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i § Mmm mná
EUR 4 7 7 5 e PART I
«¡Sit COMMISSION OF Tl
* * t f i . i , í
COMMUNITIES
(«Ρ COLLECTRONS, | | | |
POWERED NEUTRON FLUX DETECTORS
Part I : Theoretical considerations
by Ψ -li
M. GRIN
»1*1111111
Joint Nuclear Research Centre Ispra Establishment - Italy mime''
■'•'pr'
f ~xth' -fliptl'A wzVUWfr mjmMåm Materials Division
mm I f l
ρ
of the European Communities
ààt&,
AÉÜ This document was prepared under the sponsorship of the Commission
Neither the Commission of the European Communities, its contractors
nor any person acting on their behalf :
make any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method or process disclosed in this document may not infringe privately owned
"shts:or ÍES S t e p assume any liability with respect to the use of, or for damages resulting
from the use of any information, apparatus, method or process disclosed
in this document.
Uus repo
sSB S mm mm i m^WSKÊ^
m ■ wm ilPPs'eiiiiiiP Irr.
addresses listed on cover page 4
I at the price of B.Fr. 60.·
the titlt
Printed by Guyot s.a., Brussels
Luxembourg, March 1972
the best available copy.
M? •M
EUR 4775 e COLLECTRONS, SELF-POWERED NEUTRON F L U X DETECTORS Part I : Theoretical considerations by M. GRIN
Commission of the European Communities Joint Nuclear Research Centre - Ispra Establishment (Italy) Materials Division Luxembourg, March 1972 - 38 Pages - 6 Figures - B.Fr. 60.—
Collections are self-powered neutron flux detectors based on the direct collection of electric charges emitted from a short lived β emitter activated by neutron flux.
After a short description of components and measuring circuits, this report deals mainly with the theoretical sensitivity of the collections. An evaluation of self-shielding and flux depression factor and of self-absorption factor leads to the formulation of the theoretical sensitivity of the collections in normal working 'conditions. The cases of rhodium, silver and vanadium have been considered.
EUR 4775 e COLLECTRONS, SELF-POWERED NEUTRON F L U X DETECTORS Part I : Theoretical considerations by M. GRIN
Commission of the European Communities Joint Nuclear Research Centre - Ispra Establishment (Italy) Materials Division Luxembourg, March 1972 - 38 Pages - 6 Figures - B.Fr. 60.—
Collections are self-powered neutron flux detectors based on the direct collection of electric charges emitted from a short lived β emitter activated by neutron flux.
After a short description of components and measuring circuits, this report deals mainly with the theoretical sensitivity of the collections. An evaluation of self shielding and flux depression factor and of self absorption factor leads to the formulation of the theoretical sensitivity of the collections in normal working conditions. The cases of rhodium, silver and vanadium have been considered.
EUR 4775 e COLLECTRONS, SELF-POWERED NEUTRON F L U X DETECTORS Part I : Theoretical considerations by M. GRIN
Commission of the European Communities Joint Nuclear Research Centre - Ispra Establishment (Italy) Materials Division Luxembourg, March 1972 - 38 Pages - 6 Figures - B.Fr. 60.—
Collections are self-powered neutron flux detectors based on the direct collection of electric charges emitted from a short lived β emitter activated by neutron flux.
After a short description of components and measuring circuits, this report deals mainly with the theoretical sensitivity of the collections. An evaluation of self-shielding and flux depression factor and of self-absorption factor leads to the formulation of the theoretical sensitivity of the collections in normal working conditions. The cases of rhodium, silver and vanadium have been considered.
The second par t of this report which will follow will essentially consider in-pile behaviour of collections with a chapter specially devoted to a confrontation between theoretical sensitivity and experimental calibration.
The second par t of this report which will follow will essentially consider in-pile behaviour of collections with a chapter specially devoted to a confrontation between theoretical sensitivity and experimental calibration.
The second par t of this report which will follow will essentially consider in-pile behaviour of collections with a chapter specially devoted to a confrontation between theoretical sensitivity and experimental calibration.
EUR 4 7 7 5 e PART I
COMMISSION OF THE EUROPEAN COMMUNITIES
COLLECTRONS, SELF-POWERED NEUTRON FLUX DETECTORS
Part I : Theoretical considerations
by
M. GRIN
1972
Joint Nuclear Research Centre Ispra Establishment - Italy
Materials Division
A B S T R A C T
Collections are self-powered neutron flux detectors based on the direct collection of electric charges emitted from a short lived β emitter activated by neutron flux.
After a short description of components and measuring circuits, this report deals mainly with the theoretical sensitivity of the collections. An evaluation of self-shielding and flux depression factor and of self-absorption factor leads to the formulation of the theoretical sensitivity of the collections in normal working conditions. The cases of rhodium, silver and vanadium have been considered.
The second part of this report which will follow will essentially consider in-pile behaviour of collections with a chapter specially devoted to a confrontation between theoretical sensitivity and experimental calibration.
KEYWORDS
NEUTRON DETECTION SENSITIVITY NEUTRON FLUX SELF-SHIELDING ELECTRIC CHARGES RHODIUM EMISSION SILVER BETA PARTICLES VANADIUM
C O N T E N T S
1 . INTRODUCTION 5
2 . DESCRIPTION AND WORKING PRINCIPLE OP THE COLLECTRONS 6
2.1. Principle 6 2.2. Realization and component materials 6 2.3. Measuring circuit 6
3. THEORETICAL SENSITIVITY OP THE COLLECTRONS 10
3.1. Evaluation of the correcting factor K 11 3.1.1. Self shielding and flux depres- 12
sion factor K, 3.1.2. Self absorption factor K« 18
3.2. Sensitivity of the collectrons 23
4. LOSS OP SENSITIVITY AS A PONCTION OP NEUTRON DOSE 24
ANNEX 1 27
ANNEX 2 31 '
REPERENCES Í S5
- 5 -
1 . INTRODUCTION *) ι
Neutron flux measurements during irradiation tests
are normally performed by activation methods using
wires or foils of suitable materials like Co, Ag,
Ni, Au, Ih, Cu ... The amount of radioactivity
induced during a controlled exposure is a measure
of the neutron flux intensity : after exposure the
activities measured are those of the emitted fl or
Y rays. The method is simple, rather accurate but
gives only a mean value of the neutron flux during
the whole exposure.
A direct measurement of the neutron flux is possible
with fission chambers but they require a complex
electronic associated equipment and are generally
of notable dimensions : however a dealer proposes
a subminiature fission chamber of 1 mm O.d. and
10 mm long ¿f"1_7·
After the theoretical and experimental works of
MITELHAN ¿~2j, CESARELLI ¿"3_7 and HILBORN ¿~4_7,
a new type of instantaneous flux detector was deve
lopped, the "collectron" or continuous neutron flux
detector which is self powered, gives a signal direct
ly proportional to the neutron flux and can be easily
miniaturized.
*) Manuscript received on January 5> 1972
6
2. DESCRIPTION AND WORKING PRINCIPLE OF THE COLLECTRON
2.1. Principle ¿~2, 3, 4, 5, S J
The detector consists of two coaxial electrodes :
the inner electrode which is a (i emitter is isolated
from the outer electrode or collector (Pig.1).
When a neutron flux impinges on the device, the inner
electrode is activated and the emitted [2> particles
are collected by the outer electrode : it is suffi
cient to close the circuit on a suitable measuring
device to measure an electric current which, at satu
ration is directly proportional to the neutron flux.
2.2. Realization and component materials
Geometrically, the detector is composed of a ft emitter
wire, a solid dielectric material (usually high purity
AlpO,) and a sheath. The detector is then joined to a
coaxial connecting cable (Pig. 1 and 2). The measured
current is the resultant of ( emission from the emitter
and from some impurities always present in the sheath
and in the connecting wires.
2.2.1. The_emitter
The emitter must present a high crosssection to give
a good sensitivity but not too much in order to limit
the loss of sensitivity due to burn out. The radio
active half life for decay in the material should be
of the order of 4 to 5 minutes ■ for normal inreactor
applications. If other neutron absorbing isotopes are
present in addition to the principal Ô emitter they
must not produce longlived CL· emitter that could cause
slow variation of the measured current.
- 7 -
Alumina
; Stainless Steel
or inconel
Collector Coaxial connecting cable
Fig. 1 . PRINCIPLE OF A COLLECTRON
Detector
Q Ζ
\V\VA
Mono wire connecting cable
VVsS^S
Detector
Q X
JL
Two-wires connecting cable. R t 1 * R t t
Fig.2 MEASURING CIRCUITS a- normal wiring b- background compensation circuit
Emittei
ν 5 1
H h 1 «
A g 1 "
Δδ1°9
_
•ï Abundance
! *
I ι
i
'99.76
100
51.35
48.65
ι
ÎCross sec t ions ! a t 2,200 m/s !(barns) /~7 7
'· abs
jl56±7*
! 31-2
i 87-7
!
! •
ι •
¡ a c t
¡4,5^0,9
í 12 ì 2
¡140 ± 30
! 45 i 4
!3.2±0.4
J113 ¿ 13
'.Westcott
j a t 20°C j
í δ
i ¡ ι ! 1.00 ! | 1.023 i Τ ]
¡1.0044 j
! i ! 1 I ]
factors
¿SJ
s
0.0
7.255***
14.12
Capture-product ha l f l i f e
I *
3.76 m '
4.4 m
42 sec
9 9 . 9 / of 4.4 m
-*** 42 sec 2.3 m
270 j * *
24.2 s
% of 270 j
-»- 24.2 sec
Spectrum max.energy
(Mev)
•
2.6
2.44
\ 1.56
2.87
1st r e son -nance peak
(ev)
185
1.26
16.5
5.2
I
CO
I
ζ Westcott (AECL) counsels the normalized value of 150.19
13 —2 —1 xx negligible, 1 $> of signal after 1 year In a flux of 10 J n.cm s
xxx S 4
TABLE I Neutronic characteristics of /3 emitters
employed for tne realization of collectrons
9 -
The maximum energy of (b spectrum must be high to allow the β to escape from the insulant and from the emitter wire itself. All these conditions limit normally the choice of emitter to Rhodium, Vanadium or Silver whose main characteristics are summarized in Table 1.
2.2.2. The_sheath__or_collector
Normally choosen with regard to applicability (weldabi-lity, compatibility in normal working conditions) the collector is usually (as the sheath of the connecting cable) a tube of stainless steel or inconel with a thickness sufficient to stop j3 particles emitted from the central wire.
2 .2 .3 . 2°^£££ tog^cable
It is a miniature sheathed cable of the type normally employed for temperature measurements : the sheath is in stainless steel or inconel, the wires (or the wire) are in Ni, the insulant is usually MgO. This connecting cable can be the origin of spurious currents and (or) electric leakage. Theoretically a compensating wiring (see § 2.3.) can eliminate spurious currents from the coonecting. cable.
2.3. Measuring circuit
It is possible to measure either directly the current, or, more simply, the tension at the terminals of a load resistance according to Pig. 2 a . It is obvious that the cable leakage resistance R, must be large compared to the load resistance Β~.
10
Or, if R-, is very high under the laboratory conditions at the moment of the fabrication of the detector (R^^ 10 M-Ω- ) it falls drastically in normal working conditions (influence of temperature and irradiation on the insulant, see § 3.6). The connecting cable, which is always partially submitted to neutron flux can be the origin of spurious currents : in fact either the sheath or the connecting wires contain traces of β emitters and it is possible to obtain a secondary current coming from the connecting cable in addition to the main signal of the detector. Theoretically, a compensating wiring, as indicated in Fig. 2 b allows to eliminate this parasitic effect.
3. THEORETICAL SENSITIVITY OF THE COLLECTRONS
In the simplest case, when activation is due to a single emitter nuclide, the output current, as a function of time after exposure to the neutron flux, is given by :
Lj = K.N. 0. e.CT β - exp(0#693 Vl1/2).7 (1)
Ν = number of atoms with T. /?
0 = Westcott flux 0 = nv
G* = Westcott activation cross-section
^= CT0 (g + rs) — 19 e = electronic charge 1.602 10 coulomb
Tl/2 = half-life of the emitter
Κ = constant depending on geometry, beta self-absorption and neutron flux depression in the detector.
II
The response time Tp , which is the time requested to
obtain 63 fo of the saturation signal corresponding to an
instantaneous neutron flux variation, is defined as :
TE
1·
4 4 Tl/2
The equilibrium outputj or saturation signal, reached
after some Ti/2
i s "thus :
I = K.N.jZf.e.tf" (2) S
All neutron detectors gradually lose sensitivity, due
to burn out of the neutron sensing fi> emitter.
If N is the number of stable isotopes at instant 0
(beginning of irradiation), after the time t, it remains :
N = No exp ( CT . 0 . t)
and the equi l ibr ium output becomes :
I s = K±. NQ. 0. e . CT . exp (- <Γ . 0 . t )
3.1. Evaluation of the correcting factor K
In first approximation K may be considered as the pro
duct of two factors :
K = K.* . K—
where :
IL = neutron selfshielding and flux depression factor
Kp = selfabsorption factor of ß particles in the emitter
(and in the insulant)
K. and Kp must evidently take into account the geometrical
configuration of the device.
12
3.1.1. Self2Shielding_and__f¿^„degression^factor_K..
Generally speaking, the presence of an absorber of macros
copic size in a neutron flux modifies the flux : it cannot
be assumed that the flux in the absorber position is the
same as that which would exist in that position if the
absorber were absent. In general, the absorber will depress
the flux in its vicinity.
KUSHNERIUK ¿~9_J has shown that the ratio of mean flux to
surface flux in an infinite long cylindrical absorber sur
rounded by an infinite predominantly scattering medium
sustaining a thermal neutron flux, is adequately represen
ted by the following formula :
g m _!__ . JL 0O a.Ia * 2 |3
where
a = radius of the cylinder
X = macroscopic absorption cross section
β = Probability that a neutron falling in the surface
of the rod will be absorbed within it.
If 'Σ. is the macroscopic scattering crosssection and Σ + S u
the macroscopic total crosssection, |=» can be represented
by the following expression valid for a Έ. + <£. 2.5 a ,Za.P1 (a . Z a + a.Zs) a.Za+ a-£ s.P E S(aZ a + a^â)
13
where
Ρ (aüÜμ) = probability that a neutron, thrown into
the absorber from a oonstant and isotropic source density
in an infinite medium surrounding the absorber, undergoes
collision within the cylindrical absorber.'
Ppo (aZ.) » collisbn escape probability for an isotro
pic source distribution within the cylindrical absorber.
In these conditions :
j2 P.. (a.Σ«.) K s _ = 2 ï (3)
1 ^
a 2a ' ¿5P1 (a t27+ 2a.Zs.PES(a^t)
a.Z t a.Ta + a.? e
P1 and PES are given for varying (aZ.) in Pig. 3.
Moreover Mc GILL and al. ¿f"10_7 have made experimental
measurements whose results are in good agreement with
the KUSHNERIUK's computation.
Por values of a L_ up to/N/0.8 they propose the empirical
a correlation : χ (K&)
E1 — * 7-Γ (4)
1 K.a I„(Ka)
with Κ a C (Z.a)V2
a
C = 0.183 - 0.13
IQ and I1 are Bessel ' s functions,
- 14 -
i fl I!
o«l
S;^'lllllll! 1 fi ::::::
OD
Ò
ι F f f:T t Î M l l l l l l l l t~-
0 7 S i i s r ø S i : 1 SiïS^ff lSÏ:
Λ e- ^■ττττπττ_ττ
0,6 ^ i | |5 f
i::::ü±tttt:±t ι :
η «; éi i - fr îTff^f1
r - H T "ΤΗΤ'ΤΓΙ"
iiBI a:::::í:iUJÍÍaS
'ffW rft::t±4aiI.I;I ΤΓ'"ΤΤΤΤτΤΤ"ί""Γ~ o^lBfc
°·2 11 I l i i t i JL-j-1 -14-1 -1 -
■lili ο,ι Épglp ƒ U 1 ft
oEiiiliïilîj
llilill lllll 11 lililí 1 1 1! 1 4444-44+it lUll li Ü LMliffffWtfftfft
T TTTnTrTrpi ' ! t f Π t t ί ι Π Ι HT 'T ' ! ' ' 11 t ΙΤΓ^Η ΓΓ'ΤΡΓ
Slè^gniM
φ Τ||Γ[ {~Γ| r m [ I I T M Ì T
τφρτ^ΦΐΙΙ
¿■-f·'}·/ ff" J-TrTrr, f;rH* ΤΤ,ΤΤΓΤ.--
τ ì ■/! ΙΊ—Ηττι—Η" 111111 ΤΓΤΓΤ" v iTfíi!ííh
JifíliíWnTf
:
Τ—rrnvr Ι ' ' ' ' Γ ' ï ' ' ' ' Τ " τ"τ ι 11\ ΊΙΝ Μ : Ί ι '"·! ί 1' Ι τττττ"
τ^.Ι;ΐ[ίί\|ί.Ι'|.·ί·' 'l-iicHt-rR+TF
ttteljh iti
χ ìTi4p:tllS kém M X t ì: É Œ
: : : |4Ι1 , ,i:tf. pj.i,! LI-
l i p H I IBfïPf
f$íiii:
¡ 1 iff-
¿IPr" BBrml4f
~*~r I ■ ' ' " I * '——
::- -
■■ƒ *y4j+:
- |t|t|{|| 4
:: ±1111111
j 111 Η j - i-l χ - -14-4
§¡§11 TffrnTÍTffffr "TTT
14444 "fttt
ΤΓΠΤι^' Γ"-1^>^
ΤΠτΤπτΓΠΤΤ —
f* ' 'Π"'ί ΙΠΤ ΤΤΤΓ
¡^^•'-..ftf-
mm ~
f ^ f c ^ j ^
ìEÉìIsitt^^tt
- - r i * ΤΡΠΠΓΤΦΤ Γ
ΤΒΤ 111111 Ι ' Ί : '! ' Ι : 1111 Ι 11
ι Ι I ; h ι ' ! : ι 11 Ι • ■ i -ί: : : : IftTTPTfHîf f
^Γψ^ΓΠΐΗ4:ΐΙ|-ΒίΙ;Π
^B||T1!^-Îrj|?[|[|||(|[
ώβΐίΙ :::ffi:::|fcŒffl£
+111111 iti-1
'••Fifl
Ml
f-ffpF
f4
τπΒΐ
"ί'Γΐτ| ΤΤΠΉΤΠΤΠΊΤΤΤ H
j ; : uH-ftjii.LLL
IH q A 0,8 1,2 1,6 2,0 2.4 2 8
FABRICATION SUISSE
15
The experimental and calculated values are in¿cod
agreement with the computation made according to the ·.·;:;.
Kushneriuk's formula. The $ error "between the two methods
(5 to 10 <fo) is of the same order of magnitude as the expe
rimental measurements.
Values of K1 obtained "by the two methods are compiled in
Table 2.
j "~" ! ! \ ! Σ iEmitter !a(cm)Imacroscopic cross! a .2 . ! K1 ' Î I ! χ —1 ! !
{ { { section (cm" ) , , ; ! ! ! ! ! ! ! ! ! ! I ! . ! . ! +. ! ! Formula 3! Formula 4: ι ι ι a | s , t , , , 1 ■ I I I · · · ·
j ¡0.05 j ¡ | j 0.0176 J xx j 0.99 ;
¡Vanadium! !0.352!0.352!0.704! ! ! ί j .,; ¡0.025¡ | j ¡ 0.0088 | xx j 0.99 |
¡ ¡0.05 ¡ ¡ ¡ ¡ 0 .545 j 0.83 ¡ 0.82 j IRhodium ! 110.9 !0.366111.3 ! ! ! ! J ¡0.025¡ ¡ ¡ Ì 0.272 ί 0.91 ,! 0.90 j
¡ ¡0.05 ¡ ¡ ¡ ¡ 0.184 ¡ 0.88 ¡ 0.93 j
ISilver ! 13.69 !0.352! 4.04! ! ¡ ¡
j ¡0.025¡ ¡ ¡ ¡ 0.092 ¡ 0.92 ¡ 0.96 ;
TABLE 2 : Flux depression factor K1
χ from ANL 2nd edition
xx non calculated because of the magnitude of errors due
to P and PgS for values of a Z close to zero.
Values of K have been calculated for neutrons at 2,200 m/s;
they must be corrected for the Westcott flux and the neutron
temperature.
16
Preceding methods do not allow to take into account the
effects due to sheath and insulant. In fact some elabora
ted nuclear codes exist which are able to take into account
all the constitutive elements of a collectron.
The use of a WDSN code £"\\J leads to the results of Ta
ble 3· The computation was made, considering on one hand,
the shielding effect due to the single emitter and, on the
other hand, the global effect due to the whole detector
according to the geometrical conditions of Pig.4, in which
are also indicated the nuclear data adopted.
! J Γ ! Emit ter ! K. ! ! 1 ' 1 • · · I I I Ι Γ • É * i ·
! Mater ia l ! Radius ! due to the ! due to the ! 1 t ___ t 1 I
j i i s ing le emi t t e r ¡ whole de t ec to r ¡
ι ; ι t j
! ! 0.05 ! 0.99 ! 0.99 !
j V j 0.025 i 0.99 j 0.99 ;
j 0.05 j 0.71 ¡ 0.56- ¡
! mL ! 0.025 ! 0.86 ! 0.75* ! ! ! ! ! ! ! ! 0.05 ! 0.90 ! 0.83 !
j A g j 0.025 j 0.95 ' j 0.90 j • · · # · "
TABLE 3 Computation of K. according to a nuclear code,
With respect to the preceding results the only sensible
difference regards the rhodium, The additional effect of
sheath and insulant is in the range of\10 fo.
CONSIDERED GEOMETRIES ALL DIMENSIONS IN m m .
Σ, 9. 6 57
3. 2 6 9
0. 2 8 8
(2) A l2 ° J 0 . 0 0 8 8 3
Θ Θ
Vold 0
Stainless Steel 0 ,2 1 8
0 D2 3.3 10'
0 , 3 6 6
0.3 52
0 . 3 2 5
0.3 1 8
0 . 8 6 9
0.4 4 9
NUCLEAR DATA ADOPTED
I
Fig. 4 - COMPUTATION OF K1 FROM A NUCLEAR CODE
18
3.1.2. Selfabsorgtion_faetor_K2
To each monoenergetic p» particle is associated.a maximum
range X (which is normally expressed in the form of A · Ρ
in mg/cm of aluminium).
The probability for aß particle emitted from a point of a
cyclindrical source of radius a to escape from the cylin
der is a function of a'and X . För example, a β particle
formed in the center of a rod, with a range less than the
radius of the rod, would not escape.
The same β particle might escape if it were formed near
the edge of the rod, depending on the direction of travel.
If a sphere, with a radius equal to the range \ of the
considered particle is placed with its center coincident
with the origin of the (3 particle, the escape probability
is the ratio of the area of the sphere that is outside of
the rod divided by the total area of the sphere. It is
possible to make a complete computation from this model
¿~12_7» fcut it is easier to use the following formula,
valid for a cylindrical emitter the diameter of which is
low with respect to the length.
*? *= 4" ¿"0.5 F 2 (f)_7 CHJ
A = range of the particle in cm.
a = radius of cylinder in cm.
a F« (—)' integral function defined by
^ 2
F2 (4) . a/ E 3 (a)
A-* with E, (q.) = I — κ — . dp
A P
19
Values of E (q.) and F„ (q) are tabulated in Annex I
or in Ref. Ζ"14./·
The range ^ must be calculated from empirical formulas
like :
\ . ù = 0.543 E 0.16 ¿~13j valid for aluminium
between 0.8 and.3 Mev but extrapolable to
other materials,
)v . f « 0.142 E1 , 2 6 5
0.0954 log E C^J
valid f or 0.01 < E < 2.5 Mev ι J
> . f = 0.53 E 0.106 C^J
valid for E > 2.5 Mev
where Ρ is the density of the considered material.
The computation of Kp is complicated by the fact that
β particles are emitted with a continuous: energy spectrum
with energies varying from 0 to a maximum value which is
considered as characteristic of the (à emitter considered.
A general shape of a (3 spectrum of energies is given by
Fig. 5.
To each number Ν. of (* particles of a given energy must
be associated a range \. which allows to calculate a cor
responding value K. of the self absorption factor.
The value of K«, for the considered reaction, is the inte
gral of these elementary K. values. An approximate value
of the solution can be obtained in assimilating the β energy
spectrum to a sum of rectangles to each of which is asso
ciated a mean number N. and a mean energy E..
20 -
UI
>» σι k.
c
c 3
•
■o •
E Ε max.
Kinetic energy E ( M e V )
Fig. 5 β EN ERGY SPECTRUM
21
For each elementary rectangle it is possible to compute
a mean absorption factor ^ ; approached value of Kg is
then given by the sum :
2.Λ K± K2
Hi
The only difficulty consists in determining the Û, energy
spectrum of the considered reaction.
In the litterature it is possible to find values of maxi
mum energy E max. which are normally used in radiation
protection; the few experimental (3 spectra existing are
related to elements with a rather long halflife which is
evidently not the case of Rh, Ag, V. The only alternative
consists in calculating, for each considered element, the
theoretical corresponding fi> spectrum, or to use an empiri
cal formula giving, for the whole spectrum, a mean energy
value.
a) determination of K» from a theoretical fi> energy spectrum.
The computation, detailed in Annex 2, has been made in
the case of Rh . The corresponding value of Kg is 0.41.
b) determination of K2 from a mean energy value.
A mean energy value (at ± %) of a (3 spectrum can be
derived from the following formula :
Ë = 0.33 E (1 4ζΓ) <1 + Τ"") ¿~
157
where : E = maximum energy in Mev
Ζ = atomic number of the stable isotope.
22
Always in the case of Rh where E max. = 2 and Ζ = 45
this formula gives Ë = 0.967 which leads to a value of
0.43 for K2.
This value is a justification of the validity of this de
termination and the results obtained, according to the
application of this formula, are summarized in Table 4
for Rh, Ag and V in 0.5 mm diameter.
! Emitter
| Eh10*
! τ5 2
1
•
!
! Isotopie ¡abundance !.. 1o
ι 100
j 51.3 5
j 48.65
i 100
I
1
• !
•
!
!
!
1
•
Energy
E max.
2.44
1.56
2.87
2.60
I •
I 1
(Mev)
Ë
0.97
0.59
1.16
1.08
I
!
!
v .'*
0.43
0.29
0.54
0.62
TABLE 4 Values of K~ computed from mean
energy values
23
3.2. Sensitivity of the collectrons
With the values of K, and K„ it is now possible to calculate
the theoretical sensitivity of the different (ò emitters
according to formula (1). For thermal neutrons at 2,200 m/s
and neglecting the possible effects of resonance phenomena
in the epithermal region, the theoretical values of the sa
turation current are given in Table 5 for the different emit
ters considered.
K = K rK 2 I corrected s
A per neutron
0.61
0.39
0.27
0.50
0.066
1.34
II- .416
10
10'
21
■21
10 21
TABLE 5 Theoretical values of the saturation current
for various emitters of 0 0.5 mm and 10 mm long.
As neither the silver nor the rodhium follow a J_ law, it is
necessary for these elements to correct the vaïues obtained
for the Westcott cross section which depends on the condition
of irradiation. Moreover all values must be corrected for
the neutron temperature.
- 24 -
4. LOSS OF SENSITIVITY OF THE VARIOUS β EMITTERS AS A FUNCTION OF NEUTRON DOSE
Due to irradiation the number of stable isotopes diminishes with increasing neutron exposure and consequently the sensitivity of the detector diminishes with neutron exposure. In Table 6 is indicated the loss of sensitivity (in io with respect to initial sensitivity) in function of the integrated flux for various emitters.
Emitter — 2 —Ί
Integrated flux ( n. cm" .-S ) 1020 ! 5.1020 ! 1021 5.10 21 ! 10 22
V
Ag
Rh
0.045
0.44
1.5
0.22
2.2
7.5
0.45
4.4
15
! 2.25
22
!.75
4.5
44
TABLE 6 - Loss of sensitivity versus integrated neutron flux.
25
REFERENCES
¿"1_7 20th Century Electronics Ltd
New Addigton Croydon Surrey G.Β.
Chamber type FC 02 A
¿~2j G.MITELHAN, R.S.EROFEEV, N.D.ROZENBLYUM
Transformation of the Energy of ShortLived Radioactive Isotopes
Soviet Journal Atomic Energy 10 n°1 (1961)
¿~3j G.CASARELLI
A detector for High Neutron Flux Measurements
Energia Nucleare 10 n°8 (1963)
¿"4.7 J.W.HILBORN
Self Powered Neutron Detectors for Reactor Flux Monitoring
Nucleonics 22 n°2 (1964)
¿~5j W.R.LOOSEMORE, G.KNILL
Primary Emission Neutron Activation Detector for InPile Neutron Flux Monitoring
H.L. 66/1842 Harwell (1966)
¿~SJ M. GRIN
Les collectrons. Mesure en continu des flux de neutrons thermiques
EUR 3567 f (1967)
</~7_7 Neutron Cross Sections
BNL 325 2nd edition (19 58)
- 26
¿~QJ C H. WESTCOTT Effective Cross Section Values for Well-Moderated Thermal Reactor Spectra AECL 1101 (1962)
¿"9_7 S.A.KUSHNERIUK Neutron Capture by Long Cylindrical Bodies Surrounded by Predominantly Scattering Media AECL 469 (1957)
¿"10_7 I.S. Mc GILL The Distribution of Flux and Temperature in Thermal Reactor Fuel Rods DEG 319 (1961)
¿"11_7 M.F. JAMET Personal Communication
¿~Ï2J L.D. PHILIP Beta Escape Efficiency from a Cylindrical Source H.W. 81.338 Hanford (1964)
/JM J S. GLASSTONE Principles of Nuclear Reactor Engineering Mc Millan and Co Ltd P.629 p.70 (19 56)
¿~14_7 Tables of Sines, Cosines and Experimental Integral Federal Work Agency Project. WPA U.S. Government Printing Office (1940)
¿"15_7 Nuclear Engineering Handbook (1958) H.Etherington Editor, Me Graw Hill 2.34, 7.33
- 27 -
A N N E X 2
Numerical values of the funct ions :
P2 (q) = <i2 r E 3 (q)
Values of E. have been computed in CETIS (C.C.R.ISPRA) in assimilating the integral to a sum of trapezia : thus the tabulated values are slightly higher than the theoretical ones.
29 -
E0 (q) F0 (q )
to
h UI ü
< K Q. tø
■
α
υ
υ
Σ Ο
<
ω
1 2 3 1+ 5 6 7 8 9
10
π 12 13 1U 15 16 17 16 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3U 35 36 37 38 39 1+0 41 4? 43 UU 45 46 1+7 1+8 U9
I O , l o 1 . Io lo lo lo lo lo lo 2 · 2 . 2 . 2o 2 . 2 . 3 , 3o 3 · 3 . l+o l+o
u„ 1+. 5o 5 e 6o 6o 7o 7o 8o 8 · 9o
IO* lo lo lo lo lo lo lo lo 1 . 2 . 2o 2 . 2 · 2 . 2o
00000E-03 072 27E-C2 11+976E-02 23285E-02 3219UE-02 41747E-02 51991E^C2 62975E-02 71+753E-02 87382E-02 00923E-02 151+1+3E-02 310 13E-02 U7708E-02 65609E-02 84804E-02 05386E-02 271+55E-02 51119E-02 76494E-02 03702E-02 32876E-02 64159E-02 97702E-02 33670E-02 72237E-02 135 91E-02 57933E-02 051+80E-02 56463E-02 1 Π 3 1 Ε - 0 2 6971+9E-02 32603E-02 00000H-02 07227E-0 14976E-0 23285E-01 32194E-0" 41747E-0 51991E-01 62975E-0 74753E-0 87382E-0 00923E-0 15443E-01 31013E-0 ' 1+770 8 E-0 ' 65609E-0 ' 84804E-0
i+o9290E 03 1+.2813E 03 3.7183E 03 3.2291E 03 2 ·8039Ε 03 2oU3i+i+E 03 2 .1133E 03 1.8344E 03 1,5920E 03 103815E 03 1.1976E 03 1.0339E 03 9o0100E 02 7o8128E 02 6o7732E 02 5.8706E 02 5o0870E 02 •4o4069E 02 3.8167E 02 3.301+5E 02 2o8601E 02 204727E 02 2.1388E 02 U8U92E 02 U5982E 02 lo3806E 02 1.1921E 02 1.0288E 02 8o8675E 01 7o6Ul+8E 01 6o5866E 01 5,6713E 01 Uo8798E 01 Ue1957Ë 01 3·6021Ε 01 3.0922E 01 2o6522E 01 2.2727E 01 1.9455E 01 1.6625E 01 lo»+202E 01 1.2117E 01 lo0325E 01 8o7796E 00 7 o4607E 00 6 .3301E 00 5.3621E 00 4.531UE 00 3e8250E 00
l+o9290E-01 Uo9225E-01 4 .9154E-01 4 . " 0 7 9 E - 0 1 U e8999E-01 U.3913E-01 1+·8821Ε-01 U.8723E-01 4 .8618E-01 U„8506E-01 4 · 8 3 4 7 Ε - 0 1 H.8220E-01 4 ,Π034Ε-01 l+o7939E-01 4 ·7784Ε-01· 1+.7618E-01 4 . 7 4 4 2 E - 0 1 4 o 7254E-01 4 . 7 0 5 4 E - 0 1 1+.68U0E-01 1+.6613E-0-4 . 6 3 3 4 E - 0 I+.6078E-01 l+ .5806E-0
r
1+.5516E-0· 1+.5208E-0· U.U882E-0· U.I+535E-0· 4 . 4 1 3 4 E - 0 4 .3 '746E-0 ' 4 . 3 3 3 6 E - 0 ' 4 . 2901E-0 l+o2UU2E-0· U.19S7E-0' 1+.H+15E-Ö-U.0878E-01 Uo0312E-0^ 3 . 9 7 1 6 E - 0 ' 3 . 9 0 9 0 E - 0 ' 3.R1+07E-0· 3 o 7 7 2 l E - 0
-
3 .7003E-0 · 3 . 6 2 5 2 Ξ - 0 ' 3 . 5 4 4 4 E - 0 ' 3o l+629E-0-3 .3782E-0 · 3 .2901E-0 · 3 . 1 9 6 8 E - 0 ' 3 .1026E-0 ·
«I - CltANO I k « · .
30 -
E 3 ( q ) F, ( q )
(fl
H
ω υ
< α α. tø
α:
υ
ϋ
Σ Ο Η <
ω
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100
3 ·05385Ε-0 3 ·27455Ε-01 3 ο 5 1 Π 9 Ε - 0 ~ 3 .76493Ε-0 4 .03702Ε-0 4 · 3 2 3 7 6 Ε - 0 4 .64159Ε-0 4 .97702Ε-0 5 ·33670Ε-0 5 ·72237Ε-0 6ο13591Ε-0 6 .57933Ε-0 7 .05480Ε-0 7 · 5 6 4 6 3 Ε - 0 8 ο Π 1 3 1 Ε - 0 8 ·69749Ε-0 9 ·32603Ε-0
10 ·00000Ε-0 ο07227Ε .14976Ε ο23285Ε •32194Ε ο41747Ε •51991Ε .62975Ε •74753Ε .873 82Ε
2ο00923Ε 2ο15443Ε 2.31013Ε 2ο47708Ε 2.65609Ε 2·84803Ε 3.05385Ê 3ο27455Ε 3.51119Ε 3.76493Ε 4·03702Ε 4ο32876Ε 4.64159Ε 4.97702Ε 5ο33670Ε 5ο72236Ε 6.13590Ε 6.57933Ε 7ο05480Ε 7 ·56463Ε 8 . 1 Π 3 0 Ε 8.69749Ε 9.32603Ε
10.00000Ε
00 00 00 00 00 00 00 00 00 00 00 00 00 "0
ο 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
3ο2225Ε 00 2ο7092Ε 00-2.2714Ε 00 " •9011Ε 00
.5873Ε 00 ο3209Ε 00 ο0970Ε 00
9 .0829Ε-0 7ο4923Ε-0 6 . 1639Ε-0 5ο0530Ε-0 4-.1267Ε-0 3 .3550Ε-0 2 . 7 1 7 6 Ε - 0 2 · 1 9 1 4 Ε - 0 U 7 5 7 7 E - 0 U 4 0 3 5 E - 0 1 ο Π 4 6 Ε - 0 1 8ο7967Ε-02 6ο9040Ε-02 5ο3833Ε-02 4ο1667Ε-02 3ο2032Ε-02 2 .4432Ε-02 1 .8472Ε-02 U 3 8 5 0 E - 0 2 1ο0287Ε-02 7ο5609Ε-03 5 .5010Ε-03 3 · 9 5 6 8 Ε - 0 3 2*8105Ε-03 1 .9717Ε-03
i'.mmi 6 .2409Ε-04 4 .1197Ε-04 2 .6713Ε-04 L 6 9 9 0 E - 0 4 1 .0592Ε-04 6 ·4622Ε-05 3 .8521Ε-05 2ο2397Ε-05 U 2 6 8 6 E - 0 5 6 ·9859Ε-06 3 .7327Ε-06 U 9 3 0 9 E - 0 6 9ο6525Ε-07 4 β 6509Ε-07 2 ,1545Ε-07 9ο5680Ε-08 4 . 0 6 2 9 Ε - 0 8
3 .0053Ε-0 2 .9050Ε-0 2 · 8 0 0 3 Ε - 0 2 .6948Ε-0 2 .5869Ε-0 2 · 4 7 5 2 Ε - 0 2 .3634Ε-0 2 . 2 4 9 9 Ε - 0 ' 2 . 1338Ε-0" 2ο0184Ε-0 U 9 0 2 4 E - 0 1 1 .7863Ε-0 1 . 6 6 9 8 Ε - 0 ' 1 .5551Ε-01 Ί . 4 4 1 8 Ε - 0 1 .3296Ε-0 1.2207Ε-0 1ο1146Ε-0 1.0114Ε-0 9 .1267Ε-02 8 .1822Ε-02 7 .2814Ε-02 6 .4359Ε-02 5 .6440Ε-02 4 .Q063E-02 4 .2297Ε-02 3 .6120Ε-02 3ο0524Ε-02 2 ·5533Ε-02 2 .1116Ε-02 U 7 2 4 5 E - 0 2 1 .3910Ε-02 1 . Ί066Ε-02 8.67U3E-03 6 ·6919Ε-03 5 ·0790Ε-03 3 .7866Ε-03 2ο7689Ε-03. 1 .9848Ε-03 1.3922Ε-03 9 .5420Ε-04 6 ·3787Ε-04 4 .15U2E-04
' 2 . 6 3 0 2 E - 0 U 1.6158Ε-04 9 ·6100Ε-05 5 .5235Ε-05 3 .0600Ε-05 1.6298Ε-05 8 .3217Ε-06 4 .0629Ε-06
216 LINES OUTPUT THIS JOBo
31 -
A N N E X 2
Evaluation of the self-absorption factor of Rh
from the corresponding theoretical β energy spectrum
32
A.2.1. Determination of the theoretical β energy spectrum
¿"λ 1 and A 2_7
For an allowed transition, the repartition of the energies
of a ß spectrum can be derived from the law :
Ν'(ρ) dp = K.G ( ± Ζ, Σ ) ρ . Ζ ( Σ 0 Σ )2 dp
where :
ρ = particle momentum
<ΰ = energy of the (b particle expressed in relati-vistic units
2-0 = maximum energy of the spectrum
Ζ = atomic number of the daughter
G- ( - Ζ, Σ. ) = Fermi function
Ρ = (Z2-1)1/2 y E
0.512
G ( - Ζ, Σ ) is tabulated in function of '.'/"A 2 7 Ζ
Κ is a complex factor whose absolute value is not necessary for the specific case considered where the only important term is the statistic factor Σ ( Έ-Q - Σ ) which is representative of the shape Of the spectrum.
33
After integration it follows :
N = Κ· . G (t Ζ,Ζ ).ρ.Σ (Σö -Σ.):
In the specific case of rhodium we have :
E max. = 2.44 Mev
Ζ of the daughter =■ 46
Σ. β _ÌL_ + 1 = 5.765 0 0.512
It is possible to plot a theoretical spectrum, similar
to the true spectrum except for the K· coefficient.
The calculated values are tabulated in Table I and the
corresponding spectrum has been plotted in Fig. I.
A.2.2. Evaluâtion_of_the_selfabsornti£n_factor
ÏZ_ÊSËî!êZ_^!2HP.S
From the preceding values it becomes possible to calculate
a selfabsorption factor in assimilating the spectrum to
a sum of rectangles each of which corresponding to a mean
energy value and a mean number of particles (proportional
to the true number by K1 ).
By successive applications of the formula :
a good approximation of K«, extended to the whole spectrum,
can be obtained.
For a spectrum assimilated to a sum of 10 rectangles where
the mean data are those of Table I the calculation, indica
ted in Table II, leads to : 19
Σ N f K2 = ^ = 0.41
^
Ν
E (MeV)
^ ( W .2.138
2.074
.1.830
1.586
.1.342
1.098
^0.854
.0.610
.0.366
.0.122
0
r
5.765(Σ0)
5.527
5.05
4.574
4.097
3.621
3.144
2.667
2.191
1.714
1.238
Ρ=(Σ.2-1)1/2
5.44
4.95
4.46
3.96
3.48
2.98
2.47
1.95
1.39
0.73
0
Ρ Ρ
ζ 46
0.1182
0.1076
0.0969
0.0860
0.0756
0.0647
0.0537
0.0424
0.0302
0.0158
G ( ± Ζ , Ζ )
4.22
^ ο - ^
0
0.238
0.715
1.191
1.668
2.144
2¡62t
3.098
3.574
4.051
4.527
( * ο - Σ > 2
0.056
0.511
1.418
2.782
4.596
6.869
9.597
12.773
16.410
20.493
Ν.Κ·
0
7.10
53.90
122.07
191.14
244.36
271.58
266.70
230.26
164.92
78.08
0
00
4^
TABLE Ι Determination of the theoretical β spectrum of Rhodium IO4.
N
300-
250.
200.
150.
100.
50.
Ω
(arbitrary units)
-
ry—
/ © ©
/ I
©
ι
-*--'
i
, ■
•
Θ
I
Γ Τ = - = » ^ _ _ - ^
• ■
:
;
© •
1 ι
— _ " ^ _
'
©
" Ί
.Κ \~Ί χ Ι ;
• \
~~
~ Ι ι ι
¡ © ¡ © j © \ ®
Ι Ι ' ί? ¡χ*
OS
U i
0,2ΑΑ 0,438 0,732 0,976 1,22 Ι,Α6Α 1,708
Fig.I - R h, ( K
THEORETICAL β ENERGY SPECTRUM
1,952 IM
1 > -
E(MeV)
N
E
X cm
a a
F ( - ) 2 V f
Nf
1
781
0 .122
0
0
ï
2
1649
0 .366
0 .004
0 .16
6 .2
2.6.1lT
0 .08
132
l!. of c o r r e s p o n d i n g r e c t a n g l e
3
2303
0 .610
0 .014
0 .56
1.78
0 .04
0 .26
599
4
2667
0 .854
0.026
1.04
0 .96
0 .12
0 .40
1067
5
2716
1.098
0 .035
1.42
0 .70
0 .16
0 .48
1304
6
2444
1.342
0.046
1.84
0 .54
0.21
0 .53
1295
( see F i g . I I )
7
1911
1.586
0 .057
2 .28
0 .43
0 . 2 5
0 .57
1089
8
1221
1.830
0.068
2 .72
0 .37
0 .27
0 .63
769
9
539
2.074
0.078
3 .12
0 .32
0 .29
0 .66
356
10
71
2 .318
0.088
3 .52
0 .28
0.31
0 .67
47
CO en
TABLE II - Computation of the self-absorption factor from the (3> spectrum
K, -1 10
N.f
N
6.658 16.302
0.41
37
BIBLIOGRAPHY
¿"λ Λ J R.D.EVANS, Le noyau atomique, DUNOD, PARIS (1961)
¿"A 2_7 K.SIEGBAHN, Beta and Gamma Ray Spectroscopy
North Holland Publishing Co, Amsterdam (1955)
mr
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