Scholars' Mine
Doctoral Dissertations
Student Research & Creative Works
1971
Low-energy electron transport by the method of
discrete ordinates
David E. Bartine
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LOW-ENERGY ELECTRON TRANSPORT BY THE
METHOD OF DISCRETE ORDINATES
by
DAVID ELLIOTT BARTINE, 1936-
A DISSERTATION
Presented to the Faculty of the Graduate School of the
UNIVERSITY OF MISSOURI-ROLLA
In Partial Fulfillment of the Requirements for the Degree
DOCTOR OF PHILOSOPHY
in
NUCLEAR ENGINEERING
1971
j
i
ABSTRACT
The one-dimensional discrete ordinates code ANISN has been adapted
to simulate the transport of low-energy (on the order of a few MeV)
electrons.
Two different calculational techniques have been utilized
for the treatment of electron-electron collisions that result in a
small energy transfer.
One method treats such collisions by a contin-
uous slowing-down approximation, while the other method treats these
collisions by the use of a very approximate cross section.
Calculated
results obtained with ANISN are compared with experimental data for the
transmitted energy and angular distributions for 1-, 2.5-, 4-, and 8MeV electrons normally incident on aluminum slabs of various thicknesses and for 1-MeV electrons normally incident on a gold slab.
The
calculated and experimental results are in reasonably good agreement
for the aluminum slabs but are in poor agreement for the gold slab.
Calculated results obtained with ANISN are also compared with calculated results obtained with Monte Carlo methods.
iii
ACKNOWLEDGMENTS
The author would like to express his sincere appreciation to many
people at the Oak Ridge National Laboratory, particularly, Dr. Rufard
G. Alsmiller, Jr. of the Neutron Physics Division, who directed this
investigation, for his guidance and expertise.
Gratitude is expressed
to Dr. Fred R. Mynatt, to Joseph Barish and Dr. Yaakov Shima who aided
in programming the cross sections, and to Ward W. Engle, Jr. who aided
in adapting ANISN for the continuous slowing-down approximation.
This
investigation was enhanced by their many helpfUl discussions, suggestions, and continued interest.
The comprehensive review of this
dissertation by Dr. Frances S. Alsmiller and the editing by Mrs. Helen
D. Cook and typing by Mrs. Ann Keating, Mrs. Judy Spore and Mrs. Beth
Christian are greatly appreciated.
This research was carried out at the Oak Ridge National Laboratory
and was fUnded by the National Aeronautics and Space Administration,
Order H-38280A, under Union Carbide Corporation's contract with the
U.S. Atomic Energy Commission.
The author is gratefUl for the oppor-
tunity to perform this investigation at the Laboratory where the
necessary facilities were available.
Special appreciation is expressed by the author to his advisor,
Dr. Doyle R. Edwards, for his guidance and direction extended throughout the author's enrollment at the University of Missouri-Rolla, and
for his assistance in obtaining financial support for the author and
his family during this time.
The financial assistance provided by the
iv
U.S. Atomic Energy Commission in the form of a Traineeship and an Oak
Ridge Associated Universities Special Fellowship is gratefUlly acknowledged.
Finally, the author would like to express his deepest thanks to
his parents, Rev. David F. Bartine and Mrs. Elda J. Bartine, for the
awakening and stimulation of academic interest which ultimately led to
this investigation and for their continuing support, and especially to
his wife, Judith, for her willingness to leave an established life and
for her subsequent understanding, encouragement, and patience.
v
TABLE OF CONTENTS
Page
ABSTRACT . . . . .
ii
ACKNOWLEDGMENTS
iii
LIST OF ILLUSTRATIONS
vii
LIST OF TABLES . . .
I.
II.
III.
IV.
xi
INTRODUCTION .
1
LITERATURE SURVEY
3
DISCRETE ORDINATES TRANSPORT EQUATIONS .
9
TRANSPORT CROSS-SECTION DATA . . . . .
A.
. . . .
Inelast ic Electro nic Scatter ing from Atomic
Electro ns
B.
30
Elastic Coulomb Scatter ing from Atomic Nuclei
30
37
c. Bremsst rahlung (Radiat ive) Interac tions with Nuclei
and Atomic Electron s
D.
v.
VI .
Photon Interac tions
38
40
COMPARISON WITH THEORETICAL MODELS
41
A.
Elastic Multipl e-Scatte ring Angular Distrib ution
41
B.
Inelast ic-Scat tering Energy Distrib ution .
41
COMPARISON WITH EXPERIMENTAL RESULTS
51
A.
Experim ents of Rester
51
1.
1-MeV Electron s Inciden t on Aluminum .
51
2.
1-MeV Electro ns Inciden t on Gold . .
61
3.
2.5-MeV Electron s Inciden t on Aluminum .
64
vi
Page
B.
VII.
Experime nts of Lonergan et.al . . . . . . . .
69
l.
4-MeV Electrons Incident on Aluminum
69
2.
8-MeV Electrons Incident on Aluminum
72
COMPARISON WITH A THEORETICAL CALCULATION FOR AN
INCIDENT ELECTRON SPECTRUM . . . . . .
VIII.
IX.
75
A.
Transmit ted Electron Spectra .
75
B.
Transmitt ed Photon Spectra .
78
CONCLUSIONS AND RECOMMENDATIONS
80
APPENDICES . . . . . . . . . . .
82
A.
Derivatio n of the Continuou s Slowing-Down
Transpor t Equation . .
B.
High-Freq uency End-Poin t Correctio n for the
Differen tial Bremsstra hlung Cross Section
89
X.
LIST OF REFERENCES
92
XI.
VITA · · · · · · ·
97
vii
LIST OF ILLUSTRATIONS
Figure s
1.
Page
Angula r distrib ution of transm itted electro n curren t
for 1-MeV electro ns normal ly incide nt on a
2
0.0287 -g/cm -thick aluminu m slab . . .
2.
42
Compar ison of the differ ential Mpller cross section
for inelas tic electro n-elec tron collisi ons with
two differ ential analyt ic cross section s below l
MeV
3.
46
Transm itted electro n curren t per unit energy per
incide nt electro n for 1-MeV electro ns normal ly
incide nt on O.ll-g/ cm2 -thick and 0.33-g/ cm2 -thick
aluminu m slabs.
4.
.
. . . . .
48
Transm itted electro n curren t per unit energy per
incide nt electro n for 1-MeV electro ns normal ly
incide nt on 0.11-g/ cm2 -, 0.33-g/ cm2 -, and o.66-gj cm2 thick aluminu m slabs· . .
5·
50
Transm itted electro n curren t per unit energy per
incide nt electro n for 1-MeV electro ns normal ly
incide nt on 0.10-g/ cm2 -, 0.22-gj cm 2 -, and 0.32-g/ cm2 thick aluminu m slabs . . .
6.
52
Transm itted electro n curren t per unit energy per
incide nt electro n for 1-MeV electro ns normal ly
incide nt on 0.22-g/ cm2 -thick aluminu m slabs . ·
55
viii
Figures
7·
Page
Energy distributions of the transmitted electron
current at 7-5°, 47-5°, and 77-5° for 1-MeV
electrons normally incident on a O.lO-g/cm 2thick aluminum slab.
8.
59
Energy distributions of the transmitted electron current
at 7·5°, 47-5°, and 77-5° for 1-MeV electrons
2
normally incident on a 0.22-g/cm -thick aluminum
slab
9·
6o
Angular distributions of the transmitted electron
current for 1-MeV electrons normally incident
2
2
2
on 0.10-g/cm -, 0.22-g/cm -, and 0.32-g/cm -thick
aluminum slabs . . . . . . .
10.
(a)
Transmitted electron current per unit energy
per incident electron for 1-MeV electrons normally
2
incident on a 0-15-g/cm -thick gold slab.
(b)
Angular distribution of the transmitted electron
current for 1-MeV electrons normally incident on
2
a O.l5-g/cm -thick gold slab.
11.
63
Transmitted electron current per unit energy per
incident electron for 2-5-MeV electrons normally
2
2
incident on 0.31-g/cm -thick and o.62-g/cm thick aluminum slabs · · · ·
.. . . .. .... .
ix
Figures
12.
Page
Energy distribut ions of the transmitt ed electron
current at 20°, 45°, and 60° for 2.5-MeV
electrons normally incident on a 0.31-g/cm2 -
.
thick aluminum slab.
13.
.
.
67
Energy distribut ions of the transmitt ed electron
current at 10° and 20° for 2.5 MeV electrons
normally incident on a o.62-g/cm2 -thick
. . . . .
aluminum slab.
14.
.
68
Angular distribut ions of the transmitt ed electron
current for 2·5-MeV electrons normally incident on
2
0.3l-g/cm -thick and o.62-g/cm2 -thick aluminum
slabs. .
15.
(a)
Energy distribut ion of the transmitt ed electron
current at 30° for 4-MeV electrons normally
2
incident on a 1.275-g/c m -thick aluminum slab.
(b)
Angular distribut ions of the transmitt ed
electron current for 4-MeV electrons normally
2
incident on a l.275-g/c m -thick aluminum slab
16.
(a)
71
Energy distribut ion of the transmitt ed electron
current at 20° for 8-MeV electrons normally
incident on a 0.~53-g/cm2 -thick aluminum slab.
(b)
Angular distribut ion of the transmitt ed electron
current for 8-MeV electrons normally incident on
2
a 0.953-g/c m -thick aluminum slab . · · · . ·
73
X
Figures
17.
Page
Energy distribution of the incident electron
current per unit energy per incident electron
48
used by Scott
18.
as a source . . . .
Transmitted electron current per unit energy per
incident electron for a specific energy spectrum
2
(Fig. 17) normally incident on 0.50-g/cm -thick
2
and 1.0-g/cm -thick aluminum slabs.
19.
77
Transmitted photon current per unit energy per
incident electron for a specific energy spectrum
of electrons (Fig. 17) normally incident on
aluminum slabs of the thicknesses indicated.
79
Xl
LIST OF TABLES
Tables
I·
Page
Requirements for the Discrete Ordinates Calculation
Shown in Fig. 6. . . . . . . . . . . . . . . . .
57
1
I.
INTRODUCTION
The men and instruments aboard space vehicles must be protected
from the radiation encountered in extra-terrestrial flight.
A signif-
icant research effort has been under way for some time to discover the
identity, energy, and abundance of the particles involved, and to
1 2
determine their ability to penetrate shielding materials. '
A manned
space laboratory orbiting through the Van Allen electron belts would be
exposed to a large number of low-energy electrons.
A code is available
that treats low-energy electron transport by Monte Carlo methods.3
However, because of difficulties with the statistical accuracy obtained
in some cases, a nonstatistical calculational method is needed.
The
purpose of this investigation is to study the adaptability of the
method of discrete ordinates, which was developed for neutron transport, to the transport of low-energy electrons and the photons which
they produce.
Consideration here is limited to the energy range below 10 MeV,
since this is the area of primary concern for the shielding of space
vehicles.
However, this energy range is broad enough to be of general
interest.
Other problems to which the calculational method developed
here might be applied include the effect of multiple scattering on the
response of beta detectors and the effect of the energy fluctuations
resulting when a monoenergetic beam of electrons is incident on a thin
target (for example, in a device such as the electron microscope).
2
In Section II a basic discussion of electron penetration is
presented, and previous efforts made toward the solution of this
problem are reviewed.
The equations involved in the methods of solu-
tion used in this investigation are given in Section III.
The specific
forms of the cross sections and other parameters utilized in the
equations are discussed in Section IV.
Comparisons between the results
obtained from this investigation and results from other sources, both
calculational and experimental, are presented in Sections V, VI, and
VII.
A pure angular spectrum from multiple elastic scattering and a
pure energy spectrum due to electron straggling are considered in
Section V, and comparisons with experimental results are given in
Section VI.
Experimental data are limited, and none were available
to describe the penetration of an incident energy spectrum such as
that incident on spacecraft in the Van Allen belts.
A comparison with
another calculational method for the electron energy distribution
resulting from a fission spectrum source of electrons is presented in
Section VII.
Energy spectra for the photons produced by these elec-
trons and transported through large slab thicknesses are also included
here.
Conclusions and recommendations for fUrther investigation are
discussed in Section VIII.
The detailed derivation of the transport
equation with a term corresponding to the continuous slowing-down
approximation is given in Appendix A, and the derivation of the highfrequency end-point correction applied to the bremsstrahlung cross
section is shown in Appendix B.
3
II .
LITERATURE SURVEY
When electrons with energies of a few MeV penetrate matter, they
undergo a large number of collisions within a very short pathlength.
Since there are many possible energy and angular changes for each collision, this results in a distribution of electrons in terms of both
energy and direction of travel.
The most significant interactions for
the prediction of the resulting distribution by transport calculations
are elastic nuclear (Coulomb) scattering, inelastic scattering from
atomic electrons, and radiative (bremsstrahlung) interactions with
both nuclei and atomic electrons.
4
Birkhoff
described electronic inter-
actions and summarized the progress made on numerical models representing various aspects of electron transport.
More recently, Zerby and
Keller 5 presented a comprehensive state-of-the-art review of theoretical and experimental investigations in the area of electron transport.
Coulomb interactions are very frequent, resulting in an angular
distribution heavily peaked in the forward direction.
Since the mass of
the electron is minute compared to that of the nucleus, the energy loss
suffered by the electron is insignificant, and these collisions may be
considered elastic.
ing multiple -
Various methods have been developed for calculat-
6
Moliere ' 7 formu-
(Coulomb) scattering distributions.
lated a numerical function in terms of a reduced scattering angle to
describe the result of small angular deflections.
Goudsmit and
Saunderson5,S derived a Legendre series that can be evaluated for a
specific single-scattering cross section to give the distribution
resulting from angular deflections of any magnitude.
4
Electroni c collision s with atomic electrons resulting in a small
energy transfer and a correlate d small angular deflectio n are also
quite numerous .
The atomic electron involved is either elevated to an
excited state or ionized if the energy transfer is sufficien tly great.
Collision s involving a large energy transfer and angular deflectio n do
occur, but their frequency decreases as a function of increasin g energy
loss.
The secondary or knock-on electron produced in such a collision
becomes part of the transmitt ed spectrum.
High energy-lo ss reactions
are therefore particula rly important for an accurate determin ation of
the electron flux resulting at thickness es approachi ng the range of the
incident beam.
Inelastic scatterin g from atomic electrons is the pri-
mary mode of energy loss for electrons in the few MeV range.
.
.
l curve t o d escrl"b e th e
Wl"ll"lams 9 ,lO and Landaull d erlved
a unlversa
characte ristic distribut ion of energies resulting when a monoener getic
electron beam passes through a thin foil; i.e., one in which the
average energy Joss is small compared to the initial energy of the
electron.
Angular effects were not considere d.
The curve is basically
a Gaussian distribut ion centered near the most probable energy loss
with a long tail at lower energies.
Collision s involving a small
energy transfer are responsib le for the Gaussian distribut ion, while
larger energy transfers cause the tail.
.
l2 13
Blunck and Lelsegang
'
give a correctio n for the Landau theory to account for the effect of
more tightly bound atomic electrons , especiall y the K-shell electrons
for high-Z atoms.
Vavilov14 ' l5 modified the Landau distribut ion to
make it more indicativ e of the incident- particle velocity.
5
Bremsst rahlung reaction s also result in energy degrada tion,
although they are not of prime interes t in the range conside red here.
Bremsst rahlung becomes much more signific ant as the electron energy
and the atomic number of the target increas e.
However , bremsst rahlung
reaction s are crucial in the determi nation of radiatio n effects at
target depths beyond the range of the inciden t electron s.
Motz
16
Koch and
present a detailed review of the bremsst rahlung interac tion.
Various attempt s have been made to solve the complet e electron transpo rt problem by applyin g numeric al techniqu es that combine the
results of several existing theorie s.
In contras t to any of the
theorie s previou sly mention ed, such calcula tions distingu ish between
electron pathlen gth and sample thickne ss.
The moments method is a semiana lytical numeric al solution to the
transpo rt equation in which the energy, angular , and spatial dependence of the flux are describe d by a series of polynom ial expansi ons.
Spencer and Fano12 adapted the moments method to the electron transport problem .
Electro n-electr on collisio ns involvin g small energy
transfe rs were treated accordin g to a continuo us slowing- down model
which assumes that the form of the cross section s for these collisio ns
is unimpor tant as long as the correct stopping power (energy loss per
unit path length) is obtaine d.
Specifi cally, the relativ istic
M~ller 1 7 cross section for electron ic collisio ns with free electron s at
rest is assumed to be valid down to a very small fraction al energy loss
which is defined so as to give the correct total stopping power.
Spencer and Fano's method assumes an infinite , homogen eous medium and
6
includes the production of secondary or knock-on electrons.
Photon
production via bremsstrahlung reactions is accounted for, but there is
no provision for subsequent transport of the photons.
Theoretically, Monte Carlo calculations can follow each individual electron through every collision as the electrons are slowed down
and scattered through the target foil.
In practice, this is not
feasible due to the staggering number of collisions involved.
A single
electron with an initial energy of a few MeV will undergo in the
neighborhood of 10 5 collisions in the process of downscattering to the
0.1 MeV range.
Individual electronic collisions are therefore not
treated in the Monte Carlo calculations.
Instead, theories describing
various segments of the transport problem are used to group together
large numbers of collisions.
The computation proceeds by considering
successive spatial intervals, with the resulting distributions
determined by a conventional random sampling based on the suitable
multiple-scattering theories.
18
Berger and Seltzer 3 have· written a
Monte Carlo code ETRAN, in which the angular deflections can be computed by the method of Goudsmit and Saunderson, Moliere, or Fermi's
Gaussian distribution.
The spectrum resulting from energy loss is
determined by the modified Landau energy-straggling distribution or
from a continuous slowing-down model.
Collisions involving large energy
transfers can be considered separately from the continuous slowingdown model, and secondary electrons and photons are produced and transported through the target sample.
18
In general, calculations based on
7
ETRAN have shown good agreemen t with experime ntal results. 5 Nevertheless, the Monte Carlo method is restricte d by the statistic al variations inherent in random sampling.
The method of discrete ordinates offers a viable alternati ve to
Monte Carlo methods in that it can follow each electron on a collision by-collis ion basis, it does not involve random sampling, and it requires
only basic cross-sec tion data.
In the original discrete ordinates
method for slab geometry as suggested by Wick, l9, 20 the angular variable is divided into a discrete number of intervals .
The transfer
integral term in the Boltzmann transport equation is then approxim ated
by a Gaussian quadratur e formula, resulting in a set of coupled equations for the discrete- angle fluxes.
The S method is a special case
n
of the discrete ordinates method developed by Carlson. 21 Here, the
direction al flux is assumed to vary linearly between interpola tion
points in both the angular and spatial variables .
Carlson later sim-
plified and generaliz ed the S method into the current discrete ordin
22
nates method.
The flux is now stated in terms of the average values
at the midpoints of the spatial and angular intervals .
The discrete
ordinates method was developed for neutron transport and is now preferentially used for the solution of one- and two-dime nsional neutron- and
gamma-tr ansport problems in the form of codes such as ANIS~ 3 and
24
DOT.
This investiga tion is the first attempt to adapt discrete ordinates procedure s for the transport of electrons through matter.
In
principle , ANISN may be used to transport electrons by the simple expedient of introduci ng into the code the differen tial cross sections for
8
electron -nucleu s elastic collisio ns, electron -nucleu s bremsst rahlung produci ng collisio ns, and electron -electro n collisio ns.
In practic e,
however , these cross section s are quite differen t from those which
occur in neutron transpo rt, and the method has shown only partial
success in transpo rting electro ns.
The discrete ordinate s method
allows the product ion of photons and seconda ry electron s, and their
subsequ ent transpo rt through the target.
Individ ual electron ic calli-
sions are treated except in the continuo us slowing- down version of
electron transpo rt by discrete ordinat es where electron -electro n callisions that result in a small energy transfe r are handled by a continu ous slowing- down term.
Experim entally obtaine d electron -transm ission data provide a basis
with which to test theoret ical calcula tions.
The energy spectra of
electron s transmi tted through slab targets have been measure d as a function of angle. Rester and Rainwat er 25 conside red 1-MeV electron s
normall y inciden t on aluminum slabs. Rester and Dance 26 studied the
spectra resultin g from 1-MeV electron s on aluminum and gold targets .
Lonerga n, Jupiter , and Merkle 27 investig ated the transmi ssion of 4and 8-MeV inciden t-electr on beams through berylliu m, aluminum , and gold
targets .
9
III·
DISCRETE ORDINATES TRANSPORT EQUATIONS
The equat ions used to trans port elect rons and photo ns
throu gh
matte r are devel oped in this secti on.
The time- indep enden t Boltzm ann
trans port equat ion can be writt en for elect rons in a
unifo rm medium as
E
P(R, E,o) + n
!
0
E
+ n
(l)
E
T
A.!.....
- n cr (E)'#'\ R,E,O
) ,
e
and for photo ns in a unifo rm medium as
p
y
(R , E, _.0 )
dE'
E
T
,J..
_. )
- n a ( E ) ¥" (R,
E, 0
y
y
I
dE'
+ n
E
d2
er'\J-=0'\f ( I
..... I
..... )A-;::_,)
dO'
-l.-1E ,E,O
·0 '+' (-R ' El Ht
dEdO
Y
I
2
d cr
dO'
.....
y->Y (E' ,E,O' .Q)~R,E'
dE dO
.....
,0') ,
(2)
10
-+)
d 20. ( E I , E, -+1
0 ·0
l
dEd.O
J
o-~(EI) =
(
2
do-
-+)
-+1
0
E 1 ,E,O.
--~-y--=-=~---dEdO
=
2
.....
i
e£, br, inel.,
i
e£, br, inel.,
.....
dEJao d o-i (EI ,E,ol .o)
dE dO
L:i
-+ 1 -+)
2 ( 1 ,E,0
·0
E
do-.
i
= co, pe, pp,
--1--:-::::-::----
dEdO
=l:
i = co, pe, pp,
i
where
~
.....
0
a vector denoting the position of the particle;
=
dO
a unit vector in the direction of the momentum vector;
an element of solid angle;
El = the kinetic energy of the incident particle;
E = the kinetic energy of the emergent particle;
c/>(R,E,o)
=
the electron flux per unit energy;
cpy (R,E,o)
=
the photon flux per unit energy;
=
the highest kinetic energy considered;
n
=
the atomic number density:
P R,E,O)
=
the number of electrons per unit energy per steradian per
E
0
c ..
unit voLliD.e per second input at R from an external source;
11
P (R,E,O)
y
=
the number of photons per unit energy per steradian
per unit volume per second input at R from an external source (P
y
d
(R,E,O)
=
0 in the calculations
undertaken here);
2
CJ • (
l
E I ' E' 0 I
dE dO
·0)
=
the differential atomic cross section for a particle
with kinetic energy E' going in direction
0
1
to
undergo process i, after which the particle has a
for electrons,
kinetic energy D and is traveling in direction .....
o;
i = e£ (elastic nuclear scattering, for which E 1 = E),
br (bremsstrahlung scattering from both nuclei and
atomic electrons, thereby producing a photon),
inel (inelastic scattering from atomic electrons,
thereby producing a secondary electron; the
differential cross section here includes the
production of both the primary and secondary
electrons);*
*The inelastic scattering cross section used in this investigation
is the atomic cross section, and is found by multiplying the differential cross section for an inelastic electron-electr on collision by the
number of electrons per atom, z. An elastic electron-electr on collision involving an incident electron with kinetic energy E 1 results in a
primary electron with kinetic energy E and an energy loss of E 1 -E which
is imparted to the struck electron. If E 1 -E is large enough, ionization
occurs and a secondary electron is produced with kinetic energy E'-E,
neglecting the energy required for the ionization process. If E'-E is
too small for ionization, the struck electron is elevated to an excited
state. However, because of a lack of cross-section information in the
region where the energy imparted is on the order of the binding energy
or less, it was necessary in this investigation to assume that a secondary electron is produced in each inelastic electron-electr on collision,
12
so that the multip licity for such collisi ons was 2. The differ ential
cross section do.
lne 1 . (E' ,E,O' ·0) in the equatio ns in the text is
dEd,O
always assume d to include both the primar y and second ary electro n;
that
is, integr ation 0f the differ ential inelas tic cross section over energy
and angle gives the total inelas tic cross section times the multip licity,
\!
inel .
0
Tinel. (E')
f
=
E'
dE
0
f
dO
d
2
- _.
(E' ,E,o'.o)
lne 1 .
dEdO
0.
Since the multip licity is 2 by assump tion,
E'
~!
f
dE
0
dO
d
2
... __.
0.
lne 1 .
(E' ,E,o'.o )
dE dO
If the primar y electro n is defined as the resulti ng electro n with the
highes t kinetic energy , then it has a possib le range from E' to E'/2,
and crT.
(E') may also be obtaine d from
lne 1 .
E'
0'
T
(E')
inel.
=!
dE
dO
dEdO
E'
2
It is this form of
0
T.
(E') that will be used in the text.
lne 1 .
13
for photons,
= co (Compton scatterings from atomic electrons)
i
pe (photoelec tric absorption, thereby producing an
electron),
pp (pair production, thereby producing an electron
and a positron);
2
d a e--o ( E 1 , E,; 1
-~)
the differentia l atomic cross section for an elec-
dEdO
tron with kinetic energy E 1 , going in direction
0
1
to produce a photon with kinetic energy E, going
in direction
2
d cr
(
I
_.1
o,
by bremsstrahl ung scattering;
_.)
E , E, 0 ·0
_--~,y-o_e~d"""E~dO----
=
the differentia l atomic cross section for a photon
with kinetic energy E1 , going in direction
0
1
,
to
produce an electron with kinetic energy E, going
in direction 0, by photoelectr ic absorption or pair
production.
Electrons produce photons by bremsstrahl ung, and photons produce
electrons by photoelect ric absorption and pair production.
The photon
and electron transport equations are therefore cross-coupl ed and must
be solved together.
Positron coupling should also be considered since
photons produce positrons by pair production and positrons produce photons by bremsstrahl ung and annihilatio n.
However, positron transport
is not significant for the calculation s undertaken here, and so the
positron transport equation is not included.
The photon-elec tron source term in the electron transport equation
is small for the transport of incident electrons in the few-MeV range,
,
14
and will be neglected here.
The electron transport equation is then
no longer coupled to the photon transport equation in the photonelectron direction since electrons produced by photons are not included.
The electron transport equation to be solved is
....
....
(l. 'V~(R,
....
E, 0)
P(R, E,O) + n
I
E
0
dE'
I
dO'
E
(3)
T
....
-~
- ncr (E)~(R, E,O),
e
which is obtained from Eq. (l) by dropping the photon-electron source
term,
d2
dE 'Id"'
u
"Y
crre ( E ' ,E,O'·O
-- """),+.. (....
... )
dEdO
R,E',O'
Electrons still serve as a source for photons, and an electron-photon
transport case will be reported in Section VII.
The one-way coupling
scheme (electron to photon) is similar to the neutron-photon problem
that for some time has been solved by the use of various discrete
ordinates codes.
The resulting Eq. (3) for electron transport is solved by the method
of discrete ordinates.
This method was developed to solve the neutron
transport equation analogous to Eq.
sents a new adaptation.
(3),
and electron transport repre-
The code used for this adaptation,
ANIS~ 3 ,
15
has been notably successfUl in solving the neutron and photon transport
problems.
Nevertheless, the cross sections involved for electron trans-
port are so different in form from neutron cross sections that it was
not at all clear whether they would be handled correctly.
Theoreti-
cally, neutrons, photons, or electrons can be transported from an ini tia~angle and energy distribution to a final angular and energy spectrum
without a knowledge of the type particle and knowing only the probability of a reaction occurring.
ANISN is able to treat any one-dimensiona l geometry, but only slab
geometry cases are considered here.
The exact procedure for obtaining
.
the discrete ordinates form for Eq. (3) is descrlbed
elsewhere, 24 28 but
a brief indication of some of the concepts involved will be presented
here for the one-dimensiona l slab geometry case.
The energy dependence of the flux and the cross sections is
expressed in multigroup form.
Consider the energy group, G, which
extends from Eg+l to Eg' where Eg
=
Eg+l + ~EG.
The electron flux for
group G is
E
fg
.... _,.
dEcp(E,R,O) ·
E
g+l
The multigroup form of the cross section for an electron with energy
E' to produce an electron with energy E is found by integrating the
differential cross section over the energy bounds of the initial group
15
G1 from E 1
toE 1 (E 1
+ 6E 1 ) , and averaging over the initial
g +1
g
g +1
G
group and integrati ng over the energy bounds of the final group, G.
Then
d 2IJ
dE'
dE
(
e--oe
E I 'E, ...,.
0I
....:. )
.(2
dEdO
The angular dependenc e of the cross sections is expressed in an
(£max+ 1)-term Legendre series expansion in ~ , where ~
0
of the scatterin g angle.
0
is the cosine
If the Legendre expansion is defined as
£max
I:
n
£=0
where L£
(~0
) is the Legendre function. then the series coefficie nts are
given as
+l
£
M G' ,G
= 4rr(22£+ l) n
f
~0
-1
The angular variable is then expressed as a function of a fixed-coo rdinate system in which the angular variable is divided
of angles) discrete angular intervals .
into NOA (number
This is done by use of the
addition theorem for Legendre polynomi als.
The integral over angle in
17
the Boltzmann equation is replaced by a Gaussian quadrature formula
NOA
L
cf>G,D(R)wD '
D=l
where
NOA is the total number of points (angles) considered, and OD
and wD are the ordinates and weights for the Gaussian quadrature.
The weights, wD' are normalized to give a sum of one instead of 4n, so
that for one interval, D,
with
dO
==
4n wD ·
The integral over angle in Eq. (3) may then be represented by a summation
over the incident angle, D', from D'
If
~ ==
- -
...
==
1 to D' = NOA (number of angles).
0 . k, where k is the unit vector normal to the slab, then
18
The spatial region of interest is divided into specific intervals
represented by the subscript I,
where
1 to nwnber of intervals, and
I
r.
the linear distance to the beginning of interval I,
l
r.+l
the linear distance to the end of interval I·
l
The mean value theorem is then applied to each term in the transport equation
giving average flux values for each energy group, spatial
interval, and angular interval considered.
ordinates form for Eq.
0)
The resulting discrete
in the one-dimensiona l multigroup slab geo-
metry form is then
2
max
+ (r.
l+ 1
-
r.)
l
G
NOA
L
L
G'==l
D' ==1
( 4)
where
<1> G, I ,D == the electron flux integrated over energy group G,
averaged over spatial interval I and evaluated at
~D
in angular interval D,
cp G,i+l,D
==
the average electron flux in energy group G and angular interval D at spatial point r i + ,
1
T
aG
the total reaction cross section for an electron in gr0up
G,
PG,I,D = the external source in a spatial interval I for electrons
in energy group G, angular interval D·
The initial Boltzmann equation, Eq. (3), is now represented by a series
of equations similar to Eq. (4), with each equation representing the
electron balance in a so-called "phase space cell", for which ¢G,I,D
(energy group G, spatial interval I, angular interval D) is defined.
The third term in Eq . ( 4),
f
max
I:
f=O
G
I:
G'=l
NOA
D'=l
then represents the number of electrons in energy group G and angular
interval D produced in spatial interval I by electrons in all angular
intervals (D'=l to NOA), in all energy groups G' which represent
energies greater than or equal to the energies in group G.
In order to solve Eq. (4), i t is necessary to evaluate ¢G,I,D'
¢G,i+l,D' and ¢G,i,D.
First, it is assumed that the incoming fluxes,
¢G . D' are known from boundary conditions or from the calculation for
'l'
the previous interval.
Additional difference equations are then
assumed in the form
¢G,I,D::::: a¢G,i+l,D + (l-a)¢,G.
,l, D' ~ > O,
¢G,I,D::: (l-a)¢G,i+l,D +a ¢G,i,D' ~ < O,
20
2
where a is determined by a weighted difference model. 9 The discrete
ordinates form for the photon transport equation is similar to Eq. (4)
with the appropriate photon cross sections used.
As shown in Eq.
(4),
the discrete ordinates form of the transport
equation categorizes the electrons at a particular spatial point in
terms of energy groups and angular intervals.
The cross sections which
transfer electrons from one energy group to another are determined by
integrating the differential cross section over the various energy
groups, and the angular changes are described by expanding the cross
sections in a Legendre series in the cosine of the scattering angle.
If the energy and the angular changes are small enough for a particular
collision, the incident electron would be in the same energy group and
angular interval after the collision as before it.
Thus, within the
limits of the accuracy of ANISN's calculations, no change has occurred
in the energy and angular spectra.
for electrons.
Such collisions are very numerous
An approximation known as the delta-function correction
is therefore made in an effort to remove those collisions from the cross
sections.
Mynatt.
The explanation given here is similar to that presented by
26
Let the within-group Legendre series expansion coefficients be
expressed as
£
MG,G
= (2£+1)
f 1,
so that the within-group scattering angular distribution, which was
given earlier by an (£
max+1
)-term Legendre series as
(5)
21
£
max
1
4;
may now be represented as
£
max
(2Hl)f L
£
1
(~ 0 )
.
(6)
Because the cross sections are heavily forward peaked, they can be
approximated by an £max -term Legendre series plus a delta function in
the forward direction:
l
max
-l
(7)
£=0
where it is assumed that
£
max
The (£
max
+1)-term of the within-group expansion coefficients is there-
fore assumed to be a delta-function coefficient.
Since the accuracy of
the initial Legendre series representation increases with the number of
terms used in the expansion (£
max
+1), the delta-function coefficient
assumption should also be a better approximation as 1
max
increases.
I
determine f
1
and C, equate Egs· (6) and (7), multiply through by a
Legendre polynomial LN(,,... ), where N varies from 0 to £max , and then
0
integrates over
~
0
·
To
22
Since
£
+l
max
I
I:
2
(2£+1) f L (1J. 0 )LN(I-1 0
2
)~ 0
-l
£=0
(8)
= 2fN, for N s: £max'
for N > £max'
= 0
and
+l
I
N
C 5(1-J. -l)L (1-1 0 )~ 0 = C,
0
-l
then it is found that
(9)
and
c=
2fl.
max
Multiplyi ng Eq. (9) by (2£+1), and combining the result with Eq. (5)
gives the equation for the corrected coefficie nts:
£
M' £
G,G
- ( 2.Hl ) Mmax
G,G
+l
2£
max
(10)
23
The modified £
max
term Legendre series coefficie nts are used in the
scatterin g integral and the delta function is accounted for by subtracting fL
from the total cross section, where
max
==
c
2
The corrected cross-sec tion coefficie nts are commonly referred to as
P
1
The cross sections used in this investiga -
-correcte d-P£
max-1
max
tion were P -correcte d-P6 .
7
(P
1
here refers to the Legendre coeffi-
cients and is the standard represen tation.
matter of convenien ce).
L£ has been used as a
The magnitude of the delta-fun ction correc-
tion is quite large for electrons , so that it greatly facilitat es the
ANISN calculati on, especiall y since the within-gr oup cross section
determine s the number of iteration s required for convergen ce.
If the required cross sections were known, the solution could now
be obtained.
With the proper input data, ANISN could be used for obtain-
ing the transmitt ed electron spectra.
sections are not available .
However, all the necessary cross
The inelastic electron- electron atomic
cross sections that are available were derived assuming a collision in
which the energy lost by the primary electron is significa ntly larger
than the binding energy of the target electron.
Inelastic electron-
electron collision s involving a large energy transfer will be referred
to as hard coll1sion s.
No adequate cross sections are known for colli-
sions involving energy transfers of the order of the binding energy or
24
smaller.
and Fano,
Nevertheless, these cross sections were estimated by Spencer
12
.
and calculat1ons based on a similar estimate have been done
as a part of this investigation.
Inelastic electron-electron colli-
sions involving a small energy transfer per collision are very frequent
for electrons and account for a large part of the total energy degradation.
They will be termed "soft collisions."
Inelastic collisions occur so frequently that, as an approxima-
tion, electrons can be considered to undergo a continuous slowing down,
with a fixed energy loss per unit path length travelled.
This quantity
is referred to as the stopping power and is well known both experimentally and theoretically.
30
Unfortunately, the needed cross sections
cannot be derived from the stopping power alone, since the stopping
power represents an integral over the cross section.
However, the
stopping power is adequate for many applications, and one can account
for the energy loss due to soft collisions by utilizing the appropriate
stopping power in a continuous slowing-down term.
This procedure uses
the best information available and has the advantage of avoiding the
cross section for low-energy transfer collisions.
The continuous
slowing-down equation was obtained by the method used by Rossi.
31
A
complete derivation can be found in Appendix A, but an outline of the
procedure is given here.
Beginning with the electron transport equation, express Eq. (3) as
25
E
J
0
0 -v.PCR, E,ii) =
P(R E,ii) + n
E
E
+ n
f
0
d2
dE'/
rlf"'lr
~·
I
._., __.),~./__.
(Jinel. ( r
E ,E,O ·0 ~R,E
dEdO
..... , )
,o
E+I'
(11)
+ n
f
E+I'
dE'
E
d2
f
crinel.
dEdO
dO'
(E' ,E,o' .o)c/>(R,E' ,o) ,
where
0 ~e =
E-I'
ai (E) f
=
0' br.'
cr el. +
2
dE'! dO
d 0'.lne 1 .
dE' dO
(
-+I -+)
·0 ,
E,E I ,0
E
2
E
cr~(E)
=
J
dE'
I
d20
dO
, ~, ~)
inel. (
E,E ,O ·0 ,
dE'dO
E-I'
I'
=
an arbitrary value taken to be the minimum energy loss allowed
in a large energy transfer or hard collision (a collision
involving an energy transfer smaller than I' is a soft collision) ·
26
The inelastic scattering terms are now separated into terms
describing soft and hard collisions.
Let
E+I'
T = n
J
dE'
2
J
d
dO'
0
inel.
dEdO
(
-+ I -+ )rt..(-4
-4 I
E I ,E,O
·0 ~ R,E I ,o
)
E
(12)
where T now describes the scattering due to soft collisions.
Now
add and subtract
E+I
n/
dE' dcrinel. (E' ,E)f/>(R'E' ,o)
dE
E
dcr inel
dE . (E',E)
where
=I
d2
dO
a inel . ( ,
... , ..... )
dEdO
E ,E,O ·0
d2
ainel.
- (E', E,O' ·O) ·
dE dO
2
.....
.....
d a.1ne l . (E' ,E,O' ·D)
Assume
dEdO
=
2
d a.1ne l . (E', E)
--~~---
dE
That is, the soft collisions involve only an energy degradation and not
an angular change.
Then
27
E+I'
T
=n
f
do-.1ne l .
(E', E)¢(R, E' ,o)
dE
dE'
E
(13)
E
- n
dE' do-.1ne l . (E, E' )¢(R, E,o).
dE'
f
E-I'
do-.1ne l . ( , ),w
.
dE
E ,E ~R,E' ,O ) 1s
now expanded in a Taylor series, and after
defining the soft stopping power S(E) to be
E
S(E) = n
f
(E-E')
<b.
l
we . (E, E' )dE',
dE
(14)
E-I'
it is found that
(15)
T = o[S(E)c,b(R, E,O)]
oE
Substituting Eq.
(15) into Eq. (ll) gives the electron transport
equation as solved by ANISN with continuous slowing down (AWCS):
n.v<P(R, E,O)
P(R, E,0) + n
f
E
0
dE'
f
dO'
d2o~e (E' 'E,ii• .[i).p(li, E' .o•)
E
dE'
+ n
E+I'
- n
f
2
( I
-+1 ~)
1 l E , E,O ·0
ne d~dO
¢(R,E' ,o')
d cr.
dO'
cr~e (E)¢(R,E,o) - n cri(E)¢(R,E,o)
+
~E [S(E)¢(R,E,O)] .
(16)
28
The differe nce betwee n Eqs. (3) and (16) is that the inelas tic
~e
collisi ons resulti ng in an energy loss less than I' (the soft
collisi ons) are now handle d by a continu ous slowing -down term.
The
soft collisi ons are treated as part of the inelas tic and total remova
l
cross section s in Eq.
(3),
althoug h the cross section s for the soft
collisi ons are not well known.
In Eq. (16), the energy loss due to
these soft collisi ons is treated by a continu ous slowing -down term.
The stoppin g power used in the continu ous slowing -down term is not
the well-kn own energy loss per unit distanc e due to excita tion and
ioniza tion but a portio n of it, and will be describ ed in detail in the
next section .
No knock-o n electro ns are produc ed from soft collisi ons
treated by Eq. (16), and the incide nt electro n involve d suffers an energy
degred ation, but no angula r deflec tion.
To obtain the discre te ordina tes
form for T [Eq. (15)], integr ate over the energy bounds of group G.
Then
E
fg
E
{o[S(E) cp(R, E,o)])d E
\
oE
=
s(E )A.tR E
g 't'\ ' g'
o) - S(Eg+1 )r/J(R ' Eg+1' o) '
g+l
which is then incorp orated into Eq. (16) to give
(17)
29
L
~D(¢G,i+l,D- ~G,i,D)
(r~+l-
ri)PG,I,D + (ri+l- ri)
max
G
LA~
L
£=0
G'=l
*P.
X
MG, 'G
(18)
D'=l
+ ( rl. +l - rl. ) [ Sg, I (/) I d - S l I
g, '
g+ '
cpg+ l ' I ' D]
.
In Eq. (18),
¢g,I,D
= the average electron flux in spatial interval I, evaluated
at
~
0
in angular interval D with energy E ,
g
S ,I= the stopping power (energy loss/em) due to soft
g
collisions for electrons in interval I with energy E ,
a
*T
g
(E)
T
=
M*P.
G' ,G =
a* (E) + a (E),
e-+e
1
4rr r2£2+ll· n /+1 *
[
]
c;G' ....G,I
-1
=
P.th Legendre expansion term for the e-+e scattering due to
elastic, bremsstrahlung and hard inelastic collisions.
~
I D' ~ 1 I D' and ~G I D are interrelated by a weighted
g, '
g+ ' '
' '
29
difference model
similar to that used for spatial intervals.
30
IV.
TRANSPORT CROSS-SECTION DATA
Described in this section are the cross sections and other parameters used in the electron and photon transport calculations presented
in Sections V, VI, and VII.
Electrons with kinetic energies below 10 MeV undergo three significant reactions: 5 inelastic scattering from atomic electrons, elastic
or Coulomb scattering from atomic nuclei, and bremsstrahlung (radiative)
interactions with both atomic electrons and nuclei.
A.
Inelastic Electronic Scattering from Atomic Electrons
M~ller 5 ,l7 derived a relativistic cross section for an inelastic
electronic collision with a free electron at rest, which may be stated
as
2nr
=
2
0 ~z
E'2
1
--~----
(E/E' )
2
1
+ -----------2
(l-E/E')
(19)
-
~
...,..-..,...,~1-~~
K-1 12
K2
(E/E' )(1-E/E') + ~
X
'
where
daM(E' ,E,O'
dEdO
.o)
=
o[cos9-f(E' ,E)]
X
211
since in a two-body collision the scattering angle is a function of the initial and
final energy of the particle considered;
31
E'
= the
initial kinetic energy of the incident
electron;
E = the final kinetic energy of the primary
electron;
E'-E = the kinetic energy lost by the primary
electron in the collision, which is also
the kinetic energy of the secondary
electron produced by the collision;
*Z = the atomic number of the target atom;
K = (E'+m)/m;
m = the electron rest mass;
r
0
= the classical electron radius;
2
W
= m/~ ;
~
= the ratio of the velocity of the incident
electron to the velocity of light;
f(E' ,E)= coseh, E ~ E'/2;
cos9£' E < E'/2.
The resultant electron with the highest kinetic energy is defined as
the primary electron.
therefore E'/2.
The maximum energy transfer per collision is
The angle of scattering 9£ of the electron emerging
with the lower energy (secondary or knock-on electron) is given by 5
*The cross section given here is not the normal M¢ller formula,
but Z x M¢ller cross section for an electron-electron collision, since
the atomic form is used in this investigation.
32
(20)
and eh for the primary electron by
1
8
h
where coseh
~
case£
~
= [E(E'+2m)J
EE'+2m
2
(21)
'
o.
If the energy transferred by the incident electron to the atomic
electron is large enough so that the binding energy is insignificant
(hard collision), the atomic electron can be assumed to be free, and
the M~ller cross section is applicable.
When the energy transfer is of
the order of the binding energy, the collision does not fit the M~ller
cross-section criteria.
One approach used in this investigation to
treat such collisions was based on the work of Spencer and Fano.
12
This method assumes that for inelastic electron-electron collisions
involving a small energy transfer, only the rate of energy dissipation
is important.
The procedure used to determine the energy loss per unit
pathlength is similar to that formulated by Rohrlich and Carlson. 9
From Eq. (14), the stopping power for the low-energy transfer (soft)
collisions is
E'
S(E')=n/
E'-I'
dE (E'-E)
da. 1
~e . (E' ,E) •
33
The Bethe 9 '
32
theory of stopping power, in which an explicit summation
is conducted over the excitation probabilities of the atom, predicts
that for low-energy transfer collisions
(22)
where
n
e
=
o* =
the electronic number density of the target material;
33 correction for the density effect, the
the Sternheimer
mean-energy-loss reduction due to polarization of the medium;
I
I'
= the
=
average ionization energy for the target material;
the minimum energy transfer for a hard (high-energy
transfer) collision.
Analagous to Eq.
(14),
the stopping power for hard collisions may be
expressed as
E'-I'
shard ( E ' )
J
=n
dE ( E' - E )
d:~
(23)
( E' , E ) •
E'/2
Direct integration yields
(24)
2
(1
1)
K
8
+( ~
-
12
) ( 2K 1 )
I
K;
2E I 2 +
1
.en
[
E
2 ( E' -I ' )
Jfl .
34
From Eqs. (22) and (24), the total energy loss per unit pathlength due
to inelastic electronic collisions with atomic electrons is
Stotal (E')
= n (2nr2~)
0
e
I
£n [
E' (K+l)
2I 2 (E'-I')
J
+ 2 -
E'~~'
(25)
Eq. (25) differs from the result of the Rohrlich and Carlson derivation
by the presence of smaller terms (no assumption was made as to the
relative sizes of E' and I') and the inclusion of the Sternheimer 33
correction.
Spencer and Fano defined a minimum energy loss per
collision so that the correct stopping power, here given by Eq. (25),
is obtained by the use of the M¢ller cross section.
The definition may
be expressed as
E'-I'
Stotal (E') =n
J
dE(E'-E)
daM(E' ,E)
dE
E'
2
(26)
= Shard (E')
Since I' here is the minimum energy loss allowed in any collision,
there is no separate soft energy loss term.
I' is now a function of E'.
Setting Stotal [Eq. (25)] equal to Shard [Eq. (24)] and solving for I'
gives
35
I'
=
I
2
2
exp (~ +5)
2(K+l)E'
(27)
as the minimum energy loss per inelastic collision with an atomic
electron for an incident electron with kinetic energy E'.
The I'
values obtained from Eq. (27) are approximately four orders of magnitude smaller than I and therefore far below the energy loss range for
which the M~ller cross sections were derived.
However, in the absence
of an adequate differential cross section for soft collisions, the use
of the M~ller cross section down to I' in Eq. (3) does guarantee the
oorrect stopping power.
This should be sufficient if the form of the
cross section is unimportant for soft collisions and only the energy
loss matters, as Spencer and Fano assumed.
This type of calculation
will be referred to as ANISN with the M~ller cross section used to
treat low-energy transfer collisions (AWMC).
A typical first within-
group cross-section expansion coefficient, (P term of LG-G)' is of the
0
6
order of 10 before correction, but of the order of 102 after the
delta-function correction.
As an alternate treatment, these lower energy transfer (soft)
collisions were approximated by a continuous slowing-down term,
oS (E )ct>(R, E,
oE
in Eq. (16).
-by Eq . ( 22 ) as
o)
The stopping power for soft collisions, S(E), is defined
36
The only undetermined parameter in the S(E) definition is I', the
minimum energy transfer for a hard collision.
what value should be used.
cross sections to be valid.
It is not at all clear
I' must be high enough for the
M~ller
On the other hand, too high a value would
have undesirable effects on the angular distribution and electron
population since the continuous slowing-down approximation assumes
collisions are straightahead and does not account for secondary
electrons.
In general, it seems reasonable to assume that I' should
be greater than I, the average ionization energy.
The use of the
continuous slowing-down approximation to handle low-energy transfer
collisions does have some advantages as a method for calculating
electron transport.
The correct total stopping power is assured when
the hard energy loss obtained in Eq. (23) is combined with the soft
energy loss from Eq. (22).
Also, the continuous slowing-down approxima-
tion alleviates the need for a soft-collision cross section by assuming
a uniform, continuous energy loss involving no change of direction.
This type of calculation will be referred to as ANISN with continuous
slowing-down used to treat low-energy transfer collisions (AWCS).
A typical first within-group cross-section expansion coefficient is of
.
the order of 104 before correct1on,
but of the order of 10 2 after the
delta-function correction.
37
B.
Elasti c Coulomb Scatte ring from Atomic Nuclei
The differ ential cross section used here for elastic scatter ing
5 4
from a nucleu s is based on the Mott ,3 series (evalua ted as the ratio
34
of Matt-t o-Ruth erford cross section s ), with the Moli~re7 screen ing
angle and Spence r's
35 treatm ent for low-an gle scatter ing.
Goudsmit
5 8
and Saunde rson's ' expres sion for the screene d Ruther ford cross
section with Molier e's7 screen ing angle is
dcr(E,O' .
dO
o)
4
=
2rre F(Z)
[
2 2
pv
X
]
sine de
2
2
x R '
)
e
l
+
(1-cose
2 s
(28)
where
7
es = Moli~re's screen ing angle that attemp ts to accoun t for
the screen ing of the nuclea r potent ial by atomic electro ns;
2
F(Z) = z for nuclea r scatter ing;
e = the electro n charge ;
p = the relativ istic momentum of the incide nt electro n;
v =the relativ istic veloci ty of the incide nt electro n;
R =the ratio of Matt-to -Ruthe rford* scatter ing cross section s.
Spence r3
6 rearran ged the cross section in Eq. (28) to get a better
expres sion for small scatter ing angles and obtaine d
*Since the Mott series is infini te, the Matt-to -Ruthe rford ratio
can be represe nted only by an infini te series . The series repres entation was not given here becaus e it is too compli cated, bu~ a full
discus sion can be found in Dogget t and Spence r's article . 5
38
dcr(E,o.o) = [2ne 4F(Z)
dO
2 2
X
p v
sine de
(1-cose + 1
2
1
X
2 2
[1 + _rr (_f_)t3 cosljl](l-cose + 1 e ) + [R-1- --lr
2
/"2
s
/'2 137
(29)
2] 2
1 es
1-cose + 2
[
1-cose
'
where
C.
Bremsstrahlung (Radiative) Interactions with Nuclei and Atomic
Electrons
The differential cross section used to describe bremsstrahlung
37 who
interactions is given by McCormick, Keiffer, and Parzen,
recalculated the work of Racah,3
8 as
l2.....1xcxF
pI k
(30)
'
39
where
p =the relativistic final momentum of the electron;
p' = the incident momentum of the electron;
k = the energy of the emitted photon;
C = a dimensionless parameter, defined in Ref. 35, which is a
function of the initial and final electron energies;
F = a high-frequency limit correction factor, defined as
X
1 o;
= -----x
1-e
p
=X '
P = o;
where X =
2~E
p .
137
The high-frequency end-point correction is necessary so that, after
integration over angle, the cross section will not approach 0 as the
kinetic energy of the electron after the collision approaches 0.
and Motz
section.
16
Koch
present Fano's formula for the high-frequency limit cross
The derivation of the correction factor used here is presented
in Appendix B.
It should be noted that this correction is only very
approximate since it is designed to give the correct limit after
integration over angle.
There is no indication of the effect of the
correction at a specific angle.
Eq. (30) exhibits the well-known~
divergence when E'=E(K=O), so the energy integral over the differential
-4 • Therefore,
-4 MeV and
collisions that involve an electron energy loss of <10
bremsstrahlung cross section was cut off at E'=E-10
photon production with a maximum energy <10-
4 MeV are not considered.
40
D.
Photon Interactions
The photon cross sections used in the solution of the electron-
photon case presented in Section VII were taken from a photon cross-
3
section library tape prepared by MUG. 9 The Klein-Nishina approximation40 for unpolarized photon scattering from free electrons at rest
was used to account for Compton scattering.
The photoelectric and
pair-productio n cross sections were obtained from data evaluated by
McMaster et al.
41 and by Plechaty and Terrall. 42
41
V.
A.
COMPARISON WITH THEORETICAL MODELS
Ela-stic Multiple-Scattering Angular Distribution
Goudsmit and Saunderson
8
obtained an analytic expression for the
angular distribution of transmitted electrons when monoenergetic
electrons are normally incident on a sufficiently thin slab so that
the energy degradation of the electrons may be neglected.
Berger
18
used the Goudsmit-Saunderson theory to obtain the angular distribution
of the transmitted electrons resulting from 1-MeV electrons normally
incident on an aluminum slab of thickness 0.0287 g/cm 2 •
The Mott33
elastic scattering cross section, modified to account for the screening
of the nuclear charge by the orbital electrons, was used in the expansion.
The transmitted angular current of electrons calculated by
Berger is represented by the histogram shown in Fig. 1.
The ANISN
results for this case are given as the plotted points.
In the ANISN
calculation, one energy group with a range from l.Olo6 MeV to 0.9894
MeV and a midpoint of 1.0 MeV was used.
No energy degradation was
allowed and only elastic scattering was permitted; i.e., in this
calculation Eq. (3) was solved with cr.
1ne 1 •
= crb r = O.
The two calculations shown in Fig. 1 are in excellent agreement,
and thus the method of discrete ordinates can handle small-angle
multiple Coulomb scattering successfully.
B.
Inelastic-Scattering Energy Distribution
An analytic solution to the electron transport problem is
reported by Passow 43 and by Alsmiller 44 for a particular form of the
scattering cross section in which the straightahead approximation is
42
.....
..4
••
•
••
10-t
,.....
••
••
...
c::
...0
+-
••
C)
.!!
•
Q)
--...4
+c::
Q)
~
C)
c::
10-2
·-
L.
Ill
'c::
-4
Ill
...
0
C)
Q)
'ii
~.
•
+-
le
10- 3
•
..
e
-
ANISN
GOUDSMIT AND SAUNDERSON
20
40
60
80
e(degl
Fig. 1. Angular distribution of transmitted electron current
for 1-MeV electrons normally incident on a 0.0287-g/cm 2-thick
aluminum slab •
43
assumed so that there are no angular effects.
The source for mono-
energetic incident electrons of kinetic energy E is expressed as
0
~(E,O)
= N0
o(E -E) ,
0
where
N0 =the source strength, taken as 1.0 in the calculations
reported here.
The flux at distance r is
~(E,r) =
N
0
o(E -E)e-Q,r+</> (E,r) ,
0
s
(31)
where
Q = the total cross section, which has a constant value for
all energies;
<f>
s
(E,r) =the secondary electron flux.
The solution for the secondary electron flux is
•
5
(E,r)
~ Qe-QrFee(E' ,E)N g(E
0
0
1
)
2
[B(E:,E)] x r 1 [ 8frB(E0 ,E)], (32)
where
F
ee
(E' ,E) = the number per unit energy of electrons with energy E
produced in a nonelastic collision of an electron
with energy E' ,
(~)m .
_ (2-m)
-
E'
X
E
'
44
m = an input parameter;
g(E ) = E (m-1)
0
0
(2-m) dE
E
45
I 1 =the modified Bessel function of the first kind.
The differential inelastic cross section used in the analytic calculation is
daAN(E' ,E)
dEdO
= aZFee(E' ,E)
6(1-cose)
'
2n
(33)
where
a=
,
B
n
the total microscopic cross section.
A set of comparisons was made with this analytic solution in order
to verify ANISN's ability to calculate the energy distribution resulting from inelastic collisions.
Since the inelastic cross section is
proportional to (E'/E)m, a small, positive mallows downscatter over a
considerable range of E values, while a large, negative m severely
restricts the E values from a given E'.
It was therefore possible to
simulate inelastic collisions resulting in a large energy transfer and
those resulting in a small energy transfer.
With a known solution and
known cross sections, any discrepancy between the analytic results and
ANISN's calculations must be due to ANISN's method of solution.
45
=
Equatio n (3) was solved with ab r
cr
el
= o,
was replace d
and cr.
l.ne 1 •
with crAN from Eq. (33), except that
T
where cr AN= cr, a consta nt, and the multip licity
\)AN =
I
E
dE I F ( E I 'E ) '
0
becaus e the analyt ic cross section is not symme tric about ~ •
It should be noted, howeve r, that the cross section form given in
Eq. (33) is only a very rough approx imation to the M~ller cross section
given in Eq. (19).
It is diffic ult, therefo re, to predic t ANISN's
ability to handle the M~ller cross section on the basis of the
analyt ic results presen ted here.
cross section at E'
=1
MeV, 0
~
Figure 2 shows the differ ential
E
~
1 MeV, for the analyt ic cases
consid ered and for the M~ller formul a.
Since the M~ller cross section
diverg es as E ~ E' and is symme tric about E
= E'/2,
the primary
electro n from a M~ller collisi on is consid ered to have a kinetic
energy E, where E'/2 ~ E ~ E'-I', and the second ary electro n has a
The total micros copic M¢ller cross section
2
for a 1-MeV incide nt electro n in aluminum is 2.18 x 10 barns for
4
4
2
I' = 100 I (1.63 x 10- MeV), 2.29 x 10 barns for I' = I (1.63 x 10kinetic energy of E'-E.
8
MeV), and 3-53 x 10 barns when I' is determ ined by the Spence r-Fano
8
proced ure (1.07 x 10- MeV).
46
107
:
!
...--...
'l:t
(\J
0
10 4
\
... ..._ do-AN (E~E) /dE,
)(
>Q)
---- ----
~
~
E
u
..__...
m-112
0"
=10 3
10 3
:
-- --- -----t-
~
~
...
~
: I
\
~
1
\
102
~
7
-4;I
\:
'\
/
/f
\.
" ........
"""'
:
:
/
/
-
do-m(E',E)jdE
~---""
/
:
:
do-AN (E~E) /dE, m=-100;
o-= 104 :
I f
w-1
0
0.2
0.4
0.6
0.8
1.0
E (MeV)
Fig. 2. Comparison of the differential M¢ller cross section for
inelastic electron-electron collisions with two differential analytic
cross sections below 1 MeV.
47
Two separate analytic cases were run, both for 1-MeV electrons
normally incident on aluminum slabs, so that E0 = 1.0 MeV and Q =
Figure 2
The first case set m = 1/2 and a= 103.
0.0602252 x a.
shows that the analytic differential cross section for this case very
3
slowly increases from a value of 1.5 x 10 barns/MeV at E
4.7 x 103 barns/MeV at E = 0.1 MeV.
By comparison, the
=1
MeV to
M~ller cross
section is much larger near E = 1 MeV but much lower from E = 0.9 MeV
toE= 0.5 MeV.
With a total microscopic cross section of 103 barns,
this analytic case has more large energy-transfe r collisions, over a
wide range of possible transfers, than does the M~ller cross section.
The ANISN calculation used 40 energy groups from 1.0 MeV to 0.1 MeV.
2
In Fig. 3, results are plotted for aluminum slabs 0.11 g/cm thick
and 0.33 g/cm
2
thick.
Both plots show extremely close agreement
between the analytic solution and ANISN's calculated values.
This
indicates that the method of discrete ordinates can successfully be
used to calculate the results of inelastic scattering over a wide
energy range for the specific cross-section form given here.
The small
high-energy peaks appearing in Fig. 3 represent the uncollided current,
expressed as N o(E-E) e-Qr for the analytic case in Eq. (31), and the
0
electron current remaining in the source group for ANISN.
The second analytic case set a=lO
4 and m = - 100. As can be seen
from Fig. 2, this results in a large differential cross section for
small energy-transfe r collisions, with relatively few collisions below
E
=
0.9 MeV.
However, this analytic differential cross section has a
48
20
10
-ANALYTIC
• ANISN
18
•
9
1=0.11g/cm 2
c
-
I= 0.33 g /cm 2
16
8
14
7
12
6
10
5
8
4
...
0
u
..!2
Cl)
c
Cl)
'a
·u
c
>
Cl)
•
~
.......
en
c
0
-=u
Cl)
Qi
6
3
~,
\
4
\
2
~
2
r. . . . ---4
1\
\..~
.........
0
0
0.2
0.4
0.6
E (MeV)
0.8
1.0
0
0
0.2
~--
0.4
..0.6
I
0.8
1.0
E(MeV)
Fig. 3. Transmitted electron current per unit energy ~er incident
electron ~or 1-MeV electrons normally incident on 0.11-g/cm -thick and
0.33-g/cm -thick aluminum slabs.
49
finite value of 1.02 x 10
6
barns/MeV at E = 1 MeV, while the M¢ller
differential cross section diverges as E- 1 MeV.
The M¢ller cross
section then results in a great number of very small energy-transfer
collisions that are not present in this analytic case.
The total
.
. cross sect'1on, 1 04 barns, rough ly corresponds to that
m1croscop1c
obtained from the M¢ller cross section with I' =I = 1.63 x lo- 4 MeV
4
(2.12 x 10 barns), but it is much smaller than in the case where
8
8
I' = 1.07 x 10- MeV (3.53 x 10 barns). The target was a 0.66 g/cm
2
- thick aluminum slab, and results were obtained for several
depths within the slab.
In Fig. 4 the analytic transmitted energy
spectra are plotted along with ANISN's solution for 0.11 g/cm 2 ,
2
0.33 g/cm , and 0.66 g/cm 2 thicknesses. In the .ANISN calculation 8o
energy groups from 1.0 MeV to 0.1 MeV are used.
Agreement between
the analytic solution and ANISN's calculation is reasonable for all
three thicknesses, indicating that the transmitted energy spectrum
obtained from this particular cross section can be correctly calculated
using the method of discrete ordinates.
It should be noted that a
large number of smaller-energy-transfer collisions results in a
transmitted spectrum in the form of a thin spike, somewhat like a
delta fUnction distribution.
50
10
20
I
t =0.11 g /cm
50
~
9
I
-ANALYTIC
• ANISN
2
. ~
I
1=0.66 g/cm 2
18 1 - - - f=0.33g/cm 2 -
45
16
40
•
f
f4
8
~
1\
it
c::
-...
-
7
14
6
12
30
10
25
8
20
6
15
35
0
u
Q)
Qj
c::
•
Q)
"'0
·u
.E
•
5
>
Q)
:!
......
VI
c::
...0
4
u
Q)
Qj
3
1-
•
4
rw
0
0.4
10
4
2
5
2
.\
\
0.8
0.6
E (MeV)
~
0
1.0
0.2
)
I\&
0.4
0.6
E (MeV)
0
0.8
0
0.2
\
0.4
0.6
E (MeV)
Fig. 4. Transmitted electron current per unit energy ~er incident
electron for 1-MeV electrons normally incident on 0.11-g/cm -, 0.33g/cm2-, and o.66-g/cm2-thick aluminum slabs.
51
VI.
COMPARISON WITH EXPERIMENTAL RESULTS
In this section the transmitted current of electrons calculated
by the method of discrete ordinates is presented, along with the
experimentally measured spectra for cases involving monoenergetic
electrons normally incident on target slabs of varying thicknesses.
Calculational results from the Monte Carlo code ETRAN3 where available
in suitable form are also included for additional comparison.
A.
Experiments of Rester
The experimental data presented here are taken from the work of
Rester
46
and Rester and Derrickson.
calculated using ETRAN-15.
47
The Monte Carlo spectra were
Results are given for normal incidence of
1-MeV electrons on Al and Au targets and for 2.5-MeV electrons on Al
targets.
1.
1-MeV Electrons Incident on Aluminum
The points plotted in Fig. 5 show the measured transmitted electron
current per unit energy for 1-MeV electrons normally incident on
2
2
aluminum slabs of thicknesses of 0.10 g/cm , 0.22 g/cm , and 0.32
2
g/cm , respectively, roughly corresponding to 0.2, 0.4, and 0.6,
respectively, of the range of the incident electron.
The solid
curves in the figure represent the results from the discrete ordinates
calculations using a continuous slowing-down term with I'
= lOI
to
treat soft inelastic collisions, designated by AWCS (ANISN with
continuous slowing down), and the dashed curves represent the results
from the discrete ordinates calculations using the M~ller cross section
-----
~.F
11
r--
10
~i
i
~
0Q)
Q)
c
•
8
1=0.10gjcm 2
:
7
"'0
·c::;
.!:
>
Q)
::::E
6
1
4
lj
Q)
a;
3
2
t
1=0.22g/cm 2
1
0.7
3
,,
•
I
.
\
0
0.8
E (MeV)
~
0.9
0.6
\''
l¥!
'
j·
r
0.1
0.6
E (MeV)
0.8
I
:
I
0.4
j
:
j/
0.3
0.2
···..... 11
r~/
0.4
•,I
····::0:~
• 0
0.2
;£. . ..····
0.5
~W'
!I~V'
1\
~~
0.7
2
•••
•
Ld
rVJ r
•y
11
0.8
I
•
!toP ~
0.9
I
J
4
5
Ill
E
u
'
1=0.32 gjcm2
1
D.
........
c
EXPERIMENT
...Jl.... ETRAN
1------i- - - - AWCS
··············· AWMC
d
Q)
1J
1.0
•
~
9
c
5
0
0.1
0.3
0.5
~
0.7
E (MeV)
Fig. 5. Transmitted electron current per unit energy ~er incident
electron for 1-MeV electrons normally incident on 0.10-g/cm -, 0.22g/cm2-, and 0.32-g/cm 2-thick aluminum slabs.
U1
I'V
53
to deal with soft inelast ic collisio ns, designa ted by AWMC (ANISN with
The solid histogra ms represe nt the Monte Carlo
M¢ller cross section ).
calcula tions.
For the 0.10-g/c m
2
case, the experim ental results are
lower than the calculat ed results at the peak of the distribu tion, but
they are greater elsewhe re, especia lly at the higher energie s. In the
2
2
0.22-g/c m and 0.32-g/c m cases, however , agreeme nt between experimental results and theoret ical calcula tions are better along the highenergy edge of the distribu tion than at the peak or lower energy edge.
In general , the results of the discrete ordinate s calcula tions are in
reasona bly good agreeme nt with those of the experim ental measure ments.
It should be noted, however , that the results of the discrete ordinat es
and Monte Carlo calcula tions appear to be in better agreeme nt with each
other than with the experim ental measure ments.
The AWCS results are
2
somewhat higher than the AWMC results and for the O.lO-g/c m and
2
0.32-g/c m cases, are in particu larly good agreeme nt with the Monte
Carlo calcula tions.
Since the discrete ordinate s approxim ation approach es the
Boltzma nn transpo rt equation as the number of energy groups and spatial
interva ls are increase d, the accuracy of the calcula tion is depende nt
on these paramet ers.
In general , for the discrete ordinate s calcula -
tions undertak en in this investig ation, an increase in the number of
energy groups used to describ e a case causes the resultin g spectrum to
become more sharply peaked and to shift the peak of the spectrum to a
slightly higher energy.
Increas ing the number of spatial interva ls
tends todecre ase the magnitu de of the transmi tted spectrum .
Both of
these effects continue up to a point, beyond which no change is noted
54
in the transmitt ed spectrum as a result of an increase in the number
of spatial intervals or energy groups.
Unless otherwise stated, the
results presented in this investiga tion are considere d to be converged .
The number of energy groups and spatial intervals utilized in a particular calculati on is limited by the core storage capacity of the computer
and by the time required for the computati on.
Various calculati ons were
made for the case of 1-MeV electrons normally incident on a o.22-g/cm 2
thick aluminum slab in order to make a direct compariso n between the
two discrete ordinates calculati onal methods and to show the effects of
some factors.
Two AWCS calculati ons were made, one with I'
=
lOI using
160 energy groups and 100 spatial intervals and another with I'
using 166 energy groups and 155 spatial intervals .
= lOOI
In addition, two
AWMC calculati ons were made, one using 175 energy groups and 101 spatial
intervals and another using 218 energy groups and 145 spatial intervals .
The results of all four calculati ons are shown in Fig. 6 in the form of
the total transmitt ed electron current per MeV per incident electron.
The 218-group AWMC calculati on represent s a set of converged results
and gives higher values and a more sharply peaked distribut ion than
does the 176-group AWMC calculati on.
However, in order to achieve the
converged results, the 218-group AWMC calculati on requires a larger
number of spatial intervals and a much longer run time than the 176group AWMC run.
although the I'
The two AWCS calculati ons give fairly similar results,
= lOOI
calculati on requires a larger number of spatial
intervals and a longer run time than does the I'
also be noted that the I'
= lOOI
= lOI
run.
It should
calculati on gives a very strongly
55
4
I
~
•
_Jl_
EXPERIMENT
ETRAN
AWCS, !'
3
~
c:
0
"-
= 10/
············ AWM C, 218 ENERGY GROUPS
- - - AWCS, !'= 100 I
· - · - · AWMC, 175 ENERGY GROUPS
0
•
(I)
(I)
c:
->
c:
·-
~~-~
,
.tl
f=0.22g/cm2
2
I
(I)
~
..........
en
c:
"-
0
v: ft
(I)
1
~
-~
. ...
,;~
...
·~
L"'"'
·'
•. .
0
(I)
f1j
~
(I)
"'C
0
•
\
\
~I
'
·,
~
~-
0
0.1
0.2
0.3
0.5
0.4
E (MeV)
0.6
0.7
\.
0.8
Fig. 6. Transmitted electron current per unit energy ~er incident
electron for 1-MeV electrons normally incident on 0.22-g/cm -thick
aluminum slabs .
56
forward-pea ked angular distributio n for a thin case (such as 0.10
2
g/cm ), since the soft collisions are straightahe ad and the number of
soft collisions increases as I' increases.
In general, the AWCS
method requires fewer energy groups, fewer spatial intervals, less
time to produce the cross section coefficient s, and a shorter time for
calculation al than the AWMC method requires in order to obtain
converged results.
Details on the calculation s shown in Fig. 6 are
given in the following table.
IBM 360/91 computer.
The running times shown are for the
The cross-secti on production time represents
the time required to produce the cross-sectio n coefficient s used for
the particular calculation .
Other calculation s (especially for other
target thicknesses ) were frequently made with the same set of coefficients.
The calculation al time is the time required for ANISN to obtain
a solution for the problem using the previously determined cross-secti on
coefficien ts.
Table I.
Requiremen ts for the Discrete Ordinates Calculation Shown in Fig. 6
Discrete Ordinates
Calculation al Method
AWCS* (ANISN with
continuous slowingdown
I' = lOI
= 1.63 x 10- 3 MeV
Number of
Energy Groups
Number of
Spatial Intervals
Time for
ANISN
Calculation
(min.)
Time for
Cross-Secti on
Production
(min.)
160
100
11.5
18
166
155
35
15
AWMC* (ANISN with
M¢ller cross-secti on)
218
145
44
24
AWMC
175
101
14.5
16
AWCS
II
= lOOI
= 1. 63
x 10-2 MeV
*This calculation also appears in Fig. 5.
(J1
-....J
58
The calculate d and measured transmitt ed electron current per unit
energy per unit solid angle is presented in Fig. 7 for transmiss ion
2
angles of 7•5°, 47·5° and 77·5° through a 0.10-g/cm -thick aluminum
slab.
Both the AWCS and AWMC calculati ons give good agreement with
the experime ntal results at 7·5°, although the AWCS values are slightly
high, probably due to the tendency of the AWCS calculati on to produce
a forward-p eaked angular distribut ion for thin cases.
Both discrete
ordinates methods are in reasonabl e agreement with experimen t at 47·5°
It should be noted that the Monte
and in poor agreement at 77.5°.
Carlo results at 77·5° show evidences of difficult y with statistic al
accuracy.
The transmitt ed electron current for 1-MeV electrons through a
2
0.22-g/cm -thick aluminum slab at angles of 7·5°, 47.5°, and 77·5° is
shown in Fig. 8.
Both discrete ordinates calculati onal methods show
good agreemen t with the measured results at 7.5°, except at the peak
of the experime ntal distribut ion.
Agreement with experimen t is fair
at 47.5° and poor at 77·5°, with the discrete ordinates results again
The angular transmiss ion for the two discrete ordinates
2
calculati ons were a little more consisten t for the 0.22-g/cm case
being low.
than for the 0.10 g/cm
2
case.
Presumabl y this occurs because the
increase in target thickness allows sufficien t elastic angular
scatterin g so that the differenc es in the way in which the inelastic
scatterin g is handled are not extremely significa nt.
:r·>-•-"'
,'t'>,- '\ '·~:,
EU'f~~i'
10
0.20
2.0
9
1\
c
8
~
0
....
+0
Q)
1.6
~
8=
8= 71/20
c
:
6
I
~
7
Q)
+-
I 0.16
I•
•
"'·'''~"''""'·"':111•111111•••••••
•: -~-"-
EXPERIM ENT
J"LETR AN
-AWC S
············AWMC
t= 0.10 g/cm
f
2
I
D
471/20
ltl
8=771/20
0.12
1.2
•
0
c
:;;
.
-~
5
•
>
Q)
~
0.08
0.8
4
•
VI
c
0
....
+0
3
r-w
Q)
2
0.4
~
.~
0
0.6
·;.:.:."tl
lJ
.
L.._
0.8
E (MeV)
•
1.0
0.6
•
0.04
j
.
~
0
.
~
j
Q)
4
_;;;;;.
[.4
b
0.8
E (MeV)
I~
IJ
/
0
1.0
v
j
-~
0.6
0.8
~~
1.0
E (MeV)
Energy distribu tions of the transmi tted electron current
at 7.5°, 47.5°, and 77·5° for 1-MeV electron s normall y inciden t on a
O.lO-g/ cm2-thi ck aluminum slab.
Fig. 7·
(J1
(.!)
2.0
1
.o I
I
I
•
I o. 2 o
I
I
EXPERIMENT __
.sL ETRAN
--AWCS
1.6
0.8
c:
0.16
f--- ············· AWMC
t= 0.22
0
....
+-
g/c~
2
0
.!!
Q)
+-
c:
Q)
"'C
1.2
O.G
0
-....
~
1
8~ 47.,2:
n~ J
I
0.12
1
I
0
8=77'2
.
lr-i
>
Q)
~
..........
1
r'
!/)
0.4
0.8
0.08
I •
. /r\.
n_
!/)
c:
0....
+0
Q)
Q)
~2
0.4
~04
...
I'"V···....
.,;;?
~--
.... ····
0
0.6
E (MeV)
0.4
0.8
0
1 I
0.4
I
I
0.6
E (MeV)
I~
0.8
-
0
0.3
\
\
..
0.5
E (MeV)
0.7
Fig. 8. Energy distributions of the transmitted electron current
at 7.5o, 47-5o, and 77·5° for 1-MeV electrons normally incident on a
0.22-g/cm 2-thick aluminum slab.
m
0
-'"'
'
l•
~'
61
The angula r distrib utions of the transm itted electro ns per unit
solid angle resulti ng from 1-MeV electro ns normal ly incide nt on
2
I 2
0.10-g I em 2 , 0.22-g em , and 0.32-gl cm -thick aluminum slabs are
The AWCS calcula tion does show a slightl y
2
forward -peaked distrib ution in the O.lO-gl cm case, but is in good
presen ted in Fig. 9·
agreem ent with the experim ental values . The AWMC results show
2
reason able agreem ent for the 0.10 glcm slab. The discre te ordina tes
2
calcul ations are very simila r for the 0.22-gl cm case, and are in
2
reason able agreem ent with experim ent. In the 0.32-gl cm case, the
AWCS values are in good agreem ent with experim ent, while the agreem ent
for AWMC is only fair.
The higher results for AWCS are simila r to
those shown in Fig. 5 for the total transm itted electro n curren t for
2
0.32 glcm •
2.
1-MeV Electro ns Incide nt on Gold
The total transm itted electro n curren t per unit energy resulti ng
2
from 1-MeV electro ns normal ly incide nt on a O.l5-gl cm -thick gold slab,
repres enting 0.2 range, is shown in Fig. lOa.
The experim ental points
and Monte Carlo histogr am are simila r to those used earlier for the
aluminum cases, and the solid curve represe nts the results of a
discre te ordina tes AWCS calcul ation with I'
=
I(7·97x lo-
4 MeV). This
value was used becaus e it is close to the value used for the 1-MeV
aluminum runs (I'
= 1.63x1 0-3
MeV).
The discre te ordina tes results
are much lower than the experim entally measur ed values and signifi cantly
lower than the Monte Carlo calcul ation.
Additi onal work is necess ary
to determ ine if the poor agreem ent noted here is due to the cross
jl:
1
J
I!'I
62
o. 7
o.
•
I
I -,
•.. EXPERIMENT
6~
·········x.
ETRAN
--AWCS
.. ......... AWMC
·..
o. 5
•··...
0.4
··~.
\
0. 3
I I
~
~
~'•
.
"
'
0.1
0
0.3
I
I
1=0.10 g/cm 2
0,2
I
•
.
~
•
"',
•,
.
.
~
~ .
I
..
~.
0.2
1---
I= 0.22 g/cm
2
··~
""
tc...
~.
0.1
~
K.
"~ .
0
0.14 ,----.._,--...,.--,......--..---.-----,----r--....--~
..
• ...........• ~
0.12
0.
10 .........
..
····..
I
1=0.32 g/cm 2
..
""".
·············... ~-,~
.
'\.LJ-..---+--+---+- ----+----1
.........
... ~
0.08 f---+--l--~.c-+
···········...
'~
...,'r.~.....~1-----1----11---l-----l
0. 06 f---l---f----l--____: ,t--'
0.04 f---f------l---+----+ --···.p.,L'\.~..~--+--+-~
····-0., ..
·····~.
.. t-+----+----1
0.02 1-----11----+---+--+--+--+--~
····1-4 .
o~~-~-~-~-~-~-~-~~
0
20
40
60
80
ANGLE (deg)
Fig. 9· Angular distributions of the transmi~ted electron 2current
for 1-MeV electrons normally incident on 0.10-g/cm -, 0.22-g/cm -, and
0.32-g/cm 2-thick aluminum slabs.
I
I
1,6
0.....
Q)
-
Q).
1.2
-~
c:
Q)
-o
·u
c:
•
~
~ 0.8
~
..........
1/)
c:
0
.....
-
•
u
Q)
iii
~
0.4
c::
::tr.
~
0
0.3
~
y
~
/
Q)
-
Qj
•
0.24
c:
Q)
"0
u
j
-
• •
• •
~
.....
..........
(/)
/
0.16
1/)
c:
•
0
.....
u
Q)
Q)
0.08
•
(a)
0.5
AWCS
c:
0
.....
u
r
u
0.7
E (MeV)
I
• ,_.n._ ETRAN
-
•
c:
•
0.32
•
I
EXPERIMENT
~
0.9
'
•
• • •
---
...
•
~ ..............
........
• •
• • •
~ ...............
.
(b)
0
~
-
0
20
40
60
•
N
•
80
ANGLE (deg)
Fig. lO.(a) Transmitted electron current per unit energy per
incident electron for 1-MeV electrons normally incident on a
O.l5-g/cm2 -thick gold slab. (b) Angular distribution of the transmitted electron current for 1-MeV electrons normally incident on a
O.l5-g/cm2 -thick gold slab.
0">
w
r-
64
sectio ns or to the method of the calcu lation itself .
At least part
of the diffic ulty must be in the cross sectio ns, since the hard
inela stic atomic cross sectio n was determ ined by multip lying the
M¢lle r cross sectio n by Z.
This is obviou sly incorr ect since the K
ered
shell electr ons in gold are far too tightl y bound to be consid
free, but no tested correc tion factor was availa ble.
The angula r
for
distri butio n of the transm itted electr on curren t per unit angle
this case is given in Fig. lOb.
It shows the distri butio n calcul ated
by discre te ordina tes to be much smalle r at low angles than the
it
experi menta l distri butio n, as is expect ed from Fig. lOa, but that
s.
increa ses in relati on to the experi menta l points at larger angle
of
The same gener al behav ior is shown by the Monte Carlo calcul ation
at
the distri butio n, which is lower than the experi menta l measu rement
low angles but actua lly highe r at large angles .
3·
2.5-MeV Electr ons Incide nt on Aluminum
on
The points plotte d in Fig. 11 show the total transm itted electr
V
curren t per unit energy per incide nt electr on result ing from 2.5-Me
2
I em2
electr ons norma lly incide nt on 0.31-g /cm (0.2 range) and 0.62-g
(0.4 range) thick aluminum slabs.
The experi menta l points , Monte Carlo
used
histog rams, AWCS and AWMC repres entati ons are simila r to those
for the 1-MeV case.
Reaso nable agreem ent is shown betwee n the discre te
ordina tes result s and the experi menta l measu rement s in both cases.
5
2.0
•
EXPERIMENT
J""LET RAN
-AW CS
···········AWMC
~
-
1.6
4
t= 0.31
c
-
~
g/cm 2
0
....
0
Q)
Q)
c
• !::
3
Q)
~
0
->
~
~
..........
IJ)
c
1----.
r-.:
~
2
I
0
....
lf
.·
0
Q)
~
Q)
..
.~
•••.•.•. :,;,ji[··2:J
___.....-
•
0
1.7
1.5
1.9
E (MeV}
t\
2.1
•
.....
1.1
0.8
0.4
.J
v
t-
\
.
_}
..
~-
0
0.8
r
~
\
~·
•
~
1.6
1.2
,,.
'i'.,
~
2.0
E (MeV}
Fig. ll. Transm itted electro n curren t per unit energy per
ns normal ly incide nt on
incide nt electro n for 2.5-MeV electro
2
0.3l-g/ cm2-th ick and o.62-g/ cm -thick aluminum slabs.
,.,,
I
\.r.::
~-
~·.
~
cm 2
~
l
~
I= 0.62g/
1.2
.1I 1
Q)
I
I
I
en
U'1
66
hat lower
However, the discr ete ordin ates calcu lation s here are somew
the distr ibuti on,
than the Monte Carlo value s, espec ially at the peak of
calcu lation s
while agree ment betwe en discr ete ordin ates and Monte Carlo
is good for the 1-MeV cases .
ting
The trans mitte d elect ron curre nt at 20°, 45°, and 60° resul
2-thic k
from 2.5-MeV elect rons norma lly incid ent on a 0.31-g /cm
aluminum slab is shown in Fig. 12.
The calcu lated value s at 20° are
45° the agree in fair agree ment with the exper iment al point s, but at
calcu lated
ment is poor at the peak of the distr ibuti on, with the
At 60°, the discr ete
resul ts highe r than the exper iment al resul ts.
exper iment al
ordin ates calcu lation s are consi derab ly highe r than the
measu remen ts over most of the distr ibuti on.
This seems to contr adict
where the
the large -angl e calcu lation s for the 0.2 range 1-MeV case,
Fig.
calcu lated value s were low compared to exper iment (see
7).
The
reaso n for this phenomenon is not appar ent.
from
The trans mitte d elect ron curre nt at 10° and 20° resul ting
2
k aluminum
2.5-MeV elect rons norma lly incid ent on a 0.62-g /cm -thic
slab is given in Fig. 13.
Agreement betwe en the AWCS calcu lation and
butio n for
exper iment al measu remen ts is poor at the peak of the distri
both cases but is quite reaso nable elsew here.
The low calcu lation al
tly low peak
resul ts shown here are somewhat consi stent with the sligh
nt throu gh the
value s shown in Fig. 11 for the total trans mitte d curre
0.62 g/cm
2
slab.
II
0.9
4.5
A
0.8
4.0
I
c
~u
0>
3.5
o;
3.0
.,
2.5
c
c
...
"'
>
/'
lr·r1
2.0
0>
::e
.........
c
"'0
1.5
0>
1.0
::u
o;
l •
·J
....~
0.5
0
1.7
0.5
u~
tl'
0.3
·'"~
2.1
1.9
E (MeV)
0.2
~
'
I
0.4
·~
0.1
....:;:;
0.2
v
\
2.0
1.8
E (MeV)
I I I I 7:VJ 't\ ~
.....
LJ
J
1
.·l L1J
oo,l.r
r··l I I 1\1
•
i
I;;:;:?
•
1.6
£\,_
'
[./
.
.i
.
0
2.3
0.3 ~+-~-+~--~-~~
8=60°
r
0.6
•
0>
."'0
0.7
•
0.4
{'
8= 45°
8=20°
1-T
• EXPERIMENT
--AWCS
············AWMC
1=0.31 g/cm 2 --+------1
\•
2.2
1.7
1.9
E (MeV)
2.1
Fig. 12. Energy distribu tions of the transmi tted electron current
at 20°, 4~ 0 , and 60° for 2.5-MeV electron s normall y inciden t on a
0.31-g/c m -thick aluminum slab.
(J)
-....1
0.9
1.8
•
1.6
-c:
0.....
EXPERIM ENT
AWCS
2
t= 0.62 g/cm
1.4
•
• •
0.8
--
0.7
cu
cu
c:
0.6
1.2
cu
8=10°
"'0
0
c:
--.....
1.0
•
.
(/)
0.4
~
..........
(/)
c:
0.....
0.6
0.3
0.4
0.2
0.2
0.1
0
0
j
0
cu
cu
1.0
1.4
E (MeV)
1.8
I r\
I \.
1
u/
0.5
>cu 0.8
I
I
8=20°
0
I
~
•J
~
•
1.0
1.4
E (MeV)
-~
1. 8
Fig. 13. Energy distribut ions of the transmitt ed electron current
at 10° and 20° for 2.5 MeV electrons normally incident on a 0.62-g/cm 2thick aluminum slab.
en
CD
69
The angular distribu tions of the transmi tted electron current
per unit solid angle for 2.5 MeV electron s normally inciden t on
2
2
0.31-g/c m and 0.62-g/c m thick aluminum slabs are presente d in
The values calculat ed by the discrete ordinate s methods
2
for the 0.31-g/c m case are higher than the experim ental measure ments
Fig. 14.
through much of the distribu tion, as was shown in Fig. 12.
The
Monte Carlo calcula tions also follow this general tendenc y. The AWCS
2
results for the o.62-g/c m case show more reasona ble agreeme nt with
the experim ental values, although the calcula tion is slightly low at
the forward angles.
B.
~eriments
of Lonerga n et.al.
The experim ental data presente d here are taken from the work of
27 The Monte Carlo calcula tions were
Lonerga n, Jupiter , and Merkel.
3 Results
made by Edmondson, Derricks on and Peasley* using ETRAN-15.
are given for 4-MeV and 8-MeV electron s inciden t on aluminum targets .
1.
4-MeV Electron s Inciden t on Aluminum
The electron current per unit energy per unit solid angle
transmi tted at 30° from 4-MeV electron s normally inciden t on a
2
1.275-g /cm -thick aluminum slab (0.5 range) is given in Fig. 15a.
The experim ental points and Monte Carlo histogra m are plotted in the
usual manner, and the solid curve represe nts an AWCS discrete ordinate s
3
calcula tion with I' = lOI (1.63x10 - MeV). Agreement between the
discrete ordinat es results and the experim ental measurem ents is fair,
*As mention ed in Ref. 27.
70
•
EXPERI MENT
•
ETRAN
- - AWCS
0.6 r-----t-~~t--+--+·-····_···_···_:_A:::W::M:.:C~+---+---J
1=0.31 g/cm
c
-
2
0.4
0
"-
( ,)
Q)
Q)
0.2
c
Q)
~
(,)
c
"-
en
.........
en
c
0
"-
( ,)
Q)
0
0.5
0.4 •a
Q)
0.3
• ••••
...._--~4
1=0.62 g/cm
2
~ ~·
~·
0.2
"' ~ •
)'....,
0.1
0
0
10
20
30
40
50
.
~
~~
60
70
80
ANGLE (deg)
Fig. 14. Angular distribu tions of the transmi tted electron curr~nt
for 2.5-MeV electron s normally inciden t on 0.31-g/c m2-thic k and o.62g/cm2-t hick aluminum slabs.
...
''
l
71
0.44 .-----.-,...1---,-T-~J---..
EXPERIMENT
•
0.40 1---+- .J'L. ETRAN
-AWCS
2
0.361---+ ----+-- t= 1.275 g/cm
0.22
Jt\
0.20
~-
0.18
-
0.16
·c::;
0.12
c::
...
0
1
Q)
0.14
c::
...
en
.
en
c::
0...
u
.!!
Q)
Q)
~ 0.24
0.08
0.20
~ 0.16
~
/~
0.06
~
0.04
~
n..
0
0.5
1.0
~
1---+--~L-4----1----+----l
~
0.1 2 1---+-----+--f~>r-~---+----+-----1
0.04
l\.
(ol
0
i\
~"~--~-1---l
0.08 1----+--~'fr-
/....~ f
0.02
...
~ l-~---1h-\--\----+---l-------+-----J
1-
I•
Q)
~
.........
"'
u
~
0.10
>
~ 0.28 1--...l\!r-+--+-~1----+------l
lr
c::
Q)
"'0
c 0.3 2 1---+---+- ----+------ +-----l
e
I~
u
.!!
I~~
1.5
£(MeV)
2.0
2.5
~-+--+---+--lllo.'"l_-~----l
~~
(b)
o~-~--~---L---L~~
3.0
0
20
80
60
40
ANGLE (degl
100
2
Fig. 15.(a) Energy distribut ion of the transmitt ed electron
current at 30° for 4-MeV electrons normally incident on a 1-275-g/c m 2
thick aluminum slab. (b) Angular distribut ions of the transmitt ed
electron current for 4-MeV electrons normally incident on a 1-275-g/c m thick aluminum slab.
I
72
while the agreem ent betwee n the Monte Carlo values and experim ent is
However, in the summary of Ref. 27 it states "The number
2
of 4.0 MeV electro ns transm itted through 1.275-g /cm -thick slabs of
much better .
Al was 25% higher in the calcul ation.
When the calcula ted energy
spectr a and angula r distrib ution were renorm alized to the experim ental
transm ission they agreed with the measure d data."
Since the AWCS
calcul ation is consid erably higher than the experim ental results and
the Monte Carlo values at the peak and at lower energie s in the
distrib ution, it seems quite possib le that the agreem ent between the
AWCS result s and the Monte Carlo calcula tion before renorm alizatio n
might be better than that shown in Fig. l5a.
The angula r distrib ution of the transm itted electro n curren t per
unit solid angle from 4-MeV electro ns normal ly incide nt on a
2
1.275-g /cm -thick aluminum slab is shown in Fig. l5b. The AWCS values
are consid erably higher over most of the distrib ution than the Monte
Carlo results and those from the experim ental measur ements , as would
be expecte d from Fig. l5b.
It is interes ting to note, howeve r, that
the discre te ordina tes value at 0° is very close to the experim ental
point, while the Monte Carlo histogr am is much lower.
2.
8-MeV Electro ns Incide nt on Aluminum
The electro n curren t per unit energy per unit solid angle trans2
mitted at 20° from 8-MeV electro ns normal ly inciden t on a 0.953-g /cm thick aluminum slab (0.2 range) is given in Fig. 16a.
points , Monte Carlo histogr am, and AWCS using I'
previo us figure s.
= lOI
The experim ental
are shown as in
The discre te ordina tes calcula tion shows excelle nt
ijU#.
r
**-~·-
101
1.6
-4
-...
•
c
0
u
~
Q)
I
t
I
I
--AWCS
100
1.4 ~
EXPERIMENT
_n_ ETRAN
.&l
0.953 g/cm 2
c
•
,p
Q)
"'0
u
c
-...
.
en
j
u
...
Q)
~
...g 0.6
-
,.. ~r ~
..n
'\
Q)
~
en
~..I
10-2 ~
c
-o
~ 0.8
/I
c
...
0
\
"(3
/
~
en
-
1.0
\-
0Q)
>
Q)
1.2
Q)
1
~
10-1
c0
...
~
u
~
Q)
0.4
~
Q)
-
I
10-3
0
0.2
(b)
(0)
0
2
3
4
E (MeV)
5
6
7
0
10
20
30
~
40
50
60
70
ANGLE (deg)
Fig. 16.(a) Energy distribution of the transmitted electron current
at 20° for 8-MeV electrons normally incident on a 0.953-g/cm2-thick
aluminum slab. (b) Angular distribution of the transmitted electron
2
current for 8-MeV electrons normally incident on a 0.953-g/cm -thick
aluminum slab.
-...J
w
. ""
L
74
agreeme nt with the experim ental measure ments over most of the energy
range.
However, the calculat ed curve actually increase s in value below
2 MeV, while the experim entally measure d points continue to decrease in
magnitu de.
It should be noted that the Monte Carlo calcula tion,
although exhibit ing some statisti cal fluctua tion, also appears to
increase in the lower energy range.
The angular distribu tion of the transmi tted electron current per
2
unit solid angle for 8-MeV electron s normally inciden t on a 0.953-g/ cm thick aluminum slab is presente d in Fig. 16b.
The distribu tion from
the discrete ordinate s calcula tion is in excellen t agreeme nt with the
experim ental points at the higher angles, but is does not exhibit a
low-ang le peak as the experim ental distribu tion does.
The low-ang le
peak in Fig. 16b. is in sharp contras t to the high experim ental value
at 0° shown in Fig. l5b.
75
VII.
A.
COMPARISON WITH A THEORETICAL CALCULATION
FOR AN INCIDENT ELECTRON SPECTRUM
Trans mitte d Elect ron Spect ra
ble to preBecau se of the lack of exper iment al data, it was not possi
ts for the
sent a comp arison betwe en calcu lated and exper iment al resul
Using the Monte
case of an elect ron energ y spectr um incid ent on a slab.
48
3
calcu lated the trans Carlo code ETRAN of Berge r and Seltz er, Scott
a speci fic
mitte d elect ron curre nt per unit energ y for the case of
slabs , and this
elect ron energ y spectr um norma lly incid ent on aluminum
obtai ned with
theor etica l calcu lation has been compared with resul ts
in the calcu ANISN. The incid ent elect ron energ y distri butio n used
therm allation s is a repre senta tion of the spectr um resul ting from
23 5u. 4 9 This spectr um exten ds to elect ron energ ies
neutr on captu re in
17 (take n from
of the order of 10 MeV and is shown expli citly in Fig.
Ref. 48).
The trans mitte d elect ron curre nt was calcu lated by the con-
tinuo us slowin g-dow n versi on of ANISN from Eq. (16).
Inela stic col-
fer great er than
lision s with atomi c elect rons invol ving an energ y trans
I' were repre sente d by the Mplle r cross sectio n, Eq. (19)·
Those with
contin uous slowi ngan energ y trans fer less than I' were appro ximat ed by a
100 I, or 0.016 3
down term. The I' value used for this calcu lation was
to 0.15
The ANISN calcu lation used 21 elect ron energ y group s down
2
The ETRAN reMeV and 320 spati al inter vals for 1.0 g/cm thick ness.
2
Fig. 18 as a
sults for an aluminum slab 0.5-g /cm thick are shown in
s. A simil ar
histog ram and the ANISN resul ts are shown as plotte d point
2
thick .
/cm
l.O-g
slab
um
alumin
an
for
18
Fig.
in
comp arison is given
MeV.
76
• _..._
•
•
t
•
'
•
.if
>
•
Q)
~
u
Q)
1o-
2
en
•
•
N
E
(.)
'e
en
c:
•
( .)
~ 10- 3
Q)
•
•
•
•
0
2
4
6
8
10
12
ELECTRON ENERGY (MeV )
elec tron curr ent per
Fig. 17. Ener gy distr ibut ion of the incid agt
a sour ce.
as
t
Scot
unit ener gy per incid ent elec tron used by
Y"
},;.'-'
~
.
too
10_,
-
I
5
f
J
2
...,.
t
•
L'
.
~
ANISN
ETRAN J
I
2
;~
10- 2
5
i
I
10-1
u
t-"1
1-
1
N
E
l.J I
5
u
r,
>...
•
~
"'<:
u...
...
1-
....,__
h
I
(·_
'
h
r
10-2
-
-3
'
2
L......:
10
.-
...
I
L..-,
2
~
5
l
L.:
-
l
5
5
L!
•
t
2
I
It :0.5 ~/om'
2
= 1.0 gjcm 2
I
to- 4
0
!
!
2
3
l
to-3
4
5
ELECTRON ENERGY (MeV)
6
7
0
2
3
4
5
6
7
ELECTRON ENERGY (MeV)
nt
Fig. 18. Trans mitted electr on curren t per unit energy per incide
on
nt
incide
lly
electr on for a speci fic ener~ spectr um (Fig. 17) norma
0.50-g /cm 2-thick and 1.0-g/ cm -thick aluminum slabs.
-...J
-...J
78
agree ment,
In both plots the ETHAN and ANISN resul ts are in reaso nable
to 4-MeV range
altho ugh the ANISN resul ts are a little high in the 22
for the 1.0 g/cm case.
B.
Trans mitte d Photo n Spect ra
photo n curIn addit ion to the elect ron trans port calcu lation , the
d throu gh the
rent produ ced by brems strahl ung was computed and trans porte
Eq. (16) for
slab by solvi ng Eq. (2) for photo n trans port coupl ed with
elect rons.
As indic ated in Secti on III, this coupl ing introd uces pho-
the photo n
tons produ ced by elect ron brems strahl ung as a sourc e for
not introd uced
trans port equat ion, but elect rons produ ced by photo ns are
into the elect ron trans port equat ion.
The photo n calcu lation used 60
for a 50energ y group s down to 0.01 MeV and a total of 339 inter vals
elect rons are
g/cm2 -thick aluminum slab. At thick depth s the prima ry
dose at such
no longe r prese nt, and photo ns const itute the bulk of the
depth s.
photo ns
Elect rons are prese nt, produ ced by the photo ns, but the
are domin ant.
The method of discr ete ordin ates is quite capab le of
thick targe t
calcu lating the resul ting photo n curre nt, even for very
depth s, as shown in Fig. 19.
It would be very diffi cult to obtai n
by a Monte
reaso nable stati stica l accur acy in a simil ar calcu lation
Carlo proce dure.
79
1
\
2
\\
1\
'\
\
3
\
\ \'
"'v
N
~2 g/cm
2
..---20g /cm
2
50 g/cm 2
X'
X'\.
\.
1\.\ .'\
'
\.
,,
'\.
'\..
E
0
'-.\..
'\'
'\
"'"
1\.. \..'
'\ '\.
'\.
'\
\.. \..\.
i\. \ \
'\
\\
\\'
' 1\\\"
\
--
\\\
f-.---1---
~
\
l
2
6
4
8
10
E (MeV)
y per incid ent
Fig. 19. Tran smitt ed photo n curre nt per unit energ
17) norm ally
(Fig.
rons
elect
of
elect ron for a spec ific energ y spect rum
ated.
indic
s
incid ent on aluminum slabs of the thick nesse
80
VII I.
CONCLUSIONS AND RECOMMENDATIONS
prom isin g method for calDis cret e ordi nate s app ears to be a very
aluminum, but add itio nal inv esti cula ting the tran spo rt of elec tron s in
nt of its app lica bili ty. The
gati on is requ ired to dete rmin e the exte
thro ugh gold are con side rabl y
resu lts achi eved for elec tron tran spo rt
. It seems prob able that the
poo rer than the calc ulat ions for aluminum
due to the diff eren ce betw een
diff eren ce in the resu lts achi eved is
(Al) . It is not known whe ther
heav y elem ents (Au) and ligh t elem ents
due to the method of calc ulat ion
the diff icu lty exp erie nced with gold is
it seems more like ly tha t the
or to the cros s sect ions employed, but
prob lem lies in the cros s sect ions .
used to trea t low -ene rgy
Both ANISN with con tinu ous slow ing down
the Mpl ler cros s sect ion used
tran sfer coll isio ns (AWCS) and ANISN with
(AWMC) are capa ble of givi ng
to trea t low -ene rgy tran sfer coll isio ns
acce ptab le resu lts for aluminum.
At this stag e of deve lopm ent, AWCS
r ener gy grou ps to prod uce conseems pref erab le beca use it requ ires fewe
runn ing time than does AWMC.
verg ed resu lts and requ ires a sho rter
ld incl ude calc ulat ions for
Sub sequ ent area s of inve stig atio n shou
dete rmin e the rang e of app lica sev eral nonaluminum targ ets in orde r to
of the curr ent cros s sect ions .
bili ty of the meth od of calc ulat ion and
the cros s-se ctio n theo ries
An atte mpt shou ld then be made to deve lop
eme nt with the exp erim enta l refor heav y elem ents in orde r to get agre
for com pari son with exp erim enta l
sult s. Cal cula tion s shou ld also be made
stud ies of elec tron brem sstr ahlu ng.
81
could be
Calcu lation of elect ron trans port by discr ete ordin ates
the energ y-gro up
made more effic ient by a more preci se determ inatio n of
mono energ etic
struc ture requi red to achie ve conve rged resul ts for both
and energ y-spe ctrum sourc es.
The calcu lation al proce dure in ANISN
the appli catio n
could be made more effic ient for elect rons by limit ing
to those
of the conve rgenc e crite ria for a parti cular energ y group
for that group
spati al inter vals where the calcu lated elect ron curre nt
be used to reis signi fican t. In addit ion, a weigh ting funct ion might
. Weigh ting
duce the number of energ y group s requi red for a calcu lation
sectio ns to
funct ions are often used in the proce ss of treati ng cross
obtai n a multi group form
24 and norma lly invol ve the repre senta tion of
a singl e energ y
sever al group s from a norma l energ y group struc ture by
group with an avera ged cross sectio n.
ns in
The avera ging proce dure inclu des weigh ting the cross sectio
ive impor tance .
the origi nal group struc ture by some measu re of their relat
50
shows that althou gh
However, recen t work on neutr on trans port in iron
of a good
signi fican t impro vemen t can be obtai ned by the selec tion
struc ture
weigh ting funct ion, same proble ms requi re a speci fic group
in order to obtai n preci se answe rs.
~~
I
82
rl
\'
!
i
i
IX.
APPENDICES
'
83
APPENDIX A
DERIVATION OF THE CONTINUOUS SLOWING-DOWN TRANSFORT EQUATION
The derivat ion given here follows the method used by Rossi.
31
The
Boltzma nn transpo rt equation for electron s is given by Eq. (ll) as
E
n.v~(R,E,n)
= P(R,E,n )
fo
+ n
dE'
f
ciD'
E
+ n
T
2
dcr.
dE'
f
.....
.....
~ne 1
<ill'
- 0)
( E',E,0'.
dE <ill
~(R,E',n')
- n cr (E) ~(R,E,0)
l
+ n
f
(
2
E+I'
I
dE'
E
dcr.
~ne 1
d0'
.....
.....
E',E,0'. 0)
<b(R,E' ,n')
dE<ill
.....
.....
*
.....
.....
T
- n cr (E) ~(R,E,0) - ncr e-e (E) ~(R,E,0)
2
where
cr*
e-+e
=
crel + crbr '
E-I'
T
cr (E)
1
=
f
dE'
f
2
d cr.
~ne
1
<ill
..... .....
(E,E',0 '.0)
dE' <ill
E
2
E
T
cr (E)
2
=
f
E-I'
dE'
f
2
d cr.
~nel
<ill
..... .....
(E,E',0 '.0)
dE' <ill
( 34)
'~
.l
84
I
:i
,I
f
ibing
The inela stic scatt ering is now separ ated into terms descr
Defin e
large and small energ y trans fer colli sions .
T
=n
f
2
_, .....
T
........ ..
dcrin el(E' ,E,n• .n)
,n)
~(R,E
(E)
ncr
,n')
'
~(R,E
'
Cill' - - - - dECill
2
(35)
.....
-+
E+I'
dE'
f
E
Add and subtr act
where T descr ibes small energ y trans fer collis ions.
E+l'
f
n
(E',E )
dE
dcr.
~ne 1
dE'
~
_..
__.
( R,E', n )
E
where
dcr.
~ne
1
.....
2
( E' , E)
_
-
dE
Jc
_,
(E',E ,n•.n )
d cr.
1
~~----~n ----~-n_e~
dEdn
U.H
...
2
d cr.
~ne
1
_,
(E',E ,n•.n )
dEctn
Then
E+I'
T
f
= n
dE'
f
2
d cr.
ctn
~ne
I
_, _,
(E' ,E,n' .n)
1
dECill
~(R, E' ,D')
E
.....
T
.....
- n cr 2 (E) ~(R,E,n)
E+l'
+ n
f
dE'
dcr.
( E' , E)
1
~(R, E' ,n)
dE
dcr.
( E', E)
1
~(R, E' ,n)
dE
~ne
E
E+l'
- n
f
E
dE'
~ne
(36)
1
Ill
85
r
I
f.J
,d
Now assume
;i
:·:
I
2
--+
--+
dcr.1ne 1 (E',E,S1 1 .S1)
dEdn
degra datio n and not
that is, the soft colli sion s invol ve only an energ y
(36) then canc el,
an angu lar chang e. The first and last terms in Eq.
and
E+I'
T
nf
=
dcr. l (E' ,E)
~(R,E' ,n)
lnedE
dE'
E
E+I'
T
nf
=
(E',E )
dcr.
~(R,E',n)
lne 1
dE
dE'
E
E
- n
J
( E, E' )
dcr.
1
~(R,E,n) .
lne
dE'
dE'
(37)
E-I'
Defin e
=
f( E', E)
X
A
X
( E' , E)
dcr.
~(R,E',n)
lne 1
dE
=
E' - E
dx
=
E - E'
dX
A
= dE' , first term;
= -dE' , second term.
I'
Then
T
=
nj
dX f(E+x, E)
0
I'
- n
f
0
A
A
dx f(E, E-x)
(38)
~,
86
,I
I
ijj
n.
'.I
'
I'
Now let
=
f( E+x., E)
abo ut E•
Exp and ing g(E ',x.) in a Tay lor ser ies
= E whi le hold ing x. con -
stan t giv es
= g(E,x.) + K ld~' [g(E ' ,x.)] l\E'=E .
g(E• ,x.)
Then
so tha t
d
= f(E,E-x.) + K dE [f(E ,E-x .)]
or
f(E+x.,E)
=
d
f(E,E-x.) + K dE [f(E ,E-x .)] •
( 39)
A
and not ing tha t x. and x. are
Sub stit utin g Eq. (39) into Eq. (38),
dis tinc tion between them may be
var iab les of inte gra tion so tha t the
drop ped ,
I'
=
T
n
J
d
dx. K dE [f(E ,E-x .)]
0
Now set E'
= E - x., and
I'
T
=
n
f
0
dK x.
~ [do-~~~E')
cf>(R,E,n)J
( 40)
87
Def inin g the stop ping power S(E) as
I'
S(E)
=
n
J
(41)
0
or
E
S(E)
=
I
n
dE' (E-E ') do( E, E')
dE'
(42)
E-I
to find
Then sub stit ute Eq. (41) into Eq. (40) ,
T
d
= aE [S(E )
.....
.....
~(R,E,D)]
•
(4 3)
een the stop ping power and
Equ atio n (42) defi nes the rela tion ship betw
rgy tran sfer coll isio ns. As
the diff ere ntia l cros s sect ion for low -ene
ts two term s in Eq. (34). T
init iall y defi ned in Eq. (35), T repr esen
back into Eq. (34) to give
as defi ned by Eq. (43) is now sub stitu ted
E
n. V'ct»(R, E,n)
0
=
dE'
P(R, E,n) + n /
I
dn I
E
- n a*e-+e (E) ct»(R,E,n)
E
+ n
10 I
dE'
2
.....
.....
d a. 1 (E', E,n' ·D)
ct»(R, E' ,D')
lne
dn'
dEdn
E+I '
..... Cl
.....
T
- n a 1 (E) ct»(R,E,n) + ClE [S(E ) ct»(R,E,D)]
(44)
ted by~ [S(E ) ct»(R,E,n)]
The low -ene rgy tran sfer coll isio ns repr esen
elec tron s but only redu ce the
in Eq. (44) now do not prod uce knoc k-on
ener gy of the inci den t elec tron .
"•
88
Equation
(44)
fined as in Eq.
is given in Section III as Eq. (16), with S(E) de-
(42),
and is the form of the transport equation solved
by AWCS (ANISN with continuous slowing down used to treat low-energy
transfer collisions) .
89
APPENDIX B
HIGH-FREQUENCY END-POINT CORRECTION FOR THE DIFFERENTIAL
BREMSSTRAHLUNG CROSS SECTION
The bremsst rahlung cross section of McCormick, Keiffer, and
Parzen,
37 which is differe ntial in angle and energy, was given as
When Eq. (30) is integrat ed over angle, the
Eq. (30) in Section IV.
result may be expresse d as "Eq. (3BN)" in Koch and Motz:
Z
dcrk
16
2 2
ro dk p
= 137 k p0
(45)
where
L" 21n [EoE : pop-l],
E0 "
ln ( : : : : : ) '
90
E ,E
=
the initia l and final total energy of the electro n
2
in a collisi on, in mc units,
p 0 ,p
=
the initia l and final momentum of the electro n in
0
a collisi on, in me units,
k
=
the energy of the emitted photon in mc
2
units.
The bremss trahlun g cross section should have a finite value at the
high-fr equenc y limit, but Eq. (45) gives a value which approac hes 0 as
p ..... 0.
Fano's cross-s ection formul a for the high-fr equenc y limit is
given by Koch and Motz [in Eq.
3 2
dcr k
=
TT
4
dk E ~
o o
2
137 Ji (E -1) 2
Z r
(II-Y)] as:
E (E -2)
4 + _o._o.....
,.~
o
3
(Eo+l)
0
l-~l2
[
2~ 0
E
£n
0
I
l+~ J '
1-~:
(46)
where
~ ,~ ==
the ratio of the initia l and final electro n
0
veloci ty to the veloci ty of light.
A high-fr equenc y limit correc tion factor
so that dcrk [Eq · (45)]
X
F
factor was chosen to be F
= dak
F, is sought, therefo re,
The form of the
[Eq · (46)] when p=O.
X
-= ------
, so that the correc tion would
1-exp(- X)
also apply near the limit, asp ..... o. Then
F ==
X
l-exp( -X) '
F == X,
PI o
p
=0
.
91
Therefo re, Eq· (45) must be evaluate d asp- 0 so the result set equal
to Eq. (46) in order to find
x.
First multipl y Eq· (45) by the factor
X in the form X = X' Ejp, and take the limit as p - o, to get
4
3E +E30 -if0 +7E 0 +6
2 2
do-k
=
2Z ro dk X'
137 k p
Then substitu te the identiti es
0
0
(4 7)
3(E o+l)(E o- 1 )E o
2~0 ~ In(~:::)
and p 0
~ ~ 0E0 into
Eq. (46) to give
do' k
(48)
=
Now set Eq. (47) equal to Eq. (48) and solve for X', noting that
2
p o+1
=
2
Eo' so that
XI
2nZ
=
137 '
and
as given in Eq. (30), Section IV·
Tnis factor was derived to assure
the correct value as p ~ 0 after integra tion over angle, but it does
not assure the correct limit at a specific angle·
92
X.
LIST OF REFERENCES
93
REFERENCES
1.
2.
3·
Procee dings of the Symposium on the Protec tion Again st Radia tion
Hazard s in Space, Gatlin burg, Tenne ssee, November 5-7, 1962,
TID-7652, Office of Techn ical Servic es, Department of Commerce,
Washi ngton (1962) . See, for example, J. A· Van Allen, "Brief
Note on the Radia tion Belts of the Earth, " P· 1, Book 1, and
s. L. Russak, "Radi ation Dosages from Electr ons and Brems strahlu ng
in the Van Allen Belt, " p . 760, Book 2 .
Second Symposium on Protec tion Again st Radia tions in Space, Gatlin
and
autics
Aeron
al
burg, Tenne ssee, Octob er 12-14, 1964, Nation
Space Admin istrati on report NASA SP-71 (1965) . See, for example,
M. J. Berge r and s. M. Seltze r, "Resu lts of Some Recen t Transp ort
Calcu lation s for Electr ons and Brems strahlu ng," p. 437, and
G. D. Magnuson and A· W· McReynolds, "Space Electr on Radia tion
Shield ing - Brems strahlu ng and Electr on Transm ission "p. 455·
for
M. J. Berge r and S.M. Seltze r, "ETRAN Monte Carlo Code System
al
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XI.
VITA
phia , Penn sylv ania , on
Davi d Elli ott Bart ine was born in Phil adel
and seco ndar y educ atio n in
December 6, 1936. He rece ived his prim ary
Drex el Inst itut e of Tech nolo gy
Phil adel phia , Penn sylv ania . He atten ded
Bap tist Coll ege in St. Dav id's,
in Phil adel phia , Penn yslv ania , and East ern
degr ee in chem istry from
Penn sylv ania , and rece ived a Bach elor of Arts
taug ht chem istry and phy sica l
East ern Bap tist Coll ege in June 1959. He
Penn sylv ania , from Septemscie nce at Con esto ga High Scho ol in Berwyn,
phys ics and chem istry at
ber 1959 unt il June 1960. He then taug ht
unt il June 1962. In June 1962
Beav er Coll ege in Glen side , Penn sylv ania ,
scie nce teac hing from the
he rece ived a Mas ter of Scie nce degr ee in
, Penn sylv ania . From SeptemUni vers ity of Penn sylv ania in Phil adel phia
at the Montgomery County
ber 1962 unti l June 1965 he taug ht chem istry
Juni or Coll ege in Takoma Park , Mar ylan d.
In the fall of 1965 he acce pted au.
s.
Atomic Energy Commission
red the Grad uate Scho ol at
Trai nees hip in Nuc lear Eng inee ring and ente
Mis sour i. He rece ived a
the Uni vers ity of Mis sour i-Ro lla in Roll a,
ing from the Univ ersit y of
Mas ter of Scie nce degr ee in nucl ear engi neer
s. Atomic Energy Commission
Mis sour i-Ro lla in Aug ust 1966. He held au.
emb er 1968 unti l Aug ust 1969 he
Trai nees hip unt il Aug ust 1968. From Sept
in nucl ear
Atomic Ener gy Commission Spe cial Fell ows hip
Scie ntif ic App licat ions Dep artengi neer ing. He has been employed in the
Ridge Nati onal Labo rato ry
ment of the Math ema tics Divi sion of the Oak
held au.
s.
sinc e Aug ust 1969.
Shan kle of Was hing ton,
He is mar ried to the form er Doro thy Judi th
, Rebe cca, and Rob ert.
D· c. and has four chil dren , Davi d, Benj amin