Annals of Nuclear Energy 38 (2011) 590–600

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Annals of Nuclear Energy

journal homepage: www.elsevier .com/locate /anucene

Feynman-a and Rossi-a formulas with spatial and modal effects

José-Luis Muñoz-Cobo a, Carl Berglöf b,⇑, Juan Peña a, David Villamarín c, Victor Bournos d

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

a r t i c l e i n f o

Article history:Received 8 January 2010Received in revised form 7 September 2010Accepted 9 September 2010Available online 26 November 2010

Keywords:Feynman-aRossi-aStochastic transport theoryReactivity determination

0306-4549/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.anucene.2010.09.027

⇑ Corresponding author. Tel.: +46 55378204.E-mail addresses: jlcobos@iqn.upv.es (J.-L. Muñoz

(C. Berglöf), david.villamarin@ciemat.es (D. Villamarín

a b s t r a c t

Feynman-a and Rossi-a formulas including multiple a-modes are derived for stochastic and continuousneutron sources. The presented formalism is further developed to achieve spatial correction factors forthe single a-mode point kinetics representations of the Feynman-a and Rossi-a formulas. As a naturalextension of the multiple a-mode formalism, delayed neutrons are included in the Feynman-a formula.The obtained formulas are validated experimentally in a strongly heterogeneous system obeying multiplea-modes, resulting in good agreement with the presented theoretical framework.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that dual subcritical modes can appear innoise analysis applications related to subcriticality determination(Busch and Spriggs, 1994; Karam, 1964, 1965; Difilippo, 1990;Kitamura et al., 2000; Kuramoto et al., 2007). In the previousstudies mentioned, the additional a-mode has been explainedbased on a two-region point kinetics model. The main goal of thiswork is to deduce the expressions of the Feynman-a and Rossi-afunctions when several a-modes are taken into account. Anothergoal of this work is to deduce the spatial correction factor forpoint kinetics noise formulas when using continuous spallationneutron sources, Cf-252 or Pu–Be neutron sources and to includedelayed neutrons in the Feynman-a formula. Measurements ofthe Feynman-a and Rossi-a functions when using Cf-252 neutronsources have been performed at the YALINA-Booster facility(Berglöf et al., submitted for publication) and data from thoseexperiments are presented to demonstrate the applicability ofthe derived formulas.

First we will deduce the formulas for the Feynman-a and Rossi-a functions using stochastic and continuous neutron sources andthen we will compare these formulas with the point kinetics for-mulas in the fundamental mode approximation in order to deduce

ll rights reserved.

-Cobo), calle@neutron.kth.se).

the point kinetics spatial correction factors. Then the formulas forthe Feynman-a function including delayed neutrons are deducedfollowed by the experimental results. Finally the main conclusionsof the paper are given.

2. Theoretical development

2.1. The Feynman-a function

For a Pu–Be neutron source the probability generating function(pgf) for the number of counts Nc(d) in the time interval d = (t � T, t)is given by Bell (1965):

GPu=BeS ðz;dÞ ¼ exp

Z t

�1dt0Z

d3rqSð~rÞS0½ðbT SG1nÞðz;~r; t0;dÞ � 1�� �

ð1Þ

bT SG1nðz;~r; t0;dÞ ¼Z 1

0dvZ

4pdX

vSðvÞ4p

G1nðz; #; t0; dÞ ð2Þ

In these equations, S0 is the source strength or number of neutronsemitted per second, vS(v) is the source spectrum, and qSð~rÞ the spa-tial distribution of the source. Finally, G1n (z,#,t0,d) is the pgf for thenumber of counts produced by one neutron injected into the systemat the phase space point # � ð~r;X

!;vÞ at time t0 and the insertion

time of the neutron source into the system is assumed to be theremote past.

The source pgf for the number of counts Nc(d) for a californiumsource was deduced by Muñoz-Cobo and Pérez (1981), Muñoz-Coboet al. (1987) and is given by the following expression:

J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600 591

GCfS ðz;dÞ ¼ exp

Z t

�1dt0Z

d3rqSð~rÞSCf0

XIS

j¼0

eSj ðbT SG1nÞjðz;~r; t0;dÞ� 1

" # !ð3Þ

where eSj is the probability to emit j neutrons in a Cf-252 fission

event, IS is the maximum number of neutrons released per Cf-252spontaneous fission event, SCf

0 is the average number of Cf-252 dis-integrations per unit time and vS(v) is the source spectrum that inthis case is the californium one.

For the case of a continuous Poissonian source of protons withan average number of Sp(t) protons per second the pgf for the num-ber of protons injected into the system during the interval (t0, t),assuming the proton source was turned on at time t0 = �1, is givenby (Pázsit and Pál, 2008):

GSðzÞ ¼ expZ t

�1dt0Spðt0Þ½z� 1�

� �ð4Þ

On the other hand, the relationship between the pgf for thenumber of counts produced by one proton inserted into the spall-ation target at time t0 and denoted by G1p(z, t0,d) and the pgf for thenumber of counts G1n(z,#, t0,d) produced by one neutron injectedinto the system at the phase space point # � ð~r;X

!;vÞ and time t0

is given, according to Ballester and Muñoz-Cobo (2005), by:

G1pðz; t0;dÞ ¼Z

d3rqSð~rÞXIsp

j¼0

espj ðbT spG1nÞjðz;~r; t0;dÞ ð5Þ

where we have used the spallation operator defined by

bT sp� ¼Z

dvZ

d X!

fspðv;X!

tÞ� ð6Þ

that takes into account the velocity and angular distribution of thesource neutrons. In the previous equation, esp

j the probability tohave j neutrons emitted in one spallation event, and Isp is the max-imum number of neutrons produced per single spallation event. Wemust remark that for the case of a californium neutron source thespectral and angular distributions are considered as independentand the neutron emission is considered isotropic. When correla-tions are considered (Difilippo, 1991) the formulas must be modi-fied and these modifications are performed at the end of Section2.5.2. These assumptions cannot be made for a spallation neutronsource, i.e. the spectral and angular distributions are not indepen-dent and the emission of neutrons is not isotropic. For this reasonwe have included the function fspðv;X

!Þ that is defined in such a

way that fspðv;X!ÞdvdX gives the probability of a spallation neutron

to be emitted with velocity within the velocity interval (v,v + dv)and with direction inside a dX around X

!.

In addition, in a spallation interaction there are two steps. Thefirst step is a cascade process during which direct interactions takeplace and some nucleons are emitted as a result of direct collisions.The second step is the so called evaporative process. During thisprocess the highly excited nucleus emits particles isotropically dis-tributed in the center of mass system and, as a consequence of thenucleus motion after the collision; this emission is not isotropic inthe laboratory system. The assumption we have made in this paperis that the velocity and angle distribution function for the emissionof n neutrons can be factorized as fnðv1;X1; � � �vn;XnÞ �f ðv1;X

!

1Þ . . . f ðvn;X

!

nÞ. It is necessary to make this assumption to ob-

tain Eq. (5) of this paper.Next we are going to use the following theorem from the statis-

tic theory of pgfs (Medhi, 1982; Lando, 2003): Given a random var-iable Q = X1 + X2 +. . .+XY, being X = Xi"i and Y mutually independentdiscrete random variables, with probability generating functionsGX (z) and GY(z) associated to X and Y respectively, then the pgffor Q is given by GQ(z) = GY(GX(z)). The application of this theorem

considering the fact that the spallation continuous source behavesas a Poissonian source yields:

GspS ðz;dÞ ¼ exp

Z t

�1dt0Spðt0Þ½G1pðz; t0; dÞ � 1�

� �ð7Þ

where G1p(z, t0,d) is given by Eq. (5). An equation similar in structureto Eq. (7) was deduced for the number of counts produced by a Cf-252 fission source, using the probability balance method by Muñoz-Cobo et al. (1987). In this case, instead of the number of protons,they considered the number of spontaneous fission events of thecalifornium source, and instead of the number of neutrons emittedin the spallation events they considered the number of neutronsemitted in the spontaneous fission events. Assuming the normaliza-tion of the distribution function for spallation neutrons we maywrite:Z

d3rqSð~rÞ ¼ 1 ð8Þ

where qSð~rÞd3r gives the probability for one spallation neutron to be

born at a d3r located at space position~r.On account of Eqs. (4)–(7), the pgf for the number of counts

upon turning on the proton source in the system is given by:

GspS ðz;dÞ¼exp

Z t

�1dt0Spðt0Þ

Zd3rqSð~rÞ

XIsp

j¼0

espj ðbT spG1nÞjðz;~r;t0;dÞ�1

" # !ð9Þ

Eq. (9) gives the pgf for the total number of counts during thecounting interval d = (t � T, t), upon insertion of a source of high en-ergy protons into the system at the remote past, in terms of the oneneutron pgf for the number of counts.

We will perform all the deductions with Eq. (9), because thisequation reduces to Eq. (1) when we set esp

1 ¼ 1; espj ¼ 08j–1, the

spallation operator is set equal to the neutron source operator

bT S� ¼Z

dvZ

4pdX

vSðvÞ4p

� ð10Þ

and the proton strength Sp is set equal to the neutron sourcestrength S0.

To calculate the average number of counts during the countinginterval d upon introduction of the source in the remote past thatwe denote by NcðdÞ we apply the first moment operator, i.e.

½@=@z�z¼1

to the source pgf for the number of counts, given by Eq. (9). This cal-culation yields the following result:

NcðdÞ ¼ �njSsp1

� �ðdÞ

¼Z t

�1dt0Z

d3rZ

dvZ

4pd X!

Ssp1 ð#; t0Þ�nð#; t0; dÞ ð11Þ

where Ssp1 ð#; t0Þ is the source strength per phase space unit for spall-

ation neutrons at time t0 given by:

Ssp1 ð#; t0Þ ¼ Spðt0Þ�mspqSð~rÞfspðv;X

!Þ ð12Þ

and where �nð#; t0;dÞ is the first factorial moment of the number ofcounts per single neutron injected into the system at the phasespace point # � ð~r;v;X

!Þ and time t0, which according to Barlett

(1955) has been defined as:

�nð#; t0;dÞ ¼ @

@zG1nðz; #; t0;dÞ

� �z¼1

ð13Þ

Finally, the symbol ha—bi used in Eq. (11) denotes the inner productover phase space # and time of the functions a and b.

We note that if instead of applying the first moment operator toEq. (9), we apply this operator to the pgf for the number of counts

592 J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600

produced by a Pu–Be neutron source that is given by Eq. (1), we ob-tain instead of Eq. (11) the following result:

NcðdÞ ¼ �njSPu=Be1

D EðdÞ ð14Þ

where SPu=Be1 ð#; t0Þ denotes the neutron emission rate from the Pu–Be

source per phase space unit given by:

SPu=Be1 ð#; t0Þ ¼ S0qSð~rÞ

vSðvÞ4p

ð15Þ

Finally, applying the first moment operator to Eq. (3) we obtainthe average number of counts during the counting interval uponintroduction of the californium neutron source in the remote past:

NcðdÞ ¼ �njSCf11

D EðdÞ ð16Þ

where SCf1 ð#; t0tÞ denotes the average neutron emission rate from the

Cf-252 source per phase space unit that is given by:

SCf1 ð#; t0Þ ¼ SCf

0�mSqSð~rÞ

vSðvÞ4p

ð17Þ

where �mS is the average number of neutrons released per Cf-252spontaneous fission event.

According to Muñoz-Cobo et al. (1987) the first factorial mo-ment �nð#; t0; dÞ, per single neutron injected into the system obeysthe following adjoint transport integro-differential equation:

� 1v

@

@t0� Lþ

� ��nð#; t0;dÞ ¼ Sþ1 ð~r;v ; t0; dÞ ð18Þ

where the adjoint transport operator L+ is that defined by Bell andGlasstone (1970) and Muñoz-Cobo et al. (1987) and the adjointsource term Sþ1 ð#; t0Þ depends of the macroscopic detection crosssection and the efficiency for detection as follows (Ballester andMuñoz-Cobo, 2005):

Sþ1 ð~r;v ; t0; dÞ ¼ Sþ1 ð~r;vÞ � Sþ1 ðt0;dÞ

¼ gcRDc þ gsR

Ds þ gf R

Df

h i� ½Hðt0 � ðt � TÞÞ

� Hðt0 � tÞ� ð19Þ

In this equation RDx , for x = c, s, f, denotes the capture, scattering and

fission macroscopic cross sections, respectively, inside the detectorvolume, H(t0) denotes the Heaviside function, and gx for x = c, s, f de-notes the detector efficiency for capture, scattering and fissionevents. We note that Sþ1 ð~r;v; t0;dÞ will be non-zero only inside thesensible detector volume and the counting time interval, i.e.~r 2 VD and t0 2 (t � T, t).

In order to obtain the second factorial moment, lsp2 , of the num-

ber of counts, we should apply the operator

½@2=@z2�z¼1

to the source pgf, Eq. (9). This calculation yields the following result:

lsp2 ¼ ðNcðdÞÞ2 þ nðn� 1ÞjSsp

1

D Eþ mspðmsp � 1Þ

Z t

�1dt0Spðt0Þ

�Z

d3rqspð~rÞðbT sp�nÞ2 ð20Þ

where nðn� 1Þð#; t0; dÞ denotes the second factorial moment of thenumber of detected counts per single neutron injected in the sys-tem at (#, t0):

nðn� 1Þð#; t0;dÞ ¼ @2

@z2 G1nðz; #; t0; dÞ" #

z¼1

ð21Þ

bT sp�nð~r; t0; dÞ is a spectral and directional weighted importance forthe spallation neutrons born at point ~r and time t0 defined byexpression (6). Finally,

mspðmsp � 1Þ

is the second factorial moment of the number of neutrons releasedper spallation event.

The Feynman-a function is defined as the excess of relative var-iance over unity, and is given on account of Eq. (20) by:

YspðdÞ ¼ VarðNÞNcðdÞ

� 1 ¼ lsp2 � ðNcðdÞÞ2

NcðdÞð22Þ

Substitution of the second factorial moment of the number ofcounts, lsp

2 , by Eq. (20) into Eq. (22) yields the following result:

YspðdÞ ¼mspðmsp � 1Þ

R t�1 dt0Spðt0Þ

Rd3rqSð~rÞðbT sp�nÞ2 þ nðn� 1ÞjSsp

1

D ENcðdÞ

ð23Þ

The calculation of the second term in the numerator of Eq. (23), canbe expressed in terms of the first factorial moment, it is known(Muñoz-Cobo et al., 1987, 2000), that the second factorial momentfor the number of counts induced by one neutron injected at (#, t0)obeys the following adjoint transport equation:

� 1v

@

@t0� Lþ

� �nðn� 1Þð#; t0;dÞ ¼ Sþ2 ð~r; v; t0;dÞ ð24Þ

where the source term Sþ2 ð~r;v; t0;dÞ for the second factorial momentequation of the number of counts produced by one neutron, de-pends on the first factorial moment and is given by:

Sþ2 ð~r; v; t0;dÞ ¼ mðm� 1ÞRf ð~r; vÞðbT f �nÞ2ð~r; t0;dÞ ð25Þ

With the following spectral and directional weighted operator forthe fission neutrons:

bT f � ¼Z

dv 0Z

dX!0 vf ðv 0Þ

4p� ð26Þ

being vf(v) the fission neutron spectrum.To obtain a calculable expression for the second term of Eq. (23)

we use the commutation relation of the transport operator as fol-lows (Bell and Glasstone, 1970):

nðn� 1ÞjSsp1

D E¼ nðn� 1Þj 1

v@

@t0� L

� �/

� ¼ � 1

v@

@t0� Lþ

� �nðn� 1Þj/

� ¼ Sþ2 j/� �

ð27Þ

where we have taken into account that the average neutron fluxobeys the well known transport equation with the source of spalla-tion neutrons as source term:

1v

@

@t0� L

� �/ð#; t0Þ ¼ Ssp

1 ð#; t0Þ ð28Þ

where the transport operator L is given by the expression:

L� ¼ �X!: � ~r � �Rtð~r; vÞ � þ

ZdX0

Zdv 0Rsfsð~r; X

!; E

! X0; E0Þ � þvf ðvÞ

4p

ZdX0

Zdv 0�mRf ð~r0; v 0Þ� ð29Þ

where all the symbols are standard ones in neutron transport the-ory. The initial and boundary conditions specified for �nð#; t;dÞ aregiven at the beginning of Section 2.3.1. It is also assumed that theneutron flux vanished at the remote past and that also we havethe condition of non-reentrant flux at the convex boundary, i.e. /(#, t,d) = 0, for X

!�~n < 0 on~r 2 @V .

Therefore the expression for the Feynman-a function will be gi-ven on account of Eqs. (23)–(27) by:

J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600 593

YspðdÞ ¼mspðmsp � 1Þ

R t�1 dt0Spðt0Þ

Rd3rqSð~rÞðbT sp�nÞ2 þ C

NcðdÞð30Þ

where

C ¼ mðm� 1ÞZ t

�1dt0Z

d#Rf ð~r; vÞ/ð#; t0ÞðbT f �nÞ2 ð31Þ

stands for the contribution of the correlated fission neutrons.In the case of a Pu–Be neutron source, the second factorial mo-

ment of the number of neutrons released per disintegration eventis zero so the Feynman-a function simplifies to:

YPu=BeðdÞ ¼ CNcðdÞ

ð32Þ

Finally the expression for the Feynman-a function for the Cf-252 neutron source is

YCf ðdÞ ¼mSðmS � 1Þ

R t�1 dt0SCf

0

Rd3rqSð~rÞðbT S�nÞ2 þ C

NcðdÞð33Þ

Now the weighting in bT S�n in Eq. (33) is performed with the Cf-252 spontaneous fission neutron spectrum and NcðdÞ denotes theaverage number of counts produced during the counting intervalby the source insertion in the remote past.

2.2. The cross second factorial moment for the number of counts

In this case we consider one neutron detector with the twocounting intervals d1 = (t � s � T1, t � s) and d2 = (t � T2, t), dis-played in Fig. 1 Then we can define the probability to have Nc1

counts during the interval d1, and Nc2counts in the interval d2 uponintroduction of the source in the remote past.

Following the same steps as in the previous section and on ac-count of the results of Muñoz-Cobo et al. (1987), Muñoz-Cobo et al.(2008) it is found that the source pgf for the number of counts inthe counting intervals d1 and d2 when a Poissonian proton sourceis present is given by:

GspS ðz1; z2; d1; d2Þ ¼ exp

Z t

�1dt0Spðt0Þ½G1pðz1; z2; t0;d1;d2Þ � 1�

� �ð34Þ

With the pgf for the number of counts produced by one proton gi-ven by

G1pðz1; z2; t0;d1;d2Þ ¼Z

d3rqspð~rÞXIsp

j¼0

espj ðbT spG1nÞjðz1; z2;~r; t0;d1;d2Þ

ð35Þ

where G1n(z1,z2,#, t0,d1,d2) is the pgf for the number of counts pro-duced in the intervals d1 and d2 by one neutron injected into thesystem at the phase space point # and at time t0.

Proceeding as in the previous section for the Pu–Be neutronsource, it is found that the source pgf for the number of countsin the counting intervals displayed in Fig. 1 is given by

GPu=BeS ðz1; z2;d1;d2Þ

¼ expZ t

�1dt0Z

d3rqSð~rÞS0½ðbT SG1nÞðz1; z2;~r; t0;d1;d2Þ � 1�� �

ð36Þ

t - τ - T1 t - T2t - τ t t'

Fig. 1. Case of one detector with two counting intervals d1 = (t � s � T1, t � s) andd2 = (t � T2, t).

where the operator bT S�, has been defined by Eq. (10). An equationsimilar in structure to Eq. (34) holds for the pgf of the number ofcounts of a Cf-252 neutron source.

The cross second factorial moment Nc1Nc2ðd1; d2Þ, for the num-ber of detected counts upon introduction of the source in the re-mote past, is obtained by applying the operator

@2

@z1@z2

" #z1¼z2¼1

;

to the source pgf given by Eq. (34). After some calculus the follow-ing expression is obtained for this cross second factorial moment:

Nc1Nc2ðd1;d2Þ

¼Z t

�1dt0Z

d#Ssp1 ð#;t0Þn1n2ð#;t0;d1;d2Þþmspðmsp�1Þ

Z t

�1dt0Z

d3rSpðt0Þ

qSð~rÞðbT spn*

1ÞðbT sp�n2ÞþNc1ðd1ÞNc2ðd2Þ ð37Þ

where the source strength Ssp1 ð#; t0Þ per phase space unit is given by

Eq. (12) and the spectral and directional weighted importances inthe counting interval d1 and d2 for the spallation neutrons havebeen defined previously in Eq. (6). Notice that the arguments ofthese functions are T

_

sp�nið~r; t0;diÞ for i = 1, 2.The cross second factorial moment for the number of counts

produced by one single neutron obeys the following adjoint trans-port equation (Muñoz-Cobo et al., 1987; Ballester et al., 2006):

� @

@t0� Lþ

� �n1n2ð#; t0; d1; d2Þ ¼ Sþ2 ð~r;v ; t0; d1; d2Þ ð38Þ

with the following adjoint source for the cross second moment:

Sþ2 ð~r; v; t0;d1;d2Þ ¼ mðm� 1ÞRf ð~r;vÞðbT f �n1ÞðbT f �n2Þ ð39Þ

We notice that ðbT f �n1Þ r*; t0;d1Þ and ðbT f �n2Þðr

*; t0; d2Þ are functions of

the position and time of injection of the neutron, and are definedby Eq. (26).

The first term on the right-hand side of the cross second facto-rial moment can be expressed in terms of first moments on ac-count of the commutation relation of the direct and adjointtransport operators:

n1n2jSsp1

� �¼ n1n2j

@

@t� L

� �/

� ¼ � @

@t� Lþ

� �n1n2j/

� ¼ Sþ2 j/� �

ð40Þ

Therefore, Eq. (37) for the cross second factorial moment can bewritten on account of Eq. (40) and after some elementary calcula-tions as follows:

Nc1Nc2ðd1;d2Þ ¼ C12ðd1;d2Þ þ mspðmsp � 1ÞZ t

�1dt0

�Z

d3rSpðt0ÞqSð~rÞðbT sp�n1ÞðbT sp�n2Þ

þ U12ðd1;d2Þ ð41Þ

where the term

C12ðd1;d2Þ ¼ mðm� 1ÞZ t

�1dt0Z

d3rZ

dvRf ð~r;vÞ/ð~r;v ; t0Þ

� ðbT f �n1ÞðbT f �n2Þ ð42Þ

stands for the contribution from the correlated fission neutrons intime having a common ancestor, while

U12ðd1; d2Þ ¼ Nc1ðd1ÞNc2ðd2Þ ð43Þ

accounts for the uncorrelated counts coming from the last term ofthe cross covariance.

594 J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600

For the case of a Pu–Be neutron source, performing the samesteps, the following result is obtained:

Nc1Nc2ðd1;d2Þ ¼ C12ðd1;d2Þ þ U12ðd1;d2Þ ð44Þ

Finally, for a Cf-252 neutron source, the cross second factorial mo-ment for the number of counts is given by

Nc1Nc2ðd1;d2Þ ¼ mðm� 1ÞZ t

�1dt0Z

d3rZ

dvRf ð~r;vÞ/ð~r;v ; t0Þ

� ðbT f �n1ÞðbT f �n2Þ þ mSðmS � 1ÞZ t

�1dt0

�Z

d3rSCf0 qSð~rÞðbT Sn

*

1ÞðbT S�n2Þ

þ Nc1ðd1ÞNc2ðd2Þ ð45Þ

where the sub-index S, appearing in Eq. (45), denotes californiumsource properties.

2.3. Expansion in a-modes

In this section no specific mention of the source type will bementioned, only at the end of Section 2.3.2 some comments willbe made.

2.3.1. Expression for the average number of countsTo solve Eq. (18) we must take into account that �nð#; t; dÞ must

fulfill the following condition on the convex boundary @V:

�nð#; t0;dÞ ¼ 0 for X!�~n > 0 on~r 2 @V ð46Þ

Also the average number of counts is obviously zero if the neu-tron is injected at the end of the counting interval (t � T, t):

�nð#; t0 ¼ t;dÞ ¼ 0 ð47Þ

Expanding �nð#; t;dÞ in a-modes we may write:

�nð#; t0;dÞ ¼ mjuþj ð#Þf

þj ðt0;dÞ ð48Þ

where the jth adjoint eigenfunction is a solution of the equation

Lþuþj ð#Þ ¼aj

v uþj ð#Þ ð49Þ

with aj being the jth a eigenvalue and uþj ð#Þ the jth a eigenfunctionof the adjoint transport operator L+.

In the following we recall that the direct eigenfunction uj (#) isa solution of the following eigenvalue equation:

Lujð#Þ ¼aj

v ujð#Þ ð50Þ

We note that the jth a eigenvalue of the direct and adjoint transportoperators is the same. The direct and adjoint a eigenfunction satisfythe following bi-orthogonality condition (Ronen, 1986):

1v uþj ;uk

� �¼ dj;k

1v uþj ;uj

� �ð51Þ

where (a,b) denotes the inner product over the phase space vari-ables # � ð~r;v ;X

!Þ. The next step is to substitute the expansion in

a-modes given by Eq. (48), into the adjoint Eq. (18). This substitu-tion yields:

� 1v

@

@t0� Lþ

� �Xl

uþl ð#Þfþl ðt0;dÞ ¼ Sþ1 ð~r; vÞS

þ1 ðt0; dÞ ð52Þ

Now, we operate in Eq. (52), on account of the a eigenvalue equa-tion yieldingX

l

� 1v

d1lðt0; dÞdt0

� al

v flðt0; dÞ� �

uþl ð#Þ ¼ Sþ1 ð~r;vÞSþ1 ðt0; dÞ ð53Þ

Then, we perform the inner product of Eq. (53) with uj (#) and wetake into account the bi-orthogonality condition, Eq. (51). Aftersome calculus it is found that 1þj ðt

0Þ satisfy the following differentialequation:

�d1þj ðt

0; dÞdt0

� aj1þj ðt0;dÞ ¼Sþ1 ;uj

�1v uþj ;uj

� Sþ1 ðt0;dÞ ð54Þ

Solving this equation on account of the boundary condition, Eq.(47), yields the following result:

1þj ðt0;dÞ �

Sþ1 ;ujð Þ1vuþj;uj

� expðajðt�t0 ÞÞð�ajÞ

½expð�ajTÞ � 1� for t0 < t � T

Sþ1 ;ujð Þ1vuþj;uj

� 1ð�ajÞ½1� expðajðt � t0ÞÞ� for t � T < t0 < t

0 for t0 P t

8>>>>>>><>>>>>>>:ð55Þ

Therefore, the average number of counts NcðdÞ produced uponintroduction of the neutron source will be given on account ofEq. (11) for the case of a time independent source and expandingin a-modes:

NcðdÞ ¼Z t

�1dt0Z

d#S1ð#Þ�nð#; t0;dÞ

¼X

j

uþj ; S1

�Z t

�1dt01þj ðt0;dÞ

¼X

j

uþj ; S1

�Sþ1 ;uj

�1v uþj ;uj

� Tð�ajÞ

ð56Þ

where we have made use of the expansion of Eq. (48).Next, we can expand the neutron flux in the Boltzmann equa-

tion in a-modes. This calculation yields, for a subcritical systemwith a time independent neutron source introduced in the remotepast, the following result:

/ð#Þ ¼X

j

uþj ; S1

�1v uþj ;uj

� 1ð�ajÞ

ujð#Þ ð57Þ

Therefore, from Eqs. (56) and (57), it is deduced on account of thecommutation relation of the direct and adjoint operators that theaverage number of counts is given by

NcðdÞ ¼ Sþ1 ;/�

T ¼ ðgRD;/ÞT ð58Þ

This is the well known expression for the average number of countsin terms of the macroscopic detection cross section RD, the detectorefficiency g and the counting time T.

2.3.2. Expression for the Feynman-a function in terms of a-modesExpanding �nð#; t0; dÞ in a-modes in Eqs. (30), (32) and (33) for

the Feynman-a function and assuming that the system is in a sub-critical state so the neutron flux is time independent, the followingexpression for the Feynman-a function is obtained after a lengthycalculation:

Yðdðt; TÞÞ ¼ lSys2

NcðdÞþ lSource

2

NcðdÞð59Þ

The first term lSys2 is the contribution to the second factorial mo-

ment of the number of counts produced by the fluctuations in thenumber of neutrons released in the fission events that take placein the system. The second term lSource

2 is the contribution to thesecond factorial moment in the number of counts coming fromthe fluctuations in the number of neutron released per source

J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600 595

disintegration event or source spallation event. Obviously this lastterm is zero for a Pu–Be neutron source.

The contribution lSys2 to the second factorial moment is given by

the following expression on account of Eq. (48):

lsys2 ðTÞ¼mðm�1ÞT

Xk;l

Zd#Rf ð~r;vÞ/ð#Þ bT f uþk

� bT f uþl �

�Z t

�1dt01þk ðt

0;dÞ1þl ðt0;dÞ¼mðm�1ÞT

Xk;l

Ck;l1

akal

� 1þa2l ð1�expðakTÞÞþa2

kð1�expðalTÞÞakalðakþalÞT

� �ð60Þ

where the fission rate dependent coefficients Ck,l are given by

Ck;l ¼ Rf /;uþk ;uþl

� Sþ1 ;uk

� Sþ1 ;ul

� 1v uþk ;uk

� 1v uþl ;ul

� ð61Þ

where we have defined

Rf /;uþk ;uþl

� ¼Z

d#Rf ð~r; vÞ/ð#Þ bT f uþk �

ð~rÞ bT f uþl �

ð~rÞ ð62Þ

The contribution of the fluctuations in the number of neutrons re-leased per spontaneous fission event or per spallation event to thesecond factorial moment is given by:

lSource2 ¼DS�mST

Xk;l

CSk;l

1akal

1þa2l ð1�expðakTÞÞþa2

kð1�expðalTÞÞakalðakþalÞT

� �ð63Þ

where DS is the Diven factor of the neutron source

DS ¼ mSðmS � 1Þ=�m2S

and the source strength dependent coefficients CSk;l are given by

CSk;l ¼ S1;uþk ;u

þl

� Sþ1 ;uk

� Sþ1 ;ul

� 1v uþk ;uk

� 1v uþl ;ul

� ð64Þ

where, for a spallation source, we have defined

S1;uþk ;uþl

� ¼Z

d#S1ð#Þ bT spuþk �

ð~rÞ bT spuþl �

ð~rÞ ð65Þ

We notice that for the particular case of a Cf-252 spontaneous fis-sion source the operator bT sp must be substituted by bT S, that havebeen defined previously.

2.4. Expression of the Rossi-a function and expansion in a-modes

The first step to calculate the Rossi-a function is to obtain thejoint probability of detection. We remind that the joint probabilityP(2)(t1, t2)dt1dt2 of detection at two different times t1 and t2 givesthe probability that one neutron is detected in dt1 around t1 andanother neutron is detected in dt2around t2.

The expression for the joint probability of detection is com-puted performing the following limit in the cross second factorialmoment for the number of detected counts in d1 and d2:

Pð2Þðt1; t2Þ ¼ limT1T2!0

1T1T2

Nc1Nc2ðd1; d2Þ ð66Þ

To obtain this limit, the first step is to perform an expansion in a-modes in Eqs. (41), (44) and (45) for the cross covariance functionfor different types of neutron sources. The first term C12 (d1,d2) ofthe right-hand side of these equations gives the contribution fromthe correlated fission neutrons in time having a common ancestor,and after some elementary calculus one gets:

Pð2Þcorr�f ðsÞ ¼ limT1T2!0

Cðd1; d2ÞT1T2

¼ mðm� 1ÞX

k;l

Ck;lexpðalsÞ½�ðak þ alÞ�

ð67Þ

where the coefficient Ck,l is given by Eq. (61).The next step is to compute the contribution to the joint prob-

ability of detection coming from the second term of Eqs. (41) and(45). In the case of the Pu–Be neutron source, we notice that thiscontribution is zero. This contribution comes from the correlatedneutrons born in the same spallation event or the same Cf-252spontaneous fission event and we denote it by Pð2Þcorr�SðsÞ. First weexpand in a-modes �n1ð#0; t0; d1Þ and �n2ð#00; t0; d2Þ as in previous sec-tions, then we compute the limit when the counting intervals tendtowards zero. These calculations yield:

Pð2Þcorr�StðsÞ ¼ DS�mS

Xk;l

CSk;l

expðalsÞ½�ðak þ alÞ�

ð68Þ

where CSk;l is given by Eq. (64) and depends on the neutron source

strength.The third contribution comes from the uncorrelated counts

coming from the last term of the cross covariance and is given by

Pð2Þuncorr ¼X

j

uþj ; S1

�Sþ1 ;uj

�1v uþj ;uj

� 1ð�ajÞ

24 352

¼ Sþ1 ;/� 2 ð69Þ

To obtain the Rossi-a function we have computed the conditionalprobability of having one count in dt2 around t2 = t and one countin dt1 around t1 = t � s when the counting gates tend toward zero.We have deduced that the joint probability of detection at two dif-ferent times depends only on the time delays. Therefore, the Rossi-afunction is given by (Muñoz-Cobo et al., 2008):

RðsÞ ¼ Pð2ÞðsÞPð1Þðt1Þ

ð70Þ

Eq. (56) for the average number of counts, shows that the probabil-ity per unit time to detect one neutron at an arbitrary time is givenby

Pð1Þ ¼X

j

uþj ; S1

�Sþ1 ;uj

�1v uþj ;uj

� 1ð�ajÞ

¼ Sþ1 ;/�

ð71Þ

Therefore, the expression for the Rossi-a function is:

RðsÞ ¼Pð2Þuncorr þ Pð2Þcorr�f ðsÞ þ Pð2Þcorr�SðsÞ

Sþ1 ;/� ð72Þ

On account of the expression for the contributions to the joint prob-ability of detection we may write the Rossi-a function in the form:

RðsÞ ¼ Sþ1 ;/�

þ

Pk;l

mðm� 1ÞCk;l þ DS�mSCSk;l

�expðalsÞ½�ðakþalÞ�

Sþ1 ;/� ð73Þ

Eq. (73) is the general expression for the Rossi-a function for spall-ation continuous proton sources, Cf-252 spontaneous fissionsources or Pu–Be sources. We notice that in the case of a Pu–Be neu-tron source, the Diven factor DS is zero.

2.5. The single mode approximation: Comparison with point kineticsand correction factors for spatial effects

2.5.1. The single mode approximation for the Feynman-a functionThe single mode approximation for the Feynman-a function is

obtained from Eq. (59) considering only the fundamental modethat we denote by the subscript 0. In this case Eq. (59) reduces to:

596 J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600

YðdÞ ¼lSys

2;0

ðNcðdÞÞ0þ

lSource2;0

ðNcðdÞÞ0ð74Þ

The contribution to the second factorial moment of the number ofcounts from the fluctuations in the number of emitted neutronsper fission is obtained in the fundamental mode approximationfrom Eq. (60), eliminating all the mode contributions except thefundamental one:

lSys2;0 ¼ mðm� 1ÞTC0;0

1a2

0

1þ ð1� expða0TÞÞa0T

� �ð75Þ

where the coefficient C0,0 is given by:

C0;0 ¼ Rf /0;uþ0 ;uþ0

� Sþ1 ;u0

� 2

1v uþ0 ;u0

� 2 ð76Þ

being /0 the neutron flux in the single mode approximation that,according to Eq. (57), is related to the fundamental a-mode by

/0ð#Þ ¼uþ0 ; S1� 1v uþ0 ;u0

� 1ð�a0Þ

u0ð#Þ ð77Þ

The contribution to the second factorial moment for the number ofcounts comes from the fluctuations in the number of emitted neu-trons per Cf-252 source disintegration or per fission event, and it isobtained in the fundamental mode approximation from Eq. (63) byeliminating all the mode contributions except the fundamental one:

lSource2 ¼ DS�mSTCS

0;01a2

0

1þ ð1� expða0TÞÞa0T

� �ð78Þ

with the source dependent coefficient CS0;0 given by

CS0;0 ¼ S1;uþ0 ;u

þ0

� Sþ1 ;u0

� 2

1v uþ0 ;u0

� 2 ð79Þ

The average number of counts in the interval (t � T, t) in the sin-gle mode approximation is obtained from Eqs. (56) and (77) and isgiven by

Nc;0ðdÞ ¼ Sþ1 ;/0

� T ¼

Sþ1 ;u0

� uþ0 ; S1�

1v uþ0 ;u0

� Tð�a0Þ

ð80Þ

Therefore, the final expression for the Feynman-a function is ob-tained from Eq. (74) on account of Eqs. (75)–(80). After some ele-mentary substitutions the following result is obtained:

YðTÞ ¼mðm� 1ÞC0;0 þ DS�mSCS

0;0

�Sþ1 ;/0

� 1a2

0

1þ ð1� expða0TÞÞa0T

� �ð81Þ

2.5.2. Spatial correction factor for the point kinetics Feynman-a andRossi-a formulas

Let us now deduce the spatial correction factors that correct thepoint kinetics expressions for the Feynman-a function in the singlemode approximation. First, we recall that the counting rate can beexpressed as Sþ1 ;/0

� ¼ ðgRD;/0Þ ¼ eF, where e is the detection

efficiency in counts per reactor fission and F is the fission rate.Therefore, expressing /0 in terms of the fundamental a-mode, wecan write:

Sþ1 ;/0

� ¼ ðgRD;/0Þ ¼ eF ¼

uþ0 ; S1�

Sþ1 ;u0

� 1v uþ0 ;u0

� 1ð�a0Þ

ð82Þ

Next we recall the definition of the neutron mean generationtime that in the single mode approximation is given by:

K ¼uþ0 1

v ;/0

� �m uþ0 ;Rf /0

� ¼ uþ0 1v ;u0

� uþ0 ; �mRf u0

� ð83Þ

Therefore, on account of Eq. (82) we can write, noticing that /0 isproportional to u0 and that one can replace u0 by /0 in Eq. (82)to obtain:

uþ0 ; S1�

¼ ð�a0ÞeF1v uþ0 ;/0

� Sþ1 ;/0

� ¼ ð�a0Þ1v uþ0 ;/0

� �ð84Þ

Finally, on account of the definition of K given by Eq. (83), we canwrite Eq. (84) in the form:

uþ0 ; S1�

¼ ð�a0Þ�mK uþ0 ;Rf /0

� ð85Þ

Therefore, the contribution to the Feynman-a functions from theneutrons correlated in time with common fission ancestors is givenafter some simplifications taking into account Eqs. (82)–(85) by thefollowing equation:

YSysðTÞ ¼ ðmðm� 1ÞC0;0ÞSþ1 ;/0

� 1a2

0

1þ ð1� expða0TÞÞa0T

� �¼ e

D

K2ð�a0Þ2Xf 1þ ð1� expða0TÞÞ

a0T

� �ð86Þ

where D ¼ mðm� 1Þ=�m2 is the fission Diven factor and Xf the factorthat corrects spatial effect in point kinetics:

Xf ¼F Rf /0;uþ0 ;u

þ0

� uþ0 ;Rf /0

� 2 ð87Þ

The next step is to deduce the correction factor for the contributionto the Feynman-a function coming from the neutrons correlated intime because they are produced in the same source spontaneous fis-sion event, or in the same spallation event. In this case we can writeon account of Eq. (63) that the contribution of the source correlatedneutrons in the fundamental mode approximation is:

YSðTÞ ¼Ds�mSCS

0;0

Sþ1 ;/0

� 1a2

0

1þ ð1� expða0TÞÞa0T

� �ð88Þ

Eq. (88) can be recasted on account of expression for CS0;0 in the

form:

YSðTÞ ¼ DS�mS S1;uþ0 ;uþ0

� Sþ1 ;/0

� 1v uþ0 ;/0

� 2

1a2

0

1þ ð1� expða0TÞÞa0T

� �ð89Þ

that can be expressed as in point kinetics on account of Eqs. (82)–(85) with a spatial correction factor:

YSðTÞ ¼ e�mS

�mDS

ð�a0ÞKXS 1þ ð1� expða0TÞÞ

a0T

� �ð90Þ

where the factor XS corrects source spatial effects in the point kinet-ics formula:

XS ¼F S1;uþ0 ;u

þ0

� uþ0 ;Rf /0

� uþ0 ; S1� ð91Þ

Therefore, the final expression for the Feynman-a function in theunimodal approximation is the sum of both contributions i.e. YSys(T)and YS(T) and can be rearranged in the form:

YðTÞ ¼ eð�a0ÞK

DKð�a0Þ

Xf þ DS�mS

�mXS

� �1þ ð1� expða0TÞÞ

a0T

� �ð92Þ

To obtain the single mode approximation for the Rossi-a functionwe consider only the fundamental mode in Eq. (73). This calculationyields the following result:

RðsÞ ¼ Sþ1 ;/0

� þðmðm� 1ÞC0;0 þ DS�mSCS

0;0Þexpða0sÞ2ð�a0Þ

Sþ1 ;/0

� ð93Þ

that on account of the expression for C0,0 and CS0;0 can be written as:

J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600 597

RðsÞ ¼ eF þ mðm� 1ÞRf /0;uþ0 ;u

þ0

� Sþ1 ;/0

� Sþ1 ;u0

� 2

1v uþ0 ;u0

� 2

"

þDS�mSS1;uþ0 ;u

þ0

� Sþ1 ;/0

� Sþ1 ;u0

� 2

1v uþ0 ;u0

� 2

#expða0sÞ2ð�a0Þ

ð94Þ

Now, with the help of Eqs. (82)–(85) and the spatial effect correc-tion factors defined previously, we find that Eq. (94) can be recastedin the following form:

RðsÞ ¼ eF þ e2

D

ð�a0ÞK2 Xf þDS

K

�mS

�mXS

" #expða0sÞ ð95Þ

where the expressions for the correction factors Xf and XS, that cor-rect point kinetic effects, are the same as those deduced for theFeynman-a function. It is noticed that Eq. (95) reduces to the pointkinetic equation when the spatial correction factors Xf and XS are setto one.

The theory and formulas presented in this paper can be general-ized to include the velocity correlations as pointed out by Difilippo(1991), in this case the formulas in the fundamental mode approx-imation are generalized as follows. The expression:

Rf /0;uþ0 ;uþ0

� ¼Z

d#Rf ð~r;vÞ/0ð#ÞðbT f u0Þ2ð~rÞ ð96Þ

must be substituted, according to Difilippo, by the expression:

Rf /0;uþ0 ;uþ0

� c ¼Z

d#Rf ð~r;vÞ/0ð#ÞZ

dv 0Z

dX!0vf ðv 0Þ

4p uþ0 ðr;v 0;X*0ÞÞ

��Z

dv 00Cf ð~r;v 0;v 00ÞZ

dX!00vf ðv 00Þ

4puþ0 ð~r;v 00;X

!00Þ�ð97Þ

where Cf ð~r; v;v 0Þ is the correlation function for the velocities of neu-trons produced in the same fission event at point~r. Also the expres-sion for S1;uþ0 ;u

þ0

� must be corrected on account of CS(r0,v,v00) that

gives the correlation function for neutrons born in the same sourcefission events. These generalizations can be extended also to thespallation source events.

2.6. The Feynman-a function with delayed neutrons for a Cf-252neutron source

We start from the formula for the Feynman-a function with de-layed neutrons deduced by Muñoz-Cobo and Verdú (1987) for a Cf-252 neutron source for the fundamental mode approximation, andwe use in this section the same nomenclature as in the originalpaper:

YðTÞ ¼ mðm� 1ÞNc

Z t

�1dt0Z

d3rZ

dvRf ðr;vÞ/0ðr;vÞI20ð~rÞ

� ðTþðt0ÞÞ2 þ DS�mS

Nc

Z t

�1dt0Z

d3rSð~rÞI2S;0ð~rÞðT

þðt0ÞÞ2 ð98Þ

where Sð~rÞ is the average number of source neutrons emitted perunit volume by the Cf-252 neutron source. The functions I0ð~rÞ andIS;0ð~rÞ are the spectral weighted importances for the fission andsource neutrons respectively given by

I0ð~rÞ ¼ bT f /þ0

�ð~rÞ; ð99Þ

IS;0ð~rÞ ¼ bT S/þ0

�ð~rÞ ð100Þ

where /þ0 ð#0Þ is obtained from the adjoint a eigenvalue equation

considering the delayed neutrons (Akcasu et al., 1971; Muñoz-Coboand Verdú, 1987) and the time dependent function T+(t) is given bythe following expression:

Tþðt0Þ ¼ ðgRD;/0Þ1v /þ0 ;/0

� X7

k¼1

1ð�rktÞ

1

1þP6

m¼1

bmkmK ðrk þ kmÞ2

fkðt0Þ ð101Þ

being /0 the neutron flux in the fundamental mode, rk are the rootsof the inhour equation, and the functions fk(t0) are defined by:

fkðt0Þ �expðrkðt � t0ÞÞ½expð�rkTÞ � 1�; t0 < t � T

½1� expðrkðt � t0ÞÞ�; t � T < t0 < t

�ð102Þ

Next we notice that the integral of fk(t0)fl (t0) over the time interval(�1, t) yields:Z t

�1dt0fkðt0Þflðt0Þ ¼ T 1þ r2

l ð1� expðrkTtÞÞ þ r2kð1� expðrlTÞÞ

rkrlðrk þ rlÞT

� �ð103Þ

Now, performing the integrals over time in Eq. (98) on account ofEq. (103), we can write the equation for the Feynman-a functionwith delayed neutrons as follows:

YðTÞ ¼ eF1v /þ0 ;/0

� 2 mðm� 1ÞZ

d3rZ

dvRf ðr;vÞ/0ðr;vÞI20ð~rÞ

�

þ DS�mS

Zd3rSð~rÞI2

S;0ð~rÞ��X7

k¼1

X7

l¼1

1ð�rkÞð�rlÞ

1RkRl

� 1þ r2l ð1� expðrkTÞÞ þ r2

kð1� expðrlTÞÞrkrlðrk þ rlÞT

� �ð104Þ

where the coefficients Rk are defined by the expression

Rk ¼ 1þX6

m¼1

bmkm

Kðrk þ kmÞ2ð105Þ

On account of Eqs. (82)–(85) we can write Eq. (104) for the Feyn-man-a function in the form:

YðTÞ ¼ eD

K2 Xf ;0 þ DS�mS

�ma0

KXS;0

� ��X7

k¼1

X7

l¼1

1ð�rkÞð�rlÞ

� 1RkRl

1þ r2l ð1� expðrkTÞÞ þ r2

kð1� expðrlTÞÞrkrlðrk þ rlÞT

� �ð106Þ

where the spatial correction factor for system fissions is given by

Xf ;0 ¼FR

d3rFð~rÞI20ð~rÞ

�/þ0 ;Rf /0

� 2 ð107Þ

where Fð~rÞ is the fission rate per unit volume at point~r. The sourcecorrection factor is given by:

XS;0 ¼FR

d3rSð~rÞI2S;0ð~rtÞ

/þ0 ; S1�

/þ0 ;Rf /0

� ð108Þ

We note that Eq. (106) for the Feynman-a function in the funda-mental mode approximation when delayed neutrons are consid-ered, reduce to Eq. (92), when r7 ? a0, R7 ? 1 and we neglect thecontribution of the delayed neutron groups.

3. Application to experimental data

The most important formulas in this work have been applied todata from the subcritical fast-thermal facility YALINA-Booster(Berglöf et al., submitted for publication). A detailed descriptionof the experimental facility and the neutron noise experimentsperformed is given by Berglöf et al. (submitted for publication),whereas here selected results indicating the applicability of theformulas are presented.

The core of YALINA-Booster is strongly heterogeneous and con-stitutes thus a suitable experimental basis for multiple a-mode

598 J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600

determination. An overview of the facility is given in Fig. 2. As canbe seen, the core has three distinct regions: the inner fast leadzone, the surrounding thermal zone and the graphite reflector.Detectors may be located in the experimental channels labeledEC. The current set of experimental data was recorded in EC5T, lo-cated close to the fast zone, and EC6T, located in the middle of thethermal zone. Since the data from the two detectors follow thesame trends (Berglöf et al., submitted for publication) data fromonly one of the detectors are presented below.

3.1. Rossi-a and Feynman-a with multiple a-modes

Relying on Eq. (73) a fitting model of multiple exponentials wasapplied to the Rossi-a experimental data and it was found thatthree a-modes could be obtained. The data and fitting are visual-ized in Fig. 3 in both linear and logarithmic scale. Taking into ac-count the relatively deep subcriticality of the system and itsstrong heterogeneity (Berglöf et al., 2010) the presence of severala-modes is natural and shows thus similarity with earlier studies(Busch and Spriggs, 1994; Karam, 1964; Karam, 1965; Kitamuraet al., 2000; Kuramoto et al., 2007).

Concerning Feynman-a for the same data set, only two a-modescould be obtained due to dead time effects (Berglöf et al., submit-

Fig. 2. Cross-sectional view of the

ted for publication). Fig. 4 clearly shows the necessity of usingmore than one a-mode. As shown by Berglöf et al. (submitted forpublication), the Rossi-a and Feynman-a fittings give, as expected,the same results.

3.2. Feynman-a with delayed neutrons

Including delayed neutrons in the analysis of neutron noise dataaiming at finding the prompt neutron decay constant is seldomnecessary. However, it is done here to demonstrate the applicabil-ity of the presented formula. For simplicity, one-delayed neutrongroup is assumed. Based on Eq. (106) the addition of a delayed neu-tron group may be considered as another a-mode and each of theprompt a-modes appear in cross-terms with the delayed a-mode.Since two prompt a-modes are required for the prompt part, in to-tal three a-modes are applied to the experimental Feynman-a data.To include the effect of delayed neutrons, very large time gatewidths, T, must be used and consequently a very long measure-ment time is needed to obtain sufficiently low statistical uncer-tainty. In the fitting shown in Fig. 5, gate widths from 0.15 ms to2 s have been used, thus covering four orders of magnitude. Thedelayed neutron contribution can be seen as a small increase inthe Y-value after about 200 ms. The delayed a-mode, obtained

YALINA-Booster reactor core.

0 2 4 6 8 101

1.05

1.1

1.15

1.2

1.25

1.3

Time [ms]

Auto

-cor

rela

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]

Auto

-cor

rela

tion

- 1

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

Fitting: χ2 = 1.04

(a)

(b)

Fig. 3. 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 5 10 15 20-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

T [ms]

Y (T

)

Experimental data

Fitting of 1 exponential, χ2/ν=12.29

Fitting of 2 exponentials, χ2/ν=0.31Residual x 10, 1 exponentialResidual x 10, 2 exponentials

Fig. 4. Fitting of one and two a-modes to experimental Feynman-a data. Theresiduals indicate the necessity of applying at least two exponentials.

Fig. 5. Fitting to Feynman-a data including delayed neutrons.

J.-L. Muñoz-Cobo et al. / Annals of Nuclear Energy 38 (2011) 590–600 599

through fitting, has a value of around 0.1 s�1, which may well berepresented by a one-delayed neutron group decay rate.

4. Conclusions

In this paper, the Feynman-a and Rossi-a formulas with spatialeffects and multiple a-modes for any kind of stochastic or contin-uous neutron source have been deduced. In addition, the singlemode approximation has been considered and the source and fis-sion correction factors that take into account spatial effects inthe point kinetics expressions have been obtained.

When one considers only the fundamental mode in the Rossi-aand Feynman-a functions and neglecting spatial effects, the formu-las deduced in this paper reduce to the classical Rossi-a and Feyn-man-a formulas. However, an important conceptual issue is thatthe Feynman-a function cannot be deduced from the Rossi-a for-mula performing the double integration as in the papers of Souleet al. (2004) and Muñoz-Cobo et al. (2008). This method is trueonly in the single mode approximation or in point kinetics but doesnot work when several subcritical modes are considered, as in thiswork.

Delayed neutrons were included in the Feynman-a functionand it was shown that the delayed neutron groups appear as delayeda-modes and can thus be used together with the multiple prompt a-mode formula, where for the fundamental mode these delayeda-modes are the roots of the inhour equation.

The derived formulas were successfully applied to experimentaldata from the strongly heterogeneous YALINA-Booster subcriticalfacility.

Acknowledgements

This work was supported by IP-EUROTRANS contract no. FI6W-CT2005-516520, the ENRESA-CIEMAT agreement for the Transmu-tación Aplicada a los Residuos Radiactivos de Alta Actividad, SvenskKärnbränslehantering AB (SKB, the Swedish Nuclear Fuel andWaste Management Co) and the Swedish Institute through the Vis-by program.

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