ANALYSIS AND DEVELOPMENT OF THE SERIES EXPANSION METHODS IN THRESHOLD DETECTOR ACTIVATION DATA HANDLING EUR 588 e

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ALYSIS AND DEVELOPMENT OF THE SERIES

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EXPANSION METHODS IN THRESHOLD

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DETECTOR ACTIVATION DATA HANDLING

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Joint Nuclear Research Center

Ispra Establishment ­ Italy

Scientific Data Processing Center ­ CETIS

Paper presented at the Symposium on Fast and Epithermal Neutron Spectra in Reactors

Harwell, Didcot (Great Britain) ­ December 11­13, 1963

This document was prepared under the sponsorship

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E U R 588. e

E R R A T U M

Would y o u p l e a s e r e a d on t h e c o v e r p a g e :

J o i n t N u c l e a r R e s e a r c h C e n t e r

I s p r a E s t a b l i s h m e n t -

Italy-S c i e n t i f i c D a t a P r o c e s s i n g C e n t e r - C E T I Italy-S

and

EUR 588.e

ANALYSIS AND DEVELOPMENT OF THE SERIES EXPANSION METHODS IN THRESHOLD DETECTOR ACTIVATION DATA HANDLING by G. DI COLA and A. ROTA

European Atomic Energy Community — EURATOM

Joint Nuclear Research Center — Ispra Establishment (Italy) Scientific Data Processing Center (CETIS)

Paper presented at the Symposium on Fast and Epithermal Neutron Spectra in Reactors — Harwell, Didcot (Great Britain),

December 11-13, 1963

Brussels, February 1964 — 20 pages — 6 figures

The threshold detector activation data treatment by the series expansion methods has been analyzed. The numerical problems connected with the use of digital computers have been investigated. A development connected with the possibility of using detectors with quasi linear-dependent cross-sections is proposed. The series expan-sion coefficients were obtained through the Gauss method by solving a least square problem. Experimental and « test » activation data have been processed.

EUR 588.e

ANALYSIS AND DEVELOPMENT OF THE SERIES EXPANSION METHODS IN THRESHOLD DETECTOR ACTIVATION DATA HANDLING by G. DI COLA and A. ROTA

European Atomic Energy Community — EURATOM

Joint Nuclear Research Center — Ispra Establishment (Italy) Scientific Data Processing Center (CETIS)

Paper presented at the Symposium on Fast and Epithermal Neutron Spectra in Reactors — Harwell, Didcot (Great Britain),

December 11-13, 1963

Brussels, February 1964 — 20 pages — 6 figures

EUR SOO.e

EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM

ANALYSIS AND DEVELOPMENT OF THE SERIES

EXPANSION METHODS IN THRESHOLD

DETECTOR ACTIVATION DATA HANDLING

by

G. DI COLA and A. ROTA

1964

Joint Nuclear Research Center Ispra Establishment - Italy

Scientific Data Processing Center - CETIS

CONTENTS

Page

1. Introduction 5

2. The Polynomial and O r t h o n o r m a l Methods 6

3. S e r i e s Expansion Methods Generalization 7

3. 1. Input Data for n u m e r i c a l t e s t s 8

3.

Z.

The Weighting function 8

3. 3. General R e m a r k s 9

4. Relative Deviation Minimization Method R D M M 10

5. R D M M Results and Discussion 12

­ 5

Introduction

The activation measurements of the threshold detectors are frequently

used for the in­pile neutron spectra determination) this technique

has the following advantages«

­ The threshold detectors are not sensible to the high gamma fluxes

of the reactors

­ The small volume of the detectors prevent any neutron flux deformation

­ No connection with the outside of the reactor is required

­ The activation measurements are easy and not very expensive.

The experimental measurements may be expressed as A} numbers» they

indicate the reaction probability per second for one nuoleus of the

i­th isotope, when immersed into the unknown neutron flux. It follows

that: ^, es

(D

where C;

[ζ)

i s t h e d i f f e r e n t i a l neutron c r o s s ­ s e c t i o n of the i ­ t h

i s o t o p e for the r e a c t i o n i n q u e s t i o n , and ^ ( , ^ ) i s t h e f a s t n e u t r o n

d i f f e r e n t i a l f l u x (n/cm

. sec . Mev). The i n t e g r a t i o n range may "be

reduced from ( 0 , c o ) t o ( t j ,

0 5

) as

ζΓ

ν

( t ) s O

­for

E"

<

E

'^

When the A; are known, a procedure has to be established for solving

the system of equations (ï). Many methods have been proposed to attain

this solution [ 1 ] , [ 2~] . This work is a study of the group of methods

known as "polynomial" and "orthonormal" methods. This set has been

chosen for the following reasons:

1) It seems to he the more suitable for the neutron spectra description,

even if it is not reactor generated.

2) A development is possible whioh allows the activation data inter­

pretation, even if the detectors have not completely linear inde­

pendent cross­sections.

Generally this second fact prevented the use of these methods with

2. The Polynomial and Orthonormal Methods

In the polynomial method, introduced "by Uthe [ 2>~] , the spectral

shape is assumed to be a polynomial in energy, with as many terms as detectors used, multiplied by a chosen weighting funotion. This method is an extension of the simple polynomial method, in which

-κ.

(2) f l e j =

y . &Ί

E· (n - number of detectors)

' o

v

in fact the suggested form i s :

-Vf ·

(3) cp ( E ) * i*(e) Ζ ·

0

- ' . ^

1 o

where Wis) is a suitable weighting function. Originally the actual meaning of (3) was a deformation of the fission speotrum W(E) by a polynomial.

The method using a series expansion of orthonormal functions has

been suggested independently by Trice \A~\ and Hartman et al. Γ 5l ·

Here <~P (E) is given by:

■w

(4)

^ ) = λ ;

_{α ;}

Τ ^

The system of orthonormal functions is obtained using the Schmit's

orthogonalization method, beginning from the cross section functions

ere

(E).

Brownell et al. [6 J proposed the following variation of the above

mentioned method:

­ CP(E) is assumed to be described by

(5)

cflO

a

w U ) | ­ > : 4 ^

e

)

where VÌE) has the usual meaning and the set of ¡J/; functions is

obtained by the same procedure used by Hartman and Trice.

­ Many different solutions ^Ρ'(Έ) are obtained by random deviation

of the input activation data.

7

-The last considered method, due to Laming and Brown [7^ , has been

reconsidered in

γ

6 J . ^ ( E ) is assumed to be described by:

(6) Cf [e)- ^Ιε)·ΐΛ;^-ΐε)

where the /(; , i-degree polynomial, are an orthonormal set of

functions, orthogonal, in the (θ, E ^

y

) range to W(E).

3. Series Expansion Methods Generalization

Prom the above, it is easy to see that all the described methods may

be unified in one class where Q> (E) is expanded in the following

series of linearly independent functions:

(ï) CfU)-- tf/(j=) £ . a

:

y>;[e)

By a suitable interpretation of the weighting function W(E) and of

the system of functions U>(E), the expressions (2), (3)> (4)> (5)» (6)

may be obtained.

This generalization appears trivial, however it is possible to state

only one procedure for the solution of the system of equation (ï).

The appropriate ¥ function and the system

U^;(E)

may be selected on

the basis of parameters. In fact, by introducing the expression (7)

in the equations (ï), one obtains:

(8)

Al ·= G;U)|WU)L·,a.jVftle)]áE - ^ . û· S;

_J

£:

where

(9)

£ l

- 8

3·1· Input Data for numerical tests

The orthonormal methods have been tested by the following reaction data:

1) Np - 237 (n, f)

2) U - 238 (n, f)

3) Th - 232 (n, f)

4) S - 32 (η, ρ) Ρ - 32

5) Ni - 58 (η, ρ) Co - 58

6) Al - 27 (η, ρ) Mg - 27

7) Fe - 56 (η, ρ) Μη - 56

8) ΑΙ - 27 (η,« ) Na - 24

The cross section was taken from literature £_8J .

Two sets of activation data Ai have been tested:

- a "test" set, derived from the assumed O; by a "test" neutron

r -2.3 Ε _ -0.Ϊ5 E

spectrum:

cp[£.; = [e- + D.o3 e, J

o.s<-e

- an "experimental" set obtained from Ispra 1 Reactor in the PH3 position [8Ί .

3.2. The Weighting function

A series of numerical tests have been performed in order to determine the influence of the weighting function when using the orthonormal methods.

The test results show clearly that a good choice of the W ( E ) function is very important. The fig. 1 shows the results

obtained from a set of 6 threshold deteotor "test" activation data by three different weighting functions:

_ c

4.

2. a

_-E

/Oowiv (zi[ IV/AH hiiiork sptötr*«*/

Other similar results have been obtained from experimental data. For neutron thermal pile spectra the more suitable weighting

function seems to be e (see also {."j"] )«

3.3· General Remarks

The use of the above methods suggest the following observations*

1 - The expansion term number is conditioned by the number of the detectors used and vice-versa.

2 - The cross-sections CT; (E) must be linearly independent functions: if this condition is not satisfied the matrix of the system coefficients is singular. The system becomes ill conditioned if the G"; functions are quasi-linearly dependent.

In the considered sample the couples Th, U and Si, Ni appear to have similarly shaped cross-seotions. It follows that the use of the whole set is not always possible (fig. 2 ) | this is true particularly when real problems are considered, because the Ai and CT¿(E) values are affected by experimental errors, both absolute and relative to one another. This difficulty appears more evident in the methods where the orthonormal functions are generated starting from the cross-sections (see expansions (4) and (5) ).

Many numerical tests have been performed on "test" and "experimental" values (8 detectors), using the above mendioned method with

VÌE) » e and the following type of polynomials:

a) Simple polynomial.

b) Orthonormal polynomials over (O, E^A ) i) range, relative to

the specified W(E).

c) Orthonormal polynomials generated by the CT^ functions d) Laguerre polynomials.

The fig. 2 and 3 show the results respectively obtained with the "test" and "experimental" data. They are completely different from one another and general conclusions are not possible. The most of them are not good: the only reliable results appear to be the ones obtained by simple and Laguerre polynomials from "test" data, despite the fact that they show some deviations

­ 10 ­

The evidenciated troubles could be overcome by the choice of a

suitably reduced number of detectors. Numerical tests of this type

have been performed:

By using suitable sets of s A. χ detectors chosen from the eight

available, as suggested by the previous considerations, all the

methods have given reliable results. Unfortunately the spectrum

shapes are not the same for different choices of deteotors sets

as shown in fig. 4· The pronounced differences observed are a con­

sequence of the experimental errors of the Ai and vTj, (E) values.

There is no significant difference in the results obtained from

the test problem using different detectors sets, when the Ai and

<Γ·(Ε) are error free . Since for different sets of experi­

mental data (relative to the same experience) different neutron

shape spectra are obtained, the problem arises of seleoting a

representative spectrum.

The above reported remarks suggested a more general formulation of

the problem by taking into account the following statements:

1) All the available activation data, independently from the cross­

section shape, must have the possibility to be used.

2) The solution must approximate the Ai values in the best possible

way.

3) The results are desired to b6 independent from the particular

choice of the Vi/ functions.

4. Relative Deviation Minimization Method R D M M

Suppose the searched flux may be expanded in the series:

(10)

cp|e) = Wie) ·

Ζ- °~¿

4 ^ I

e

)

t h e b e s t aproximation Q

7

(E) t o Ci? (E) i s assumed t o be the one

minimizing t h e q u a d r a t i c form:

A· ­ j y , r l 5 ) ^ U ) d 5 '

(11)

Q(/VW, < ^ , < ^ , . · · ^ ) = Ζ |

tu

i

A;

11

-0

2

> S U

6

) = V</le)Z

c

Q-'^ Ψ-ΙΌ

I

The quadratic form (11) represents the sum of the squares of the

relative deviation of the A. and, according to the Legendre least

square postulate, the obtained solution gives, in the sense of

least square, the best aproximation to the exponential data.

The search of the minimum of Q, once m is fixed, is performed by

the Gauss Method.

The requirement of the minimum of Q imposes the conditions:

(13)

1£L

s

o

k . l , ...,/vvu

and hence leads to the linear system in the m unknowns a., a_ ...

a :

_m

m m

(14) S

1

S a - S e

where:

θ

S

3?U,

Κ

1)

s

:]

-

-L js-.-UJuKe)^;^)^

τ

S indicates the transpose of the matrix S. Among the solutions

CP. , Cp

t

,

^f/ito.>

"the one that gives the minimum value of Q

is chosen.

When m « nthis method is actually equivalent to the oollooation

method [ 9l » where the simple system S a - e is solved. The last

one is the method used in all the procedures described in eeot. 3/

that may be regarded as particular cases of rhe RDMM, once imposed

12

-5« R D M M Results and Discussion

In order to carry out the data processing by the RDMM a FORTRAN code for the standard IBM-709O has been prepared.This oode has been used to process the "experimental" and "test" data with different ohoices of polynomial expansions. In all the tests performed the weighting function W(E) - e was used.

The more significant results are shown in fig.s 4 to 6.

The spectra obtained by using the reduced sets of deteotors (with the collocation method: see 3·3), as well as a spectrum obtained by the complete set of deteotors are shown in fig. 4· The comparison of these curvs indicates that the speotrum obtained by the RDMM may be regarded as an average between the ones obtained by the collocation method.

The fig. 5 shows the results obtained from "experimental" data (complete set of 8 detectors) by the RDMM using the polynomials a), b ) , d) of sect. 3·3 and the Chebyshev polynomials. Significant differences exist for experimental data only at high energy values of speotrum. These differences do not appear when "test" data are used. This fact indicates that the RDMM results may be considered inddpendentjfrom the particular ohoioe of the IP functions.

In the test performed with 8 detectors the values fo the quadratio form Q (11) were not very different from eaoh other when m - 5,6,7· This faot results in very similar spectrum shapes, as may be seen

in fig. 6. This figure shows the ourves Cf> , γ ^ , (γ-, obtained by

13

-B i b i o g r a p h y .

1 P . D e l a t t r e - C.E.A. 1979 (1961)

2 "Neutron d o s i m e t r y " — P r o c e e d i n g s of t h e Symposium of Harwell 10-14 Dec. 1962 IAEA 1963, 2 v o l l .

3 P.M. Uthe - WADC-TR-57-3 / CNE-9 ( O c t . 1957) 4 J . B . T r i c e - APEX-408 (1957)

5 S.R. Hartmann, A.D. A s t i a - 142029,WADD (1957) / WADC 57.375 6 G.L. B r o w n e l l , M.R. Rajn, R.A. Rydin, R . I . Schermer —

r e f e r e n c e 2, p a g . 70

7 W.P. Lanning, K.W. Brown - WAPD-T-I38O ( S e p t . I 9 6 I )

8 M. B r e s e s t i , A.M. Del Turco, A. O s t i d i c h , A. Rota, G. S e g r e , r e f e r e n c e 2, p a g . 49

9 L. C o l l a t z - "The Numerical Treatment of t h e D i f f e r e n t i a l E q u a t i o n " Springer—Verlag, B e r l i n I960, p a g . 28-29

10 F . B . H i l d e b r a n d - " I n t r o d u c t i o n t o Numerical A n a l y s i s "

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