KTE TLIANERSY OF MICHIGAN INDUSTRY PROGRAM OF'THE COLIEGE OF F-GIMERTNG THE MASREDTYN OF DYNAMIC!UCITAR REACTOR PAM1 TERS BY MUTIODS OF STOCHASTIC PROCESSES Robext W. \3lbrecht A dissertation sutbmitted in partial fulfillment of the zrequirements for the degree of Doctor'of Philosophy in the tJniversity of Michigan 1961 JSunee 1961 P-.522

ACKNOWLEDGMENTS The guidance and assistance given the author by members of his doctoral committee was very helpful. The author is especially grateful to Professor William Kerr, chairman of the committee, who gave freely of his time and energy in all phases of this efforts Aid in the understanding of the theoretical aspects of this effort was given by Professors R. K. Osborn and D. Darling. Many people participated in the execution of the experiment. N. Barnett made the experiment possible by lending the author the tape recorder which is normally in his use. W, Dunbar and the members of the reactor operating staff were very cooperative throughout this experiment which required special effort on their part. P. Herman gave advice freely on the instrumentation requirements. Assistance with the analytical work was given by the computing center staff, especially Greg Smith who worked with the author on the nonlinear estimation program. For all of the above aid the author expresses his appreciation. The opportunity for the author to participate in the Nuclear Engineering Program at the University of Michigan would not have been possible without the aid of three years financial support by the Oak Ridge Institute of Nuclear Studies, For this support the author is most grateful. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENT.................... o o o o o o. o o o. o o o. o o. ii LIST OF FIGURES..... V o o o o o o o oo o ooo oo V LIST OF TABLES... o o o o o,, o o o o o o o O o o o ii ABSTRACTT.... viii I. INTRODUCTION...................................00..00 0... 1 Ao Introductory Remarks......................... oo............ 1 B. Organization of Text....................... 3 II. STOCHASTIC PROCESS THEORY................................... 5 A. Definition and Notation......,.. o....o........... 5 B. Relationships Between Stochastic Signals in Linear Systems.............................. o 7 Co White Noise...................................o....... 9 D. System Driven by White Noise............o o.oo.o oo o..o 11 E. Relationship Between Variance to Mean Ratio Over an Interval and Autocorrelation Function..................... o o......... 11 III. MATHEMATICAL MODEL o, o O O.................... o. 14 A. System Approach..o.....oooooo.o.oooooooooo.OOOOO 14 B. Physical Approach................oooo.oo oo.... o 22 IV. EQUIPMENT o o o o o o o o......o o o o o o o o o o o. 30 Ao Experimental Setup.... c..............o o o o o o o. 30 B. Description of Equipment.............................. 30 C. Operating Characteristics.........,o................ o 44 V. EXPERIMENT.,.O O................ OOOO,.oooo............45 A. Experimental Conditions..000....... o........... 45 B. Equipment Limitations,... o...o.................. o... 52 C. Experimental Procedure... 00.0...............00.....0.. 60 VIo DATA ANALYSIS.......................................... oo 66 A. Data Handling...................................... 66 B. Estimation............................................ 68 VII. RESULTS...... o............................................ 85 A. Experimental Results................................... 85 B. Mathematical Model (General) 0 o o..o o o o o. o o..... 90 iii

TABLE OF CONTENTS CONT ID Page C. Mathematical Model (2-group)..00000000000.00.000000000. 93 D. Presentation of Results............................ 95 E. Theory vs. Experiment................................. 103 VIII. CONCLUSIONS..............................0...00.00OO 106 A. Introduction.......................................... 106 B. Comparison of Conclusions with Preceding AuthOrs..s..60..........00OOOOOOO 0ooooooooooooo.o.o...o 106 C. Independent Conclusions...OOOOOOO OOOOOOOOO oo...o...... 108 D. Relevance to Future Experiments......................... 109 APPENDICES.......... o o o o o o o o oo o o o oo o o o o o o o o o o oo o o oo 111 BIBLIOGRAPHY.....................ooooo.....oo..o ooo.... o120 iv

LIST OF FIGURES Figure Page 2.1 Block Diagram of Linear System.......................... 7 3.1 Noise Source Only Signal and Noise..ooo-.....o.....................oo...... 16 3.2 Noise Measurement Block Diagram. o..........o o. o o. o. o 17 3.3 Possible Chains for an Accidental Pair of Counts......... 23 3.4 Possible Events Leading Up to a Coupled Pair of Counts....,,,......................... 25 4.1 Data Taking Configuration............................. 31 4.2 Data Transcription Configuration...................... 32 4.3 Core Configuration for BF3 Tube Experiments............. 33 4.4 Core Configuration for Fission Chamber Experiments...............o o o o.. o.oo. o.. 33 4.5 Circuit Diagram of Pulse Shaper.............o....... o 37 4.6 Simplified Block Diagram of Gate Scaler................ 39 4.7 Information Converter Parallel "Staircase" to Serial "Digital. "... o o,,o o O......... 41 4.8 Power Supply for Converter......... 42 5.1 Gamma Background Measurement....,................... 49 5.2 Relationship of True Events to Registered Events in Paralyzable and Non-paralyzable Counters............... o.............................. 53 5.3 Loss in Mean Count Rate Due to Dead Time Losses for Paralyzable and Non-paralyzable Cases...................................... o o........... 55 5.4 Ratio of Observed to True Variance Verses Np for Paralyzable and Non-paralyzable Counter *ooo oo ooeooo*o...,, o, oooo *o..... oooo o o.ooo.....o o..o... 55 5.5 Ratio of Measured to True Variance to Mean Ratio Versus Np -...oo..-. o-.,...o..o...o......o.,, 57 ~

LIST OF FIGURES CONTID Figure Page 5.6 Wow and Flutter Measurement on Ampex 307 Tape Recorder..,..................... o o o o o o 59 5.7 Differential Rod Calibration........................... 62 5.8 Integral Rod Calibration..,,,........... o........, 65 6.1 Observation Space..c..... oe, ooo ooooo oooooo.oooo o o. 69 6.2 Parameter Space................ o..o............ o. 70 6.3 Observation Space....,... o....., ooo.. ooooooooo. 71 6.4 Observation Space....,.... o..... oo, oo o...... oo 72 6.5 Parameter Space...................... o......... o 72 6.6 Sum of Squares vs........................... o o....... o 75 6.7 Observation Space........, o o,,...,,,o. 75 6.8 Parameter Space.,.,....o_,......... 76 6.9 Linearized Parameter Space.....,..,,,oooy.,o 76 6.10 Parametric Curves for 1f); 1X350..... 82 6.11 Parametric Curves for 1(7); Oo OC:2...,o........ o.oo 85 7.1 Data and Curve Fit:Run BF -1....o.,o...oooo,... 86 3 7.2 Data and Curve Fit: Run BF -2............................ 0 87 7.3 Data and Curve Fit- Run BF -3............................. 887 3 7.3 Data and Curve Fit: Run BF3-3..oo o o,,.o ooooo.,.o 88 7.4 Data and Curve Fit: Run Fission Chamber,o...... oo. o89 7.5 Comparison of BF3 Experiments Fitted Curves...o,.oo..... 91 7.6 Confidence Contours BF -1..........,.......,,,....... 96; 3 7.67 Confidence Contours BF -21....,o.,....,,9,.......,, 97 3 7-7 Confidence Contours BF 3- 2.... o.oo~ 97 7.8 Confidence Contours BF -3.o.....,,o o o o.,,,,,,,, o....., 98 vi

LIST OF TABLES Table Page 7.1 Physical Group Constants............................ 99 7.2 Reduced Group Constants........................... 99 7.3 Results BF3-1.......................................... 100 7.4 Results BF3-2............................................ 101 7.5 Results BF3-3.......................................... 102 vii

ABSTRACT Mathematical models are developed for the statistic variance/ mean in a point, unloaded reactor. An experiment which measures variance/mean for counting times between one millisecond and ten seconds is discussed. Comparisons between the mathematical model and the experiment are made. The results show that the delayed neutron contribution to the measured variance/mean is significant, that these delayed neutron effects should be accounted for when using this method for determination of prompt neutron lifetime, and that these techniques may be used to measure dynamic reactor parameters in steady state. The results of this experiment establish the promt neutron lifetime of the Ford Nuclear Reactor. This experiment is the first reactor noise experiment to be performed in such a way that delayed neutrons make a significant contribution to the measured quantities. Hopefully this investigation will lead to a wider use of this technique as a diagnostic tool in nuclear reactor analysis. viii

I. INTRODUCTION A. Introductory Remarks When a nuclear reactor is operating in steady-state its neutron population is not constant. When observed with a sensitive instrument, the neutron population is seen to fluctuate about some average value. These fluctuations reflect some of the characteristics of the reactor. In particular, the dynamic characteristics of the reactor are reflected in these fluctuations. This is as it should be since fluctuations are indeed dynamic phenomena. Careful measurements of quantities which describe these fluctuations in neutron population should yield information about the dynamic reactor parameters which control the fluctuations. Obviously one must first formulate some mathematical model for fluctuations in terms of reactor parameters so that one can know what to expect from his measurements, over what regions to measure, and what is the required reactor configuration to yield optimum results. One must also be sure to know how to distinguish "signal" from "noise," That is, when one is measuring in a reactor, the counting system may produce electrical impulses which are not caused by the detection of reactor neutrons and are not distinguishable from reactor neutrons, Possible sources of these impulses are source neutrons, gamma rays and electrical transients, It therefore becomes necessary for the theory to take the noise into account since it may well be a substantial part of the measured fcluctuations. -1

W2Since flucuations in neutron population are by no means regular fluctuations such as a sine wave, it is necessary to take a large amount of data in order to distinguish the contribution of dynamic parameters with any accuracy, This dictates that a statistic which is amenable to data collection and data processing by automatic techniques should be measured, In the following, mathematical models will be developed which describe neutron population fluctuations, the difference between "signal" and "noise" will be delineated, and an experiment which measures these fluctuationsy and thus dynamic parameters, will be described, One might ask, "Why measure dynamic parameters by stochastic process techniques?" The answers to this question are many. Some of the most important reasons for making stochastic process measurements of nuclear reactor dynamic parameters are the disadvantages of making the measurements by other techniques. Other techniques employ some method of introducing a perturbation into the reactor and observing the resultant response of the mean neutron population. Disadvantages of these techniques are that a perturbation must be introduced which inevitably alters the dynamic parameters, the perturbation may have to be large so that the mean neutron population is easily observed over the fluctuations and. may thereby drive the reactor beyond the applicability of the mathematical model, it may be very difficult to insert into the reactor the required perturbing mechanism, or it may be dangerous to drive the reactor through transients because of thermal shock or other possible damage.

-3Of course other, and more direct, reasons for making stochastic process measurements exist, One of these is to compare mathematical models with experiment, This is important since only a few stochastic process measurements have been made to date and the theoretical-experimental comparisons are as yet imcomplete, To the author4s knowledge no experiments have, as yet, been performed for correlation times such that delayed neutrons are important. In the following, delayed neutrons will be included in the mathematical model, measurements will be made over times for which delayed neutrons are important, and comparisons will be made between the observed and predicted effect of delayed neutrons, Investigators who have preceded the author in this area have made fundamental contributions to the theory and experimental techniques of stochastic processes, Feynman, DeHoffman, and Serber(l) used these techniques to measure the distributions of the number of (16) neutrons from fission, Luckow 6 measured prompt neutron lifetimes by measuring fluctuations as did Cohn, 8) Velez(2324) developed an equation for the autocorrelation function and attempted measurements on the Ford Nuclear Reactor, Theoretical work in this area was done by Feynman, DeHoffman, and Serber (), Brownrigg and Littler Frisch and Littler(12) Feiner, Frost and Hurwitz 10, Moore(17 18,19) Bennett(1), Courant and Wallace(9), and several others, B. Organization of Text Chapters II and III deal with the development of the mathematical model for the statistic of interest here; the ratio of the variance

"4 to mean number of neutron counts over an interval of time, Chapter III applies this theory to determining the mathematical model for stochastic processes in a nuclear reactor. Chapters IV and V describe the experiment which was performed to test the models. Chapter IV describes the equipment which was used. Chapter V describes the experimental technique used to make the measurements, Chapter VI describes the techniques employed to process and analyze the data. Chapter VII discusses the results of the experiment with respect to the mathematical model. Chapter VIII discusses the results and formulates conclusions based on these results, The original contributions in this dissertation are contained mainly in the experiment. Heres the effects of delayed neutrons on the stochastic process measurements are shown clearly for the first time, Some credit is also claimed for deriving the mathematical model in a way which is, in the author's opinion, more illuminating than those derivations which appear in the literature. The techniques of nonlinear estimation developed for application to this problem represent a more powerful method of analyzing the results than has been used before,

II. STOCHASTIC PROCESS THEORY This chapter will be devoted to illuminating the concepts of stochastic process theory. Several statistical functions will be defined. Notation will be adopted for these statistical functions which will be used consistently hereafter, The: fundamental relationships between these statistical functions will be demonstrated. These relationships will be exploited to show the connection between stochastic processes and general system equations. For a more detailed description of this material see Laning and Battin(5) or Newton et alo(20) A. Definiti aotaion and Notation In the following, consider x(t) and y(t) to be fluctuating functions of time and xi(t) and yi(t) to be the i-th member of an ensemble of fluctuating functions of time. 1. Mean of x(t) The time average of x(t) = x(t) T im-T xlt dt (2.1) T T The ensemble average of x(t) x(t) N i i=l -=/,O 2. Mean Square Value of x(t) The time average mean square value of x(t) = x2(t) = liam 2T X (t)dt (2.3) T-.co -D T + -5

-6= — Ensemble average mean square value of x(t) x2(t) N =/im N -x t) (244) 1=! 3. Variance of x(t) The time average variance of x(t) var(x(t)) 2 _= X tl t -x */t) (2. 5) The ensemble average variance of x(t) ar(x(t)) X2 2 = X (tl.-x(t) (2.6) 4. Autocorrelation F muction of x(t) The time averaged autocorrelation function of x(t) xx(T) =Xt)x(l t+T T r) ) (2~7) The ensemble average autocorrelation function of x(t) t* xx(tl, ) / I/ _N_____ N 5, Cross Correlation Function of x(t) and y(t) The time averaged cross correlation function of x(t) and y(t) =- txy(T) =x(t)y(t+T' =im 2 x(t)(t-r)dt (2.9) The ensemble average cross correlation function of x(t) and y(t) s fxy(tljy) N =1xtln+-i =lim ), t,+r) (2,10) 1= /

.76, Power Density Spectrum It is useful to introduce a frequency function which is defined as 1/2it times the Fourier transform of the time-average correlation function in order to ultimately deal with transfer functions in the frequency domain. It can be shown(l5 20) that this frequency function will be the power density spectrum. That is, it will be a function which will measure the power density of the signal as a function of ~requency. The integral of this power density spectrum over all frequencies will then be the total power in the signal. The power density spectrum corresponding to the time-averaged correlation function is denoted Tx, x 7xV, or Ho depending on which correlation function 4rx rxy or ryy is being considered. For example, the power density spectrum corresponding to the cross correlation function defined in (2,9) is. Relationships Between Stochastic Signals in Linear Systems) B. Relationships Between Stochastic Signals in Linear Systems Consider a linear system which is described by a Green's function g(t), and has a driving function x(t) and a response y(t). This system can be represented by the block diagram of Figure 2.,1 Input x(t) Output y(t) LINEAR SYSTEM -- g(t) Figure 261 Block Diagram of Linear System

-8We seek the relationship between the autocorrelation function of the input and the autocorrelation function of the output. The relation between the input and the output is given by the convolution integral as Yrt)]= 9[ t,]x t-t,|dt, (2.12) and similarly, r00 y(t+ ( 9tz)xtr-T-i dtz (2.13) These two expressions can be substituted into the definition of the autocorrelation function to obtain 7-' y J IniM Y t 4Jj+7)dt 7T'-r~ / 2 7.r. T-00 J-r.(2.14) -,z~ T/.dt id t~,/x^f —t,)d, g[)xlr-,-r-~ -7. -c T-. —,>, By interchanging the order of the limit process and the other integrations so that we integrate with respect to t first, we get,_Th fT)r - /dt4 gt dtx( t-t)x(,t -)-t (2.15) Recalling the definition of autocorrelation function, Equation (2415) is written 7' ylr^fld7$;9t,9[tglA~l~x~> }U^\ ^(2.16) This is the relation between the autocorrelation function of the output and the autocorrelation function of the input,

-9In a similar way, an expression for the cross correlation function between input and output signals may be derived. Substitute the equation o, 9ItT) = d to g(t )x (t+ T- ) (2 17) into the definition of cross correlation function, -7gxyi- f odtx^f Y(tT) (2.18) to obtain r7' rC E i/g (-/x( f/.Xr X (2. 19) T-^o QO Again interchange integration and the limit process to get d y(r)L) t (T- )_ (2.20) The relation between the input and output power density spectra may be found by Fourier transforming both sides of the relationship between input and output autocorrelation functions. This leads to e c 08 re I rye' tj(i.Jl t 4 (2.21) dr7,1,,1dr-e eT9(t;dt+9(t)2XJrf t ta) & _-a -a0 n _, The order of integration may be changed and the arguments of the exponentials adjusted to give ra i -/a r t ^ ra -I0 tt 1 -/.z 222) e dr., edt r: e d.Je -t, 2 The system's transfer function is defined as the Fourier transform of the Greents function and denoted T(ic)o Substituting this into

.10~ Equation (2.22) and recalling the definition of the power density spectrun E quation (2~22) may be written \yy(w)= T(wu)T-/wU\ jw) \ (2.23) C, White Noise A random process, x(t), possessing a constant power density spectrum is referred to as white noise. The physical origin of this term is in the concept of white light: a light that possesses all frequencies in equal amounts. If x(t) is such that x(t) O; xUh()o, > A (2.24) then, the process x(t) can be considered to be white noise in a range of 1 frequencies 0 < o << ~ 0 In this case, the autocorrelation function for white noise can be approximated by a delta function, ^it(1) - c6(A) (2.25) so that the powner density spectrum can be found by Fourier transforming to be Y lw ) =eC (2.26) It should be noted that, in a strict sense, white noise is a physically unrealizable phenomenon since it is a random process having

-11 an infinite average powero This follows from the fact that the total power is given by 4 x w dw = 0 }d = (2.27) In spite of this fact, white noise is a useful concept both for certain theoretical purposes and as a practical approximation to noise of a very broad bandwidth. In many problems a noise spectrum may be known to be substantially constant over the frequency range of interest. When this is true, the use of a constant power density spectrum for all frequencies often simplifies mathematical manipulation without introducing significant inaccuracy in the result. Do System Driven by White Noise A system driven by white noise has a constant input power density spectrums C, so that the output power density spectrum will be proportional to the square of the modulus of the transfer function. This follows from substituting C for Txx(j) in Equation (2.23) to get: jlu) = cT(iw )TH i) (2.28) E. Relationship Between Variance to Mean Ratio Over an Interval and Autocorrelation Function In certain systems, of which a nuclear reactor is an example, it may be more feasible experimentally to measure the accumulated value of y(t), the output noise, over an interval than to measure either the autocorrelation function or power density spectrum directly. The ratio of the variance to mean of this accumulated random process for intervals

-12of length T is related. to the autocorrelation function and thus the power density spectrum of y(t), To derive this relationship, consider y(t) to be a stationary stochastic signal. The variance of the integral of y(t) over an interval of length T will be related to its autocorrelation function. Let x(T) be the integral of y(t) over the interval T. The relationship between x(T) and y(t) is 7r X(|T|r) | Ytdf (2.29) According to relation (2.7) the autocorrelation function of y(t) is written y4-trto) (-ytJ f riiz (2.30) Integrals of this autocorrelation function are investigated over the triangle 0 < t1 < t2, 0 < t2 < r, Integrating once, tar~~~~~ t ) jVyy~t-r.]df = yfi)y(tdD (2,31) Y-tz x(t}

-15Integrating again (by parts) f f yy (l;-yld d r, Y $ 1z d z XZ7 - 27^ (2432) -x"(rl 2 2 The ratio of the variance to the mean of x(-) is the quantity which is measured. in the experiment to follow. This quantity can be formed from Equation (2.32) yieldJing V~~~da~~_r_ _________y= X ( 2 (7)rdr 7(2.433) Ao=?o x ((233 It is to be noted that the variance/mean over an interval measures acc'umulated. correlation. That is, it is related, to the double integral of the autocorrelation function and. hence to the system equations through Equation (2.16)4

III. MATEEMATICAL MODEL The mathematical model used here for stochastic processes in nuclear reactors operating at steady state is based. upon a point reactor model Two approaches to the derivation of this mathematical model are presented here, One approach will be termed the "system approach" which will use the results of the previous section to derive the power spectral density, autocorrelation function, and count-rate variance for a nuclear reactor by operations upon the square of the modulus of the subecritical reactor transfer function, The other approach to be discussed will be the "physical approach" in which details of the multiplying processes in the reactor are followed showing the mechanisms which give rise to correlation. The validity of this model will be examined in relation to an experiment in Chapter VII. A. System Approach in Chapter II it was shown that a linear system being driven by a white noise source has an output power density spectrum proportional to the square of the modiulus of the system transfer function. In the system approach to the problem of predicting the characteristics of reactor noise we assume that a reactor operating in steady state is a linear system driven by a white noise source, 1 WhTite Noise Source According to definition (2,24) a white noise driving force must be such that if x(t) is the white noise source, then x(t) = 0 -14

-15and xx(X) = x(t)x(t+X) = O for X s greater than some o which is << X the correlation time of interest in the system. The assumption of white noise is used in the: noise analysis of many systems when it can be argued that the possible origins of the noise are characterized by the conditions above. In a reactor, a possible origin of the noise is in the fluctuation of source neutrons from both external sources and the sources inherently existing in a reactory Another possible source is in the fluctuations of the reactor parameters themselves. (17918) In the following, white noise will be assumed as the source noise for the derivation of the mathematical model by the system approach, The characteristics of the possible noise sources mentioned above are such that this assumption seems to be warranted, It may be possible to investigate the character of the noise source by an experimental technique, By applying a known perturbation to the reactor and observing the output with and without this perturbation we may be able to gain information as to the characteristics of the inherent driving force. For example, if yn (.) is the output power density spectrum due yy' to the inerent noise alone and sn(o) is the output power density YY spectrum due to signal plus noise (where signal is denoted Ts (I) and xx the noise is yn (x(). An experiment may be performed to measure nyy(m) and Yyyi(O)o Consider the systems depicted in Figure 3olo If the signal and the noise are uncorrelated, then

-xx IT(.)l 2 MA |T(<I I 2 Noise Sowce Only Signal and Noise Figure 341 SO r n ^M-T-l7jj I [txl r<(W)] (3#1W) so thats eliminating the transfer- function sn n ar ^'s: ^() ~'vr'snf,) and. n (m) axe measured quantities and Y. (M) iS a YyY Yy xx' / lnown pertutrbation, Thas by perfmomri the two experiments ouatlined above$ one may be able to infer from the results the shape of the input noise power density spectramz t (t). and establish the shape of the unknown driving force~ 2. Output Noise A point, unloaded, nuclear reactor is now to be characterized'by a linear system with a white noise dxiving forzce such that the input power density spectrum is a constanmt.% (Y ) = Co We can normalize the problem by assigning the value 1 to Co

-17In a measurement of the output noise from a reactor some of the neutrons which we measure may be source neutrons or other neutrons which have been previously characterized as part of the assumed white noise input. This effect is taken into account by assuming that some of the noise which is measured has not been operated on by the reactor transfer function. The system under consideration is shown in Figure 3.2 where n(t) is the assumed noise source, G(t) is the reactor s Green's Function, y5(t) is that portion of the noise which has been operated on by G(t), portions of n(t) are shown driving both output and input and y(t) is the measured output which includes both y (t) and n2(t)o n2(t n(t: _ ON G(T) (t nl(t) y,(t) y(t) Figure 3,2 Noise Measurement Block Diagram' It has been shown in Chapter II that the power density spectrum of the output noise of a linear system is proportional to the square of the modulus of the transfer function so that, in this case, Z snce we have assumed, n(t) is white noise and nn1(3.3)() s^.nce we have assumed n-i(t) is white noise and YI ^ (co) s= lo

The measured output, y(t), is equal to the sum ys(t) + n2(t)It is then necessary to find the power density spectrum of y(t). The autocorrelation function of y(t) may be written T - T y(Tr) =Iim 2 — (t)y(trdt The above equation show s that the au coelation uncton of rT \ysn fT 2 2 -7- -T +unItrn r7' P()a4d' /? 2/ T2- J6co The above. equation shows that the autocorrelation function of y(t) is equal to the autocorrelation function of ys(t) plus the autocorrelation function ofn2(t) plus the sum of the cross-correlation functions r (T) and ). gsT Ysn2

-19If it is assumed that n(t) and ys(t) are uncorrelated then Yn{ (t) =r f/ ct)y (t+0)14T 2 T T-SCo T' j< Taking the Fourier Transform we see that -7- ) (37) And substituting for (n o)and Ty y (w), we get XJiyy )= A1(f) (3.8) where we have normalizedy assn.n() antat 2n2() to 1 %yW-^ (f ^\'T ~~t/ ~~3 +j\J(3.6) Taking the sbriti a pointsfnloaded reactor And substituting for. -, we get/. ~ /u;,. -f —

~20where.j are the roots of the inhour equations,, and Xj are the fraction and decay constant respectively for each delayed neutron precursor, p is the reactivity, and n is the number of delayed neutron groups. Using the above, the power density spectrum is given by 71-/ rIn - (( / +A 4 XAg-L ) (3.10) We have not, as yet, taken into account the average neutron level in the reactor. To do this, we define k to be the average neutron level and introduce the term k~26() into the power density spectrum 2 indicating that when w -> 0 the power density spectrum is to go to ke Using this, the measured power density spectrum is written +yw) = / 6^ + J X(3. 11) j-D key Inverse Fourier transforming, the autocorrelation function becomes tU"fy= t)~lf(7e) h-h _S;0&.(3-12) +1X- flt.fu/Y s Tf Is co r n f o a e (24) The form of this autocorrelation function agrees with Velezo (2 The inversion indicated above is straight forward if care is taken in choosing the path of integration in the complex plane so that no positive exponential occurs. The result is

-21nl+ n+l A i T n + =,, 6(r)+ke+ZArB eUI(3.12a) I=61rI)+k+)4 j n+/ where j, s AJ. +(4 k-/ J R The count rate variance to mean ratio is related to the autocorrelation function by Equation (2.33) repeated below with x replaced by kT. Va r l nrl )-k 7.r.. /.Y. d r/T'"d'"d'T (3.13) Mean k T- Yy Performing the indicated operations the equation for variance to mean ratio becomes Va r T2 - k - n+ A, BerY j2 AI_ _ J -(3 K kW =l ]' This is the quantity of interest in the experimental work to be detailed later. The above' derivation- although straight-forward, does not make explicit the characteristics of the multiplication processes which contribute to the form of the final equation. The derivation to follow will deal with these multipllcaton processes d nd also will show how the detecting system affects the result.

-22B. Physical Approach The count-rate variance to mean ratio may be derived by consideration of the multiplication processes in a nuclear reactor without reference to the stochastic process functions defined earlier. Here, no white noise source is hypothesized; the argument follows from basic probabilities to be defined. This derivation takes into account the efficiency of the detector which is used to measure the output noise. 1. Definitions Consider two time intervals, Atl and At2. An equation for the expected number of neutrons detected in these two non-overlapping intervals will be developed by following the fission chains. Define: p(mAtl) = probability of m detections in Atl p(n,At2) = probability of n detections in At2 p(m,Atl;n,At2) - joint probability of m detections in At1 and n detections in At2 p(n,At2pm,Atl) = conditional probability of n detections in At2 on the hypothesis of m detections in Atl <c(tl)c(t2)AtlAt2> = expected number of counts in Atl and At2 2. Physical Derivation In order to compute the count-rate variance to mean ratio, the expected number of counts in a pair of intervals of time is first computed. This is done by observing that for small intervals the expected number of counts is approximated by the probability of a count, thus: lim c (t )c(t2)AtlAt2 l mp(mAt; nX;nAt2) Ati1 jo0nAt (3.15) = <c(tl)c(t2)dtldt2> = p(l,dt1;l,dt2)

-23This probability can be separated into two terms, one term representing those events for which the probability of a count in At2 is independent of whether or not a neutron was detected in Atl, the second term representing those events for which the probability of a count in At2 depends upon whether or not a neutron was detected in At1 /t>(/,,; / 4 ) -^ (/,; /, )+4f (/,d t X/f,^ ) (3.16) p, ),, p44) + p(, t,)p(/, /,, 1/0) The first term is identified as the probability of an "accidental" pair of counts and the second term is identified as the probability of a "coupled" pair of counts. Accidental pairs of counts arise from the deteption of a pair of neutrons which have no common fission as an ancestor; coupled pairs arise from the detection of two neutrons having a common ancestor fission. This will become more evident in the following development, a) Accidental Pairs Accidental pairs of counts arise from detection of neutrons which do not belong to a common fission chain, Figure 3.3 illustrates possible events leading to an accidental pair of counts _x _ _// I x x 1' — Detection Photo in dt2 Fission v Detection SourJ_ _ __ _ / in dtl me Neule Neut:~ X -- 0_ dt1 dt2 Figure 3.3 Possible Chains for an Accidental Pair of Counts,

.24. To derive an expression for the probability of an accidental pair of counts, define: F = average fission rate - /fl y (Ec nf (,r) zrc rc/E 9 (3.17) E 3 where n(Egr)d rdE P the expected number of neutrons in d3 r about r with energy in dE about E, Zf(Er) ^ the probability per unit path for small paths that a neutron with energy E at space point r suffer a collision which induces a fission. v = neutron speed corresponding to energy E e average counter efficiency Ir~'0Ep (E, t/7rJf(E r) 03OI 4' (E jwr)J( ) /c/E E r where E' the probability per unit path for small paths that a neutron with energy E at space point r suffers a collision which results in a detection. Using these definitions, the probability of a detection in an interval of time dt is p( / c/Olt)9 f (3-.18) Thus, the probability of an accidental pair of counts in dt1 and dt2 is given by p(6 dfi)p (/ d ) p V / (3,19)

-25b) Coupled Pairs Figure 3 4 illustrates the possible ancestry of a coupled pair of counts, q fissions occur |-|t between t & t2. _ Detection in dt2 Common fission at t p fissions S0 Detection.... occur between in dt Time t t 1 tt 1t2 t dtl dt2 Figure 3 4 Possible Events Leading Up to a Coupled Pair of Counts. As indicated in Equation (3-16), the quantity must be computed. First, p(ldtl) can be written as (see Equation (3.18)) P (/1 dfJ = AsCc/7;; (3.20) To compute p(ldt2 |ladtl), the following quantities are defined: V m the number of neutrons emitted in a fission N1 = the probability per unit time for small times that a neutron born at time t have a progeny (including itself) in the system at time tlo N2 the probability per unit time for small times that a neutron born at time t have a progeny (including itself) in the system at time t2.

-26Using the above, the probability of one count in dt1 from a progeny of a fission which emits v neutrons at time t is -Fd-nl 6 dI, The probability of a count in dt2 from a progeny of the same fission at time t which produced a detection in dt1 is ->/)~ /2 df So3 the probability of a pair of counts in dtl and dt2 due to the progeny neutrons of a common fission at time t which emitted v neutrons is gLFrE/ tof/V/ Q/7 d7; A fission at tine t may emit any of several numbers of neutrons; so, considering the average common ancestor fission we average over the V s. The probability of one detection in dtl and one detection in dt2 due to any common ancestor fission is given by integrating over all past time so that the probability of a coupled pair of counts is written, ~/.d4),)6pe/7 //dt) J=-f&2(I -7)<t/df V/V oafA (3. 21) c) Probability of a Pair of Counts Equations (3o15), (3o16), (3o19) and (3o21) may be combined to give the expected number of pairs of counts in intervals dt, and dt, due to both accidental and coupled pairs, yielding: C (it)c( t C (oztc/& Qt; A c/72 /v) (3.22) = FL tc-k t + F(F z-M7) d4,o't z A/f It -GO

-27It remains to determine N1 and Np2. From the reactor kinetic equations, let us suppose that N,-) Aje Nt^)A~l (3.23) J=l /=/ where the Ai are the numerator terms of the reactor transfer function in the form of Equation (3.9) and., are roots of the inhour equation. At this point, it is not obvious that the expressions (3.23) truly define the required probabilities per unit time. These identifications, however, lead to the sanme result as the system approach, thus we will conclude that the quantities N1 and. N2 are properly defined and identified here. Now, the expected number of counts in dat, and dt2 may be written P& ^/ t 1 <c c ^)cft) fzdtf, -.t =,F 6 F6f CO;J(324)) Doing the inttegral, we get <C (/)C(tZ /OlV -t / A/tc/i^ __(3.25) Now we d.o the sum over j, identifying d 3 y-3 - -5. (3. 26) 6"/~~~~~~

-28The equation now becomes <C^ ^t,^(t)c(7)a7>Wl f /fc/?>- ^/fi ( 1)/fz Be t (3.27) /=/ In the experiment, we measure over an interval of length T which includes t1 and t2. If we have c covits in an interval, the nmber of pairs of counts in the interval is c(c-l)/2 and the expected number of pairs of counts is c(c-l)/2. The epected number of pairs of counts in an interval of length T is related to the expected number of pairs of counts in dtl and dt2 by cr-,j f c ( )c(tC d tc/ftz (3.28),=o 0 =o Applying this relation to Equation (3.27) and doing the integrals, we get c c -/) FZZ F/Th </7A8 t 1] (3.29) The temi FeT can be identified as c so that the above equation may be written __ -_+ _ ^S /C B He1 (3-30) Rearranging terms, the eq.uation becomes: Ia ___ — li /-_- I (3.31) cMgean / 72' ____ [ ___ -C

-29c-2 where the identification of - (T) with mea (r) has been made from Of _: mean the definition of variance to mean ratio. The introduction of the detector into this derivation has introduced the counter efficiency, e, into the equation for variance to mean. Also. the study of the fission chains has introduced the constant v2 which does not appear in Equation (3.14). The normalization implicitly assumed in this derivation is not necessarily the same as that assumed in the derivation leading to (3.14) so it may be expected that the two results would differ by a constant factor, which they do,

IV. EQUIPMENT A. Experimental Setup Two experimental configurations were used in the experiment. One configuration was used to record data on the tape recorder and the other configuration was used to count pulses over a gate time and record the count on IBM cards. These two configurations are shown in Figures 4.1 and 4,2 The figures are self-explanatory. Each piece of equipment used is discussed below. Those pieces of equipment which are standard items are discussed briefly with reference to manufacturer's data while the special pieces of equipment are discussed in more detail. Bo Description of Equipment lo Reactor The Ford Nuclear Reactor is a one megawatt swimming pool reactor using 90% enriched U235 MTR type fuel elements. The critical mass, with a carbon reflector on the vertical faces is approximately 3kg. The reactor is light water moderated and cooled. It is used primarily as a neutron source and as a training reactor. The detector was placed at the core-face, in the reflector, for each of the experiments. The core configurations during the experiments are shown in Figures 4,3 and 4.4. 2. Detectors Manufacturer's specifications for the two detectors used are listed below: -50^

HIGH VOLTAGE SIGNAL A FPULSE TAPE GATE AMPLIFIER SHAPER RECORDE SCALER DE-COUPLER PLAYBACK OF ELEMENT SIGNAL CHANNNEL (DISPLACED BY AT) ~I PRE- AMPLIFIER WATER- PROOF CONTAINER DETECTOR REACTOR Figure k.i Data Taking Conxfiguration.

COUNTS IN INTERVAL, T PULSE 7(SERIAL DIGITAL) PE GECONVERTER CARD TAPE GATE ( PARALLEL CARD RECORDER SCALER STAIRCASE TO PUNCHER SERIAL DIGITAL) COUNTS IN INTERVAL, T -/ (PARALLEL STAIRCASE) Figure 4,2 Data Transcription Configura-tion.

-33KEY: 0 EMPTY CHANNEL O Oo S WITH PLUG 0 H 0 H H EMPTY CHANNEL WITH SAMPLE HOLDER S MOVABLE Po-BE SOURCE T DETECTING TUBE CARBON REFLECTOR gB ELEMENT E FUEL ELEMENT SPECIAL ELEMENT WITH Figure 4.3 Core Configuration CONTROL ROD(CR) OR for F Tube ExperSHIM ROD(A,B,C) io be En itsen.s ~ FISSION CHAMBER FOR REACTOR INSTRUMENTATION H H CR B MT T C A Figure 4~ 4 Core Co'figuration for Fission Chamber Experiments.

a) Fission Chamber Manufacturer Westinghouse Electric Co. Model Noo WL-6376 Mechanical data overall length 117/ 8 diameter (maxo) 2-3/32" net weight 1-3/4 lb. insulating matIls polystyrene and alumina sensitive length 6" body mat l aluminum neutron sensitive mat2l 2 mg/cm2 U308 fully enriched in -235 filling A-N2 at 1 atm, Operational ratings operating voltage (app ) 300 volts sensitivity 0.7 count/neutron/cm2 neutron flux range 2. 5 to 2, 5x105 neutrons/cm2/sec operating plateau 200 to 800 volts output 200M volts, rise time 0.2O sec b) BF3 Tube Manufacturer N, Wood Counter Labo Model No G-10-12 Mechanical data overall length 17" diameter 1" net weight not specified insulating matIls glass sensitive length 12" body mat'l aluminum -neutron sensitive matl enriched BF3 (96% B10) filling BF3 and quenching gas at 40 cm Eg Operational ratings operating voltage (appo) 2000 volts sensitivity not specified neutron flux range not specified operating plateau length of plateau = 300 volts output not specified The detectors were mounted in waterproof cans, The BF3 tube had its pre-amplifier in its can, The pre-amplifier for the fission chamber was outside of the pool, The BF3 tube had a lead gamma ray shield, 1/4" thicke wrapped around its water-proof can, The high voltage leads were enclosed in a one inch diameter "Tygon" tubeo

-35 3o Pre-Amplifiers The pre-amplifier used with the BF3 tube was a transistorized cathode follower with a 15 volt battery as a power supply. This preamplifier was used for impedance match; the gain from its input to the end of the twenty foot cable leading to the de-coupler was approximately 0,3. The pre-amplifier was built in a 1" diameter, 6" long aluminum tube with a connector of the EF3 tube at its input and a connector to the signal cable at its output, No provision was made to turn the pre-amplifier power supply on and off, Operating continuously, the battery life was approximately two weeks. The pre-amplifier used with the fission chamber was the standard pre-amplifier used with a model 218 linear amplifier. The manufacturer's specifications are given below: Manufacturer Baird Atomic Inco Model No, 219A Operating characteristics Maximum gain 30 Bandwidth 2mc Rise time,025pt sec maxo Fall time 10 sec mino Input polarity positive or negative Input impedance 1000 megohm Line arity 2% 4. Amplifier The amplifier was used on the delay line bandwidth with the output of the pulse height selector. Therefore, only the specifications corresponding to this operating condition are given: Manufacturer Baird Atomic Inc. Model 218 Operating characteristics Bandwidth A2mc Rise time Oe.2j sec Decay time 0. 8j sec

536Maximum gain 1600 Input polarity negative Input impedance 2000 ohms Output imedance (PHS) 4700 ohms Linearity 1% Amplitude (PHS) 60 volts negative 5. Pulse Shaper To tape record pulses successfully it was necessary to match the output impedance of the amplifier to the input impedance of the tape recorder and to supply the tape recorder with pulses of optimum shape and size. In order to investigate pulse shape and size a combination one-shot multivibrator, cathode follower circuit with variable height and width output pulses was built. The pulse used to drive the tape recorder was optimized on this pulse shaper with the optimum pulse being a 20 volt, 8pt sec nearly square pulse. The circuit diagram of the pulse shaper is shown in Figure 4o 5, 6. Tape Recorder Manufacturer Ampex Corpo Model Noo 307 Operating characteristics NO. of channels 1 Tape speed 60"'/sec Drive Capstan, friction drive Wow and flutter o01% rms Tape width 1/4" Maximum size reel 10 1/2" Maximum length of tape 3600 9 lmil thickness Bandwidth 80 kc 7. Gate Scaler Manufacturer Dymec Corp. Model No. 2500 Operating characteristics Range 1 cps to 100 kc Accuracy tl colu-nt, ~stability of crystal Stability 1 part in 105 Registration 5 places, colaumnar display) to 99,999

B+ B+ B+ B+ 682,u.I F V[X |64 -t |6 6.8 KQ S 2.2.001ol I. F 6C4 B + megQ ( 5.6Ka 25'IF 6.8Ka INPUT 5 FOK e6C4 -1- ~ ~~I Keg' l <+ 5,ur-L0 b F L12A^ ^ """Z^T7 """ 3.3KQlj OUTPUT:? 3K5Q INVERTER MONOSTABLE MULTIVIBRATOR CATHODE FOLLOWER Figure 4,5 Circuit Diagram of Pulse Shaper,

9 9 9 9 SIGNAL GATE 9I8 8 INPUT A a_ 7 a7a7 7 7 77 7 6 6 6 6 6 55 5i 5 5 RESET 44 44 4 3 3 3 3 3 @ —--------- ^& —--------- t 131 i i i 22 22 2 DIGITAL DISPLAY COUNTE INPUT B FREQ.STD, ^ ___oSWITCH PRESET COUNTER EXT. - ~@@~RESET L INT. I UT MULTIPLIER TISPLM u TIME CIRtCUI INTERNAL TIME BASE Figure 4.6 Siwplif ed Block Diagram of Gate Scaler.

-40o the preset counter has reached its preset count and the signal gate has been closed. After this."display time" it resets both the display and present counters back to zero and a new count cycle begins, 8, Converter In order to punch the. numbers displayed on the gate scaler onto IBM cards using an IBM-024 card puncher, it was necessary to build a machine to convert the "parallel staircase" output of the gate scaler to the "serial digital" input required by the card puncher~ This machine is called the "converter" and its circuit diagrams are shown in Figures 4 7 and 4.8, The operation of the converter is as follows: a) Initially, the stepping relay is in the no signal position while the gate circuit of the gate scaler is closed and the display scaler is counting, b) When the scaler displays its count, a signal is sent to the convertert which simultaneously resets the stepping relay to position 1 and opens the circuit of the stepping relay wiper so that no signals are admitted to the converter, c) The signal circuit of the converter is closed and a signal representing zero (140 volts) is admitted. The bank of ten relays converts this voltage into a closed circuit corresponding to the activation of the zero key in the card puncher and a zero is punched. When the card puncher punches the zero, a signal is fed back from it to the converter which makes the stepping relay simultaneously open its signal circuit and step to position 2.

',,-ITT-FaC,, TeTIS oq ^,, S IBoXirTS TSTT J -cI9 -lq-sAuoo UO'T3 JOJuioTl'ILtj s.InfgT 0 / Z?.;'9., 9S f) zO l'/ g / o- [1 — -T - -~- [ -} - i- --' _no i_.__!P_/ i _-n, LJ I., 2 0' 4; L -$ H q <6@wr 6>fJ ZT - -- AOZ- u — ___ 1 I _ I _ — 0 ~~L _ ^A, 10 1/2- / X XJ t',/zZ <A O L__-__? O,-, T-o o'? L,, — / —k_~oOzj,,-/, ^0!.1J d - " ~: - M o -;IT, J < T 0-63 48 tooo OA A — -- OL-',- ] <'-6,,'.,.?0! < ^ _= —Kg M I I I ~' I fr' -- ^J __ II ~o-'C ~. I^I ___ - -i1"^. [ _M-o A ~\ />OL- i -i- y^ -- 6 -' MO? ~ L M I -' AOL -7 *~ ~ MO? I - T'o _ i i <1.M~,I I'-_w T\_ -I i ~' ~'z I so /.=-I —c 1 L ~'"'1~~~~~~~~~~~~~LUc/

6AS7 CI~~ Ijy^^ 1 so~~~~~~~~~~~~~~~~~~~~~~~-.I/H~ 4 4 z~ 5-5;2 4~~~~~~~~~~~~~~~~ Is ___i7 __v(/\ v 45c0 4c70K'OOiL w /00 _a 6AU/6 Cn, 4 \^ i-i J -- - -- -- -? —- - -- ^ ~70K< - (' 633 V u. \ \ —---—,|| ---------- ] —--— \ l~~~2ci - 3' 6. i'"" T ~ X ^ "'______;"" 12 V D. C. 1/ ~ ~ ~ ~ ~02 Vr /<7F \ /O W /OK( -------- ---— 40^^. 4 0^r f \~ 6. 3Y J^ ---------- ^ ------------— ^ -------. —-------------- )2 \ D. C. /26~~~0 V/2 v Figure 4.8. Power Supply for Converter. 60 ^ ~~~~Figure 4.8. Po~-er Supply for Converter.

.43d) The signal circuit on the stepping relay closes, admitting a voltage corresponding to the i-th digit on the display counter (50v to 140v representing 9 to zero in 10 volt steps). The bank of ten relays translates this voltage into an appropriate closed circuit and the corresponding number is punched, Punching of the number gives a feedback which opens the stepping relay signal circuit and steps it to the next position. e) Step d) is repeated for i = 1 to 5o f) When the stepping relay reaches the no-signal position the scaler resets and a new count is begun. The sequence of events a through f above takes about one second, Every sixth digit punched on an IBM card is a zero. This helps in editing the cards and checking for errorso 9. IBM 024 Card Puncher The IBM 024 card puncher is designed to be a hand operated key punch with a typewriter keyboard. This typewriter was modified so that the closing of relays could be controlled by the converter instead of by the depressing of keys. This was accomplished by wiring the converter relay contacts in parallel with the keyboard contacts (see Figure 4, 7). Another modification of the card puncher was required to make it release the card after it had punched 72 columns and go on to the next card. This modification consisted of a cam which was mounted on the control drum shaft and used to trip a microswitch which controlled. the automatic feed mechanism, When column 72 was punched the microswitch released thhe nesen card and ented te by means of the IBM 024 card. feed mechanism.

C. Operating Characteristics The operation of the above experimental equipment was routine after several bugs had been worked out. Some trouble was incurred in the stability of the D. C. voltages in the system but this was minimized by using a regulated power supply and occasionally checking bias adjustments. No relay failure was encountered although some of the relays in the converter operated over two million times,

V. EXPERIMENT Two significant experimental runs were accomplished on the Ford Nuclear Reactor at the University of Michigan. Data was taken in run #1 using a fission chamber and the data in run #2 was taken using a BF3 tube. In each of these experiments several tape recordings were made at different reactivities. In Appendix A the reactor parameters and data points for these experiments can be found. This section discusses the experimental conditions which must be met to insure success and the steps which were taken to meet these conditions. A. Experimental Conditions In order to perform an experiment to measure nuclear reactor parameters by methods of stochastic processes on a reactor such as the Ford Reactor it is necessary to optimize the experimental technique on several considerations. The competing factors which lead to an optimization are the need for a high efficiency detector with small sensitivity to gamma radiation, the need for putting the detector in a region of low neutron entropy while being limited in count rate by the resolving time of the equipment, and the need to apply point reactor theory to the analysis while locating the detector as close to the reactor as possible. Recall that the form of the equation for count-rate-variance to mean ratio is V (T) = 1 + E (correlated terms) (5.1) M The first term on the right hand side of Equation (5o1) arises from the accidental pairs of counts as shown in Chapter III. The second -45

w46term arises from the coupled pairs. The problems mentioned above arise primarily from the need for the correlated part to'be significant compared to one; indicating that E should be as large as possible and that the correlated terms should. also be as large as possible, The discussion of experimental conditions leading to large values of e will be discussed under "counter efficiency" and those leading to high values of correlation will be discussed under "correlated termso," l, Counter Efficiency In the theoretical development, counter efficiency e, has been defined as the average number of neutrons detected per fission in the reactor. To obtain a ratio of V/M significantly different from one, we need this efficiency to be at least 10 and preferably of order.4 10. This need for high efficiency immediately suggests using a large counter which will intercept many neutrons, a sensitive detector which will count a large proportion of the neutrons which it intercepts, and placing the counter in a position of high neutron densityo Each of these topics is discussed below. a) Size of Counter Tihe size of the counter used is limited by the practical dimensions of counters, the available space in which to put a counter, and the requirements for resolving time put upon the instrument. In this experiment, a No Wood Lab G-10-12 BF3 tube and a Westinghouse-WL 6376 fission chamber were used. Each of these counters, when enclosed in a waterproof container, was of such a size as to fit into a fuel element channel in the Ford Nuclear Reactor. The size of the counters

-47was found to be satisfactory in all respects; the dimensions are given under "equipment. b) Counter Sensitivity In a reactor which has been operated at high power, high detector sensitivity is especially hard to obtain since, in general, a detector that is sensitive to neutrons is also sensitive to gamma radiation. Thus, the problem of achieving high detector sensitivity is intimately related to the problem of eliminating the gamma background. The BFO tube used was considerably more sensitive to both neutron and gamma radiation than was the fission chamber. For this reason, the BFo tube could not be used in the initial experiments in 3 which severe precautions to eliminate gamma ray background were not taken. Using the fission chamber for these experiments, E was small, and thus the correlated parts were not as large as in the later experiments using the BF3 tube. To achieve a higher sensitivity, it was necessary to use the BF3 tube and to take precautions against detecting gamma radiation, This was accomplished by first determining what the effects of a gamma ray background were on the characteristics of the counter9 and then taking steps to eliminate these effects. Qualitative measurements of the effect of gamma ray background on the BF3 tube were made. The effect on the voltage traverse of increased gamma background is to both shorten the length of the plateau and to shift the location of the plateau to a higher voltage. The corresponding effect on the discriminator traverse is to shorten the length of the discriminator "plateau" and to make the slope more steep.

.48The experimental set-up used to make these measurements is shown in Figure 5.1. The BF3 tube was kept at a constant distance from the neutron source but moved relative to the gamma-ray source (in this case the shut down reactor). Observations of pulse-height on the oscilloscope were made simultaneously with the voltage and discriminator traverses. It was observed, that with increasing gamma ray intensity the pulse height of the neutron pulses decreased while the pulse height of the gamma pulses appeared to remain approximately constant. The decrease in pulse height of the neutron pulses is attributed to the lowering of the effective interelectrode voltage by the gamma ray field. This is caused by the fact that the ions produced by the gamma rays provide a space charge opposed to the interelectrode voltage which has the effect of lowering the effective interelectrode voltage resulting in smaller pulses since the gas multiplication of the proportional counter. is a strong function of the voltage. The behavior of the pulses due to gammas appears to be due to competing effects. One would expect the gamma ray pulse height to decrease due to the lowering of the gas multiplication in the proportional region in the same way as the neutron pulse heights decrease. However, an increase in the gamma ray field also gives rise to an increase in simultaneous gamma ray pulseso Apparently in this case} the effective increase in gamma ray pulse height due to simultaneous events is of the same order of magnitude as the decrease in size of gamma ray pulses due to less gas multiplicationo

-49LOCATIONS OF DETECTOR FOR GAMMA BACKGROUND -_ MEASUREMENTS p, / I I ~/ I: SOURCE CORE (COLD ELEMENTS) REFLECTOR Figure 5. 1 Gamma Background Measuremeit.,

-50One way to minimize this gamma ray effect is to minimize the probability of simultaneous events. This indicates a need for the shortest possible resolving time in the equipment. Again, however there are competing csaionsiderations. To make the resolving time as short as possible one would use the smallest instrument possible. This cannot be done because of the requirement for a large size counter for high efficiency. Thus, the minimum resolving time is determined by the minimum detector size consonant with efficiency considerations. Of course, one may also optimize the detector configuration by placing the electrodes as close together as possible. In making this type measurement one should always optimize the performance of the BF3 tube by operating at proper voltage and discriminator values. The foregoing discussion indicates that special attention must be paid to the level of the gamma ray background in determining optimum voltage and pulse height selection since this optimum is strongly effected by the magnitude of the gamma ray backgrounds The fission chamber used had a low operating voltage, lower sensitivity, and small gas mutiplication and was therefore able to operate in considerably larger gamma-ray fields than the BF3 tube. c) Location of Counter From the standpoint of achieving high counter efficiency it is apparent that with a thermal neutron detector it is most desirable to place the detector in a position of high thermal neutron flux,

512, Correlated Terms The magnitude of the correlated terms depends strongly upon the number of correlated events, The relative number of correlated events depends upon the operating configuration of the reactor and the location of the detector. If the reactor is operating at a very large negative reactivity, the multiplication is small. Since the pairs of detections traceable to a common fission are the ones responsible for correlated terms, one would suspect that small multiplication would yield small correlation. This is born out both by the point reactor theory and the relative magnitude of the variance/mean curves taken at different reactivities with the BF3 tube. Therefore, to achieve a high degree of correlation it is necessary to operate the reactor at very nearly critical. Although no space dependent model for stochastic processes has yet been found one may make some intuitive guesses about the magnitude of the correlated terms as a function of spaceo Since the core of the reactor is the source of correlated events, one would expect that the relative number -of correlated events detected would decrease as the detector is moved away from the core. Making an analogy to entropy, this is tantamount to saying that the entropy of the:- neutrons increases with increasing distance from their source, This implies that there is some point of minimum neutron entropy in the core and that entropy increases as we move away from that point. According to the above arguments, the optimum operating configuration for the experiment is with the reactor nearly critical and the detector close to the core,

-52B. Equipment Limitations One must operate within the confines of equipnent limitations when meeting the above experimental conditions. The most serious limitations on the experiment are the resolving time and the time base stability. In this experiment, the performance of the system with respect to these two considerations is limited by the tape recorder. The effective resolving time of the tape recorder is approximately 5Li secs and its time base stability (wow ad flutter) is such that 0, 2% inaccuracies may be expectedo 1. Resolving Time The effects upon the mean number of counts per interval and the -variance of the number of counts per interval due to finite resolving time are not the same, The differences are discussed belowo a) Mean Number of Counts Per Interval The number of events registered in an interval of time must be less than or equal to the true number of events occurring in that interval of timeo This is due to -the finite resolving time of any physical system. Two extreme cases may be considered with the expectation that real life is represented by some intermediate caseo The two cases considered are that of the paralyzable counter and the nonparalyzable counter. For a comparison of these two conditions see Figure 5 2~ The notation used is: p dead time following occurrence of an event n mean observed count rate N mean true cont rate

-53PARALYZABLE COUNTS TRUE EVENTS NON - PARALYZABLE COUNTS Figure 5.2 Relationship of True Events to Registrated Events in Earayrzable and.Ton-paralyzable Counters,

— 54. 1) Paralyzable Counteer - The paralyzable situation is defined by a system which is unable to provide a second output pulse unless there is a time interval of at least p between two successive true events. For this case the expected mean observed count rate is 2) Nonpa.ralyzable Counter - The nonparalyzable situation is defined by a system which is unaffected by events occurring during the recovery time, p. For this case, the expected mean obseraved count rate is given approximately by /7 O N-1 (5.3) i N/3 for large N and small po A curve showing these effects is shown in Figure 5<.3 b) Variance of nmber -of Coants Per Interval The following notation is employed in addition to that used under a)o -1 = variance of measured number of counts in interval os2 variance of true number of counts in interval T length of interval,S n mean number of tue events in intewval of length T nT' mean number of measured events in interval of length T 1) Paralyzable Counter - Making the assumption that N is large, the distribution is normal~ and p is small, the measured vari ance is given approximately by

-551.0 NO DEAD -- TIME LOSS~ - NON- PARALYZABLE Ic 0.5 /'o, P yzabe PARALYZABLE Ca ~~"^~~,~CASE - 0 0.5 1.0 1.5 2.0 N P 1.0'i' 0.5 — / PARALYZABLE NON- PARALYZABLE - 0 Figure 5- 4 Ratio of Observed to True Varoiance Verses Np for Paralyzable and Non-Paralyzable Counter. - \ ^ - - -- -- - -- - -- -

-562 2 -2AO (~4) 2) Nonparalyzable Counter - Making the same assumptions as in the paralyzable cases the measured variance is given approximately by i/2^ e^"Z (5,5) / ^ f/a^)3 A cuxre showing theese effects is shown in Figure 5A, 4 c) Variance to Mean Using the above results^ the measurecd ratio of variance to mean for the two cases is: 1) Paralyzable V' ^!e- (5.6) 2) ITonparalyzable ^/ _ __ __ _ (57) T- N$T (,~ )Zo A -curre showing these effects is shown in Figu-re 5T50 Using these resultsl it is found. that the counting losses to the variance to mean ratio limit the counting rate to about 2000 counts per second for a 5% loss in variance to mean, (Paralyzable predicts 3% loss_ nonparalyzable predicts 5.5% loss. ) In accordance with these results, the average couxrt rate was kept below 2500 CPS whenever possible,

-571.0 - i+ 0.5 ii - NON- PARALYZABLE.. _E 0i ii imi,! i ii iii 0 0.5 1.0 1.5 2.0 N P Figure 5.5 Ratio of Measured to True Variance to Mean Ratio Versus Np.

.582, Time Base Stability The stability of the time base used in these measurements was determirned by the stability of the tape transport system since it was by far the most unstable factor in the overall time base4 The crystal oscillator in the scaler was much more stable,^ Figure 5 6 shows a sample measurement of the wow and flutter characteristics of the tape recorder. These effects can be analyzed as follows: If ni are the number of observed events in a tine interval T, then if the time interval varies by an amount cT arodund T, the experimental -variance to mean ratio can be written,2 (n;tEh - (Z IZ 2, >) (V3Lr) =, ftS (/=)-/}__ __ _ (5.8) expt? ~ci Z6 /=1 / where p is the number of intervals measured. The theoretical variance to mean ratio (taking into account the resolving time but not the time base instability) is (Var e________________ O thas 7 —-an],,- (5. 9) evar Thus expanding (mean)expt. and collecting terms

0.3 0.2 RECORDED COUNT RATE (ON TAPE) - *.*-.-. PLAYBACK COUNT RATE -00000 Z 0 _o 9 I o 0 0 S o %ef%% 0 o ob~o~o~~oo'8 oO d~d o o o~.'O. -0.21 Figure 56 Wow and. e easemen on Ampex 07 Tape Recorder -0.2 oO a.. o o -0.3 TIME -- Figure 5,6 Wow and Flutter Measurement on Amrpex 307 Tape Recorder.

-60\Va = / __ Var + Var 62 Var A/?ea7 f6/te 1/lea/n / E t/yea 1~6 MSea/ (5.10) I2 + 6 /eaX hAeoy sign- = /-26+Z ( Var I/ - 6 Ml Z/t^ heo9y s in — The rms wow and flutter of the tape recorder was measured to be about 0o2%. The corresponding exror from the above is (5.11) /-. oo R+C/~./o-~. 0 Thus, the expected variation in mean expt due to time base inaccuracies is approximately 0X 24 o. Note that second order terms begin to enter the piacture when E becomes large. Co Experimental Procedure To insure that the reactor was operating in a suitable manner for data taking, the following experimental procedure was adoptedo The objects of the procedure are to establish the reactivity dauring the experiment- obtain an optimum count rate, and make preliminary investigations into the character of the data. The first step in the procedure is to eliminate the gamma ray background as much as possible, Since the Ford Nuclear Reactor

-61has been operated at high power for several years, this background can be very large. The major contribution to eliminating the gamma ray background was the unloading of the fuel elements which had been operated at high power and the reloading of the core with fuel elements which had never operated at above approximately ten watts. Further reduction in gamma ray background was achieved by moving the reactor core to the thermal column position where the activity from the structure is much lower since the reactor is seldom operated there and the maximum operating power at the thermal column position had been 100KW. Not only was the maximum amount of gamma ray background removed, but, in the case of the BF3 tube, the detector was also shielded with 1/4 inch of lead. The optimum operating voltages were determined as outlined in (5-A). When the above conditions were met, the control rod was calibrated by the period method. The control rod calibration curves are shown in Figures 5<7 and 5.8. Appendix B contains the data for the control rod calibration in the BF3 tube experiment. During control rod calibration, care was taken to assure that the reactor power did not rise to extreme levels. Having the control rod calibrated, criticality was established by establishing a steady state with no (external) source present and at a fairly high level (-N10 watts). This steady state was achieved by manipulating shim rods with the control rod at the upper limit, This enabled the control rod to determine the degree of subcriticality at which the measurements were to be taken.

-622.0 _ _ 1`20 *, __ _ _________ x 15OD PO N. I-re rential Rod Calibration. -J I I a0.5 Ia: 0 36 34 32 30 28 26 24 22 20 18 16 14 12 ROD POSITION,INCHES Figure 5.7 Differential Rod Calibration.

-632.6 —. —- - 2.4 2.2 — XBF-3 — 2.0 1.8 "~~~~S <iX 2~ ~BF -2 0.8 z 0.6 0.4 0.2 12 14 16 18 20 22 24 26 28 30 32 34 36 ROD POSITION, INCHES Figure 5.8 Integral Rod Calibration. Figure 5.8 Integral Rod. Calibration.

;64o Having established. criticality, the control rod. was dxriven into the core an amount determixned by the initial experimental re. activity which was selected at 50% insertion. The power level of the reactor was allowed. to decrease until the detecting instrment was recording at approximately its optimum count rate. At this time, the neutron source was moved closer to the reactor until the cousnt rate leveled off. (Due to the fact that the source was hung into the pool from a crane, this operation had to be adjusteda several times until the proper steady state was reachedo ) The count rate was moniteed. for- more than thirty minutes in every instance to be certain that a steady state was achieved before any data was taken. At this point, the reactivity was determined, the optimum count rate was achieved, and a subcritical steady:state was obtained. As a final check on the operating condition, a series of fifty gates of 1/2 second duration were counted directly from the scaler. The ratio of variance to mean for this sample was determined using a desk calculator. If this ratio was significantly gyeater than. one, the tape recorder was prepared. for recording and. a tape recording of reactor noise was then made. The mechanical dJrive on the tape recorder tape transport system was warmed up prior to taking data so that the expansion of the friction drive from heating would not shift the time base at playback. The recording heads and drive were cleaned prior to recordingD During recording, the output of the pulse shaper driVen by the nonoverlpading amplifier was suplying signal to the tape recorderO

-65The output of the tape recorder was connected to the gate scaler and the gate was set at ten seconds. The number of counts over each of these ten second intervals was recorded by hand for later analysis to be sure that steady state had been achieved and that the distribution of these counts was as expected. Care was taken to be sure that spurious electrical transients were kept to a minimum during the experiment by avoiding use of light switches and not recording during the time when the IBM clocks reset. (The clock setting impulse in the building was suspected of introducing an impulse.) After the recording was made, the control rod was moved to a new position and the procedure repeated. The playback of the recorded pulses and subsequent data handling are discussed in Chapter VI.

VI. DATA ANALYSIS The analysis of the data obtained in the experiment had to be made as efficient as possible since there was a large bulk of data to process. The transcription of data from the tape recording to IBM cards was accomplished in a semi-automatic fashion by modifying an IBM 024 card punch as described in Chapter IV. In this section, the procedure for obtaining the data on IBM cards is discussed under Data Handling and the processing of these cards is discussed under Estimation, theory and practice. A. Data Handling The tape recorder functions as a simulated Ford Nuclear Reactor. The tape record is considered to be a representative sample of reactor steady state. The transcription of data from the tape to IBM cards is accomplished by playing the tape recorder to the gate-scaler with the gate set at various gate times. When the gate scaler displays its count, this count is punched on an IBM card as described in Chapter IV. This process is continued until either a group of 600 gates for that gate time are punched or the tape has been run through four times whichever comes first. Harris(3) has shown that this number of measurements leads to satisfactory estimates of variance to mean ratio. For short gate times, the group of 600 gates was punched and for long gate times the tape was run four times and the number of gates punched was less than 600, The reasons for the above criterion were that Harris(l3) showed that samples of this size led to an unbiased estimate of variance/mean and it was observed experimentally that larger samples did not significantly reduce -66~

-67the spread in the data. Increasing the number of gates per data point would not have a large effect on the quality of the data, but would greatly increase the time required to transcribe data. For the longest gate times, approximately 75 nonoverlapping gates per traverse of the tape could be measured. This number of gates leads to an unbiased estimate of the variance to mean ratio. By rerunning the tape some new information is obtained for counts recorded in the gaps where counts were not previously recorded. In this way, a second traverse of the tape adds to the accuracy of the measurement. As the tape is traversed again somewhat more new information is added; however, the returns become less and less significant as the tape is re-run more times for the same gate time. The above criterion balances the small gains from further tape runs against the time required to make these runs. When a quantity of data was punched, representing a spectrum of gate times and a total number of gates of about 40,000, the cards were edited for double punches and omitted punches on the IBM-101 card sorter. It was found that the average number of cards having such a defect was approximately one per 500 which represents an average number of defects per gate of one per 6,000 since there are twelve pieces of data per card. After editing, the cards were arranged according to the input format of the calculational program, and the mean, variance, and variance to mean ratio was computed for each gate. These computations are listed in Appendix A.

-68B. Estimation 1. Estimation of Parameters - Theory Since the parameters which enter the mathematical model for nuclear reactor stochastic processes enter in a nonlinear fashion, the estimation of these parameters must be accomplished by methods different from standard multiple linear regression. No direct technique exists for handling nonlinear estimation; thus, it is necessary to modify linear regression to deal with the problem. In this section, the method of nonlinear estimation will be developed by a geometrical argument which will then be translated to the analytical development. Having made an estimate of the parameters, it is necessary to assess the validity of these estimates. These procedures are also discussed in this section. a) Geometrical Development For identically distributed, independent, Gaussian random variables the "least squares" criterion provides maximum liklihood estimates of parameters. This is the criterion used in the nonlinear estimation procedure to be discussed. Geometrically, the least squares criterion estimates the point on a sub-space generated by the function of interest which is closest to the point in observation space determined by an experiment. This geometrical development is based on work by Box(6) To illustrate this, consider an experiment for which there are three observations of the dependent variable corresponding to three observations of the single independent variable. This makes the observation space a three space. Consider the function of interest to be one

-69which has two parameters so that the function sub-space is a sub-surface in this three-dimensional space. (See Figure 6.1) Observation Point.Function (YlyjY3J) Subspace / / //' / Figure 6.1. Observation Space The following notation is used: Function'-=flI, X 2) (6.1) Observed values of independent variable X =X,X2 3 (6.2) Observed values of dependent variable /y-,, j2, j3 /(6.3)

-70Parameters 0=(090 (6.4) Maximum liklihood estimate of parameters A A 0=0610 (6.5) Sum of squares 3 2 s=f (y,-0,) (6.6) i= / In order to generate the function subsurface, the parameters 01 and 02 are varied over their possible ranges in parameter space (see Figure 6.2). < ^ ~-2 -Range of Definition of Parameters / Parameter Space / Point Q1 i 1 Figure 6.2. Parameter Space Note that a particular point $1 in parameter space maps into a particular point 01 in the function sub-surface of observation space.

-71For identically normally distributed random variables, the leastsquares criterion minimizes the distance between the observation point y and the function sub-surface 9. Thus, a point is determined corA responding to O which are the maximum liklihood estimates of the parameters. Minimum Distance to Surface y / = f(x,$, 2) /./ - Surface Figure 6.3. Observation Space A\ A The algorithm used to find the points 0 and 9 is as follows: 1. Select initial parameters G~ which determines an initial point 0o in the function subsurface (see Figure 6.5). 2. Construct a tangent plane to the function subsurface at the point 0. (This corresponds to a linearization of the function about the point ~o) (see Figure 6.4).

d72Minimization of Distance to 0~T Plane Tangent. 2 - ~7 X Plane T 2 Sub-surface Figure 6.4. Observation Space Figure 6.5. Parameter Space 3. Perform a multiple linear regression in the tangent plane oT~ to find the point AdC in the tangent plane which is a minimum distance - oT from y. This determines new values of parameters G in parameter space (see Figure 6.4). 1I~T 1 4. Using values of 9G = 91, find the corresponding in the subsurface. Now repeat setps 2, 3 and 4 replacing all superscripts (i) by (i + 1). Continue this until a convergent G is determined. 5. Plotting the sum of squares Si against Gi we see that a minimum S is being approached. (see Figure 6.6)

-73S 00 ( l(eS *' (02,S2) Figure 6.6. Sum of Squares vs. G Since the function subsurface may be in general nonlinear, the only claim that may be made for the parameters G which are computed by the above procedure is that they satisfy a local minimum for the length of the line y - 7. Certain questions concerning the confidence one may A have in the parameters G determined by the above method must be answered. These are: 1. For a certain degree of confidence, over what range may the parameter estimates vary? 2. What is the relative confidence that can be placed in the determination of several parameters? 3. Is the local minimum sum of squares found by this method the true absolute minimum? 4. How successfully may one use linear methods for answering questions 1, 2 and 3? The questions raised are answerable by applying methods of statistical analysis discussed below.

-741. A confidence region can be determined which defines a contour surrounding the least squares estimates of the parameters. The formulation is based on an approximation to the function 0 denoted T which is linear in the parameters. If the approximation were correct and the errors were independently and identically normally distributed, the confidence region would include the true values of the parameters with a confidence coefficient of (1 - a) where a is the significance level of the Fisher F-distribution at the appropriate degrees of freedom. The sum of squares on the confidence contour is generated from S = Sm( p p (5. 7) where S = contour sum of squares Sm= minimum sum of squares P = number of parameters N = number of observations F = Fisher's F-distribution value a = significance level The geometrical significance of this confidence contour is as follows. Consider the observation space. The value of ~ determined by least squares lies in the function subsurface. Construct a tangent plane to this subsurface XT. Now, any XT in the tangent plane will be a greater distance from y than T. Applying the F-test, the contours at fixed distances from y in the (T plane are swept out. Call these contours F. (see Figure 6.7)

-75/ T z-^ Figure 6.7. Observation Space This contour also sweeps out a contour in the linearized T parameter space G which will be an ellipse. In general, however, the projection of FT on the function subsurface, called F, will not be an ellipse, but will be some non-ellipsoidal contour. Likewise, the contour in the linearized parameter space GT will be an ellipse, while the contour in parameter space will be non-ellipsoidal (see Figures 6.8 and 6.9).

-76Ellipse Non-ellipse 2 T Figure 6.8. Parameter Space Figure 6.9. Linearized Parameter Space The contours found in the T tangent plane are used to establish the region over which the parameters may be varied for a specified confidence in their estimation. 2. The relative confidence which one can place in the estimation of parameters can be judged by the elongation of the ellipse in normalized, linearized parameter space. That is, all of the parameters are given the same dimension by a normalizing factor so that if the two parameters 91 and G2 are equally well determined, the confidence contour in linearized parameter space will be a circle. If one parameter or one linear combination of parameters is better determined than another, the contour in parameter space will become elliptical with the poorest determined parameter or linear combination of parameters having the largest projection of the ellipse and the most

-77well determined parameter or linear combination of parameters having the smallest projection of the ellipse. 3. Determining whether the distance'Sm from y to 0 is the true absolute minimum distance from y to the 0 subsurface is in principle only possible by calculating -the sum of squares corresponding to each point on the surface. This task is a near impossibility in most cases, thus alternative methods are employed. One method of gaining information about other possible minima is to re-do the problem several times, each time starting from a different initial guess of parameters and seeing if we always return to the same minimum; or, if another local minimum is found, the values of the sums of squares are compared to ascertain which is better. Another method of attacking this problem is to generate larger and larger contours in the tangent 0T plane about the point 0T and calculate the sum of squares along the projections, F, of these contours on the 0 subsurface, looking for a sum of squares smaller than S. If by several applications of the above tests there is a consistent preference for one minimum, one may infer with some degree of certainty that that is the true minimum. 4. Information may be gained concerning the applicability of linear methods to question 1, 2 and 3 by comparing the contour in parameter space to the elliptical contour in linearized parameter space. If the two contours are fairly similar, one may infer that the function is approximately linear in the parameters in the region of 0 and thus these linear methods are (to some approximation) applicable. If the

-78two contours are highly dissimilar, one is suspect of the linear methods for dealing with questions 1, 2 and 3. b) Analytical Development The following will be a fairly terse analytical development of the same problem dealt with in the geometrical development with the exception that now generalizations to k independent variables, n observations and p parameters will be made. In this section, little descriptive material will be included with the thought that the preceding geometrical development sufficiently illuminates the concepts. Mathematical Model 0=f(x, *..Xk;., ) (6.8) Experimental Data Observed values of dependent variable ys\:.zi-~~ It~:3~ ~(6.9) Corresponding values of independent variables (/ zX X= | o (6.10) \Xnl w XnK Computed Values 0= tot- =:' /..xl;J )e(6.11) _\fn f X,,/ —'XQ.^ Op\

-79Algorithm Linearization by first-order Taylor's expansion 0 o fT( ~~~~~~T\ *~P~t (612 Definitions f ~(6.14) ET 0 -f fn 0 -FnI f \6~p fo= - (6.15) \f + \4,d (f D (OO-eo) (6.16) Of? fT f=^k (6.17) Multiple Regression Fit (f-fo) t(gy-f0) Odoa^ ^O =,y-f") D~~~D0130.D'(gfo) ~ ~ ~ ~ ~ (613 D'Q^~=oV' Df(Od,)8 D&J (6.18) 30 (D6D0)15jca

-80Iteration Formula gW + D*DiDW (6.19) 2. Estimation of Parameters - Practice The application of the theory of nonlinear estimation to the problem of estimating the 2n+2 parameters appearing in the point reactor stochastic process model is accomplished with the use of the IBM 704 computer (where n is the number of delayed groups). The program used to perform the indicated operations is a modification of a program written for nonlinear estimation by the mathematics and applications department of International Business Machines Corporation (2, 3, 4, 5). The function required to be fit to the data is 7' =/ L / ex tJ(6.20) Here, the parameters ai enter nonlinearly and the range of T is over four decades in time (0.001 sec - 10 sec). The magnitude of the Oi varies by a factor of 104. To aid convergence it is best to make the original parameter estimates as close as possible to the "true values" of the least-squares parameters. To do this, parametric curves function -OCr G(b/ / —--— e (6.21)....

_81were generated on the'704.' These are plotted on 4-cycle semi-log paper for small T and large a in Figure (6.10) and on linear paper for large T and small a in Figure (6.11). Comparing these curves to the experimental points, the approximate values of the parameters ai may be estimated along with the approximate values of the Kis. This estimation procedure is accomplished with the theoretical values of ai and Ki corresponding to the experimental conditions. That is, for particular Ak, 9, e, Xj, and Pi, particular values of Ki and ai are determined. Since Xi's and Pi's are well known, it is certainly most desirable to have the initial estimates of parameters in the range given by theoretical considerations. Using these considerations, a best "eye-fit" to the data can be made for the original parameter estimates. The algorithm discussed in section VI-B will successfully converge upon the "least squares" parameters as long as the D'D matrix (Equation 5.18) is well conditioned. The conditioning of the matrix is a measure of how near the matrix is to being a singular matrix. If there are approximate linear dependencies existing, the matrix will be nearly singular and the digital computer will have difficulty in matrix inversion. This difficulty was encountered several times, and the regression sum of squares diverged instead of converged for cases where poor conditioning was indicated. To test the conditioning of the moment matrix, D'D, the eigenvalues of the moment matrix were calculated prior to entering the iteration stages of the program. A large ratio of the largest to the smallest eigenvalue was indicative of poor conditioning. The conditioning can be changed by adjusting the parameter estimates,

1.0 0.8 0.6 / I 0.4 IQk/ - 0.2 -0.2 -... 0.001 0.01 0.1 10 T IN SECONDS Figure 6.10. Parametric Curves for 1 (T); 1 C0T350

1.0T IN.8.6 Jl 03 0.2.. 0 I 2 3 4 5 6 7 8 9 T IN SECONDS Figure 6.11. Parametric Curves for p1(T); 0.1i-~_ 2

-84adjusting the size of increments used in finding partial derivatives, and reformulating the functional expression. Interpolation A modification of the estimation procedure is termed interpolation. If it is found that iteration is successful in reducing the sum of squares so that the slope is greater than E, we suspect that it will be profitable to continue investigating the function in this direction so we activate "interpolation." The algorithm for interpolation is as follows: 1) Let 90 and 91 be the vector values of the parameters before and after iteration respectively. Call the corresponding sums of squares So and S1. Divide the distance between 0 and 01 by two and call this point 01 with a sum of squares Si. -2 2 2) Sl is compared to S1. a) If S1 is less than or equal to S1, 2 the interval Gi - go is divided by two and Si calculated for 01. -2 -0 4 b) If Sl is greater than S1, 92 is calculated along with S2. 2 3) Either procedure a) or b) is followed until the sum of squares increases, in which case a Lagrangian fit to the last three points is made or until it is determined (by the program) that the gain by interpolation is too small, in which case the last 9 is kept along with the last S. The non-linear estimation program follows the general outline given for the theory of nonlinear estimation. The results obtained are listed for each experimental run in the "Results" section showing the parameters estimated, the minimum sum of squares obtained, the confidence regions, the relative confidence in parameters, and the degree of nonlinearity in which the parameters enter.

VII. RESULTS A. Experimental Results The experimental points obtained for the three runs with the BF3 tube (designated BF3-1, BF3-2, BF3-3) are shown in Figures (7.1), (7.2) and (7.3). The data from which these points are plotted is tabulated in Appendix A. Figure (7.4) shows the experimental points for the fission chamber run. The data for this run is also tabulated in Appendix A. The fitted curves which are the least squares fit to the data are shown along with the contribution to these curves from each term of an equivalent two-delay group mathematical model. Note that the contributions of each term are not shown for the fission chamber experiment since theory and experiment were in considerable disagreement on this experiment. The data and curve for the fission chamber experiment are included here for illustrative purposes. The effect of having a lesssensitive detector is shown clearly by comparing the value of the ordinates for the fission chamber experiment to those for the BFo tube experiments. The maximum value of V/M for the fission chamber experiment was not enough different from one to adequately show the effect of correlation. That is, in this experiment, the accidental terms contributed a significantly large amount to the result and thus the estimation of reactor parameters, which are a function of the correlated terms, was insufficiently accurate. Obviously, a detector which is only slightly less efficient than the fission chamber used would show no correlation at all. Thus, -85

60............ m 50 0 z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ <% 30 I~~~~~~~ —' ~~w ~~ ~ ~ ~ ~~ z 20 _ _ _' TERM I <[ ~~~~~~~~~~~~~~~~~~~~~TERM, I 0 -- - -- - -__ —___ -^^ TEJRM3 20__ —-_-_-__ _ ___ ^__ _ _ ^ - - _ _. -.0001.01 O.I I 10 GATE TIME, SEC Figure 7.1. Data and Curve Fit Run: BF3 - 1

60 -i-i —i —-l-,ll-,|i||| —ll- I |||| --------—.I I I,- - ll - -I —I -40 —. —---. 50 40, SE0. W ~ 2 30 < 20 QQ10 I QO I Q I I 10.001 0.01 0.1 I I0 GATE TIME, SEC Figure 7.2. Data and Curve Fit Run: BF -2

.60..____-__ 50 40 < * m 30 IAJ < 20 TERM I GATE TIME, SEC Figure 7.3. Data and Curve Fit Run: BF3 -3 Run: BF3 -3

z < __ 4__ _ __ _ __ ___ _ ___ __ ___ _ -__ _ A__ _ Zb ~ GATE TIME (SEC.) Figure 7.4. Data and Curve Fit Run: Fission Chamber

-90the comparison of the results of the fission chamber experiment with the BF3 tube experiments illustrates the need for a high efficiency detector. The problems associated with using a high efficiency detector are discussed in Chapter V. The three BF3- tube runs shown were all accomplished with the reactor operating in the same operating configuration with the exception that the reactivity was changed from run to run. Run # BF3-1 had the greatest multiplication, BF3-2 had a somewhat lower multiplication and BF3-3 had a still lower multiplication. Figure (7.5) shows a comparison between the fitted curves for these three experiments which illustrates clearly the effect of multiplication on the magnitude of the correlated terms. As the multiplication increases the correlation increases. This is in accordance with the mathematical model. Note also that as multiplication is increased the curve becomes less flat in the region between where the statistic V/M is being controlled by prompt neutrons and the region where it is being controlled by prompt and delayed neutrons. That is, as multiplication is increased, the prompt and delayed neutron contributions become less distinguishable. This is also in accord with the mathematical model which predicts this behavior. B. Mathematical Model (general) The mathematical model developed in Chapter III is rewritten here for convenience (Equation 7.1). (-) +ZK [/- ] (7.1) //

60 50 - 40 -- --- F z w 30 0 20.0001.01 0.1 I 10 GATE TIME (SEC) Figure 7.5. Comparison of BF Experiments Fitted Curves

-92In attempting to estimate the parameters in this model which are best least square estimates it is necessary to employ the techniques of nonlinear estimation which are outlined in Chapter VI. One of the important points which must be decided is what value of n to use. It was at first decided to use n = 6 which is the number of physical delay groups measured by Keepin and Wimett(l4) for example. However, when n = 6 was used in the mathematical model it was found that either the computation would not converge, or, if it did converge, that the values of the parameters Ki and O(i obtained bore no resemblence to the theoretical parameters. It was found that one of the primary reasons for divergence was that the moment matrix, D'D, used in the regression analysis was poorly conditioned and the algorithm no longer was valid. Erroneous results were produced even when the analysis converged because the data was insufficient to correctly predict fourteen parameters. Several other values of n were tried. It was found that n = 3, 4 or 5 gave the same result as n = 6 and were therefore unsatisfactory. On the other hand, it was found that n = 1 converged but that the parameter estimates for the one equivalent delay group were not at all in correspondence with the parameters used in the usual one delay group model. This result is obtained because the one delay group model is not a good approximation to the many group model. Firstly, the one delay group model is not capable of preserving some of the characteristic sums of the six group model. That is, for the usual

-953one group model 6 6 A =V A; X,B *ZARr il f- i 1 a, A; 1A2;i 2 (7.2) where P is the equivalent one group fraction and A is the equivalent one group decay constant, pi is the fraction of the ith precursor and Xi is the decay constant of the ith percursor. Thus, we see that the mathematics of the one group model are somewhat inadequate. Also, the one-group model is especially poor in describing reactor behavior in the high frequency (short time) region which is the region in which the measurements were made. We see then, that the one delay group model is incapable of predicting the experimental results, so we are left with only the two group model (n = 2). All comparisons between theory and experiment which will be given here will use the two group model. For this reason a discussion of this model follows. C. Mathematical Model (2-group) The two group model discussed here is based on the work of Skinner and Hetrick(21) and Skinner and Cohen(22) These authors showed that reduced group constants Ai and Ai which satisfy the

-94asymptotic behavior of the transfer function must satisfy the relationships 6 6 Z A;=Z; ZAA=; 6dA 7=! /=1 z -Ai zAi /=/ /i=/ / l E1 7;=1; / =,2 I (7.3) where Ai and Ai are the relative abundance and decay constant for the ith reduced group;'ci and Pi are the relative abundance and decay constant for the i-t physical delay group. Obviously Equations(7.3) can be uniquely satisfied by equivalent two group constants since we have four equations and four group constants. If 6 7_,7,= /=/ 6!-l (7.4) -/ ~ X,

-95then the two group constants are given by A= Ct CD A~, 7C -+-C-D (7.-5) A,, -3 y +- C -- where a/ 7. -Zj) For the physical constants of Keepin and Wimett(14), table (7.1), the group constants shown in Table (7.2) were calculated. These group constants lead to values of Ki and ai in the mathematical model, Equation (7.1), which may be compared with the Ki's and i's obtained from the least squares fit to the experimental points. D. Presentation of Results Tables (7.3), (7.4) and (7.5) show the comparisons between the experimental results and the theoretical model. Confidence contours as described in Chapter VI are shown for the projections onto the Ki-ai and K2-a2 planes in Figures (7.6), (7.7) and (7.8) for the three BF3 tube experiments whose experimental points and fitted curves are shown in Figures (7.1), (7.2) and (7.3) respectively. In Tables (7.3), (7.4) and (7.5) the theoretical parameters are shown for two different types of computation. In the first line

-9699% 160 Contour Sum of Squares 409 140 S1 = 408 S2 = 1160 9 % CONf. 3 4 S3 = 410 S4 = 1744 120 F~'""~> 99% CONF' 99.9% (S, ~t \ "S. ~ Contour Sum of Squares \ ^/^^r^^ ^~ s1 ~ ~ 459 100 Grid Sums of Squares 1a00 \\ ^ ^ st = 456 = 4527 80 S' = 463 s = 10627 3 4 46S0 3 ON F.iMinimum Sum of Squares 99.99% 5 = 376 40 ONF. 20 17 18 19 20 21 22 23 K, Projection on K1 - al Plane.24 i A2.22 99% I22FAX<W^ CContour Sum of Squares x S S - 99 % CONF. 409.20 Grid Sums of Squares S = 430 S2 = 409 ___ _ ___ \US'I s _ ___ S3 = 442 S4 = 409.18 3 99.9%.A S,~~~ Contour Sum of Squares e$N X Qf * 459 0.16 _ -99.9% _, SCONF. Grid Sums of Squares.14 S Sl = 618 S = 460.12 ___ ____ ____ ____ ____ ________ ___ ___5' = 647 ^ = 458 K2 Projection on K2 - 52 Plane Figure 7.6. Confidence Contours - BF3 -1

-9799% Contour Sum of Squares 344 160 - Grid Sums of Squares ji^""- - S1 = 342 S2 = 1157 /\f ^s, S3 = 344 S4 = 2135 140 I 3 4 \ e \t\ ~~S; KContour Sum of Squares 120 386 10k\ \' \ S' \ Grid Sums of Squares Ko 1,00 -^\^. S \- -- s{ =383 S) = 4624 co,_. \ XS = 391 S' = 1523 CONF. 40 /__ I Minimum Sum of Squares 40 4 S 5 S315 14 15 16 17 18 19 20 21 K, Projection on K1 - 01 Plane 99% Contour Sum of Squares.25 - 344.25!,., Grid Sums of Squares sl = 372 S2 = 344.23 Si 37 L S3 =379 S =343 99.9.21 Contour Sum of Squares 386 Grid Sums of Squares.19 S' = 599 S = 387 17 -99%CONF. S S 5 = 596 S4 385 N 99.9 % CONF. S3.15.13.11.09 - 30 35 40 45 50 55 60 65 70 Kz Projection on K2 - a2 Plane Figure 7.7 Confidence Contours BF -2.

-98200 --- 99% Contour Sum of Squares'2 252 180 \ \ Grid Sums of Squares S1 251 S = 366 s3 = 254 S = 482 1 60 2 99.9% 2 | &\ \Contour Sum of Squares 140 2\ \\ 99.9 % 283.S \ O CONF. Grid Sums of Squares 120 2 St _ - = 280 S3 = 881 0 2 __-,0_s,_ s'=289 SI = 2137 80 CONF. 40 - II 12 13 14 15 16 17 18 K; Minimum Sum of Projection on K1 - C~ Plane Squares S = 231.36.34 S 99% (<2 Contour Sum of Squares.32 \- 252 \99%CONF. \Grid Sums of Squares -\. 99 % CONF. S1 = 254 S2 = 252.30 S3 = 263 S4 = 252 \\\S' s^3 9 4 99.9% S! \ \ \\'Contour Sums cf Squaresi 28 - 283.28 X~^S Grid Sums of Squares 99.9%.26 S6 S~~ "-CONF. S' = 305 S' = 284.S'\\\\ 1 2.24 SI = 344 s = 284.22.20.18 25 30 35 40 45 50 K2 Projection on K2 - a2 Plane Figure 78. Confidence Contours BF3 -3

-99TABLE 7.1 PHYSICAL GROUP CONSTANTS (KEEPIN & WIMMETT) Xi(sec-1) ai ai 1.27x10-2 2.47x10-4.038 3.17x10-2 1.385x10-3.213 1.15x1O-1 1.222x10-3.188 3.11xlO-1 2.646x10-3.407 1.40 8.32x10-4.128 3.87 1.69x10-4.026 TABLE 7.2 REDUCED GROUP CONSTANTS Ai(sec_1) l 0.507 0.663 0.0209 0.337

-100TABLE 7.3 experiment number BF3-1 reactivity -.172 dollars detector BF curve of data and fit Figure 7.1 minimum sum of squares 376 standard deviation of residuals 2.03 theoretical delay group constants Table 7.2 mathematical model Equation (7.1); n=2 PARAMETERS K 1 K2 K, K2 _2 K3 53 theoretical parameters 19.92 83.42 96.5.240 2430.000126 $ = 91.6 p sec theoretical parameters 19.92 96.3 99.240 2400.000126 i = 80.2 t sec experimental parameters 19.92 83.91 73.87.172 > 400 <.002 experimental parameter 18.3 60 62.138 - ranges, 99% 21.5 110 86.204 confidence experimental parameter 17.45 46 55.5.120 ranges, 99.9% 22.4 122 92.5.223 confidence % diff. expt. and theory o% 0% 23.4% 28.4% =& 91.6 p. sec % diff. expt. and theory o% 13.4% 25.3% 28.4% R = 80.2 p. sec

-101TABLE 7.4 experiment number BF3-2 reactivity -.2400 dollars detector BF3 curve of data and fit Figure 7.2 minimum sum of squares 315 standard deviation of residuals 1.88 theoretical delay group constants Table 7.2 mathematical model Equation(7.); n = 2 PARAMETERS K1 1_ K2 22 K3 3 theoretical parameters 17.19 102.76 70.2.256 1050.000134 i = 78.6 p sec. theoretical parameters 17.19 101.9 70.6.256 1048.000134 Q = 80.2 t sec. experimental parameters 17.19 102.5 47.19.173 > 400 <.002 experimental parameter 15.7 72 37.130 ranges, 99% 18.7 135 57.215 confidence experimental parameter 14.6 52 31.5.100 ranges, 99.9% 19.5 154 64.5.240 confidence % diff. expt. and theory 0% 0o 32.6% 32.4% = 78.6 [i sec % diff. expt. and theory 0% 1% 33.1% 32.4% - a = 80.2 p. sec

-102TABLE 7.5 experiment number BF3-3 reactivity -.3400 dollars detector BF3 curve of data and fit Figure 7.2 minimum sum of squares 231 standard deviation of residuals 1.61 theoretical delay group constants Table 7.2 mathematical model Equation (7.1); n = 2 PARAMETERS K1 l 1 K2 Q2 K3 _3 theoretical parameters 14.17 122.08 45.7.277 425.000148 = 71.5 4 sec theoretical parameters 14.17 110.1 46.8.277 409.000148 i = 80.2 X sec experimental parameters 14.17 122.4 35.43.267 > 400 <.002 experimental parameter 12.85 88 31.228 ranges, 99% 15.4 161 40.309 confidence experimental parameter 12.15 64 28.203 ranges, 99.9% 16.1. 182 43.332 confidence % diff. expt. and theory 0% 0o 22.5% 3.62%o R = 71.5 - sec % diff. expt. and theory 0o 9.8% 24.3o 3.62%o Q = 80.2 J. sec

-103under "parameters" the theoretical parameters were computed by normalizing the value of K1 to the experimental value and choosing a lifetime,, which would make al theoretical agree with al experimental. In the second line under "parameters" K1 was again normalized as above, but the value of al was computed using the average value of prompt lifetime obtained in the experiments. The material above the "parameter" tables is self-explanatory; it gives reference to the pertinent operating data, experimental data, and statistical data for that experiment. Note that only qualitative estimates are shown for parameters K3 and O3. Note also that no confidence is stated for these estimates. The reason for this becomes apparent by looking at the contribution of the third term to the total V/M curve. Since the contribution of these terms is small, no significant estimates of the parameters could be made. The only conclusion that could be made was that the contribution of this term was only at long times, and the contribution was not large. The parameters governing this third term could be estimated if data was taken for gate times longer than ten seconds. E. Theory vs. Experiment The pertinent points concerning the comparison of the model with experimental results can be obtained from the tables and figures mentioned in the previous section. It is apparent that experiment and theory agree in a qualitative way. However, we see that the model and the experiment may differ by as much as 30% in some parameters and that the differences appear to be more severe in experiments BF3-1 and BF3-2 than in BF3-3.

-104. The reasons for the disagreements observed could be either that more experimental errors occur as the multiplication of the reactor is increased or that the model is less accurate as the multiplication is increased or both. It can be argued that "both" is the correct statement. 1. Experimental Errors V As multiplication increases the M ratio increases (FigV V ure 7.5). It is also true that as M increases the variance of M increases V so that to get results with the same certainty at large M as those with V smaller M we need more data. In the experiments shown the amount of data per experiment is constant so we would expect less experimental accuracy for large multiplication. 2. Model Inaccuracies It has already been observed that the effects of prompt neutrons and delayed neutrons become less distinguishable as criticality V is approached. Indeed, the model predicts M -> o for all gate times, T, V when ZK- 0. Obviously, if AK is very close to zero and M is very large everywhere the estimation of individual parameters would be nearly impossible since many possible combinations of parameters would yield the same result. Thus, we expect that the estimation of parameters becomes less accurate as criticality is approached. It is also true that the model is only a point reactor model and it is suspected that spatial effects become more important as criticality is approached. The reason for this hypothesis is that a larger portion of the events detected are correlated events as the reactor approaches criticality. Correlated events arise from common fissions

-105(Chapter III) which are spatially distributed in the core. It is apparent that the spatial distribution of the ancestors of accidental events is unimportant since no matter where they originate they are defined to be V uncoupled events and give rise to a M of 1. It is not at all apparent, however, that the effect of correlated events is independent of the location of the common ancestor fissions. In fact, it is most likely that the spatial distribution of ancestors will make itself felt in the shape of V the M curve for correlated events. So, since we have more correlation at higher multiplication we might expect ctstronger spatial effects and thus larger inaccuracies in the point reactor model.

-lo6VIII CONCLUSIONS A. Introduction Some of the conclusions which may be drawn from the preceding are in disagreement with conclusions made by other investigators in this area. In this section some of the points of agreement and disagreement will be discussed and resolved. Other conclusions arise from the work reported here and have not been discussed by other authors. These conclusions will be discussed here along with the possible relevance of this experiment to future experiments in this field. B, Comparison of Conclusions with Preceding Authors Bennett(l), having derived essentially the same mathematical model that is used here, plotted the model as a function of prompt neutron lifetime. This plot illustrated the effect of delayed neutrons on the shape of the curve. Bennett pointed out that one cannot use the measurement of variance to mean ratio to infer the prompt neutron lifetime of a reactor without taking into account the effect of the delayed neutrons. This measurement supports the above conclusion. It is not in general possible to measure the term controlled by prompt neutrons without also having some contribution from delayed neutrons. Other authors have used prompt neutron theories to find prompt neutron lifetimes by this method without taking delayed neutrons into account'') Obviously their measurements are in error by an amount controlled by the delayed neutron contribution. For reactors with very short lifetimes, one may be able to neglect delayed neutrons with only small errors occurring.

-107Velez(24) concludes that "theoretical derivation and experimental measurement of the correlation functions can be two independent lines of research, but regarding applications, in the author's opinion, the most urgent need is for experiments to check the formulas already obtained". The research presented here has been directed toward fulfilling part of this urgent need. With respect to the equations derived by Velez, the work done here has revealed some important information. 1) In the derivation of the autocorrelation function Velez introduced the term "m'ean time to fission" and defined N1 as the expected number of neutrons in the system at time ti from one ancestor neutron at t. This definition would be satisfactory so long as there is only one class of neutrons (prompt neutrons for example). However, the proper way to define terms in the case where delayed neutrons are important is to include a characteristic time in each group by defining N1 to be a frequency function as was done here. Velez's formulation leads to a discrepancy with experiment of a factor of 1010 at long gate times using the six delayed group model. 2) Velez illustrates the relative magnitudes of the prompt and delayed terms using a lumped one-group model. This work shows that the one lumped delay group model is quite grossly inaccurate in predicting the results. Luckow(16) made measurements of the variance to mean ratio for gate times p1 sec in ZPR IV and ZPR V at Argonne National Laboratory. Data points which did not follow the prompt neutron model were termed "wild points" and the following comment was made concerning them. "These

_108_ points have the disconcerting feature of not appearing in every series. Any physical explanation is therefore open to question...oIt is thought that these wild values reflect the contributions of the delayed neutrons for long measuring (not gate) times." The measurement discussed in this document shows clearly that the "wild points" observed by Luckow were contributions of delayed neutrons. However, the contributions were for long gate times and not measuring times as Luckow suggested. Also, a physical explanation is not very open to question since Luckow's data shows that more "wild points" were seen for experiments performed closer to critical. Using parameters characteristic of ZPR IV and ZPR V it can be shown that the model used here predicts that the delayed neutron effect should be less significant as multiplication is reduced. Thus, the model which includes delayed neutrons is in agreement with Luckow's experiment. C. Independent Conclusions This work has corroborated a point reactor mathematical model describing the experimentally observable stochastic processes in nuclear reactors in measurements in which delayed neutrons are important. In Chapter VII comparisons between the model and the experimental results are made which lead to the conclusion that the two-delayed group model is more accurate for more sub-critical situations. It has been demonstrated that the use of a tape recorder to simulate reactor fluctuations is a desirable alternative to either prolonged reactor operation or a more sophisticated measuring system. It has been shown that the contributions to the measured variance to mean ratio from all physical delay groups can not be determined by this type of measurement unless a great deal more data is taken.

-109This experiment has demonstrated the feasibility of measuring nuclear reactor dynamic parameters by stochastic process techniques. The prompt neutron lifetime of the Ford reactor was found to be 80+10L sec. D. Relevance to Future Experiments In order to significantly extend the amount of information which the experiment will yield it is necessary to have a considerably more sophisticated data taking technique. The extension of the data taking technique to a system with shorter resolving time could be accomplished by using a video tape recorder instead of the audio tape recorder used here. The resolving time of the scaler and other circuitry would also have to be improved, but the tape recorder is the most critical since it is the limiting piece of equipment in the present set-up. Obviously a shorter resolving time leads to higher permissible count rates and better statistics. A many channel recording and transcription system would also lead to improved measurements. In order to analyze data at times greater than 10 seconds it would be advisable to eliminate the operation of punching data on IBM cards and arrange for the tape recorded information to be entered into a computer directly. Future experiments may well be directed at many of the problems raised by this experiment. One line of investigation would be to refine the measurements in the delayed neutron region and achieve more accurate comparison between the model and the experiment.

-110l Another direction which may be profitable is to investigate further the effect of reactivity on the comparison between theory and experiment. Perhaps further information concerning this phenomenon would yield information which would be valuable in improving the description of the process. Spatial effects are open for investigation both theoretically and experimentally as are energy dependencies. Both theoretical and experimental investigation of the assumed white noise source used in the model could be pursued.

APPENDIX A DATA Fission Chamber Number Variance Mean Variance/Mean Gate Length Of Gates -1 (sec) 600 2.3247 2.2567 0.030.001 600 5.9086 4.7850 0.235.002 600 8.4899 7.3150 0.161.003 600 11.394 9.2317 0.234.oo4 600 11.565 11.565 0.000.005 600 16.598 13.892 0.195.006 600 20.470 16.083 0.273.007 600 23.138 18.643 0.241.008 600 31.363 23.195 0.352.010 588 41.420 30.034 0.379.013 600 54.768 38.732 o.414.017 600 68.784 45.778 0.502.020 576 72.496 53.365 0.358.023 600 95.633 57.108 0.675.025 600 93.338 61.685 0.513.027 600 96.543 68.802 0.403.030 600 131.22 80.762 0.625.035 600 139.63 92.320 0.512.040 588 160.59 103.16 0.557.045 600 178.61 113.66 0.572.050 600 224.17 137.38 0.632.060 600 280.58 159.85 0.755.070 600 337.13 184.o4 0.832.o80 600 397.81 208.40 o.909.o90 600 387.60 229.07 0.692.100 600 442.82 252.88 0.751.110 576 540.12 277.30 0.948.120 600 531.26 300.37 0.769.130 6oo 564.63 321.21 0.758.14o 6oo 697.94 368.67 o.893.160 600 862.78 414.25 1.083.180 600 956.48 460.29 1.078.200 600 1055.10 506.34 1.084.220 600 1158.30 566.48 1.047.250 600 1264.80 620.76 1.038.270 600 1682.80 684.72 1.458.300 600 1821.00 716.43 1.542.310 516 1441.10 760.43 o.895.330 552 1815.70 831.36 1.184.360 600 1574.70 895.84 0.758.390 600 1803.50 969.41 0.860.420 600 2330.80 1059.40 1.200.460 6oo 2371.00 1145.60 1.070.500 600 2573.60 1265.70 1.033.550 600 2967.00 1367.30 1.170.600 -111

K112Number Variance Mean Variance/Mean Gate Length Of Gates -1 (sec) 600 3134.6 1497.7 1.093.650 600 3147.1 1602.6 0.964.700 600 3510.8 1732.9 1.026.750 600 3910.3 1830.2 1.136.800 600 4466.5 2052.3 1.176.900 600 4684.0 2282.8 1.052 1.000 600 5598.9 2531.8 1.211 1.100 600 5752.6 2745.4 1.095 1.200 600 6078.8 2991.9 1.032 1.300 588 6657.0 3223.3 1.065 1.400 600 7508.6 3423.3 1 193 1.500 600 8532.4 3685.9 1 315 1.600 600 9173.6 3877.6 1.366 1.700 600 9350.8 4114.0 1.273 1.800 588 9735.8 4370.9 1.227 1.900 6oo 10271. 4561.1 1.252 2.000 552 10516. 4832.4 1.176 2.100 6oo 12287. 5068.7 1.424 2.200 504 13206. 5285.3 1o499 2.300 600 14280. 5525.3 1.584 2.400 540 13488. 5742.8 1o349 2.500 6oo 15620. 5987.8 1.609 2.600 588 15788. 6210.7 1.542 2.700 600 15015. 6447.3 1.329 2.800 576 16852. 6664.6 1.529 2.900 600 17102. 6842.6 1.499 3.000 588 18199. 7129.3 1.553 3.100 6oo 19672. 7362.2 1.621 3.200 576 21599. 8064.7 1.678 3.500 552 23196. 8743.9 1.653 3.800 420 27044. 9436.2 1.866 4.100 432 29551. 10359.0 1.853 4.500 420 36993. 12208.0 2.030 5.300 384 40839. 13127.0 2.111 5.700 432 48433. 14278.0 2.392 6.200 408 46535. 15409.0 2.020 6.700 324 54427. 16581.0 2.282 7.200 336 61749. 17715.0 2.486 7.700 300 69394. 19333.0 2.589 8.400 288 77055. 20947.0 2.678 9.100 324 86785. 22995.0 2.774 9.999

-113BF3 -1 Number Variance Mean Variance/Mean Gate Length Of Gates -1 (sec) 588 4.9011 2.5068 0.955.001 600 12.533 4.8883 1.564.002 600 45.120 10.147 3.447.004 600 86.058 14.825 4.805.006 600 134.77 19.715 5.836.008 600 189.67 24.022 6.896.010 600 301.96 34.132 7.847.014 600 480.88 43.310 10.10.o18 600 659.67 53.867 11.25.022 588 760.84 66.364 10.46.028 600 1389.1 87.670 14.84.035 576 1508.6 105.87 13.25.043 600 2169.0 125.06 16.34.051 588 2353.2 146.44 15.07.060 576 3253.3 176.28 17.46.071 600 4418.o 211.80 19.86.085 564 4869.8 239.17 19.36.100 588 5473.2 295.09 17.55.120 600 8091.3 366.82 21.06.150 588 9093.4 441.59 19.59.180 588 10367. 507.79 1942.210 6o0 12206. 577.50 20.14.240 600 15013. 663.22 21.64.270 6oo 15568. 727.82 20.39.300 588 18786. 854.58 20.98.350 6oo 22247. 971.07 21.91.400 588 23901. 1104.2 20.65.450 6oo 28821. 1217.9 22.67.500 600 32336 1386.6 22.32.570 600 36428. 1580.5 22.05.650 600 43152. 1769.7 23.38.730 600 57275. 2009.5 27.50.820 600 57696. 2219.0 25.00.910 588 65869. 2452.9 25.85 1.000 600 75979. 2701.2 27.13 1.100 600 79018. 2948.9 25.80 1.200 600 87347. 3181.0 26.46 1.300 600 103392. 3443.4 29.02 1.400 588 113116. 3686.2 29.69 1.500 600 129516. 3935.3 31.91 1.600 588 126016. 4165.6 29.25 1.700 600 138212. 4419.1 30.28 1.800 600 144516. 4648.9 30.09 1.900 588 157357. 4919.4 30.99 2.000 600 162904. 5184.1 30.42 2.100 600 163083. 5415.2 29.12 2.200 600 173056 5655.9 29.60 2.300 516 188159. 5912.2 30.83 2.400 564 201228. 6131.8 31.82 2.500 600 239031. 6381.5 36.46 2.600 600 308080. 6643.4 45.37 2.700 600 243236. 6887.1 34.32 2.800

-114Number Variance Mean Variance/Mean Gate Length Of Gates -1 (sec) 600 256914 7116.5 35.10 2.900 564 263306. 7328.8 34.93 3.000 552 255058. 7597.4 32.57 3.100 540 308993. 7854.2 38.34 3.200 504 288741. 8081.4 34.73 3.300 528 302654. 8372.8 35.15 3.400 516 348975. 8594.0 39.61 3.500 516 411817. 8909.1 45.22 3.600 468 360856. 9120.0 38.57 3.700 504 392816. 9335.3 41.08 3.800 528 406665. 9616.5 41.29 3.900 468 420116. 9883.0 41.51 4.000 492 423017. 10089. 40.93 4.100 492 414984. 10322. 39.21 4.200 468 461338. 10542. 42.77 4.300 432 474295. 10880. 42.59 4.400 468 448443. 11089. 39.44 4.500 480 477901. 11300. 41.29 4.600 456 472160. 11547. 39.89 4.700 444 521301. 11767. 43.30 4.800 444 523476. 12038. 42.48 4.900 456 542311. 12288. 43.13 5.000 408 569815. 12540. 44.44 5.100 432 572401. 12760. 43.86 5.200 420 620139. 13042. 46.55 5.300 396 633293. 13265. 46.74 5.400 396 632115. 13562. 45.61 5.500 396 664113. 13783. 47.18 5.610 384 708852. 14070. 49.38 5.730 372 742006. 14437. 50.40 5.860 372 743248. 14720. 49.49 6.000 372 726245. 15123. 47.02 6.150 360 804746. 15500. 50.91 6.310 348 812204. 15889. 50.11 6.480 336 779075. 16363. 46.61 6.660 336 831676. 16815. 48.46 6.850 324 878255. 17322. 49.70 7.050 312 949975. 17908. 52.05 7.300 312 978582. 18682. 51.38 7.600 288 1028044. 19515. 51.68 7.950 288 1105216. 20494. 52.93 8.350 276 1254602. 21606. 57.07 8.800 264 1377585. 22853. 59.38 9.300 228 1447419. 24506. 58.06 9.999

-115BF3-2 Number Variance Mean Variance/Mean Gate Length Of Gates -1 (sec) 600 3.3148 1.7900 0.852.001 600 10.611 3.7167 1.855.002 600 34.793 7.3083 3.573.004 600 55.046 10.548 4.18.006 588 89.388 13.995 5.387.o08 600 137.06 18.288 6.494.010 600 242.75 24.138 9.057.014 588 436.58 38.660 10.298.022 576 683.82 50.384 12.57.028 396 747.29 6. 11.21.035 600 1129.2 75.733 13.91.043 588 1295.8 91.535 13.16.051 564 1845.1 109.99 15.77.060 600 1850.5 128.23 13.43.071 600 2400.6 150.22 14.98.085 600 2944.6 176.70 15.66.100 600 3806.3 212.31 16.93.120 600 4452.4 269.86 15.50.150 600 5213.0 315.02 15.55.180 600 6952.6 374.90 17.55.210 600 7377.5 417.20 16.68.240 600 9263.8 486.55 18.04.270 600 10304. 534.98 18.26.300 600 13573. 620.16 20.89.350 588 15655. 715.36 20.88.400 600 17004. 802.08 20.20.450 600 19195. 889.oo 20.59.500 600 20660. 1015.6 19.34.570 552 25468. 1170.0 20.77.650 564 23948. 1289.7 17.57.730 600 31296. 1458.1 20.46.820 600 39549. 1785.1 21.16 1.000 600 43228. 1961.0 21.04 1.100 600 46434. 2136.6 20.73 1.200 600 53864. 2300.7 22.41 1.300 600 60738. 2486.2 23.43 1.400 600 59089. 2678.1 21.06 1.500 600 69003. 2860.2 23.12 1.600 600 73716. 3027.4 23.35 1.700 600 80749. 3215.2 24.11 1.800 576 84476. 3398.5 23.86 1.900 576 89383. 3557.1 24.13 2.000 540 93176. 3739.9 23.91 2.100 576 93526. 3910.9 22.91 2.200 54o 102877. 4105.0 24.06 2.300 564 109487. 4281.4 24.57 2.400 600 114406. 4441.4 24.76 2.500 600 112030. 4636.9 23.16 2.600 600 125097. 4814.9 24.98 2.700 576 135841. 4977.3 26.29 2.800

-116Number Variance Mean Variance/Mean Gate Length Of Gates -1 (sec) 564 142163. 5159.3 26.56 2.900 564 149663. 5334.4 27.06 3.000 600oo 187881. 5545.8 32.88 3.100 600 162620. 5701.3 27.52 3.200 600 166985 5880.9 27.39 3.300 588 165495. 6073.0 26.25 3.400 576 176659. 6228.8 27.36 3.500 564 200534. 6410.9 30.28 3.600 564 216027. 6580.8 31.83 3.700 552 208740. 6768.8 29.84 3.800 528 242743. 6963.9 33.86 3.900 528 206264. 7106.8 28.02 4.000 468 292097. 7300.5 39.01 4.100 480 234248. 7475.3 30.34 4.200 444 234228. 7637.7 29.67 4.300 420 228560. 7841.9 28.15 4.400 4o8 260018. 8017.4 31.43 4.500 468 254538. 8179.5 30.12 4.600 432 279849. 8382.9 32.38 4.700 444 266729. 8537.3 30.24 4.800 4o8 290489. 8719.4 32.32 4.900 420 331803. 8929.3 36.16 5.000 444 320039. 9084.5 34.23 5.100 420 320904. 9272.0 33.61 5.200 420 303987. 9450.0 31.17 5.300 408 341436. 9633.4 34.44 5.400 384 356519. 9806.0 35.36 5.500 360 351485. 9976.6 34.23 5.610 360 341815. 10191.0 32.54 5.730 384 349885. 10445.0 32.50 5.860 324 421929. 10678.0 38.51 6.000 348 410246. 11253.0 35.46 6.310 348 436353. 11552.0 36.77 6.480 336 452038. 11864.0 37.10 6.660 336 469225. 12200.0 37.46 6.850 324 459810. 12563.0 35.60 7.050 324 497646. 13003.0 37.27 7.300 300 529836. 13548,0 38.11 7.600 288 574739. 14175.0 39.55 7.950 264 611611. 14903.0 40.05 8.350 264 668707. 15682.0 41.64 8.800 252 741850. 16583.0 43.73 9.300 228 658282. 17794.0 35.99 9.999

-117BF -3 Number Variance Mean Variance/Mean Gate Length Of Gates -1 (sec) 600 44.2748 2.1750 o.965.001 600 10.543 4.0817 1.583.002 600 29.470 8.5467 2.448.004 588 59.195 13.162 3.498.006 600 95.520 16.978 4.636.oo8 600 141.10 21.083 5.693.010 600 277.09 30.397 8.116.014 600 365.18 38.120 8.578.018 600 506.71 46.545 9.886.022 600 718.80 61.168 10.75.028 600 897.33 73.052 11.28.035 600 1233.2 90.205 12.67.043 600 1224.4 105.89 10.56.051 588 1852.5 127.76 13.50.060 600 2216.6 147.68 14.01.071 600 2458.9 175.81 12.99.085 600oo 2879.9 211.21 21.64.100 600 3416.8 254.22 12.44.120 600 4486.9 319.33 13.05.150 600 5707.2 374.30 14.25.180 600 7224.4 444.05 15.27.210 600 7557.7 506.87 13.91.240 576 8554.o 565.98 14.11.270 576 9603.9 633.73 14.15.300 588 12416. 733.43 15.93.350 600 16240. 845.53 18.21.400 600 16126. 949.06 15.99.450 600 18782. 1052.1 16.85.500 600 21144. 1206.6 16.52.570 600 25991. 1359.8 18.11.650 564 27626. 1536.9 16.98.730 588 32119. 1723.9 17.63.820 600 39106. 2104.6 17.58 1.000 600 45314. 2313.3 18.59 1.100 600 47028. 2521.2 17.65 1.200 600 57538. 2741.5 19.99 1.300 588 63610. 2940o1 20.64 1.400 600 69048. 3163.7 20.83 1.500 600 80198. 3565.6 21.49 1.700 600 87746. 3788.7 22.16 1.800 600 94826. 3999.1 22.71 1.900 588 99989. 4205.8 22.77 2.000 600 101119. 4417.3 21.89 2.100 564 99918. 4620.4 20.63 2.200 564 120631. 4848.5 23.88 2.300 600 125558. 5044.3 23.89 2.400 600 133882. 5270.8 24.40 2.500 600 146853. 5470.0 25.85 2.600 600 140980. 5698.8 23.74 2.700 600 143798. 5888.8 23.42 2.800

Number Variance Mean Variance/Mean Gate Length Of Gates -1 (sec) 600 169132. 6081.5 26.81 2.900 564 162910. 6326.5 24.75 3.000 600 161667. 6526.4 23.77 3.100 600 173704. 6727.0 24.82 3.200 564 180121. 6958.2 24.89 3.300 552 187628. 7154.7 25.22 3.400 516 197352. 7363.6 25.80 3.500 540 192995. 7583.1 24.45 3.600 528 224183. 7773.6 27.84 3.700 468 210592. 8015.8 25.27 3.800 456 236398. 8212.2 27.79 3.900 480 237680. 8618.1 26.58 4.100 444 248552. 8848.0 27.09 4.200 456 272048. 9072.6 28.99 4.300 468 269963. 9281.9 280o8 4.400 480 288333. 9475.7 29.43 4.500 456 353850. 9729.8 35.37 4.600 432 332909. 9933.0 32.52 4.700 420 382351. 10151.0 36.67 4.800 456 303159. 10314.0 28.39 4.900 432 305533. 10532.0 28.01 5.000 432 358595. 10757.0 32.34 5.100 408 334156. 10948.0 29.52 5.200 420 350528. 11153.0 30.43 5.300 420 327887. 11349.0 27.89 5.400 396 372725. 11580,0 31.19 5.500 384 354685. 11803.0 29.05 5.610 396 359175. 12060.0 28,78 5.730 372 402668. 12332.0 31.65 5.860 384 433353. 12624.0 33.33 6.000 372 440865. 12940.0 33.07 6.150 360 436544. 13289.0 31.85 6.310 360 449541. 13631.0 31.98 6.480 312 505731. 14000.0 35.12 6.660 324 461988. 14412.0 31.05 6.850 324 494748. 14836.0 32 35 7.050 324 589045. 15380.0 37.30 7.300 312 542913. 15950.0 33.04 7.600 300 609144. 16758.0 35.35 7.950 288 641285. 17575.0 35.49 8.350 276 727448. 18513.0 38.29 8.800 264 703853. 19592.0 34.93 9.300 252 795052. 21055.0 36.76 9.999

APPENDIX B CONTROL ROD CALIBRATION Position Position Pos. Position Position Position CR CR Period Reactivity Position Average No. Rod A" Rod B" Rod C" Initial Final " T sec. Increment Position -4 1 13 83 13.62 13.98 35-7/16 30-3/16 189.5 sec 4.41 x 10~4 5-1/4.84 x 10-4 32-15/16 2 13.68 13.64 14.60 32-13/16 27-27/64 138.5 5.745x 10-4 4-63/64 1.15 x 10O4 29-29/32 -43 14.05 14.05 13.95 30-3/4 26-17/64 162. 5.o42x 10 4-31/64 1.13 x 10 28-1/2 4 14.19 14.og09 14.25 28-7/8 23-7/16 87.3 8.28.x 104 5-7/16 1.53 x 26-1/8 5 14.43 14.38 14.39 25-15/32 21-5/8 123. 6.33 x 1O4 3-7/32 1.645x 1O- 23-35/64 P 6 14,38 14.85 14.44 23-27/64 19-9/32 125. 6.247x 10-4 4-9/64 1.510x 10-4 21-11/32 7 14.90 14.11 15.10 22-3/16 16-15/16 127. 6.17 x 10-4 5-1/4 1.175x 10-4 19-9/16 8 14.90 14.81 14.95 18-15/16 12 188. 4.44 x 10-4 6-15/16.64 x 104 15-15/32

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