DETERMINATION OF THE THERMAL NEUTRON FLUX IN THE CORE OF THE REACTOR. Measurement guide

Transcription

1 BUDPEST UNIVERSITY OF TECHNOLOGY ND ECONOMICS Institute of Nuclear Techniques (BME NTI) DETERMINTION OF THE THERML NEUTRON FLUX IN THE CORE OF THE RECTOR Measurement guide uors Máté Szieber Éva M. Zsolnay pproved by Szabolcs Czifrus

2 Contents 1. Introduction Theory Neutron spectrum and neutron flux distribution in ermal reactors Determination of e ermal neutron flux by activation meod Principle of e meod Determination of e ermal neutron flux from e activity of e irradiated detector Determination of e absolute value of ermal neutron flux Determination of e relative distribution of ermal neutron flux Objective of e measurement Equipment and materials needed Description of e apparatus to measure e wire activity Procedure Measurement of e relative distribution of e ermal neutron flux Measurement to determine e absolute value of ermal neutron flux Evaluation of measurement data Evaluation of e relative distribution of e ermal neutron flux Error analysis of e relative flux measurement Error analysis of e absolute value of e ermal neutron flux Estimation of statistical error Systematic error Knowledge check References HUVINETT 1

3 1. INTRODUCTION The nuclei of certain naturally occurring isotopes can be transformed into radioactive ones by exposing e material to neutron radiation, and e activity of e radioactive products produced can be measured by means of appropriate counter system. In addition to e factors determined by e conditions of measurement, is activity is affected only by e neutron flux in e point of irradiation and by e activation cross section of e target material which in e present case, is e neutron detector. Provided e activation cross section is known, e neutron flux can be determined by measurement of e activity of e sample irradiated. s e activation cross section of a number of materials depends on e neutron energy in different way is meod will yield information also on e energy distribution of neutrons. In e present measurement e ermal neutron flux will be determined using activation detectors.. THEORY.1. Neutron spectrum and neutron flux distribution in ermal reactors Fast neutrons, e energy distribution of which can be seen in Fig. 1., are produced in fission reactions. In ermal nuclear reactors, e fast neutrons gradually slow down due primarily to e collisions wi e atomic nuclei of e moderator, and a neutron spectrum schematically depicted in Fig.. is formed. Let e density of e neutrons wi energies between E and E+dE near a point r of e detector be n(r,e)de. Thus, e definition of neutron flux is as follows:,, r E vn r E, (1) where v is e velocity of a neutron wi energy E. s far as is measurement is concerned, e most important part of e neutron spectrum is e ermal spectrum, which can approximately be described by e Maxwell distribution: r, E E E exp, E E ( kt ) kt () where E is e more or less arbitrarily chosen upper limit of e ermal region (usually.65 ev), k is e Boltzmann constant, and T is a quantity wi dimension of temperature characteristic of e ermal neutron spectrum. 1 By ermal neutron flux we mean e following integral: E r, r E de, (3) which is often written in e following form: 1 It is usually called neutron temperature, which is slightly higher an e absolute temperature of e reactor. HUVINETT 1 3

4 r v.n r, (3a) where n is e density of e ermal neutrons around point r, and v is e average velocity of e ermal neutrons (see below). H a s a d á s i s p e k tr u m neutr on flux (arbitr ary unit) s p e k t ru m (ö n k é n y e s e g y s é g ) E ( M e V ) Figure 1. The energy spectrum of e fission neutrons r e a k to r s p e k tr u m 1 ermal neutrons t e rm ik u s n e u t ro n o k (M a x w e ll s p e k t ru m ) n e u t ro n f lu x u s neutr on flux (arbit rary unit) ( ö n k é n y e s e g y s é g ) 1.1 1/E spectrum fission neutrons 1 / E s p e k t ru m h a s a d á s i n e u t ro n o k resonances re z o n a n c iá k h a t á s a. 1 1.E- 9 1.E- 8 1.E- 7 1.E- 6 1.E- 5 1.E- 4 1.E- 3 1.E- 1.E- 1 1.E + 1.E +1 E ( M e V ) Figure. The schematic energy spectrum of a ermal reactor In ermal reactors, e spatial change of e neutron flux is significant. In case of a uniform bare (wiout reflector) slab reactor core, e spatial distribution of e ermal neutron flux, according to e reactor eory, is given by e following relationship [1]: HUVINETT 1 4

5 where Φ (4) ( x,y,z πx πy πz ) Φ (,, ) cos( ) cos( ) cos( ), a b c a a δ, b b δ, c c δ, a, b, c actual physical dimensions of e core, δ extrapolation leng. ccordingly, e distribution of neutron flux along axis z at a given place x, y can be written: where Φ Φ πz,z ) Φ cos( ) Φ ( z ), (5) c ( x,y πx πy Φ (,, ) cos( ) cos( ). a b The non-uniform neutron flux distribution is unfavorable, among oers as it results in a nonuniform burn-up of e fuel rods, e central rods burning up soon due to e high flux density while rods arranged in e edge of e core are hardly burned up. This problem may be eased by using a reflector surrounding e reactor by a material scattering a considerable part of neutrons leaking rough e reactor boundary back into e core. s a result, e ermal neutron flux increases wi explicit reflector peaks in e ermal neutron flux distribution on e edge of e reactor (Fig. 3.) N e u tr o n f lu x c / C / R e f le c t o r R e a c to r c o r e R e f le c t o r Figure 3. Thermal neutron flux distribution in a reflected reactor Use of a reflector improves economy of e reactor construction, as less fuel is required to achieve e critical mass. The so-called reflector saving means e decrease of e linear dimensions of e critical reactor due to e reflector. Neglecting e extrapolation leng it can be determined HUVINETT 1 5

6 at a good approximation by fitting a function (in e present case a cosine function) to e neutron flux distribution developing in e core of e reactor and extrapolating e curve to e reflector area. The extrapolated boundary of e bare critical reactor equivalent to e reflected reactor i.e. which would produce e same neutron flux distribution wiin e core as does e reflected reactor would be at e zero value of is curve (i.e. at zero flux (Fig.3.)). Hence, reflector saving is understood as e difference between e linear dimensions of e equivalent bare critical reactor and e reflected critical reactor: C c, (6) where C distance between e zero value points of e cosine curve fitted to e neutron flux distribution inside e core, c actual physical dimension of e core... Determination of e ermal neutron flux by activation meod..1. Principle of e meod Irradiating an activation detector in ermal neutron flux e reaction rate during activation is expressed by R Φ σ N act T, (7) where Φ ermal neutron flux [m - s -1 ], σ act microscopic activation cross section [m ], N T number of target atoms in e sample. The rate of change of radioactive atoms during irradiation is e difference between e rates of production and decay: d N ( t ) Φ σ N λn ( t ) act T, (8) d t where decay constant of e radioactive nuclei produced [s -1 ], t time [s]. Be N ( ) at e start of irradiation. Then e solution of e differential equation (8) using (7): R t N ( t ) (1 e ). (9) The activity of e detector at e end of irradiation enduring time T: ( T ) T N ( T ) R (1 e ). (1) The activity at time after finishing e irradiation: T ( T, τ ) R (1 e ) e. (11) These relationships will, however, be accurate only if each atom of e sample is irradiated by e same neutron flux. In case of a foil of non-negligible ickness, e average neutron flux will be ), due to neutron absorption. Their ratio is e self- lower inside it ( Φ ) an on e surface ( shielding factor: Φ HUVINETT 1 6

7 G Φ Φ. (1) Finally, e time function of e detector activity is obtained by using e above formulae: ( T T, τ ) Φ G N (1 e ) e act. (13) T... Determination of e ermal neutron flux from e activity of e irradiated detector In e ermal neutron energy range, e velocity distribution of neutrons follows e Maxwell- Boltzmann distribution according to which e most probable velocity of e neutrons is: v kt, (14) m where k Boltzmann constant, T neutron temperature [K], m mass of neutron [g]. t room temperature ( C = 93 K) v = m/s which corresponds to an energy of.5 ev. gain, according to e Maxwell-Boltzmann distribution, e average velocity of neutrons: v v. (15) π The reaction rate defined by (7) describes e activation process for irradiation in a monoenergetic neutron flux. s e ermal neutron velocities are different and e cross sections are dependent on e neutron velocity v, e activation process would be correctly described by: R Φ ( v ) σ ( v ) N d v act T. (16) Generally, is would be seriously complicate procedure and, erefore, e integral above is substituted by e product in (7) where e values Φ and σ act will be determined in concordance wi each oer in such a way at eir product yields e actual reaction rate according to (16). For instance, it is customary to define neutron flux Φ n v, where v is e most probable velocity of neutrons according to (14), n is e total density of ermal neutrons. In is case, e reaction rate for a 1/v detector ( ~ 1/v) and non-distorted Maxwell-Boltzmann spectrum will be obtained from: R nv N, act T, (17) where, act activation cross section belonging to e most probable velocity of Maxwell- Boltzmann distribution. Defining e ermal neutron flux as neutrons according to (15)), involves a cross section and a neutron temperature Tn is given by: n Φ n v (where v is e average velocity of ermal v, act which, in case of a cross section 1/v ( T ) v, act n, act (18) and T = 93 K. The number of target nuclei NT : T T HUVINETT 1 7

8 N T m L where m mass of sample [g], abundance of target isotope, mass number of target isotope [g], L From e detector activity in (13), only e activity induced by ermal neutrons shall be taken into account. Irradiating e detector at a place where also a significant epiermal flux is present and/or e epiermal activation cross section of e detector is significant compared to e ermal activation cross section, en a part of e activity obtained is due to e epiermal neutrons. Therefore a correction has to be applied to determine e contribution of e epiermal neutron flux to e activity produced. This is possible because e neutron absorption cross section of cadmium is very high at ermal energies (Fig. 4.), while negligible in e epiermal energy range. Hence, if e detector covered wi cadmium of appropriate ickness, en ermal neutrons will be practically fully absorbed by e cadmium so at e activity of detector will be induced only by epiermal neutrons of energies above about.4 to.7 ev, depending on ickness of e cadmium cover. The activity induced by e ermal neutrons is simple to determine by irradiating and measuring e activity of two uniform detectors, one bare and e oer covered wi cadmium, in e same circumstances: (19) b Cd, () where b activity of e bare detector, Cd activity of e cadmium-covered detector. It is also customary to define e so-called cadmium ratio, e ratio of e activities of bare and cadmium-covered detectors: b R Cd. (1) Cd On e basis of () and (1), e ratio of activity induced by ermal neutrons to e total activity of e detector: 1 b 1. () R Cd In case of detector materials wi activation cross section following e 1/v law rough e whole neutron energy range of interest, e value of RCd is very high an us, according to (), e share of epiermal neutrons in e activity induced in e detector may be neglected. HUVINETT 1 8

9 1.E +5 1.E +4 ( n, ) h a t á s k e r e s z t m e ts z e t (b a rn ) 1.E +3 1.E + 1.E +1 1.E + D y C d u E- 1 1.E- 1.E- 1.E- 1 1.E + 1.E +1 1.E + 1.E +3 E ( e V ) Figure 4. Total microscopic cross section of Dy, u and Cd vs. neutron energy Using a bare detector and a cadmium covered detector of identical masses, e ermal neutron flux is calculated from relationships (13), (18), (19) and () as follows: Φ = 1 e λτ N T σ v, act (T n )G [ 1 e λt b (τ ) Cd (τ )]. (3)..3. Determination of e absolute value of ermal neutron flux The following series of reactions will take place in gold irradiated by neutrons: 197 u 79 ( n, u ( n, 79 u ( n, 79 u (, T (, (, = h T = d T = 4 8 m in 1 / 1 / 1 / m 8 Hg 8 Hg 8 Hg ( T 1 / = 4 m in 199 Hg 8 Of is multi-step reaction series, actually it is sufficient to take e first one into consideration because e activation of 198 u by neutron capture can be neglected because of e experimental conditions (, act, irradiation time etc., see later). In e range of ermal neutron energies, e activation cross section of 197 u follows e law HUVINETT 1 9

10 1/v but it has a number of resonance peaks in e resonance region (see Fig. 4.) stressing e role of cadmium correction according to (19). The absolute value of e ermal neutron flux can be determined by gold activation. The absorption cross section of e 197 u isotope is also depicted in Fig. 4. The following reaction series takes place in gold irradiated by neutrons: 197 ( n, ) 198, ( 64, 68 ó ra ) u 79 u 8Hg s depicted in Fig. 5., β and γ particles are formed simultaneously in e hour half-life decay of 198 u. (More precisely, 95.6% of e β decays is followed by e emission of a kev γ-photon.) Thus, e absolute activity of gold can be determined wi a coincidence counter. Figure 5. The decay scheme of 198 u [5] In order to achieve is, an instrument is used at is equipped bo wi a scintillation beta detector and a scintillation gamma detector. The outgoing signals of e beta and e gamma channels are connected to a counter and to e two inputs of a coincidence circuit. Thus, e number of e beta particles and e gamma photons is recorded, as well as e number of e coincidence events occurring during e time of measurement. Let nβ, nγ and nco denote e number of beta particles, gamma photons and coincidence events per second, respectively. Using ese symbols, e following relations can be written: n= ; n = kγ ; nco = kγ (4a) (4b) (4c) HUVINETT 1 1

11 where ηβ and ηγ are e efficiency of e beta and e gamma detectors, kγ is e frequency of e γ- line, and is e number of decays per second in e foil, i.e. e activity of e sample. The following expression can be derived: (Bq) = n β. n n co γ It can be seen at e activity of e sample can be obtained wiout knowing e efficiency values. The measuring instrument is composed of a beta and a gamma scintillation detector, two single channel analyzers (differential discriminators), ree counters and a coincidence circuit. The signals from e detector arrive in e single channel analyzers, wi e aid of which e required range can be selected bo from e beta and e gamma spectra. (This is important for reducing e background.) The coincidence circuit, also called e ND gate, has (at least) two inputs and one output. pulse only appears on e output if signals arrive at e inputs simultaneously. Of course, e time difference of e temporal coincidence of e signals has to be specified: signals arriving at e coincidence counter wi a time difference smaller an a given time interval τ are detected as simultaneous; τ is called e resolving time of e instrument. Two of e ree counters are connected directly to e output of e single channel analyzer and count nγ and nβ, while e ird one counts nco. The counters are synchronized so at ey can be turned on simultaneously and count for e same amount of time. During e measurement, e fact at e beta detector is to some extent also sensitive to gamma radiation has to be taken into consideration. This is e so-called gamma sensitivity of e beta detector, which can be determined by placing a beta absorber (a plastic foil) on e side of e sample at faces e beta detector. The counting rate n in e β channel obtained in is way is e gamma sensitivity of e beta detector. noer correction has to be applied because of random coincidences. The number nrnd of e random coincidences per unit time is: nrnd = n n (6) where τ is e unlock time of e coincidence instrument, while nβ and nγ are e counting rates in e beta and e gamma channels. ccordingly, e activity of e foil is: (5) = ( n n n n )( n n h ) β βγ γ γ ko rnd β, γ, co n n n co (7) where hγ is e counting rate of e gamma background (cps), nβγ is e counting rate measured in e beta branch when applying e beta absorber (is includes e value of e beta background hβ as well), n,,co is e coincidence counting rate when applying e absorber, while e primed quantities are e corrected beta, gamma and coincidence counting rates, respectively. In e case of e usually applied ND type coincidence circuits, resolving time τ is e sum of e duration of e signals on e input of e coincidence circuit. Since e signals, which are.5 μs long standard logic pulses (TTL), arrive at e inputs from e single channel analyzers, resolving time can be considered.5 μs. Based on expressions (4c) and (5), it can be easily seen at if e condition were met, e number of random coincidences would be equal to at of true coincidences. In our case, we would need a.51 MBq activity source for is to be true. Since e activity of e source HUVINETT 1 11

12 we use is lower an at by several orders of magnitude, nrnd can in most practical cases be neglected. In e range of ermal neutron energies, e activation cross section of 197 u follows e law 1/v, but it has a number of resonance peaks in e resonance region (see Fig. 6.). The highest resonance occurs at 4.47 ev, where σc = 989 barn. Therefore, cadmium correction according to () plays an important role here. Thus, e activity of an irradiated foil has to be determined bo wi and wiout a Cd cover. Table 1. summarizes e measurements to be performed in order to determine e absolute value of e ermal neutron flux. Table 1. The measurements to be performed in order to determine e absolute value of e ermal neutron flux using e coincidence meod wiout β-shield wi β-shield β γ coincidence β γ coincidence bare foil nβ nγ nco nβγ nβγ,co Cd-claded foil no foil nβ nγ nco nβγ nβγ,co hγ..4. Determination of e relative distribution of ermal neutron flux The following reaction is applied to determine e relative spatial distribution of e ermal neutron flux: 164 Dy 66 ( n, 66 Dy ( T 1 / = 1.3 m in 165 Dy 66 ( T 1 / =.3 5 h 165 Ho 67 The activation cross section of dysprosium is very large and approximately 1/v-dependent roughout e whole neutron energy range. ccordingly, e contribution of epiermal neutrons to its activity is negligible. Hence, e activity of such detectors is proportional to e ermal neutron flux and, erefore, e counts during e measurement can be directly used to calculate e relative ermal neutron flux distribution. The simplest way to provide identical experimental conditions (identical volume of samples, identical irradiation time etc.) is to use a wire sample to be activated and measuring e activity of wire sections. However, e activity of e sample decreases continuously in e course of measurement, which has to be compensated during e measurement or correction has to be applied for decay in evaluating e results. In is exercise, we prefer compensation. The activity of e detector at e start of e measurement can be written on e basis of HUVINETT 1 1

13 formula (13): ( T z, τ ) Φ ( z ) G N (1 e ) e act. (8) T Provided e activity of e detector is not reasonably large compared wi e dead time of e counting apparatus, e measured intensity I() will be proportional to e activity of e sample: I ( τ ) η ( τ ) (9) where efficiency of counting apparatus. The measurement delivers e total number of counts during a finite measuring period tm: or B t m I ( t ) d t (3) B t m η ( τ ) e t d t (31) Hence: B η ( τ )( 1 e t m ). (3) The activity of e sample: ( τ ) 1 B λ. (33) e t m If e measuring period is short ( tm ), en B () tm, and () B / ( tm). The number of counts during e measurement can be determined using formulas (8) and (33): B ( z, τ ) Φ G T act T t m ( z ) 1 e e. (34) N (1 e ) The subproduct in figure brackets is e same for each point of e wire. In order to obtain values at are proportional to e ermal neutron flux from e number of counts, e values of e function t m m 1 t F t, t e e have to be determined for every single data point. For is purpose, tm is fixed at an appropriate value at e beginning of e measurement, and e time of e end of each measurement is recorded when scanning e parts of e wire. Thus, knowing e half-life of 165 Dy (based on [6] it is.334±.1 hours), e correction according to (35) can be calculated. 3. OBJECTIVE OF THE MESUREMENT In is exercise, e relative distribution of e ermal neutron flux will be determined along a vertical axis of e core, using e meod described in chapter..4. Then, e vertical reflector savings are determined. wire of Dy-l alloy containing 1% of Dy is used for e measurement of e neutron flux distribution. The absolute value of ermal neutron flux is determined by gold activation detector. (35) HUVINETT 1 13

14 4. EQUIPMENT ND MTERILS NEEDED Reactor + pneumatic rabbit system for sample delivery Plexiglass rod to hold e wire Coincidence instrument Wire activity measuring apparatus (controlled by a PC) Wire of Dy-l alloy Foil of Dy-l alloy Gold foils l and Cd capsules for covering e foils Radiation protection equipment (rubber gloves, pincers etc.) 4.1. Description of e apparatus to measure e wire activity wire of Dy-l alloy is used to determine e spatial distribution of e ermal neutron flux. The activity as a function of position along e Dy-l wire is determined wi a so-called wire activity measuring apparatus. Since e wire is positioned along e axial coordinate z, instead of writing B(r,t) in expression (34), we now write B(z,t). The wire activity measuring apparatus consists of e following parts: too-racked mounting rail in which e wire is fastened lead tower rough which e wire holding rail can be pushed scintillation beta measuring head fitted into e lead tower, and a copper collimator below it so at it sees only a few mm part of e wire stepper motor which makes forwarding e rail below e detector possible stepper motor control unit connected to a computer via a USB port digital spectrum analyser (Canberra DS 1) for processing e signals of e beta measuring head, also connected to e computer computer wi data acquisition software (GenieMot) HUVINETT 1 14

15 Figure 6. The cross section of e reactor core a automatic control rod; b1, b safety rods; c fast pneumatic rabbit system; d ermal rabbit system; e vertical irradiating channels in water; f fuel assemblies; g graphite reflector elements; h vertical irradiating channels in graphite; i vertical irradiating channel in fuel assembly; j neutron source; k manual control rod. 5. PROCEDURE 5.1. Measurement of e relative distribution of e ermal neutron flux In preparing e measurement, e wire is placed in a groove of e plexiglass rod. The moderation properties of plexiglass are similar to ose of water, and us e disturbance of e flux distribution of e active zone will be negligible. The plexiglass rod is en placed by e operator to core position E6 (neighbouring D5) of e non-operating reactor, and e irradiation is performed. Simultaneously, e Dy foil is also activated in e rabbit system in core position D5. Since e wire used is made of Dy-l alloy, e HUVINETT 1 15

16 samples have to rest for at least minutes after irradiation to permit e interfering activity of l to decay. (The half-life of e 8 l is.4 minutes.) Then e wire is removed from e plexiglass rod and placed into e wire activity measuring apparatus. In e measurement, β - particles from e decay of 165 Dy are detected. First e Dy foils irradiated wi and wiout a cadmium cover are measured wi e aid of e beta measuring head. By comparing e number of counts, it can be verified at e assumption at only a negligibly small part of e activity originates from e reactions induced by e epiermal neutrons is true. Thus, e activity distribution of e wire irradiated wiout e cover will indeed yield e distribution of e ermal flux. Subsequently, e holding rail has to be pushed into e starting position and e measurement points and measurement time have to be set on e computer. The spectrum range in which e numbers of counts are summarized has to be set as well. t e end of e measurement an SCII file is obtained, e rows of which contain e position, e number of counts acquired during e time of measurement in e pre-selected spectrum range, and e time of completion of e measurement in e given position. 5.. Measurement to determine e absolute value of ermal neutron flux In e irradiation channel in core position D5 two gold foils of identical size and mass one bare while e oer in cadmium cover are irradiated. The foils in a polyeylene sample holder are forwarded to e operating reactor by means of e pneumatic rabbit system joining e irradiation channel. Record e time of e end of irradiation and measure time t elapsed until e measurement of e activity of e individual foils. Take e activity of e sample corresponding to e bare and Cd values in () wi e coincidence counter. 6. EVLUTION OF MESUREMENT DT 6.1. Evaluation of e relative distribution of e ermal neutron flux ccording to expression (35), e decay time correction has to be performed on e numbers of counts measured in e individual points. The position dependence of e corrected data points obtained in is way is proportional to at of e ermal neutron flux distribution (cf. (34)). Plot e data points, en determine e axial reflector saving using formula (6) by fitting a cosine function to e central section of e measured curve according to formula (5) Error analysis of e relative flux measurement It is known from probability eory at e distribution of e number of counts follows e Poisson distribution. Consequently, e variance of e individual numbers of counts is equivalent to eir expected value. The latter is approximated by e numbers of counts, so e following expression is usually true: D B i M B i B i. (36) This approximative formula is valid wi e same condition as e measurement meod itself: dead time and e effects of e background can bo be neglected. The simplest way of correcting dead time is providing e time of measurement tm as live time. However, it is important to make sure at dead time does not exceed a few percent even in e case of e highest numbers of counts. Usually a suitable curve fitting software is needed to perform is task. HUVINETT 1 16

17 6.. Error analysis of e absolute value of e ermal neutron flux Error analysis of e absolute value of e ermal neutron flux is considerably more difficult an at of e relative distribution. First of all, two types of errors need to be differentiated between: e statistical error and e systematic error Estimation of statistical error On e assumption at e distribution of e number of counts follows e Poisson distribution, e variance of e activities calculated based on expression (4) can be written using e well-known formula for e propagation for uncertainty (e.g. [6]), after neglecting nrnd: From is relation, relative standard deviation can be written in e following form: (37) (38) It can us be seen at e relative standard deviations of e individual numbers of counts are added up. Since of e ree measured values will expectedly be an order of magnitude smaller an e oer two, it will affect e statistical error of e activity e most. Therefore, e time of measurement has to be chosen so at will also be sufficiently high. Wi e exception of Nd e quantities in formula (3) are calculated quantities, e uncertainties of which are systematic errors (see below). The accuracy wi which Nd is known is practically equivalent to at of e mass M of e foils (cf. formula (19)). 4 Thus, e relative standard deviation of e ermal neutron flux is: D D M M D ( bare ) D bare Cd ( Cd ). (39) 6... Systematic error By performing e measurement carefully, e statistical error expressed in relation (3) can be reduced to 1%. It is more difficult to reduce systematic errors, which can be as high as 1%. The most important errors are e following: - Self-shielding factor G is hard to calculate. This is on e one hand due to e fact at formula (1) is just approximative (especially in e case of e cadmium-covered foil), on e oer hand cross section σt in e formula is difficult to calculate. - similarly great number of assumptions are needed for e calculation of e product v d in formula (17). few examples to illustrate e problems which may arise: e neutron spectrum 3 In maematical statistics ese terms are known as standard deviation and bias. The terms above are colloquial and not accepted in maematics. 4 The masses of e bare and e cadmium-covered foil may be different. In is case e formula for calculating e error is much more complicated. For simplicity, we assume at eir masses are equivalent. HUVINETT 1 17

18 to be used in formula (16) is only known from calculations, e cross section of e ratio is only known wi a finite accuracy, etc. - Decay constant λ in formula (13) is known wi a finite accuracy only. However, e effect of is is only in e order of magnitude of one ousand (e.g. e half life of 198 u based on [6] is.6948±.1 days). Furer inaccuracies stem from e fact at e activation factor containing T assumes at at e beginning of activation e sample gets in e irradiation position, and back, instantaneously. This can be corrected if instead of equation (8), a differential equation considering e real time dependence of e neutron flux during e movement of e sample is solved. However, if e time of activation lasts 1 s or longer, is correction is below 1%. In e end, it can be seen at because of e numerous systematic errors we cannot actually speak about measuring e neutron flux. What we can achieve is e approximative determination of e neutron flux, e value of which is strongly dependent on e assumptions made during e evaluation. 7. KNOWLEDGE CHECK 1. How do we define e scalar neutron flux? What is its dimension?. Characterize e spatial distribution of ermal neutron flux in a bare and a reflected reactor! 3. What is e principle of determination of e ermal neutron flux by activation meod? 4. Why do we use cadmium cover for one of e gold samples and how do we define e cadmium ratio? 5. Why is it not necessary to use cadmium cover for e dysprosium (Dy-l alloy) wire? 6. How is e absolute value of e ermal neutron flux determined? 7. What statistical and systematic errors occur when determining e value of e ermal neutron flux? 8. REFERENCES [1] Duderstadt, J. J., Hamilton, L. J.: Nuclear Reactor nalysis, John Wiley & Sons, New York, [] Bell, G., Glastone, S.: Nuclear Reactor Theory, Litton Educational Publishing, Inc., 197. [3] Szatmáry Zoltán: Bevezetés a reaktorfizikába, kadémiai Kiadó, Budapest () [4] Csom Gyula: tomerőművek üzemtana I., reaktorfizika és technika alapjai, Műegyetemi Kiadó (1997) [5] NuDat database, National Nuclear Data Center, Brookhaven National Laboratory, [6] Jagdish K. Tuli: Nuclear Wallet Cards, National Nuclear Data Center, Brookhaven National Laboratory, Upton, New York, U.S.. () HUVINETT 1 18

19 [7] Szatmáry Zoltán: Mérések kiértékelése, egyetemi jegyzet, BME Természettudományi Kar, HUVINETT 1 19