton and Christophorou (1967)

Schutten, de Heer, Moustafa, Boerboom

and Kistemaker (1966) ; Maerk and Egger

(1975)

Glupe and Mehlhorn (1971)

Moddeman, Carlson, Krause and Pullen

(1971)

Opal and Beaty (1972)

1
Altogether 27 possible modes of excitation were taken into consideration, and use was made of the very convenient parameterization of the excitation cross sections by Green and collaborators. The Opal-Beaty ionization cross section,

differential in the secondary electron energy, was available only at 500 eV.

It was fitted at that energy by a simple analytical formula which is a modification of the Miller cross section (for scattering by free electrons) and

which takes into account exchange effects). Following Kim (1975) it was assumed that the spectrum of energy transfers in glancing collisions is

determined predominantly by the oscillator-strength distribution, and is

nearly the same at all energies. The simple analytical formula was therefore

retained, but was renormalized at each energy to obtain agreement with the

experimental total ionization cross section. At spectral energies from 200

to 600 eV the ionization cross section was made to go over into the Miller

cross section. Ionization was assumed to be possible from five orbitals, with

binding energies of 12.6, 14.7, 18.4, 32.2 and 540 eV. The relative probability of electron ejection from the K-shell (followed by emission of an Auger electron) was estimated from experimental data; the relative probabilities

of ejection from the other orbitals was estimated from a Weizsacker-Williams

calculation by Seltzer (private communication)

.

RESULTS

We summarize here certain significant conclusions that can be drawn from

the analysis of results performed so far, and back them up with information

in tabular form.

1. As shown in Table 1, the energy degradation spectra calculated

with the use of detailed inelastic collision cross sections are in

good agreement at high and intermediate energies with the results

of the Spencer-Fano (1954) theory. The latter assumes energy

transfers to be governed by the Miller cross section, adjusted

with a cut-off on low-energy transfers to give the correct stopping

power. It turns out that this approximation gives good results for

spectral energies down to 200 eV; below this energy the SpencerFano theory is no longer applicable. This presumably holds not only

for water vapor but also for other gases of similar atomic number.

Below 200 eV, the new Monte Carlo results are the appropriate

extension of the Spencer-Fano theory into the low-energy region of

interest for radiation chemistry yield studies.

2. A new and useful scaling law has been discovered, which applies at

low spectral energies, say, below a few hundred eV, and which greatly

reduces, and at low energies practically suppresses the dependence

of the energy degradation spectra on the initial electron energy.

It has been customary to represent the energy degradation spectrum

in terms of the differential tracklength distribution y(T,T ),

defined such that y(T,T )dT is the average tracklength of electrons

with energies between T and T+dT. The new scaled quantity is

T^ = T(100/T

o )(Na/p)y(T,T

o

)

.

2
Here T is the spectral energy and T

q

th^ source energy, both in

eV; N is the number of molecules per cm^, o is the total cross section for inelastic collisions, in cm , and p is the density of

the medium, in g/cm

£ y (T,T ) is the differential tracklength

distribution in g/cm per e?. The quantity dG/dT can be interpreted as the differential yield of inelastic collisions per unit

spectral energy, and is normalized to an energy input of 100 eV.

The dimensionless quantity T dG/dT is given in Table 2 for initial

electron energies T between 100,000 and 1000 eV. It can be seen that, as claimed, the dependence of T dG/dT on T becomes progressively

weaker as the spectral energy T decreases. This suggests that energy

degradation spectra for values of T much higher than 100 keV could

be obtained without much trouble by using the Spencer-Fano theory

down to 200 eV, and adding on the low-energy part of the spectrum

assuming a universal spectral shape. Finally, we note that the yield

G^(T

q

) of events of type i can be calculated from the expression

T

o G . (T ) = f

(dG/dT) (o./a)dT

1 o J 1

o

where a. is the cross section for the event of type i. Again, the

yield is normalized to an energy input of 100 eV.

3. Auger electrons emitted after ionization from the K-shell make a significant contribution to the energy degradation spectrum at

spectral energies below 500 eV. The magnitude of this contribution

is given in Table 3. Many previous treatments of the energy degradation spectrum, including that of Spencer and Fano, have omitted

the Auger electron contribution. Therefore the comparison given

in Table 2 does not include Auger electrons.

4. In the course of the Monte Carlo calculation of the energy degradation spectra, the yields of various events have also been

obtained, for example, the yield of ionizations. Table 4 gives

the mean number of ionizations as a function of the initial electron

energy, as well as the ionization fluctuations as summarized in

terms of the Fano factor, which is the ratio of the mean square to

the mean number of ionizations. If the fluctuations were governed

by a Poisson distribution, the Fano factor would be unity. As can

be seen from Table 4, this is the case only quite close to the

ionization threshold (12.6 eV) . At higher energies the Fano factor

assumed a limiting value of 0.25.

5. The value of W (mean energy needed to produce an ion pair in water

vapor) is shown in Table 5. It can be seen that W is almost

constant down to energies of a few hundred eV, whereupon it rises

at first slowly and then quite rapidly at lower energies. Between

10 keV and 100 keV there is a very slow rise of 1.1%. It would not

be altogether surprising if this rise continued as the electron

3
energy is further increased. Table 6 compares our value of W with

that obtained in other calculations, with differences presumably

arising mainly from the different assumed cross sections, but to some extent also from the calculational techniques. The experimental W-value for water vapor is given as 29.9 eV by Adler

and Bother (1965) and as 30.1 eV by Booz and Ebert (1963).

These values differ by 1.3% and 0.7%, respectively, from the result

at T =1 keV of the present calculation, differences which are well

within the combined limits of experimental and theoretical uncertainties.

6. The contribution of ionization events from different spectral

regions of the energy degradation spectrum is indicated in

Table 7. It can be seen that a significant contribution to the

ionization is made by high-energy as well as by low-energy electrons. Moreover, the relative contributions from different

spectral regions are seen to be rather insensitive to the initial

energy of the primary electron, if scaling by T is used for the

top half of the table.

7. From the inelastic scattering cross sections compiled for the

Monte Carlo calculation, one can also derive the electron stopping

power. Table 8 compares the results thus obtained with the values

given by the Bethe stopping-power formula (assuming a mean excitation energy of 71.7 eV for water vapor). Between 100 keV and

20 keV, there is no difference. At lower energies, the present

results fall below the results from the Bethe theory, by 1.3% at

10 keV, 6.8% at 5 keV, 8.3% at 2 keV and 11% at 1 keV.

8. The CSDA (continuous-slowing-down-approximation) range has been

computed from the new stopping-power values in Table 7, and is given

in Table 9. The range is computed as the pathlength which the

electron would travel until it reaches a cut-off energy T , assuming

that the energy loss along the entire path is always equaS to the

mean loss, i.e., the stopping-power. The results in Table 9 are for

a cut-off T = 12.6 eV, equal to the lowest ionization potential in

water vapor. Depending on the chosen value of T , different range

values are obtained; for example:

CSDA range, g cm

T , keV

o T = 4.5 eV

c T = 12.6 eVc T = 20.0 eV

c

100 1.522 x 10“ 2 1.432 x 10" 2 1.423 x 10~ 2

10 1.246 x 10' 2 2.828 x 10" 42.599 x 10“ 4

1 0.926 x 10“ 47.156 x 10" 6

6.500 x 10~ 6

9.

In radiological physics, it is often of interest ot know the "local"

energy deposition in terms of restricted stopping power, i.e., a mean energy loss per unit pathlength from events in which the energy

transfer from the electron to the molecule is smaller than some chosen

cut-off energy A. In cavity ionization theory, the A is typically

4
10 to 15 keV, and the Bethe theory for the restricted stopping power can be applied above ~ 10 keV in water. In track structure theory

in radiobiology, cut-off energies A as low as 100 eV are considered.

As is shown in Table 10, the cross sections assumed in the present

work then lead to significant departures from the predictions of the

Bethe theory, even for electron energies as high as 100 keV.

REFERENCES

Adler, P. and Bothe, H. K. , (1965), Zeit. Naturforsch 20a , 1700.

Beenakker, C.I.M. , De Heer, F. J. , Krop, H. B., and Mohlmann, G. R.

,

(1974) Chem. Phys. b_, 445.

Compton, R.N., and Chr istophorou , L.G., (1967), Phys. Rev. 154 , 110.

Booz, J. and Ebert, H.G. (1963), Strahlentherapie , 120 , 7.

Glupe, G. and Mehlhorn, W. (1971), J. Phys. 4_, 40.

Kim, Y. K. (1975), Radiat. Res. 61^, 21; 6A_, 205.

Kutcher, G. J. and Green, A.E.S., (1976), Radiat. Res. 67, 408.

Moddeman, W.E., Carlson, T.A. , Krause, M.O., and Pullen, B.P. (1971),

J. Chem. Phys. J15, 2317.

Olivero, J.J., Stagat, R.W. , and Green, A.E.S., (1972), J. Geophys. Res.

JJ_, 4797

Opal, C.B., Beaty, E.C., and Peterson, W. K. , (1972) Atomic Data 4_, 209

Paretzke, H. W. , (1976), Proc. 5th Symposium on Microdosimetry, Euratom

Publication 5452 d-e-f.

Schutten, J., De Heer, A. J. , Moustafa, H.R. , Boerboom, A.J.H., and

Kistemaker, J. (1966), J. Chem. Phys. 44_, 3924.

Spencer, L.V and Fano, U., (1954), Phys. Rev. 93^, 1172.
.

'
TABLE 1

fy(T,To

) s ( t)] j

[y (T »Tq

) F(T,T

o )]

Spencer-Fano

1 T/T

° T = 100 keVo 50 keV 20 keV 10 keV 5 keV 2 ke\

l/2-

l/6 1.048 1.059 1.088 1.098 1.075 1.117

l/2- 2/6 0.988 0.996 1.012 1.003 0.983 1.012

l/2-3/6 0.966 0.992 0.989 0.983 0.979 0.986

l/2‘4/6 0.963 1.000 0.988 0.989 0.982 0.968

l/2-5/6 0.972 1.005 0.994 0.997 0.971 0.966

1/2 0.983 0.993 0.996 0.997 0.962 0.970

1/4 1.013 1.013 0.985 0.986 0.997 0.981

1/8 0.991 0.998 0.999 0.991 0.995 0.953

1/16 1.001 0.970 0.983 0.997 1.008

1/32 0.964 0.966 0.997 1.010 0.890

1/64 0.970 0.972 1.031 0.921

1/128 0.964 1.001

1/256 0.997 1.050

1/512 1.068

Comparison of energy-degradation spectra from the present work and the

corresponding spectra from the Spencer-Fano theory. T

q

is the source

energy, T the spectral energy, y(T,T

Q

) the differential tracklength

distribution, S(T) the electron stopping power, and F(T,T

q

) the quantity

which plays the role of a stopping power in the Spencer-Fano theory.
TABLE 2 /

T dG ( T , T )/dT

o

T

T

o=i00,000 50,000 20,000 10,000 5000 2000 1000 ey

(eV)

50,000 1.03

20,000 0.462 0.822

10,000 0.274 0.475 1.08

5000 0.217 0.300 0.589 1.14

2000 0.200 0.230 0.345 0.552 1.00

1000 0.222 0.243 0.309 0.405 0.613 1.35

500 0.272 0.286 0.327 0.376 0.485 0.838 1.52

200 0.513 0.509 0.529 0.522 0.550 0.653 0.882

100 0.684 0.692 0.691 0.69 7 0.707 0.740 0.832

50 1.00 1.00 1.01 1.01 1.02 1.03 1.06

20 1.60 1.61 1.61 1.62 1.64 1.65 1.66

10 1.88 1.88 1.89 1.90 1.89 1.91 1.92

5 1.73 1.73 1.74 1.75 1.77 1.78 1.80

Scaled energy <degradation spectra in water vapor. The quantity dG/dT

is the differential yield, per spectral energy interval, of inelastic

collisions produced by the primary electrons and all their secondaries.

The differential yield is normalized to an energy input of 100 eV

into the medium.
TABLE 3

ENERGY PERCENT CONTRIBUTION OF AUGER ELECTRONS

INTERVAL

T

(eV)

= 100

o 50 20 10 5 2

500 - 490 55.0 50.8 44.1 35.4 25.1 8.4

490 - 480 39.3 35.3 30.4 22.4 12.7 4.6

400 - 380 27.9 26.0 22.4 15.6 11.5 3.4

300 - 280 20.9 19.2 16.1 11.1 8.9 3.3

200 - 190 13.3 12.3 11.3 8.1 5.4 2.4

100 - 96 7.1 6.5 6.1 5.2 3.6 2.0

50 - 48 5.7 5.1 4.4 3.0 2.4 1.3

30 - 28 5.0 4.6 4.0 2.8 2.3 1.0

20 - 19 4.9 4.4 3.9 3.0 2.4 1.1

10 - 9.6 4.9 4.4 4.0 2.8 2.4 0.8

5 — 4.8 4.9 4.4 3.8 2.6 2.4 1.1

Percent contribution by Auger electrons (and their secondaries) to the

energy degradation spectrum. The contributions shown are averaged

over the indicated energy intervals.
TABLE 4

T

o

n.

l

2 / n. / n.

i l

,12.6 0.0 1.00

13 0.037 0.96

14 0.139 0.86

15 0.213 0.79

20 0.347

i 0.65

50 1.225 0.40

100 2.822 0.33

200 6.152 0.28

500 16.24 0.25

1000 32.98 0.24

2000 66.33 0.24

5000 165.9 0.24

10,000 330.7 0.25

20,000 659.1 0.25

50,000 1639. 0.25

100,000 3270. 0.25

2 Mean number of ions produced, n^ , and Fano factor f = n^ / n_^

,

for electroas of initial energy T

q

slowed down in water vapor.
TABLE 5

T W

o (eV) (eV)

20 57.67

30 47.07

50 40.81

100 35.43

200 32.51

300 31.54

500 30.78

1000 30.32

2000 30.15

3000 30.17

5000 30.14

10,000

20,000

30.000

50.000

100,000

30.23

5

30.34

5

30.44

5 3Q.51

5

30.58

Average energy needed for the production of an ion pair, W, for

electrons with initial energy T

q

in water vapor.
TABLE 6

T

o

w. eV

(eV) OSG KG P Present

Work

20 73.6 82.5 44.6 57.7

40 33.3 43.2 36.9 43.2

60 26.9 37.9 34.6 39.0

80 25.4 35.8 33.4 37.0

100 25.0 34.9 32.7 35.4

200 25.0 33.2 31.2 32.5

400 25.0 31.7 30.6 31.0

600 25.0 30.9 30.5 30.6

800 25.0 30.6 30.4 30.4

1000 25.0 30.0 30.4 30.3

Value of W, the mean energy needed to produce an ion pair,

as calculated by various authors.

OSG: Olivero, Stagat and Green (1972)

KG: Kutcher and Green (1976)

P: Paretzke (1975)

The results of Kutcher and Green are for liquid water; all the

others are for water vapor.
TABLE 7

Spectral energy T, keV

Percent

100 100.0 10.0 1.0

90 81.1 8.41 0.921

80 56.6 6.12 0.765

70 29.5 3.96 0.588

60 7.43 1.95 0.438

50 0.802 0.662 0.300

40 0.249 0.244 0.175

30 0.101 0.100 0.0921

20 0.0494 0.0490 0.0481

10 0.0276 0.0274 0.0270

0 0.0126 0.0126 0.0126

T

o 100 10 1 keV

Cumulative

/contributions Qf ionization events from various spectral regions of

the energy degradation spectrum. For example, for an electron with

initial energy T

q = 100 kEV, 50 percent of the ionization is produced

by electrons (primary and secondaries) with energies less than 0.802 keV.
TABLE 8

T, keV dE/ dx,

(a)

_2

MeV/g cm

(b)

T, keV dE/ dx

,

(a)

_2

MeV/ g cm

(b)

100 4.14 4.14 0.3 200. 230.

50 6.65 6.65 0.2 221. 257.

• 30 9.73 9.73 0.1 221.

20 13.3 13.3 0.05 130.

10 22.5 22.8 0.03 67.7

5 36.3 38.8 0.02 27.8

3 53.5 56.9 0.01 2.86

2 70.4 76.2 0.005 0.035

1 110. 122. 0.003 0.0021

0.5 165. 183. 0.002 0.013

Stopping Power for Electrons in Water Vapor

(a) Present work; derived from detailed

cross sections for ionization and

excitation.

(b) From Bethe stopping-power formula,

evaluated with mean excitation

energy I = 71.7 eV.
TABLE 9

ELECTRON CSDA ELECTRON CSDA

ENERGY RANGE ENERGY RANGE

(eV) (g cm

2

) (eV) (g cm

- 2

>

100,000 1.432 X io

" 2 1000 7.156 X io

- 6

80,000 9.723 X Hf3

800 5.446 X Hf6

60,000 5.913 X Kf3

600 3.980 X io

- 6

50,000 4.303 X io

-3

500 3.343 X io

- 6

40,000 2.911 X io

" 3

400 2.764 X io

' 6

30,000 1.756 X io

“ 3

300 2.236 X io

-6

20,000 8.635 X io

'4

200 1.761 X io

' 6

15,000 5.234 X io

~ 4

150 1.540 X i<f

6

10,000 2.828 X io

-4

100 1.319 X io

" 6

8000 1.799 X io

~ 4

80 1.223 X io

- 6

6000 1.096 X io

- 4

50 1.037 X 10‘6

5000 8.095 X io

“ 5

40 9.509 X io

' 7

4000 5.615 X io

" 5 30 8.333 X io

' 7

3000 3.538 X io

' 5

20 6.103 X io

" 7

2000 1.890 X io

" 5 15 3.241 X io

-7

1500 1.240 X io

" 5 12.6 0.0

Electron range in water vapor.
TABLE 10

A

100 keV 50 keV

(eV) (a) (b) %; DIFF (a) (b) % DIFF

400 0.6940 0.6672 3.9 0.7167 0.6855 4.4

200 0.6467 0.6001 7.2 0.6649 0.6160 7.3

100 0.5993 0.5267 12.1 0.6128 0.5408 11.8

50 0.5519 0.4551 17.5 0.5607 0.4674 16.6

A

30 keV 20 keV

(eV) (a) (b) %; DIFF (a) (b) % DIFF

400 0.7355 0.6982 5.1 0.7522 0.7088 5.8

200 0.6797 0.6265 7.8 0.6929 0.6352 8.3

100 0.6237 0.5499 11.8 0.6331 0.5574 11.9

50 0.5675 0.4750 16.3 0.5731 0.4812 16.0

A

10 keV 5 keV

(eV) (a) (b) 2; DIFF (a) (b) % DIFF

400 0.7854 0.7393 5.9 0.8262 0.7854 4.9

200 0.7191 0.6616 8.0 0.7518 0.7029 6.5

100 0.6518 0.5804 11.0 0.6752 0.6168 8.6

50 0.5840 0.5006 14.3 0.5974 0.5316 11.0

Ratio of the restricted to the total electron stopping power in water

vapor, as a function of the cut-off energy A .

(a) Calculated according to the Bethe theory

(b) Derived from the excitation and ionization cross sections used in the present work.

Note that the restricted stopping power excludes energy transfers

in which the secondary electron acquires a kinetic energy greater than A
^BS-IIAA (REV. 7-73)

i U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 77-1283

2. Gov’t Accession

No.

3. Recipient’s Accession No.

4. TITLE AND SUBTITI H

Spectra from Energy Degradation of Fast Electrons in Water

5. Publication Date

6. Performing Organization Code

7. AUTHORISE

I Martin J. Berger

8. Performing Organ. Report No.

). PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2401437/2401439

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP

)

Office of Naval Research, Arlington, Va. 22217

U.S. Energy Research & Development Admin., Wash., D.C. 20545

13. Type of Report & Period

Covered

Interim report - Oct. 1976-Julv 1977

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less [actual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report gives data for energy degradation spectra in water vapor.

Calculations utilized the Monte Carlo method and included Auger electrons.

Results include 1) a new energy scaling principle at low energies, 2) values

difference source energies for W, the mean energy loss per ion pair formed,

3) data on the electron range calculated by continuous-slowing-down approximation, and 4) data on stopping powers and restricted stopping powers as compared with results from the Bethe theory.

7. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Charged particle transport; electron energy degradation; electron penetration; slowing-down spectra; stopping power in water; W-value in water.

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