Annals of Nuclear Energy 38 (2011) 194–202

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Annals of Nuclear Energy

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Auto-correlation and variance-to-mean measurements in a subcritical coreobeying multiple alpha-modes

Carl Berglöf a,*, Manuel Fernández-Ordóñez b, David Villamarín b, Vicente Bécares b,Enrique M. González-Romero b, Victor Bournos c, José-Luis Muñoz-Cobo d

a Department of Reactor Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Swedenb Nuclear Innovation Group, CIEMAT, Avda. Complutense 22, 28040 Madrid, Spainc Joint Institute for Power and Nuclear Research, National Academy of Sciences, Akad. Krasina str. 99, Minsk, Belarusd Dept Ingeniería Química y Nuclear, Universidad Politéchnica, Camino de Vera 14, 46022 Valencia, Spain

a r t i c l e i n f o

Article history:Received 11 January 2010Received in revised form 10 November 2010Accepted 11 November 2010Available online 13 December 2010

Keywords:Neutron noise experimentAuto-correlationRossi-aVariance-to-meanFeynman-aReactivity determination

0306-4549/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.anucene.2010.11.009

⇑ Corresponding author. Tel.: +46 55378204.E-mail address: calle@neutron.kth.se (C. Berglöf).

a b s t r a c t

Neutron noise measurements based on the Rossi-a and Feynman-a methodologies have been performedin a heterogeneous subcritical system. It is shown that the traditional single alpha-mode formulations ofthe Rossi-a and Feynman-a methods are not applicable due to the presence of higher alpha-modes. For-malisms taking into account multiple alpha-modes are applied resulting in satisfactory results. Threealpha-modes could be identified using the Rossi-a method, whereas only two could be obtained usingthe Feynman-a method. In the Feynman-a case, the possibility to obtain the fastest decaying alpha-modewas diminished due to detector dead time effects. It was found that the slowest decaying alpha-modedoes not exactly correspond to the prompt decay found in pulsed neutron source measurements, whichconfirms the results of previous studies. Strengths and weaknesses of the multiple alpha-mode Rossi-aand Feynman-a methods observed in this study are pointed out.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction neutron noise data from the strongly heterogeneous subcritical

Neutron noise methods are important tools for reactivity mea-surements of subcritical core configurations. The most appliedmethods are the Rossi-a (Orndoff, 1957) and Feynman-a (Feynmanet al., 1956), which have been used in many reactor systems overthe years: (Busch and Spriggs, 1994; Hashimoto et al., 1996; Karam,1964, 1965; Kitamura et al., 2006; Kuramoto et al., 2007; Souleet al., 2004; Wallerbos and Hoogenboom, 1998) to mention a fewstudies. An often encountered problem is the presence of more thanone alpha-mode, which has been explained on a semi-empiricallevel based on a two-region point kinetics approach (Kistner,1964; Spriggs et al., 1997; Kuramoto et al., 2007). In the case ofRossi-a, it is known that an arbitrary number of alpha-modes canbe incorporated as a series of exponentials by formulating thetime-dependent Boltzmann equation as an eigenvalue problem(Borgwaldt et al., 1967). A model based on stochastic transporttheory was recently proposed, allowing a thorough analytical treat-ment of multiple alpha-modes for both the Rossi-a and theFeynman-a method (Muñoz-Cobo et al., 2010). In this work, theapplicability of the multiple alpha-mode approach is verified on

ll rights reserved.

core YALINA-Booster at various subcriticality levels.

2. Rossi-a and Feynman-a with multiple alpha-modes

The Rossi-a and Feynman-a methods are frequently applied toprompt subcritical systems to obtain the prompt neutron decayconstant, a, defined as:

a � 1n

dndt: ð1Þ

In this equation n is the mean neutron density in the system. Thus,information concerning the subcriticality of the system is obtainedsince the prompt decay is closely related to the reactivity:

a ¼q� beff

K; ð2Þ

with symbols frequently used in the literature. The Rossi-a method,or the auto-correlation, describes the correlation of the signal withitself and is traditionally given in textbooks (Uhrig, 1970; Williams,1974) in the following form:

pðtÞdt ¼ C1eatdt þ C2dt; ð3Þ

where C with subindex indicates constants. Since the prompt decayconstant here is defined negative in a prompt subcritical system, it

C. Berglöf et al. / Annals of Nuclear Energy 38 (2011) 194–202 195

is clear from Eq. (3) that the second event probability density, p, de-creases exponentially with time approaching a constant level. Thehigher probability to obtain a second event close in time is causedby detected neutrons originating from the same fission chain. Onthe other hand, the probability to obtain an uncorrelated event isconstant in time and completely random. The constant level C2 isequal to the count rate of the detector, which will be of importancelater.

The presence of correlated counts affects the variance of the sig-nal as compared to a pure Poisson signal, which is utilized in theFeynman-a method. It has been shown that the deviation fromunity, Y, of the variance-to-mean ratio depends on the gate width,DT, used when constructing the count histogram (Feynman et al.,1956). This relation, traditionally given for a single alpha-mode,depends on the reactivity through the prompt neutron decayconstant:

YðDTÞ ¼ C 1þ 1� eaDT

aDT

� �; ð4Þ

where C is a constant.Dual alpha-modes have previously been observed (Busch and

Spriggs, 1994; Karam, 1964, 1965; Kitamura et al., 2000; Kuramotoet al., 2007) and have been explained by a two-region point kinet-ics approach (Kistner, 1964; Kuramoto et al., 2007; Spriggs et al.,1997).

A methodology for introducing multiple alpha-modes into thestochastic transport equations has been proposed (Muñoz-Coboet al., 2010). The proposed formalism by Muñoz-Cobo et al. in-cludes an arbitrary number of alpha-modes and introduces, inthe case of Feynman-a, second order cross-terms not included inprevious studies. The Rossi-a formula for N alpha-modes ofMuñoz-Cobo et al. has the same form as the one derived byBorgwaldt et al. (1967):

RðtÞ ¼ C þXN�1

k¼0

XN�1

l¼0

Ck;leal t

�ðak þ alÞ; ð5Þ

whereas the corresponding formula for the Feynman-a case takesthe following form:

YðDTÞ ¼ C0

XN�1

k¼0

XN�1

l¼0

Ck;l1

akal1þ a2

l ð1� eakDTÞ þ a2kð1� ealDTÞ

akalðak þ alÞDT

� �: ð6Þ

In these equations C with subindex are constants, not necessarilythe same for the two methods (Muñoz-Cobo et al., 2010). The Ros-si-a formula, can easily be rewritten to a single sum of exponentialswhich is its expected form. In the case of Feynman-a, cross-termsappear that are not present in models based on the two-regionpoint kinetics approach. It is observed that, for a single alpha-mode,represented by subindex 0, Eqs. (5) and (6) break down to Eqs. (3)and (4) respectively.

3. Experimental setup: the YALINA-Booster

The YALINA-Booster is a subcritical fast-thermal core (Fig. 1).The core consists of a central lead zone (booster), a polyethylenezone, a radial graphite reflector and a front and back biologicalshielding of borated polyethylene. The fast spectrum lead zoneand the thermal spectrum polyethylene zone are separated by aso called thermal neutron filter, or valve zone, consisting of onelayer of 108 pins with metallic natural uranium and one layer of116 pins with boron carbide (B4C), which are located in the outer-most two rows of the fast zone. Hence, thermal neutrons diffusingfrom the thermal zone to the fast zone are likely to be absorbedeither by the boron or the natural uranium. In this way, a couplingof mainly fast neutrons between the two zones is maintained.

The booster has two sub-zones, the inner and outer booster,with different fuel pin pitch. Both booster sub-zones can be loadedwith 36% enriched uranium oxide fuel, but in the inner subzonemetallic uranium of 90% enrichment can be used as well. The ther-mal zone is loaded with uranium oxide of 10% enrichment. Thethree core loadings described in Table 1 and Fig. 2 were includedin this study.

Three B4C-control rods (CR) can be inserted in the thermal zoneand allow changing the reactivity of the system by about 0.5 $.Hence, the sensitivity of the two reactivity measurement methodscan be tested.

There are seven axial experimental channels (EC1B-EC4B andEC5T-EC7T) in the core, two axial (EC8R, EC9R) and two radialexperimental channels (EC10R, EC11R) in the reflector. In addition,there is one neutron flux monitoring channel in each corner of thecore (MC1-4). In this work, the channels EC5T and EC6T were usedonly. In these channels two long 3He detectors (Canberra 12NH25/1) could be used with satisfactory count rates. In addition, a stronghigher alpha-mode is expected in EC5T due to the close location tothe fast booster zone and EC6T can be used as reference due to itscentral location in the thermal zone.

The neutron noise measurements were performed with a 252Cfneutron source located in the centre of the core with the neutrongenerator beam tube still in place. The source strengths of the neu-tron sources used are given in Table 2. For some of the measure-ments a weaker source, referred to as INK5, was used in additionto the strong source, referred to as INK7.

The signal from the detectors were acquired by a counter/timercard NI-6602 that wrote the arrival time of each event to disk witha precision of 12.5 ns. After the experiment any data analysis couldbe performed off-line since a complete record of the measurementwas kept. The measurement time varied from 1 h to 17 h.

A complete description of the core design is given by Kiyavitskayaet al. (2007).

4. Experimental results: auto-correlation

In the definition of the Rossi-a function, Eqs. (3) and (5), theconstant term represents the mean count rate of the detector. Con-sequently, the dead time correction can be performed in a straight-forward manner, since each bin represents a count rate (providedthe type and length of the dead time is known). After correction,the histograms are normalized to unit level to enhance compari-sons. Such normalized Rossi-a histograms are displayed in Figs.3–5 for all configurations in both linear scale and log-scale. Inthe log-scale histograms the unit level has been subtracted to visu-alize the exponential decay.

As can be seen in the log-scale histograms, the Rossi-a histo-grams do not obey a single exponential decay as described bythe traditional model. The following model, based on Eq. (5), wasused for the fitting:

f ðtÞ ¼X2

i¼0

Aieai t þ Leadt; ð7Þ

In the fitting procedure it was found that three exponentials areneeded to obtain a good fit for configuration SC0 and SC3a and twoexponentials for configuration SC6. L was in all cases found to bevery close to unity and the decay rate of the delayed neutrons,ad, was in all cases negligible. An example of fitting can be foundin Fig. 6.

The results from the fittings are summarized in Table 3 togetherwith values from pulsed neutron source (PNS) measurements(Bécares et al., submitted for publication). For the cases with highcount rate, the reduced v2 is depressed due to dead time effects.The dead time introduces a shift of the Poisson distribution andthe uncertainty is calculated as the square root of the uncorrected

Table 1Core configurations.

Zone and fuel enrichment Expecteda keff

Inner booster Outer booster Thermal zone

90% 36% 36% 10%

SC0 132 – 563 1141 0.977SC3a – 132 563 1077 0.950SC6 – 132 563 726 0.850

a Expected value based on Monte Carlo simulations (X-5 Monte Carlo Team,2005).

Fig. 1. Schematic cross-sectional view of the YALINA-Booster reactor core (SC0).

196 C. Berglöf et al. / Annals of Nuclear Energy 38 (2011) 194–202

data, thus causing a reduction in the v2 after applying the deadtime correction.

Due to the limited time the higher alpha-modes are active (lessthan 1 ms), the accuracy of the fittings is low. On the other hand,from Table 3 it can be found that the highest alpha-mode, here de-noted a2, seems to be the same for all configurations at a specificdetector position (EC5T or EC6T). In the same way, it seems likethe second highest alpha-mode, here denoted a1, is the same for

configurations SC0 and SC3a at the detector positions EC5T orEC6T. With these assumptions, the fitting can be performed collec-tively to obtain results with considerably higher accuracy. Theseresults are displayed in Table 4. In the collective fitting it was as-sumed that ad is zero. This assumption could be made withoutincreasing the value of the reduced v2.

Firstly it should be made clear that the set of higher alpha-modes (a1 and a2) obtained in the two detector positions are notnecessarily the same. It is natural that some higher alpha-modeshave higher amplitude in some parts of the core and lower, ormay even be invisible, in other parts, as discussed by Rugama etal. (2002). This applies, in particular, to source driven cores withstrong heterogeneity such as the present one. Thus, the parametersa1 and a2 should be interpreted as the first and second higher al-pha-mode observed in the particular detector position and donot necessarily correspond to the same alpha-mode in the twopositions. In this respect it is important to emphasize that thesehigher alpha-modes are easily identified since they are driven byspace effects rather than reactivity as for the fundamental mode,which can be seen in the fitting results (Table 4).

Fig. 2. Core configurations considered for the measurements.

C. Berglöf et al. / Annals of Nuclear Energy 38 (2011) 194–202 197

A first observation is that although some of the measurementswere lengthy (up to 17 h) and good statistics was achieved, theexponential fitting was cumbersome. As can be seen in the figures,the correlated part is most of its active time occupied by higheralpha-modes. The fundamental mode becomes visible only close

to the uncorrelated background. Therefore, obtaining good resultsfor the fundamental mode is difficult.

Concerning the fundamental mode, there is a noticeable devia-tion for the deep subcritical configuration, compared to the PNSmeasurements. A similar trend may be discerned among those re-sults of the less subcritical configurations having low uncertainty,however, the effect is less pronounced. As proposed by Kuramotoet al., 2007, the relation between the alpha-modes and the reactiv-ity is non-linear at deep subcriticality. They have shown that atsome level of subcriticality the lowest alpha-mode do no longerobey Eq. (2) in a heterogeneous system. Hence, obtaining the glo-bal reactivity from noise measurements with non-single alpha-modes is not a straightforward task. If not being careful, interpret-ing the obtained alpha value as a global quantity, faulty conclu-sions regarding the reactivity might be drawn. In this specificcase, the keff would have been under-estimated. Therefore, caremust be taken when analyzing Rossi-a data from heterogeneoussystems during core loading procedures and it is important toidentify possible higher alpha-modes. Otherwise, the analysis willcause a faulty perception of the core reactivity.

A consequence of the difficulty to obtain low uncertainties ofthe fundamental mode is that, despite lengthy measurements, itwas not possible to see any significant difference when havingthe control rods withdrawn or inserted. A reactivity change ofapproximately 0.5 $ was thus not possible to follow, which indi-cates a remarkably low performance of the methodology whenseveral alpha-modes are present.

5. Experimental results: variance-to-mean

The Y-value of the Feynman-a formula is calculated from exper-imental data as

YðDTÞ ¼ VðcðDTÞÞ�cðDTÞ � 1; ð8Þ

where V is the variance of the counts c when measured with a gatewidth DT. The uncertainty of Y is given by the variance of the var-iance (Cramér, 1966) according to

r2ðYÞ ¼ ð1þ YÞ2 1þ Y�cnþ 2

n� 1

� �; ð9Þ

During some of the measurements it was found that the countrate was not fluctuating around a stationary value during the mea-surement. There was a slowly decaying gamma background causedby activation of construction materials during preceding measure-ments performed at higher power. The count rate decreased in theworst case only by about 1% in 17 h, but the effect was large en-ough to cause a significant bias to the result. In those cases, thedata was divided into sub-parts with semi-stable count rate. TheY-values were evaluated for each sub-part independently accord-ing to Eq. (8). The final Y-value was then calculated as a mean valueand its uncertainty as a standard error.

When analyzing the Feynman-a data it was noticed that a cor-rect treatment of the system dead time is essential. By studying thetime interval distribution (TID) of the counts, describing the ex-pected waiting time for the subsequent count, it is possible to drawconclusions about the dead time of the electronics chain, both interms of magnitude and type. The two ideal types of dead times,extending (or paralyzing) and non-extending (or non-paralyzing),give characteristic shape of the TIDs as described in for instanceICRU Report 52 (1994). Such a TID histogram is shown in Fig. 7in dashed line for one of the electronic chains. The first observationis that there are no events occurring within 3.4 ls from the previ-ous count, thus giving us the magnitude of the dead time. Concern-ing the dead time type, conclusions must be drawn from the shape

Table 2Neutron source strengths.

Configuration Approximate source strength[neutrons per second]

Strong source (INK7) Weak source (INK5)

SC0 (1.64 ± 0.17) � 106 (1.17 ± 0.12 � 105

SC3a (1.52 ± 0.15) � 106 (1.09 ± 0.11 � 105

SC6 (1.37 ± 0.14) � 106 –

198 C. Berglöf et al. / Annals of Nuclear Energy 38 (2011) 194–202

of the TID. As can be seen, there exists some non-ideal effects dis-turbing the experimental TID. For an ideal E-type (extending) deadtime sE, the TID should be flat for sE < t < 2sE, but in this case theshape is fuzzy. However, to a good approximation concerning thecount rate the signal might be assumed to be affected by an E-typedead time of 3.4 ls. Concerning the variance this is not necessarilytrue.

0 1 2 3 4 5 6 7 80.9

1

1.1

1.2

1.3

1.4

1.5

Auto

-cor

rela

tion

SC3a: CR Out, EC5T, Cf-252(INK5)SC3a: CR Out, EC6T, Cf-252(INK5)SC3a: CR In, EC5T, Cf-252(INK7)SC3a: CR In, EC6T, Cf-252(INK7)

Time [ms]

(a) (

Fig. 4. Rossi-a histograms for confi

0 1 2 3 4 5 6 7 80.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

Time [ms]

Auto

-cor

rela

tion

SC0: CR Out, EC5T, Cf-252(INK5)SC0: CR Out, EC5T, Cf-252(INK7)SC0: CR Out, EC6T, Cf-252(INK7)

((a)

Fig. 3. Rossi-a histograms for confi

There are two fundamental approaches to include the dead timecorrection in the analysis of Feynman-a data. One way is to intro-duce the dead time in the stochastic transport theory, which hasbeen done both for NE-type (non-extending) (Kitamura et al.,2000; Yamane and Hayashi, 1995) and E-type (Degweker, 2000)dead times. This allows the dead time to be included directly inthe fitting model when processing the data. Another approach isto apply corrections to dead time affected Y-values, Yd, to obtainthe true values on which the fitting analysis is based (CastroMadero et al., 1968; Difilippo, 2002; Hazama, 2003). The latter ap-proach has an advantage in the sense that combinations of deadtimes may be applied instead of just single E or NE dead time types,as described by Hazama (2003).

A dead time correction of the variance-to-mean ratio requires asignal having well known dead time characteristics on which thecorrection can be applied. A common way to get around problemswith unknown dead time mechanisms is to introduce a NE-typedead time to the experimental data in the software post-processor

0 0.5 1 1.5 2 2.5 3 3.5 410-4

10-3

10-2

10-1

100

Time [ms]

Auto

-cor

rela

tion

- 1

SC3a: CR Out, EC5T, Cf-252(INK5)SC3a: CR Out, EC6T, Cf-252(INK5)SC3a: CR In, EC5T, Cf-252(INK7)SC3a: CR In, EC6T, Cf-252(INK7)

b)

guration SC3a (5 ls time bin).

0 0.5 1 1.5 2 2.5 3 3.5 410-4

10-3

10-2

10-1

100

Time [ms]

Auto

-cor

rela

tion

- 1

SC0: CR Out, EC5T, Cf-252(INK5)SC0: CR Out, EC5T, Cf-252(INK7)SC0: CR Out, EC6T, Cf-252(INK7)

b)

guration SC0 (5 ls time bin).

0 2 4 6 8 101

1.05

1.1

1.15

1.2

1.25

1.3

Time [ms]

Aut

o-co

rrela

tion

SC3a: CR Out, EC5T, Cf-252(INK5)

Fitting: χ2/ν = 1.04

0 0.5 1 1.5 2 2.5 3 3.5 4

10-4

10-3

10-2

10-1

Time [ms]

Aut

o-co

rrela

tion

- 1

SC3a: CR Out, EC5T, Cf-252(INK5)

Fitting: χ2 = 1.04

(a) (b)

Fig. 6. Rossi-a histogram and fitting in linear and logarithmic scale (5 ls time bin). The contribution from individual exponentials is indicated by dashed lines.

0 1 2 3 4 5 6 7 80.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07Au

to-c

orre

latio

nSC6: CR Out, EC5T, Cf-252(INK7)SC6: CR Out, EC6T, Cf-252(INK7)

0 0.5 1 1.5 2 2.5 3 3.5 410-4

10-3

10-2

10-1

100

Time [ms]Time [ms]

Auto

-cor

rela

tion

- 1

SC6: CR Out, EC5T, Cf-252(INK7)SC6: CR Out, EC6T, Cf-252(INK7)

(a) (b)

Fig. 5. Rossi-a histograms for configuration SC6 (5 ls time bin).

C. Berglöf et al. / Annals of Nuclear Energy 38 (2011) 194–202 199

(Müller, 1991). The imposed dead time must be long enough to re-move possible unknown contributions but at the same time asshort as possible not to increase too much the statistical uncer-tainty. In this case, it was found that after imposing a NE-type deadtime of 12.0 ls the TID is close to identical to a TID of an E- and aNE-type ideal dead time in series. A comparison of the experimen-tal TID after the introduction of the NE-type dead time and a MonteCarlo simulation of a Poisson signal of the same count rate isshown in Fig. 7 and indicates satisfactory agreement.

After imposing the NE-type dead time, the correction method ofHazama (2003) could be applied as follows:

YðDTÞ ¼ YdðDTÞ þ 1� k2

k3 : ð10Þ

k is the fraction of counts from a true count rate n surviving aNE-type dead time (Müller, 1973):

k ¼ 11þ nsNE

: ð11Þ

As suggested by Hazama, k is estimated by extrapolating thedead time affected Feynman-a data linearly to zero, as shown inFig. 8. The fitting interval must be chosen carefully to avoid biasdue to the strong dead time effect at low DT. Knowing the valueof Yd(0) and the relation Y(0) = 0, k is achieved from Eq. (10). Inthe paper of Hazama, before applying Eq. (10) to all data, k was cor-rected according to the true dead time effect (NE–E series deadtime). In our case the true dead time is E–NE in series having atransmission factor given by (Müller, 1973):

kE�NE ¼1

ðsNE � sEÞnþ esEn: ð12Þ

Inserting the result from the linear fitting yields

Table 3Experimental results based on one fitting per data set. For the higher alpha-modes the relative error is given in parenthesis.

EC Conf. CR Source a2 [s�1] a1 [s�1] a0 [s�1] aPNS [s�1] v2/v

SC0 Out INK5 �13000 (26%) �2740 (15%) �694 ± 22 �638 ± 3a 0.97INK7 �9400 (100%) �2200 (100%) �707 ± 145 0.02

EC5T SC3a In INK7 �8900 (28%) �2100 (52%) �1096 ± 356 �1081 ± 15 0.22Out INK5 �10800 (5%) �2300 (10%) �1102 ± 63 �1053 ± 9 1.04

SC6 Out INK7 �11400 (12%) – �2997 ± 150 �2614 ± 13 0.79

EC6T SC0 Out INK7 �7300 (100%) �2300 (100%) �700 ± 155 �638 ± 3a 0.03SC3a In INK7 �9600 (57%) �2700 (25%) �1124 ± 132 �1091 ± 3 0.15

Out INK5 �6100 (9%) �2200 (20%) �1101 ± 96 – 1.02SC6 Out INK7 �9100 (15%) – �3249 ± 140 – 0.96

a Estimation from measurements in EC1B and EC2B.

Table 4Experimental results based on collective fitting.

EC Conf. CR Source a2 [s�1] a1 [s�1] a0 [s�1] aPNS [s�1] v2/v

SC0 Out INK5 �674 ± 10 �638 ± 3a

INK7 �2442 ± 50EC5T SC3a In INK7 �11340 ± 426 �1094 ± 77 �1081 ± 15 0.48

Out INK5 �1114 ± 8 �1053 ± 9SC6 Out INK7 �3094 ± 113 �2614 ± 13

EC6T SC0 Out INK7 �746 ± 115 �638 ± 3a

SC3a In INK7 �2656 ± 71 �1156 ± 86 �1091 ± 3 0.28Out INK5 �6821 ± 251 �1163 ± 7 –

SC6 Out INK7 – �3042 ± 62 –

a Estimation from measurements in EC1B and EC2B.

0 5 10 15 20 25 30 35 401.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2x 104

Time [μs]

Cou

nts

[s-1

]

τE

τNE τE

Experimental TIDExperimental TID after imposing τNE = 12.0 μs

Monte Carlo simulation: τE = 3.4 μs, τNE = 12.0 μs

Fig. 7. Time interval distribution diagram (TID) for experimental data before and after imposing a non-extending dead time of 12 ls and a simulated case based on anextending dead time and a non-extending dead time in series.

200 C. Berglöf et al. / Annals of Nuclear Energy 38 (2011) 194–202

kE�NE ¼1

k�1 � 1� sEnþ esEn; ð13Þ

which may then be used in Eq. (10) for correction of Feynman-adata. In this particular case, when the dead time is known and theextending contribution is small compared to the non-extending,Eq. (12) may be used directly in Eq. (10) to a good approximation.In the above relations, the true count rate was obtained iterativelyfrom

m ¼ kE�NEn; ð14Þ

where m is the observed count rate.The correction method was verified by performing the same

measurement twice, but at two different count rates using thetwo 252Cf-sources with different activity. Data obtained whenusing the strong source (INK7) should coincide with data fromthe weak source (INK5) after correction. As seen in Fig. 9, the sameresults are obtained after correction although the original Feynman-a

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.43

-0.42

-0.41

-0.4

-0.39

-0.38

-0.37

-0.36

-0.35

-0.34

-0.33

Time [ms]

Y

Fig. 8. Linear extrapolation of Feynman-a data to obtain Yd (0).

Fig. 9. Feynman-a diagrams for the same detector and position but different sourceintensity, before and after dead time correction.

Fig. 10. Feynman-a data for all configurations at detector position EC6T. Fittingsare indicated by solid lines.

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

ΔT [ms]

Y(ΔT

)

SC3a, CR In, EC5TSC3a, CR In, EC6T

Fig. 11. Feynman-a data for configuration SC3a, EC5T and EC6T. Fittings areindicated by solid lines.

C. Berglöf et al. / Annals of Nuclear Energy 38 (2011) 194–202 201

data sets deviate strongly from each other. It should be stressedthat such good agreement could not be obtained without imposingthe NE-type dead time.

The introduction of an additional dead time has the drawback ofcausing data loss, but gives, in this case, better characteristics ofthe remaining data. In the case of the Rossi-a analysis the NE-typedead time was not introduced since the count rate could still be ob-tained and the data loss reduced the statistical uncertainty toomuch. The same results could be obtained with the imposed deadtime, but with larger error bars, in particular for the higher alpha-modes.

Another important aspect to keep in mind is that Eq. (10) is va-lid only for DT� s (Hazama, 2003), whereas the methods based onstochastic transport theory is less strict; DT > s (Degweker, 2000;Yamane and Hayashi, 1995). This causes some limitation when try-ing to obtain possible higher alpha-modes using the method ofHazama, as will be seen later.

Corrected Feynman-a data are displayed in Fig. 10 for all config-urations and the fittings are included as solid lines. The most obvi-ous difference between the configurations is the amplitude of the

Table 5Results from Feynman-a analysis based on Eq. (6).

EC Conf. CR Source a1 [s�1] a0 [s�1] aPNS [s�1] v2/m

SC0 Out INK5 �2300 ± 100 �647 ± 26 �638 ± 3a 0.35INK7 �2100 ± 1400 �690 ± 39 1.71

EC5T SC3a In INK7 �2290 ± 50 �1127 ± 13 �1081 ± 15 0.63Out INK5 �3500 ± 400 �1165 ± 39 �1053 ± 9 0.37

SC6 Out INK7 – �3140 ± 270 �2614 ± 13 0.42

EC6T SC0 Out INK7 �2400 ± 750 �715 ± 22 �638 ± 3a 2.42SC3a In INK7 �2700 ± 240 �1138 ± 26 �1091 ± 3 0.78

Out INK5 �3700 ± 570 �1210 ± 38 – 0.31SC6 Out INK7 �3200 ± 350 – 0.55

a Estimation from measurements in EC1B and EC2B.

Table 6Results from Feynman-a analysis based on two-region model.

EC Conf. CR Source a1 [s�1] a0 [s�1] v2/m

EC5T SC0 Out INK5 �2350 ± 1200 �647 ± 62 0.35INK7 �2100 ± 1200 �699 ± 42 1.69

SC3a In INK7 �2700 ± 100 �1177 ± 68 0.64Out INK5 �3100 ± 800 �1137 ± 55 0.36

EC6T SC0 Out INK7 �3000 ± 1500 �757 ± 33 2.45SC3a In INK7 �2250 ± 120 �1038 ± 34 0.74

Out INK5 �3800 ± 900 �1235 ± 49 0.31

202 C. Berglöf et al. / Annals of Nuclear Energy 38 (2011) 194–202

Feynman-a function which is larger closer to criticality, but alsothe faster approach to the asymptotic level at deeper subcriticalitycan be distinguished by eye. The small effect of the control rods canbe distinguished in the plot for SC3a. The amplitude differs signif-icantly between the two detector positions as can be found inFig. 11, but not the a0, which can be seen in the numerical resultsfrom the fittings (Table 5). In contrast to the Rossi-a results onlytwo exponentials could be found. The fastest decaying exponenthas partly become hidden by the imposed dead time. Moreover,the condition DT� s puts a lower limit on the fitting start point,thus reducing the possibility to obtain the fastest decaying al-pha-mode. Nevertheless, the two exponentials obtained agreeswell with the results from the Rossi-a analysis and the bias as com-pared to the PNS results is again observed.

For comparison, the same fitting analysis was performed a sec-ond time without the cross-term, thus using the two-region modelof Kistner (1964), Spriggs et al. (1997), and Kuramoto et al. (2007).No major differences could be found in the results (Table 6) and, infact, any of the two models may be used at least when applied totwo alpha-modes.

6. Conclusions

In the Rossi-a analysis of data from three subcritical configura-tions of YALINA-Booster it was found that multiple alpha-modeswere present. The higher alpha-modes made the fitting procedurecumbersome and obtaining good results of the fundamental modewas challenging. The slowest decaying alpha-mode deviatedslightly from the prompt decay obtained from pulsed neutronsource measurements. The deviation was increasing with subcriti-cality and was thus strongest in the deep subcritical configuration.Consequently, the obtained alpha values cannot be used in a singlemode model to obtain the reactivity, meaning that care must be ta-ken when applying the Rossi-a method during for instance coreloadings in heterogeneous systems since the reactivity may be un-der-estimated.

It was found that, in contrast to the Rossi-a method, the Feyn-man-a method was very sensitive to non-ideal conditions in termsof dead time and non-stationary count rate. However, the prob-lems could be handled to the cost of the highest alpha-mode. Thetwo alpha-modes obtained with the Feynman-a method agreedwell with the Rossi-a results.

Although long measurement times were spent, the accuracy ofthe fundamental mode after fitting was not high enough to allowthe observation of a 0.5 $ reactivity difference. This indicated thatwhen higher alpha-modes are present, the measurement timemust be considerably prolonged in order to obtain results withhigh accuracy as compared to situations where only one alpha-mode is dominating.

A comparison of the Feynman-a formula for multiple alpha-modesand the two-region formula was carried out for two alpha-modes.It was found that for two alpha-modes the cross-term appearing inthe multiple alpha-mode formula is of second order importance.

Acknowledgements

This work was supported by IP-EUROTRANS contract no.FI6 W-CT2005-516520, the ENRESA-CIEMAT agreement for theTransmutación Aplicada a los Residuos Radiactivos de Alta Actividad,

Svensk Kärnbränslehantering AB (SKB, the Swedish Nuclear Fueland Waste Management Co) and the Swedish Institute throughthe Visby program.

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