- -~ - . \ - - I ‘ -- This report war prepared osa scientific account of Govern- ment-sponrored work. Neither the United States, nor the Com- mission, nor any perron acting on behalf of- the,Commission makes any worrnnty or representation. express or implied, with respect to the accuracy, completeness, or usefulness of the in- formation contained in this report, or thot the ure of any infor- 1 motion, apparatus, method, or process disclored in this report may not infringe privatelyowned rights. The Commission assumes no liability with respect to the use of,or from damages resulting 1 from the use of, ony infomotion, apparatus, method, or procerr I disclored in this report. DAm: 3/28/49 $,-so $0 ,:.‘ ‘4 * Cherni ck _.* -’ “This documenitc ontain T’. . Initial Xperinents <, of its contCnL5 in prohibited ar:d ma penalties under applicable Federal on the Brookhaven Reactor. f r-D- - So 3-0 _ _ ‘ 111, ANALYSIS OF THE CIINTOPT CilITICU ZPBI?GlP In the present renort,we cokute the fundamental pile constants of the Clinton reactor from their critical emeriment (F.P.R. Vol. V, Chay. . 23) These constants include the critical size, mul t i d i mt ion factor, migration area and criticpl Lmlacian. In addition, we have used their data to obtrJIin the sFontaneous neutron yield of naturP1 uranium. W e get 14.0 neuts./kg sec. which agrees well with the value of 14.3 neuts./kg ; sec. determined by Yermi from an exponential experiment (C-26). The accu- racy of our determinc7tion is estimated at 10%. The critical size cpn be determined from the Clinton data with good accuracy. (Zven better rcsu1,ts would have been obtained if they had used * . more loadings in the neighborhood of criticplity) We get ’217.80 0.04 cm for the actual loading radius by extramlating from below critical and 218 .LO 0.05 cm by extranoiating from above critiwl. These vc7lues are _. close to those cited in‘the P.P.R. The difference of 0.6 cm must be attri- buted to some small sggstemtic error; nossibly to the necessity of correct- ing in an arbitrary manner for the presence of P smll absorber Dlaced in the pile a t the first loading above criticality. ‘ For the remaining Constants, M2, idy, k e -1, it is easy to obtain 1 - agreement within 10% of the ’theore tically predicted values. Better agree- to emect at rresent due to various systemtic DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. errors. Among the latter are: 9 I 1. The conver9ion of nile Deriod to excess k. l . The latter figure has changed by 15% since the wblication of the Clinton data. In the analysis of a critical emerirnent the accuracy of the determination of PI2, the migratlon area, is n; better than the accu- racy of this conversion factor. ? . - 4 2. The conversion of observed activities to therrral flux. Our determination of the spontaneous neutron yield of uranium is no * I better than this figure, or within 10% of the correct value. I 3. The reflector savings. . The latter is intrinsically difficult to determine from the OakRidg data, the reason being that the reflector savings is always smll compared' to the loading radius. Our attempt to obtain this paramter showed that i the data was not sensitive to changes in the reflector savings of 20 cm 4- 4%- or more, The reflector savings is, however, reauired in the determination t 6. Of the critical Laplacianb 'qe can hoDe to imrove on the Clinton critical i experiment by running transverses at various loadings and extranolating for the augmented pile radius and herice for the reflector savings. 4 4. Other systematic errors. t I Here we can l m s uch variables as temerature and barometric effects, unknown f o i l weights and cadmium ratios, irregularities of the loading, etc. An example of the latter was the non-loading of three central channels z;. in the Clinton pile, IVe can hope to avoid some of these systematic errors by keeping a record of temerature and pressure, weighing the foils etc. 1 b - f 1 1- Theore tical predictions From data published in the Chicago Handbook ( C h a ~ .I V,E)w e find the i: .. 1.- following values for the fundamental constants of the Clinton pil?e-*, - = = f 0.890 fa? 0.004 * = p 0.886 1.029 ~ .. . - . . .. . .... ,' = a. - 'W From these data The value of I., the diffusion length, assumed for the AGOT graphite of the xtile is 50.3 em, ilensity 1.6. The zverage age of fission neutrons in grhphite at indium resonance energy (1.44 ev) is taken as 334 em2 and . 2 eo the age at thermal energy is obtained by adding an incremeat of 53 cm The latter increment is admittedly uncertain ih view of nossible crystal- line effects at low energies. The value of 8, is thus taken at 387 cm2. To obtain Lo, the diffu-sion length of thermal neutrons in the pile, we have Lo2 f modL3L . However, one must correct L2'for the increased temperature of pile neutrons and the presence of empty cylindrical channels. The corrected expressions are rj- P 4 where LJ, L11 are the Dile diffusion lengths respectively perpendicular - to and parallel to the channels, T = 370' is the absolute temperatwe of I = the pile neutrons, Q 0.034 is the fraction of pile volume taken up by empty channels, a i- 2.1 is the channel radius andrlth = 2.7 is the trans- port mean free path for thermal neutrons, The neutron age is li'kewise incras by the Dresence of channels . * and also by the nresence of uranium since the latter is much less efficient as a slobrinp dmm medium. @n the other hand the age is decreased by in- elastic scattering in the uranium rods. The corrected age for a nile is h .. . . ir- * - - where P 0.15 is the probability of a fast neutron m'king an inelastic the = collision in a metal rod of radius 1.4 cm, V,/V1 0.015 is rztio of the 42 volume of metal to that of granhite and Jf 3.5 cm is the mean free 2 path of a fast neutron. Insertion of the data gives - .L ' 0, - 389.1 M , = 703.7 '-J? = -95 .z x 10- 6 A. L112 = 3z.4 811 = 396.7 Mi12 = 719.1 3 11 = -93.2 x 10-6 :* " i I As merage values over the nile, then l0 = 317 cmz, Oo 2 393 cm 2 . The fundamental constants vredicted for the Clinton nile we therefore: 3 iv? = 710 cmz, A = -?L x 10-6 c d , IC>=, = 1.067. C - m3merinentp1 Find inps In the analysis of the ewerimental data, the Piutonium Project qe- f A port takes tbe reflector savings at U.5 cm. '5th this vqlue they get t M? 630 cm7-, /3 -1.045 x 10' G , kciz 1.066 f For a conversion factor, 1 inhour was taken as e-uivalent to 3 x in k. If the more recent figure of 2.6 x lou5 is used, we get = 142 546 cm2, A= -1.045 x lom6 , k = 1.057 It is clear that most of .the trouble lies in taking too low a value for . the reflector smings Re-evaluation of the Clinton Data -. Ve shall re-evaluate the Clinton data by using a more reamnable value for the reflector savings. Before doing so, it is of interest to m u ,c. try to obtain the reflector savings direct1.v from the data together with a vecision estiw te , Before criticality, the formula to be fitted is where -4, is the observed sctivitv, rx the ''observed'' 1o;lding radius, rc -4- the critical loading radius, b the reflktor swings, Pnd c a DroDortion- I ality constmt which can be relnted to the snontanenus neutron yield of i. uranium. The three constRrlts, e, 6 and r , are to be obtained on the C * basis of 5 ~ ~ o i n t sZ.o uatim (1) is deliberately left in non-iinear form so that the resulting least siuare solutiim will weipht most the points I near criticnlity where the activi?;ies are lprgest. The least snuare tech- - 4 nique has been fully described in B.N.I. Log if C-820. Taking c - ,0073, - - 0 z 42.5 ern and r - 217.4 cn! for ‘an amrosimate set of solutions and 1i ne- C arizing eQu8t ion (1) by Teylor series emansion we get for our least scrunre solvtiw,. c . These results show thqt il( - 1. The reflector savin,ys can not 3e determined from the Clinton data with # any nrecision whr7tever. In the comlete absexe of informtion on refiector snvir@5!, the criti- 7. cal radius is determined to a wecision of c7bout n.?‘%. 3. Under the Same conditions, the nronortiomlity constant c is correct to about 20%. With a better estirmte of the reflector savings the me- cision can of course be greatly improved. . - It was therefore thought advisable to take the theoretical reflector * savings and combine this with the exnerimentnl dztta. The diffusion length of the Clinton gaohite is about 50 crn. ’The reflector swings according ‘.i to enuation (8) of B.N.L. Log-& C-2853 is then 53 cm. There are theoreti-. cal uncertainties in the~latterfi ’gure but it is certainly to be Dreferred- to the fklmerimental va1u.e of 4?.5 cm given in P9.9. Vol. V. 11 . W,p&--T and determine the-constants c and rc by least souares. Ye get c z 0.006453 0 .000110 - p I 217.54 0.12 4 rC the internal consistency of the data now resulting in a standard error i in the critical loading radius of 1 part in 2000 and an error of 2% in the proportionalitv constant'. This takes no account, of coume, of system- atic errors. b Harmonic Corrections One systemtic error that can be elimi&ted is that due to the non- 2 linearity of the curves of l/As against 1/Ii for small loading radii. We can refine our results by aploljring harmonic corrections to the observed activities. The corrections can be obtained from o w f ormu-la (B.BJ,L.Log i# c-2853) The following table gives the required harmonic corrections: Ob served IXarmoni c Corrected Act ivit y Correc ticn Activities ' . 620 - 1.081 6 70 I . 1180 1.047 1235 3918 1.019 2973 6344 '1.012 , - 6420 10558 '1.007 , 10632 Repetition of our nreviou.s work now yields = , ' . I c 0.006762 Y .000036 .r = ?17.80 6 0.04 .. rc The increpsed precision of the f t is reflected in smaller standard ,errors _I The value of apqlying hrrmonic correcti s to the datcq is thus confirmed,. The critic31 radius corresponds to a loading of 360.9 channels as against -6- 360 clmmelq ,@-\.en in P .P .E. Smnteneous Yeutron Yield By use of the fins1 equation of 3 .N.L. Icy C-?c?53, we cm correhte the pronortional constant c which we have just obtained with the natural neutron yield of uranium. W e hme where K is the fador required to convert bpre indium activity to therm1 flux. Taking Ap = 2.71, p = 0,886, 1/12 687, L2 = 317, c = .00676, we find - - Q 0.0424 K neuts/cn? To estimate K, we use the formula - ' nv - 0.103 A, I 1 - 1.07 i C.R. 1 for stmdard indium foils. Ve do not how the cdmiun rptio for the Clin- ton nile, but taking this as 6, we, get K - 0.0837 = 3 and Q 3.55 x 10-3 neuts/cm sec. From the lattice constcants md the fact tht there were 65 4" slugs per c-hannel, we can obtain the density of mtural wrnium in the pile. We find 10-3 94 I- .. kg/m3 ' Hence we get 14~;On euts/kg sec w i t Pn estimated accumcy of 10%. Our ignorance of the cadmium ratio introduces an error of about 2%. The possible error in the ratib ~'/i is likewise a few percent. The chief uncertainties lie in the conversion of z'ctivity to flux 2nd in the experi- mental value of the txonortion2lity constmt which, as we have seen, de- Dends mon the assumed reflector savings and m2y be subject to other system- atic errors. . .. Reduction of Data Above Crit icalitz The basic data from loadings zbove critical ,we the observed pile . 'periods and loading r adii . For a thema1 nile i . ' = with H the augmented Pile height 2nd Rc the augmented nile radius. Also, if the effective multivlication constant of the nile, keff, * * is close to unity kx: Combining these relations we have for the excess k of the pile, On the other hand kex is connected with the pile Deriod T through - A i the forrriulzs We can therefore mite 2) and Aetermine the-constants C and R, from the data. There is a conplete analom between equation (2) and ec&tion (1) which holds before critical. In the form in which (2) hzts been set m, the Doints near critical which correspond to the longest periods w i l l receive the most weight. The ob- servational data are .tabulated below: