A fast neutron spectrum unfolding method using activation measurements and its application to restoration of a thermonuclear reactor blanket neutron spectrum

Nuclear Instruments and Methods 203 North-Holland Publishing Company

403--407

(1982)

403

A FAST NEUTRON SPECTRUM UNFOLDING METHOD USING ACTIVATION MEASUREMENTS AND ITS APPLICATION TO RESTORATION OF A THERMONUCLEAR REACTOR BLANKET NEUTRON SPECTRUM V.M . NOVIKOV, A.A. SHKURPELOV,V.A. ZAGRYADSKY, D.Yu. CHUVILIN and Yu .V . SHMONIN 1. V. Karcharoa Mnmae of Atomic Energy. 123182 Mos-r, USSR

Received

8

Octoher

1981

and in revised form

25

January

1982

This article describes a fast neutron spectrum unfolding program. The programtakes into a-xount a priori information about the neutron spectrum, the experimental values of activationt it tegrals errors andactivation detector cross sections errors, 'the usefulness of the unfolding program was demonstrated by its application to the determinAtion of neutron spectra from I to 0 MeV in the molten-salt blanket mode: of a thermonuclear reactor . The details of the fast neutron energy spectrums in a thermonuclear reactor blanket i; of interest both for more precise calculation methods and choice of nuclear constants as well as for engineer purposes, for example . estimation of materials strength under radiation. The activation method is one of the simplest and most accurate methods for neutron spectroscopy . It gives small perturbations of neutron spectrum and permits neutrons measurements in the interval 0.5-14 MeV accompanied by intensive y-radiation . From the mathematical point of view the neutron spectrum unfolding method with the help of activation integrals is not a correct problem. So it may lead to more than one answer. Therefore, there is a need to use additional a priori information to obtain the physically simplest solution . The activation integrals could be written with help of quadratures, using effective nuclear data : n A,,-ae(E)T(E)dE=Zfork=I ... .. M x u t- t

f

where y~ =

J

E "'q) (E) dE.

f~Da,,(E)4'(E) dE+ E(E'É t '$(E) dE E, t uf

where4,(E)= f~tp(E')dE'

Graph of a,,(E) obtained svilh the help of oA(E) .

and (a(E): A ( I 1 11

(1)

To calculate effective nuclear data an analysis of the form of thenuclearcross sections a,t (E) was made . This shows the possibility of drawing aF(E) as a,v.(E) linear function and ;:onstank value (see fig. 1). We shall writethe activation integral with the help of ,,(E) as :  . A,-

I ;g. I .

(E)-

ati(Et) (2)

0167-5087/82/0000-0000/$02 .75 , 1982 North-Holland

E ,E-F.,

.R(E,) .

Et

-E. : < x.

Taking into account, that theintegral neutron flux ss(F ) is a smooth function, one can calculate the second integral on the right hand side of (2) with sufficient precision using Gauss quadrature formula and taking only three terms in thequadrature suns . Et - Eu

E

Et-E"

E r.

=

f:,r'~(E)dE= ~

`k(F')

E,+Eu E,-E  2 + 2

2 f
1 ,4,~(E"~):

.s :, t

404

V.Al. Novikor er ul. / Spocrrarn unfelding rnerh .d

where PL. are roots of the Legendre polynomials (A, = A,=5/9 ; A2=8/9) and (I
tained by calculations [7] .

f t'

f(E) dE then

if E,=E; t1 ; if E,

The calculations shows, that replacement of the integrals (1) by thequadrature formulas results in an error of less than 1%. The 24-group nuclear constants often used for activation detectors are listed in table 1 . A priori information about the investigated neutron spectrum was presented by a set of the integral neutron fluxes 0 _ (i~j) = fé4p(E') dE', which includes the calculated spectra, the results of measurements using other methods, etc. This set describes approximately the neutron spectrum being investigated. Let us choose from this set a reference spectrum 0l°), which has an intermediate form between other spectra of the set.

EI kl ;

then

A,=f xojL (E)T(E)dE t.,

.1

_J

-1

~ I A-(E)f(E)dE

f

x

M,,m(E, )+ E oo,(E)T(E)dE.

(4)

The calculation shows, that one can make a choice of so that the contribution to the integral (4) from Jo,t(E) will be less then 10%. This permits one to average the function Aa,(E) inside the J E, Ej, , ] with help of some theoretical thermonuclear spectrum, ob-

Table 1 Effective nuclear data (mb) E,-E,

~', NR~n

Rh, .',,

Iuo,~,o

200 eV-1 keV 1 keV-0.1 MeV 0.1 MW-0.3 McV 0.3-0 .7 0.7-LO5 1 .05-1 .35 1 .35-LR 1 .8-2.0 2,0-2.85 2.85-3.0 3.0-3.8 3.8-4.2 4.2-5.0 5.0-5.35 5.35-5 .8 5.8-8.5 8.5-112 11 .2-13 13-14 14-14.1 14.1-15 15-16 16-17 17-20

0 0 27 .51 962.3 1862 1997 1625 1669 1692 1661 1610 1550 1519 1493 1526 2065 2338 2306 2315 1935 2369 2438 2579 2673

0 0 0 0 847 837 819 815 754 1174 1104 1064 1075 1185 1225 1311 1253 700 405 336 304 280 258 238

0 0

]M'V]'1

°1

0

0 12 .2 82 .2 82 .2 347 347 358 358 358 358 357 354 343 288 162 76 .76 63 .4 58.6 55.8 52.3 50.1

Energy in MeVexcept when specified otherwise.

U,2"",s

0 0

0

9 19 182 495 586 563 548 564 547 556 578 624 936 1032 1032 1085 1175 1229 1277 1333 1371

N

P~r

0 0 0

0.96 1 .204 5.892 16 .83 7.936 178 .4 210.2 169.' 764 492 481 .2 628.7 648 664 591.7 490 448 423 373 272 .7 188 .3

0 0 0

0 0 0 2.69 28.02 38.53 134.4 104.3 101 121 .7 119 .9 130 .8 133 .2 134 108.6 93.76 87.58 83.91 78.6 69.84 58.76

Zn`,'

0 0 0

0 0 0.057 2.07 6.25 56.97 61.31 64 .82 258.2 163 .6 160 .3 276 251 .9 286.1 272 243 226.8 216.3 198.2 166.8 141 .6

g;,,

0 0 0

0 0.005 0.09 1 .865 3.21 88.17 86.53 358.5 337.7 257.4 239.1 286.6 319.8 366.8 344 285.6 258.4 241.9 221 .3 189 .5 147

Fe,P)

Cline

0 0

0 0

0

0 0 0 0 44.4 43 .9 32 .5 36 .4 511 503 473 459 597 620 543 413.7 375 349 314.9 255.7 206.2

0

0 0 0 0 0 1 .572 5.328 43 .96 94 .38 107 .9 94.42 98 .38 107.2 113.8 115 115 115 115 115 115 115

405

V. M. Novikoo et al. / Spectrum unfolding method

(Pi = maxJ,)/Ao ;

We introduce the matrix i

~(ul i

j=l_ .,N;

v= I, . . .,L.

Using the method of ref. 5 it is possible to define some vectors j1 p), which describe general features of the integral fluxes of the set. Q

Q

,-- I= Y, XrIIr .i-AJ : J=l, . . . .N, f, =! 'Pi,o1 vP= j

where Ay_, .,. >max(A(j )), and XP must be determined from experiments and (1). Let us write the errors Ai and Sk, where Sk are the errors in the A k in (1),

Ah~rr

sigp)

0 0 0 0 0 0 0 0 0 0 0 0 0 27.3 27.3 27.3 74.7 88.3 82.8 82 .34 78 .61 76 .22 74.3 65.7

0 0 0 0 0 0 0 0 0 0 0 0 '7.73 27.9 25.1 486 617 433 390 369 357 336 297 256

Sk

(dk"

- -k .I -1 )\ 1 + 4k

where Q< M, M - the number of measurements, Alj" - errors of the expansion. It is assumed, that the investigated spectrum could be written with help of the vectors (rIp).

and

K=I, . . .,M) .

v= 1, . . ., LI ,

P =I

=

1746,

0 0 0 0 0 0 0 0 0 0 0.0086 0.0708 0.675 3.11 6.22 38 .24 82 .9 I08 .2 114 113 111 107 95.8 80.%

w,S

Mgiô)

Al ;',,,

r Im&~r

0 0 0 0 0 0 0 0 0 0 0

0 0 t) (1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 311 1879 1541 1599 1627 1667 1733 1702

a

0 0 0.4972 73.43 161 .8 206.9 205.5 197 191 180 161 140

0

0 0 0.0596 0.695 0.984 29.05 84.9 154 123 121 119 113 97 .6 81 .51

Sk =min8;

The solutions 4i and XP canbe obtained by ntinintizing the following functional

j= 1. . . .,N1 ,

" ft, A("i i 1= I X(r'Irlr~i -~

Ai =PJA

A o =maxNvl;

+

_

fi

Sz Az -. t 11 h'

Q Y Xrllr .r~ /P,

p=1

r ~ or - 1 . $

(10)

i

The error A was varied from A = A(1 with constant steps until the solution began to satisfy the system (1) within the limits of the measurement errors, the corresponding Awascalled theoptimum A. The error matrix can be obtained by means of the minimizing of the functional (10), using optimum A. This unfolding method was used for determining the

Cu°n.w

Mai~zm

0 0 11 u

0 0 0 1

0 0

0 0 0 0 0 0 0 0 0 0 2.02 615 922 755 796 816 865 877

1, 0

0

0 0 0 0 0 0 0 3.902 721 1101 886 928 970 1030 1069

0 1

Fvi,»

Caihm

Nli~_  v

2 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6.32 193 .1 249A 547 567 .6 554 650 720

0 0 11 0 0 0 0 tl tl tl 0 0 0 0 0 0 0 0 24.9.5 15.38 17.45 36 .71 43 .54 49.8

0 0

0 0 0 11 0 0 0 0 0 0 0.0085 0.954 66 .18 34 .29 35 .61 65 .82 65 .21 78.06

V AL Noeiioo et al / Spe<4rnm unfolding method

Table2 Saturation activity (5 min) - ' for three different points Type of reaction

n)' t `mlu z"yim,p) °" Co "Fe(n .p)5 "Mn "A4n,a)'N . "Nb(n.2n)° 'Nb ">'Ag(n,2n)"Ag '5 Cu(n, 2n)`'4Cu ' °F(n,2n)t"F "'Cu(n,2n)"'Cu " Ni(n.2n)"wi

Saturation activty (X 10 4) ai distance rfrom the model centre r=76 mm

r= 124 mm

r= 155 mm

(19 .74-0.64) (34 .33w 1.08) (8 .31 "-0.19) (7 .65-"0.37) (31 .31=1.44) (41 .72=_0 .89) (51 .23= 1 .09) (2.90~0.08) (20.38 "--0 .60) (1 .72=-0.08)

(9.67 "_-0 .41) (12.94~: 0.58) (2.62 0.07) (2.35=0.17) (10.51=0.23) (12.39- 0.31) (14.99 _= 0.34) (0.84-0.03) (6.05 w 0.16) (0.49=0,03)

(6.69 1-0.30) (8.45 ' 0.27) (1 .55 r0.04) (1 .27--0.11) (5.32 ^0.15) (6.38=0.18) (7.97=0.23) (0.41= 0.02) (3.06 "-0.09) (0 .26-`0 .02)

fast neutron spectrum in the molten-salt blanket model of a thermonuclear reactor. The molten-salt model and the activation integral measurements were described in refs, 2 and 4. The activation integrals for three different points, located 76 mm, 124 mm and 155 mm from the neutron source are shown in table2. The fast neutron spectra determined with the help of the unfolding program and calculated with the help of the Monte Carlo program BLANK [71 are shown in fig . 2. The group neutron fluxes both obtained by the unfolding program (experiment) and the Monte-Carlo program (calculation) were normalized on the of 14 .1-14.2 MeV group flux at the salt-air boundary (60 mm).

This flux really depends on the distance from the neutron source only, so the normalization allows one to compare the unfolded neutron spectrum and the calculated one. A priori information about the neutron spectrum was chosen to be a set of the integral neutron fluxes, calculated with the help of the Monte-Carlo program BLANK for some space points of the moltensalt blanket model. The errors of the activation integrals were 7 to 9% . According to [61, the errors of the activation detectors cross section were 20-30% . All this led to about 38% maximum error of the unfolded neutron spectrum. The total statistical error of the calculate- was about 5%.

Fig. 2. Fast neutron spectra determined using the unfolding program andcalculations from the Monte-Carlo program BLANKwithin the r fatten-salt blanket modelof a thermonuclear reactor. (a) 76 mm from the modelcentre ; (b) 124 mm from the centre; (c) 155mm frog. thecentre. The solid lines show experimental values andthe broken lines, calculated values.

V. M.

Nooikoo et ul. / Spectrum unfolding meths!

As one can see from fig . 2, the maximum deviation between the experimental and the calculated spectra was about 40%. This deviation was less than sum of the experiment and calculation errors. Thus we can conclude, that a fast neutron spectrum unfolding program with the help of the activation integrals and a set of a priori integral neutron fluxes has been developed. The possibiitties of this unfolding program were demonstrated by its application to the prob. lem of the neutron spectrum in the range I to 14 MeV in the molten-salt blanket model of a thermonuclear reactor.

407

References A.A. Shkurpelmet al., Atomic energy 44 (1978) 352. V.M. Novikov et al.. Atomic energy 48 (1980) 332 . V.M. Novikov et al., preprint IAE-3165, M. (1979). V.M. Novikov et al ., Nucl. Instr. andMeth. 173 (19811) 449 . V.A. Wagavtt, Computing mathematics and mathematical physics magazine 11 (1971) 289 . [61 Themetrology of the neutron measurements in the reactors. The materials of the I All-Union school (Riga, 1976) vol. I (1976) no. 2. [71 S.V . Marinet al.. preprint IAE-2832, Nt. (1977). I] [2] [3] [41 151