Equations for neutron leakage multiplicity correlations were derived up to septuplets using a ... rates of multiplicity correlations for U-Pu mixed dioxide increased ...
Characteristics of Neutron Leakage Multiplicity Correlations up to Septuplets Derived Algebraically Takashi HOSOMA Technology Development Department Tokai Reprocessing Technology Development Center Sector of Decommissioning and Radioactive Waste Management Japan Atomic Energy Agency (Received Abstract Equations for neutron leakage multiplicity correlations were derived up to septuplets using a probability generating function, in which a spontaneous fission and a (α,n) reaction were considered as a starting event of a neutron leakage chain resulting from a fission cascade. The coefficients were too large, thus a new definition ηm = νm /m ! was introduced where νm was the conventional factorial moment. Count rates of multiplicity correlations for U-Pu mixed dioxide increased acceleratedly depends on leakage multiplication and detection efficiency, which was more significant in higher order multiplicity. This tendency results in reverse of normal order of counts, i.e. doubles becomes smaller than septuples, in a sample of large leakage multiplication. Similar tendency had been reported for Pu metal in the United States, therefore obtained tendency was compared to the report by converting leakage multiplication to effective 240Pu mass supposing a sphere. An example of leakage multiplication and detection efficiency was shown to set the order of counts within an appropriate range. (158 words) Keywords: probability generating function; neutron leakage multiplication; fission cascade; multiplicity; neutron coincidence counting; U-Pu mixed dioxide; materials accounting; safeguards. Introduction Equations for neutron leakage multiplicity correlations up to triplets are frequently referred in papers for neutron coincidence/multiplicity assay system, and they have been used for nuclear materials accounting and for safegurds to determine plutonium weight. Several assay systems for small and unshaped U-Pu (U/Pu = 1) mixed dioxide powder (ex. in waste and in glovebox) have been installed and improved 1, 2, 3 at the Plutonium Conversion Development Facility (PCDF) adjacent to the Tokai Reprocessing Plant. There are two kinds of correlations, i.e. true correlations resulting from one primary fission followed by induced fissions and accidental correlations resulting from independent events. The former was called ‘coupled pair’ and the latter ‘accidental pair’ regarding doublets by Hoffmann 4 in 1949. The frequently referred equations are associated with the true correlations, the derivation is explained in the paper by Hage and Cifarelli 5 in 1985. Another derivation up to doublets is explained in the paper by Croft 6 in 2012, however it is difficult to apply these methods to heigher correlations greater than quadruplets. There is another way using probability generating function (PGF) discussed by Böhnel 7 in 1985, which might be applied to heigher correlations but not so easy to understand nor to calculate manually. In this paper, derivation by PGF was rewritten using standard expressions of stochastic process then equations up to septuplets were derived algebraically assisted in doing mathematics by Mathematica. The purpose is to investigate a new way to observe/analyze the distorted profile of neutron multiplicity distribution comes from overlap between two spontaneous fission nuclei ex. Pu and Cm . It is sure that counting technique for high order correlations has not been established because of large uncertainty of counting statistics due to thermalization and smal leakage multiplication of a sample. However, it could be possible in theory to observe the correlations based on the characteristics of the equations, if sample could have a large leakage multiplication and counting technique could be established.
Methods Preliminary Information 8 Probability generating function GX for descrete stochastic variable X (non-negative integers) is defined as a function of continuous variable |z| < 1 : def
GX (z) = z X =
∞ �
(1)
z n PX (X= n)
n=0
�∞ where PX is probability mass distribution of X, thus n=0 PX (X= n) = 1 . As an example of PX , neutron multiplicity distribution Pν is suitable. The k th derivatives of GX regarding z are expressed as:
GX (z) =
∞ �
z n PX (X= n) = PX (X= 0) +
n=0 �
GX (z) =
∞ �
∞ �
z n PX (X= n)
n=1
nz
n−1
PX (X= n) = 1 PX (X= 1) +
n=1 ��
GX (z) =
∞ �
∞ �
n z n−1 PX (X= n)
(2)
n=2
n(n−1) z n−2 PX (X= n) = 2 PX (X= 2) +
n=2
∞ �
n(n−1) z n−2 PX (X= n)
n=3
Therefore, PX is expressed using the k th derivative of GX at z = 0 as: (k)
(k)
GX |z=0 = k! PX (X= k)
→
PX (X= k) =
GX |z=0 k!
(3)
The k th factorial moment is expressed as the k th derivative of GX at z = 1 as: (k)
GX |z=1 =
∞ �
n=k
n(n−1) · · · (n−k+1) PX (X= n) = X(X−1) · · · (X−k+1)
(4)
It is possible to express PGF using the k th factorial moments (proved by binomial expansion): GX (z) =
(k) ∞ � MP X ( z − 1 )k k!
k=0
(k) def
(k)
where MPX = GX |z=1
(5)
For independent descrete stochastic variables X1 , X2 , · · · XN , the composite PGF is given as a product of each PGF as : GX1 +X2 +···XN (z) = GX1 (z) GX2 (z) · · · GXN (z)
(6)
For example, N is the number of eggs spawned and Xi = {0, 1} has the distribution PXi of hatching and growth. In case of no difference between eggs ( X1 , X2 , · · · XN are assumed to be equivalent), the PGF for S = X1 + X2 + · · · + XN is given as: GS (z) = [ GX (z) ]
N
= [ GX (z) ]
N
= GN (GX (z))
(7)
This example could easily be associated with a fission cascade where N is the number of neutrons yield from an induced fission according to Pν and Xi has the distribution of staying alive without leakage nor capture and causing a fission. However, we have to focus attention not on ‘neutrons multiplication’ inside a sample but on ‘leakage multiplication’ to derive equations for neutron multiplicity correlations.
If X is a conditional descrete stochastic variable corresponding to condition j having the probability pj , the total PX and GX are expressed as: �
PX (X = n) =
all j
GX (z) =
�
�
where
pj PX (X = n | j)
(8)
pj = 1
all j
(9)
pj GX (z) | j
all j
PGF for a neutron leakage chain Figure 1 shows a neutron leakage chain resulting from a fission cascade where a spontaneous fission or a (α,n) reaction is the starting event. The average number of spontaneous fission neutrons νs1 and the average number of induced fission neutrons νi1 are used where the subscript ‘1’ indicates a singlet that means neutrons are counted individually, whereas a doublet ( νs2 , νi2 ) used in coincidence counting means neutron pairs come from spontaneous or induced fission. The usage of symbols are normal in conventional papers regarding neutron coincidence/multiplicity counting, together with the ratio of the number of neutrons yields from (α,n) reaction to the number from spontaneous fission αr , the leakage probability l and the induced fission probability p in Figure 1:
l
p
l
if
νs1 or αr νs1
+
νs1 or αr νs1
This chain stops when all neutrons leak.
if
νi1 ×p νi1
l
p
+
νi1 ×p νi1
•••
=
(νs1 or αr νs1 ) × MT
Figure 1 Neutron leakage chain resulting from a fission cascade Neutron leakage multiplication ML and total multiplication MT are defined as a: ML = l MT = l [1+p νi1 +(p νi1 )2 . . . ] =
l 1−p−pc 1−p = ≈ 1−p νi1 1−p νi1 1−p νi1
(10)
where pc is neutron capture probability, ex. by (n,γ) reaction, however it is not so large (several percent of p ) thus l ≈ 1−p is normally supposed. Now, let us suppose the PGF of leakage neutrons in Figure 1 when the chain starts from one neutron (not one fission/reaction). From the equations (1), (7) and (9), the PGF for a neutron is written as:
Gh (z) = (1 − p)
1 �
n=1
zn 1 + p
� ∞ �
n=0
Gi ( Gh (z) )
�� n
�
[ Gh (z) ] Pνi (νi = n)
(11)
where Pνi is neutron multiplicity distribution of induced fission and νi = νi1 . The first term on the right side of the equation means direct leakage, thus X = {1} and PX = 1 . The second term means leakages through induced fission as a composite of leakages in each νi , which results in the nest of the PGF for a neutron. The subscript ‘h ’ comes from the one in Böhnel's equation (9) where the PGF is explained as ‘the PGF for the number of neutrons of this first and all successive generations that leave a system’.
a.
ML − 1 p ML − 1 ML νi1 − 1 = = p MT , = p, = MT are useful transformations. νi1 − 1 1 − p νi1 ML νi1 − 1 νi1 − 1
Solving k th derivatives The equation (11) is simplified to: Gh (z) = (1 − p ) z + p Gi (Gh (z))
(12)
The k th derivatives are solved algebraically assisted in doing mathematics by Mathematica: �
Gh =
1−p � 1 − p Gi
��
Gh =
�� (1 − p)2 p Gi � 3 (1 − p Gi )
(3) Gh
(1 − p)3 p = � (1 − p Gi )4
(4) Gh
(1 − p)4 p = � (1 − p Gi )5
(5)
Gh =
�
�
(3) Gi
(4) Gi
�� p 2 + � 3 (Gi ) 1 − p Gi
�
�� �� p p2 (3) 10 G + G + 15 (Gi )3 � � i i 2 1 − p Gi (1 − p Gi )
�
� � (1 − p)5 p � (5) p (4) �� (3) 2 15 G G G + 10 (G + ) � � i i i i (1 − p Gi )6 1 − p Gi
�� �� p2 p3 (3) 2 105 G 105 (Gi )4 + (G ) + � � i i 2 3 (1 − p Gi ) (1 − p Gi )
(6)
Gh =
� � (1 − p)6 p � (6) p (5) �� (4) (3) 21 G G G + 35 G + G � � i i i i i (1 − p Gi )7 1 − p Gi � � �� p2 (4) (3) 2 �� 2 210 G + (G ) + 280 (G ) G � i i i i (1 − p Gi )2
�
�� �� p4 p3 (3) 3 1260 G (G ) + 945 (Gi )5 + � � i i 3 4 (1 − p Gi ) (1 − p Gi )
(7)
Gh =
(13)
�
� � (1 − p)7 p � (7) p (6) �� (5) (3) (4) 2 28 G G G + 56 G + G + 35 (G ) � � i i i i i i (1 − p Gi )8 1 − p Gi � � �� p2 (5) (4) (3) �� (3) 3 2 378 G (G ) + 1260 G G + G + 280 (G ) � i i i i i i (1 − p Gi )2 � � �� �� p3 (4) (3) 2 3 2 3150 G (G ) + 6300 (G ) (G ) + � i i i i (1 − p Gi )3 � �� �� p4 p5 (3) 4 6 17325 Gi (Gi ) + 10395 (Gi ) + � � (1 − p Gi )4 (1 − p Gi )5
The following transformations are applied later to the above equations: (k)
Gi |z=1 = νi (νi − 1)(νi − 2) · · · (νi − k + 1) = νik
(14)
1−p 1−p = ML � = 1 − p νi1 1 − p Gi
(15)
p p ML − 1 def = MLν = � = 1 − p νi1 νi1 − 1 1 − p Gi
(16)
Rewrite the PGF considering the source of a neutron The starting event of Figure 1 is a spontaneous fission or a (α,n) reaction. The PGF for a neutron is rewritten using Gs (z) for spontaneous fission where Pνs is used instead of Pνi as: GH (z) =
αr νs1 1 Gh (z) + Gs (Gh (z)) 1 + αr νs1 1 + αr νs1
(17)
where the subscript ‘H ’ comes from the one in Böhnel's equation (13). The coefficients of the first and the second terms on the right side of the equation, i.e. αr νs1 /( 1 + αr νs1 ) and 1/( 1 + αr νs1 ) means probabilities of a neutron from a (α,n) reaction or a neutron from a spontaneous fission, because (α,n) reaction occurs αr νs1 times more frequently than spontaneous fission. Same as the equation (14), the next ransformation is applied later to the above equation: G(k) s |z=1 = νs (νs − 1)(νs − 2) · · · (νs − k + 1) = νsk
(18)
The k th derivatives are too long to be shown here, however it is possible to enclose the right side of each derivative by the term (1 − p)k (1 + αr νs1 )−1 (1 − p νi1 )1−2k after the equations (13), (14) and (18) are applied. Finally, the equations (15) and (16) are applied to eliminate 1−p and p , followed by replacing (ML − 1) /(νi1 − 1) by MLν to express equations in short form. Conventional equations for S, D and T and matching probability mass distribution of leakage neutrons The conventional equations for neutron multiplicity correlations S, D and T , i.e. counts per second of singles, doubles and triples are shown below for a spontaneous fission and (α,n) reactions. In these equations, the number of spontaneous fissions per unit mass and time Fp , effective mass of spontaneous fission nuclei (238Pu, 240Pu and 242Pu) meff , counting efficiency �n (counts per neutron), characteristic values depend on counter setting fd / ft , and well known factorial moments νs1 , νs2 , νs3 , νi1 , νi2 , νi3 defined in the equation (22) are used as: (19)
S = Fp meff �n (1 + αr ) νs1 ML Fp meff �2n fd D= 2
�
νs2 + νi2 (1 + αr ) νs1
Fp meff �3n ft T = 6
�
νs3 +
where
def
νs/i m = m !
max �
�
ML2
ML − 1 [ (1 + αr ) νs1 νi3 + 3 νs2 νi2 ] νi1 − 1 � � �2 ML − 1 + 3 (1 + αr ) νs1 νi2 2 ML3 νi1 − 1
�
νs/i =m
ML − 1 νi1 − 1
νs/i m
�
Pν = νs/i (νs/i −1) (νs/i −2) · · · (νs/i −m+1)
(20)
(21)
(22)
To compare with these equations, it is necessary to express the probability mass distribution of leakage neutrons derived from PGF for a spontaneous fission event. Therefore, the equations are: (k)
G |z=0 Pm (m= k) = (1 + αr νs1 ) H k!
(23)
Obtained probability mass distribution of leakage neutrons for a spontaneous fission Results of the equation (23) up to k = 7 is enclosed by MLm /m ! which is same as the conventional equations. The terms in the enclosure are collected by νs1 , νs2 , νs3 · · · followed by collection by 2 3 MLν , MLν , MLν · · · . The final expressions are shown as: (24)
Pm (m= 1) = ML (1 + αr ) νs1 Pm (m= 2) =
ML2 [ νs2 + νi2 MLν (1 + αr ) νs1 ] 2
Pm (m= 3) =
� � � ML3 � 2 νs3 + 3 νi2 MLν νs2 + νi3 MLν + 3 νi2 2 MLν (1 + αr ) νs1 6
(25) (26)
These three equations are exactly the same as conventional equations (19), (20) and (21) in the condition that the term Fp meff �m n (1/fd /ft ) are removed to pick probability up for a spontaneous fission, though P the sum of m is not one (not normalized). The final equations for higher order correlations have very large coefficients, so the new ‘combination’ based definition transfromable from the factorial moment is proposed as the equation (27), which is the expectation of receiving m from ν with distribution Pν put omnidirectionaly from a fission shown in Figure 2:
def
ηs/i m =
max �
�
νs/i =m
νs/i m
�
Pν =
νs/i (νs/i −1) (νs/i −2) · · · (νs/i −m+1) m!
Sample
(27)
Detector
Neutrons from a fission
Figure 2 ‘Combination’ based definition transfromable from the conventional factorial moment Obtained distribution derived algebraically from PGF for m ≥ 4 for the first time will be right: ML − 1 def = MLη = MLν ηi1 − 1
(28)
Pm (m= 1) = ML (1 + αr ) ηs1
(29)
Pm (m= 2) = ML2 [ ηs2 + ηi2 MLη (1 + αr ) ηs1 ]
(30)
� � � � 2 Pm (m= 3) = ML3 ηs3 + 2 ηi2 MLη ηs2 + ηi3 MLη + 2 ηi2 2 MLη (1 + αr ) ηs1
(31)
Pm (m= 4) = ML4 [ ηs4 + 3 ηi2 MLη ηs3 � � 2 + 2 ηi3 MLη + 5 ηi2 2 MLη ηs2 � � � 2 3 + ηi4 MLη + 5 ηi3 ηi2 MLη + 5 ηi2 3 MLη (1 + αr ) ηs1
(32)
Pm (m= 5) = ML5 { ηs5 + 4 ηi2 MLη ηs4 � � 2 + 3 ηi3 MLη + 9 ηi2 2 MLη ηs3 � � 2 3 + 2 ηi4 MLη + 12 ηi3 ηi2 MLη + 14 ηi2 3 MLη ηs2 � � � 2 + ηi5 MLη + 6 ηi4 ηi2 + 3 ηi3 2 MLη � � 3 4 + 21 ηi3 ηi2 2 MLη + 14 ηi2 4 MLη (1 + αr ) ηs1
(33)
Pm (m= 6) = ML6 { ηs6 + 5 ηi2 MLη ηs5 � � 2 + 4 ηi3 MLη + 14 ηi2 2 MLη ηs4 � � 2 3 + 3 ηi4 MLη + 21 ηi3 ηi2 MLη + 28 ηi2 3 MLη ηs3 � � � 2 + 2 ηi5 MLη + 14 ηi4 ηi2 + 7 ηi3 2 MLη � 3 4 + 56 ηi3 ηi2 2 MLη + 42 ηi2 4 MLη ηs2 � 2 + ηi6 MLη + ( 7 ηi5 ηi2 + 7 ηi4 ηi3 ) MLη � 3 � + 28 ηi4 ηi2 2 + 28 ηi3 2 ηi2 MLη
(34)
� � 4 5 + 84 ηi3 ηi2 3 MLη + 42 ηi2 5 MLη (1 + αr ) ηs1
Pm (m= 7) = ML7 { ηs7 + 6 ηi2 MLη ηs6 � � 2 + 5 ηi3 MLη + 20 ηi2 2 MLη ηs5 � � 2 3 + 4 ηi4 MLη + 32 ηi3 ηi2 MLη + 48 ηi2 3 MLη ηs4 � � � 2 + 3 ηi5 MLη + 24 ηi4 ηi2 + 12 ηi3 2 MLη � 3 4 + 108 ηi3 ηi2 2 MLη + 90 ηi2 4 MLη ηs3 � 2 + 2 ηi6 MLη + ( 16 ηi5 ηi2 + 16 ηi4 ηi3 ) MLη � 3 � + 72 ηi4 ηi2 2 + 72 ηi3 2 ηi2 MLη
(35)
� 4 5 + 240 ηi3 ηi2 3 MLη + 132 ηi2 5 MLη ηs2 � � � 2 + ηi7 MLη + 8 ηi6 ηi2 + 8 ηi5 ηi3 + 4 ηi4 2 MLη � � 3 + 36 ηi5 ηi2 2 + 72 ηi4 ηi3 ηi2 + 12 ηi3 3 MLη � � 4 + 120 ηi4 ηi2 3 + 180 ηi3 2 ηi2 2 MLη � � 5 6 + 330 ηi3 ηi2 4 MLη + 132 ηi2 6 MLη (1 + αr ) ηs1
Multiplicity correlations are obtained by multiplying Fp meff �m n (1/fd /ft · · ·/f7 ) to these equations. The word ‘correlation’ is used because probability mass distribution of leakage neutrons is observed like a correlation of neutrons on the time axis of coincidence/multiplicity counter. The author thinks that the new definition (27) is preferable because it is physically natural. It is also good that the term 1 /m ! in equations (20), (21), (25), (26) are eliminated.
Results Change in the multiplicity correlations as a function of m With reference to the measurement conditions to count S, D and T of clean U-Pu mixed dioxide powder at PCDF, the parameters Fp , �n , fd , ft · · ·f7 and αr were supposed to be 474 g-1s-1, 0.4, 0.67, (0.67)2 … (0.67)6 and 0.7, respectively. For ηs/i m , 240Pu and 239Pu for 2 MeV neutron (due to little moderation) were selected because of their large contribution on neutron yield, then ηs/i m were calculated using Pν consolidated/proposed by Zucker and Holden 9, 10 in 1984 - 1986. They are shown in Table 1 and 2. 240
Pu and 239Pu (2 MeV neutron)
Table 1 Pν of ν=0
1
2
3
4
5
6
7
0.0632
0.2320
0.3333
0.2528
0.0986
0.0180
0.0020
0.0006 a
Pu (2 MeV) 0.0063 0.0612 0.2266 0.3261 0.2588 11 a. Estimated by Gaussian fit of the distribution based on Terrell in 1957.
0.0956
0.0225
0.0026
240
Pu
239
Table 2 ηsm of
240
Pu and ηim of 239Pu (2 MeV neutron)
m=1
2
3
4
5
6
7
ηsm
2.1534
1.8933
0.8672
0.2186
0.0300
0.0020
0.0001
ηim
3.1591
4.1058
2.8583
1.1653
0.2852
0.0407
0.0026
Here, the multiplicity correlations for 4 ≤ m ≤ 7 are named Qr , Qt , Sx and Sp (quadruples, quintuples, sextuples and septuples). Obtained multiplicity correlations for meff = 1 g are shown in Figure 3 as a function of ML . The values are shown in Table 3. Counts�s 10 000.
ML = 1.6 ML = 1.5
1000.
ML = 1.4 100.
ML = 1.3
10.
ML = 1.2
1.
ML = 1.1 0.1
0.01
0.001
ML = 1.0 0.0001
0.00001 S
D
T
Qr
Qt
Sx
Sp
1
2
3
4
5
6
7
m
Figure 3 Multiplicity correlations for one gram spontaneous fission source ( �n = 0.4, αr = 0.7, fd = 0.67 )
Table 3 Multiplicity correlations for one gram spontaneous fission source ( �n = 0.4, αr = 0.7, fd = 0.67 ) ML =1.0 ML =1.1 ML =1.2 ML =1.3 ML =1.4 ML =1.5 ML =1.6
S (m = 1)
D (m = 2)
T (m = 3)
Qr (m = 4)
Qt (m = 5)
Sx (m = 6)
Sp (m = 7)
694 764 833 902 972 1,041 1,111
96.3 159 241 342 466 615 790
11.8 42.4 102 205 371 621 984
0.806 12.5 51.7 150 363 776 1,519
0.0306 4.23 29.5 124 399 1,087 2,628
0.00064 1.54 18.1 109 470 1,632 4,871
0.000004 0.603 11.7 101 579 2,563 9,450
Italic face indicates the minimum value of each ML . A particular trend is found that m corresponds to the minimum changes from 7 → 4 → 2 according to the increase of ML , in other words, the order of magnitude changes according to ML . If αr = 5 (ex. high impurities sample), values become larger but the trend does not change as shown in Table 4, though αr is a term constituent of the probability mass distribution. It should be noted that the equations are reduced to (1 + αr ) ηs1 and ηs2 , ηs3 , · · · at ML = 1 , which results in the same values at ML = 1 in Table 3 and 4 excluding S . The trend does not change but shifts to lower ML regarding the change of fd as shown in Table 5, which is a corollary of the equations. Table 4 Multiplicity correlations for one gram spontaneous fission source ( �n = 0.4, αr = 5.0, fd = 0.67 ) ML =1.0 ML =1.1 ML =1.2 ML =1.3 ML =1.4 ML =1.5 ML =1.6
S (m = 1)
D (m = 2)
T (m = 3)
Qr (m = 4)
Qt (m = 5)
Sx (m = 6)
Sp (m = 7)
2,450 2,695 2,940 3,185 3,430 3,675 3,920
96.3 268 498 796 1,168 1,621 2,164
11.8 76.8 223 496 955 1,673 2,739
0.806 23.1 114 365 939 2,096 4,240
0.0306 7.83 65.1 301 1,032 2,940 7,342
0.00064 2.85 39.9 265 1,216 4,414 13,617
0.000004 1.10 25.7 245 1,499 6,940 26,442
Table 5 Multiplicity correlations for one gram spontaneous fission source ( �n = 0.4, αr = 0.7, fd = 0.85 ) ML =1.0 ML =1.1 ML =1.15 ML =1.2 ML =1.3 ML =1.4 ML =1.5 ML =1.6
S (m = 1)
D (m = 2)
T (m = 3)
Qr (m = 4)
Qt (m = 5)
Sx (m = 6)
Sp (m = 7)
694 764 798 833 902 972 1,041 1,111
122 202 251 305 434 591 780 1,002
19.1 68.2 109 164 331 597 1,000 1,583
1.65 25.6 55.3 106 307 742 1,585 3,102
0.0793 11.0 31.8 76.5 321 1,034 2,817 6,808
0.0021 5.05 19.6 59.4 358 1,543 5,362 16,009
0.00002 2.51 12.9 48.7 420 2,412 10,684 39,401
From the Figures and Tables, it became clear that: i) The counts increase acceleratedly depends on ML ,% which is more significant in higher order m; ii) The normal order of counts ( S > D > T · · · ) for a small sample reverses at larger ML , ex. ML ≥ 1.4
at fd = 0.67 or ML ≥ 1.3 at fd = 0.85 for �n = 0.4 . There is little effect of αr on the order;
iii) Ratio of the highest count to the lowest count fits within two orders of magnitude, ex. ML ≥ 1.2 at
fd = 0.67 or ML ≥ 1.15 at fd = 0.85 for �n = 0.4 . We have never experienced such a large ML .
Change in the order of multiplicity correlations as a function of ML An example of the second item of the list had been reported by Ensslin 12 in 1997 for Pu metal within the range 20 g ≤ meff ≤ 80 g , where D > T > Qr changes to D > Qr > T then to Qr > D > T and finally to Qr > T > D according to the increase of meff . Similar change/reverse in multiplicity correlations by PGF for the U-Pu mixide dioxide is shown in Figure 4 and 5, though the x-axis is not meff but ML . Counts�s 1000
Sp
Sx Qt Qr
800
600
400
D
200 T
1.0
1.1
1.2
1.3
1.4
1.5
1.6
ML
Figure 4 Change in the order of multiplicity correlations ( �n = 0.4, αr = 0.7, fd = 0.67 ) Counts�s 1000
Sp
Sx Qt Qr
800
600
400 D
200
1.0
T
1.1
1.2
1.3
1.4
1.5
1.6
ML
Figure 5 Change in the order of multiplicity correlations ( �n = 0.4, αr = 0.7, fd = 0.85 ) The relationship between meff and ML is not so simple, because ML depends on the induced fission probability p which depends on sample size, shape, density, elemental and isotopic composition, energy spectrum of neutron and its cross-sections. However, it would be true that ML correlates roughly with meff , so the trend is consistent with Ensslin's report.
Discussion Estimation of meff satisfying the order Qr > T > D in Ensslin's report According to the obtained equations, ML corresponds to the final order Qr > T > D in Ensslin's report (Pu metal, �n = 0.5, αr = 0 ) is evaluated to be larger than 1.45 . The value p is calculated to be 0.126 from the footnote one supposing νi1 as 3.159. Moreover, p can be expressed according to Figure 6 (reprinted from JAEA report 13 ) using a probability of neutron leakage without collision PE and a ratio of fission cross section to total cross section for all elements rf /t , where probability of capture or (n,2n) reaction are supposed to be suffciently small for 2 MeV neutron. Therefore, PE is calculated to be 0.476 when rf /t is supposed to be 0.15 for the Pu metal composed of 60wt% 239Pu and 25wt% 240Pu . l (1 − PE )(1 − PE − rf /t ) PE
(1 − PE ) PE
PE
1st collision 1 − PE
2nd collision
(1 − PE )(1 − PE − rf /t )
(1 − PE ) rf /t
(1 − PE )(1 − PE − rf /t )2 PE
(1 − PE )(1 − PE − rf /t )2
(1 − PE )(1 − PE − rf /t ) rf /t
p=
3rd
(1 − PE )(1 − PE − rf /t )2 rf /t
(1 − PE ) rf /t PE + rf /t
Figure 6 Relation between p, l and PE supposing elastic/close-to-elastic collisions (reprint from Ref.13) For 239Pu and 2 MeV neutron, half of total cross section is elastic cross section and the rest is composed of fission cross section and inelastic but close-to-elastic cross section half and half as shown in Figure 7. Therefore, rf /t % is roughly supposed to be 0.6 × 0.5 × 0.5 = 0.15. 5
10
4
10
3
Cross-section [barns]
10
tot el non f inl g
total elastic total-elastic fission inelastic gamma
Pu-239 (n,tot) Pu-239 (n,el) Pu-239 (n,2n)
Pu-239 (n,non) Pu-239 (n,f)
Pu-239 (n,n+f)
Pu-239 (n,inl)
2
10
Pu-239 (n,g)
1
10
0
10
-1
10
-2
10
0.001
0.01
0.1 Energy [MeV]
1
Figure 7 Cross-sections of 239Pu for fast neutron
10
In the case of a sphere, it is possible to estimate the radius r of the Pu metal from the PE and mean free path lm of neutron using an average length L from an arbitrary point inside a unit sphere to an arbitrary point on its surface, which is a new concept proposed in the JAEA report. Therefore, r is calculated to be 1.65 cm when lm is supposed to be two cm in Pu metal of density 19.8 g cm-3. As a result, the weight of Pu metal mg and meff corresponds to ML = 1.45 is estimated to be 372 g and 93 g. Though such a simple method, this result is, fortunately, in good agreement with Ensslin's report. Average length L from an arbitrary point inside a unit sphere to an arbitrary point on its surface x=1 Pi
θ L
x=0
Ps
Pi
Arbitrary point inside a unit sphere
Ps Arbitrary point on its surface 2
L2 sin2 θ + (L cos θ + x) = 1 � � 1 1 π L = 0.90143 L= L dθ dx π 0 0
For a sphere of radius r and mean free path lm: � � rL PE = exp − lm
Figure 8 Relation between PE , lm and L for a sphere of radius r (reprint from Ref.13) Estimation of meff corresponds to ML = 1.2 of U-Pu mixed dioxide handled at PCDF In the same way, the value p for ML = 1.2 satisfying the ratio of the highest count to the lowest count fits within two orders of magnitude b at fd = 0.67 and �n = 0.4 is calculated to be 0.0717. Then, PE is calculated to be 0.51 when rf /t is supposed to be 0.0875 for the U-Pu mixed dioxide. This supposion is based on 44wt% U and 43wt%Pu composed of 60wt% 239Pu and 25wt% 240Pu. Then, r is calculated to be 2.98 cm when lm is supposed to be four cm in the dioxide of true density 11.2 g cm-3. At last, the weight of dioxide mg and meff corresponds to ML = 1.2 is estimated to be 1241 g and 133 g. In case of powder density (2.2 g cm-3) actually handled at PCDF, lm grows to 20.4 cm, therefore r , mg and meff also grows to 15.2 cm, 32.2 kg and 3.45 kg, respectively. It should be noted that lm and r are inversely proportional to the ratio of density change but mg and meff are inversely proportional to the square of the ratio. As a result, such a large size sphere is difficult to be handled at PCDF (probably in other nuclear fuel facilities, too) from the view point of criticality control. Estimation of Feynman variance According to Croft 6, relation between S, D and Feynman varianceY is expressed as: D=
1 fd S Y 2
(36)
where w2 of Croft's paper is assumed to be one supposing sufficient measurement time, and fd is the fraction of gate effectiveness to count doubles normally configured at 0.60-0.75 at PCDF. As a result, estimated Y value for ML = 1.2 at fd = 0.67 for �n = 0.4 is 0.86. This is several times larger than the one calculated from S and D of actual U-Pu mixed dioxide powder (difference of �n is adjusted because Y is proportional to �n ), which is consistent with the fact that the largest ML we have experienced is less than but close to 1.1 ( ML is obtained from S and D )14. A review of criticality control is desired. b.
The ratio of the highest (ν=2) to the lowest (ν=7) of Pν is the order of 100. To restore profile of neutron multiplicity distribution from multiplicity correlation, it would be necessary to achieve the same order of magnitude.
Subjects to be investigated to observe high order multiplicity correlations Obsevation of high order multiplicity correlation is very challanging. To fit the ratio of the highest count to the lowest count within two orders of magnitude, the target ML will be set at ML = 1.2 at fd = 0.67 or ML = 1.15 at fd = 0.85 for �n = 0.4 (more ML for lower �n or less ML for higher �n ). There will be four major subjects:
• • • •
Criticality control of a sample, especially for lower �n ; Count high order correlations; Select appropriate neutron energy range; Satisfy both high efficiency and small uncertainty.
The efficiency �n = 0.4 had been achieved for thermal neutron and 3He tubes, however uncertainty of counting statistics becomes large according to the increase of multiplicity m in high order multiplicity measurement at present. The uncertainty could be reduced if die-away time of moderator is smaller and faster neutron is utilized 15, however the counting efficiency would seriously decrease. A new detector and counting technique are desired to the challanges to observe high order multiplicity correlation and also to solve 3He shortage problem 16. Conclusion Referring to Böhnel's paper, equations for neutron leakage multiplicity correlations were derived up to septuplets using a probability generating function, in which a spontaneous fission and a (α,n) reaction were taken stochastically as a starting event of a neutron leakage chain resulting from a fission cascade. The coefficients of the equations were too large, thus a new definition ηm = νm /m ! (expected number of neutron pairs) was introduced where νm is the conventional factorial moment. Next, count rates from singles up to septuples were evaluated supposing the assay system for the U-Pu mixed dioxide powder we have handled at PCDF, although quadruples or more is very difficult to be counted at present. As a result, count rate of neutron correlations increased acceleratedly depends on leakage multiplication ML and detection efficiency �n , which was more significant in higher order multiplicity. This tendency results in reverse of normal order of counts, i.e. doubles becomes smaller than septuples in a large ML sample, ex. ML = 1.4 at fd = 0.67 for �n = 0.4 , if septuples could be properly counted. Similar tendency had been reported for Pu metal in the United States, therefore obtained tendency was compared to the report by converting ML to effective Pu mass meff supposing a sphere assisted by the recent JAEA report because the relationship between ML and meff was not so simple. Fortunately, the result was in good agreement with the prior report. The above encouraged me to estimate meff corresponds to ML = 1.2 at fd = 0.67 for �n = 0.4 which satisfies the ratio of the highest count to the lowest count be within two orders of magnitude. However, the result was larger than the one that is approved from the view point of criticality control. Feynman variance was also estimated and the result was 0.86 which is several times larger than the normal one but still in deep subcriticarity. Therefore, to observe/analyze high order multiplicity correlations, a new detector and counting technique together with a review of criticality control are desired. References 1.
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10. M S ZUCKER and N E HOLDEN, Energy Dependence of the Neutron Multiplicity Pν in Fast Neutron Induced Fission of 235,238U and 239Pu, BNL-38491, 1986. 11. J TERRELL, Distributions of Fission Neutron Numbers, Physical Review 108(3), pp.783-789, 1957. 12. N ENSSLIN, A GAVRON, W C HARKER et al., Expected Precision for Neutron Multiplicity Assay using Higher Order Moments, LA-UR-97-2716, 1997. 13. T HOSOMA, Essentials of Neutron Multiplicity Counting Mathematics - An example of U-Pu Mixed Dioxide, JAEA-Research 2015-009, Section 4 and 5, 2015 (in Japanese). 14. N ENSSLIN, Chapter 16: Principles of Neutron Coincidence Counting, LA-UR-90-732 (Passive Nondestructive Assay of Nuclear Materials, 1991. 15. S A POZZI, J L DOLAN, E C MILLER et al., Evaluation of New and Existing Organic Scintillators for Fast Neutron Detection, Proceedings of INMM Annual Meeting, 2011. 16. M M PICKRELL, A D LAVIETES, V GAVRON et al., The IAEA Workshop on Requirements and Potential Technologies for Replacement of 3He Detectors in IAEA Safeguards Applications, JNMM XLI, pp.14-29, 2013.
