The one-dimensional discrete ordinates code ANISN was adapted to transport low energy (a few MeV) electrons. Calculated results obtained with ANISN were compared with experimental data for transmitted electron energy and angular distribution data for electrons normally incident on aluminum slabs of various thicknesses. The calculated and experimental results are in good agreement for a thin slab (0.2 of the electron range), but not for the thicker slabs (0.6 of the electron range). Calculated results obtained with ANISN were also compared with results obtained using Monte Carlo methods

Alsmiller, R. G., Jr.

Barish, J.

Bartine, D. E.

Engle, W. W., Jr.

Mynatt, F. R.

English

NASA Technical Reports Server

LOW-ENERGY ELECTRON TRANSPORT WITH THE METHOD OF
DISCRETE ORDINATES I
D. E. Bartine, R. G. Alsmiller, Jr.,
F. R. Mynatt, W. W. Engle, Jr.,
and J. Barish
Oak Ridge National Laboratory
Oak Ridge, Tennessee, 37830
The one-dimensional discrete ordinates code ANISN has been adapted to
transport low-energy (_ a few MeV) electrons. Calculated results obtained
with ANISN are compared with experimental data for the transmitted electron
energy and angular distributions for 1-MeV electrons normally incident on
aluminum slabs of several thicknesses. The calculated and experimental
results'to date are in good agreement for a thin slab (0.2 of the electron
range) but are not in good agreement for thicker slabs (0.6 of the electron
range). Calculated results obtained with ANISN are also compared with cal-
culated results obtained using Monte Carlo methods.
INTRODUCTION
The transport of low-energy (of the order of a
few MeV) electrons through matter is important in
the shielding of manned space vehicles that pass
through the Van Allen electron belt. A code that
treats this transport by means of Monte Carlo meth-
ods is available (ref. 1), but because of the poor
statistical accuracy which can be obtained in some
cases, a nonstatistical method of calculation is
needed. To fill this need, the method of discrete
ordinates has been adapted to the transport of low-
energy electrons. The method of calculation is
discussed in the next section, and the calculated
results are presented and compared with experi-
mental results and other calculated data in the
last section.
METHOD OF CALCULATION
In principle, the discrete ordinates code
ANISN (ref. 2) may be used to transport electrons
by the simple expedient of introducing into the
code the differential cross sections for electron-
nucleus elastic collisions, electron-nucleus brems-
strahlung-produciug collisions, and electron-
electron collisions. In practice, however, these
cross sections are quite different from those which
occur in neutron transport where the method of dis-
crete ordinates has been used extensively, and the
IThis work was funded by the National Aeronautics
and Space Admnistration, Order H-38280A, under
Union Carbide Corporation's contract _rith the
U. S. Atomic Energy Commission.
816
method has shown only partial success in transport-
ing electrons.
In the Monte Carlo treatment of electron trans-
port, the individual electronic collisions are not
considered, but rather the theories of multiple
Coulomb scattering and continuous slowing down are
used to group together large numbers of collisions
(ref. 1). In the work reported here, the individ-
ual electronic collisions are treated except that
those electron-electron collisions which result in
very small energy transfers (of the order of the
average ionization potential of the atom) are
treated using the continuous slowing-down approxi-
mation. The equation solved is the Boltzmann trans-
port equation with a continuous slowing-down term
added:
_'V_(_,E,_) = P(_,E,_)
E d2Oe,.e(E',E._'._ )
+ n f o dE' f d2' ¢(_,E',_')
E dEd_
E d2Oin(E',E,_'-_ )
+ / o / d_' (_,E',_')
E+I' dEd_
- n oT(E) _(R,E,_)
+_[s(_) _(_,E,_)], (i)
https://ntrs.nasa.gov/search.jsp?R=19720010060 2020-03-17T03:46:18+00:00Z
d2Oe_(E',E,_''_) d2Oel(E',E,_'._)
dEdO dEd_
d2Obr(E',E,_'._)
+
dEd_
OeT_e(E') = f dE f d_
d2Oe÷e(E',E,_''_)
dEd_
d2Cin(E',E,_''_)
T (E')=fdEfd_
°in dEd_
oTCE ') = a_+eCE') + o_nCE') ,
where
g=
d_ =
E=
* (R,E,_) =
E --
0
n =
P(R,E,_) =
d2gel(E',E,_'._)
a_.d_
d2abr(E',E,_''_)
dEdn
d2_in(E',E,_''_)
dNd_
a vector denoting the position
of the particle;
= a unit vector in the direction
of the mc_entumvector;
an element of solid angle;
the kinetic energy of an electron;
the electron flux per unit energy;
the highest kinetic energy con-
sidered;
the atomic number density;
the number of electrons per unit
energy per steradian per unit
volume input at R from an ex-
ternal source;
the differential cross section
for an electron (E') going in
direction _' to undergo elastic
nuclear scattering (Coulomb scat-
tering) to (E,_);
the differential cross section
for an electron (E',_') to under-
go bremsstrahlung (radiative)
scattering with both nuclei and
atomic electrons to (E,G);
the differential cross section
for an electron (E',_') to under-
go inelastic scattering from
atomic electrons to (E,_), there-
by producing a secondary electron
with energy E'-E;
I' = an arbitrary value taken to be
the minimum energy loss allowed
in an inelastic electron-
electron collision;
S(E) = the stopping power (energy loss
per unit path length) due to in-
elastic electlon-electron col-
lisions resulting in an energy
loss smaller than I' (these col-
lisions are presumed to be
straightahead).
The differential cross sections for electron-
nucleus elastic collisions and electron-nucleus or
electron-electron bremsstrahlung production are
taken from standard sources (refs. 3 to 5). The
differential cross section given by Moller (refs.
3, 6) is used to describe electron-electron colli-
sions which result in large energy transfers (_ I').
The stopping power used to treat electron-electron
collisions resulting in small energy transfers
(< I') is obtained in the manner outlined by
Rohrlich and Carlson (ref. 7) with the density ef-
fect correction taken from Sternheimer (refs. 8, 9).
The I' value used in the cases reported here is
0.0163 MeV, i.e., 100 × I where I is the average
ionization potential for aluminum, 163 eV (ref. 8).
RESULTS AND DISCUSSION
Goudsmit and Saunderson have obtained an ana-
lytic expression for the angular distribution of
transmitted electrons when monoenergetic electrons
are normally incident on sufficiently thin slabs so
that the energy degradation of the electrons may be
neglected (ref. I0). Using the Goudsmit-Saunderson
theory, Berger obtained the angular distribution of
the transmitted electrons resulting from I-MeV elec-
trons normally incident on an aluminum slab of thick-
ness 0.0287 g/cm 2 (ref. ii). The transmitted angu-
lar current of electrons calculated by Berger is
represented by the histogram shown in figure 1. Also
shown in the figure are the results obtained for
this same case with the discrete ordinates code ANISN.
In the ANISN calculation, one energy group with a
range from 1.0106 MeV to 0.989h MeV and a midpoint
of 1.O MeV were used. No energy degradation was
allowed and only elastic scattering was permitted;
= _ =i.e., in this calculation equation 1 with Oin ° r
S(E) = 0 was solved. The two calculations shown in
figure 1 are in excellent agreement, and thus the
method of discrete ordinates can handle satisfac-
torily small-angle multiple Coulomb scattering.
817
ORNL-DWG "l'f- _4@,i
0 20 40 60 80
8(deg)
FIGURE l.--Angular distribution of transmitted
electron current for I-MeV electrons normally
incident on a 0.0287-g/cm2-thick aluminum slab.
Rester and Rainwater (ref. 12) and Rester and
Dance (ref. 13) have measured the transmitted elec-
tron current per unitenergy fro_monoenergetic
electrons normally incident on aluminum slabs of
varying thicknesses. The solid histograms in fig-
ures 2, 3, and h show the measured transmitted
electron current per unit energy for aluminum slabs
of thicknesses of O.11 g/cm 2, 0.22 g/cm 2, and
0.33 g/cm 2, respectively. Also shown in the fig-
ures are calculated results obtained with ANISN
(solid curves) and calculated results obtained with
the Monte Carlo transport code ETRAN of Berger and
Seltzer (ref. 1). In figure 2, the agreement be-
tween the ANISN results and the experimental data
is very good except at the very highest energies,
and in this case, i.e., for a 0.11-g/cm2-thick slab,
the ANISN results are in slightly better agreement
with the experimental data than the results given
by ETRAN. In figure 3, neither ANISN nor ETRAN is
in good agreement with the experimental data. In
figure h, which shows results for a relatively
thick slab, 0.33 g/cm 2 (0.6 of the electron range),
the Monte Carlo calculation is in very good agree-
ment with the experimental data, while the ANISN
calculation is in very poor agreement. The reason
for the progressive failure of the continuous
slowing-down ANISN calculation as the thickness in-
creases is not known.
12 --
o_0
c
®8
>
_6
0
0..3
ORNL-OWG 70-t13t3
T T l
TARGET MATERIAL: AI ) . _ I ] ]
TARGET THICKNESS: = 0.11 g/cm z
lii [ i
J-t RESTER-DANCE EXP I _ i I I
--!'_ BEIRsGER-SELTZER --
'
0.4 0.5 0.6 0.7 0.8 0.9 t.0
ENERGY (MeV)
FIGURE 2.--Transmitted electron current per unit
energy per incident electron for l-MeV electrons
normally incident on a 0.11-g/cm2-thick aluminum
slab.
@
c
_2
>.
:E
%
2
|
o
0.2
] ! }
TARGET MATERIAL: AI
TARGET THICKNESS =
-- 0.22 g/cm2
EO= I MeV
J-LRESTER- DANCE EXP
.:'LBERGER- SELTZER
ANISN
.2"£',_
0,5 0.4
-i
_r/
0.5 0.6
ENERGY (MeV)
ORNL OWG FO-_._S_4
..J
,i
..L
0.7 0,8 0.9
FIGURE 3.-- Transmitted electron current per unit
energy per incident electron for I-MeV electrons
normally incident on a 0.22-g/cm2-thiek aluminum
slab.
818
f.4
4.2
l
o_
o.B
>
_ 0.6
_ 0.4
0.2
ORNL- DWG 70- H315
I I I /
TARGET MATERIAL: AI
TARGET THICKNESS = 0.53 ]/cm 2
EO=I ieV [_ RESTER-DANCE EXP
BERGER- SELTZER ..i.._ _-,..4,
-- ANISN . ,--' ,
r--=.,
I"
i,
i,
w _ 'lii
-] J '1
] / \12
.rg" i
I ,y
J !
!b, "
0 0,t 0.2 0.3 0.4 0.5 0.6
ENERGY (MeV)
O.7
FIGURE h.--Transmitted ezec_ron current per unit
energy per incident electron for I-MeV electrons
normally incident on a 0.33-g/cm2-thick aluminum
slab.
The calculated and measured transmitted elec-
tron current per unit energy and per unit solid
angle is shown in figure 5 as a function of energy
at several angles for a 0.11-g/cm2-thick aluminum
slab. The calculated results agree fairly well
with the experimental data at all angles, thus in-
dicating an acceptable energy-angle correlation.
Because of the lack of experimental data, it
was not possible to present a comparison between
calculation and experiment for the case of an elec-
tron spectrum incident on a slab. Using the Monte
Carlo code ETRAN of Berger and Seltzer (ref. 1),
Scott (ref. l_) calculated the transmitted electron
current per unit energy for the case of a specific
electron energy spectrum normally incident on alu-
minum slabs, and this theoretical calculation was
compared _Ith results obtained with ANISN. The
incident electron energy distribution used in the
ealeulations is a representation of thespectrum
resulting from thermal neutron capture in 235U
(ref. 15). This spectrum extends to electron
energies of the order of l0 MeV and is shown ex-
plicitly in reference l_.
5
2 --
i0 o
_ s
® 2
_ 5
,o-2
_5s _J
ORNL- DWG 71-t482
I 1 CALCULATE{) ANLSN -- "7 71/2 deg-
-- ,,lr- EXPERIMENT OF RESTER /.AND RAINWATER
p-; \i
/ 11
_ , r
J--
S --
2 ----
t0-4
0,3 0.4 0.5 0.6 0.7 0.8 0.9 '1.0
E {MeV)
_IGURE 5.--Energy distribution of transmitted elec-
tron current at the specific angles indicated for
1-MeV electrons normally incident on a 0.11-g/cm 2-
thick aluminum slab.
The ETRAN results for an aluminum slab 0.5 g/
cm 2 thick are shown in figure 6 as a histogram and
the ANISN results are sho_n as plotted points. A
similar comparison is given in figure 7 for an
aluminum slab 1.0 g/cm 2 thick. In both figures 6
and 7 the ETRAN and ANISN results are in very good
agreement. Thus, ANISN is apparently capable of
treating accurately the transport of low-energy
(< l0 MeV) electrons through relatively thick
shields in some cases, e.g., the cases considered
in figures 6 and 7, but not in all cases, e.g.,
flgures 3 and _.
819
ORNL- DWG 69-12206AR
_0"
5
2
10-2
10-3
O ! 2 3 4 5 6 7
ELECTRON ENERGY(MeV)
FIGURE 6.--Transmitted electron current per unit
energy per incident electron for a specific energy
spectrum (ref. 15) normally incident on a 0.5-
g/cm2-thick aluminum slab.
t0-2
5
2
o
10-3
2
10.4
o
4"-" •
! m ,
I
• ANISN
ETRAN
ORNL" OWG 69-12205R
t.J !" -- --
! •
i
"1
t
i
-- TARGET =0.t g/cm 20FAI I
| 2 3 4 5 6 7
ELECTRON ENERGY (MeV)
FIGURE 7.--Transmitted electron current per unit
energy per incident electron for a specific
energy s_ectrum (ref. 15) normally incident on a
1.0-g/cm -thick aluminum slab.
REFERENCES
1. BERGER, M. J.; and SELTZER, S. M.: ETRAN, Monte Carlo Code System for
Electron and Photon Transport Through Extended Media, National Bureau
of Standards Documents NBS-9836, 1968, and NBS-9837, 1968.
2. FaNGLE, W. W., Jr.: A Users Manual for ANISN, a One-Dimensional Discrete
Ordinates Transport Code with Anisotropic Scattering, Computing
Technology Center, Union Carbide Corporation, Document K-1693, 1967.
3. ZERBY, C. D.; and KELLER, F. S.: Nucl. Sci. Eng., vol. 27, 1967, p. 190.
4. SPENCER, L. V.: Phys. Rev., vol. 98, 1955, p. 1507.
5. McCORMICK, P. T.; KEIFFER, D. G.; and PARZEN, G.: Phys. Rev., vol. 103,
1956, p. 29.
6. MOLLER, C.: Ann. Physik, vol. lh, 1932, p. 531.
7. ROHRLICH, F.; and CARLSON, B. C.: Phys. Rev., vol. 93, 195h, p. 38.
8. BERGER, M. J.; and SELTZER, S. M.: Tables of Energy Losses and Ranges
of Electrons and Positrons, NASA Document NASA-SP-3012, 1964.
9. STERNHEIMER, R, M.: Phys. Rev., vol. 103, 1956, p. 511.
lO. GOUDSMIT, S. A.; and SAUNDERSON, J. L.: Phys. Rev., vol. 57, 1940, p. 2h;
and Phys. Rev., vol. 58, 19hO, p. 36.
ll. BERGER, M. J.: Monte Carlo Calculation of the Penetration and Diffusion
of Fast Charged Particles, p. lh9 in Methods in Computational P_sics,
Vol. I, eds. B. Adler, S. Fernbach, and M. Rotenberg, Academic Press,
New York, 1963.
12. RESTER, D. H.; and RAINWATER, W. J.: Investigations of Electron Inter-
actions with Matter, NASA Document CR-33h, 1965.
13. RESTER, D. H.; and DANCE, W. E.: Investigation of Electron Interaction
with Matter, LTV Report No. 0-71000/6R-18, 1966.
lb. SCOTT, W. W.: Comparisons of the Results Obtained with Several Electron-
Penetration Codes, Oak Ridge National Laboratory Document ORNL RSIC-28,
1970.
15. CARTER, R. E., et al.: Phys. Rev., vol. 113, 1959, p. 280.
820


Low-energy electron transport with the method of discrete ordinates

One-dimensional code adaptation for electron transpor

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Low-energy electron transport with the method of discrete ordinates

Abstract

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