I
TEE SILWING
DOWN AND DIFFUSION
INITIALLtY MONONRNIETIC
UTRBONSI
OF
HYDROCNOUS MDIA
by
LW3C~
JAMES MARKS
5MhITTi!D IN PARTIAL JULILIK
}RE(JIRWi T
S
O
0OF
THE DEG
OX
BAOHELOR OF SC3BXiN
at the
MASSA0HSETTS
IWSTITUTi
OF
ECHNOLOGY
(1950)
Signature of Author .... c
Oertified by ..........
Thesis Supervisor
'I
TABLI
O
. ****
..... .
.
COsNT]TS
Pag
Abstract
. .·
·
··
Acknowledgment
I.
II.
Introduction.
xperimental
. .
·.
. .
.
.
· .
.
. . . . . .. . .
· . . . .
· · ·. · .
Technique
.
. .
iii
. .
iv
. . . .
1
.
. . . .
.
.
3
Figure 1: Initial Saturation Activities versus
Radius, Sand plus Water .........
Figure 2:
7
Initial Saturation Activities versus
Radius,
Dry
Sand
. . . . . .
. . . .
9
III.
The Neutron
IV.
Diffusion Theory and Slowing-Down
Theory . . . . ..
17
V.
Correlation of Theory with Experiment ..
23
Figure 3:
'igure 4:
Figure 5:
FPigure 6:
Figure 7:
Figure 8:
Source
. . . . . . . . . . . .
. .
. . . .
11
(Slowing-DownDensity) x (Radius Squared)
versus Radius,Dry Sand...
....
. *.
25
Slowing-DownDistance versus Energy,
Sandplus Water ..
..........
27
(Slowing-DownDensity) x (Radius Squared)
versus Radius, Sandplus Water . . . .
30
(Slowing-DownDensity) x (Radius Squared)
versus Radius, Sandplus Water . . . .
32
(Slowing-DownDensity) x (Radius Squared)
versus Radius, Sandplus Water . . . . .
33
(Thermal Neutron Density) x
Sand
plus
Water
.... .. ..
(Radius Squared) versus Radius,
36
TABLE OF CONTENTS
..... . .
Page
VI.
Summary .
Appendix...
.
.....
. .....
Bibliography...
.....
.....
.
. .
......
.....
ii
.
.
.
42
........
·
49
ABSRACT
Thermal and indium resonance neutron distributions
mixed with water and in dry sand are given.
The intensity
in sand
of the
neutron source is determined to be 1.241O6 neutrons per econd
on Pebruary 1, 1950.
The experimental distributions are comparedwith the predictions
obtained.
of the elementary
Age Theory."
Qalitativ
agreement is
The agreement is better when a "spherical harmonic"
correction is applied.
More advanced theories are briefly discus-
sed,
iii
AOCOWIX)DGMwST
The author wishes to express his dept and his gratitude to
Professor Clark Goodman. Doctor Gardner A. Norton gave assistance
at
every turn and is warmly thanked.
Thanks also are due Miss Joan Sullivan for assisting with
the calculations, to Mr. Sidney Millen for his help with the resolving time apparatus, and to Miss Dorothy E, Blotner, who typed most
of this thesis,
and to many oShrs who gave assistance.
iv
I.
ITaRODUCTION
The slowing down and diffusion
of neutrons in hydrogenous
mixtures have been studied extensively at this laboratory 1 and elsewhere. Correlation of the experimentally determined spatial distributions of slowed-downneutrons with theoretical predictions of these
was undertaken particularly by Tittle,
distributions
able asccess.
However, the acquisition
2
with consider-
of a new monoenergetic neutron
source of comparatively low energy suggested a re-examination of the
application
of elementary slowing-down theory to the spatial
distribu-
tions of neutrons in substances containing hydrogen. Such a study
was expected to result in a better determination of the extent of
validity
of the approximate NAge Theory' in this case.
Age theory
is so muchsimpler to apply than more rigorous theories that complete
knowledgeof its applicability
Distributions
is important.
of indium resonance (1.44 ev.) neutrons and
of thermal# (cadmiumdifference) neutrons in dry sand and in sand
saturated with water have been determined experimentally.
rom the
resonance neutron data, the strength of the source has been evaluated.
Muchof the difficulty
from the finite,
nonnegligible
in applying age theory here arises
size of the neutron source.
Since the
'Dacey, J. Z., Paine, R. W., and Goodman,C., Laboratory for Nuclear
Science and Engineering Technical Report No. 23, 1949. Hereafter
referred to as T.R. 23.
Tittle, C. ., aul, H., Secrest, B. L., and Goodman,C., Laboratory
for Nuclear Science and Engineering Technical Report No. 21, 1949.
Hereafter referred to as T.R. 21.
2 T.R.
21, pp. 30-33, 48-53.
point source solution is unsatisfactory, distribution functions corresponding to various idealized sources, which approximate the actual source,
have been compared to the observed distribution.
The results
indicate
that a straightforward application of age theory to this relatively lowenergy neutron source gives results valid to somewhatgreater distances
from the source than is the case for neutrons in the Mev. range.
3
II.
3lZXPRiNTAL iT0ENIQI3
The experimental procedure followed in this work is similar
to that developed by Amaldi and Fermi 1 and employed with various
ifications by manyother investigators.
mod-
2
Thin indium foils 3 in the slowing-down mediumwere exposed
to the neutron flux.
Neutron absorption by indium leads to a radio-
active decay which can be observed in a counter:
I 115+ n
I 116
n
In1 1 6
(2-1)
n
54 min..
I
8l
+ -(0.85
Mev. max.) + gammas.
(2-2)
Exposures were made in a four-foot cubical steel tank4 filled
sand. 5
with silica
The Sb24 _ Be photoneutron source
the bottom of a thin, snugly fitting
3.8
as lowered to
aluminum tube (outer diameter
m.) and rested about 21 inches below the sand surface.
During
most of the runs, an aluminum tube containing the same mediumas that
in the tank was inserted into the source holder above the source.
1Amaldi,
., and Fermi,
2Amaldi, 3.,
Hafstad,
.,, Phys.
L. B., and
ev. i0, 899-928 (1936).
u"ve,
896-912 (1937).
O'Neal,
. D., Phys. Rev.
,
1-4 (1946).
Munn and Pontecorvo, Cam. J. Res. O,
Tittle,
. A., Phys. LRev. .1,
157-167 (1947).
Dacey, J.
. W., M.I.T. Course VIII Ph.D. Thesis, Jan. 1948, and T.R. 21.
., Paine, R. W., and Goodman, C., T.B. 23.
Rush, J. H., Pys.
Rev. 27, 271-273 (1948),
3T.2. 23, pp. 18-20.
4 T.R.
23, p. 15.
5 T.R.
21, p. 14.
and numerous others.
The foils were enclosed in holders of cadmium or aluminum,
described by Dacey and Paine.
flattened
aluminum
These were inserted
into
tubes extending down into the sand
depth as the source holder.
fell on the same straight
thin-walled,
to the
same
The tubes were so arrangedthat no two
line from the source.
The distance from
each tube to the nearest wall of the tank was equal to or greater
than its distance from the source.
Accuracy of distance measure-
ments from source to tube (center to center)
After exposure, the foil activity
time schedule.
is estimated at t0.
2
cm.
was counted on a standard
The front a-d the back of each foil were exposed
separately for counting because the beta rays were being counted.
Initial
activity for a foil is here defined as the sumof the
initial
activities
for the front and back of the foil.
series of routine corrections
A tedious
must be applied to the raw counting
data to reduce it to an initial saturation activity that is proportional to the neutron flux.
ization
of counters
Details of these corrections, standard-
and counting procedure
are discussed by
aul,2
and by Dacey and Paine.*
10p.cit.,
2 Yaul,
3
pp. 23-24.
H., M.I.T. Ph.D. Thesis,
0p.cit.,
Course XII, May, 1949, pp. 30-34.
pp. 26-27, 35-38, 45-52.
5
Unfortunately, when the author began this study he did not
knowthat the foils had been found to have a constant residual activity
of their own. The details are given in an unpublished report by Faul
and armer, which was not resurrected until long after all the data
had been taken.
A series of background counts was subsequently made, some
with and somewithout foils in the counters.
A difference of 1.4-0.6
counts per minute was found; this is of doubtful significance but time
limitations prevented taking more data.
The initial
assuption
saturation
were recomputed on the
activities
that the true background rate was higher by the above amount
than the background observed at the time of the counting.
Obviously,
this is not very satisfactory.
The backgroundwas observed to fluctuate greatly, especially
during the day.
Since these fluctuations
could not be exactly compen-
sated for, it was assumed that the probable error of the observed background rate was twice the statistical
This procedure should give
error.
a better estimate of the uncertainty in the measurement.
For correcting
theral"
activities,
Tittle's
most recent
modified version of Bothe's relation for the foil drain correction
factor has been used:
Bothe's formula:1
f =
1.(2-3)
_.
,
, ,,
12'(A
+ k +3L
1 T.R.
2,
p. 48.
Tittle's formula:1
1
f
1+ a. (k
2
i.e.,
(2.4)
(-4)
3L
-1)
r 2rk + 3L
transport meanfree path is used instead of scattering mean
free path.
In the present work the 4.1 hour activity 2 of In 115 need
not be considered because the neutrons emitted by the source used
here have less energy than the reaction threshold.
Of the two types of exposure made, those in which the
foils were in aluminm holders produce a foil activity which is due
to both thermal neutrons and neutrons of 1.44 ev. energy, since aluminumis essentially transparent to neutrons of these energies.
The cadmiumholders,
on the other hand, absorb neutrons of
less than 0.3 ev. energy, so that foils exposed in them are activated
almost completely by resonance absorption of 1.44 ev. neutrons.
The
difference betweenthe two initial saturation activities is, therefore,
due principally
to thermal neutrons, and is denoted
thermal activity."
In figure 1, the average initial saturation activities for
thermal and for resonance absorption in a mediumof sand saturated with
water is given as a function of distance from the center of the source
as of February 1, 1950. The composition of the medium was:
gm/cm3
Sand:
1.62O.03
Water:
0.387+o0.01 gm/cm
'Unfortunately, no completely accurate published reference to Tittle'ts
work is available for reference.
2 T.R.
3cf.
23, p. 44.
p. 3.
jc,5
V_
INITIAL SATURATION ACTIVITIES
vs. RADIUS
DATE: FEB. 1,1950
MEDIUM:SAND, SATURATED WITH WATER.
INDIUM RESONANCE:CORRECTED FOR CADMIUM ATTENUATION.
THERMAL:
lo
CORRECTED
FOR FOIL
DRAIN.
I
i
3
10
z
n
i
w:
F-:
Iz
M
doe
10:
10_
N
I
i
I
\35.4
1.0
0
- - ----
--
5
--
I
10
-
I
I
15
20
r (CM.)
-
25
-I
30
-I
30
or, alternatively,
Oxygen:
1.21 gm/cm 3
Silicon:
0.757 gm/cm3
Hydrogen:
0.043 g/cM 3
In Figure 2, the resonance and thermal activities
on
Febrnary 1, 1950, as a function of distance are given for a slowingdownmediumof dry sand.
The initial
The density of the sand was 1.62 gm/cm.3
saturation activities
of the foils,
given
above, are proportional to the corresponding neutron fluxes to
which the foils had been exposed.
The foils and counters have been
calibrated against a standard neutron flux at Oak Ridge. The details
are described by Dacey and Paine;
(nv)i I
s (neutrons/cm -sec)
(nv),(neutrons/cm
In theories
The density
2 -ec)
0.038(A )s(counts/min)
0.192(A)Ti(Ooutt
of slowing downand diffusion,
sidered is not the neutron flux.
treated.
the results are:
s/min)
the quantity con-
For thermal neutrons, the density is
is found by dividing the flux as found above
(Equation 2-6)by the standard velocity of thermal neutrons,
2.2 x 105 cm/sec.
For neutrons in the process of being slowed down, the quantity
of interest
1
Op.cit.,
is the slowing-down density,
pp. 60-67.
q.
This may be obtained from
(2-5)
(2-6)
0qt
U)
U)
J
c-
Q
D
a
N
o0
.11
aO
0
0
LL
.
I)
aWI
-J
.4
I
w
'I
Ir
W
I'
P
5
Uz0
O0U)z01
oW
I-.
Oo
-0-
D
W
to
1
-. 1
C i*.r
o.9
co xZ
-_O
D
U,
-J
W
WE
in x
-4
zZ
z
0
00
N
08
0
8I
00D
(NIW /SINnOo)Sv
00
It
00
°O
0
-0
the foil activity by the relation
=
0.338(A )S
(No8 ~ )
(2-7)
where the quantity No ~ is the "slowing-down power of the medium."
The "slowing-down power" is discussed in Section IV.
Subsequent graphical presentation of the data will be in
the form of neutron density or slowing-downdensity, as maybe
appropriate.
4A
III.
The Neutron Source
In order to obtain a flux of initially
monoenergetic neutrons,
a photoneutron source was employed. Thereactions leading to the production of neutrons are:
Sbl24
60 days
B99 +B
Sl4
8
i
(3-1)
(l.78 Mev)
+ nl1._
Q was thought to be 1.63 Mev.; this
(32)
is now known to be wrong.
Assuminga value for the binding energy of the deuteron of 2.23 ev.,
rather than 2.187 Mev., Q takesthe value Q = 1.673 Mev.
The energy of the emitted neutron depends on the source
according to the relation
*1
A-
1 (:E
- Q) +
[2(
cos e
where A is the mass number of the initialnucleus.
Accordingly,
at 90 degrees,
n
= 31
ev. and the maximum
energy spread
1/2
/ 2 Y =
2(. .
1)
Q(3-4)
= 3 Kev.
lSeaborg, G.
2Bell
and
., and Perlman,
lliott,
I., Rev. Mod. Phys.
Phys. Rev. 74,
value has not yet been assigned.
3Banson, A. 0., Phys. Rev.
jQ,
1948, p. 615.
1948, p. 1552 L. A final, definitive
, 1949, pp. 1794-1799.
(3-3)
Therefore, E n
31+-!. Kev. assuming the particular value
of Q given above.
The value of the spread in energy is of prime interest.
It
shows that while the source is not strictly monoenergetic, it may
in comparison with the
reasonably be considered so, especially
R -
B
a
e
sourcespreviouslyused in this
laboratory,
which
probably have a continuous distribution of energies from several Mev.
to a few hundred Kev.
Experimental determinations
of neutron energy from this type
of source do not agree among themselves.
29 and 35 Kev.
The measurements by Hanson
2
probably
give the most
Hanson measured the maximumpulse height
accurate value yet obtained.
of recoil
Wattenburg quotes values of
protons in a hydrogen-filled
counter.He finds
proportional
a value for the average neutron energy of 2;3
Kevy. Considering the
various measurements, a value for the average initial energy of 26 Key.
seems reasonable.
At first,
it was hoped that the present experiments would give
a good check on the initial
neutron energy of the Sb
was hoped these experiments would measure the initial
-24-B
e
source.It
neutron energy.
This
the reasons are discussed in Section V. A 2.04
provedto be impossible;
Mev. 3 gamma ray from
b 124 has been reported.
This gamma ray is very weak
and perhaps even doubtful; it probably does not contribute significantly
to the neutron flux.
T.R.
Other gammas are given
21, pp. 56-57.
T.R. 23, pp. 14-15.
2Op.
cit.
Seaborg and Perlman, op.cit.
off; these, however, have lower energies than the reaction threshold
for neutron emission.
Unfortunately,
inquiries
to Oak Ridge have not yet been
fully answered, so that the precise internal structure of the source
is not known.
However,the following is quoted:l
"AUtimonyslugs, coated with beryllium and sealed
in aluminum ackets,
may now be irradiated
to
produce
antinony-beryllium neutron and distributed
to off-project users.'
The outer dimensions of the cylinder containing the source
are given as:
diameter, 3.02 am.
length, 3.02 cm.
An aluminum plate projects
from one end, so the over-all
length is
approximately 4.45 cm.
from the above, it seemsprobable that most of the interior
of the source is taken up by the antimony slug, with a relatively thin
layer of beryllium.
A knowledgeof the absolute source strength is necessary for
this work and is desirable, in general, for any
applications
whatsoever.
The experiuental results, giving foil activity (hence neutron
flux) as a function of distance, permit an evaluation of the source strength,
Qn'if the neutron distribution is assumed spherically symmetric (this is
probably valid except close to the source).
lIutogel.
U.S.A..O.,
Catalogue and Price List No. 3, July,
Oak Ridge, ennessee, p. 29.
1949.
Isotopes
Division,
In the sand-water mixture, capture is completely negligible
above 1.44 ev.
Therefore,
0(dV-5)
=
- 4A(0.038)(7 No. ) JARS
x r2 dr
(3-5)
q is the slowing-down density referred to on Page 8.
To calculate the slowing-down power at 1.44 ev.:
Melkonian 2 gives the free particle scattering cross
sections of hydrogen and oxygen as
((
)R = 20.36
(8)O
=
3.73
barns
barns
Goldsmith, Ibser, and Foeld3 give for silicon
rt = 2.25
barns.
Using Tittle's value of ac = 0.1410.4 barns,4
(C) Si - 2.1 barns
From these cross sections and the composition of the medium
(page
8), it is found
o'
that
= 0.546 cm-1
at 1.44 ev.
1T.R. 23, pp. 67-68.
2
Melkonian,
., Phys. Rev. 76 (1949), pp. 1744-1759.
3 Goldsmith,H.
(1947),
4
., Ibser, H. W., and Feld, B. T., Rev. Mod. Phys, 9
p. 272.
Tittle (Thesis), op.cit., p. 149.
Hence, 0n = 0.2606
Sr
dr
1.9 cm
The integral
Simpson's Rule.
was evaluated
numerically
To obtain the contribution
from 2 to 30 cm. by
from more distant
neutrons,
it was assumed that far from the source the activity would vary as
A2,
Ae
- r
(3-6)
r
z
the empirical constants being determined from a curve of Ar2 versus r
as
j
A = 1.66 x 108 (cots/mn)(cm
2
)
Lr = 2.83 cm.
r
Theresult is:
J
fAr2 &r
4.77 x 106
2 cm.
on February 1, 1950.
The contribution to the integral of the region inside 5 cm.
and the one outside 30 cm. is about 16 per cent.
These contributions
may easily be in error by 20 per cent or more; on the other hand, the
contribution from 5 to 30 cm. is better known,probably to within
10 per cent.
A reasonable value of qn is
. = (.24l0.15)
as of
t
ebruary 1, 1950.
x l10neutrons/ec.
(3-7)
Qn for other dates may be found by using the 60-day half-life
1
of Sb
b
24
to correct (-7).
17
IV.
DI
SION
TBEORY AND SLOWING-DOWl
THEORY
The steady-state diffusion of thermal neutrons, in a
medium in which both scattering and capture may take place, is
described approximately by the differential equation
2 A
n
n = neutron density (neutrons/unit volume)
= mean life of thermal neutrons against capture
s = rate of production of thermal neutrons per unit volume
L
diffusion length.
The diffusion length, L, is related to the properties of the
medium; before defining it, some preliminary discussion is necessary.
If maand as are the atomic cross sections, for absorption
and for scattering of neutrons, of a nuclear species in the medium,
andN is the atomic concentration
of the species, the macroscopic
scatteringcross sectionsare Z Na(cm
1 ),
and j Neo-(cm1) .
The suma-
tions areover the nuclear species present in themedium.
The reciprocals of the above quantities are the scattering
and absorption mean free paths:
"
a
(cm)
1
S1, "(cm)
5xoa
a
The transport mean free path is defined as
(4-2)
(4-3)
18
htr P_
=
where
e
[co
ave
B
(cm)
2.
(4-4)
is the mean cosine of the scattering
-
angle
(in the laboratory coordinates) of neutrons colliding with a nucleus of
mass number A.
In terms of the quantities defined above, the diffusion length
is defined as
L2
2
3
The diffusion
equation (4-1) may be obtained by several methods,
but the most fruitful procedure is to obtain it as an approximation to the
Boltzmanntransport equation; using this procedure,2 it is evident that
the neutron density must vary slowly with distance (fractional change in
density in one transport meanfree path is negligible), and that the
neutron flux is almost isotropic.
In particular, the absorption cross
section must be small compared to the scattering
distance from sudden discontinuities
cross section, and the
in the medium, and from concentrated
sources, should be greater than a transport meanfree ath.
The above restrictions
are not very severe, and, in general, are
satisfied in this work.
1Marsbak,
R.
2Marshak, .
pp, 10-20.
.,
ev.
od.
., Brooks,
hys.
19(1947),
p. 188.
., and Hurwits, H.,
cleonics
4,
(May, 1949),
i9
The elementary theory of neutron slowing-down may also be
derived as an approximation to the transport equation. 1
Energy of a neutron is usually specified in terms of
u
=
log
E
i
, where 3E is the initial energy of the neutron.
This
quantity is of interest because in a collision between a neutron and
a nucleus the fractional energy loss of the neutron is characteristic
of the collision.
Let n(r, u)
numberof neutrons per unit volumeper unit
logarithmic energy range (i.e.,
per unit of u).
Then the slowing-down
density is defined as
= f n(r, u)v(u)o(u)
(4-6)
is the mean logarithmic energy loss per collision.
If a
single nuclear species is present,
(M
1)2
4M
2
M
1
It is seen that the dimensions of q are (neutrons/cm2-ec).
Actually,
q is
the density of neutrons slowing down past a given energy
per unit t ime.
In terms of q, the differential
V 2q a
-Q(
(L 5 2 )
'Marshak, op.cit., pp. 213-216.
2Ibid., p. 188.
equation for slowing downis
(L2 )
(4-8)
1
20
The term on the right is a source term (neutrons
for the steady state.
of energy 3 produced per unit volume per second).
L , termed the slowing-downdistance, is a characteristic
length of the neutron distribution,
V
2
1
3
_0
I
defined for Iquation (4-8) as
_I_- ...
=
'
3
L 2 is also sometimes termed the
age" of the neutrons, and
Equation (4-8) is called the age equation because of its formal resemblance to the time-dependent heat flow equation.
q may also be con-
sidered as the density of neutrons per unit age interval.
A numberof assumptions are associated with 3quation (4-8).
Firstly,
the assumptions concerning the underlying transport theory are:
(a) Inelastic scattering is negligible.
(b)
lastic scattering is S-wave.
(c) Effect of chemical binding is negligible.
These assumptions are valid in this work downto indium
resonance energy (1.44 ev)- but assumption (c) is not complied with in
slowing downto thermal energies.
Secondly, Age Theory is valid under the following limitations:
(d) The theory is applied only to distances from the
source less thana distance on the order of
(e)
The average number of slowing-down collisions
is
large.
If)
The fractional rate of change of mean free path
per collision is small.
21
As a rule, none of these restrictions are obeyed by hydrogenous media; the distribution beyond L82 /X s is important, the average
number of collisions is from ten to twenty, and in slowing down from
several Mev. the hydrogen cross section changes by a factor of about
ten.
However, in this work, the initial
neutron energy is so low
that the cross section increases by only about 15 per cent in slowing
This is not unreasonable from the standpoint of age
down to 1.44 ev.
theory.
solution
A typical
source in an infinite
Q3
of £quation (4-8) is that for a point
medium:
3/2
(4iT)
r2
£
L
(4-10)
L(
This distribution exhibits a gaussian behavior throughout
space.
However, the distribution
of neutrons that have madeno col-
lisions follows an exponential law, so that at large distances q
should tend towards an exponential behavior.
It
was the above consideration
that motivated the assumption,
in calculating the source strength, that the activity decreased exponentially
beyond
thirty centimeters, with a characteristic "relaxation
length," Lr (Equation 3-6). In fact,
of a curve
to
it influenced the original
fit the data on activities.
choice
F
due to a
In determining the thermal neutron distribution
source of fast neutrons, age theory is used to obtain the distribution
of neutrons that are just becomingthermal; that is, a distribution
appropriate to the source, such as Equation (4-10), with the particular
value of L corresponding to thermal energy, 0.025 ev.
This distribu-
tion is then used as the source term in the diffusion equation (4-1).
The value of LS 2 for slowing down to thermal energies is
given by Equation (4-9).
This procedure is rather doubtful,
because
chemical binding is important and produces a rapid increase in the
hydrogen scattering cross section with decreasing energy.
It is known
that when the hydrogen cross section increases rapidly age theory tends
to underestimate seriously L .
On the other hand, any inelastic scattering that occurs will
tend to counterbalance the error.
1
T.R. 21, Fig. 20, p. 34.
I
V
CORRILATION O
TOY
WITH
RiL
IM[IT
The previous material will be summarized. Indium resonance
and thermal foil activities
are knownfor dry sand and for sand satu-
rated with water; from these, neutron density or slowing-downdensity
may be found. 1
available.
An elementary theory of slowing down and diffusion
A probable value of the initial
is
energy of the neutrons
(26 Kev.)has been decided upon.
Fora mediumcontaining several nuclear species, the mean
logarithmic energy loss,
e
, is a linear combination of the
'
of the various nuclei;
iloS
2
-
(51)
Eqgation (4-9) then takes the form3
L2 = l/3
IJ
(
(5-2
/
f )( £5o
7 otr)
The indium resonance distribution
sidered first.
Using the cross sections given on Page 14, one finds
No s
.
-
.
c.f. pp. 8, 10.
'2.R.
3Ibid.,
21, p. 65.
p. 66.
in dry sand will be con-
- 1 , or
0.158 ocm.
_No
0.0173 cm.
ENOtr
= 0*152 ocm.1
=
6.32 cm.
-
·. ·
Assuming the cross sections are independent of energy,
L 2_
s
127 log
1.44
For 26 Kev., L is 35.4 cm.
For such a large L the source certainly is effectively
a point source.
quation (4-10) gives the slowing-down density,
for a point source in an infinite medim. In figure 3, q
r
q,
is
plotted against radius, for this distribution function and for the
observed distribution.
From the appearance of
igure 3, it would seem that a
larger value of Ls (i.e., larger I ) is called for.
However,two
factors would tend to depress the observed q. The cross section of
silicon probably decreases at high energies,1 but its value is not
well known. This would give a larger L .
Still
more important,
tank of dry sand is not even approximately an infinite
the
medium. The
top and bottom of the tankare each about 1.5 slowing-down distances
from the foils2 and the sides will also have an effect as well.
Since
the extrapolated slowing-downdensity will vanish a short distance outside the tank, the density inside will be depressed to an extent that
cannot be accurately determined. Thus, the curves in igure 3 are not
necessarily
inconsistent
with an initial
energy of 26 Key. At small
distance from the source, the curves merge, as they should.
lGoldsmith, Ibser, and Feld, op.cit.,
2c... p. 3.
p. 272.
M3S
(SLOWING- DOWN DENSITY) X (RADIUS) 2 AT -1.44 E:vs.
RADIUS
IY
DATErFEB. I, 1950
MEDIUM?
DRY SAND
.
,lo
z
2
...
,h,_~~~,
X
I)
i
In
I,
z
W
z
.
'
'
Cr
10
I
0
5
10
15
(CM) 20
r (CM)
25
30
35
Equation (4-10) can be fitted
by using an L of 40 cm.
approximately to
This would require
s
igure 3
E to be about 450 Kev.,
0
a fantastic value.
These
results illustrate the defects in this experiment as
a method of measuring initial
to Eo,
but is roughly a linear
neutron energies.
function
of
L is very insensitive
If the silicon cross
s. *
section as a function of energy were as well known as
that
of oxygen,
a verycarefulexperiment with a very intense source might give a
rough check on the initial energy.
Enough sand would have to be used
to approximate an infinite medium; this would require probably at least
five or six times the linear dimensions of the present tank.
distribution
The whole
need not be found; the part of it out to the peak of the
qr curve would suffice.
The peak would occur at 2L , or 71 cm. Age
theory in this case would be good out to about
L2 s
~
Z
200 cm.
In the case of the sand-water mixture, the variation in the
hydrogen cross section must be considered in finding L
Goldsmith, Ibser, and Feld
(of)
The data in
are represented approximately by
= (20.362 - 0.943
A) barns
(3 in Kev.) up to 50 Key.
Using for oxygen and silicon
Equation
(4-9) gives L
Op.cit., p. 261.
2Melkonian, op.cit.
.
the cross sections on Page 14,
at 1.44 ev. as a fnction
of Eo between 15 Kev.
-0
-
#01
cr
a:
W
Z
w
-J
o
p-
z
U,
a:
w
-
w 4
It
3:
_-
qt
-Owy
n,-
0
p-
z0ZW
C
a:
4-
0
LLJ
Z
z0o(I) z4
0
0
u,
LZ
z
-0
w
o
-J
cI)
,
6
Nb
roj
0
N)
a:
28
and 40 Kev.
30 ofI 26
The results are given in Figure 4.
For a value of
ev., L is 4.05 cm.
In the age theory point source solution (4-10), the slowing-
downdistance is imply related to a quantity called the "meansquare
distance,"
r 4 qdr
r/
(5s3)
r 2 qdr
0
Therelationship is
r2
6 L2
(54)
The mean square distance of the experimental distribution
is given by
J
A
r4 dr
r2 =- -0o
--
(55)
A r 2 dr
0
The denominator in (5-5) has already been given.l
The
numerator was evaluated the same way,with the result
(1)I
p. 15.
n
res. = 157
.2
(5)
On the other hand, 6L22 is 98.4 c. 2 The difference
between these two
values indicates a serious breakdownin age theory, or that the source
cannot be considered a point source, or both.
In Figure 5 are plotted the observed q
q
i
r for a point source with L - 4.05 c.
r 2 and the theoretical
The theoretical curve is
seen to have a muchhigher peak, nearer the source than that for the
experimental curve, and then falls away very rapidly.
The theoretical
curve has to rise above the experimental one near the source because the
areas under the curves are normalized to the source strength.
Distributions corresponding to extended sources will nowbe considered.
As has been stated,l the source probably consists of a relatively
thin beryllium jacket around the antimony slug.
aluminum tube of outer diameter 38 2 cm.
The source rests in an
for aluminum is 12.5 cm.,
so that the aluminumcan probably be neglected.
that the
It is assumed
cylinder of radius 1.9 cm. can be represented by a sphere of 2.0 cm.
radius.
Wallace and LeCaine3 give, for an Infinite Mediumwith
Spherical avity, Point Source at Centre of Cavity,"
22
arW
where a is the radius of the cavity and the error function is tabulated,
4
'P. 13.
2 P.
3.
3 Wallace,P.
R., and LeCaine,J.,
Elementary Arnroximations
in the Theory
of Neutron Diffusion N.R.C. No. 1480 (Chalk River, Ontario, p. 70.
4
as
Tables
of Probability
Functions,
Volume I, National Bureau of Standards, 1941.
IF1i_
'
'
'
/(SLOWING:DOWN
~....
'' "-":
I?,-
..'.-..:..:
.
......
DENSITY). X(RADIUS)
...AT
L44.E.--vSt..
RADWS ..
t.
. .
MEDIUMSAND .SATURATED...
BI95- . DATE-FE
--...........
.. ... .
4
1~-vi-RQW--.-TES.
·.
...
.~.....
......
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:WITHWATER
I....................... :..
.
.
.........
I
01
Z
:I~~~~~~~1
...
I
.
'
.
...................
......
,
-....
cm,
x
to
.I
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'
. . .......
.
.~~~~~~~
.'...
r,)
0z
LEGED
_. ... ...........D:I-...:_:
' . -'" ~ -:-·-- .--.i....i:
t~EGE
..........
..........X.E%
\
- :---
.L.
:. ..~
......
-
..I....
: .. ·
..
.--. ' · -- I-- .- :
.....
'.1..
.....
.... .r-. ~
.-:.
.....
..... :_
~..:....
.....
...
. ·.....i.
·-...
'aR :
...........
NT .....E :.-..:
..~~~~~~~~~~~~
. .
.-
................
.......
'
(POINTSOUR..
...
GE...
LJ
~~~~~~~~~~I
.-.
1.
·
.....\ ..
..:.. .....:.....~
. .-:- j..... · · ~~~~~~~~~~~~~~~~~~~~~~~~~~~
I:
'. :-
.- . ....
~~~~~~~~~~~~~~~~~~~~~~~~~~~~\-:····-.--
-.. -.
-. ..·. .....
"' ..... ~~~~~~~~~~~~~~~~~~~:
I
i :
.
.
'
~
............
·-,.........',..~.........i..'...
..... ...........
:................
...............................
~.............................
...............
................
....................
..........
~
.......
.......................
I0~
...
....
'
0
5
10
15
r CM)
20
25
30
31
Actually,Wallaceand LeCaine apply the boundary condition:
q
0 at r
a.
=
Therefore, any spherically
cavity is described by (5-7).
In particular,
symmetric source in the
the source maybe a spher-
ical shell at or near theboundary.
In Figure 7, the function q * r 2 for this
as well as the experimental curve.
but
not
very
much.
source is plotted,
There has been some improvement,
The peak of the curve is shifted outward and
flattened somewhat,but it appears that the major part of the difference
between the empirical r 2 and 6 L 2 is due to the failure
of age theory.
L2
In this case =
164 = 12 cm.
1.36
1
Wallace and LeOaine give, for a spherically
symmetric shell
source of radius b,
(r-
a) 2 /4L 2
-(r
a) 2 /4Ls 2
t3/2- ,,
3
(5-8)
In this case the space inside the shell contains the same
material as the space outside.
Therefore, the real source probably
falls somewherebetween these two (Equations 57 and 5-8).
In Figure 6, qr2 for (5-8) is plotted, as well as the experimental curve.
The peak of the curve is a little
nearer the origin, and
the curve falls off faster than in Figure 7. Distributions (5-7 and
5-8) seem to be a fair approximation to q out to about 15 cm., or almost
twice the distance from source to peak of curve.
1
Op.cit., p. 57.
2 T.l.
21, p. 51. Figure 27.
Tittle et al 3 find that
J...'..;.......1.../
:I~4
.
I* 4
(SLOGWtNG--H-D
DO-W-DENS-TY1 X(RADIUS)z
-
" AT
~. 1,.95O
DAE:f
9~a~
I)A~:~~~,
..........
:' .:....
...
.. .i.":',2. ... MEDIUM:
S'AN"D,
SATURA'TED. WITH
/.......I
~~~~.1
T.":4:E
:RADIUS
vs
WATER......
VI
4
.........
x..
V
104
LIO
0
n
o~
I~~~~~~~~~~~~~~~~~~~
.....
...- .............
.........
·
r
r
-'
L^··--·-·~-......:. .·......
. ......
_.I~.--
P'...
..........
A,
......· _
.....
zIi0X.
cn
z
I-i
1.
LEGEND
o
.. 0
: .
-·-
-----
EXPEMMENTAL:rnr
HEORETICAL
......
......
(SPHERICALSHELL SOURCE)
:: . _ ... .........
. ........... i...'
'.............
.......
;', ......'........- .....
! .....'=.......: ............~~~~~~~~~~~~~~~~~~~~~~~~~~~:..
'.. L..:..; ......... .' ........ ' .....:..~..... .....
, .........
L.. ........
.
' ':' :- ' 'i'
· LEGE'ND~~~i
V
:
· :''~~~
......
:'''. .. '''"'-"~~~''':~'
j."'
'---'.THEOR:ET~~~~~
'7.i'-.i'.:.[."'"""...
f'"';........ EXE'IME~AL~ ' ......... :'." '"'~'~-"'~~
.,
I" '
o0
5
' ..
"'......~~~~~~~~~~~~~~~~
::'t "~ .....
i~,.:.,:':.::::
{SPHERCALS'E~"ELSOURC)".,~::el?:,:::.
.10
15
r(CM)
20
25
30
'
35
.
I
"'i4
·-
/--"
-
~ ............
'~ i
':' *~
~~~~~~~~~~~~~~~~~~"'~''"~
~"~ I
.i :
-
`'' ~"".~.'~T .`..·...;"';""'.'...-'"(SLOWING
DOWND--ENSITY
..·...-
..f.(RADIUS
..... AT-4.44 EV:
v s --
--DATE'-FEB"I,195O
............
·-. .·-. . ·`· · -··-·--- ·- ·-
-:...
"M."EDMc S......SATURATE
-···
-UIIU
1UAt0E
I
iI
..
.-.
...
\V.~~~~t
.....
.
io
..
C-)
....
w
u
~~~~~~~~~~~~~~~~...
-i
".....
.:.
:'-......
....
~",....... ...........-. ' .........""'i";
. *:'"'.. . ....
.!'':
.....
·'"
.:"-.....
'.':".
.. -:'!"-::;-;
......
;"*"
.....
:*'!"':''.'"
......
;*.**=
Iz
'Iii
N
.~~~~~~~~~~~~~~~~~~~~~
10
\I~~~~~~~~~~~
LEGEND:
:
...................:..
: ...
EXPERMENTAL...... .... .
THEORETICAL'
(SPHERICAL
I0
CAVITY:
\R=2:CM.
:
--- THEO RE.TICAL(SPHERIAL. CAVTY
...
AGE 'THEORY CORRECTED)'
0
5
10
15
r (CM)
20
...-
25
30
35
34
for high energy neutrons in a similar medium,age theory begins to
fail badly at a distance of 1.5 times the distance of the peak of the
qr2 curve.
In that case, Flugge's theory was applied using an age
theory value of L . Thus, (5-7) and(5-8) for this low-energy case
give about equal results to a straightforward application of Flugge's
theory to the high-energy case.
Marshak, Hurwitz,
spherical harmonic method."
theory by their
improvements on age
and Brooks1 obtain
For a point source in
hydrogen, it can be shown an improvement on (4-10) is
r
q(r)e
2
,1+-(7- s +- r )
If themultiplicative
correction factor in (5)
(5-9)
is pplie to an
If the multiplicative correction factor in (5-9) is applied to an
extended source in a mixed medium,it loses its logical compulsion,
but the results are interesting.
A corrected curve for a spherical
cavity is given in Figure 7; the result is seen to be a very much
better approximation to the experimental curve than anything heretofore obtained.
Its mean square distance is probably not muchless than
the experimental curve.
In any case in which a source not too different
from a point source is in a hydrogenousmedium,multiplication of the
age theory distribution function by the correction term will probably
produce better result s*
35
Wallace and LeCaine1 give the thermal neutron density due to
cavityas
a point" source of fast neutrons in a spherical
s/a ra
Sige
n
2
2
1
4nr(L - a)
L2/L2
+ We
..
a- L
r
_
-
{1+
+a
a
L
2L
a +(5-10)
L
a
erf
a+L
Here, L
2Ls
r- a
L
L {1 - erf(r
+
8nrL
r
- erf(r -
(
2L
-
)L
corresponds to an energy range from N
to the thermal
energy, 0.025 ev.
To obtain the increase in L
in going from 1.44 ev. to 0.025
ev.,
Melkonia'ls data 2 for the hydrogen cross section below one ev. is approximated by ()
T081)b.
he othercross sections
= (20.36+
were taken the
same as those on Page 14.
Then L 2 = 6.29 cm.Z from 1.44 ev. to 0.025 ev. or L 2(26 Kev. to
0.025 ev.) = 16.4 + 6.3 = 22.7 cm."
The diffusion
believed to be in error.
length,
L.
length
L was calculated
Since mean square distances add,
r 2th is found in the usual way to be 352 cm.2
Op.cit., p. 103.
20p.cit., p. 1758.
This is
There is another way to estimate the diffusion
r 2 th = r 2 th source + 6 L2.
1
to be 7 cm.
· ·U.'
TRON "IDEN'STY)X(RADIUS)2 v s'RADIUS
(THERMAL"
......
~~
D.F :S
E,-,.5...:A
::..........
...W:.--A;.:
.
MEDIUM: SAND,SATURATED
..............
DATE~~~~~~~~~~~~~~~~~~~~~:
i.WITH
.ER1,9
......... ..............
'......
.WATER''"
~~~~~~~~~~~~~~~~~~~~~,
.
/
i~~~~
/r
I
00-2.
.
:, -..-
"I
LEGEND;..~~~.~......
x2
.. ..
/
..
I..
-..........
~~~
~~~
~~~
~~~~~~~~~~~~~~~~~~~
. ... . .~
. ..
oU
U,
0zr
:::)
D
w
z
C
0.1 .
-'----L
-----L
EXPERIMENTAL
5.CMI .. EOR
-OCM(S'
H..RICALCAi ........
-L...,.CA THEOR ETI c·AL'(SP HE R,11. A.-........ .T....
0(')
.. 0
5
i~~~~~~~~~~~~~~~~~~~~~~~~~~
.
50
r (M)
20
25
30
35
Using the indium resonance distribution
to the thermal
source,
as an approximation
6 L2 = 352 - 157 m.
or L = 5.7 cm.
In Figure 8, distributions
corresponding to (5-10) are
given for diffusion lengths of 5.7 cm. and 7 cm. It is evident that
the normalization is not correct.
This, however, might result from
inaccuracies in calculating the large-radius end of the distribution
from equation (5-10).
Many of the more complicated formlas
by Wallace and Le0aine, such as (10),
either
given
involve differences
between large numbers that are almost equal, or else products of very
large numbers by very small ones.
conveniently obtained.
interpolation
In either
case, high accuracy is not :;.
In interpolating between tabulated values,
formulas may be needed to obtain sufficient
This limits the utility
accuracy.
of anysolutions that are found in Wallace
and Lecaine.
Something should be aid about more advanced theories.
These treat the transport equation in a more sophisticated way,
making fewer aummptions that limit the validity of the theory;
of course, the theory usually is then harder to apply.
A typical
example is the "Exponential
1
!a. cit
a
-n.
2.-O420..
iI
i~~ ~~x__,,|
1
__~_
Theory" expounded by
s38
In effect, Marshak identifies the square of the slowing down
distance with one half the mean square distance of a neutron distribution from a plane source. He then solves the transport equation in
such a way that he obtains the mean square distance with high precision.
The slowing down distance thus found, when put into the age theory
which
plane source or point source solution, will give a distribution
has the sae 'meansquare distance as an observed distribution,
though the theoretical and experimental distributions
ily coincide from point to point.
al-
don-'t necessar-
However,when the exponential
theory value of L 2 is used for someother type of source, the identity of the mean square distances no longer is necessarily true.
Of course, if the L
from exponential theory differs sharply from
that given by (494), the former is still probably a better value.
Exponential theory is difficult
to apply to a mixed medium
and it is knownthat for low-energy neutrons it gives the samevalue
of L 2 as (4-9) for a water medium. Hence it was not used in this
work.
Another rigorous treatment of the problem is that by
Verde and Wick.
They treat the transport
equation by the same
integral equation methods as Marsak, with a different kind of result.
An expression for the slowing downdensity is obtained in the
form exp (-r/))
A in this expression is a function of r, and is
found by numerical integration.
For water, at small distances X ia
almost constant, so that the slowing downdensity is a gaussian.
At large distances it is approximatelyproportional to r, so that
the behavior of the density is essentially exponential.
'Verde,
K., and Wick, i. C., Phys Rev. is
(1947),
In other
pp. 852-864.
39
words, it predicts in detail, successfully, the transition from a gaussian to an exponential.
Unfortunately the mathematical development
of the theory depends upon the assumtion that the hydrogen oross section varies as a power of neutron energy (i.e.,
in u).
as an exponential
This is a good approximation
for neutrons of several Mevo
energy, but is completely false for the source used in this work.
VI.SIS2Lk
For high energy neutrons in hdrogenous materials, Tittle,
et al,
found absolutely no agreement between experiment and a
straightforward application of age theory. For F1ugge's theory
plus age theory they found fair agreement using age values of Ls.
Usingexponential values of Ls , the agreement was still
better.
In the present work it was found that age theory, straight-
forwardly applied, gavequalitative agreement well out past the peak
of the qr 2 curve.
The somewhatbetter
results
obtained by unmodified
age theory in this case are directly traceable to the fact that in
the energy range of the neutrons used here, the hydrogen scattering
cross section did not vary as much. Muchbetter agreement was found
when the spherical harmonic correction was applied to age theory.
The author feels that for low energy neutrons in hydrogenous materials
the spherical harmonic correction
to age theory should be
applied whenever the experimental setup remotely resembles the
conditions under which the correction was derived.
Age theory is of only incidental interest
shielding because hydrogenous materials
in neutron
are so commonlyused.
In
such a case age theory gives a qualitative picture of what happens
near the source, and it can instill
whole topic.
somephysical feeling for the
However,in shielding what is of interest is the inm-
probsle event, the neutron that goes great distances
1 T.
R. 21, p. 51.
from the source
without much slowing down. Only more developments along the lines of
Verde and Wick can give information about this.
In using measuredmean square distances to characterize
shielding properties of various materials, care should be taken
if the sige of the source is appreciable comparedto the root mean
square distance of the neutron distribution.
-fif all the neutrons
get a head start, the results are likely to be misleading.
op. cit.
Initial saturation activities, counts per minute,
of individual exposures.
Not corrected for cadmium attenuation or foil
depre ssion.
I
L
::
4
-4
0
0
+l
to
C
+I
O
+3
O
u0
+3
*0
0
0
e
n
I0
n-4
0o
d·0r
O0
o0
a:
0
o0
o
10
U)
-
0
rA
o
0 O
r
U-
O
00
0
6
0
*,r
u
0
CD
CE)
+1I
a
O
0'
CO
H a
1-4
C'.
H
CQ
4.)
0
104
0
oin
0
a2
A
ceV us
0
*
0
O
C';2
CE
*
-
1n
'-
H
0
1-4
oE
1
0
+ S
to
to
CN H
+ I +I
4)
E-4
0H
00
rH
o10
OD
to
1-
a
u
+
10
0-
n
+I
P ne
1
to0
+1
C0
M
k
r
1
at0
r
m
44
o_
v
Ci-
H
-r
*4
03
1
H
to
0'
n
H
CX
C'
10
"4
'
0C +
(O
E
a
M
00
s
+10
C')
-4
C
'..
o
6
Q
_
fil
D.0
*
n
c*
cn *0
a,
cw
0
H
C
C
M
44
i
i
I
EI
_s0
0
to Fio
+ 1
C)
0
'.-4
t-4
H
0)
N
+I
to *
+1
+t1
0)
o1
n
+1
+
C)
V)
H
L
+
L-
C
0
'v
0C
5
a,
P
tO
;
0' 0
io
* +1I.
CD
0 o
rn
'-4
'-4
44
H
+coa'
10
~0
o0
+
00
aC V
+t .
LO
0D
. +1 +1*
+1
10
0L
O
H
t
0
0
q4
a')
·
to
C)
r- V I-4
00
_-
10
0
0
00
0
0
t
H
H
r
C')
V3
V
C')
HH
Co
C
C
'-q
0
0
m^
Et
Cl) '-4
4
H
44
rQ
£O o O
o10 X
r9
C'
m
o040
C')
_
El
:4
*4
a,
oo
O
E-1
CD
.. .1
O
m
cq
o_
v
_C)
Sa
+
+t
o
E
++
LO
r-
.
ad
+
+1+1
v
4
'
0
0O
H
H
0O
H
c'.
)
t4)
45
*H
0
r4
IO
a)
H
·
r·
A
LO
L3
·
I
rlH H
@
)
%%%
0
Y0
4,o
C)
0
*
_u
0
0
N
O
to
t)
H
H
-
h
n
Em
A
*4
a sn E-
C
t'-4
M
C
c
p4;
0o
uf-s
0*
H
0
0
H
*
aCQ
a)
V)
0
:.
o
0
0
0
0
>4
m4
rj)
4p
Hi
E
P4P·
a
c· a
03
U
o1
C)
02
u2
_
U
1
H
E-r
H
d
0
0
0
L0
*
c-
C0
10
0
10
C
L
t0
0
*
*
V)
H
o
1O
w
0
X
·;
H
F4
d
n
03
n
>.A
C
H
b
(D
C
r0
o
_
|-
y
_s
oz
H
0
4
**1
C
I
.
p
4,6.
.
i
a
_
10
4
0
P
H
0
I.
ar
-4
i
I
4'
* C* 0C°*00 0 00 00
i
i
t43
SS
-4
B
0
0
risr
*
:M
+
I
I
UN0
H
+4 +' s+9
I
14
_
o
I
I
la
'
C)
20
*1t
OD
0.)
0D4 '0f
'
+5 +aI 4
90'4o
.
to
+3t +A I
S
E-'
*
44
5
%t
O 0
0
0 0
0
to
to
o0
o
-a4
v
aI.o
-4
0
'v
8
H
4-
E4
0 a0
c0) 10 0
o0
t0
H
H
mN
to
H
$3
11-
E-
-H
0_h
0
~zq
a) H H
- CO
N
U
w
c
'
M
H
.I-'
:
ar
t_
O
_
S4
' o o a o o
o oI Hq cn0
H 1N r'0 to
t
*
*
*0
*
*
*
t
4
-El
co
atCQ
4' 0OL
H
to
to
V)
0V
.4
C)
0
14
'4
a,
sH
H
H
CO
0
o
oa
to
H
rCQ tO
to
O
tO
H
H
I
d5
a
0to
I-I
C,
"I
1U
.64
*1
6-4
6-4
to
XH
H a
+1
+
U
1-1
0H
to
X
in
+
co co
to
at
+d+
+
a
9
E-
-4
Ga
H
6.
sto
1
H
o>
c
at
t
ato
mo
to
to
L
To
PE4
. -I
0
4
0
tD
0*
to
CQ
C)
I
lo
9
*4%
Xr
'4
to
o
r-I
aE o0
t
c
-4
C
t
CQ
to
t
V4
co
U
-
I
1-0
Ot)COV
H
o
to
H
H
H
m
0
H
H
v+1
+
+1
OH
+41
4
.-%
54
*r
:
a
ul
'-4
I
W
0
a
"0
a
C,
o
OrI)
Q
,H
0 00 m
00
~
H
cq
H
o
aw
O3
co
r
H
N
H
O
H
H
-~
O
to
0 0O
aa8
-4S
1-4
O
So
00
C)
Y
t o
m
W
_
e
1-4
cra
r.
P
9
a
V14
W4
i
R
1-4
W
1-4
-P
0
o
ii
E4
R
H
c;
rA
1-4
U
P1-4
E
As
a ;
0
at
OC
0
H
co
co
-
-
0
-4
C0
H
CD
O
4
C
0
c
z
C;
·
H
44
h
'4
H
H o
0
Hi
C 0
0
0
ar)
0
H
0
0)·
t
L)
o
.9BLIOQiRPHI
Amaldi, ., and Fermi, E.,
Slow Neutrons,"
Amaldi, 3.,
Hafstad,
Artificial
Bell,
.
HX(n,
(1948),
On the Absorption and the Diffusion of
Phys. Rev. 5i (1936), pp. 899-928.
L.
., and Tuve,M. A.,
Sources," Pbys.Rtev.
., and
lliott,
L. G.,
Neutron Yields from
(1937), pp. 896-912.
Ganma-Raysfrom the Reaction
)D3 and the Binding Energy of the Deuteron,"
pp. 1552-1553.
Phys. Rev. 74
Dacey, J. ., Paine, R. ., and Goodman,C., Shielding Properties
of Various Materials against Neutrons and GammaRays,' L.N.*S.
Technical
Report No. 23 (1949).
Faul, H.,
Neutron WellLogging,' M.I.T. Ph.D. Thesis, Geology
Department,
1949.
Goldsmith, . H., Ibser, H. ., and Feld, B. T., Neutron Cross
Sections of the leents,"
Rev. Mod. Phys. 11 (1947), pp. 259-297.
Hanson, A. O.,
"nergies
of Neutrons
from X T - D, L
and b - B Photo-Neutron Sources,'" Phys.
v.
- D, Y-
(194),
B,
pp. 1794-
1798.
Sarhalc,R. B., ITheory of the Slowing Downof Neutrons by Elastic
Collisions
238.
with Atomic Nuclei,
Marshak, R
N
Rev.
od.
., Brooks, H., and Hurwitz,
hys. 19 (1947),
H., NIntroduction
Theory of Diffusion and Slowing Downof Neutrons," I-IV,
(May-ugust, 1949).
pp. 185to the
ucleonic
Melkonian, E., "A Precise Determination of the SlowNeutron Cross
Section of the Free Proton," Phys. Rev. Z6 (1949), pp. 1744-1749.
Melkonian 1., "SlowNeutron Velocity
N2 , A,
pectrometer Studies of 02,
2 , H, 0 and Seven Hydrocarbons,'
Phys. Rev. 7 (1949),
pp. 1750175.
Munn, A.
., and Pontecorvo, B., Spatial
Distributions
in Hydrogenous Media Containing Bismuth, Lead, and Iron,'
25 (1947), pp. 157-167?
OtNeal, R. D.,
II.
The Slowing Down of Low Energy Neutrons
of Neutrons
Can. U. Res.
in Water."
Determination of Photonentron Energies," Phys.Rev. 70 (1946),
pp. 14.
Seaborg, G. T., and Perlman,
(1948), pp. 585-667.
Tables of Probabillty j
I.,
"Table of Isotopes,"
Rev. Mod. Phys. 20
inctions, Volume I, National Bureau of Standards,
1941.
Tittle,
C. W.,
The Slowing Down and Diffusion
of Neutrons
in Hydrogenous
Media," M.I.T. PhD. Thesis, Physics Department, 1948.
Tittle,
C. .,
aul, H., Secrest,
C., "she Slowing
. L., and Goodman,
DownandDiffusion of Neutrons in Hydrogenous Media," L.N.SJ.
Report
No. 21,
Technical
1949.
Verde, M., and Wick, G. C., Some Stationary Distributions
in an Infinite
Medium," Phys.Rev. UL (1947), pp. 852-864.
of Neutrons
, and LeOaine, J., "Elementary Approximtions in the
Wallace, .
Theory of Neutron Diffusion," N.R.CO.. No. 1480 (Chalk River).
Weisskopf, V. F.,
Neutron Diffusion,"
juclefar
Power, Vol. I.(Adison-Wesley
I.
TheSciene and r.ninserin
Press,
Cambridge, 1947).
of