با . . . : -... . . { I OF ORNLP 1110 : s. :: . . . M . TTEFEEST 1 . & · ni MICROCOPY RESOLUTION TEST CHART NATIONAL AUREAU OF STANDARDS - 1963 . . . . . - LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representa- tion, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, appa- ratus, method, or process disclosed in this report may not infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus. method, or process disclosed in this report As used in the above, “person acting on behalf of the Commission" includes any em- ployee or contractor of the Commission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employ- ment or contract with the Commission, or his employment with such contractor. ORNA PT110 CONF-654-89 APR 27 1968 RECENT DEVELOPMENTS IN THE PROTON-RECOIL SCINTILLATION NEUTRON SPECTROMETER W. R. Burrus and V. V. Verbinski Oak Ridge National Laboratory Oak Ridge, Tennessee INTRODUCTION The ideal neutron spectroineter would have a high and known effi- ciency, good resolution with a clean line shape, wide energy range, low sensitivity to gamma radiation, and requine simple electronics. As is evident from other papers given today, there has been a steady advance toward these goals from a number of different directions. Recoil proton scintillators have been used for more than a decade as neutron spectrom- eters, but they have been plagued by two difficulties. The first is that -LEGAL NOTICE - The report wuo prepared um accomat of Government Korond work. Nolder the Onlued Cocuniadton, nor any porma acting as bolal the Commissioo: A. Makes my warranty or mprotalco, apruned or implied, with respect to the accu- I rusy. coepletree, or urutalnu od be information catalood in this roport, or that we we al may talareuation, apparatu, matbod, or procon doclovodla to mport may not infring B. Annamon say Howlithes with repect to the ur, of, or lor dumugon routing frog the une of my taformation, apparatus, wethod, or procesu discloud in this report do wnd lotsbon, "persoa acttog oa bedel of the Coomlaoo" laclodas my plogue or coatractor of the Comission, or mmployee of mucha contractor, o the extent that del comployw or contractor of the Commission, or vaplogue of such contractor prepuro.. diectantes, or pronous accue to, wy Information parhaat to do neployedt or contract with the Codeminion, or Hs employment with each contractor. 1 pirately owned ride; or hatou, mor the scintillators are sensitive to gamina radiation as well as to neutrons, and the second is that the response is not i single line, but rather a complicated smear which more nearly looks like a rounded rectangle than a peak. Receniily methods have become available for overcoming both these traditional shortcomings, and the proton recoil scintillation spectrom- eter now appears more attractive for general use. The gamma sensitivity problem is greatly reduced by the use of pulse shape discrimination to reject unwanted gamma pulses. The second problem of poor line shape is Research sponsored by U. S. Atomic Energy Commission under contract with the Union Carbide Corporation and partially funded by U. S. Army Nuclear Defense Laboratory under order of NDL-23-63. INENT CLEARANCE OBTAINED. RELEASE TO TE PURUC IS APPROVED. PROOEDURES ARE ON EKLE IN THE REDUVING SECTION. reduced by the use of an unscrambling code which uses accurately measured pulse-height distributions for monoenergetic neutrons. 2-4 In this paper, we will elaborate on the details of the pulse-shape discrimination circuit which we are using, and on the details of the unscrambling code. The companion paper by Verbinski et al.' will con- sider the problem of ineasurement and/or computation of the required pulse-height distributions for monoenergetic neutrons. SCINTILLATOR CHARACTERISTICS V We have investigated NE-213 and Stilbene scintillators. Stilbene has better resolution, but is not readily avuilable in sizes larger than 2 by 2 in. and is nonisotropic so that its calibration depends upon the orientation of the crystalline axes with the direction of the incoming neutrons. This nonisotropy affects the location of the upper edge of positions of the bumps on the pulse-height distribution due to alpha reactions and due to recoil of carbon atoms. Since the nonisotropy is much reduced for gamma radiation (which is usually used for calibration in the absence of a monoenergetic neutron calibration source), it is difficult to take the norisotropy into account properly in the calibra- tion procedure. The pulse-height resolution was investigated by one of us since this directly limits the energy resolution in neutron spectroscopy with an unfolding computation. The fractional resolution was determined by dividing the difference in pulse height be seen the 88% point and the 12% point of the upper edge of the pulse-height distribution (in suitable pulse-height units) by the pulse height of the 50% point. A 5- by 5-in.- diam NE-213 scintillator with factory applied white reflecting paint yielded the worst value of 15% at 14.7 MeV. A 2- by 2-in. NE-213 scintillator yielded a value of 11%. Removing the paint from the out- side of the glass capsule and substituting a layer of household aluminum foil (shiny side in) reduced this value to 6.5%. Apparently the optical contact made by the white reflecting paint is detrimental to the total was reflection of light by the glass capsule at angles greater than the critical angle. A l- by l-in.-diam. NE-213 detector gave a value of 6.2%, while a 15 by 12-in.-diam. detector gave an anomalously high value of 8.2%. This value was reduced to 5.4% when the snug aluminum foil reflector was replaced by a loose fitting aluminum foil reflector spaced about in. from the glass capsule. A l- by l-in. Stilbene scintillator with a packed Mgo reflector* gave a fractional resolution of 3.6%. A 1- by 2-cm-diam Stilbene detector with a nachined aluminum reflecto:: Ini M showed a comparable resolution of 3.2%. At 3.1 MeV, measurements show much less difference between Stilbene and NE-213. It appears that Stillbene is the choice scintillator in the energy range near 14 MeV, if one is willing to cope with the variations of response function with crystal orientation that was mentioned above. If not, the 2 - by 1-in.-dian scintillator of NE-213 with a loose fitting aluminum foil reflector (about the size of the RCA 6810-A photocathode) would be a good choice. Where high efficiency is an important considera- tion, the 2- by 2-in.-diam NE-213 with aluminun foil reflector is indi- cated. It is doubtful that the 5- by 5-in.-dian NE-213 would find much This scintillator was loaned to us by llarshaw Scientific for the purpose of making these studies. -4- use for obtaining neutron spectra via unfolding techniques. On the other hand, it has proven extremely useful for time-of-flight measure- ments with a LINAC because of its size. PULSE-SHAPE DISCRIMINATOR The pulse-shape discriminator used is 2.odificution of Forte's circuits in which both the 14th dynode and the unode of an RCA 6810-A photomultiplier tube are used to derive a pulse-shape discrimination signal. The diagram of the photomultiplier circuit is shown in Fig. 1. The anode is clipped and supplies the "fast component" of the scintilla- tion signal. This fast signal is mixed with the signal from the 14th dynode in a resistive summing network so that the negative fast component substracts from the positive dynode signul. A small excess of fast neg- ative signal from the ancde is fed to the sunner so that the slow com- ponent from a gamma ray cannot effectively overcome the negative signal from the anode, while the larger slow component from a recoil proton results in a net positive signal which triggers the pulse-shape-discrim- ination discriminator. The IN270 diodes proved superior to several inore recent types for this application. A linear signal is taken from the 10th dynode for pulse-height analysis. It passes through a conventional preamplifier and double clipped linear amplifier. A timing signal is also taken from the zero crossing of this signal. Because of the wide range of pulse heights (400:1) which must be recorded to cover the neutron energy range 300 keV to 15 MeV, the linear channel was operated at several different gains so RCA 6810-A PHOTOMULTIPLIER ORNL-DWG 65-1246 CATHODE GRID T- DYN. 1H 350k DYN. 2H DYN. 3H DYN. 4H DYN. 5H Wowowhow DYN. 6H 3 27k DYN. 7H 33k DYN. 8H7 39K DYN. 9H 347k DYN. 101T 362k 0.01 THOUTPUT LINEAR OUTPUT DYN. 1117 + k + 375k DYN. 12H 391k DYN. 1317 475 31:06 DYN.14 H ANODE FORTE PSD OUTPUT 391k . . ': . . toit To . . . . 20k 2/2 in. HH2200 DELAY CABLE M tu 1.5 K IN270 . - ... ' HV ~+2150 VOLTS +21sovours doo ; - - . . . . . . . Fig. 1. Forte pulse-shape discriminution circuit. . as to cover the range of .02 to 8.0 pulse-height units with at least 2 chinels per resolution width. * The pulse-shape discriminator signal from the photomultiplier suinning circuit goes to a conventional preamplifier and to a single RC clipped vucuum tube amplifier (Modified A-8, ORNL drawing number Q-1819). The overload and pulse-shape characteristics of this amplifier are critical for operation at neutron energies below 800 keV. Using this wa amplifier, it was possible to find a single setting (with a high voltage of 2170 V and discriminator Level = 2.5 V) so that it was possible to accept pulses from .03 to 8 light units (i.e., 300 keV to 15 MeV neutrons) with a 100:1 rejection ratio for Co gamma rays and from .05 to 8 light units (1.e., 600 keV to 15 MeV neutrons ) with 1000:1 rejection ratio. The 100:1 ratio is common for accelerator experiments, while the 1000:1 figure is more coinon for work with reactor neutrons , The pulse-height analyzer is gated by both the pulse-shape discrim- inator signal and the linear channel zero crossing signal (with suitable time delays ) with a coincidence resolving time of about 1.5 hsec. This coincidence requirement eliminates false counts due to delayed positive overshoots from large negative overload pulses in the pulse-shape dis- criminator channel, since these overloads occur several microseconds following the zero crossing of the linear channel. The discrimination ability degrades at high counting rates due to base-line jitter and pulse pile-up. Our circuit is designed for a maximum count rate of 1000 to 2000 counts/sec. The resolution width was taken as the full width at half maximum (FWHM) of the Gaussian distribution which was used to smear the Monte- Carlo pulse-height distribution. Figure 2 shows the effect of the discriminator coincidence requirement . in a 20 mnr/nr field of "Co gamnas with a l- by 2-cin Stilbere scintil- . quinas . . . lator. The top curve shows the pulse-height distribution due to gammas - .... a. . . "ON WELS . alone without any pulse-shape requirement. The triangles give the pulse-height distribution of 2.85 MeV neutrons plus 20 mr/hr oco gammas with a pulse-shape requirement. The square points are the same except with the "Co source removed. The two distributicns are identical but . . .... for some gamma leakage in the channel 10-15 region. An amount of tolerable gamma contamination decreases with the larger scintillators because on the higher counting rates for the same flux. A 2- by 2-in. NE-213 scintillator can tolerate only to o as much gamma field (2-3 mr/hr), but of course, has 15 times greater neutron detection efficiency. . .. . ; -8 UNCLASSIFIED ORNL-DWG 64-6779 . Азис и са counts per channel/sec WITHOUT PULSE-SHAPE DISCRIMINATION • 20-mr/hr 6°CGAMMA RAYS WITH PULSE-SHAPE DISCRIMINATION * 2.85-MeV NEUTRONS + 20-mr/hr 6°C GAMMA RAYS • 2.85-MeV NEUTRONS o 20 40 60 80 100 CHANNEL NUMBER 120 140 160 180 muns Fig. 2. Pulse-height distributions obtained in 5-nin runs with a lo by 1-cm-diam cylindrical Stilbene scintillator. -9- INSTRUMENT RESPONSE The pulse-height distribution of a multichannel scintillation spectrometer is related to the unknown neutron spectrum (E) by JA, (E) (E JE = [1 = 1, 2, ..., NR. (1) where: A: (E) = probability that a unit intensity source with energy E will produce a count in channel i. This is the same as the counting efficiency vs energy of a single channel analyzer which accepts pulses over just the width of the ith channel. 4(E) DE = neutron spectrum, i.e., the number of neutrons with energy between E and E + DE. cy = average number of counts in channel i, i = 1,2, ..., NR. The actual observed number of counts c, will be statistically distributed about C. In the common case of Possion statistics, the standard devia- tion of c4 is given ipproximately by std (c) = wc. We assume that the standard deviation is known, and denote this value by sz (i = 1,2, ..., NR). The efficiency functions A, (E) can be measured at selected energies, or can be computed by a Monte-Carlo computer program as described in the companion paper by Verbinski, et al.' DESIRED RESPONSE Due to various instrumental distortions, the pulse-height distribu- tion : (i = 1, 2, ..., NR) is often not an adequate approximation to the -10- unknown spectrum (E). Suppose that we possessed an ideal multichannel analyzer whose response was given by PL = SOW(E) (E )LE, k = 1, 2, ..., NW. (2) where the ideal response functions are shown in Fig. 3 for several values of E. Note that the ideal analyzer is shown as yielding a single Gaussian distribution for each neutron energy, and that the centroid of the Gaussian distribution is linearly proportional to the incident neutron energy. Figure 3 also shows a slice through the surface of W, (E) parallel to the E direction. Then, if we had such a multichannel analyzer, the counts themselves, Pre (k = 1,2, ...,NW), would constitute a useful ap- proximation to the neutron spectrun Q(E). UNFOLDING METHOD The basic idea of the unfolding inethod is to estimate the counts Pv (k = 1,2, ..., NW) which would have resulted in an ideal multichannel analyzer with response functions W, (E), using as data the counts c; (i = 1,2, ...,NR) which were observed in the actual multichannel una- lyzer with response functions A, (E). Figure 4 is a graph of A, (E) for sevtral values of E for an actual 2- ky 2-in.-dium NE-213 recoil proton scintillation spectrometer as described in lef. 5. Note that the pulse- height distributions for a monoenergetic neutron are not peaks but are roughly rectangular distributions extending from zero pulse height up to a maximum. The unit of pulse height on Fig. 4 is such that à 1.27 Mev electron source would produce a pulse height of about 1 unit. The position ... стадым.Усм . .. . . . . . . . . . . 5. UNCLASSIFIED ORNL-DWG 64-tor24 2.0 Mev .. .. ... --.--.- ЦЛАА. т . 1 - . 1 NEUTRON ENERGY (Mev) PEAK POSITION VS NEUTRON ENERGY RESPONSE Fig. 3. The response surface W (E) for an ideal neutron spectrometer. UNCLASSIFIED ORNL-DWG 64-!0722 PULSE HEIGHT = 0.3 mm O 2 2 4 6 8 NEUTRON ENERGY (MeV) !!! ÚHIHIIHIMITRIJHIMANNHAM H 1:1:11HHH O !! COUNTS/PULSE HEIGHT UNIT - NEUTRON 8.12 MeV NUDIN HARTINICO -12. UN ANY . KWL 5.97 MeV HAHHAH AHIHHINIMAS V L 3.96 MeV NEUTRON ENERGY (MeV) 0.2 MeV os 1.25 MeV 0.96 MeV 10.6 Mer 10 - 2.98 Mer - ------ PULSE HEIGHT “EDGE“ VS NEUTRON ENERGY < 1.91 MeV . 2.5 PULSE HEIGHT o 15 20 --PULSE HEIGHT DUE 1.27 MeV ELECTRONS Fig. 4. The response surface A, (E) for the actual neutron spectrometer. -13- of the upper edge of the pulse-height distribution as a function of E is given by the solid line and is a nonlinear function of E. A slice through the surface of A, (E) is shown for a particular value of pulse height. In the special case that Q(E) were discrete with less than NR com- WS ponents, the classical method of solving the above estimation problem would be to set up a system of NR equations relating C, to the discrete components, and solve by the method of least squares for the intensities of the various discrete components. Then the required estimates of phy would be calculated from Eq. (2). The classical method based on ordinary least squares fails, however, if there are more than NR discrete com ponents in the spectrum, Q(E), or if the spectrum has continuous com- ponents. Even if the spectrum could be approximately represented by only NR discrete components, the results would be very sensitive to small sta- tistical fluctuations in c, unless the W (E) were exceedingly carefully chosen, and the computations were performed to high precision. Several weve nonclassical ad hoc techniques have been partially successful previously in solving the unfolding problem.<-> These methods, however, usually can not produce a valid estimate of the errors in unfolding, and are difficult to rigorously justify. The FERDO method which we will describe below is a simplification of the SLØP methodº for unfolding instrument response. The simplified version is presented here for brevity, and because it yields nearly the same results as SIØP for the neutron scintillation spectrometer unfolding problem. Wys ܚܙܐ[- Mo INTEQUALITY CONSTRAINED ESTIMATIÓN In order to have a meaningful solution to the estimation problem, we must determine an interval of uncertainty for each estimated quantity BK (k = 1,2, ..., NW) which has a specified confidence level. The final results of our estimation method 18 NW airs of numbers, (Pxe, Bee) k = 2,2, ..., NW. Each pair gives a univariate confidence interval for which Probability Cocos Dias Burong 2 .68 . The 68% confidence level is taken to roughly correspond to the usual convention of reporting count results "plus/minus" one standard deviation. The nonclassical element which is used in the analysis is that we make implicit use of the physical fact that Q(E) 2 0. It can be shown that the problem of determining the smallest possible confidence interval for Pe has the following extremal formulation: Se on p607 3. S 4263) 968 2 DE 50 A (B) (8)28 - oqje/} Tempo - 0607670 (%(8) 668 dB ISA (8) @CELE - oplos 11= where the value of x (among other things ) depends upon the confidence level desired. These extremal problems can be restated verbally as follows: "From all conceivable spectra Q(E), consider only those which ere nonnegative, and which are consistent with the experimental data when substituted into Eg. (1). Then select that spectrum -15- which yields the largest possible value of Ple from Eq. (2), und also select that spectrur which yields the smallest possible value of Pk These two extremal values of Pi form a valid confidence interval for the true Py Interestingly, the above extremal formulation without the inequality Q(E) 2 0 gives the same results as the ordinary least squares method S when applied to a discrete spectrum. The interval obtained from Eq. (4) is said to be optimal in the sense that no smaller interval is possible based upon the experimental data and the a priori constraint that Q(E) 2 0. FERDO, however, is not an optimal method because of the simplifying assumptions which have been made to speed it up. Thus, the confidence intervals from FERDO will be wider than the best possible theoretically. The efficiency of FERDO is still quite good for scintillation spectrometers, and typically confidence in- tervals are only a few percent wider than optimal. THE FERDO ESTIMATION METHOD The FERDO viewpoint of the inequality constrained estimation problem is that we try to find a certain nearly optimal combination of slices through the response functions' A, (E) (such as shown in the upper corner for Fig. 4 for a particular value of k) so that their sum approximates" the response of the kth channel of the desired ideal analyzer, or NR W, (E) = { 0, A. (E) i=1 as shown in Fig. 3. where the uz (i = 1,2, ..., NR) are a set of coeffi- cients. Then an estimate of the desired PE = SW (E) (E JE 1s given by -36- NR @= SUA, (FE i=1 NR (E JE = £ Wece (5) 1=1 91°1 where we have made use of Eq. In principle, one needs to fit each desired response function W, (E) (k = 1,2, ..., NW) with a different com- bination of funtions A, (E) using different sets of coefficients so as to get the best result for every estimate, free but fortunately FERDO is able to do most of the numerical work for all the fits at once. Since each fit- ting problem is treated essentially independently, we are going to drop the subscript k from W, (E), and write only W(E) in the remainer of this section. After we have derived a method for just one such function, we can then apply the same method, in turn, to each of the NW functions. STATISTICAL CONFIDENCE REGION WTO Some Although is an estimate of Pk, we wish to determine an uncertainty for Ôn due to the statistical errors in ce. Even with no statistical errors (s; = 0, i = 1, 2, ..., NR) there will generally still be some uncertainty in @re due to the finite number, NR, of actual channels. The FERDO confidence interval, therefore,, includes both the statistical uncertainty and the uncertainty due to the finite number of actual chan- nels. NR The statistical error in @s = { u,C4 is given by the customary 1:1 - formula for the standard deviation of a sum of terms, sta (on) = std NR E u,c, lidl d NR = / E ums; V 1.:-1 (6) -17- NR However, there is another source of uncertainty if Eu, A, (E) E W(E). 1=1 4 For www.binuous functions A, (E) and W(E), it is usually not possible to choose u, (i = 1,2, ...,NR) to get an exact match at all values of E, with just a finite NR. 'Thus, we recognize that a discrepancy inay exist and take it into account in urriving at the final uncertainty. The known nonnegativity of Q(E) allows us to easily find a weak upper bound, 0, such that NR 8 = upper bound Jo[w(E)- EUA, (E) Q(E)]dE (7) i=1 The final confidence interval could be given by : Boy = fk - stu(ik) - 0 k + sta (@ny) + 8. It is then easy to see that the probability that comes prisonez .68. The 2 sign rather than an equality sign holds here since 8 is only an upper bound and may substantially overestimate the actual uncertainty. MINIMIZATION SCHEME The HERDO problem consist of two parts. First we must determine a suitable expression for 8, Then we must nunerically determine the coef- ficients u, so that the resulting confidence interval is as narrow a6 . possible. The first step is to find an expression for 8. The key rela- .. .. .. . tion is the inequality: .-** NR NR Aj*(3)(E;) W(Eg) - 474(Eg) s WIE,) - 4.A, (E;)! ----- A+ (3) (E;) (8) -18- which holds at a finite set of values, E, (j = 1, 2, ..., NC)for any i* such that Aq* (E.) € 0. We will later see that there is a particular way to select 1* vs j so as to obtain the narrowest confidence interval. This inequality would also hold at values of E between the set E,, j = 1,2, ..., NC so that NR W(E) - E 44 (E) s W(Eg) - NR J (9! i i=1 *1*(]}(E) 4*(3)(E,] 1 ) if the functions A, (E) and W(E) were linearly varying between values of Ey, j = 1,2, ..., NC. We now introduce the only essential assumption of the inequality constrained methods, that the values of E, can be chosen sufficiently close together so that the inequality of Eq. (9) holds, even though the functions W(E) and A, (E ) may not be quite linear between the set of points E,, j = 1,2, ..., NC. One may of course, check this assump- tion after the completion of the problem, by substituting the functions and the coefficients into E d checking that the inequality holds smea for all E and not just for values of E, In practice for functionis W(E) and A, (E) which have been smeared by a Gaussian smoothing function, 2 to 3 energy points, E., per smoothing width (full width at half maximum of the smoothing function) suffices to · insure that Eq. (9) will hold at all intermediate values of E. If too few points are taken, the inequality may not be valid at intermediate points, and if too many points are taken, the computation will be slower and the inequality becomes weaker. Usually the pulse-height distribution smoothing function due to plotomultiplier statistics, etc., has a width which varies with energy. This indicates that the most efficient way to choose the minimun number of vulues of E, is to choose the spacing -19- nonuniformly so that the distance between points 16 proportional to the smoothing width in the neighborhood of the point. Remember that the smoothing width in energy and in pulse height may not be simply related. Now from Eq. (1) and the inequality of Eq. (9), NC E j=1 0 = *() NR [ ) - (10) i=1 u, A, E, || Az*(3) (Eg) NR 1=2 is an upper bound to So[WE) -E 4; 4, (E) °(E) jar. Since ēq të not avail- able, we use ce +6, instead and choose 1*(3) to minimize ( O which follows from (ta - 6/7)2 = 0, and the well known Schwartz M soveli 1,7 mle. (13) Then 6 SG'= NC Nitra 2/C+* + 8 *-+ (werte) 10,1 - 144,60;119 mit au) 41 87 + .NR (NC/72) E WE,) - j=1' (24) = 1 1 A+ (Eg) The value of the parameter 1 is free and can be chosen for best results. It is found that the results are relatively insensitive to values of a from about .01 to 1.0. A value of 7 = 0.4 18 used in FERDO. This e' may be readily minimized by setting the derivativee with respect to u, (1 = 1,2, ..., NR) equal to zero, and solving the resulting equations. In matrix notation, the solution 18 4 - [AR2A" + (12/NC)827 --AR?w (15) where: . Ang A(B min Q is 'a diagonal matrix with S 18 a diagonal matrix with Sy = 84, and ; WCE). -21- We now have a valid confidence interval [ộk - 6', Ek + e'], or going back to Eq. (11), we obtain the narrower interval [ête - €, @+ €]. CALCULATION OF TRANSFORMATION MATRIX The bulk of the numerical work in FERDO is the computation of the transformation matrix HT = [AQ?A" + (T?/NC)s?) --AQ2. This matrix is then multiplied by the "window" vector for each function W. (E) (k = 1,2, ..., NW) to determine the coefficients u. The confidence interval is then evaluated from Eq. (14), -or Eq. (11). Equatior (14) is used in FERDO although Eq. (11) yields somewhat sharper results. The computation of the transformation matrix may proceed formally, but due to accumulation 0 of round-off errors, double precision is necessary. Alternatively, a non- direct method can be used which results with less round-off error. One TOU "TOY such method is based on the observation that the least squares solution of the equations I with weight matrix (NC/72)22 ME! L 0 52 is identical to Eq. (15). Then an orthonormalization least squares method an e can be used to obtain HTwithout the necessity for forming the explicit -22- product AQ"A", thus yielding greatly reduced round-off error. Note that HT always exist cince [AQ-A" + (74/N)SC) is never singular as long as 8.20, and I t 0. In FERDO, a Gramm-Schmidt-Hilbert orthogonalization routine, GINV, 1s utilized, with successive reorthogonalizations to avoid gradual loss or orthogonality due to round-off errors. The routine GINV is entered with (s-AQ) and returns R = [(s--AQ) (s-ZAQf+ (72/NC )I]-2 (5--AQ). Then the desired HT 18 given by (s-RQ). The result of these precautions is that a single precision calculation for HT is entirely adequate. FØRTRAN PROGRAM A listing of the FØRTRAN-IV/FØRTRAN-63 programs for FERDO and GINV is given in the APPENDIX, and the notation closely follows that of the text. The user mist supply three additional subroutines to read in or create the matrices, Az y = (E,), Wks = W (E;), and the vectors cy and S4. Since FERDO is a Lexiographic Algorithm and identifies functions merely by their indices, it has proved a great comfort and convenience to include a Lexiographic Key in the FERDO output which associates phys- ical labels with each index. A short numerical example is given so that a user may check his code for proper functioning. When setting up a large problem from scratch, it is usually desirable to have graphical displays of various intermediate results, which were omitted from FERDO for brevity. It is hoped, however, that the FERDO code will give prospective users an idea of the method until such time as more glorious versions can be pub- lished with more user oriented helpful features, and with more extensive documentation. en que -23- COMMENTS ON FERRO Many numerical methods for the solution of integral equations make some assumption about the form of the unknown function Q(E) and obtain a corresponding numerical integration formula which reduces the integral no e equation to a matrix equation. FERDO makes no assumption about the form of p(E) and contains no nunerical integration formula. It is the implicit use of Q(E) 2 0, and A, (E) 2 0 that allows this departure. If the reader should be sorely tempted to set Aqy equal to the value of A, (E) averaged about E, - rather than the value just at E, – in the expectation that there is probably some quadrature formula implied and that such a procedure will reduce the error, then FERDO is probably not the code he should use, This unfortunately means that an old problem usually has to be reformulated for FERDO. · · -- -. .-. .. . .. . . ..-. - .. ---. incrimininku...*. na to predictions imao su a w -24. RESULTS The pulse-height distribution shown in Fig. 5a is that for a discrete line spectrum of neutrons from the Be (a, n)12c reaction in which the neutron leaves the c nucleus in either the ground state or one of the several well-separated excited states (1.e., 4.43, 7.6, or 9.6 Mev). This pulse-height distribution was obtained for a single amplifier gain. In Fig. 5b is shown the spectrum obtained as a result of analyzing the EV WOS data of Fig. 1 with the unfolding code. The ground level and two excited levels are obtained with about 15% resolution, which is a limit imposed by the scintillator used for this series of measurements, Ir. fig. 6a is shown a composite pulse-height distribution obtained from two pulse-height distributions, one at low gain (as in Fig. 1) and one at 5x this gain. For each of these two pulse-height distributions, a current integrator was used to measure the total number of alphas striking the beryllium target. The current integrator readings and the gain ratio facilitate proper normalization of the two individual pulse- height distributions. In practice, one simply uses the two (or more) unnormalized pulse-height distributions as inputs to unfolding code, along with the amplifier gain settings and current integrator readings. The output of the unfolding code is shown in Fig. 6b. No break is observed in the spectrum at 4 Me V, where the two pulse-height dis- tributions join together. In Fig. 7a is shown a pulse-height distribtuion for the beam from the ORNL Tower Shielding Reactor. This is an example of a continuous spectrum. The spectral intensity obtained from this pulse-height dis- tribution by analysis with SLOP is shown in Fig. 76. Along with this is -25- shown the same spectrum as obtained with a 'Li sandwich "shielded diode spectrometer" described in an earlier talk. The agreement in shape and magnitude is fairly good considering the combined uncertainties of the two spectrometer techniques. All the unfolded spectra were obtained with an early set of response functions which were no good below 1 Mev. The scintillator used was a 2- by 2-in.-diam glass-encapsulated NE-213 liquid scintillator painted at the factory with a white reflective coating. As was explained previously, this turned out to be a rather bad choice. CONCLUSIONS We believe that the two traditional disadvantages of the proton recoil spectrometer (gamma sensitivity and non-Gaussian response ) have been mitigated, and that it will enjoy increasing use for measurement of neutron spectra in the 0.5 to 14 MeV range. - . . . . . -... more than the formationen bestemmelser s i mobile animatio -26- UNCLASSIFIED ORNL-DWG 64-11082 ܗ ܗ |_ Be (a,ni"?c, "2c* Ex=7.3 Mev =0° TA ܛ $1E) (neutrons/cm. MeV) ܚ vill ܢ ܝ HIINIUMMUNITIIL 1 ܘ ܂ . 2 4 6 8 10 NEUTRON ENERGY (MeV). 12 14 COUNTING RATE X 10 (a) X1/100 3 4 PULSE HEIGHT Fig. 5. Pulse-height distribution (counts/grouped bin width) and unfolded neutron spectrum (neutrons/cm - MeV) for the "Be (a,n)ac reaction with E = 7.3 MeV and angle of observation 8 = 0°. -27- ܗ UNCLASSIFIED ORNL-DWG 64-11081 TT °Be (arn)"?c, 12* Ex=10.1 Mev A=170° ܗ ܛ $(E)(neutrons/cm. MeV) --- ܚ_ _ ܝ VIINIL ܘ 1 2 du Wuustumosfera 4 6 8 10 NEUTRON ENERGY (MeV) 12 14 f -------- HIGH-GAIN DATA LOW-GAIN DATA S - PLN COUNTING RATE x 10 (a) LX 1/100 - - 3 PULSE HEIGHT Fig. 6. Pulse-height distribution (count/grouped bin width) and unfolded neutron spectrum (neutrons/cm - MeV) for the 'Be (a,n)tac reaction with E = 10.1 MeV and angle of observation, 0 = 170°. -28. UNCLASSIFIED ORNL-DWG 64-11083 72 x 2-in. NE-213 SCINTILLATOR $(E)(neutrons / cm. MeV). NEUTRON SPECTRUM FROM TSR-II WATER MODERATED REACTOR OLI SANDWICH DETECTOR Doo 4 6 8 10 NEUTRON ENERGY (MeV) 12 14 LOW GAIN - - MEDIUM GAIN 0.8 H-HIGH GAIN 1x 10 COUNTING RATE 0.2 - vurro" - - - OO 5 6 3 PULSE HEIGHT remembe... ' No Mew Fig. 7. Pulse-height distribution (counts/grouped bin width) and unfolded neutron spectrum (neutrons/cm - MeV) for neutron beun emerging from a bear hule through the shield of the water-moderated Tower Shielding Reactor-II. The point of measurement was 142 in. froin the core surface. home *. -29- REFERENCES 1. M. Forte, A. Konsta, and C. Maranzana, "Electronic Methods for Discrimination of Scintillation Shapes", Paper NE-59, IAEA Conf. on Nuclear Electronics, Belgrade, May 15-20, 1961. 2. B. Brunfelter, J. Kockum, and Il. Zetterstrom, Neutron Dusimetry II IAEA, 351 (1963). 3. G. During, R. Jansson, and N. Starfelt, Arkiv För lystk, Band 26, Nr. 19, pp. 293-307 (1964). 4. B. Banville, Compteur Absolu Pour Neutrons Rupides, (Thése), Univ. de Montreal, (October 1960). 5. V. V. Verbinski et. al., following paper of these proceedings. 6. W. R. Burrus, Neutron Phys. Div. Ann. Progr. Rept. Oct. 1963, ORNL-3499, Vol. II. pp. 120-24. 7. W. R. Burrus, Utilization of A Priori Information in th: Statistical Interpretation of Measured Distributions, (Dissertation) The Ohio State University (1964), ORNL-3743. -30- APPENDIX FERDO CODE AND DEMONSTRATION PROBLEM -31- - AIN DIMENSION A169,71) RIO?!), S169) W169,7 ;) HT 169,71), 01711,UT 1691, TUGIN(71,71), ATEMP171), IDENTI I 4) MO#69 PEAN 15,905) TINENT(I), [#1,141 WRITE 16,9061 IDENTIJI, J#1, 14 ) ņran 15,9011 NP, NC,NW 'PITE 16,9721 NRNC, NW 7 4*0.4 CALL ARFANGA, MR, NR, NC, NWI CALL ACREANIR,S,MR, NR,NCI CALL WREADOW, MR, NC, NIW!) C ! CALCULATE O VECTOR 2217 JUINC (J)#1.77?5 .7 M II, NR ¿F (All, Jil12,21,10 I MINICAMAXI/B(1),0,0) + S(III/A(I,JI IFIQIJI-AMIN)20,20,15 15 QIJI HOMIN 20 CONTINUF Clu CALCULATE HT DO 25 I#1, NR DO 25 J#INC HTII, JI RIJ)*(I,JII/S(I 25 CONTINUE CALL GINVIHT, MR, NR,NC TAW/SQRT (FLOATINCIDUGIN, ATEMPI no 50 I#,NR DO 5?? J#INC HT(I,J) #IQI JI*HTII,JII/SII) 50 CONTINUE Cuu BEGIN WINDOW LAP WRITS 16,9161 II ENT(J), J#1,14) WRITE 16,977) no 510 KW, NW no on I #1, NP UT(I)#0.0 6? CONTINUE DO 100 J#INC. IF(W(K, J)171,190,70) 777 DO 80 1#i,NIR UT( I )#UT(II+W(K, J) *HTII, JI 80 CONTINUE -32- 100 CONTINUE USSUM#0.0 UTB#0.07 no 150 I#INR UTR#UTR+UT(I)*RCIT USSUM#USSUM+IUTI I J *SI) 1**2 150 CONTINUE WUAQ#0.0 DO 200 J*INC SUM#0.0 DO 175 I #INR SUM#SUM+UT(I)*ACIJI 175 CONTINUE WUAQ#WUAQ+QIJI*QIJI*(WIK, JI-SUM)**? 200 CONTINUE TTSQ#SQRT (1.0+TAW#TAWI RAD#SQRTIUSSUMEWUAQI*TTSQ PHIUPHUTB+RÁD PHILOHUTB-RAD ERRI #TTSQ*SQRTIUSSUM ERROTTSQ*SQRT (WUAQ) WRITE 16,904) K, ERRI , ERR2,PHILO,PHIUP 500 CONTINUF 901 FORMAT(8110 902 FORMATI TH0,5X, 19HNUMBER OF ROWS #,15/6X, 1 9HNUMBER OF COLUMNS #, 115/6X, 19HAUMBER OF WINDOWS #,15) 904 FORMAT CIH , 18 ,4617.71 905 FORMAT (13A6,A2 j 906 FORMATITHI,13A6, 1A2) 907 FORMAT( IHO, 2X, 6HWINDOW, 6X, 4HERRI,13X, 4HERR2,13X, 3HPLO, 14X, 3HPUPI STOP END -33- SUBROUTINE GINVIA,MR, NR,NC,TAU,U, ATEMPI - -- - - - UPON ENTRY A A MATRIX WITH NR ROWS AND NC COLUMNS AFTER THE RETURN THE ORIGINAL AIS DESTROYED AND THE ARRAY A CONTAINS THE TRANSPOSE OF T THE TRANSFORMATION MATRIX IX # THAI WHICH SALVES THE PAIR OF EQUATIONS : . A*X # B AND TAU I*X # 0 IN THE LEAST SQUARE SENSE. VUUUUUUUUUUUUUUUUUUU MR # IST DIMENSION NO. OF ARRAY A IN THE CALLING PROGRAMS TAU IS A NONNEGATIVE CONSTANT WHICH CONTROLS THE ERROR PROPIGATION OF THE TPANSFORMATION MATRIX. IF TAU # D. THEN THE RESULTING TRANSFORMATION MATRIX IS THE ORDINARY LEAST SQUARES TRANSFORMATION MATRIX (THE INVERSE IF NR. NC).. ATEMP AND U ARF USED FOR WORKING SPACE BY THE ALGORITHM AND DO NOT NECESSARILY CONTAIN ANY RELAVANT NUMRERS AT THE CONCLUSION. ATEMP MUST BE DIMENSIONED AT LEAST NC. BY THE CALLING PROGRAMS U MUST BE DIMENSIONED AT LEAST NC*NC BY THE CALLING PROGRAMS NO. OF MULTIPLICATIONS # NC**? 15/? NR + 2/3 NOI DIMENSION A(MR,NCIU (MR, NCI , ATEMP INCI TAUSQ # TAU** 2 PLACE UNIT MATRIX IN U DO – I # INC. i0 4 INC UlIJI 0.0 U(I,II # 1.0 ter nnnnnn ORTHOGONALIZE COMPINED MATRIX (A ABOVE UI BY GRAMM-SCHMINT-HILPEPT METHOD WITH FIRST NR ROWS WEIGHTED WITH 1 AND THE OTHER NC ROWS WEIGHTED WITH 1/TAU. THEN REORTHOGONALIZE TO LESSEN ROUND OFF ERROR.. DO 20 I # 1, NC II # 1 - 1 IF (II) 2,11,2 DO 10 LL # 1? DO 10 J # II MOT # 0.0 DOT2 # 0.0 O 3 K # 1, NR DOT # Alk, I)*A( KJ) + DOT DO 6 K # 1,j DOT2 # UK, I)*U(K, J) + DOT2 DOT # DOT + DOT2* TAUSQ DO 8 K # lj UIK, I) # UIK, I) - DOT*U1K, JI -34- DO 10K # 1,NR Alk, l1 # Alk, Il - DOT *A (K, J) ili NORMALIZE THE COLUMN 1 OF THE COMBINED MATRIX DOT # non NOT2 # 0.0 no 12 K # NR DOT # Alk, 1 ) *AIK, I) + DOT DO 14 K # 1, DOT2 # Ulk, [)00K, Il + NOT2 MOT # nnt + DAT?*TAUSO MOT # SORTIDOT) no 17 K # 1, 1 Ulkoil #VIK, Ilinnt no 19 K # 1 N'R Alk, I) # Alk, ill not 20 PONTINUE กากา CALCULATION OF THE TRANSPOSE OF THE TRANSFORMATION MATRIX TITRANS.) # A# UITR.) Do 50 no 1 # 1,NR. 45 J # i NO ATEMPI JI # 0.0 DO 45 K # J, NC ATEMPIJI # All,K) *UlJ,K) + ATEMPIJI CONTINUE. 50) J # 1 NC A(1,Jl # ATEMPIJI CONTINUF 45 no 50 RETURN END F FERDO DEMONSTRATION TEST CA . . . .. . . - . 2 3 4 5 6 7 8 - . RESPONSE FUNCTIONS A(E), 1:1, 2, ... 9 - ---- jiminin I .. - .. Eco-- ... ni.... . .... .... ici: . c. ko 1 2 3 1 ,4 .... 1 WINDOW FUNCTIONS ..... ! W (E), k = 1, 2, ... 61 .... :::.. T-Add-.- --- ir : : : ili .... ..... 2:1:ņ ņģ ī nga gël.:.:12. :3 .4 5:36. 7. 8. 9,20 ... 11 12 13 . COMPARISON POINTS A 2.. ::. 0 :2 2 2 3 4 5 6 7 8 9 10 11 12 ...:.::... Partiole Energy (Mev) ::. 13 14 15 . .. Front Awesom wi :.. . .. . - .. ... . .in . ...... .... FERDO DEMONSTRATIÓN TEST CASE RESPONSE MATRIX 10 11 12 13 14 4 15.0 20.0 1 1.0 0.0 0.0 0.0 5 6.0 15.0 2 15.0 6.0 1.0 0.0 6 1.0 6.0 7 0.0 1.0 9 0.0 8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3 20.0 15.0 6.0 1.0 1.0 0.0 0.0 15.0 20.0 15.0 6.0 0.4 0.0 0.0 0.0 0.0 6.0 15.0 20.0 15.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0 10 o wa uh w Now 0.0 0.0 0.0 0.0 0.0 0.0 -36- 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.0 1.0 0.0 0.0 0.0 15.0 6.0 1.0 0.0 0.0 20.0 15.0 6.0 1.0 0.0 10.4 18.4 18.4 10.4 3.0 1.0 6.0 15.0 20.0 15.0 1.0 6.0 15.0 20.0 15.0 0,0 0.0 0.0 1.0 6.0 15.0 0.0 0.0 4.0 3.0 10.4 1.0 6.0 0.0 - - - - - - FERDO DEMONSTRATIÓN TEST CASE WINDOW MATRIX i 2 3 4 al u Fwn H 1.0 0.0 0. 0.0 0.0 0.0 1 2 15.0 13.0 10.4 0.0 0.0 0.0 3 20.0 19.2 18.4 0.0 0.0 10.0 4 15.0 17. 0 18.4 0.0 0.0 10.0 5 6.0 8 .4 10.4 0.0 0.0 10.0 6 1.0 2.4 3.6 20.0 1.0 10.0 7 0.0 0.2 0.4 0.0 5.0 10.0 8 0.0 0.0 0.0 0.0 18.4 10.0 0.0 0.0 0.0 0.0 15.0 10.0 9 10 0.0 0.0 0.0 0.0 0.0 10.0 11 0.0 0.0 0.0 0.0 2.0 10.0 12 0.0 0.0 0.0 0.0 1.0 0.0 13 0.0 0.0 0.0 0.0 0.0 0.0 14 0.0 0.0 0.0 0.0 0.0 0.0 -37- om 1 Ĉ 3 4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 RIGHT HAND SIDE AND STANDARD ERRØRS 7 6 9 1.0 -38- FERDO DEMONSTRATION ONE NUMBER OF ROWS NUMBER OF COLUMNS NUMBER OF WINDOWS 6 LEXIOGRAPHIC KEY RESPONSE MATRIX ROW LABEL PFAK AT E # 3.0 MEV PEAK AT E # 4.0 MEV PEAK AT E # 5.0 MEV PEAK AT E# 6.0 MOV PEAK AT E # 7.0 MEV PEAK AT E # 8.0 MEV PEAK AT E # 9.0 MEV PEAK AT E #10.0 MEV PEAK AT E #11.0 MEV WINDOW MATRIX ROW LABEL GAUSSIAN AT # 3.0 MEV GAUSSIAN AT E # 3.25 MEV GAUSSIAN AT E # 3.50MEV TRIANGLE AT E # 6.0 MEV GAUSS.(DISC. AT E # 10.0) TRAPAZOIDAL (E # 2.0-12.5) COMPARISON POINTS COL 2 EGG G 676 LABEL E # 0.0 MEV E # 2.0 MEV 3.0 MEV 4.0 MEV 5.0 MEV 6.0 MEV E # 7.0 MEV E # 8.5 MEV # 10.01 Double point to reproduce discontinuit in window No. 5. E #10.0(+) E #12.0 MEV E #12.5 MEV E #13.0 MEV E #14.0 MEV FERDO DEMONSTRATION ONE WINDOW ERRI 0.1041030E-01 0.1211000E-01 0.1055172E-01 0.73644541E-01 0.6393450E-01 0.1562105E-00 ERR2 0.5057427E-05 0.1517981E-01 0.1952416E-01 0.5856968E 00 0.3407444E-00 0.4108813E-00 PIO 0.98956111 00 0.1009622E 01 0.9956193E 00 -0.3248182E-00 0.57678702 00 0.1.1599521 01 PUP 0.101.012E 01 0.10181159E 01 0.10398117E 01 0.8557993E OO 0.1270168E 01 0.2039086E 01 END : 1. TT ' . . - S . + - DATE FILMED 18/20/65 * -