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ATOMIC ENERGY COMMISSION
EXPONENTIAL EXPERIMENTS ON
LIGHT WATER MODERATED 1 PER
CENT U-235 LATTICES
H. J. Kouts
November 28, 1952
Brookhaven National Laboratory
Upton, New York
Technical Information Service, Oak Ridge, Tennersse
J'r L Z*SSIIED
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disclosed in this report.
Date Declassified: October 31, 1955.
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Issuance of this document does not constitute authority
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ington 2T, D. C.
Exponential Exp*eriments on Light Water Moderated, 1 Per Cent U-23r Lattices
Report Written by
H. J. Kouts
Work Done by
H. J. Kouts
November 28, 1952
Work performed under Contract No. AT-30-2-Gen-16
Brookhaven idotional Labora~tory
Upton, New Ylork
Dlcg'lzed by the Inte- nel AIChive
in 2012 with funcllng Irom
University of Florida, George A. Smathers Libraries with sulpport from LYRAASIS andj me Sloan Foundation
I. Introduction, 2
II. Experimental Facilities. 2
III. Buckling MIeasurements. 6
A. Theory 6
B. Experimental Techniques 8
C. Analysis of Measurements and Results 11
D. Cadmium Retios 13
IV. Anrisotropy Measurement. 1
A. Introduction 13
B. Theory 14e
0. Experimental Methods 16
D. Results 17l
V. Migration Area Measuremaents, 27
A. Lu~troduction 27
B. Theory 27
C. Experimental Methods 28
D. Analysis of Data and Resulta 29
VI. Intracell Flux Traverses. 38
B. Experimental Procedure 38
0. Analysis of Data and Results 40
VII. The Effect of Absorber Rode. 43
IIII. The Effect of a "Gap" on1 the Azil Relazation Length. 45
Eyponential Experiments on Light Water M3oderated. 1 Per Cent U-235 Lattices
At the request of the Division of Reactor Developmnent of the UI.S.A.E.C.,
the Phsics Division of the N~uclear Engineering Department at Brookhaven under-
took a program of exponential experiments on ordinary water moderated, slightly
enriched uranian lattices. The purpose of the measurements was to provide nuclear
design information for the core of a reactor for the production of plutonium and
power. In particular, nuclear data were to be provided on tile basis of which the
Atomic Energy Division of the H. K. Ferguson Co. now the Walter Kidde Nuclear
Laboratory could prepare a feasibility report on such a reactor. Consequently,
th~e program of the expe~riments was planned in close cooperation with members of
the WKNJL and cooperation was maintained during the course of the experiments.
The present report is a sumrary of the exponential experiment neasure-
ments which were made during the period November 15, 1951 to June 30, 1952, as
part of a broader Reactor Components Testing Project. The experiments to be des-
oribed include measurements of the critical buckling, the migration arse, intra-
cell flux traverses, effectiveness of control rod materials, the effect of re-
moving uranium rode from the assembly, ete.
II. Experimental. Facilities.
The facility for the exp~onential assembly was planned with great care in
view of the difficulties met in earlier experiments on ordinary water moderated-
natural uranium lattices at Oak Ridgel.
The measurements were carried out in a water tank placed on a thermal
column which occupies a portion of the top of the` Brookhaven pile shielded The
thermal column consists of five one1Boot thick layers of graphite,. stepped from
la 4' x 4' size at the bottom to 5' x 5' at the top. This~ stepped construction
prevents fast neurtron leakage up the side of the thermal columnj and the well-
machined surfaces of the graphite blocks which manke up the structure provide
tight packinggyhich ensures a well-moderated source of neutrons at the. top sur-
face. The cadmium ratio of this source as measured with five mil thick indium
foils was better than 105.
The water tank is cylindrical, six feet in diameter and six feet high.
Surrounding it is a makeshift shield of three-inch steel armor plate and assorted
blocks of Brookhaven concrete.
Inside the large water tank is a perforated aluminum cylinders four feet
in diameter =nd four feet high, which serves as a vertical support for the top
tube plate system, from which the uranium rods are suspended. This top is a' ring
of three inch thick arnor plate which holds the circular aluminum top tube plates.
The inner tank with a tube plate is shown in Fig. 1.
The tube plates are made of 28 aluminum, the upper plate being 1*5n thick,
and the lower plate *75" thick. In eadb are drilled 275 holes for the uranium
rods, and 20 holes for the insertion of foils. C~are was. taken to locate these.
holes accurately, the tolerance being Os -*5 mils for each hole dianeter, and
-5 mils overall for the distance between holes. These low tolerances were main-
tained largely because we did not know the effect of variations of individual rod
positions on, reactivity and the positioning of foils, and we preferred working
with conditions which could be trusted
The uranium was prepared at KC-2r, and was rolled into rods of .750n dia.
at the St. Louis plant of the iMalinekrodt Chem~ical Corporation. Th~e Brookhaven
metallurgy group then straightened the rods to within 15 nile lateral deviation
in four feet of length, and clad than wi-th 30 mil thick aluminum. Most rods are
single four-foot lengths of uraniun. A few are slightly under four feet in lengths
and a few are made of shorter aeotions matched to provide approximately four foot
Analyses of the uranian composition were made at Oak Ridge. The average
composition is 1.027 per cent f .001 per cent by weight.
Distilled water purchased commnercially was used throughout the experiment.-
Occasional spectroscopic tests of purity were supplemented by tests of reproduci-
bility of the data. In fact, the criterion of reproducibility was used constantly
as a check: on the possibility that other factors might be influencing the measure-
nents. For a large part of the time the water contained boric said in solution.
At all tines the tank contained cadmiumn from the shutter (see below). There was
always the possibility that boron or cadmium coatings might form on the aluminum
cladding of the roday thus changing the reactivity. For this reason, particularly,
reproducibility of the data was held to be important. This reproducibility was
mraintained throughout the excperimenmts.
Best results from an exponentia~l experiment are obtained if the source of
2. Lack of accurate positioning of rods and lack of straightness of rods have
been suggested as the source of errors in the Oak Ridge measurements (see
Best results from an exponential experiment are obtained if the source
of thermal neutrons feeding the reproducing lattice is made about the sam~e size
as the lattice itself. In this event the decay vertically of the flux harm~onics
is most rapid, and the region of flux harmonies corrections is smallest. For
this reason the bottom of the large water tan~k was covered with a cadmium sheet
having a central circular orifice. The hole-in the cadmium had a radina about 7 st
larger than thg) of the max~i~umu uraniua loading. This defines the source size
most suitable for rapid decay of harmonica, because the radial extrapolation dis-
tance of the thermal neutron flux in the reproducing lattice was found to be
close to seven centimeters for those loadings investigated.
A shutter system of oadmium covered aluminum plates was designed to fit
under the lattice assembly, to make possible turning off tihe source flux when
irradiations were not being mazde. Because of corrosion and working under the
weight of' the uranium and support assanbly, this shutter system never worked prop-
erly. In practise, the source flux was always turned off by the dropping of a
saduium sheet into the water tank, over the disc source, after the inner tank
with its uranium loading was removed by crane.
The water tank contains a system of steam pipes which allow heating of
the water. Thus the equipment has provision for finding temperature coefficients
of the various quantities mleasured. No temperature coefficients have yet been
measured, but it is planned to determine some in the near future.
III. Buckling Measurements .
We have used throughout the course of the measurements a one-group view
of the diffusion problem in the lattice. Accordingly, we suppose the neutron
flux satisfies the relation
(1) 2q +B2dy = 0,
and the critical equation is simply
(2) kno=1 + 29 2
The standard exponential experiment involves obtaining the Laplacian by
differentiation of the measured thermal flux distribution. For a cylindrical
array such as we used the space dependence of the flux can be written as a
series of Bessel functions:
where r and z are cylindrical coordinates in the lattice.
h is the reflector savings, and a, are the successive roots of
(4) Jo(r) = 0
The coefficients f, have the form
(5 )a, En 3n ex (g) / s+D s -[i (n ) B 1/
the exponents being real for a subcritical assembly. C, is small compared with
Dn,, since the first term is only important near the upper boundary of the lattice.
Furthernores since the rate of decay of the higher harmonies is greater than that
of the fundamental, above some value of z', tle, flux can be repressented closely
by the fundamental alone:
(6) ~ sIg f~), a>2s '
Ordinarily, one would measure the radial and axial flux distributions
in the region of validity of (6). The radial distribution would be fitted to
Jo(al r/ro), the axial distribution fitted to 01 ep s/L + D1 expjBF -s/ ad
the buckling wquld be
(7) B /o2-1L
Unfortunately, thle small loadings we were working with did not permit
accurate measurements of the radial flux distribution. Syamnetry conditions mrade
only five positions available for flux measurtsments on any one radius, and the
outenrost of these was so close to the essentially infinite water reflectors as
to make it useless. TIhe effect of the reflector can be seen from a typical
radial traverse, shown in Fig. 2. Thlus only four points were. available along
any one radius, and a curvo-fitting to these points was not accurate enough for
On the other hand, the axial flux distributions showed in every case the
behavior characteristic of the presence of the fundamental only, from about 10 amn
3. The decision to use reflected lattices was made mainly because of the small
amount of uranium available for the oxperimont. .The reflector sa~ving~s was
equivalent to about 0.7 tons of extra uranium.
L. The lack, of accuracy does not rise simply from the fewness of points. Rather,
it is caused by the feet thlat these points are on the flat pa~rt of the Jo.
curve, and ame~ll experimental errors in thea values of the foil activities cause
largo variations in the extrapolated end-point.
upward from the bottom of the lattice. Mbreoverl the end corrections were small
below about 70 as front the bottom of the lattice. Thus there existed about 60 an
of variation of a over which the flux decaiy was simply exponential, and so quite
good measurements of L could be made. He therefore decided to try basing mteasure-
mIents of B2 fin measurements of L alone.
For such a determination of B2, one would nee~sure L for a wide range of
rod loadings. Each such loading would be idealized to a cylinder with the sese
loaded radius the case would have if the uranium were uniformly distributed at
the same water-to-metal ratio. Least squares fits would then be made to X and B2
( 8) ( al/R+1)2 L/2 -1 =B
Such an analysis contains of course an assumption that A and B2 have the
same value for all loadings used. This is perhaps no worse than the assumption
ordinarily made in an exponential experiment on an inhonog~eneous lattice that
the results can be interpreted in terms of fudged calculations liodeled after homno-
geneous reactor theory, with a value of the buckling which does not depend on the
size of loading. Nevertheless we have kept in mind the fact that this assumption
underlies the measurement, and later in this report present experimental evidence
that the assumption is probably not far wrong. This problem is also being in-
vestigated from a theoretical viewpoints
B. Experimental Techniques.
The measurements of neutron flux distributions were made with indium foils,
0.220" in diameter and five mile thick. Each foil was counted in from three to six
counters, to a total of more than ten thousand counts,, (1 per cent statistical
accuracy) per fail. Counter resolving times were found by the two-source mesthod,
and in all counts resolving-time corrections were kept below 3 per cent. He
believe that in th-is vwa we have kept to about .1 per cent of the total counts
per fail the error introduced by having to make resolving time corrections. Count
times were controlled by a preset thmer which was accurate within .3 seconds (at
most *5 per coalt per count), about half systematic and half random. The estimated
error due to timling is ~uns about .25 per cent. The overall estimated error in
the activity of an individual foil is then about 1.3 per cent about the means when
thd conditions for accuracy are worst.
The foils were positioned in foil rods inserted through the top tube
plate. At first thin-walled aluninue tubes were used, with slots to hold the
foiled punched at ten centimeter intervals. The foiled were placed in thin aluminum
foil covers, each being held together with Duco cement. The cement was removed
by two washings in acetone before foil counting was begun. The meadata error in
vertical positioning of the individual foils was about 1 rmn. For thel shortest
Tolaixation length we measureds the maimrum error in foil. activity measured was
thus about 0.7 per cent f'rom this source. Each foil ves used only once, so there
were no errors from the presence of long-lived activities.
The aluminum foil positioning tubes were later replaced by machined lucite
rods. Positioning error was thus reduced (to about 0.2 per cent at most). The
use of Duco canent was avoided, and the replacenant rate of foil rods which had
been large because of bending of the aluminum and rapid enlargement of the holes
for foils, was naterially reduced.
Because of large local variation of flux, it was necessary that all foils
be centered accurately in a lattice cell. Ths was accomplished through the use
of lucite spacers located at one foot intervals in height throughout the lattice.
The foils were weighed to within 0.1 ag each. Their weights averaged
0.027 gma, with not aauch variation about the mean. Measurements of activity as
a function of foil weight showed that corrections to the activity due to weight
variation are about 20 per cent of the weight correction. MYatched sets of foils,
with weight variations limited to ?: .1 mg, were used at every stage. Th~e esti-
mated error caused bty individual, variationns in foil weight was at vost about .05
per cent. This contribution to the total error is negligible.5 The value of the
indium half life was taken as 54.05 minutes for foil activity calculations. At
the end of the buckling apoperien~ts a careful determination of the half-life was
masdes and gave a result T1/2 5i4.14 .0 m4linutes. The error in all the bucklings
introduced by this discrepancy would be about 0.3 per cent. A measurement of the
indiumn half-life was suggested to us by A. Mattenberg, who pointed out the lack
of agreement in previously published values.
The compounding of all these errors indicates inaccuracies of individual
foil activities of at mnost 1.5 per cent about the nean. Statistical variations in
5* The variation with weight of the weight correction to the activity was measured.
It was found that al1 per cent weight variation was accompanied by only a 0.2
per cent change in foil sensitivity. This is much less than the direct pro-
portionality usually easumeod.
the mleasured quantities were consistent with this estimated error.
C. Analysis of Messuirements and Results.
The relaxation lengths listed in Table 1 are obtained from least squares
fits of excponentials to the axal fliix traverses. Water-to-metal ratios of 1.5:1,
1.75:1, 2:1, 2*5:1, 3:1, end 1:1 are reported. These ratios are really those of
water plus alumrinus to uraniuml. Also given in Table 1 are the residuals of the
least squares fits of B2 and h.
Table 2 gives .the values of B2 and X which resut from the least squares
fits. Fig. 3 is a plot of B2 vs wsat~r-to-metal ratio fron Table 2. Fig. 4 is a
plot of X vs water-to-metal ratio from Table 2.
The estimated errors in every case refer to reproducibility of data. Thus
they do not include systematic errors or limitations to the validity of the method.
The effect of having ignored end corSrections was considered. To determine
the magnitude of this effect the following procedure was used. Values of L vere
calculated from flux mecasuremnents over a range of 60 an, beginning 10 an fron the
bottom of the lattice. Least squares fits of B2 and A were made for this set of
data. The calculation was then redone leaving out the points at 70 as from the
bottom of thle lattice. Since end corrections are greatest at this points the
changes in computed values of B2 and X may be attributed to than. Small abanges
in B2 were observed; they everagod about 0*5 per cent, in the direction consistent
with the presence of end corrections. Since these changes are of an order of
magnitude smaller than the accuracy of the measurements, it is felt that end cor-
rections are not of sufficient importance to warrant further investigation. The
values given in Tables 2 and 3 refer to the six point axial attenuation Ieasure-
ments, which have the smaller and effect corrections.
At no stage were arUy effects observed which might be attributed to thle
presence of the harmonies in (3).
Since the method used to find the buckling is not standard, it was con-
sidered advisable to make some radial traverses in order to be able to compare
results with those which would normally have been obtained. TIhe best radial
traverses were made with the 3:1 water-to-metal ratio, at a loading of 265 rods.
At this loading the lattice was approximately an elliptic cylinder, the distance
to the edge of the metal loading being different for the two directions along
which radial traverses were made. The effective radii vere found by fitting tho
measured foil activities to Jo(al ig i)* The values of the reflector savings for
the two radii were then found as the difference betwoon values of Reff and the
actual distance fromu the lattice center to the edge of the letal loading. The
average of six radial traverses gave in this way a value of 6.65 + *50 an for the
reflector savings. This is to be compared with the value of 6.9L f .11 a~n de-
termnined by the method of exial bucklings only. The agreement of these values
to within three millimeters is strikingly good.
a single measurement of thle reflector savings was carried out in the LS5:1
lattice, in connection with the anlisotroply measura~nant discussed later in this
report. The measured value of k in this case was 7.67 anu, with somewhat uncertain
accuracy. The value given in Table 2 is obtained by the method of axial bucklings
is 7.71 i .14 an. Agair the agreement is exceptionally good.
As a further check on the validity of tle assLumption of constant B2 and 1,
the buckling for the 3:1 lattice was recoq.luted, using only loadings between 181
and 263 rods. B2 and h woro unchanged within the expericantol error roportod in
Thus the orperiDontel data available tends only to bear out the accuracy
of the assumption underlying the method we have used. It would still be interest-
ing to see thes results of critical experiments made on these lattices.
If the slowing-dowsn and diff~usion of neutrons does not take place iso-
tropically in the reactor core, thG exponential experiment must be modified con-
sidera~bly. The effect of such anisotropy has been discussed by Young and Wheelor
they showed that if the anisotropy is large it is possible to draw grossly in-
accurato conclusions froll a simple exponential exper~iment.
To a certain extent, information on the energy distribution of low energy
neutrons m~yr be detorcined front the coduium ratios. Therefore accurate usesure-
nocnts of tho codduin:. ratio were made at thle center of a lattice coll, .uaing indiun
and gold fails. The indium: foils were five uils thick; the gold foils were 1.2
mnils thick. Moosured values of the cadmaiun ratio are listed in Table 3 and.plotr
ted in Fig. 5*
IV. Anisotrop~y M\Ioouronente
As will be seen in Part V of this report, the migration area which we have
measured is essentially a constant over the range of water-to-metal volume ratios
investigated, and is in this region quite close to the-m~igration area of fiesion
6. C-90* -13-
neutrons in water alone. TIhus it appears that because of high inelastic scatter-
ingr uranium in water may be considered quite closely oquivalont volumnewisG tOn
the average) to the water it displccos, for the purposes of neutron alo~ing-down.
It might be suspected that this condition mafkes the~ existence of anisatropic of-
foots unlikely. Nevertheless, the situation socluod to warrant a sac~reh for such
an anisotropic character of th :;igrction area.
Young and Whoolor also roportod and analysod the sugestion by Wigner
that a doubleo exponential exporimant" would yield the degree of anisotropy. It
is such a double exponential experiment which vo have carried out for the 1.5:1
water-to)-metal volume ratio lattice, which~ should have the greatest anisotropy
of the lattices we have used.
For the general theory of the double exponential experiment, one should
see the p~aper by Young and Whleelor. We give here a discussion suitable for :.me
We consider our subcritical lattice in the shape of a rectangular paratl-
lelopiped, with edges Px$ y The fulel rods are supposed to lie in the a
direction. If the ~thermal neutron source activating this array is placed at oneJ
end of the lattice (the side with odges 1x and a ) thu ono-group critical
equation will. bc
(9) : -1 = M (B~ 8 ) MB .i
k_ is of course the infinite pile preproduction factor.
(10) Bx=~ 7 2
(11) Bs = 1/L .
L is the relaxation length for decay of the fundamental in the st direction, X is
the reflector savings, and M~ and M have physical meaning aassciated with the
mean equere distanco a fission neutron travels ~in dirootions parallel to the rods
and normal to the rods, respectively.
If, on the other hand, the source is placed on one of the side of the
lattice (that defined by y and a), we shall have in place of (9)
(12) kz-1 = M~b~ M (b2 2)
(13) bj = R
s 5/2 + X)2
(14) b = B
(15) b = 1/(Lt)2
L' is now the relaxationl length of' the fundamental in the x direction, neasured
in this geometry. From (9) and (12) we obtain
H2 (B2 + B ) MB2a = M 2 + M2 (2
(16) M /M2 = (B2 + b2)/(Bx ~)
In practise, we took tkbo first geometry to be cylindrical, so that instead
of (9) one has
(17) k -1 = ZB 11 B2
(18) B1 = (a R+b)2.
al being the first root of the Beasel function of order soro, and R boing tho
radius of the loading cylinder. The relation which corresponds to (16) is do-
rived fron (12) and (17):
(19) 11~ /di = (Bq + b2)/(B2 + b2~- ~
0. ~ExJporiuentall MeIthods.
B1 and B2 can be found froml the buckrling noTauroancnts in cylindrical
geonotry described earlier. Thus to find the anisotropy it was only nocossary
to carry out the second u~ousurcment abovo the determination of b2t, by2 and b2.
F~or this purpose a set of lucite support piooos was constructed; these
permitted a horizontal layered construction of the lattice in a 16 x 16 rod
array thus a loading of 256 rods. Support was provided by the lucito at four
points along the length of the rods, to prevent sagging, and the foils which were
used to measure the neutron distribution were placed in maachined holos in the
surface of the lucite support structure. The cadmina which defined the shape of
the thermal neutron source had out in it a rectangular hole about 14 c7 larger
on each side than that of the array. This source shape was most suitable for the
appearance only of the fundamontal.
The gold foils which were used throughout were .220 inches in dienneter,
and were about one mil thick. They wore chosen as a set matched in eight to
about .5 per cent. The foil counting techniques are the same as those described
Equation (17) does not refer to any particular radius of the lattice, and
Sas a result B2 andi B m~ay be taken in (19) for any loading. We have chosen
Prather arbitrarily to base the analysis on tbo 235 rod loadings, the results
ij. of which are given in T'able 2. Here, L is 17.72 an by mloasuromuent. Analysis
of the least squares meth~ods used to obtain L show an expected accuracy of about
.10. We allog, for each of- the two muoasurements double this error, so SL (the
error in L) is assumlod about i .14. The loaded radius at 265 rods is 23.087 cm.
The reflector savings given by buckling measurements is 7.71 emn. Allthough this
number is here being questioned (its derivation is based oh use of the isotropic
critical Gquation rather than on (17)), it is doubtful that it is wrong by more
than *: 5 em. Thus vo may take Rth = 30.80 r ,50 can with some confidence. So
far we have
(20) B = 1/L2 = 3.18 x 10"3 & .052 x 10-3 cmd"
B2 = (al/R+1) = 6.09 x 10-3 t .055 x 10-3 ad"^.
Wiith the lattice placed horizontally, a set of seven foils was placed in
the~ yr direction, to provide a determination of the flux distribution in that
direction* A least squares fit to A cos bx gave b = .0513 t .0046, or,@/2 + h =
30.62 ex f 2.75 cn. Since Py = 32*95 on, we have h = 7.67 an 2*75 an. Inr-
spection of thG least squares fit to the plot of the flux distribution given in
Fi6* 6 shows that the cosine curve was actually shifted somewhat from the assumed
center of the lattice. If all foils are translated slightly in the same direction,
:. the error is considerably reduced. During the acasurement such a displacement
seems to have occurred. We prefer therefore to accept the above value of X, but
vith an error closer to about *5 an. Thus
(21) = 2.63 x 10-3 *043 x 10-3 an".
The flux values obtained from this traverse are listed in Table 4.
ALlso, be can be obtained accurately enough from just the rod length (four feet]
and this reflector savings; we have then
2 3 .a 03 -m2
(22) be = 53x1 .0 0 c
The thirteen foils which were used to neassure bx2 seemed to separate into
two groups) the syaPnetry of the triangular lattice cell. in which a given foil was
placed determining its group. Those in lattice colle with a vertex upward lay
on a different curve from those with a vertex downward. These two curves seemed
to have quite different end corrections, but apparently could be fitted byr the
same value of L'. Least squares fits to oeah of A exp x/L B oxp -x/L; agred
fairly well with LI = 100 f 30 an. The large error is caused b7 the presence of
sizceable and corrootions, the thicknzess of the lattice in the x direction being
only 45.90 emn. The results of the x traverse are given in Table 5 and shown in
Fig. 7. From the value of L' we have
(23) = 10"A t 104~ am"^
Substituting the values from (20), (21), (22), and (23) in (19), we obtain
(24) M~ /~ = 1.039 .
An estimate of accuracy can be obtained by differentiating (19). Putting
for the inoment
dB2 + db2 -JdB2, +d2 -d2)
The uncertainties in the various values can be considered as randy; the un-
certainty in 5 is then given by
(dB2) + db2. 2 2 dB2)2 + (dbp2)2 (dbY) 1/ .2
(25) af =
-B + b b
Insertion of the uncertainties given in (20), (21), (22), and (23) gives
(26) AT~ ~ .0as
The anisotropy indicated by (26) is just barely outside the error limits
set by (26); we conclude that within experimental error it is negligible. Thus
there appears to be too little anisotropy to influence the validity of the ex-
ponential experiment too greatly.
Another search for anisotropy could be made, based on the recasting of
(19) in the form
(27) =1 3B2 -B~
with a thre~e constant fit of 51, and k-1/M~ to measurements of B2~ as a function
of loading The agreasont of values of X obtained fron the two constant fit with
the values obtained froml radial traverses indicate though that such a procedure
would also show negligible anisotropy.
Relaxetion Loggthes L (an~)
In Water-81ightly-Enriched Uranina Lattices
Rosiduals of L (an)
let nrn 2nd run
L (1st run)
L (2nd nrn)
No. of Rods
Table 1. (Continuad)
Residuals of L (an)
let run 2nd run
No. of Riods
L (1st run)
L (2nd run)
Residuals of L (anI)
let rnm 2n~d run
No. of trods
L (1st ru)
L (2nd nru)
- .379 -.069
Table .1. (Continued)
Results of Loast Squares Fits to B2 and X
Latice 1stRun 2nd Run Avon 1st Run 2nd Rlun Avseraa
1.5:1 2.85 *62.94 + .05 2.89 +- .05 7.56 & .14 7*56 &t .11 7.71 +- .14
1.75:1 *~ 3.453 2.034 3.486 t.053 3.L70 af c033 7.16 .08 7.15 .12 7.16 &t .10
2:1 3.65 &.09 3.86 *.07 3*75 1.08 7.23 .26 6.75 .20; 6.94 *23
2*5:1 3.700 t.055 3.647 ',025 3.673 f.048 6.81 &.17 6.99 &.09 6.90 & .16
3:1 3.304 2.028 3.271 +.006 3.28g &.018 6,82 .11 7.05 f .09 6.94 f .11
:11.76 + .10 (1.88 + .04 [1.86 + .06+ 6.83 F *52 6.37 f .19 6.42 *f .22
# Aveircages for this lattice are weighted because one mecasureme~nt is Iluch poorer than
Gold Gadmiumu Ratios
2.~292 -+ .038
2.88, t .025
3.307 &t *045
Indium Cadmuiumn Ratios
2*576 2 .014
3.044 ;t .16
4*30 & .033
5*7462 t .092
Neutron Flux Moasuremeonts fronl y-Traverse
SThe residuals are those resulting fray fitting column 2 to A cos Bx. The best
values, on which the residuals are based, are
A = 819.9 28.0
B = .0513 j: .00~46 ca-1
Neutron Flug Moasure m
( A\ activity)
"Columnu 2 was fitted to A e'-aX + B o+0%. Although
points made boat values somewhat uncerttain, these
the scattering of tho few
GROUP 1 2083
GrouP 2 2162 -1041
IV. Migration Area iMeasurem~ents.
The values of the migration area are based on weasuranlents of the buckling
in lattices with boron poisoned water, as a function of the boron concentration.
The theoretical basis for the rmoasurem~ent is discussed in the next section.
We ma g express he in two well-known ways, either by the critical equations
(1) 1= 1 + B2(T + L2)
or by the fourfactor formula
( 2) ~= f pE P
B2 is the buckling, 7 the age to thermal of fission neutrons, L2 the thermal dit-
fusion area, so 7 + L2 9i2 is the adpgration area; f is tbo thermal utilization,
'1the number of noutrona captured in uranian; which cause fission (per neutron
cycle), & is tho fast fission factors and p is the resonance escape probability.
If the moderator is mado anna neutron absorbing bry means of the addition
of a poison, those quantitios in (1) and (2) which will be changed are ~, B22
f, and L2, the other remaining constant; 20~ msain effect of the poison is to de-
crease f. Hence, if the measured values of B2 are plotted against f, a straight
lino should bo obtained, the slope of whidb gives Mi2. For fs theoretical values
calculated by the Kidde group, were used to get preliminary values of M2. These
will eventually be corrected by the use of nexperimeontaln' values of f obtained
from measurements of intracell flux distributions.
0. Experimental Mothods.
A typical mlecauromeont of the migration area involved dissolving succas-
sively increasing amounts of 8203 in the water, and Ileasuring the values of the
buckling of a lattice in these poisoned mcoderators. At each stako the boron con-
celntration was determined by the analytical group in the Brookhavon chemistry
department. The bucklings were mleasured at three different boron concentrations
for each lattice. When the measuroments with unpoibroned water are included, the
dependence of B2 on boron concentration is thus found at four values of the boron
concentration for each water-to-metal vrolumeo ratio.
The buckling was found by measuring the axial relaxation length as a
function of toe number of rods loaded, a method described earlier. For each
poisoned lattice, measurements were made at only five loadings, because of the
large number of balcklings which had to be measured in a short tiuo.
Thermal neutron flues were measured with foils of five mil thick indian
.220 inches in diameter. The techniques used to position foiled in the lattice
and to measure their activities are described in Part III of this report.
Because of the possibility that boron might plate out of solution and
onto the aluminum cladding of the rods, tests of reproducibility wore made when
possible. Thoso consisted of neasuronents of the relaxantion length in an un-
poisoned lattice, and comparison of the buckling it indicated with previous
measuranants. The results of theso test exials are given in Appendix 1. They
lead to the conclusion that no important change in reactivity of the lattice
occurred during the buckling and emigration area measure-ments.
An earlier effort to Yoasure the migration area by another means was un-
successful. In this attempt, the uranium rods were covered with cadmian sleeve,
!:to suppress neutron multiplication. A measurement of the thermal neutron flux
~distribution due to a known fission source was to have yielded the uigration aron.
SLow values of the flux fro, tbo fission source kept this measurement from being
successful. This method of determining the migration area was used at Oak Ridge
I in an ocrlior investigation of this reactor typo. The Oak Ridge result differ
a;~iguificantly from ours, particularly at the lower water-to-metal ratios. These
discrepancies will be discussed in a later section
D. Analysis of Data and Results.
The values of the relaxation lengths, as found from least aquarea fits to
the axial flux measure~ments, are given in Table 6. Also given are the residuals
resulting from last squares fitting to the buckling and reflector savings. The
best values of B2 snd h are listed in Table 7. The boron concentrations; calou-
lated values of f, measured values of! B and-deduced values of M12 are listed in
Table 8, for wator-to-motal volume ratios of LS5:1, 2:1, 3:1, and 11:1. The
values of B2 when plotted against f for a given lattice yielded good straight
linas; the value of M~2 was obtained by a least squares fit. A typical plot of
B2 against f is shown in Fig. 8. Also shown in Table 9 are the values of k indi-
cated by the values of N12 and B2 for the four lattices.
The values of migration area and a~given by this report aust be considered
as tentative, because they are based on calculated values of f. There is some
uncertainty about the results of the calculations because of the simplifying
assumptions about the geomnetrzy, the angular distribution of the flux, and the
creas-sectiona. The value of the.mitration area as obtained from a poisoned
lattice experiment is quite sensitive to the values of f used and it appears
that accurate measured values of f must be obtained before the migration areas
reported here can bon~aldeo trustworthy. A preliminary estimate of the "experi-
mentala values of for the unpoisoned lattices is given in section VI of this
report. These values differ significantly from the values provided by Kiddes
and theoretical work on this problon is in progress at BNL. For these reasons,
the errors eited with the values of the emigration area must be considered to
represent only measuoes of the internal consistency of the experiments and the
possibility of a systematic error must be borne in mind.
The most striking result of these experiments is therefore the apparent
constancy of M2 over the range of value of the water-to-metal ratio studied.
This result is quite different from that predicted by present theory, as can be
seen front Fig. 9. When calculated values of L2 are enbtractod from M2 (and L2
is small in these lattices), the age is obtained. The resulting ages are plotted
against water-to-metal volume ratio in Fige 9, and compared with the theory of
Soodak and Forman.
Measured Values of Axial Relaxation Lenetth for Various Boron Concentrations.
Nwuber of Rods Rlelayation Length L~an) L Residual (an)
1.rj:1 Lattice. .216 Boron atoms/103 water molecules
262 16.898 -.023
229 15.686 .063
193 14.21) -.011
157 12*735 -.085
*121 11.430 .056
1.5:1 Lattice, .563 Boron atomus/LOS water molecules
263 14.960 .101
229 13.879 -co054
193 12.823 -.090
157 11.710 -.124
121 10.828 ,l60
1.551 Lattfoo, .860 Boron atoms/103 water muoleculos
263 13.677 -.002
229 12.976 .020
211 12* 538 -.015
193 12.121 -.013
175 11.710 .013
2:1 Lattice, *359 Bornn atoms/10 water molecules
265 18*744 *073
229 16*750 -.254
193 15.288 .438
157 13.597 .181
121 12.085 -*059
2r1 Lattice. .590 Boron atomafl0 water molecules
265 16.367 .061
229 14.867 -.261
193 13.899 .392
157 12.568 -.090
121 11.205 -,108
Relazation Longth L(cm)
---~~~~~~~~~~~ -~-- '-- ----~- ~ ~ S -~Ei
L:1 Latticel r071 Boron atoas/10 water molecules
Table 6, (Continued)
Numnber of Rods
2:1 Lattice, .8214 Boron atomaf/l03
3:91 Lattice. .171 Boron atoms/103
229 19.97 8
193 17 .794
157 15.7 67
1 :3 Lattice .145 Boron atens(103
3:1 Lattice. .512 Boron atoms/LOS vater molecules
L Residual (an)
Table 6. (Continued)
Numiber of Rods Relaxation Longth L(em) L Residual (cm)
4:1 Lattico, .146 Boron atoms/103 water molecules
265 18.895 -*008
229 17 .626 .033g
193 16 .203 .022
157 14*571 -.136
121 13.202 .091
4:1 Lattice, .218 Boron atoms/103 water molecules
265 17.097 .020
229 16.187 .064
193 15045 -.001
157 13.745 -.114
121 12*595 *073
Buckling and Reflector Savingeo qe a Function of Boron Concentration
Boron C neentration
Lattice (B atoms/10~ 190 molecules) 82 -2) x 103 Acam)
1.5:1 .216 2.164 i .082 7 .61 i .18
1*5:1 563 1.107 a .217 7.61 i .46
1*S:1 .860 .24;3 t .05 7.75 2 .12
2:1 *359 2.045 i .255 7 .45 .67
2:1 .590 1.212 *314 7.11 .78
2:1 .826 .326 i .094r 7.32 t .22
3:1 .174 2.086 i .052 6.79 i .20
3:1 *345i 1.137 1 .057 6*37 -+ .19
3:1 *512 .019 ;r .049 6.59 t .17
4:1 .074 1.103 i .050 7.16 i .26
1:1 .146 .596 i .067 6.60 *30
4:1 .218 -.054 f .070 6.81 i .32
IlhY1__ I_ YC
Values of the Migr a
Determined by the Poisoned-Lattice Method
(atomes per 1000
Lattice molecules of water)
(an" x 10':
30.06 1: 1.21
28.69 & 1.23
28.47 & 1*413
Preliminary Values of 6
Results of Test of Axal Hensasuremnts of Reproducibili~ty
The poisoned lattice measurements were performed on the 2:1, 4:1, 1.5:1,
and 3:1 lattices, in that order.
Immediately after the 2:1 poisoned lattice measurements, a single axial
measurement was carried out with the Ic:1 lattice and pure water. Use of the
measured rdiaxation length and the reflector savings from reference 1 gave a
value of B2 = 1.816 x 10"3 om-2, compared with the previously measured best
value of 1.79 x 10
Lanediately after the 1.5:1 poisoned lattice measurementJ a single clean
axial was measured with the 3:1 lattice. The value of B2 thus. indicated was
3.23 x 10 3, compared with. the best valuel of 3.287 x 10
Just after the 3:1 poisoned lattice runs, a clean axial flux measurement
was made with the 2:1 lattice. This gave B2 = 3.62 x 10"3s compared with the
best value of 3.75 x 10-3,
Within the accuracy of the measurements, these results imply reproduci-
bility of the data throughout the course of buckling and migration area mueasure-
VI. Intracel Flux Traversea.
The values of the migration area we have reported earlier depend on cal-
culated values of the thermal utilization f. Since there is some uncertainty
about tbo theoretical method used to find these values, an attempt at measuring
them seems useful. He have used the direct approach of measuring the thermal
neutron flux distribution in the water and in the fuel raday thus thle values of
f we obtain are still uncertain by an amount depending on the crosa sections used
and hence on the assumed neutron temperature.
Evaluation of f of courao also has motivation from the desire to check
experimentally the methods need to calculate k through the four-factor formula.
B. Ejrperimental Procedure.
Intracell flux distributions were measured for l*5:1, 2:1, 3:1, and 4:1
water-to-mretal volurme ratios. These are actually ration of water plus aluminan
to uranium. Because these lattices are so tightly packed, it was necessary to
nee very small detector foil, not only in the roda but also in the water, if
any detail in the flux distribution was to be obtained.
~The foiled used were 1*5 miLImbeter diameter diace of lucite containing
dysprosium oxide, about *5: millinueters thick. D~yaprosina vas considered ideal as
a detector, because its lack of low energy resonances makes possible the measure-
muent of thermal fluxes without the need for making cadmium difference measurements.
Foils of this type have been used by the reactor physical group at Argonne, with
The foils were punched out of a small sheet masde in a hot press from a
powder mixture of lucite polymer and dysprosiumn oxide. Because the activity de-
aired (T 140 minutes) was not reported in the literature with an accuracy
sufficient for our purposes, a measurement of the half-life was carried out. The
procedure and result, 51/2 = 139.17 14 minutes, are reported as a letter to
the editor in the Physical Review. During the course of this measurement, it
was found that no other activity which would interfere with the one desired was
present in detectable amounts. Mass spectrographic analysis of the dysprosiuml
showed only trace impurities, due mostly to,other~ rare earths.
Flux traverses in the uranium were made using a split fuel rod, with nine
Holes of about 1.6 millimeter milled on each of two diameters. A drawing of a
iicross section of this rod is shown in Fig. 10. From the figure it is seen that
the foils are quite close together compared to their diameters. They are re-
placing a medium (uranium) with nearly the same absorpjtions however, and so the
influence of neighboring foils on each other is small.
The use of finite-sized foil of course introduces an error arising from
the fact that the flux is not constant over their area. For tihe foil aises used
and the fluxes observed, it can be shown that the observed activity -can be at-
tributed to the flux at the fail center, with an error of at most .2 per cent.
Since corrections for this effect would be anall compared to errors in measurement,
they have not been made.
The foila used in the water were positioned in a piece of lucite which
ran through the lattice. The horizontal positions of these foils relative to those
in the fuel rod were known to be within about five mils (.013 an).
Foils La the uranium were not placed at the same vertical level as those
in the water. The difference in elevation was -measured, however, and the known
axial relaxation length made it possible to correct the observed activities to
those they would have had if they had been at the same height. The error intro-
duced by this procedure is considered negligible.
Each foil used was counted to about seven thousand counts in each of six
end window counters. Thus there vae no need to make counter efficiency corree-
tions, the activity of a given foil being determined from the total counts ob-
served in all counters.
In all, six complete intracell flux traverses were measured, one in the
1.5:19 one in the 2:1, two in the 3:1, and two in the 4s1 lattices. an additional
partial traverse was also obtained (in the metal) in the l*Ssi lattice.
C. Analvaia of Data ga :Results.
The experimental data obtained from the intracell flux traverses are shown
in Figs. 11 16. In view of the triangular syimmettry of the lattice the neutron
flux in the moderator was measured along; two lines, the first joining rod centers
and the second along: the median of the triangle. The most complete moderator data
was obtained at the higher moderator to fuel volume ratios. Only a few points
could be obtained at the 1*5:1 volume ratio because of the tightness of the lattice.
The flux data uco the fuel rod were first fitted to the Bessel function
lo(Ko~r) where r is the radial position of the foil from the rod center and Ko =
1/Lo is the reciprocal diffusion length. The function lo(Kior) represents the
symptnotic form of the flux distribution in the fuelsl but in order to obtain a
good fit of the flux data, a value of Ko considerably higher than the theoretical
value must be assumed. The results of the least square fit of the data are given
in Table 10 below.
Bessel Function Pit of Flux Traverse in the Uranium Rod
Ratio Ko (in-1) L, (an) F
l*5:1 2.970 0.855' 4*7 per cent 1.147
1.Fr1 2. 818 0.901 ~t6.0 per cent 1.133
2:1 2*585 0.982 8.0 per cent 1.113
3:1 2.659 0.955 5.4t per cent 1.119
3:1 2. 644 0.961' 7.0 per cent 1.118
4ll 2.988 0.850 54 per cent 1.149
4:1 3.152 0.806 2.9 per cent 1.165
The ratio F of the flux at the surface of the fuel rod to the average
flux in the rod is given in the final column of Tablo 10. Thle values do not
very significantly with moderator to fuel volume ratio. The average value of
F=1.135 which corresponds to Ko = 1.116 em1 ausdipltnghefu uv
lo(Kor) for the fuel rod in Figs. 11 16.
The values of F obtained from the BNL experiments ear much higher than
would be expected from elemnentary diffusion theory. These results are in con-
fornity with those obtained with uranium rods at other laboratories, notably
at North Anerican Aviation. Kidde has estimated the thermal utilization of our
water lattices (HKF-lk92D-151, Mar. 4J 1952) by the use of an elementary diffusion
theory formula with the value of F corrected by means of a spherical harmonics
calculation. Their estimates are given in Table 11.
Vol, Ratio _g
These values of f were used in the preliminary estimates of th~e migration area
(Section V of this report).
The theoretical flux curves in the water obtained by the use of elementary
diffusion theory are shown in Figs. 11 16,1 corresponding to a value of F = 1.135.
The lower ourve in each figure corresponds to the currently acceptedl value of L1 3
2.85 cm for the diffusion length in water. On the basia of these flux dlistribu-
tions the relative absorption of thermal neutrons in fuel, cladding and moderator
is shown in Table'12.
Thermal Neutron Absorption in a Unit Cell
Ratio f fal ~fmod
1:1 .9169 .00)50 .0781
2:1 .8873 .0048 .1079
3:1 .8298 00045 .1657
4:1 37759 .0042 .2199
Although the values of f agree with the K~idde estimates it may be seen tbat the
theory greatly underestimates the neutron flu~x in the inderator. The reason for
this discrepancy is clear from the exp~erimen~tal flux data. Extrapolation of these
data to the moderator fuel interface would indicate a discontinuity in flux in
this region. The apparent discontinuity arises fran the neglect of the non-
asymptotic solutions of the transport equation which are important near such an
The required refinements of reactor lattice theory are not within the
scope of this report. However, to obtain a better fit of the experimental neutron
flux curves in the moderator and hence more realistic estimates of the thermals
utilization, we need to modify the value of the diffusion length 1 lsi the miod-
erator. Thle modified diffusion theory curves are also shown -in Figa. 11 16.
The corresponding values of the thermal utilization are given in Table 13.
.iodified Values of the Thermal Utilization
Vol. Ratio f
C ~1*5:1 .910o
These values of f are preliminary and are being refined by a more exact analysis
VII. The Effect of Absorber Rods.
The effectiveness of a central absorber rod in lattices of the type under
investigation was studied in another set of experiments. Cadhnium tubes of dif-
ferent dianmeters were tested; the change in the axial relaxation length caused
by replacing the central uranium rod by an absorber rod was measured. Hollow
cadniurm rods of different diameter were used; the wall thickness was 0.056 indb
in each casse. In m~ost of the experiments, the rod was filled with water; in one
case the rod contained a steel cylinder. The fractional change of the axial
buckling was also calculated fran the relaxation lengths.
To test the effectiveness of an absorber rod as compared to that of the
removal of a fuel rod, the central fuel rod in the 3:1 lattice was removed, and
the change in relaxation length measured. The value of the relaxation length
was 32*32 a~n with 254~ roda in the lattice; when the central rod was removed, theo
relaxation length decreased to 32.25 em, or a decrease of 0.07 an. This decroase
Corresponded to a fractional decrease in the axial buckling of 0.0)022 r 0).0017, or
about 0.2 per cent. In the other experiments, the change in relaxation length
corresponds to the replacement of the central fuel rod by the absorber rod.
The results of the experiments are given in Table 14. Estimates of the
precision has been shown in two cases, which seaned to be typical. No attempt
has been made at BNL to treat the data from a theoretical standpoint.
The Effect o~f Absorber Rods on the Asial Rolaxbtion Longth
Rod Outer Dieamter of Type of Relazation Fractional Change in
Lattice Loading Cadniumn Tube (in) Filling Length: L(on) the Azial Buckling~
1.5:1 255 No tube --- 18.96
0.81 H20 16.93 0.253
2:1 263 No tube --- 26.89
0.95 H20 23.27 0*335' o .010
3:1 254 No tube --- 32.32
0.81 H20 26.77 0.461
253 No tube --- 32025
0.65 R20 26.88 0.439
1+:1 253 No tube --- 23.15
0.95 "20 21.11 0.203
0.81 H20 21.71 0.138
263 No tube --- 25.18
0*75 Steel 22.40 0.2522 f 0.004,
VIII. The Effect on the Agial Relaxation Lonath of a "Gap" in the Lattice.
In the design considered by the Walter Kidde Co. for a power-plutonium
reactor, the lattice is to be divided into several groups of rods. WJhen the
fuel is ready for processing, the sub assanblies are to be separated before being
removed fran the reactor. In this case, it is desirable to know what change in
reactivity may be expected when two sub-assemblies are separated, and a series
of experinent~p was made in order to obtain some information of this kind. In
these experiments. the relazatioin lengths were measured for two different arrange-
ments of 160 rods; this was done for each of the four usual lattices. In the -first
arrangement, the 160 rods were in a rootangle with 16 rows, each containing 10
rods. To form the second arrangement, tho rodls in one of the two central rows
were removed and placed at the end of the rectangle. This resulted in 2 rectangles
each of 8 rows with 10 rods per row, the two rectangles being separated by one
row without rods. The second lattice may be regarded as a split lattice separated
by a water gap. The axial relaxation length was measured for each arrangement;
the results ear listed in Table 15. The fractional change in the exial buckling
is plotted against water-to-metal ratio in Fig~. 17. The fractional change in
the axial buckling is defined as
L2 12 L1
where the subscripts refer to the two arrangements.
Gau_ on the Relaxation Lenp~th
Effect of a Central
Fractional Change in
in Agial Buckling:
0.0367 (Gain in
In three of the four lattices the effect of the separation was to de-
orease the relaxation length (decrease the reactivity). In the tightest lettica,
1*5:1, there was a slight, about 2 per cent,, increase in relaxation length, indi-
cating that the separation caused a alight increase in reactivity.
Si~ ,~, *i
i f;g :
Fi@lre 1 Ficture of Tsn~. Ne-ati,e ~~o. ~-?15-2
The inner tank with a tube plste -nd fuel rods.
TYPICAL RADIAL FLUX TR AV ERS E
IN 2:1 LATTICE, SHOWING FLUX
m RISE NEAR REFLECTOR
|~~ | 1 I
O 0 0O
I I I I |
REFLECTOR SAVINGS VS. WA~TER-TO-
METAL VOLUME RATIO
.75" RODS OF 1.027% 25
URANIUM, LIGHT WATER MODERATOR
WATER-TO-METAL VOLU ME RATIO
CD ~O M (U
QIIVt~ ~nlWQ~3 -51-
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b ~L b~Lb
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FOIL POS TIONS
FO R INTRA CELL FLUX TRAVERSE
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