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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
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ORNA - 0-2,672
Conf-66.001. 34
HC .00; mu_so
NOV 2 9 1966
TUR
CALCULATIONS USING THE ALBEDO CONCEPT OF THE SUBCADMIUM FIUX.ALONG THE
CENTER IINE OF A STRAIGHT, A TWO-LEGGED. AND A THREE-LEGGED SQUARE.
CONCRETE DUCT ARISING FROM INCIDENT SUBCADMIUM NEUTRONS AND
COMPARISON WITH EXPERIMENT,
R. E. Maerker and F. J. Muckenthaler
Oak Ridge National Laboratory
Oak Ridge, Tennessee

RELEASED FOR ANNOUNCEMENT
IN NUCLEAR SCIENCE ABSTRACTS
LEGAL NOTICE
This report was prepared as an account of Government sponsored work. Nelther the United
States, nor the Commission, nor any person acting on behalf of tha Commission:
A. Makes any warranty or representation, expressed or implied, with respect to the accu-
racy, completeness, or usefulness of the information contained in this report, or that the use
of any information, apparatus, method, or proceso disclosed in this report may not tafrirge
privately owned righto; or
B. Assumes any liabilities with respoot to the use of, or for damages resulting from the
use of any information, apparatus, method, or procesı disclosed in this report.
As used in the abovo, "person acting on behalf of the Commission" Includes any cm-
ployee or contractor of the Commission, or employee of such contractor, to the extent that
such employee or contractor of the Commission, or employee of such contractor prepares,
disseminates, or provides access to, any information pursuant to his employment or contract
with the Commission, or his employment with such contractor.
*Research sponsored by the U. S. Atomic Energy Commission urider contract
with the Union Carbide Corporation and funded by the Defense Atomic
Support Agency under Orders E0-802-65 and E0-80-66. '
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INTRODUCTION
2.
This paper is vlie fifth in a contemplated series of six papers which deal
with the problem of calculating neutron transmission through large concrete-
walled rectangular ducts. Since the general approach to this problem makes
use of the albedo concept, whereby the reflecting properties of the concrete
are first determined in great detail, the first three papers in this series
present the results of Monte Carlo calculations and experiments of the dif-
ferential angular albedos for concrete.,994. The last three papers present
the results of incorporating these albedos into a general Monte Carlo duct
code which calculates fluxes and dose rates at detectors located arbitrarily
within the duct and comparisons of these results with experiment. The first
of the duct comparisons deals with fast-neutron dose rates.? The present
paper treats thermal-neutron fluxes along the duct center line arising from
an incident beam of thermal neutrons, using the previously determined dif-
ferential angular albedo data for thermal neutrons."
In the next section a brief description of the duct code is presented
and the assumptions inherent in utilizing the albedo concept described. The
following section describes the results of the calculations. A description
of the experiments conducted at the Tower Shielding Facility appears in the
succeeding two sections, followed by a comparison of the experimental and
calculational results. Some important conclusions as a result of this work
appear in the final section.
DESCRIPTION OF THE RANDOM-WALK DUCT CODE
..
The thermal-neutron (1.e., subcadmium, E S 0.5 eV) differential angular
current albedo for concrete di (00,0,0) was incorporated into a Monte Carlo
duct code originally written by Cain and modified by Maerker.? Here ho
.......
...
...
.... - "
IL-
L
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.
is the incident angle and the reflected polar angle both measured with
respect to an inward drawn normal to the surface, and o is the conventional
reflected azimuthal angle (p = 0 corresponds to a direction of the reflected
neutron which has the same components parallel to the surface as that of the
incident neutron). The albedo data are used in three forms:
(1) Ash (00,0,0) itself, used in the statistical estimation procedure;
(2) As a(@o) by integration over 0 and sp, and applied in the random-walk
procedure at every wall reflection as a factor multiplying the incident neu-
tron statistical weight;
(3) As conditional probability density functions, so 100,0,)dp/a(@.)
describing the probability of reflection at the angle e per unit cose
for a given 0o and (60,0,0)/DAR (100,0,0) dep describing the probability
of reflection at the angle o per radian for a giver og and e, and which
are also used in the random-walk procedure.
Since the semiempirical curve fits for .. (80,0,0p) and a(@o) reported
· in (3) described so well the results of the albedo calculations and measure-
ments, they were used without change in the duct code. The conditional prob-
ability density functions were sampled ty techniques described by Kahn.
The use of the curve fits and the sampling techniques applicable to these
curve fits saved many cells of fast memory storage, but at the expense of
somewhat longer computer times. An additional advantage of the curve fits,
however, lies in thefact that do is used as á continuous variable and is not
"quantized" to the values 0, 45, 60, 75, and ~go degrees used in the original
albedo calculations. Further, e and o are also picked continuous ly over.
their allowed ranges and no interpolation schemes are necessary.
Figure 1 illustrates the geometry involved in a typical neutron wall-
scattering event. From a previous reflection, point As the thermal neutron
18 proceeding with a statistical wei.ght Wo in a direction indicated by the

ORNL-DWG 66-5697
*
.
-
-
. -
1
Fig. 1. Typical Geometry of a Wall Reflection. The neutron is proceeding iron point A to
point B. The probability that a reflection at 's produces a detector response at D is to be calculated.

t
i
n g or do as tool to save the idea is that
o
*
***
vector Rp. The coordinates of the intersection of Rag with a wall surface,
point B, are determined, and eo and po, the polar and space-fixed azimuthal
angles of incidence, respectively, are calculated. The coordinates of a
detector at point D then determine 0 and op, the reflected polar and space-
fixed azimuthal angles, respectively. The contribution to the flux at D .
from a thermal neutron incident at the angl.e eo at B 18 proportional to dalan
evaluated at the reflected angles 0 and Q = Qo - 08• The statistical esti..
mate from the reflection at B is thus
0(D) = Wo (00,0990 - OBD) Rep
.
The random walk is then continued. For the particular eo at point B,
O' is chosen from the probability density function in A and op' from the
probability density function in o for the particular e chosen. The space
fixed azimuth after reflection 3.3 chosen as po to' or po -p' with equal :
probability. The new statisticaì weight of the neutron proceeding from
point B is
W = a (eo)Wo .
The neutron history then proceeds from point B with the new statistical
weight W and direction cosines Yxo Yao Y, which are determined from e', p;
Po, and the orientation of the wall containing point B. Thus, for the wall
Illustrated in Fig. 1,
Y = sine 'cos(40 tope),
Yy = sine'sin(oo + 4'),
Yg = |cose'l,
depending on which sign was chosen. The neutron history is terminated when
the neutron fails to survive a game of Russian roulette when its weight falls

-5-
below a predetermined value or, if the ends of the duct are open, when the
neutron escapes. .
The source geometry and angular distribution are completely arbitrary
and for source thermal neutrons the initial statistical weights are unity.
An option in the code allows tabulation of the contribution to the fluxes at
each of several detectors from each order of reflection. Another option
allows a mild bias of the reflected angular distributions. This bias affects
only the random walk, and is used to force more neutrons deep into the duct
by sempling from an isotropic distribution in /cose i for the first few
reflections rather than from the natural distribution, which varies approxi-
mately
This bias is particularly useful when fluxes are desired
at detectors located far down a duct leg.
· The code can also calculate simultaneously with the neutron fluxes
secondary gamma-ray doses arising from wall capture. The data for this
phase of the calculation is taken from the secondary gama-ray curve fits.
of reference (3). The code does not follow multipie-wall reflections of these
gamma rays since garma-ray dose albedos are small, and only estimates the
detector dose contributions whenever the therma). neutron interacts with a
wall. Following Fig. 1, such a contribution would be
Dose (D) – Wo ek (100,0) /

6.
From the discussion involving Fig. 1, it is to be observed that the
albedo concept, in which the sole effect of the wall is assumed to be a
reflection of the neutron back into the duct at its point of incidence, 18
. strictly adhered tôių the random walk. For the case of a straight duct, this
assumption probably tends to overestimate the flux because of the "trapping"
of neutrons near wall intersections In the real situation, i.e., the probabil-
ity of reflection from an infinite plane (the geometry used in generating the
altedos) 18 higher than would actually occur near the edges of a duct since
more wall material must be penetrated. For multilegged ducts, the above
effect, which is enhanced in the bend since more wall intersections occur
here, and the effect of neglect of .corner in-scattering and direct trans-
mission through the corner lip of the bend tend to cancel one another. The
assumptions inherent in the albedo concept should become better the larger
the cross-sectional dimensions of the duct, since the probability of
"trapping" in the vicinity of the wall intersections and corner lip pene-
tration effects both decrease with increasing duct cross-sectional area.
From the albedo calculations cited in references 2 and 4, the average dis-
placement of a nonthermalized reflected neutron from its point of incidence
is a few inches, and this value is not very sensitive to incident neutron
energy. An estimate of the displacement of a reflected thermal neutron
from its point of incidence can be obtained indirectly from reference 3,
in which calculations indicate that the thermal-neutron albedo saturates at
about 4 inches thickness of concrete. Thus the average displacement of a
thermal neutron on the incident surface can be assumed to be also of the
order of a few inches, and this value compared to the duct cross-sectional
dimensions is a measure of the magnitude of the "trapping" effect. A measure

T
.
of the magnitude of the corner lip in-scattering and transmission effect is
the relative increase in cross-sectional area of the duct produced by an
additional mean-free-path in the wiath. For thermal neutrons in concrete,
· this additional width is of the order of an inch. For thermal neutrons,
then, the albedo concept should apply very well to a multilegged duct of
cross-sectional dimensions of the order of a foot or larger. The albedo
concept should also apply to the wall-capture secondary gamma rays for ducts
of comparable size.
RESULTS OF THE CALCULATI ONS
Calculations were carried out for three configurations of a 3-ft by
3-ft-square concrete duct. One case involved a straight duct 65 ft long,
the second a two-legged duct with a right-angled bend located 15 pt down
the first leg, the second leg being 50 ft in length, and the third a three-
leg, and the second 15 ft down the second leg, the third leg being parallel
to the first leg and 35 ft in length (see Fig. 2). In each case the source
of thermal neutrons was assumed to be incident at an angie of 450 with
..
.
I
respect to a side wall normal, tightly collimated to lie within an area
approximately one-eighth that of the mouth and centered approximately at
the geometric center of the mouth, The 45° tightly collimated geometry was
used to serve as a rigorous test of the calculation, because the fluxes at
detectors located greater than several feet from the duct mouth should be
due entirely to wall-reflected radiation. The rather unusual source spatial
distribution was used because it conformed to the results of the experimental
beam mapping.
ra".
ORNL-DWG 66-8670..
OPEN FOR TWO-LEGGED DUCT MEASUREMENTS
CLOSED FOR THREE-LEGGED DUCT MEASUREMENTS
SEs RacerDecoracao
-o- oor-
4
OPEN FOR THREE-LEGGED DUCT MEASUREMENTS
CLOSED FOR TWO-LEGGED DUCT MEASUREMENTS
45° TIGHTLY
COLLIMATED SOURCE
OPEN FOR TWO- AND THREE-LEGGED DUCT MEASUREMENTS
CLOSED FOR STRAIGHT DUCT MEASUREMENTS
wat
oľ-
K-3 ft
- 15 ft —
OPEN FOR STRAIGHT DUCT MEASUREMENTS
CLOSED FOR TWO- AND THREE-LEGGED DUCT MEASUREMENTS

FIG. 2. Top View of Source and Duct Geometry.

.:
- -
The results of the calculations, normalized to the total number of sub-
cadmium neutrons per second per watt Incident on the source wall in the
experiment are shown in the last four columns of Tables I, II, and III and
Sul e plotted as points in Fig. 3. One interesting feature of the calculations!
is shown in the last two columns of the tables, and this is .
that the thermal neutrons undergo many wall reflections in the course of their
transmission through the duct. For the straight duct, the singly reflected
fractional contribution exhibits a minimum in the vicinity of 15 ft down the
duct, and approaches an asymptotic value for large distances down the duct.
This behavior is similar to that of the fast-neutron dose rate for the same
source geometry, but the relative importance of the kingly reflected neutrons
is not as great for the the rmal neutrons. For detectors located in the
second and third legs of a multileggeâ duct the relative contribution of
the two- or three-times reflected theral neutrons to the total flux is
almost insignificant, and orders of reflection considerably higher than
two or three become quite important. Figure 4 11lustrates the quantitative
behavior of the effect of the bends plus 15 ft of duct length on detector
fluxes located at three symmetrical positions within the three-legged duct.
From this figure, one can see that thermal neutrons reflected about 4 times
contribute most to the first leg detector, 9 times to the second-leg detector,
and 15 times to the third-leg detector. Comparing these values with similar
breakdowns in the thermal fluxes in the vicinity of 27 ft down the straight
duct and 27 ft down the second leg leads to the conclusion that each right-
angled bend seems to require about 4 additional reflections to navigate.
The calculations and experiments, shortly to be described, also indicate
that there is little effect on the thermal flux "upstream" from the bend
..
..
......
....
ORNL DWG. 66-9410
· TABLE I. COMPARISON OF MONTE CARLO AND EXPERIMENTAL RESULTS FOR THE STRAIGHT DUCT
Center Line
Distance
o th Singly
(neutrons.am-2.sec-1. wottal
th
Measured
Calculated
Monte Carl
Percent
Std. Dev.
Max No, Reflections
That Must Be Followed
Reflected
9th
To Produce 90% of oth
1.0
1.4
0.435
.
2.52 x 10°
1.84 x 10°
1.13 x 10°
7.50 x 10-
5.22 x 10-
2.70 x 10-7
1.52 x 10-
0.188
1.5
1.2
1.2
3.1.
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0.096
.
2.46 x 10°
1.68 x 10°
1.06 x 100
7.06 x 10'
4.90 x 10-7
2.57 x 10-
1.53 x 10'
8.7 x 10-2
3.95 x 10-2
1.36 x 10-2
5.4 x 103 .
2.45 x 103
1.30 x 10-3
4.0
..
...
0.069
27 -
8.7 x 10-2
4.! x 10-2*
1.33 x 10-2
5.4 x 10-3*
.
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4.5
4.5
7.0
6,1
10.3
11.4
0.080
*
0.099
30.
2.34 x 103
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.
11.2
1.23 x 10-3
8.2 x 10-4*
5.5 x 10-4
8.6 x 10-4
0.122
0.149
0.174
0.176
11.6
14.3
.
6.45 x 10-4
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12
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*Interpolated.

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ORNL DWG. 66.9411
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TABLE II. COMPARISON OF MONTE CARLO AND EXPERIMENTAL RESULTS FOR THE SECOND LEG
Center Line
Au (neutrons.cm-2sec - watt! Monte Carlo oth Doubly Max No. Reflections
Distance
Percent
That Must Be Followed
(ft)
Reflected
Measured
Std. Dev.
Calculated
- To Produce 90% of oth
.
..
...
5.1
15
16.5
18.5
0.092
0.095
*10*2*
4.4 x 10-2
3.8 x 1024
1.32 x 10-2*
5.5 x 10-3
2.3 x 10-3*
0.046
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4.6 x 10-2
3.8 x 10-2
1.32 x 10-2
5.5 x 100%
2.6 x 103
1.23 x 103
7.8 x 10-4
4.8 x 10-4
3.3 x 10-4
1.71 x 1074
1.07 x 10-4
6.0 x 105
0.017
0.009
0.009
0.006
0.009
1
.
5.2
5.5
7.1
8.4
12,2
13.5
14.1
15.1
20.5
22.7
20.0
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1.09 x 10 %.
6.8 x 104
4,2 x 10-4
2.9 x 10-4
1.48 x 1024
87 x 105
5.8 x 109
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0.009
0.008
0.011
19
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*Interpolated.
.
ORNL DVG. 66-9412
*
- *** -*
TABLE III. COMPARISON OF MONTE CARLO AND EXPERIMENTAL RESULTS FOR THE THIRD LEG
Center Line
Din (neutrons.cmsec-.watt- Monte Carlo oth Triply Max No. Reflections
Distance
That Must Be Followed
(ft)
Calculated
Std. Dev.
Percent
wed.
Reflected
Measured
th
To Produce 90% of oth
9.3
0.023
0.024
0,022
0.027
0.005
0.005
8.6 x 10-4*
7.9 x 10-4*
6.8 x 10-4
6.0 x 10-4
3.4 x 10-4
1.48 x 10-4
7.8 x 10-5
4.4 x 10-5
2.6 x 10-5
1.67 x 10-5
1.11 x 10-5
8.1 x 106
6.6 x 106
5.1 x 106
-12-
1.08 x 103
8.7 x 10-4
7.8 x 10-4
6.7 x 104
3.75 x 10-4
1.48 x 10-4
7.7. x 105
3.7 x 10-5
2.0 x 10-5
'1.37 x 105
8.8 x 10-6
•
4.1 x 10 9
3.3 x 10-6 .
11.4
13.8
11.9
16.1
8.3
10.5
11.2
12,7
15.6
15.6
21.9
0.002
0.002
6.4 x 10-6
0.002
....
16.3
48
50
17.6
0.002
ratione its caminte
*Interpolated.
b
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..-13-
ORNL-DWG 66-8674

-SMOOTHED MEASUREMENT -
• CALCULATED, STRAIGHT DUCT
O CALCULATED, TWO-LEGGED DUCT -
CALCULATED, THREE-LEGGED DUCT
THERMAL NEUTRON FLUX (neutrons.cm-2. sect. watt)
4 . 10 16 22 28 54 40 46
CENTER LINE DISTANCE FROM MOUTH (ft)
FIG. 3. Center Line Thermal Flux as o Function of Center Line Distance
for Incident Thermal Neutrons.
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0.12
• 12 ft DOWN STRAIGHT DUCT -
• 12ft DOWN SECOND LEG
A 12 ft DOWN THIRD LEG
Τ
Ι
Ι
FRACTIONAL THERMAL FLUX DUE TO EXACTLY N REFLECTIONS
Ι
Ι. Ι. Ι
Ι
Hur
0
5
10
15
20
25
30
35
.
Fig. 4. Fractional Thermal-Neutron Flux Due to Exactly N Reflections as a
Function of N for Three Detector Locations,
-15-
when a right-angled bend is introduced into the duct. For example, the
center line thermal fluxes from the duct mouth to about 14 it down the
straight duct are unaffec'ced when the first bend 18 placed at a point 15
· feet down the straight duct. This does not necessarily mean that "downstream"
perturbations have little effect on "upstream" thermal fluxes as 18.prob-
ably the case for fast-neutron dose rates” but rather that the relative effect
on the "upstream" thermal flux of covering an opening in a straight leg and
uncovering an opening of equal size in a side wall near the original opening
is small.
The secondary gamma-ray dose rates caused by wall cupture of the thermal
neutrons were also calculated for the three duct geometries, and the results
can be summarized by the following equation:
Du(rads hr-2) ? 108 opth (neutrons sec"em2)
which reproduces most of the Monte Carlo values to within +10 percent.
One correction was applied to the Monte Carlo calculated thermai-
neutron fluxes. This was m multiply the flux at the detector D by
exp(-4(D) A450), where l(D) is the center line distance in feet from the
mouth of the duct to the detector D. This factor represents a minimum
correction to account for nitrogen capture (1.e., thermal neutron dis-
appearance) in the duct when fillled with air, and is relatively insignificant,
amounting to a few percent at most. However, since the experiments
where conducted in air the approximate corrections were applied and the
results in Fig. 3 and Tables I, II, and III have this factor included.
The results presented are for 4000 histories and no bias out to 25 ft
for the straight duct (running time on a CDC-1604 computer approximately
one hour) and 4000 histories with bias fram 25 ft to 40 ft (about 40 min.).

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For the second leg 20,000 histories were used with blas (about 2 hrs.),
and for the third leg 50,000 histories were used with blas (about 3 hrs).
Preliminary runs on the IBM-360/75 computer have indicated factors of
around 5 in reducing these rather long computing times.
.
DESCRIPTION OF THE EXPERIMENT
A series of measurements were made at the ORNL Tower Shielding Facility
of the subcadmium flux center line distributions for the same source and duct
geometries as those employed in the calculations. For this experiment the
TSR-II, a water-moderated spherical reactor, was placed inside a spherical
lead and water shield which contained a cylindrical, stepped opening in the
horizontal midplane of the shield from which a collimated beam of reactor
· associated radiation emerged. The diameter of the opening was 15 in. for the
outer 2-ft section and 10 in. for the inner 2-ft section. The beam was .
further collimated by placing a 30-in.-long water shield containing a cylin-
drical 3-in.-diam air duct inside the outer 2-ft section. The center of the
mouth of the duct was located at a point 5 ft from the outer edge of the
reactor shield.
The concrete ducts were made of the same steel-reinforced, low-silicon
content, normal construction grade concrete capable of developing an ultimate
compressive strength of 3000 psi that was used in the previously reported
albedo studies? The ducts were made in 5-ft and 10-ft lengths, having an
inside cross section 3-ft square with 9-in.-thick walls, the top wall: of
each section being removable. Corner sections were built for conversion of
the straight duct to a duct containing one or two bends. A sufficient number
of straight sections was available to form a straight duct 65 ft long.
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The first 10-ft section of the duct was placed on its side with its
axis lying in the horizontal plane at an angle of 45° with respect to and
passing through the beam center line. The beam was thus incident on the
removable lid of the first section (the "source wall"), and this arrangement
made it extremely easy to take measurements both with (foreground) and without
(background) the source wall present. The remainder of the sections were
placed upright with the lids acting as ceilings. The entire duct structure
was supported on iron stands approxima tely 80 in. above the ground, so
that the duct center lines always lay in the same horizontal plane as the
center line of the tightly collimated Incident bean.
A cylindrical water shield representing a "beam catcher" was placed in
the bean just beyond the removable source wall to reduce the background in
the duct due to air scattering. The inside surface of this shield was
lined with borated plastic and its mouth covered with 30-mil cadmium to ..
minimize the backscattering of thermal neutrons into the duct. Additional
shielding was placed between this shield and the duct to attenuate neutrons
scattered from the beam catcher.
Despite the care taken in reducing the contribution to background from
the collimated incident beam it was also found necessary to place additional
shielding around the duct, particularly tetween the reactor and the duct
second leg. Further shielding in the form of water, oil drums, and concrete
blocks were added on each side of the duct. Two photographs of the complete
experimental arrangement for the two- and three-legged ducts are shown in
Figs. 5 and 6.
Measurements of the subcadmium flux incident on the source wall and
along the duct axes were made using a 2-in.-diam spherical BFS proportional
counter. A conversion factor from count rate to subcadmium flux for bare

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-20-
-
cadmium covered detector readings was obtained by exposing bare and cad- ..
mium-covered gold foils in conjunction with the BFs to the same reactor flux.
The reactor power was monitored by another BFs and two 3-in.-diam fission
chambers placed in a water-filled porthole, on the opposite side of the reactor
from the bean port. Daily calibrations of all instruments were made using a
previously measured subcadmium flux generated by enclosing a PoBe source in
a lucite container.
MEASUREMENTS
The subcadmium flux incident on the inside surface of the source wall
was mapped as a function of the horizontal and vertical position of points
on the wall, but with the source wall removed. The detector was mounted on
a traversing mechanism and measurements made forizontally along the plane
of the wall for eleven different elevations above the floor of the duct with
the detector bare and cadmium covered. The difference in the counting rate
between the bare and cadmium covered detector readings was multiplied by
the calibration factor to give subcadmium fluxes. By plotting and cross-
plotting the fluxes as functions of detector position, Q(x,z), the number
n of subcadmium neutrons incident on the source wall could be determined
from the expression
n = (3.f)JJ (x,z)dxdz = 1.65 x 104 neutrons sec-2 watt-1,
source
wall
in which 3 is a unit vector directed along the wall normal away from the
duct, î is a unit vector along the beam centerline, and 3 • f = cos 45º.
The plots of p(x,z) show that a negligible fraction of the incident
neutrons hit the floor and the roof of the duct, and all but a few percent
impinge on the first 5-ft section of the source wall.

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-21-
: Four different measurements were required at a particular detector
location on the duct center line to effect a determination of the subcadmium
flux ilue to incident subcadmium neutrons striking the source wall. These
were bare BF3 detector readings for the cases of a bare and cadmium covered
reactor beam in conjunction with the presence and absence of the source wall.
If we let
A = detector reading with reactor beam open and no source wall, but
with the missing source wall area covered with cadmium,
B = detector reading with reactor beam cadmium covered and no source
wall, but with the missing source wall area covered with cadmium,
C = detector reading with reactor beam open and source wall in place,
D = detector reading with reactor beam cadmium covered and source wall
in place,
then
C. Da detector reading due to subcadmium neutrons reflected initially
. 'from the source wall, the floor and roof of the duct, and air
scattering from the reactor collimator,
and
A - B = detector reading due to subcadmium neutrons reflected initially
from the floor and roof of the duct and air scattering from the
reactor collimator,
so that
.:(C-D) - (A - B) = detector reading due to subcadmium neutrons reflected
initially from the source wall.
These four measurements were made along the duct center line with the BF3
attached to an equally long traversing mechanism whose movement was remotely
controlled. Thus it was necessary to reposition the mechanism after its
length of travel for the longer legs of the duct. The traverser was set so
that each 13.5-ft span overlapped the preceding span by 2 ft. thus providing
several check points between runs. The results of these individual measure-
ments are presented in Tables IV, V, and VI, and the quantity (C-D) (A-B)
converted to flux units, using the calibration factor of 1.23 x 10*2/cm2/sec
-22-
Table IV. Measurements Along the Center Line of Straight Duct to
Determine Subcadmium Fluxes from Subcadmium Neutrons Incident
Initially at 450 to Source Wall
Counts •min-2.w-2-monitor-2
Center Line
Distance from
Mouth
(ft)
C
D
A
9.5 x 102
1.00 x 102
9.3
5.1 x 102
3.2 x 102
2.4 x 101
1.7 x 202
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.0 x 100
X 100
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3.42 x 102
3.33 x 102
3.07 x 102
2.43 x 102
1.60 x
1.12
8.07 x 102
6.10 x 102
4.64 x 101
3.52 x 101
2.65 x 101
1.66 x 101
x 102
x 102
x 102
x 102
x 201
2.75 x 201
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5.7 x 10.
4.5 x 10°
3.5 x 10°
2.7 x 10°
2.1 x 100
2.24 x 101
1.67 x 102
1.28 x 101.
7.9 x 10
4.87 x 10
2.97 x 100
1.92 x 10°
33 x 10°
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1.11 x 10^2 5.5 x 10-2 1.6 x 10-2
x 10^2 4.54 10 2 1.43x 10-2
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10-2
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2.2 x 10-2
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1:3 x 10-2
1.1 x 10-2
9.0 x 10°3
7.0 x 1003
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5.5 x 10°3
43.5
-23-
2
Table V. Measurements Along the Center Line of Second leg of the Two-
Legged Duct to Determine Subcadmium Fluxes froin Subcadmium Neutrons
Incident Initially at 45° to Source Wall
Counts•min-1•w-1.monitor -1
Center Line
Distance from
Mouth
(ft)
10
TE
B
USE
.
7.0 x 10*1
6.6 x 7.0"
$.
275 x 200
1.95 x 100
8.00 x 100 4.10 x 100
7.60 x 200
3.77 x 100
5.10 x 10°
3.40 x 10°
2.35 x 10° 1.32 x 20°
1.67 x 100 9.7
1.24 x 20°
10-1
7.10 x 10^2 4.23
4.25 x 10*1 2.60 x
62 x 102
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10°1
2001
10-2
7.3
103
10º1
10º2
20-2
.2092
2.8 x 102
2.6 x 10°3
1.9 x 10-1
1.3 x 10°3
1.0 x 10*1
7.0 x
102
102
3.3 x 10-2
2.1 x 10-2
1.4 x 10-2
•Ox 102
7.0 x 10-3
.9 x 20
.7 x 103
3.0 x 2003
2.5 x 1003
2.2 x 103
1.9 * 1003
1.7 x 1023
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1.1 x
8.1 x 10
N
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2.20 x 102
1.74 x 10?
10-2
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3.2 x 1023
2.7 X 2003.
2.4 x 103
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Tabie VI. Measurements Along Center Line of Third Leg of The Three-
Legged Duct to Determine Subcadmiun Fluxes from Subcadmium
Neutrons Incident Initially at 45° to Source Wall
Counts•min-2.w-Z.monitor-1
Center Line
Distance from
Mouth
(ft)
.
D
10*2
20*2
x 102
.0
7.0
N
10-3
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x 1003
x 10-3
x 1003
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1.47 x 20 1
8.80 x 10
5.70 x 10
4.00 x 10
2.90 x
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9.0 x 10-2
5.60 x 10 2
3.70 x 102
2.60 x 10-2
1.87 x 10-2
1.40 x 102
1.07 x 103
8.3 x 10-3
5.25 x 10°3
3.40 x 10 3
2.25 x 1023
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2092
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7.80 x 10-3
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2.35 x 10 3
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4.3
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-25.
per count/min/monitor is shown for most of the detector locations used in
the Monte Carlo calculations in the second columns of Tables I, II, and III,
and as the solid curves in Fig. 3..
COMPARISON OF CALCULATED AND EXPERIMENTAL RESULTS
2
The calculated results per incident thermal neutron multiplied by
r
1.65 x 104 (the total number of incident subcadmium neutrons per second
per watt in the experiments described in the previous section) and the
experimental results are compared in Tables I, II, and III and in Fig. 3.
The agreement is remarkably good, being within a few percent throughout the
entire range of better than five decades of attenuation. The trend develop-
ing in the third leg of the measurements overtaking the calculated values
may be due to the increasing importance of air scattered subcadmium neutrons
entering the open end of the rear of the duct in the experiment.
The effect of a bend is adequately treated by the calculation, and the
slight overestimate of the flux provided by the calculation in the immediate
vicinity of the bends and the second leg is consistent with the trend antici-
pated by employing the albedo concept, as was discussed in the beginning
of this paper.
· CONCLUSIONS
Perhaps the most remarkable feature of this work is the accuracy of the
calculation. Thus at least two important conclusions can be drawn. First,
the albedo concept applies very well to the treatment of thermal-neutron
diffusion through a torturous duct of cross-sectional dimensions of the
order of a few feet, and the approximations inherent in its use seem. to be
:
every bit as good as in the treatment of fast neutrons.? Second, the fact
that the center line thermal fluxes are due to high orders of thermal-neutron
reflection (in the middle of the third leg, for example, the contribution
HEL
from thermal neutrons suffering between 10 and 25 wall reflections makes
up better than 80 percent of the detector response) indicates the accuracy
to which the total thermal-neutron albedos, alco), must be known, since
the statistical weight of the neutron 18. reduced by a(eo) at every reflec-
tion. If the total albedos are consistently in error by X percent, the
fluxes after N orders of reflection would be in error by approximately
NX percent due to th3.8 cause alone. A reasonable conjecture from the above
discussion 18 that the a(Ao) values used are probably a few percent high
for some ranges of eo, and a few percent low for other ranges of Do, so
that the errors in the statistical weighé of a thermal neutron after n
reflections tend to cancel. Certainly this is a more realistic conclusion
than a claim of one or so percent accuracy throughout the range of 2. for
a(eo). It is equally conceivable that the alco) values used apply better
to the duct geometry than they would for infinite plane geometry and in a
sense may include the average effect of "trapping" near the wall. Inter-
sections. The differences in alco) for these two geometries is very small,
and only the cumulative effect over a large number of reflections would be
observable. Whatever the explanation as to which geometry the al00) as used
really represent, they apply with high accuracy to duct geometry when used in
conjunction with the albedo concept.
From an inspection of Fig. 3, the attenuation along the third leg is
seen to be sensibly the same as that along the second leg, so that an extrapo-
lation to further legs and bends should not incur serious error. Thus, it is
18 sufficiently flexible to be applied to a wile variety of rectangular multi-
legged duct geometries. For those situations in which thermal neutrons incident
upon the mouth of a concrete duct comprise an important source of the dose deep
.
.
.
. -2-
inside the duct (1.e., thermal-neutron multicollision dose and secondary
gamma-ray wall capture dose), the technique of employing the thermal-neutron
differential angular current albedos and secondary gamma-ray differential
angular dose "albedos" described in l'eference 3 in conjunction with a Monte
Carlo random-wall duct code such as is described in reference 7 is suffi-
ciently accurate to give good results.
ACKNOWLEDGMENTS
"The authors wish to acknowledge the assistance of all the personnel.**
of the Tower Shielding Facility who participated in the duct measurements
and in particular, J. L. Hull, J. J. Manning, K. M. Henry, G. M. Argo,
C. C. Barringer, J. N. 'Money, and F. E. Richardson, should be mentioned.
We are also indebted to c. E. Clifford of this Laboratory for his invaluable
advice and suggestions concerning both the experimental and theoretical
phases of this work.

-28-
References
1. Work funded by Defense Atomic Support Agency under Order E0-804-66.
2. R. E. Maerker and F. J. Muckenthaler, "Calculation and Measurement of
the Fast-Neutron Differential Dose Albedo for Concrete," Nucl. Sci. Eng.
22, 4(1965).
3. R. E. Maerker and F. J. Muckenthaler, "Measurements and Single-Velocity
Calculations of differential Angular Thermal-Neutron Albedos for Concrete,"
Nucl. Sci. Eng. 26, 3(1966).
4. W. A. Coleman, R. E. Maerker, F. J. Muckenthaler, and P. N. Stevens,
"Calculation of Doubly Differential Current Albedos for Monodirectional
Beams of Epicadmium Neutrons Incident on Concrete and Comparison of the
Reflected Subcadmium Component with Experimental Results," Nucl. Sci.
Eng. (to be published).
5. R. E. Maerker and F. J. Muckenthaler, "Monte Carlo Calculations Using the
Albedo Concept of the Fast-Neutron Dose Rates Along the Center Lines of:
One- and Two-Legged Square Concrete Open Ducts and Comparison with Experi-
ment," Nucl. Sci. Eng. (to be published)..
6. V. R. Cain, "Calculation of Thermal-Neutron Flux Distributions in Concrete-
Walled Ducts Using an Albedo Model with Monte Carlo Techniques," Oak Ridge
National laboratory Report ORNL-3507 (1964).
7. R. E. Maerker and V. R. Cain, "AMC: A Monte Carlo Code Utilizing the Albedo
Approach for Calculating Neutron and Capture Gamma Ray Distributions in
Rectangular Concrete Ducts," Oak Ridge National Laboratory Report, ORNL-3964
(to be published).
8. Herman Kahn, "Applications of Monte Carlo," The Rand Corporation, AECU-3259
Revised (1956).


END
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12/ 23/ 66



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