>./4-t-7.-ri^y VL - 20^ ASSIFIED i UNCLASSIFIED BNL-2094 Subject Category: PHYSICS UNITED STATES ATOMIC ENERGY COMMISSION EXPONENTIAL EXPERIMENTS ON LIGHT WATER MODERATED 1 PER CENT U-235 LATTICES By H. J. Kouts J. Chernick I. Kaplan November 28, 1952 Brookhaven National Laboratory Upton, New York Technical Information Service, Oak Ridge, Tennessee Date Declassified: October 31, 1955. This report was prepared asa scientific account of Govern- ment-sponsored work. Neither the United States, nor the Com- mission, nor any person acting on behalf of the Commission makes any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the in- formation contained in this report, or that the use of any infor- mation, apparatus, method, or process disclosed in this report may not infringe privately owned rights. The Commission assumes no liability with respect to the use of, or from damages resulting from the use of, any information, apparatus, method, or process disclosed in this report. This report has been reproduced directly from the best available copy. Issuance of this document does not constitute authority for declassification of classified material of the same or similar content and title by the same authors. Printed in USA, Price 1^0 cents. Available from the Office of Technical Services, Department of Commerce, Wash- ington 25, D. C. BKL-209^ Exponential E xperiments on Light Water Moderated, 1 Per Cent U-235 Lattices Report Written by Work Done by H. J. Kouts H. J. Kouts J. Chemiok K. Downes I„ Kaplan R. Sher G. Price V. Walsh November 28, 1952 Work performed under Contract No. AT-30-2-Gen-l6 Brookhaven National Laboratory Upton, New York ill Digitized by the Internet Archive in 2012 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://archive.org/details/exponentialexper9891broo Contonts I . Introduction • 2 II » Experimental Facilities. 2 III. Buckling Measurements. 6 A. Theory 6 B. Experimental Techniques 8 C. Analysis of Measurements and Results 11 D. Cadmium Ratios 13 IV. Anisotropy Measurement. 13 A. Introduction !3 B. Theory 1^ C. Experimental Methods 16 D. Results !7 V. Migration Area Measurements. 27 A. Introduction 2 T B. Theory 27 C. Experimental Methods 28 D. Analysis of Data and Results 29 VI. Intracell Flux TraverseSo 38 .*. Introduction B. Experimental Procedure C. Analysis of Data and Results 38 38 VII. The Effect of Absorber Rods. ^3 VIII. The Effect of a "Gap" on the Axial Relaxation Length. +5 Exponential Experiments on Light Water Moderated. 1 Per Cent U-235 Lattices I. Introduction . At the request of the Division of Reactor Development of the U.S.A.E.C, the Physics Division of the Nuclear Engineering Department at Brookhaven under- took a progran of exponential experiments on ordinary water moderated, slightly enriched uranium lattices. The purpose of the measurements was to provide nuclear design information for the core of a reactor for the production of plutoniun and power. In particular, nuclear data were to be provided on the 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, the program of the experiments was planned in close cooperation with members of the WKNL and cooperation was maintained during the couroe of the experiments. The present report is a summary of the exponential experiment measure- 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- cribed include measurements of the critical buckling, the migration area, intra- cell flux traverses, effectiveness of control rod materials, the effect of re- moving uranium rods from the assembly, etc. II. Experimental Facilities . The facility for the exponential assembly was planned with great care in view of the difficulties met in earlier experiments on ordinary water moderated- natural uranium lattices at Oak Ridge • 1. CP-2842. 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 shield. The thermal column consists of five one-foot thick layers of graphite, stepped from a 4 1 x 4' size at the bottom to 5 1 x 5' at the top. This stepped construction prevents fast neutron leakage up the sides of the thermal column, and the well- machined surfaces of the graphite blocks which make up the structure provide tight packing which 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 10-*. 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 cylinder, four feet in diameter D 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 armor plate which holds the circular aluminum top tube plates. The inner tank with a tube plate is shown in Fig. lo The tube plates are made of 2S aluminum, the upper plate being 1«5" thick, and the lower plate •75" thick. In each are drilled 275 holes for the uranium rods, and 20 holes for the insertion of foils. Care was taken to locate these holes accurately, the tolerance being 0, -»5 rails for each hole diameter, and -5 mils overall for the distance between holes. Thes« 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 2 with conditions which could be trusted * The uranium was prepared at K-25, and was rolled into rods of e750" dia. at the St. Louis plant of the Malinckrodt Chemical Corporation. The Brookhaven metallurgy group then straightened the rods to within 15 mils lateral deviation in four feet of length, and clad them with 30 mil thick aluminum. Most rods are single four-foot lengths of uranium. A few are slightly under four feet in length, and a few are made of shorter sections matched to provide approximately four foot lengths. Analyses of the uranium composition were made at Oak Ridge. The average composition is 1*027 per cent - oOOl por cent by weight. Distilled water purchased commercially 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- ments. For a large part of the time the water contained boric acid in solution. At all times the tank contained cadmium from the shutter (see below). There was always the possibility that boron or cadmium coatings might form on the aluminum cladding of the rods, thus changing the reactivity. For this reason, particularly, reproducibility of the data was held to be important. This reproducibility was maintained throughout the experiments. Best results from an exponential 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 reference l) . -A- Best results from an exponential experiment are obtained if the source of thermal neutrons feeding the reproducing lattice is made about the same size as the lattice itself. In this event the decay vertically of the flux harmonics is most rapid, and the region of flux harmonics corrections is smallest. For this reason the bottom of the large water tank was covered with a cadmium sheet having a central circular orifice. The hole- in the cadmium had a radius about 7 3m larger than that of the maxiumum uranium loading. This defines the source size most suitable for rapid decay of harmonics, 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 cadmium covered aluminum plates was designed to fit under the lattice assembly, to make possible turning off the source flux when irradiations were not being made. Because of corrosion and working under the weight of the uranium and support assembly, this shutter system never worked prop- erly. In practise, the source flux was always turned off by the dropping of a cadmium 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 measured. No temperature coefficients have yet been measured, but it is planned to determine some in the near future. III. Buckling Measurements » A. Theory. 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 (i) 2 q * B 2 q> = o, end the critical equation is simply (2) k^= 1 + W 2 B 2 The standerd 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: n=l where r and z are cylindrical coordinates in the lattice . X is the reflector savings, and a n are the successive roots of U) JoW ■ The coefficients f n have the form (5) % ■ z' . Ordinarily, one would measure the radial and axial flux distributions in the region of validity of (6). The radial distribution would be fitted to J (ai r/r ), the axial distribution fitted to C]_ exp jz/Lj+ D^ expj-z/LJ , and the buckling would be (7) B 2 = ( ai /r ) 2 - 1/L 2 . Unfortunately, the small loadings we were working with did not permit accurate measurements of the radial flux distribution. Symmetry conditions made only five positions available for flux measurements on any one radius, and the 3 outermost of these was so close to the essentially infinite water reflector as to make it useless. The effect of the reflector can be seen from a typical radial traverse, shown in Fig. 2. Thus only four points were available along any one radius, and a curvo- fitting to these points was not accurate enough for our purpose • 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 cm 3- The dacision to use reflected lattices v/as made mainly because of the small amount of uranium available for the experiment. The reflector savings was equivalent to about 0.7 tons of extra uranium. 4.. The lack of accuracy does not arise simply from the fewness of points. Rather, it is caused by the fact that these points are on the flat part of the J curve, and smell experimental errors in the values of the foil activities cause large variations in the extrapolated end-point. -7- upward from the bottom of the lattice. Moreover, the end corrections were small below ebout 70 cm from the bottom of the lattice. Thus there existed about 60 cm of variation of z over which the flux decay was simply exponential, and so quite good measurements of L could be made. We therefore decided to try basing measure- ments of B on measurements of L elone. For such a determination of B , one would measure L for a wide range of rod loadings. Each such loading would be idealized to a cylinder with the seme loaded radius the case would have if the uranium were uniformly distributed et the same water- to-metal ratio. Least squares fits would then be made to X and B^ in (8) ( ai /R + X) 2 - 1/L 2 = B 2 . Such an analysis contains of course an assumption that X and B^ have the same value for all loadings used. This is perhaps no worse than the assumption ordinarily made in an exponential experiment on an inhomogeneous lattice - that the results can be interpreted in terms of fudged calculations modeled after homo- 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 thst the assumption is probably not far wrong. This problem is also being in- vestigated from a theoretical viewpoint. B. Experimental Techniques. The measurements of neutron flux distributions wore made with indium foils, 0.220" in diameter end five mils thick. Each foil was counted in from three to six -8- counters, to a total of more than ten thousand counts, (1 per cent statistical accuracy) per foil. Counter resolving times were found by the two- source method, and in all counts resolving- time corrections were kept below 3 per cent. We believe that in this way we have kept to about .1 per cent of the total counts per foil the error introduced by having to make resolving time corrections. Count times wore controlled by a preset timer which was accurate within .3 seconds (at most .5 per cent per count) , about half systematic and half random. The estimated error due to timing Is thus ebout «25 per cent. The overall estimated error in the activity of an individual foil is then about 1.3 per cent about the mean, when the conditions for accurecy are worst- The foils wers positioned in foil rods inserted through the top tube plate. At first thin-walled aluminum tubes were used, with slots to hold the foils punched at ten centimeter intervals. Tho foils 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 maximum error in vertical positioning of the individual foils was about 1 mm. For the shortest relaxation length we measured, the maximum error in foil activity measured was thus about 0»7 per cent from this source. Each foil was 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 cement was avoided, end the replacement rate of foil rods, which had been large because of bending of the aluminum and rapid enlargement of the holes -9- for foils, was materially reduced. Because of large local variation of flux, it was necessary that all foils be centered accurately in a lattice cell. This 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 i>ig each. Their weights averaged 0.027 gms, with not much 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. Matched sets of foils, with weight variations limited to - .1 nig, were used at every stage. The esti- mated error caused by individual variations in foil weigha was at most about .05 per cent. This contribution to the total error is negligible. The value of the indium half life was taken as 5A»05 minutes for foil activity calculations. At the end of the buckling experiments a careful determination of the half-life was made, and gave a result T^/2 = 5A»M * »07 minutes. The error in all the bucklings introduced by this discrepancy would be about 0.3 per cent. A measurement of the indium half-life was suggested to us by A. Wattenberg, who pointed out the leek of agreement in previously published values. The compounding of all these errors indicates inaccuracies of individual foil activities of at most 1.5 per cent about the n&sn. Statistical variations in 5. The variation with weight of the weight correction to the activity was measurod. It was found that a 1 per cent weight variation was ecconpanied by only a 0.2 per cont chenge in foil sensitivity. This is much less than the direct pro- portionality usually assumed. •10- the aieasured quantities were consistent with this estimated error. C. Analysis of Measurements and Results. The relaxation lengths listed in Table 1 are obtained froia least squares fits of exponentials to the axial flux traverses. Water- to-metal ratios of 1.5:1, 1.75 si, 2:1, 2.5:1, 3*1* ?nd 4.sl are reported. These ratios are really those of water plus aluminum to uranium. Also given in Table 1 are the residuals of the 2 least squares fits of B and X. Table 2 gives the values of B* and X which result from the least squares fits. Fig. 3 is a plot of B^ vs water- to-metal ratio from 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 methodo The effect of having ignored end corrections was considered. To determine the magnitude of this effect the following procedure was used. Values of L were calculated from flux measurements over a range of 60 cm, beginning 10 cm from the 2 bottom of the lattice. Least squares fits of B and X were made for this set of data. The calculation was then redone leaving out the points at 70 cm from the bottom of the lattice. Since end corrections are greatest at this point, the changes in computed values of B and X may be attributed to them. Small changes in B^ 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 -11- values given in Tables 2 and 3 refer to the six point axial attenuation measure- ments, which have the smaller end effect corrections* At no stage were any effects observed which might be attributed to the presence of the harmonics 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 heve been obtained. The best radial traverses were made with the 3 si 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 were found by fitting the measured foil activities to J (a]_ g ). The values of the reflector savings for the two radii were then found as the difference between values of R Q ff and the actual distance from the lattice center to the edge of the aietal loading. The average of six radial traverses gave in this way a value of 6065 - .5O cm for the reflector savings. This is to be compared with the value of 6.94 ~ »H c 21 de- termined by the method of axial bucklings only. The agreement of these values to within three millimeters is strikingly good. A single measurement of the reflector savings was carried out in the 1.5.1 lattice, in connection with the anisotropy measurement discussed later in this report. The measured value of X in this case was 7.6? era, with somewhat uncertain accuracy. The value given in Table 2 is obtainod by the method of axial bucklings is 7 .71 - .14. cm. Agair. the agreement is exceptionally good. -12- As a further check on tho validity "of the assumption of constant B 2 end X, the buckling for the 3 si lattice was recomputed, using only loadings betwoen 181 and 263 rods. B and X wore unchanged within the experimental error roportod in Table 3. Thus the experimental data available tends only to bear out the accuracy of the assumption underlying the aethod we have used. It would still be interest- ing to see the results of critical experiments made on these lattices. If the slowing- down and diffusion of neutrons does not take place iso- tropically in tho reactor core, the exponential experiment must be modified con- siderably. The effect of such anisotropy has been discussed by Young and Wheeler ; they showed that if the anisotropy is large it is possible to draw grossly in- accurate conclusions frou a simple exponential experiment. D. Cadmium Ratios. To a certain extent, information on the energy distribution of low energy neutrons may be dotemined fron the cadniun ratios. Therefore accurato uocsure- uents of tho cadniun ratio were made at the center of a lattice coll, using indiun and gold foils. The indium foils wore five nils thick; the gold foils were 1.2 mils thick. Measured values of the cadmiun ratio are listed in Table 3 and plotr- ted in Fig. 5. IV. Anisotropy Measurement. *>. Introduction. 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 -siigration area of fission 6. C-90c _ 13 _ neutrons in water alone. Thus it appears that because of high inelastic scatter- ing, uranium in water may be considered quite closely equivalent volumewise (on the average) to the water it displaces, for the purposes of nGutron slowing-down. It might be suspected that this condition makes the existence of anisotropic ef- fects unlikely. Nevertheless, the situation soouod to warrant a search for such an anisotropic character of tnc Migration aroa. Young and Whoolor also roportod and analysed the suggestion by Wigner that a "doublo exponential experiment 11 would yield the dogree of anisotropy. It is such a double exponential experiment which wo 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. B. Theory . For the general theory of the double exponential experiment, one should see the paper by Young and Wheeler. We give here a discussion suitable for ur purposes. We consider our subcritical lattice in the shape of a rectangular paral- lelepiped, with edges Jl ~ Ji'> JL„* The fuel rods are supposed to lie in the z direction. If the thermal neutron source activating this array is placed at one end of the lattice (the side with odges X- x and JL ) the one-group critical equation will bo (9) ^- 1 = I "^ B x- B y } m A B z • k is of courso the infinite pile reproduction factor e ■H- (10) B x = B y" = —f- 2 7 U x /2 + X) 2 (11) b 2 - iA 2 • L is tho relaxation length for decay of the fundamental in the z direction, X is 2 2 the reflector savings, ond M end M have physical meaning associated with the H ■* mean square distanco a fission noutron travels in directions parallel to tho rods and normal to tho rods, respectively. If, on the other hand, the source is placed on one of the sides of the lattice (that defined by y and z), wo shall havo in place of (9) (12) k^-1 = M 2 b 2 -*M^ (bj- b 2 .) whera (13) l 2 w z b z - (iz/2 + X) 2 b z " :E 2 y ■ V(L') 2 • (14) (15) L' is now the relaxation length of the fundauentel in tho x direction, neasured in this geometry. From (9) and (12) we obtain li l (B x + B P " M „ B i = M « b z + M f ( b y " b x> or (16) M* /fcj = ( B 2 + b 2)/ (B 2 + tj) . In practise, we took the first geometry to be cylindrical, so that instead of (9) one has •15- (17) V-l = MfBf - V$l with (18) B* = (a-j/R+A) 2 , a-i being tho first root of the Bessel function of order zero, end R being tha radius of the loading cylinder. Tho relation which corresponds to (16) is de- rived frou (12) and (17): (19) ii? AlJT = (Bf + b|)/(B2 + b 2 - b 2 ) . C. Exporiuental Methods. 2 2 B and B£ can be found froa tho buckling noasur orients in cylindrical geometry described earlier. Thus to find tho anisotropy it was only necessary 2 2 2 to carry out tho socond uocsurciient abovo - tho determination of bj, by and b z « For this purpose a set of lucite support pieces 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 machined holes in tho surface of the lucite support structure. The cadmiun which defined the shape of the thermal neutron source had cut in it a rectangular hole about 14 cm larger on each side than that of the array. This source shape was most suitable for the appearance only of the fundamental o The gold foils which were used throughout wore <>220 inches in diameter, and were about one mil thicke They wore chosen as a set matched in weight to about .5 per cent. The foil counting techniques are the same as those described earlier. -16- D. Rosults . Equation (17) doos not refer to any particular redius of the lattice, and 2 2 as a result B z and B, may be taken in (19) for any loading. We have chosen rather arbitrarily to base the analysis on the 235 rod loadings, the results of which are given in Table 2<. Here, L is 17»72 cm by measurement. Analysis of the least squares methods used to obtain L show an expected accuracy of about * .10. We allow for each of the two measurements double this error, so $ L (the error in L) is assumed about - .1^. The loaded radius at 265 rods is 23.087 cm. The reflector savings given by buckling measurements is 7«71 cm. Although this number is here being questioned (its derivation is based on use of the isotropic critical equation rather than on (17)), it is doubtful that it is wrong by more than - .5 cm. Thus we may take R+X = 30.80 - ,50 cm with some confidence. So far we have (20) . Bg = 1/L 2 = 3.18 x 10" 3 * .052 x 10" 3 cm" 2 B 2 = (a x /R+X) 2 = 6 o 09 x 10~ 3 ± 0O55 x 10~ 3 cm" 2 . With the lattice placed horizontally, a set of seven foils was placed in the y direction, to provide a determination of the flux distribution in that direction, a least squares fit to A cos bx gave b = .O5I3 - 0OO4.6, or JL /2 + X = 30.62 cm ± 2,75 ca. Since £ y = 32.95 en, we have X = 7<-67 cm ± 2.75 cm. In- spection of the least squares fit to the plot of tho flux distribution givon in Fig. 6 shows that the cosine curve was actually shifted somewhat from- the assumed center of tho lattice. If all foils are translated slightly in tho same diroction, the error is considerably reduced. During the measurement such a displacement seems to have occurred. Wo prefer therefore to accept the above value of X, but -17- with an error closer to about .5 cm. Thus (21) h 2 . - 2.63 x 10' 3 * .043 x 10" 3 cu~ 2 . The flux values obtained from this traverse are listed in Table 4» 2 Also, b z can bo obtained accurately enough from just the rod length (four feet; and this reflector savings; wo have then (22) b 2 = .523 x 10" 3 * .004 x 10" 3 cm" 2 . 2 The thirteen foils which were used to measure b^ seemed to separate into two groups, the symmetry of the triangular lattice cell in which a given foil was placed determining its group. Those in lattice colls 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 by the same value of L 1 . Least squares fits to each of A explx/!']- B oxpl-x/LM agreed fairly well with L 1 = 100 ± 30 an. The largo error is caused by the presence of sizeable end corroctions, the thickness of the lattice in the x direction being only 4.5«90 cm. The results of the x traverse are given in Table 5 and shown in Fig. 7« From the value of L 1 wo have (23) b 2 = 10~4 * 10" 5 cm" 2 . Substituting the values from (20), (21), (22), and (23) in (19), we obtain (24) "x^J = 1# ° 39 ' An estimate of accuracy can bo obtained by differentiating (19) . Putting for the inoaent 1=M 2 /M 2 we get -18- dB 2 + db 2 -5(dB 2 + db 2 - db?) dl = B I + b x " *p The uncertainties in the various values can be considered as random; the un- certainty in 5 is then given by ( 2 5) A * - .2 v 2 B, + b. {(dB 2 ) 2 * (db 2 ) 2 - 3 2 [(dBf) 2 * (db 2 .) 2 * (db 2 ) 2 ]} 5 1 + b x - 4 .1/2 Insertion of the uncertainties given xn (20), (21), (22), end (23) gives (26) Al ~ .028 The anisotropy indicated by (24) is just barely outsido 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) fr = * B ~ B z with a thre.e constant fit of "5, X, and k-l/M^ to measurements of B 2 as a function of loading Tho agreemont of values o.f X obtained from the two constant fit with the values obtained from radial traverses indicate though that such a procedure would also show negligible anisotropy. -19- Tablo 1. Relaxation Lengths: L (en) In Water-Slightly -Enriched Ursniun Li jttices Residuals of L (en) No. of Rods L (1st run) L (2nd run) 1st run 2nd run 1. 5:1 Lattice 271 19.559 19.793 -.156 .126 253 18.733 18.492 .042 -.132 235 17.722 17.723 .009 .093 217 16.892 I6.564 .121 -.111 199 15.865 15.741 .006 -.012 181 15.154 14*896 .183 .040 163 H.143 13.997 .043 .020 145 13.200 13.129 -.037 .020 127 12.271 12.259 -.104 .014 109 11.282 -.094 91 10.435 -.191 73 9.665 .095 55 8.801 1. 75:1 Lattice .071 253 21.935 22.168 235 20.287 20.391 217 18.811 18.757 199 17.666 17.744 181 16.285 16.623 163 15.215 15.237 145 14.257 14.203 127 13.118 13.062 109 12.071 12.134 91 11.060 II.O45 73 10.012 10.030 .097 .161 -.026 -.057 -.105 -.268 • 044 .034 -.125 .143 -.045 -.080 .097 -.002 .024 -.068 .023 .060 -.037 .019 .065 .068 •20- Toblo 1. (Continued) Residuals of L (en) No. of Rods L (1st run) L (2nd run) 1st run 2nd run 2;1 Lattice 271 29.087 -.188 265 253 235 217 199 181 169 163 145 127 109 245 235 217 199 181 163 145 127 109 91 73 27.297 -.122 25.955 26.539 .262 .282 23.424 23-947 -.010 .180 21.551 21.510 .080 -.135 19.631 19.612 -.100 -.180 18. 252 .113 16.815 -.372 16.781 .145 15.109 14.905 -.266 -.341 14.557 14.102 •453 .156 13.028 2.5 Jl Lattice .145 30.472 30.021 .352 -.011 27.800 28.091 -.312 -.097 24.737 25.062 -.332 .041 22.886 22.763 .333 -.032 20.330 20.688 -.073 -.037 18.761 18„U4 .249 -.152 16.950 16.867 .137 .214 15.147 15.317 -.108 .154 13.793 13.932 .002 -.037 12.341 12.400 -.060 -.003 II.O58 11.119 .022 -.045 -21- Table 1. (Continuod) Residuals of L (cm) No. of Rods L (1st run) 3:1 L (2nd run) Lattice 1st run 2nd run 265 36.647 36.774 -.153 .300 253 33.592 33.437 .145 -.180 235 29.448 29.460 .016 -.199 217 26.152 26.296 -.104 -.280 199 23.754 23.669 .134 •H5 181 21.577 21.419 .211 .139 163 19.547 19.438 .164 -.100 145 17.311 17.653 -.287 -.022 127 15.821 16.016 -.139 -.174 91 13.006 .051 55 9.933 4:1 Lattice -.127 267 24.845 25.521 -.347 -.021 253 24.396 24.333 22.808 .350 .017 -.019 235 22.546 22.888 -.105 .062 217 21.315 21.390 -.001 -.027 199 20.342 20.150 .307 .076 181 19.019 I8.58O .224 -.203 163 17.525 17.690 16.589 .15.999 -.060 .160 .284 -.307 145 16.016 16.237 -.379 -.069 109 13.936 .049 -22- Table 2. Results of Least Sauares Fits to B end X B 2 (cn- 2 )(x 103) X (en) Lattice 1.5:1 1st Run 2.85 ± .06 2nd Run AvorafiQ 2.89 - .05 1st Run 2nd Run 7.56 * .11 AverajTG 2.94 ± .05 7 .86 ± .14 7.71 ± .14 1.75:1 3.4-53 ±.034 3.486 i.053 3.470 ±0 033 7.16 * .08 7.15 * .12 7.16 * .10 2:1 3 «65 * .09 3.86 ± .07 3.75 ± .08 7.23 * .26 6.75 ± .20 6.94 * .23 2.5:1 3.700 ±.055 3.647 *.025 3.673 ±-048 6.81 * .17 6.99 ± .09 6.90 * .16 3:1 3.304 ±.028 3.271 ±.006 3.288 ±.018 6c82 * .11 7.05 * .09 6.94 ± .11 4sl 1.76 i .10 1.88 ± .04 ,1.86 * .06* 1 6.83 * .52 6.37 * .19 6.42 * .22 * ivercgos for this lattice are weighted because one measurement is much poorer than the other. ■23- Table 3. Cadmium Ratios Lattice Gold Cadmium Ratios Indium Cadmium Ratios 1.5:1 1.918 ± .052 2.576 * .oh 2:] 2.292 * .038 3.O44 * .16 3:1 2,834 ± .025 4.30 i .033 4*1 3.307 * .045 5.742 * «092 ■24- Tablo 4. Neutron Flux Measurements frou y- Traverse Distanco From Assumed Center (en) Measured Foil Activity (units arbitrary) 583.3 Loss' Fits (A; t Squares Residuals Hetivity) -15-776 17-9 -10.039 718.1 4.6 - 4.303 798.3 - 1.8 1.434 821.2 3.5 7.171 760,2 - 4.7 12.908 634.9 -11.6 18. 645 464.7 - 8.0 * The residuals are those resulting from fitting column 2 to A cos Bx« The best values, on which the residuals are based, are A = 819.9 * 28.0 B = .0513 ± .OO46 cm" 1 -25- Tabic 5 G°OUP 1 GROUP 2 Neutron Flux Measurements from x- Traverse Di stance From Measured Lea: it Squares First Foil Foil Activity Fit Residuals (en) (arbitrary units) JA Activity) 0.0 1160 - 8* 4.970 1028 5 9.940 882.5 2.7 H-910 758.3 21.6 19.880 570.5 23.5 24.85O 446.9 - 4.5 29.820 316.6 6.9 0.0 1107 -15 4.970 962.9 20.4 9.940 Lost 14.910 652.1 .7 19.880 490.2 - 9.2 24.85O 350.7 2.9 *Column 2 was fitted to A o~ ax + B c +ax . Although the scattering of the few points made best values somewhat uncertain, theso are about A B a(cnf- GROUP 1 2083 -911.5 .010 GROUP 2 2162 -1041 .010 -26- IV. Migration Ares Measurements* A. Introduction. Tho values of the migration area are based on -measurements of the buckling in lattices with boron poi3onad water, as a function of the boron concentration. The theoretical basis for tho measurement is discussed in the next section. B. Theory. We may express k^ in two well-known ways, either by the critical equations (1) 1^ = 1 + B 2 (t + L 2 ) or by tho four-factor formula (2) ip af V P B 2 is tho buckling, t tho age to thermal of fission neutrons, L 2 the thermal dif- fusion area, so T + L 2 = M 2 is the -migration areaj f is tho thermal utilization, •^ the number of neutrons captured in uranium which cause fission (per neutron cycle) , £ is tho fast fission factor, and p is the resonanco escape probability. If the moderator is made more neutron absorbing by means of tho addition of a poison, those quantities in (1) and (2) which will be changed are 1^ , B 2 , 2 f, end L , tho others r«toaining constant; the main effwet of tho poison is to do- crease f . Hence, if the measured values of B 2 are plotted against f , a straight lino should bo obtainod, tho slope of which gives M 2 . For f, theoretical values calculatod by tho Kidde group, were used to get preliminary values of M 2 . These will eventually be corrected by the use of "experimental" values of f obtained from measurements of intracell flux distributions. -27- C. Experimental Mothods. A typical measurement of tho migration aroa involved dissolving succes- sively increasing amounts of B2O3 in the water, and measuring tho vo.luos of tho buckling of a lattice in these poisoned moderators. At each stago the boron con- centration was determined by the analytical group in the Brookhavon chemistry department. The bucklings were measured at throo differont boron concentrations for each lattice. When the measurements with unpoisoned water are included, the 2 dependence of B on boron concentration is thus found at four values of the boron concentration for each water- to-metal volume ratio. The buckling was found by measuring tho axial relaxation length as a function of the 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 bucklings which had to bo measured in a short tino. Thermal neutron fluxes were measured with foils of five mil thick indium, .220 inchos in diameter. Tho techniques usod to position foils in tho lattice and to measure their activities are deBcribod in Part III of this report. Because of the possibility that boron might plate out of solution and onto the aluminum cladding of tho rods, tests of reproducibility wore made when possible. Thoso consisted of neasurononts of tho relaxation longth in en un- poisoned lattice, and comparison of the buckling it indicated with previous measurements. The results of theso test axials arc given in Appendix 1. They load to the conclusion that no important change in reactivity of the lattice occurred during the buckling and migration area measurements. -28- An earlier effort to measure the "migration area by another means was un- successful. In this attempt, the uranium rods were covered with cadmium sleeves, to suppress neutron multiplication. A measurement of the thermal neutron flux distribution due to a known fission source was to have yielded the uigrntion area* Low values of the flux fron tho fission source kept this measurement from being successful. This method of determining the migration area was used at Oak Ridge in en oarlior investigation of this roactor typo. The Oak Ridge results differ significantly 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 squares fits to tho axial flux measurements, aro given in Tablo 6. Also given are the residuals resulting from least squares fitting to tho buckling and reflector savings. The 2 best values of B and X are listed in Tablo 7« The boron concentrations, calcu- 2 2 lated values of f, measured values of B , and deduced values of M are listed in Table 8, for water- to-metsl volume ratios of 1.5:1, 2:1, 3*1* end 4:1. The values of B^ when plotted qgainst f for a given lattice yielded good straight lines; tho value of M"^ was obtained by a least squares fit. A typical plot of B against f is shown in Fig. 8. Also shown in Table 9 are the values of k indi- cated by the values of tr and B^ for the four lattices. The values of migration area and k given by this report must 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 -29- assumptions about the geometry, the angular distribution of the flux, and the cross- sections. Tho value of the migration area as obtained from a poisoned lattice experiment is quite sensitive to tho values of f used and it appears that accurate moasured values of f must be obtained before the migration areas reported here can bo tuade trustworthy. A preliminary estimate of the "experi- mental" values of f for the unpoisoned lattices is given in section VI of this report. These values differ significantly from the values provided by Kidde, and theoretical work on this problom is in progress at BNL. For these reasons, the errors cited with the values of the migration area must be considered to represent only measures of the internal consistency of the experiments and the possibility of a systematic error must bo borne in mind. The most striking result of these experiments is therefore the apparent constancy of M^ over the range of values of the water- to-metal ratio studied. This result is quite different from that predicted by present theory, as can be seen from Fig. 9. When calculated values of L are subtracted from M^ (and L is small in these lattices) , the age is obtained. The resulting ages are plotted against water- to-metal volume ratios in Fig. 9, and compared with the theory of Soodak and Forman. -30- Tafalo 6. Measured Values of Axial Relaxation Length for Various Boron Concentrations. Number of Rods Relaxation Length L(cn) L Residual (cm) 1.5:1 Lattice, , .216 Boron atoms/10^ water molecules 262 16.898 -.023 229 15.686 .063 193 U.21i -.011 157 12.735 -0O85 121 U.430 .O56 1.5:1 Lattice. i .563 Boron atoms AO^ water molecules 263 U.960 .101 229 13.879 -0O54 193 12.823 -.090 157 11.710 -cl2A 121 10.828 ,160 1.5il Lattice, , .860 Boron atoms AO^ water molecules 263 13.677 -.002 229 12.976 .020 211 12.538 -.015 193 12.121 -.013 175 11.710 .013 2:1 Lattice, . ►359 Boron atoms^lCK water molecules 265 18.744 .073 229 16.750 -.254 193 15.288 .438 157 13.597 -.181 121 12.085 -.059 2:1 Lattice, , ■590 Boron atoms/10^ water molecules 265 16.367 .061 229 U.867 -»26l 193 13.899 .392 157 12.568 -.090 121 11.205 -.108 •31- Tsble 6, (Continued) Number of Rods Relaxation Length I -(cm) L Residual ( cm) 2:1 Lattice, .824 Boron atomd/lCp water mol( scules 265 1A.700 -.003 229 13.902 .O64 193 12.862 -.O48 157 11.8A8 -.057 121 IO.85O .061 3:1 Lattice, .17 A Boron atoms/10^ water moli scules 263 22.519 .O84 229 19.978 -.131 193 17.794 -.041 157 15.767 .086 121 13.579 .005 3:1 Lattice, .3AS Boron atoms /lO-^ water mol< scules 263 18.128 -.043 229 16.825 -.001 193 15*467 .089 157 13.910 .028 121 12.225 -.074 3:1 Lattice, .512 Boron atonisAO-* water moli scules 263 15.659 .036 229 14.856 -.063 193 13.874 .013 157 12.794 .024 121 11.582 -.008 4:1 Lattice, .07A Boron atoms/lCr water molecules 265 21.287 -.005 193 17o790 .069 157 16.010 .101 121 13.939 -.034 -32- Table 6. (Continued) Number of Rods Relaxation Length L(cin) L Residual (cm) &:1 Lattice, •1A6 Boron atoms/10^ water molecules 265 18.895 -.008 229 17.624 .053 193 16.203 .022 157 U.571 -.136 121 13.202 .091 a:1 Lattice, .218 Boron atoms/10^ water mole jcules 265 17.097 -.020 229 16.187 .064 193 15.045 -,001 157 13.745 -.114 121 12o595 -073 -33- Table 7 Buckling and Reflector Savings as a Function of Boron Concentration Lattice Boron Concentration (B atoms/lO-* HoO molecules) B 2 (cm- 2 ) KlO 3 Jjcm) 1.5*1 ,216 2.164 * .082 7.61 ± .18 1.5*1 • 563 1,10? ± e217 7.61 * .46 1.5:1 ,860 .243 * .05 7.75 * .12 2:1 • 359 2.045 * .255 7.45 * -67 2:1 .590 1.272 * .314 7.11 * .78 2:1 .824 .326 * .094 7.32 * .22 3:1 .174 2.086 * .052 6.79 * .20 3:1 .345 1.137 * .057 6.37 ± .19 3:1 .512 .019 * .049 6.59 * .17 4*1 .074 1.103 ± .050 7.16 * .24 4:1 .146 .596 * .067 6.60 ± .30 4:1 .218 -.054* .070 6.81 ± .32 -34- Tabla 8. Values of the Migration Area Determinod bv the Poisoned-Lettic ;e Method Lattice Boron Concentration (atoms per 1000 molecules of water) Thermal Utilization (Theoretical) Buckling (cm"* x 10 3 : experimental) Migration Area (cm2) 1.5*1 0.216 • 563 .860 0.917 .900 .874 .852 2.86 2.20 1.19 0.33 28.99 t 0.37 2:1 0.359 .590 .824 0.889 085I .828 08O8 3-75 2.05 1.27 0.33 30.06 i 1.21 3:1 0.174 e345 .512 0.828 .803 .779 .758 3.29 2.09 1.14 0.02 28.69 * 1.23 4:1 0.074 .146 .218 0.774 .761 .748 .735 1.86 1.10 0.60 -0.05 28.47 ± 1.43 -35- Tablo 9» Preliminary Values of Ko Lattice Buckling (cm -2 x 103) 2.86 Migration Area (cn2) Ko 1.5:1 29.0 1.083 2:1 3*75 30.1 1.113 3:1 3.29 28,7 1.094 4:1 1.86 28.5 1.053 -36- APPENDIX 1 Results of Test of Axial MessuTeaients of Reproducibility The poisoned lattice "measurements were performed on the 2:1, l^il, 1.5 si, and 3 si lattices, in that order. Immediately after the 2j1 poisoned lattice measurements, a single axial measurement was carried out with the 4:1 lattice and pure water. Use of the measured relaxation length and the reflector savings from reference 1 gave a value of B = 1.816 x 10"^ cm" , conparod with the previously measured best -3 value of 1.79 x 10 . Immediately after the lo5*l poisoned lattice measurement, a single clean axial was measured with the 3 si lattice. The value of B thus indicated was -3 1 -3 3.23 x 10 , compared with the best value of 3.287 x 10 J . Just after the 3 si poisoned lattice runs, a clean axial flux measurement was made with the 2:1 lattice. This gave B^ = 3.62 x 10"3, compared with the best value 1 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 measure- ments. -37- VI. Intracoll Flux Traverses. A. Introduction. The valuos of the migration area we have reported earlier depend on cal- culated values of the thermal utilization f . Since there is some uncertainty about the theoretical methods used to find these values, an attempt at measuring them seems useful. We have used the direct approach of measuring the thermal neutron flux distribution in tho water and in the fuel rods; thus tho values of f we obtain are still uncertain by an amount depending on the cross sections used and hence on the assumed neutron temperature. Evaluation of f of courso also has motivation from the desire to check experimentally the methods used to calculate k through tho four- factor formula. B. Experimental Procedure. Intracoll flux distributions were moasurod for 1*5:1, 2:1, 3:1, and 4:1 water- to-metal volume ratios. Theso are actually ratios of water plus aluminum to uranium. Because theso lattices are so tightly packed, it was necessary to use very small detector foils, not only in tho rods but also in the water, if any detail in the flux distribution was to be obtained. The foils used were 1*5 millimeter diameter discs of lucite containing dysprosium oxide, about *5 millimeters thick. Dysprosium was considered ideal as a detector, because its lack of low energy resonances makes possible the measure- ment of thermal fluxes without the need for making cadmium difference measurements. Foils of this type have been used by the reactor physics group at Argonne, with considerable success. -38- The foils were punched out of a small sheet -made in a hot press from a powder mixture of lucito polymer and dysprosium oxide. Because the activity de- l/2 sired (t — H.0 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, tV = 139.17 - *1A 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 dysprosium 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 cross section of thi3 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 absorption^ however, and so the influence of neighboring foils on each other is small. The use of finite- sized foils of course introduces an error arising from the fact that the flux is not constant over their area. For the foil sizes used and the fluxes observed, it can be shown that the observed activity can be at- tributed to the flux at the foil center, with an error of at most .2 per cent. Since corrections for this effect would be small compared to errors in measurement, they have not been made. The foils 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 cm). -39- Foils in tho 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 observod activities to those they would have had if they had been at the same hoight. The error intro- duced by this procodure is considered negligible. Each foil used was counted to about seven thousand counts in each of six end window counters. Thus there was no need to make counter efficiency correc- tions, the activity of a givon foil being determined from the total counts ob- served in all counters. In all, six complete intracell flux traverses were measured, one in the 1.5:1, one in the 2:1, two in tho 3tl, and two in the 4*1 lattices. An additional partial traverse was also obtained (in the metal) in the 1.5:1 latticeo C. Analysis of Data and Results. The experimental data obtained from the intracell flux traverses are shown in Figs. 11 - 16. In view of the triangular symmetry of the lattice the neutron flux in the moderator was moasured 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 in the fuel rod ware first fitted to the Bessel function I (K r) where r is the radial position of the foil from the rod center and Kq = 1/L is the reciprocal diffusion length. The function I (K r) represents the asymptotic form of the flux distribution in the fuel, but in order to obtain a -40- good fit of the flux data, a value of Kq 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. Tablo 10., Bessel Function Fit of Flux Traverse in the Uranium Rod Ratio K p ( irrl ) kp (cm) p 1.5sl 2.970 0.855 * 4«7 per cent 1.147 1.5:1 2.818 0.901 ± 6.0 per cent 1.133 2:1 2,585 0.982 ± 8.0 per cent 1.113 3:1 2o659 0.955 ± 5-4 per cent 1.119 2.644 0.961 ± 7.0 per cent 1.118 2e988 O.85O ± 5o4 per cent 1.149 3.152 0.806 ± 2.9 per cent I.I65 3:1 4:1 4:1 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. The values do not vary significantly with moderator to fuel volume ratio. The average value of F = 1.135 which corresponds to Kq = 1.116 cm -1 was used in plotting the flux curve I (K r) for the fuel rod in Figs. 11 - 16. The values of F obtained from the BNL experiments are much higher than would be expected from elementary diffusion theory. These results are in con- formity with those obtained with uranium rods at other laboratories, notably at North American Aviation. Kidde has estimat&d the thermal utilization of our water lattices (HKF-1492D-151, Mar. U> 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. •41- Table 11. Volo Ratio f 1.5:1 2:1. 3:1 4:1 0.917 0.889 0.828 0.775 These values of f were used in the preliminary estimates of the 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, corresponding to a value of F = 1.135« The lower curve in each figure corresponds to the currently accepted value of Li = 2.85 cm for the diffusion length in water On the basis of these flux distribu- tions the relative absorption of thermal neutrons in fuel, cladding and moderator is shown in Table 12. Tablo 12. Thermal Neutron Absorption in a Unit Cell Vol. Ratio f f al f mod 1.5:1 e9l69 .OO5O ,0781 2:1 .8873 .OO4.8 „1079 3:1 .8298 ,0045 .1657 4:1 ,7759 .00^2 ,2199 Although the values of f agree with the Kidde estimates it may be seen that the theory greatly underestimates the neutron flux in the moderator. The reason for this discrepancy is clear from the experimental flux data. Extrapolation of these data to the moderator fuel interface would indicate a discontinuity in flux in this region. The apparent discontinuity arises from the neglect of the non- asymptotic solutions of the transport equation which are important near such an interface. -42- 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 thermal utilization, we need to modify the value of the diffusion length L]_ in the mod- erator. The modified diffusion theory curves are also shown in Figs. 11 - 16. The corresponding values of the thermal utilization are given in Table 13 • Table 13. ■iodified Values of the Thermal Utilization Vol, Ratio f 1.5:1 .910 2:1 =871 3:1 .819 4:1 .755 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. Cadmium tubes of dif- ferent diameters were tested} the change in the axial relaxation length caused by replacing the central uranium rod by an absorber rod was measured. Hollow cadmium rods of different diameter were usedj the wall thickness was 0«056 inch in each case. In most 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 from the relaxation lengths. -A3- 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 measuredo The value of the relaxation length was 32.32 cm with 254. rods in the latticej when the central rod was removed, the relaxation length decreased to 32.25 cm, or a decrease of 0.0? cm. This decrease corresponded to a fractional decrease in the axial buckling of o 0022 - 0.0017, or about 0.2 per cent. In the other experiments, the chenge 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 seemed to be typical. No attempt has been made at BNL to treat the data from a theoretical standpoint. Table U. The Effect of Absorber "Rods on the Axial Relaxation Length Rod Outer Diamter of Type of Relaxation Fractional Change in Lattice Loading Cadnlum Tube (in) Filling Length; L(cn) the Axial Buckling 1.5:1 255 No tube 18.96 0,81 H 2 16,93 0.253 2:1 263 No tube 26.89 0.95 H 2 23.27 O.335 i 0.010 3:1 254 No tube 32.32 0.81 H 2 26*77 0,461 253 No tube 32 s 25 O.65 H 2 26.88 O.439 4.:1 253 No tube 23.15 0.95 H 2 21.11 0,203 0.81 H 2 21.71 0.138 263 No tube 25. 18 0.75 Steel 22.40 0.252 2 * 0.004. 2 -LA- VIII. The Effect on tha Axial Relaxation Length of e "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. When the fuel is ready for processing, the sub assemblies are to be separated before being removed from 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 experiments was made in order to obtain some information of this kind. In these experiments, the relaxation lengths were measured for two different arrange- ments of 160 rods) this was done for each of tha four usual lattices. In the first arrangement, the 160 rods were in a reotangle with 16 rows, each containing 10 rods. To form the second arrangement, the rods 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 j the results are listed in Table 15. The fractional change in the axial buckling is plotted against water- to- metal ratio in Fig. 17. The fractional change in the axial buckling is defined as (l?"^ 2 ) * Z? 7-5 > L l where the subscripts refer to the two arrangements. -45- Table 15 Effect of a Central Gap on the Relaxation Length Lattice lo5:l 2:1 3:1 Relaxation Fractional Change in Arrangement Length (cm) in Axial Bucklinc No Gap Gap 13.26 13.51 0.0367 (Gain in Reactivity) No Gap Gap 15.70 14-80 0.124 (Loss) No Gap Gap 17.73 15.37 0.331 (Loss) No Gap Gap 16.48 14*16 0.354 (Loss) In three of the four lattices, the effect of the separation was to de- crease the relaxation length (decrease the reactivity). In the tightest lattice, 1.5:1, there was a slight, about 2 per cent, increaso in relaxation length, indi- cating that the separation caused a slight increase in reactivity. -46- Figure 1 - Picture of Tank. Negative No. 8-215-2 The inner tank with a tube plate and fuel rods. 47- 12 TYPICAL RADIAL FLUX TRAVERSE IN 2:1 LATTICE, SHOWING FLUX RISE NEAR REFLECTOR 10 8 cr 4 UJ X EDGE 0F| LOADING 10 15 20 CM FROM CENTER FIG 2 25 -1+8- LU 2 2 3 K3-I o > a: tr _J CO o rr LU 2 $5 LU Q 1 i^- o O c\j S 1- o I • cc a: LU LU Ll. f- 1- O ^ < ?S ^ CO CO > o o tr H X C3 "3* IS S2 O h- Z> < — ro OJ LU o LU O h- I or LU < 'KH o lO O \T) O 8 < CO QC O »- 9. 6 -CL 5 2 3 WATER-TO-METAL VOLUME RATIO FIG 4 -50- to in _l Ll_ — / h- — -z. / U_ o (T f> C/) / LU h- / cc ~3 / < LU / 3 2 / f CO Ll ? -O co 1 < — LU \ LU if) \ _J OC -UJ \ 1 — > \ < % • cr — >- — I 1 o C\J m o o o IT) cr LU 00 o ■ E o o I in 1 o o 00 o o CO O O o o CM O^ (SlINfl AdVdliadV) A1IAI10V ~IIOd -52- C9 u_ — I I " i i / • 7 r — Ll O X / / 3 / — LU _l 4- — CO Ll_ / / cr / — CO UJ •z. / Q_ 1 -TRAV o q: i- UJ o/ / / / / Z> o CD Q_ z> o cr — X Z / / / / Ll_ Ll — f / / / CO CO LU — QC a: < < / / Z> => / / o ~. ° // 1 CO CM CO // f/ Q_ O i- co < Z> CO o < // cr LU cr lu m w e? _l o _l \ \ 1 o 1 I l + l l 1 o o o o o o o o o o o o 00 o 00 CD sfr OJ o ro 10 O OJ — ID O u_ CO o £ — u_ o cr u_ E o — in — o Oi (siiNn Advuiiaav) aiiaiiov nod -53- 1 1 1 1 I z o _l \ 5 UJ \ - \- :e \ — z UJ o \ c_> 1- \ z o s cc \ _ o U) \ 3 1- \ z Z < \ o < $ \ CO or o 3 1 _ Q_ 1 Q CO CO UJ 3 X CO o CO or cr or Q UJ Ul z O — > 1- < UJ or z $ >3 < _l h- X — o 3 X o "in \ o _J r^ CD _l CO O 1 1 1 1 I \ o ro o ro in 00 X (M E O o CO ~— 00 "Z _J u. in o 3 CO - in o O o co 00 o CD 00 O 00 o 00 CO o o oo NOIlVZniin "1VIAIU3H1 -54- 1 ' 1 1 ro 1 2 ii - (/) UJ 2 - r- 1 NEUTRON AL LATTIC a FORMAN 3/4/44 WATER ED URAN •ERIMEN 81 i- I < 1 leftn P ~ ti !i 1 o zu *2 I x o * I O — LU 1 h- 2 > 1 1 E OF FISSIO A WATER M NRY SOODA S. DM CFI875 TA FROM LI IGHTLY ENF PONENTIAL >- CD < o z ui ac < -1 X 1 < £ x u. 0(0 liJ 1 — _) CD — — 5 J e / K> 1 — o5 / ^ / < / o / lot 1 o / in/ ■ — •' 1 1 1 1 1 1 in - _l < UJ u. — <\j cr UJ 5 o o o o o O to in - CD cp z tn in Q nj Q Z> o \ < o ~~ _l 0^ O 1^ - \ + _l UJ O 1 \ E \ cr 3 — u \ o < ti- ll. ID \ - CE o - _J — \ UJ Q O s UJ 5 _J d o tn O \ u. o o > GO • • 9 o ~ UJ 2 5 \ 5 Q 3 _l Q _i UJ < O 13 _1 > U. O 1 i i i i X E UJ Q O cr o °° uj o o Q < a. xnid NoainaN 3AI±V13d -57- CO UJ o 2 3 Z < CC I CC UJ I- < L. O CO z o I- 3 m co o x 3 UJ o < 1 i l 1 cr < V) _i ■" o 3 E o E o cr o 2 - o in 2 cr - O ro UJ in UJ _ 5 t- u. o *0 i< UJ CD z < - i — i \ UJ 2 Q UJ - u \ 2 o o o 2 2 O O _l _l < < - ! ° \ s o o CVJ © -58- i i 1 1 i 1 'e n £ o — in •si- ro a o* ii CO _ UJ o □>\ 1- 1- < _l _ 2 a \ O 3 Z < cc °\ _ 3 1 \a \ q: LlI i- < 5 - U- o o z z o ~- r E* CC \° 1- co _ a X 3 i \ E vCD U. o (0 - _l - ^ _l UJ ro II o o ii te! < op a: — o — \- 1- z < - a: _i o > i 1 i i 1 q 3 T - ,- to e _ CNJ cvj O -59- CO o l- < cr 3 < 5 ii o cm to ri ^ 7 o h- Z 3 .1) 00 CE cr i- co Q X =3 UJ o < or 1 1 o 1 1 1 1 1 1 e _ q _e - □ 2 » - - o\ °\ - - D X - \ ° - ii O < T \ E u S9 ii o M O O \oo - - o > 1 i 1 1 1 \oO - o in CM in CM CM r- o in in CM e - u •- U. CM CM O CM -6o- 1 1 1 1— D 1 1 w a LU o H- , 1- i □ < _l s O D 3 _ Z < CC \ ^ 3 I T \ E\ UJ o \ if) \ - (- G \ ro \ < T \ ii" \ 5 E N o o \ W \ u. o rO o — OT Z ■i z — a \. \^ 2 z M O ^^ N. t- 3 3 CC CD _ cc O ^^^v H w a " X K 3 ^^^^H ■ E li. * o ID _l _l it °\ ^ LU o \ ° ~ O h- \ » < < CC a: H Z _i o > 1 1 i 1 1 o ■3- o 01 o 6 O -61- □ c 1 D _ — in Z < D ro cr o " 3 1 V V« cr ~" III E < \o $ * - Li. O « n CO o co b z z o — CD * o ^ cr 1- - — Q X oo\ 'e ZJ o _l (D u. - _l « _l — cd\ o ■* < " cr o — z 1- < Oh _l o > i 1 1 1 1 3 - F — o O *- u_ -62- •iKX ■ or < o ^ CO CD — ) F x x < -2 UJ > - > ID- en a. ce Q 3 UJ UJ o x o X o O UJ UJ o UJ CO o o o z o — h- — »- iii rr 1- 1- < < >; < co to 9 ui _J h- _l X Q <-> o < -L CD _J CD _J CO lO d s< KH J L 3AI10V3U SS31 l I I I o> co d 6 f- CD d d in sj- d d ro co — O 6 O 3AI10V3U 3dOW l_l L o ro < UJ 0J lO o I- cc Ul 9Nn>