- Email: [email protected]
Models for a three-parameter analysis of neutron signal correlation measurements for fissile material assay
550
Nuclear Instruments and Methods in Physics Research A251 (1986) 550-563 North-Holland, Amsterdam
MODELS FOR A THREE-PARAMETER ANALYSIS OF NEUTRON SIGNAL CORRELATION MEASUREMENTS FOR FISSILE MATERIAL ASSAY D.M. CIFARELLI Unmersab L. Bocconi, Istituto dt Metodi Quantitatiui, 20136 Milano, Italy
W. HAGE
Commission of the European Communities, Joint Research Centre - Ispra Establishment, 21020 Ispra (Va), Italy Received 19 December 1985 and in revised form 14 May 1986
For the nondestructive assay of Pu-containing fuel, interpretation models are derived to obtain its spontaneous fission rate or its Pu-mass equivalent. The factorial moments of the frequency distribution of neutron signal multiplets in signal and randomly triggered time intervals are used as measurement results.
1. Introduction The fissile mass of special fissile materials is determined via the measurement of the spontaneous fission rate knowing the isotopic composition of the fuel . Corrections are required for the neutron emission rate due to (a, n) reactions and neutron multiplication caused by primary neutrons. The neutron signal correlation technique permits the determination of the number of fissions generating a signal pulse train containing signals from (a, n) reactions. This distinction is possible because fission neutrons are emitted in groups per fission event, whereas (a, n) reactions produce mainly single neutrons. With this technique, neutrons emitted by a test item are slowed down in a moderator, diffuse there as thermal neutrons and are partially absorbed by neutron detectors incorporated in the moderator. Absorbed neutrons inside these detectors are transformed into electrical signals leading to a signal pulse train. Several methods [1-5] have been developed to analyze this pulse train giving two independent measurement results: the total counts and a quantity related to correlated signal pairs. In many practical applications, a two-parameter analysis is not sufficient if neutron multiplication corrections are required with an unknown (a, n) neutron emission rate, or if neutron multiplication is insignificant, but the test item modifies the neutron detection probability. The first case occurs with Pu-containing fuel with unknown age or contamination with light nuclei, the second when monitoring small test items with internal moderation (small waste packages or solutions) . A three-parameter analysis can be performed [6], analyzing the pulse train such that the number of signal events inside fixed observation intervals is registered during a measurement time TM . The use of moments [7] of the respective probability distributions leads to simple interpretation models, especially with the factorial moments [8,9]. For the determination of three unknown parameters characterizing a test item, the factorial moments [9] and the model for fast fission [10,11] are applied. 2. Theory Consider a signal correlator device which registers the number of signals existing inside fixed time intervals T. These intervals are triggered either randomly or by each neutron signal [6,12] . The quantities 0168-9002/86/$03 .50 C Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
D.M. Cifarelh, W. Hage / Three-parameter analysis of neutron signal correlation measurements
551
obtained experimentally are the background multiplets Bx(T) and the trigger multiplets N,( ,r) (x = 0; 1, 2, 3, - - ), defined by [8,9], respectively, Bx(T) = number of events with x signals inside K randomly triggered inspection intervals; Nx(T) = number of events with x signals inside inspection intervals T obtained during an observation time TM. Each signal triggers an inspection interval T . This leads to the frequencies bx (T)
(1)
Bx(T)1K,
=
a d n, (T)=Nx(T)INT,
(2)
where NT is the number of total counts collected during the measurement time TM . bx(T) and n x (T) are the probability counterparts to the frequencies bx(T) and n' (T). The factorial moments 00
Y_ =
x
l1
(x)bx(T)Y_ IL x
cc .) Mn(, x= ~L
= bt
( xtt ) bx (T) ,
(3)
00
(FA.)nx( T )
(i1,)nx(T),
(4)
x=~
as defined in eqs . (3) and (4) are the effective numbers of signal multiplets with t. signal events inside the observation intervals T . These effective numbers are generated by x >__ tt signal events inside randomly and signal-triggered intervals, and are referred to as randomly and signal-triggered factorial moments of order ,u, respectively . In ref . [9] recurrence formulas for the factorial moments M,, (,,) and Mn(u) were derived . Both moments are not normalized on ,u! in contrast to Mb(,,) and m . OA) of eqs . (3) and (4), respectively . With the identity Mb (P) and mn(,)
"'b(it)
_ Mn(W) , ~~
[he equivalent recurrence formulae of ref. [9] for the normalized factorial moments become /A-i
and
L
M n(q) M h(u-q) ,
M n(P)
=
Mb(O)
= 1 and
q=0
with mn(0)
= mn(O) = 1 .
The quantities m*(m) and m*(rn) are analytical expressions and depend on the characteristics of the test sample, the moderator-detector assembly and the operational conditions of the multiplet analyzer. The following assumptions are introduced : 1) the test sample has point geometry, 2) fast neutron multiplication only takes place in the test sample, 3) no neutrons return from the detector head to the test sample, 4) induced fissions occur at the same time as the primary neutron emissions, 5) the time response function of the moderator detector assembly is a pure exponential function with decay constant X,
552
D.M. Crfarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements
6) the primary neutron energy has no influence on the thermal neutron detection probability E, 7) the start of each observation interval -r of the trigger multiplets is delayed by the time T for each trigger signal, 8) no dead time losses of signals occur. With these conditions the following holds : mb( ") =w(NL)E"T[S« {(1-P)811,+va(r~) P P } +FSvs( w ) p ],
(7)
and mn( ") =f"E
"+1 T [Sa((1-P)so
Mb(1)
with W GL ) =
"-1
and
f=
K= O
_
(1-~1)(-1)x l K
e -AT( 1 -
" +v«(" +1) P P) +Fsvs( " +1) P ],
( 8)
-ATK
XrK
e-AT) .
(9)
For convenience the following quantity is introduced R( ") -E"TM[S«f (1-P)s1" +v~( ") P P} +Fsvs( ") p ] .
(10)
From eqs. (7) and (8) it then follows m b(") TM
__
m
n(" -1 )
NT for g> 1 .
(11)
The symbols used are : S. = (a, n) neutron emission rate of test item, FS = spontaneous fission late of test item, = probability that a neutron generates an induced fission event, P E = probability for detection of a neutron, T = observation interval, T = delay between trigger signal and start of observation interval r, A = fundamental mode decay constant of moderator-detector assembly, PMP) = probability for the emission of v fast neutrons per prompt fission caused by a primary neutron generated by reaction j (j = a or I then (a, n) reaction or induced fission ; j = s for spontaneous fission), = g.th factorial moment of the P,v(p) distribution V'(") (P) 1 for n v (12) V,I,ITP E (p JP), 11 0 for n v=" Eqs. (5)-(12) permit a determination of the spontaneous fission rate FS in the presence of (a, n) reactions and neutron multiplication . Analytical expressions [8] for P,,(p) lead to rather complex expressions even with the fast fission approximation . Moreover, recently analytical expressions were given in two independent publications [10,11] for the factorial moments PJ(" ) p of this probability distribution . Using these relations and the fission escape multiplication M where M=
1 -P
1 -PVt (I )
,
(13)
D. M. Cijarelh, W. Hage / Three-parameter analysis of neutron signal correlation measurements
553
the following expression for vl(u) (p) is obtained from ref . [10] or [11] after some algebraic operations : l'1(w) P - Mi"
E
k=1
a1PKM k-1 ,
(14)
with Y 1 YI 2 l( ) ( ) - 1 YI(1) Y 1 YI 3 l( ) ( )
;1(1) - 1
a131 =v1(3)
a122
=
a132 =
Y 2 Y1 2 l( Y ( ) - 2_ - 1 PI(1)
+i
J()p2(vI(1)
_ 1) 2,
yl(1) Y 1(2) vI(1)
-
1 ,
yl(1) 11(3)
_
Y I(1)
P
1(2) _ 1) + 2 (P
P
1(2)
4
2 v V l(1) 1(2) ( _ 1)2 , YI(1)
2
a133
_2
(1 5)
Y1(1)YI(2) =2 _ 1)2 . YI(1) -
vl(r') is the factorial moment of order u of the PI distribution for induced fission caused by primary fission neutrons. For j = s, vl(IA) is the factorial moment of order p, of the PS,, distribution for spontaneous fission of the test sample. Within the uncertainty of the existing data for the P distribution by fast neutron fission, it can be assumed that PI([')
(16)
= Y«(w)-
With eqs. (14) and (15) the interpretation of the neutron correlation technique requires as nuclear data only the moments of the P1 distribution of the isotopes present in the sample for spontaneous (j = s) and induced fission (j = a = I) processes . 2.1.
The factorial moments
of the P
distribution
For spontaneous fission reactions there exists a large number of publications from which the Ps, distributions for the various spontaneous fission isotopes can be taken. A recent data evaluation has been performed in ref . [13]. For induced fission events mainly data for thermal neutron induced fissions [13-15] exist for the various fissile isotopes present in fuel. There is one exception : Fr6haut et al. have measured the P,(E) distribution of 239Pu at various fission neutron energies E between 1 and 10 MeV . These data are at present evaluated at BNL [16]. In the analysis of the neutron signals the TI(,,) values for an average fission neutron spectrum are required. The average value of 00
YI(1) _
PP'
v=1
increases with neutron energy for the fissile isotopes leading to an even more pronounced increase of the moments with higher order p. The procedure adopted to overcome this difficulty was to plot v,(2) as a function of v,(1) and vl(3) as a function of vI(2) for the isotopes with measured P1 distributions from thermal fission . This resulted [13-16] in a parabolic dependence of vI(2) and v,(3) on PI(1) and v1(2),
D.M. Crfareld, W. Hage / Threeparameter analysis of neutron signal correlation measurements
554
7
7
Fm-257
Fm-257 I7 MeV
I le ,/
6 MeV I Î
Cf 252
5
`5 MeV Cf-250 ~4 MeV
5
4
Cm-244
3
U-233
4
Pu-238
2
~~Cm-248
I
3
Cm-246 Pu-239
Pu-239~
2
Cm-242
U-235
U-235
2.5
3.0
3.5 iN(~)
Fig. 1. v(2)
as a
Pu-242 I` Pu-236
4.0
function of v(1) .
oMeasured value o For Pu-239 at different energies
0--, U-238 1
~~ Cf-248 p--j1 MeV Pu-241 Cm-246
U-233
Pu-240
oMeasured value o For Pu-239 at different energies
2 MeV
Cm-244
Pu-238
Pu-236
\Pu-242
3 MeV
Cf-246
Pu-240
U-238
2.0
-250 '/Cf 4 MeV
4
P-2 MIV
1 MeV Pu-241
Cf-252 5 MeV
3 MeV Cf-246
o--6 MeV
2
3
Fig. 2. v(3)
4 as a
5
-0 function of v(2) .
6 v2)
respectively (figs . 1 and 2). Determining with a Watt fission spectrum f(E) [17] and the fission cross section of (E), the value of PI(1) Ef [18] for a fission spectrum is obtained for each fast fissioning isotope :
f
00
f(E)vf())(E)of(E)
f "of(E)of(E) dE
dE .
(17)
Inserting the value of PI(1) Ef into fig. 1 gives v,(2) Ef and using this value in fig . 2, vj(3) Ef is obtained as an approximation to a fission spectrum average. In many cases vj(1) E can be determined as the average value of the test sample with nuclear codes. From figs . 1 and 2 then follow the corresponding values for the higher factorial moments . The application of this procedure is valid as long as the neutron slowing down time inside the test item is much smaller than 1/X . 3. Formulae for typical cases In the following, models for an interpretation of correlation measurements with three unknown parameters are derived . The cases considered are : 1) Absolute determination of spontaneous fission rate FS with unknown (a,n) neutron source S. and unknown neutron multiplication factor M with known instrumental parameters E, X, T and T. 2) Relative determination of FS with unknown S. and M, using a reference sample . 3) Absolute determination of FS with unknown S. and E and known M, a, T and T.
D.M. Cifarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements
555
4) Absolute determination of Fs, E and M with known a, X, T and T, a being the ratio S,,/Ps(1)Fs . 5) Determination of M for fissile material with negligible spontaneous fission rate (FS = 0), unknown Sa and E . For all these methods the quantities R1 , R 2 and R 3 are determined from the randomly triggered multiplets applying eqs. (5) and (11) : R1 = R2 = R3=
TM
mb(1) - NT,
(18)
TM 1 2 [ mb(2) - 1 M(1)] , Tw(2)
(19)
Tw(1)
TM
(
[ mb(3)
Tw(3)
-
1 3 l mb(2)mb(1) + -mb(1)1 ,
w(2) = 1
w(1) = 1,
AT
(20)
(1 - e - X T ),
_1 _ 4 e -J" + e-2AT), w(3) = 2%~T (3 -
(21)
or from signal triggered multiplets with eqs. (6) and (11) : R1 = mn(0)NT = NT, R2=
N Î
IMn(1) -
( 22 ) ( 23 )
Mb(I) ],
R3 - f2 [ m n(2) - Mb(l) ( mn(1) - mb(1) )
jl
+
w(2) ~
1
Mb(l)
(24)
il
or by combining both types of multiplets [8]. The Ru (1 _< tt< 3) obtained from the measured data can be expressed by the instrumental data of the detection head and the physical and nuclear data of the test item using eqs. (10), (14) and (15) . They are: Rl = E[b11Fs+b12Sal,
(25)
+ b22Sa~,
(26)
R3 = E 3 [b31Fs+b32Sa1+
(27)
R2
= E
2 ~b21Fs
with b11 = TMMYS(1), M-1 l b21 = TMM2 'is(2) + - 11 vs(1) "1(2) I, v 1(1) ~ )%S(J)v21i(2) _ _
I,
b12 = TMM, b22 = TMMZ
632 = TM M
M-1 _ Y1(2), Y 1(1) -
3 M-1
M-1 +2 ~_1(3) Y 1 - 1 v1(1) ~ v1(1)
(28)
556
D.M. Cifarelh, W. Hage / Three-parameter analysis of neutron signal correlation measurements
Eqs. (18)-(28) are used in order to derive the interpretation models for the abovementioned five cases. In eq . (28) PI(I) is the spectrum average value of vj(I) Ef defined in eq . (17) . With known v,(I) the numerical values of PI(2) and v,(3) are obtained from figs . 1 and 2, respectively. Case 1 : absolute determination of FS , Sa and M. For the absolute determination of F, with an unknown (a, n) reaction rate Sa and neutron multiplication factor M, the instrumental data E, X, and nuclear data vs(I4) and vüw) of the test sample have to be known with sufficient precision. Eliminating in eqs. (25), (26) and (27) F, and Sa and using the coefficients of eq. (28), an algebraic equation of order 3 in M is obtained [19] : ao + a,M+ a 2 M 2 + a3 M3 = 0,
(29)
with E R2 IXl =E2
Ys(3) v1( 2) 2 -1 PI(I) ~ vs(2) v I(2) vs(3) 2v1(2) 1 _ RI R2_ IX2 -1 I - vI(3) I + E2 - 1 vI(I) vs(2) vI(I)
E
R 2 2v I(2) a3 = (30) - a2 . E 2 (vt(I) - 1)
The neutron multiplication factor M is determined from eq . (29) by taking the solution with the smallest possible value of M > 1 . With the known multiplication M a solution of eqs. (25) and (26) with respect to the two unknowns Fs and Sa gives: 1 TM
and S
-
I
I
R2 - RI V I(2)( M 1) E2M2 vs(2) EMvs(2)(vI(I)-1)
_ 1
RI
Ti,,I
EM
1 +
vs(I)v1(2)(M- 1 )
R2
-
vs(2)YI(I)-1)
(31)
,
vs(I)
(32)
E 2M2 vs(2)
The spontaneous fission neutron emission rate is mainly determined by R2, i.e . the measured signal doublets . As long as the (a, n) reaction rate is not the dominating contribution to RI, the influence of the singlets is less important due to the M-1 weighting. Larger errors will occur in Sa because the first and second term of eq . (32) are of the same order of magnitude. Case 2 : relative determination of FS .
For the relative determination of the spontaneous fission rate in the presence of (a, n) reactions and neutron multiplication, at least one well characterized reference sample must be available. Determining the R v values of the test sample and those of the reference sample R ß ,(0) (1 <-_ tt <- 3) the instrumental parameters E, X and T and T are eliminated completely if the factorial moments of randomly triggered intervals T are used (see eqs. (18), (19) and (20)). This is also true when combining the randomly and signal triggered factorial moments [8] . It is no longer true when using the signal triggered factorial moments. Applying eqs. (25), (26) and (27) for the test and reference samples, the following system of equations is obtained : r~,
=
R Rw~O)
[Fs (0)bt, I (0) +
Sa(0)bw 2(0)]
= Fs bl, I +
Sabw 2
1 < ~. < 3 .
(33)
D.M. Cifarelh, W. Hage / Threeparameter analysis
of neutron signal correlation measurements
557
b,,,(0) and b,,2(0) are the known coefficients of eq .
(28) of the reference sample with the known multiplication factor M(0), the known nuclear data P.,( ,, )(0) and vI(w)(0) and measurement time TM. The coefficients b,.I and b', 2 of the unknown test sample contain the unknown multiplication factor M and the known nuclear data vs(',) and Eliminating by means of eqs. (31) and (32) Fs and S., an equation of order 3 for the determination of the multiplication factor M is again obtained : 3 = 0. aô + a1M + a*M 2 + a *M 2 3
(34)
The coefficients a* now depend on rt which is proportional to the ratio R',/R,,(0) and to of the reference sample. They are : ao =
vs(3) ~
Ps(2)
-2 _
v1(2)
,
_1
vM)
v v s(3) t(2)
1
a* =ri -
and Sa(0)
-r3,
ai = r2
aj
Fs(0)
Ps (2)
2 vI(2) r2 _ -1 vI(I)
_v1(3)
2vu2) +r2 _ _1 , vt(I)
a*,
(35)
with ri =
R
(~) FS (0) Y1(I)CI(0) >
(36)
CI (0) = M(0) [1 + a(0)] ,
(37)
r2 - R
(38)
c2(o)
r3
F, (0) vs(2)C2 (0),
(0)
= MZ(o) f 1 + (M(o) -1)(1 + a(o)) R3
R3(0)
Ys(I)Y1(2) vI(I)
1
~,
(39)
(40)
Fs(0)vs(3)C3(~),
C3 (0) = M3 (0)I 1 + 2(M(o)
+(M(0) - 1)(1
-1) -
ys(2)"1(2)
Ys(3)( v I(I)
1)
-
+ a(0)) [M(o) - 1]
vs(I)vi(2)
v vs(3) ( I(l)
+
vs(I)VI(3) Ps(3) ( vI(I) -
With known multiplication factor M, the spontaneous fission and (a, n) reaction rates respectively, are given by : F -_
s
S =
- ri (M
r2
TMM ri TMM
TMM
vs(2)
~l
+
- 1)
V
Y I(2) s( I )
"s(2)(vI(I)-1)
v1(2) vs(2)(vI(I)-
1)
(M- 1) -
>
(41)
1) F,
and S.,
42
r2
_s(I)
Tm M 2 Ps(2)
(43)
558
D .M. Cifarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements
In the case that there are several reference or test samples with known Fs, Sa and known multiplication factor M, the constant CN of eqs . (37), (39) and (41) serve to correct the R,, values (1 < [L< 3) for these two contributions . The a and M corrected value of RA is: RF,c = e"Fsvs(w ) TM = Rf,/CF, .
(44)
RtLc has to lie within the error margins on a straight line with the given spontaneous fission rate
FD (or the z4° Pu-mass equivalents) as abscissa, provided that the limitations of the model are valid and the nuclear data correctly chosen . Case 3: absolute determination of FS with unknown S. and e.
A spontaneous fission and (a, n) neutron emitter surrounded by moderating material influences the neutron detection probability of the detector head. A part of the source neutrons is slowed down in this moderator and is absorbed in the thermal neutron filters of the detector head. Such conditions exist when monitoring small waste packages with moderating material like polythene, paper, rubber, etc. In this case a determination of F5 , S. and e is required assuming that the neutron multiplication is either negligible or known . First the neutron emission rate ratio S
(45)
a= _ a Ds(1)Fs
is determined forming the ratio R2
ß = R,R3 2 .
(46)
This ratio becomes independent of e for a point source and independent of Fs by using eq. (45). Replacing R1, R 2 and R3 in eq. (46) by eqs . (25), (26) and (27), the following solution of a quadratic equation in a is found : a=
B2 +4AC -B
(47)
2A
with 2 2
A = ßblibai - b22vs(1), B = ßb11( bal + b32vb(1)) - 2 b21b2A(l),
(48)
C = b21 2 - ßbllb3l .
The coefficients b,, (1 < i :< 3 and 1
( 49 )
is independent of e. It is used knowing the a-ratio for the determination of the spontaneous fission rate Fs: FS =
y(b21 + ab22vs(1 » Tmbil(1 +
a)2
(50)
.
With known a and FS the neutron detection probability e follows from eq. (25). It is: e
_
Ri
FSvsMMTM (1 + a)
.
(51)
D.M. Cifarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements
559
Where there is no neutron multiplication, the following relations are valid : M= 1, a=
z Ps (2)
-1,
s(l)vs(3)
Fs=Y
"s(z) v(1)(1+a)
2
> TM
and R1
e
.
Fsvs(1) (1 + a)TD,1
(52)
Case 4 .: absolute determination of FS , e and M with known a-ratio. In some cases the a-ratio of waste items with internal moderation is better known than the neutron multiplication factor M. This occurs especially when measuring fuel fabrication waste of known isotopic composition and no or a known contamination with isotopes having a large (a, n) reaction cross section. First a relation for the unknown neutron multiplication factor is obtained using eqs. (25), (26), (27) and (28), forming the ratio :
ß=
z Rz
(53)
RIR3 .
After some algebraic operations, a quadratic equation for M is found with the solution : M_
Q+
Q z -4PR 2P
(54)
'
where
z P = ß alc3 _ bz,
Q=2a z b z -ßa 1 b3 , R = ßalas _ a2z
(55)
and aI
= vs(1) (1 + a),
az = vs(z) bz
v1(1) -
= vs (2) - a z ,
a3 = vs(3) -
1
1+a
1 1 l2vs(z)vI(z) + (1 + a) + C3, v I(1) vs(1)PI(3)1 _
b3 _ vs(3) - a3 - C3,
C3 = 2(1 + a)
z Vs(l)VI(2)
( v1(1)
-
1)
z
(56)
Having determined M, Fs follows directly from eq . (50) . Case -5: no spontaneous fission primary neutrons. In some practical applications, fuel has no spontaneously fissioning isotopes but (a, n) primary neutrons . In these conditions the neutron multiplication M, the (a, n) reaction rate S. and the detection
560
D.M.
Cifarelli, W. Hage / Threeparameter analysis of neutron signal correlation measurements
probability e can be determined . This leads to the simple case that expression for M is obtained: M_
The ratios Sa =
y1(3) ( y1(1) -
ß
v2I(2)
1 -ß ß
1)
bll = b21 = b31
= 0. From the ratio ß an
+1.
(57)
and y serve to derive an equation for the determination of Sa:
_Y
TM
ß 1-~
With known Sa and
v 1 (3) YI(2)
M,
(58)
.
the detection probability e follows simply from eq . (25) .
4. Test of theory by experimental data 4.1. e
not modified by sample
Part of the derived analytical expressions was tested with experimental data using a 8.45 g metallic Pu sample exposed in a hollow cylindrical detector head of 1 m height and 32 3He proportional counters . By means of a computerized registration system [20], the factorial moments of the frequencies to have x signals inside, randomly and signal triggered observation intervals T were measured. Table 1 gives the average values and the standard deviations of these moments obtained in eight different runs, each with a measurement time of 8 min. The predelay T for the three observation intervals Tl = 25 .6 ,us, T2 = 51.2 Ets and T3 = 102.4 ,us was 6.4 [,s in each case. The decay time of the detector head was 51.3 ,us and its neutron detection probability e = 0.27. The latter quantity [20] was determined in a separate experiment from an unknown 252 Cf spontaneous fission neutron source, using the nuclear data for v,(,) and Ps(2) and R, and Table 1 Moments of randomly and signal-triggered observation intervals
ORDER OF MOMENT m 0 1 2 3 4
Moments Mb(m)of Randomly Triggered Observation Intervals i - 25 .6 /is 1 .0000 6 .61483 3 .74852 1 .67851 5 .2053
10 -3 10 - " 10 -5 10 -7
t 3 .9 t 2 .25 t 4 .85 t 1 .35 t 1 .5
i - 102 .4 its
i- 51 .2 its 10 -5 10 -5 10 -6 10 -6 10 - '
0 .99999 1 .32301 1 .31250 1 .02862 6 .80980
10 10 10 10
-2 -3 -4 -6
t 7 .99 t 6 .97 t 2 .52 t 7 .96 ± 1 .79
2
10 -1 10 -5 10 -1 10 -6 10 -6
1 .0000 2 .64591 4 .12235 4 .88466 4 .84310
10 -2 10 -3 10 -4 10-5
t 1 .47 t 1 .36 t 6 .10 t 2 .54 t 6 .60
10 10 - ' 10 -5 10 -5 10 -6
Moments m n (m) of Signal Triggered Observation Intervals -r 0 1 2 3
1 .00004 9 .52149 10 -2 5 .85962 10 -3 4 .27510 10 - " NT = 258.4 s -'
t t t t
2 .10 1 .59 4 .47 9 .06
10 -3 10 -3 10 -4 10 -5
NB = 2.5 s -1
1 .00003 1 .54135 10 -' 1 .53349 10 -2 1 .25952 10 -3
t 2 .27 10 -3
t 1 .84 10 -3 t 7 .01 10 -4 f 1 .63 10 -4
9 .99779 2 .17687 3 .09289 3 .48763
(count rate with and without sample).
10 - ' 10 -' 10 -2 10 -3
t t t t
2 .32 3 .50 2 .77 1 .04
10 -3 10 -3 10 -3 10 -3
D.M. Cifarelli, W. Hage / Three parameter analysis of neutron signal correlation measurements
561
Table 2 Determined M -1, vs(,) F, and Sa
RANDOMLY TRIGGERED INTERVALS T 2 = 51 .2 lts Ts --- 102 .4 l.ts
i, = 25 .6 gs
M-1
0 .0183 + 0 .0036
0 .0219 ± 0.0028
vs(,)Fs[1/s] Sa[1/s1
0 .5
6.6 ± 13
930 ± 6
920 , 6
0 .0206 ± 0.0026 924 ± 6 5 .1
SIGNAL TRIGGERED INTERVALS M-1
0 .0256 + 0.0043
vs(,)F,[l/s] Sa[1/s1
4 .3
920 ± 10
0 .0223 + 0 .0015 923 ± 4
0 .0180 ± 0.0021 930 + 5
4.7 ± 9
0.4
SAMPLE DATA M-1
vs(,)Fs [1/s] Sa[1/s]
0 .0240
REMARK TI MOC
918 ± 24 0
values derived from signal and randomly triggered observation intervals. In order to obtain the neutron multiplication M, the spontaneous fission rate FS and the (a, n) neutron emission rate S«, the formulae of case 1 are used, in connection with the factorial moments Mb(,) (j = 1, 2, 3) of table 1. First are determined the quantities R, using eqs . (18), (19) and (20), calculating w(j) (j = 1, 2, 3) according to eq. (21). With known R ., e, vt(t) and vs(,), the coefficients a, (i = 0, 1, 2, 3) of an equation of third degree in M are obtained . For each gate length the neutron multiplication factor M is determined from the root of eq. (29) for M> 1 . With known M, the spontaneous fission rate FS, and the (a, n) neutron emission rate S. follow from eqs. (31) and (32), respectively . A similar procedure was applied using as measured data the factorial moments m,,(,,) (p = 0, 1, 2) of the three signal triggered observation intervals T. Table 2 gives the results for M- 1, i~,), FS and S. for both measurement methods and the three observation intervals. The uncertainties quoted are the standard deviations of the eight runs each analyzed individually . The results for the spontaneous fission neutron emission rates agree very well with the values known from the test sample [21]. The uncertainty in PS 1 F of the test sample is mainly due to the uncertainty of the spontaneous fission neutron half-life of 2 Pu [22]. The determined quantities are within their error limits independent of the duration of the observation intervals and independent of the type of trigger interval used . The smallest statistical error appears to be for observation intervals T close to the decay time of the detector head . R 2 and R 3 can be obtained by combining the factorial moments of both signal and randomly triggered observation intervals. In this case the measured Mb(l) and Mb(2) [8] are taken to calculate R2 and R 3 . The values obtained for M, v, (1) F, and S. lead to almost the same numerical values as found with the signal triggered factorial moments. R2
4.2. e modified by test sample Monitoring small waste packages of material containing hydrogen, the Pu mass is frequently so small that the neutron multiplication factor M becomes close to 1 . In this case the spontaneous fission rate Fs , the (a, n) reaction rate S« and the detection probability e can be determined according to the formulas derived for case 3 by putting M = 1 . However, if the Pu content is of the order of grams, then the determination of the spontaneous fission neutron emission rate v t1 ,F, depends strongly on the neutron multiplication factor M.
562
D .M. Cijarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements 1000
vsi,~Fs 900
Se v,,, > Fs 800
700
NEW,"-
0 .25
1
1 .01
1 .02 -i M
Fig. 3. Neutron emission rate as a function of M .
1 .03
0 .2
1
1 .01
1 .02 10
M
1 .03
Fig. 4. Detection probability c as a function of M .
Such conditions are presented in fig. 3 for the 8.45 g metallic Pu sample. This figure is based on the experimental data obtained from randomly triggered factorial moments with an observation interval ,r = 51 .2 ps . Not knowing the characteristics of the sample, it is only possible to state that the multiplication factor is below 1 .0219 since otherwise S. would become negative. Knowing the isotopic composition of the fuel it is recommendable to determine an approximate value of the a-ratio, and to calculate with the formulas of case 4, FS , M and e . By this type of analysis the uncertainty in the determination of v5(1) FS can often be kept within smaller limits . Fig. 4 gives the neutron detection probability as a function of M for the same conditions as in fig. 3. The same tendency of vs(l) F and e as a function of M has been found in ref. [23] when analyzing the frequency distribution of a signal pulse train for randomly and signal triggered observation intervWls.
5. Conclusions The interpretation model has been tested successfully with a small Pu-metal sample. In the near future, tests will be carried out on the validity of the point model and the model for neutron multiplication with smaller and larger PUOZ samples. Dead time corrections of the factorial moments are still a problem for further theoretical studies.
Acknowledgements Many thanks are due to Mr . L. Bondar for providing some measured data . Much appreciated were discussions with Mr . N . Barrett, Mr . L. Bondar, Mr . R. Dierckx and Mr . A. Prosdocimi . Gratefully acknowledged is the assistance of Mrs. K. Caruso for performing the calculations and preparing the figures, and to Mrs. G. Tenti and Mrs. M. Van Andel for typing the manuscript .
References [1] G. Birkhoff, L. Bondar, J. Ley, R. Berg, R . Swennen and G . Busca, EUR 5158e, Commission of the European Communities, Joint Research Centre, Ispra (1974).
D.M. Cifarelle, W. Hage / Threeparameter analyses of neutron signal correlation measurements
563
[21 K. B61mel, KfK-2203, Kernforschungszentrum Karlsruhe (1975) . [31 N . Ensslin, M.L . Evans, H .O. Menlove and J .E . Swansen, Nucl. Mater . Man . 7 (1978) 43 . [41 E.J . Dowdy, C.N . Henry, A.A . Robba and J .R. Pratt, Proc . Int. Symp . on Nuclear Material Safeguards, Vienna, SM-231/69 (IAEA, Vienna, 1978) . [51 M .O . Deighton, Nucl . Instr. and Meth . 165 (1979). [6] L. Bondar, Proc . Int . Meeting on Monitoring of Plutonium Contaminated Waste, Ispra, Commission of the European Communities, Ispra (1979) . [7] N . Shenhav, Y. Segal and A. Notea, Nucl . Sci . Eng. 80 (1982) 61 . [8] R. Dierckx and W . Hage, Nucl. Sci. Eng. 85 (1983) 325 . [9] W. Hage and D .M . Cifarelli, Nucl. Sci. Eng . 89 (1985) 159 . [101 K . Böhnel, Nucl . Sci . Eng. 90 (1985) 75 . [11] W . Hage and D .M . Cifarelli, Nucl . Instr . and Meth . A236 (1985) 165 . [12] M .S. Krick and J.E . Swansen, Nucl. Instr. and Meth . 219 (1984) 385 . [131 M .S . Zucker and N.E . Holden, 6th ESARDA Symp. on Safeguards and Nuclear Material Management, Venice, Italy (1984). [141 J .W . Boldeman and M .G. Hines, Nucl . Sci. Eng . 91 (1985) 114. [15] N .E. Holden and M .S . Zucker, 7th ESARDA Symp . on Safeguards and Nuclear Material Management, Li6ge, Belgium (1985) . [16] M .S . Zucker (BNL), private communication (1985) . [171 Proc . Consultants, Meeting on Prompt Fission Neutron Spectra, IAEA, Vienna (1971) . [18] F . Manero and V.A. Konshin, At . En . Rev . 10 (1972) 637 . [19] W. Hage and D .M . Cifarelli, Proc. 26th INMM Ann. Meeting Albuquerque (1985). [20] L . Bondar, Joint Research Centre, Ispra, private communication (1982). [21] J .F . Gueugnon and K . Richter, European Institute for Transuranium Elements, Karlsruhe, Fabrication report K0282049 (1982) . [22] C. Budtz-Jergensen and H.H . Knitter, Nucl . Sci . Eng . 79 (1981) 380 . [23] L . Bondar et al ., ESARDA Proc, Venice, Italy (1984) .
Nuclear Instruments and Methods in Physics Research A251 (1986) 550-563 North-Holland, Amsterdam
MODELS FOR A THREE-PARAMETER ANALYSIS OF NEUTRON SIGNAL CORRELATION MEASUREMENTS FOR FISSILE MATERIAL ASSAY D.M. CIFARELLI Unmersab L. Bocconi, Istituto dt Metodi Quantitatiui, 20136 Milano, Italy
W. HAGE
Commission of the European Communities, Joint Research Centre - Ispra Establishment, 21020 Ispra (Va), Italy Received 19 December 1985 and in revised form 14 May 1986
For the nondestructive assay of Pu-containing fuel, interpretation models are derived to obtain its spontaneous fission rate or its Pu-mass equivalent. The factorial moments of the frequency distribution of neutron signal multiplets in signal and randomly triggered time intervals are used as measurement results.
1. Introduction The fissile mass of special fissile materials is determined via the measurement of the spontaneous fission rate knowing the isotopic composition of the fuel . Corrections are required for the neutron emission rate due to (a, n) reactions and neutron multiplication caused by primary neutrons. The neutron signal correlation technique permits the determination of the number of fissions generating a signal pulse train containing signals from (a, n) reactions. This distinction is possible because fission neutrons are emitted in groups per fission event, whereas (a, n) reactions produce mainly single neutrons. With this technique, neutrons emitted by a test item are slowed down in a moderator, diffuse there as thermal neutrons and are partially absorbed by neutron detectors incorporated in the moderator. Absorbed neutrons inside these detectors are transformed into electrical signals leading to a signal pulse train. Several methods [1-5] have been developed to analyze this pulse train giving two independent measurement results: the total counts and a quantity related to correlated signal pairs. In many practical applications, a two-parameter analysis is not sufficient if neutron multiplication corrections are required with an unknown (a, n) neutron emission rate, or if neutron multiplication is insignificant, but the test item modifies the neutron detection probability. The first case occurs with Pu-containing fuel with unknown age or contamination with light nuclei, the second when monitoring small test items with internal moderation (small waste packages or solutions) . A three-parameter analysis can be performed [6], analyzing the pulse train such that the number of signal events inside fixed observation intervals is registered during a measurement time TM . The use of moments [7] of the respective probability distributions leads to simple interpretation models, especially with the factorial moments [8,9]. For the determination of three unknown parameters characterizing a test item, the factorial moments [9] and the model for fast fission [10,11] are applied. 2. Theory Consider a signal correlator device which registers the number of signals existing inside fixed time intervals T. These intervals are triggered either randomly or by each neutron signal [6,12] . The quantities 0168-9002/86/$03 .50 C Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
D.M. Cifarelh, W. Hage / Three-parameter analysis of neutron signal correlation measurements
551
obtained experimentally are the background multiplets Bx(T) and the trigger multiplets N,( ,r) (x = 0; 1, 2, 3, - - ), defined by [8,9], respectively, Bx(T) = number of events with x signals inside K randomly triggered inspection intervals; Nx(T) = number of events with x signals inside inspection intervals T obtained during an observation time TM. Each signal triggers an inspection interval T . This leads to the frequencies bx (T)
(1)
Bx(T)1K,
=
a d n, (T)=Nx(T)INT,
(2)
where NT is the number of total counts collected during the measurement time TM . bx(T) and n x (T) are the probability counterparts to the frequencies bx(T) and n' (T). The factorial moments 00
Y_ =
x
l1
(x)bx(T)Y_ IL x
cc .) Mn(, x= ~L
= bt
( xtt ) bx (T) ,
(3)
00
(FA.)nx( T )
(i1,)nx(T),
(4)
x=~
as defined in eqs . (3) and (4) are the effective numbers of signal multiplets with t. signal events inside the observation intervals T . These effective numbers are generated by x >__ tt signal events inside randomly and signal-triggered intervals, and are referred to as randomly and signal-triggered factorial moments of order ,u, respectively . In ref . [9] recurrence formulas for the factorial moments M,, (,,) and Mn(u) were derived . Both moments are not normalized on ,u! in contrast to Mb(,,) and m . OA) of eqs . (3) and (4), respectively . With the identity Mb (P) and mn(,)
"'b(it)
_ Mn(W) , ~~
[he equivalent recurrence formulae of ref. [9] for the normalized factorial moments become /A-i
and
L
M n(q) M h(u-q) ,
M n(P)
=
Mb(O)
= 1 and
q=0
with mn(0)
= mn(O) = 1 .
The quantities m*(m) and m*(rn) are analytical expressions and depend on the characteristics of the test sample, the moderator-detector assembly and the operational conditions of the multiplet analyzer. The following assumptions are introduced : 1) the test sample has point geometry, 2) fast neutron multiplication only takes place in the test sample, 3) no neutrons return from the detector head to the test sample, 4) induced fissions occur at the same time as the primary neutron emissions, 5) the time response function of the moderator detector assembly is a pure exponential function with decay constant X,
552
D.M. Crfarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements
6) the primary neutron energy has no influence on the thermal neutron detection probability E, 7) the start of each observation interval -r of the trigger multiplets is delayed by the time T for each trigger signal, 8) no dead time losses of signals occur. With these conditions the following holds : mb( ") =w(NL)E"T[S« {(1-P)811,+va(r~) P P } +FSvs( w ) p ],
(7)
and mn( ") =f"E
"+1 T [Sa((1-P)so
Mb(1)
with W GL ) =
"-1
and
f=
K= O
_
(1-~1)(-1)x l K
e -AT( 1 -
" +v«(" +1) P P) +Fsvs( " +1) P ],
( 8)
-ATK
XrK
e-AT) .
(9)
For convenience the following quantity is introduced R( ") -E"TM[S«f (1-P)s1" +v~( ") P P} +Fsvs( ") p ] .
(10)
From eqs. (7) and (8) it then follows m b(") TM
__
m
n(" -1 )
NT for g> 1 .
(11)
The symbols used are : S. = (a, n) neutron emission rate of test item, FS = spontaneous fission late of test item, = probability that a neutron generates an induced fission event, P E = probability for detection of a neutron, T = observation interval, T = delay between trigger signal and start of observation interval r, A = fundamental mode decay constant of moderator-detector assembly, PMP) = probability for the emission of v fast neutrons per prompt fission caused by a primary neutron generated by reaction j (j = a or I then (a, n) reaction or induced fission ; j = s for spontaneous fission), = g.th factorial moment of the P,v(p) distribution V'(") (P) 1 for n v (12) V,I,ITP E (p JP), 11 0 for n v=" Eqs. (5)-(12) permit a determination of the spontaneous fission rate FS in the presence of (a, n) reactions and neutron multiplication . Analytical expressions [8] for P,,(p) lead to rather complex expressions even with the fast fission approximation . Moreover, recently analytical expressions were given in two independent publications [10,11] for the factorial moments PJ(" ) p of this probability distribution . Using these relations and the fission escape multiplication M where M=
1 -P
1 -PVt (I )
,
(13)
D. M. Cijarelh, W. Hage / Three-parameter analysis of neutron signal correlation measurements
553
the following expression for vl(u) (p) is obtained from ref . [10] or [11] after some algebraic operations : l'1(w) P - Mi"
E
k=1
a1PKM k-1 ,
(14)
with Y 1 YI 2 l( ) ( ) - 1 YI(1) Y 1 YI 3 l( ) ( )
;1(1) - 1
a131 =v1(3)
a122
=
a132 =
Y 2 Y1 2 l( Y ( ) - 2_ - 1 PI(1)
+i
J()p2(vI(1)
_ 1) 2,
yl(1) Y 1(2) vI(1)
-
1 ,
yl(1) 11(3)
_
Y I(1)
P
1(2) _ 1) + 2 (P
P
1(2)
4
2 v V l(1) 1(2) ( _ 1)2 , YI(1)
2
a133
_2
(1 5)
Y1(1)YI(2) =2 _ 1)2 . YI(1) -
vl(r') is the factorial moment of order u of the PI distribution for induced fission caused by primary fission neutrons. For j = s, vl(IA) is the factorial moment of order p, of the PS,, distribution for spontaneous fission of the test sample. Within the uncertainty of the existing data for the P distribution by fast neutron fission, it can be assumed that PI([')
(16)
= Y«(w)-
With eqs. (14) and (15) the interpretation of the neutron correlation technique requires as nuclear data only the moments of the P1 distribution of the isotopes present in the sample for spontaneous (j = s) and induced fission (j = a = I) processes . 2.1.
The factorial moments
of the P
distribution
For spontaneous fission reactions there exists a large number of publications from which the Ps, distributions for the various spontaneous fission isotopes can be taken. A recent data evaluation has been performed in ref . [13]. For induced fission events mainly data for thermal neutron induced fissions [13-15] exist for the various fissile isotopes present in fuel. There is one exception : Fr6haut et al. have measured the P,(E) distribution of 239Pu at various fission neutron energies E between 1 and 10 MeV . These data are at present evaluated at BNL [16]. In the analysis of the neutron signals the TI(,,) values for an average fission neutron spectrum are required. The average value of 00
YI(1) _
PP'
v=1
increases with neutron energy for the fissile isotopes leading to an even more pronounced increase of the moments with higher order p. The procedure adopted to overcome this difficulty was to plot v,(2) as a function of v,(1) and vl(3) as a function of vI(2) for the isotopes with measured P1 distributions from thermal fission . This resulted [13-16] in a parabolic dependence of vI(2) and v,(3) on PI(1) and v1(2),
D.M. Crfareld, W. Hage / Threeparameter analysis of neutron signal correlation measurements
554
7
7
Fm-257
Fm-257 I7 MeV
I le ,/
6 MeV I Î
Cf 252
5
`5 MeV Cf-250 ~4 MeV
5
4
Cm-244
3
U-233
4
Pu-238
2
~~Cm-248
I
3
Cm-246 Pu-239
Pu-239~
2
Cm-242
U-235
U-235
2.5
3.0
3.5 iN(~)
Fig. 1. v(2)
as a
Pu-242 I` Pu-236
4.0
function of v(1) .
oMeasured value o For Pu-239 at different energies
0--, U-238 1
~~ Cf-248 p--j1 MeV Pu-241 Cm-246
U-233
Pu-240
oMeasured value o For Pu-239 at different energies
2 MeV
Cm-244
Pu-238
Pu-236
\Pu-242
3 MeV
Cf-246
Pu-240
U-238
2.0
-250 '/Cf 4 MeV
4
P-2 MIV
1 MeV Pu-241
Cf-252 5 MeV
3 MeV Cf-246
o--6 MeV
2
3
Fig. 2. v(3)
4 as a
5
-0 function of v(2) .
6 v2)
respectively (figs . 1 and 2). Determining with a Watt fission spectrum f(E) [17] and the fission cross section of (E), the value of PI(1) Ef [18] for a fission spectrum is obtained for each fast fissioning isotope :
f
00
f(E)vf())(E)of(E)
f "of(E)of(E) dE
dE .
(17)
Inserting the value of PI(1) Ef into fig. 1 gives v,(2) Ef and using this value in fig . 2, vj(3) Ef is obtained as an approximation to a fission spectrum average. In many cases vj(1) E can be determined as the average value of the test sample with nuclear codes. From figs . 1 and 2 then follow the corresponding values for the higher factorial moments . The application of this procedure is valid as long as the neutron slowing down time inside the test item is much smaller than 1/X . 3. Formulae for typical cases In the following, models for an interpretation of correlation measurements with three unknown parameters are derived . The cases considered are : 1) Absolute determination of spontaneous fission rate FS with unknown (a,n) neutron source S. and unknown neutron multiplication factor M with known instrumental parameters E, X, T and T. 2) Relative determination of FS with unknown S. and M, using a reference sample . 3) Absolute determination of FS with unknown S. and E and known M, a, T and T.
D.M. Cifarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements
555
4) Absolute determination of Fs, E and M with known a, X, T and T, a being the ratio S,,/Ps(1)Fs . 5) Determination of M for fissile material with negligible spontaneous fission rate (FS = 0), unknown Sa and E . For all these methods the quantities R1 , R 2 and R 3 are determined from the randomly triggered multiplets applying eqs. (5) and (11) : R1 = R2 = R3=
TM
mb(1) - NT,
(18)
TM 1 2 [ mb(2) - 1 M(1)] , Tw(2)
(19)
Tw(1)
TM
(
[ mb(3)
Tw(3)
-
1 3 l mb(2)mb(1) + -mb(1)1 ,
w(2) = 1
w(1) = 1,
AT
(20)
(1 - e - X T ),
_1 _ 4 e -J" + e-2AT), w(3) = 2%~T (3 -
(21)
or from signal triggered multiplets with eqs. (6) and (11) : R1 = mn(0)NT = NT, R2=
N Î
IMn(1) -
( 22 ) ( 23 )
Mb(I) ],
R3 - f2 [ m n(2) - Mb(l) ( mn(1) - mb(1) )
jl
+
w(2) ~
1
Mb(l)
(24)
il
or by combining both types of multiplets [8]. The Ru (1 _< tt< 3) obtained from the measured data can be expressed by the instrumental data of the detection head and the physical and nuclear data of the test item using eqs. (10), (14) and (15) . They are: Rl = E[b11Fs+b12Sal,
(25)
+ b22Sa~,
(26)
R3 = E 3 [b31Fs+b32Sa1+
(27)
R2
= E
2 ~b21Fs
with b11 = TMMYS(1), M-1 l b21 = TMM2 'is(2) + - 11 vs(1) "1(2) I, v 1(1) ~ )%S(J)v21i(2) _ _
I,
b12 = TMM, b22 = TMMZ
632 = TM M
M-1 _ Y1(2), Y 1(1) -
3 M-1
M-1 +2 ~_1(3) Y 1 - 1 v1(1) ~ v1(1)
(28)
556
D.M. Cifarelh, W. Hage / Three-parameter analysis of neutron signal correlation measurements
Eqs. (18)-(28) are used in order to derive the interpretation models for the abovementioned five cases. In eq . (28) PI(I) is the spectrum average value of vj(I) Ef defined in eq . (17) . With known v,(I) the numerical values of PI(2) and v,(3) are obtained from figs . 1 and 2, respectively. Case 1 : absolute determination of FS , Sa and M. For the absolute determination of F, with an unknown (a, n) reaction rate Sa and neutron multiplication factor M, the instrumental data E, X, and nuclear data vs(I4) and vüw) of the test sample have to be known with sufficient precision. Eliminating in eqs. (25), (26) and (27) F, and Sa and using the coefficients of eq. (28), an algebraic equation of order 3 in M is obtained [19] : ao + a,M+ a 2 M 2 + a3 M3 = 0,
(29)
with E R2 IXl =E2
Ys(3) v1( 2) 2 -1 PI(I) ~ vs(2) v I(2) vs(3) 2v1(2) 1 _ RI R2_ IX2 -1 I - vI(3) I + E2 - 1 vI(I) vs(2) vI(I)
E
R 2 2v I(2) a3 = (30) - a2 . E 2 (vt(I) - 1)
The neutron multiplication factor M is determined from eq . (29) by taking the solution with the smallest possible value of M > 1 . With the known multiplication M a solution of eqs. (25) and (26) with respect to the two unknowns Fs and Sa gives: 1 TM
and S
-
I
I
R2 - RI V I(2)( M 1) E2M2 vs(2) EMvs(2)(vI(I)-1)
_ 1
RI
Ti,,I
EM
1 +
vs(I)v1(2)(M- 1 )
R2
-
vs(2)YI(I)-1)
(31)
,
vs(I)
(32)
E 2M2 vs(2)
The spontaneous fission neutron emission rate is mainly determined by R2, i.e . the measured signal doublets . As long as the (a, n) reaction rate is not the dominating contribution to RI, the influence of the singlets is less important due to the M-1 weighting. Larger errors will occur in Sa because the first and second term of eq . (32) are of the same order of magnitude. Case 2 : relative determination of FS .
For the relative determination of the spontaneous fission rate in the presence of (a, n) reactions and neutron multiplication, at least one well characterized reference sample must be available. Determining the R v values of the test sample and those of the reference sample R ß ,(0) (1 <-_ tt <- 3) the instrumental parameters E, X and T and T are eliminated completely if the factorial moments of randomly triggered intervals T are used (see eqs. (18), (19) and (20)). This is also true when combining the randomly and signal triggered factorial moments [8] . It is no longer true when using the signal triggered factorial moments. Applying eqs. (25), (26) and (27) for the test and reference samples, the following system of equations is obtained : r~,
=
R Rw~O)
[Fs (0)bt, I (0) +
Sa(0)bw 2(0)]
= Fs bl, I +
Sabw 2
1 < ~. < 3 .
(33)
D.M. Cifarelh, W. Hage / Threeparameter analysis
of neutron signal correlation measurements
557
b,,,(0) and b,,2(0) are the known coefficients of eq .
(28) of the reference sample with the known multiplication factor M(0), the known nuclear data P.,( ,, )(0) and vI(w)(0) and measurement time TM. The coefficients b,.I and b', 2 of the unknown test sample contain the unknown multiplication factor M and the known nuclear data vs(',) and Eliminating by means of eqs. (31) and (32) Fs and S., an equation of order 3 for the determination of the multiplication factor M is again obtained : 3 = 0. aô + a1M + a*M 2 + a *M 2 3
(34)
The coefficients a* now depend on rt which is proportional to the ratio R',/R,,(0) and to of the reference sample. They are : ao =
vs(3) ~
Ps(2)
-2 _
v1(2)
,
_1
vM)
v v s(3) t(2)
1
a* =ri -
and Sa(0)
-r3,
ai = r2
aj
Fs(0)
Ps (2)
2 vI(2) r2 _ -1 vI(I)
_v1(3)
2vu2) +r2 _ _1 , vt(I)
a*,
(35)
with ri =
R
(~) FS (0) Y1(I)CI(0) >
(36)
CI (0) = M(0) [1 + a(0)] ,
(37)
r2 - R
(38)
c2(o)
r3
F, (0) vs(2)C2 (0),
(0)
= MZ(o) f 1 + (M(o) -1)(1 + a(o)) R3
R3(0)
Ys(I)Y1(2) vI(I)
1
~,
(39)
(40)
Fs(0)vs(3)C3(~),
C3 (0) = M3 (0)I 1 + 2(M(o)
+(M(0) - 1)(1
-1) -
ys(2)"1(2)
Ys(3)( v I(I)
1)
-
+ a(0)) [M(o) - 1]
vs(I)vi(2)
v vs(3) ( I(l)
+
vs(I)VI(3) Ps(3) ( vI(I) -
With known multiplication factor M, the spontaneous fission and (a, n) reaction rates respectively, are given by : F -_
s
S =
- ri (M
r2
TMM ri TMM
TMM
vs(2)
~l
+
- 1)
V
Y I(2) s( I )
"s(2)(vI(I)-1)
v1(2) vs(2)(vI(I)-
1)
(M- 1) -
>
(41)
1) F,
and S.,
42
r2
_s(I)
Tm M 2 Ps(2)
(43)
558
D .M. Cifarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements
In the case that there are several reference or test samples with known Fs, Sa and known multiplication factor M, the constant CN of eqs . (37), (39) and (41) serve to correct the R,, values (1 < [L< 3) for these two contributions . The a and M corrected value of RA is: RF,c = e"Fsvs(w ) TM = Rf,/CF, .
(44)
RtLc has to lie within the error margins on a straight line with the given spontaneous fission rate
FD (or the z4° Pu-mass equivalents) as abscissa, provided that the limitations of the model are valid and the nuclear data correctly chosen . Case 3: absolute determination of FS with unknown S. and e.
A spontaneous fission and (a, n) neutron emitter surrounded by moderating material influences the neutron detection probability of the detector head. A part of the source neutrons is slowed down in this moderator and is absorbed in the thermal neutron filters of the detector head. Such conditions exist when monitoring small waste packages with moderating material like polythene, paper, rubber, etc. In this case a determination of F5 , S. and e is required assuming that the neutron multiplication is either negligible or known . First the neutron emission rate ratio S
(45)
a= _ a Ds(1)Fs
is determined forming the ratio R2
ß = R,R3 2 .
(46)
This ratio becomes independent of e for a point source and independent of Fs by using eq. (45). Replacing R1, R 2 and R3 in eq. (46) by eqs . (25), (26) and (27), the following solution of a quadratic equation in a is found : a=
B2 +4AC -B
(47)
2A
with 2 2
A = ßblibai - b22vs(1), B = ßb11( bal + b32vb(1)) - 2 b21b2A(l),
(48)
C = b21 2 - ßbllb3l .
The coefficients b,, (1 < i :< 3 and 1
( 49 )
is independent of e. It is used knowing the a-ratio for the determination of the spontaneous fission rate Fs: FS =
y(b21 + ab22vs(1 » Tmbil(1 +
a)2
(50)
.
With known a and FS the neutron detection probability e follows from eq. (25). It is: e
_
Ri
FSvsMMTM (1 + a)
.
(51)
D.M. Cifarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements
559
Where there is no neutron multiplication, the following relations are valid : M= 1, a=
z Ps (2)
-1,
s(l)vs(3)
Fs=Y
"s(z) v(1)(1+a)
2
> TM
and R1
e
.
Fsvs(1) (1 + a)TD,1
(52)
Case 4 .: absolute determination of FS , e and M with known a-ratio. In some cases the a-ratio of waste items with internal moderation is better known than the neutron multiplication factor M. This occurs especially when measuring fuel fabrication waste of known isotopic composition and no or a known contamination with isotopes having a large (a, n) reaction cross section. First a relation for the unknown neutron multiplication factor is obtained using eqs. (25), (26), (27) and (28), forming the ratio :
ß=
z Rz
(53)
RIR3 .
After some algebraic operations, a quadratic equation for M is found with the solution : M_
Q+
Q z -4PR 2P
(54)
'
where
z P = ß alc3 _ bz,
Q=2a z b z -ßa 1 b3 , R = ßalas _ a2z
(55)
and aI
= vs(1) (1 + a),
az = vs(z) bz
v1(1) -
= vs (2) - a z ,
a3 = vs(3) -
1
1+a
1 1 l2vs(z)vI(z) + (1 + a) + C3, v I(1) vs(1)PI(3)1 _
b3 _ vs(3) - a3 - C3,
C3 = 2(1 + a)
z Vs(l)VI(2)
( v1(1)
-
1)
z
(56)
Having determined M, Fs follows directly from eq . (50) . Case -5: no spontaneous fission primary neutrons. In some practical applications, fuel has no spontaneously fissioning isotopes but (a, n) primary neutrons . In these conditions the neutron multiplication M, the (a, n) reaction rate S. and the detection
560
D.M.
Cifarelli, W. Hage / Threeparameter analysis of neutron signal correlation measurements
probability e can be determined . This leads to the simple case that expression for M is obtained: M_
The ratios Sa =
y1(3) ( y1(1) -
ß
v2I(2)
1 -ß ß
1)
bll = b21 = b31
= 0. From the ratio ß an
+1.
(57)
and y serve to derive an equation for the determination of Sa:
_Y
TM
ß 1-~
With known Sa and
v 1 (3) YI(2)
M,
(58)
.
the detection probability e follows simply from eq . (25) .
4. Test of theory by experimental data 4.1. e
not modified by sample
Part of the derived analytical expressions was tested with experimental data using a 8.45 g metallic Pu sample exposed in a hollow cylindrical detector head of 1 m height and 32 3He proportional counters . By means of a computerized registration system [20], the factorial moments of the frequencies to have x signals inside, randomly and signal triggered observation intervals T were measured. Table 1 gives the average values and the standard deviations of these moments obtained in eight different runs, each with a measurement time of 8 min. The predelay T for the three observation intervals Tl = 25 .6 ,us, T2 = 51.2 Ets and T3 = 102.4 ,us was 6.4 [,s in each case. The decay time of the detector head was 51.3 ,us and its neutron detection probability e = 0.27. The latter quantity [20] was determined in a separate experiment from an unknown 252 Cf spontaneous fission neutron source, using the nuclear data for v,(,) and Ps(2) and R, and Table 1 Moments of randomly and signal-triggered observation intervals
ORDER OF MOMENT m 0 1 2 3 4
Moments Mb(m)of Randomly Triggered Observation Intervals i - 25 .6 /is 1 .0000 6 .61483 3 .74852 1 .67851 5 .2053
10 -3 10 - " 10 -5 10 -7
t 3 .9 t 2 .25 t 4 .85 t 1 .35 t 1 .5
i - 102 .4 its
i- 51 .2 its 10 -5 10 -5 10 -6 10 -6 10 - '
0 .99999 1 .32301 1 .31250 1 .02862 6 .80980
10 10 10 10
-2 -3 -4 -6
t 7 .99 t 6 .97 t 2 .52 t 7 .96 ± 1 .79
2
10 -1 10 -5 10 -1 10 -6 10 -6
1 .0000 2 .64591 4 .12235 4 .88466 4 .84310
10 -2 10 -3 10 -4 10-5
t 1 .47 t 1 .36 t 6 .10 t 2 .54 t 6 .60
10 10 - ' 10 -5 10 -5 10 -6
Moments m n (m) of Signal Triggered Observation Intervals -r 0 1 2 3
1 .00004 9 .52149 10 -2 5 .85962 10 -3 4 .27510 10 - " NT = 258.4 s -'
t t t t
2 .10 1 .59 4 .47 9 .06
10 -3 10 -3 10 -4 10 -5
NB = 2.5 s -1
1 .00003 1 .54135 10 -' 1 .53349 10 -2 1 .25952 10 -3
t 2 .27 10 -3
t 1 .84 10 -3 t 7 .01 10 -4 f 1 .63 10 -4
9 .99779 2 .17687 3 .09289 3 .48763
(count rate with and without sample).
10 - ' 10 -' 10 -2 10 -3
t t t t
2 .32 3 .50 2 .77 1 .04
10 -3 10 -3 10 -3 10 -3
D.M. Cifarelli, W. Hage / Three parameter analysis of neutron signal correlation measurements
561
Table 2 Determined M -1, vs(,) F, and Sa
RANDOMLY TRIGGERED INTERVALS T 2 = 51 .2 lts Ts --- 102 .4 l.ts
i, = 25 .6 gs
M-1
0 .0183 + 0 .0036
0 .0219 ± 0.0028
vs(,)Fs[1/s] Sa[1/s1
0 .5
6.6 ± 13
930 ± 6
920 , 6
0 .0206 ± 0.0026 924 ± 6 5 .1
SIGNAL TRIGGERED INTERVALS M-1
0 .0256 + 0.0043
vs(,)F,[l/s] Sa[1/s1
4 .3
920 ± 10
0 .0223 + 0 .0015 923 ± 4
0 .0180 ± 0.0021 930 + 5
4.7 ± 9
0.4
SAMPLE DATA M-1
vs(,)Fs [1/s] Sa[1/s]
0 .0240
REMARK TI MOC
918 ± 24 0
values derived from signal and randomly triggered observation intervals. In order to obtain the neutron multiplication M, the spontaneous fission rate FS and the (a, n) neutron emission rate S«, the formulae of case 1 are used, in connection with the factorial moments Mb(,) (j = 1, 2, 3) of table 1. First are determined the quantities R, using eqs . (18), (19) and (20), calculating w(j) (j = 1, 2, 3) according to eq. (21). With known R ., e, vt(t) and vs(,), the coefficients a, (i = 0, 1, 2, 3) of an equation of third degree in M are obtained . For each gate length the neutron multiplication factor M is determined from the root of eq. (29) for M> 1 . With known M, the spontaneous fission rate FS, and the (a, n) neutron emission rate S. follow from eqs. (31) and (32), respectively . A similar procedure was applied using as measured data the factorial moments m,,(,,) (p = 0, 1, 2) of the three signal triggered observation intervals T. Table 2 gives the results for M- 1, i~,), FS and S. for both measurement methods and the three observation intervals. The uncertainties quoted are the standard deviations of the eight runs each analyzed individually . The results for the spontaneous fission neutron emission rates agree very well with the values known from the test sample [21]. The uncertainty in PS 1 F of the test sample is mainly due to the uncertainty of the spontaneous fission neutron half-life of 2 Pu [22]. The determined quantities are within their error limits independent of the duration of the observation intervals and independent of the type of trigger interval used . The smallest statistical error appears to be for observation intervals T close to the decay time of the detector head . R 2 and R 3 can be obtained by combining the factorial moments of both signal and randomly triggered observation intervals. In this case the measured Mb(l) and Mb(2) [8] are taken to calculate R2 and R 3 . The values obtained for M, v, (1) F, and S. lead to almost the same numerical values as found with the signal triggered factorial moments. R2
4.2. e modified by test sample Monitoring small waste packages of material containing hydrogen, the Pu mass is frequently so small that the neutron multiplication factor M becomes close to 1 . In this case the spontaneous fission rate Fs , the (a, n) reaction rate S« and the detection probability e can be determined according to the formulas derived for case 3 by putting M = 1 . However, if the Pu content is of the order of grams, then the determination of the spontaneous fission neutron emission rate v t1 ,F, depends strongly on the neutron multiplication factor M.
562
D .M. Cijarelli, W. Hage / Three-parameter analysis of neutron signal correlation measurements 1000
vsi,~Fs 900
Se v,,, > Fs 800
700
NEW,"-
0 .25
1
1 .01
1 .02 -i M
Fig. 3. Neutron emission rate as a function of M .
1 .03
0 .2
1
1 .01
1 .02 10
M
1 .03
Fig. 4. Detection probability c as a function of M .
Such conditions are presented in fig. 3 for the 8.45 g metallic Pu sample. This figure is based on the experimental data obtained from randomly triggered factorial moments with an observation interval ,r = 51 .2 ps . Not knowing the characteristics of the sample, it is only possible to state that the multiplication factor is below 1 .0219 since otherwise S. would become negative. Knowing the isotopic composition of the fuel it is recommendable to determine an approximate value of the a-ratio, and to calculate with the formulas of case 4, FS , M and e . By this type of analysis the uncertainty in the determination of v5(1) FS can often be kept within smaller limits . Fig. 4 gives the neutron detection probability as a function of M for the same conditions as in fig. 3. The same tendency of vs(l) F and e as a function of M has been found in ref. [23] when analyzing the frequency distribution of a signal pulse train for randomly and signal triggered observation intervWls.
5. Conclusions The interpretation model has been tested successfully with a small Pu-metal sample. In the near future, tests will be carried out on the validity of the point model and the model for neutron multiplication with smaller and larger PUOZ samples. Dead time corrections of the factorial moments are still a problem for further theoretical studies.
Acknowledgements Many thanks are due to Mr . L. Bondar for providing some measured data . Much appreciated were discussions with Mr . N . Barrett, Mr . L. Bondar, Mr . R. Dierckx and Mr . A. Prosdocimi . Gratefully acknowledged is the assistance of Mrs. K. Caruso for performing the calculations and preparing the figures, and to Mrs. G. Tenti and Mrs. M. Van Andel for typing the manuscript .
References [1] G. Birkhoff, L. Bondar, J. Ley, R. Berg, R . Swennen and G . Busca, EUR 5158e, Commission of the European Communities, Joint Research Centre, Ispra (1974).
D.M. Cifarelle, W. Hage / Threeparameter analyses of neutron signal correlation measurements
563
[21 K. B61mel, KfK-2203, Kernforschungszentrum Karlsruhe (1975) . [31 N . Ensslin, M.L . Evans, H .O. Menlove and J .E . Swansen, Nucl. Mater . Man . 7 (1978) 43 . [41 E.J . Dowdy, C.N . Henry, A.A . Robba and J .R. Pratt, Proc . Int. Symp . on Nuclear Material Safeguards, Vienna, SM-231/69 (IAEA, Vienna, 1978) . [51 M .O . Deighton, Nucl . Instr. and Meth . 165 (1979). [6] L. Bondar, Proc . Int . Meeting on Monitoring of Plutonium Contaminated Waste, Ispra, Commission of the European Communities, Ispra (1979) . [7] N . Shenhav, Y. Segal and A. Notea, Nucl . Sci . Eng. 80 (1982) 61 . [8] R. Dierckx and W . Hage, Nucl. Sci. Eng. 85 (1983) 325 . [9] W. Hage and D .M . Cifarelli, Nucl. Sci. Eng . 89 (1985) 159 . [101 K . Böhnel, Nucl . Sci . Eng. 90 (1985) 75 . [11] W . Hage and D .M . Cifarelli, Nucl . Instr . and Meth . A236 (1985) 165 . [12] M .S. Krick and J.E . Swansen, Nucl. Instr. and Meth . 219 (1984) 385 . [131 M .S . Zucker and N.E . Holden, 6th ESARDA Symp. on Safeguards and Nuclear Material Management, Venice, Italy (1984). [141 J .W . Boldeman and M .G. Hines, Nucl . Sci. Eng . 91 (1985) 114. [15] N .E. Holden and M .S . Zucker, 7th ESARDA Symp . on Safeguards and Nuclear Material Management, Li6ge, Belgium (1985) . [16] M .S . Zucker (BNL), private communication (1985) . [171 Proc . Consultants, Meeting on Prompt Fission Neutron Spectra, IAEA, Vienna (1971) . [18] F . Manero and V.A. Konshin, At . En . Rev . 10 (1972) 637 . [19] W. Hage and D .M . Cifarelli, Proc. 26th INMM Ann. Meeting Albuquerque (1985). [20] L . Bondar, Joint Research Centre, Ispra, private communication (1982). [21] J .F . Gueugnon and K . Richter, European Institute for Transuranium Elements, Karlsruhe, Fabrication report K0282049 (1982) . [22] C. Budtz-Jergensen and H.H . Knitter, Nucl . Sci . Eng . 79 (1981) 380 . [23] L . Bondar et al ., ESARDA Proc, Venice, Italy (1984) .