Hembd, H.

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EUR 3275.e EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM THE CONNECTION BETWEEN THE TIME-MOMENTS AND THE FUNDAMENTAL DECAY CONSTANT OF A THERMALIZED NEUTRON PULSE by H. HEMBD . 1967 -•uhi · Joint Nuclear Research Center Ispra Establishment - Italy Reactor Physics Department Reactor Theory and Analysis LEGAL NOTICE This document was prepared under the sponsorship of the Commission of the European Atomic Energy Community (EURATOM). Neither the EURATOM Commission, 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 rights ; or 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. This report is on sale at the addresses listed on cover page 4 at the price of FF 4.—- FB 40. DM 3.20 Lit. 500 Fl. 3.— When ordering, please quote the EUR number and the title, which are indicated on the cover of each report. Printed by Vanmelle Brussels, February 1967 This document was reproduced on the basis of the best available copy. EUR 3275.e THE CONNECTION BETWEEN THE TIME-MOMENTS AND THE FUNDAMENTAL DECAY CONSTANT OF A THERMALIZED NEUTRON PULSE by H. HEMBD European Atomic Energy Community — EURATOM Joint Nuclear Research Center — Ispra Establishment (Italy) Reactor Physics Department — Reactor Theory and Analysis Brussels, February 1967 — 14 Pages — FB 40 The decay of a thermalized neutron pulse in a large moderating sample is studied by the time moments method applied to the Boltzmann equation. The time moments define the mean emission time or lifetime for neutrons before escaping or being absorbed at a certain energy. The decay constant of the neutron pulse is derived as a mean value of the inverse emission time. The first three coefficient^ in the expansion of the decay constant in terms of the geometrical buckling are identical with those obtained by Nelkin in hi« eigenvalue approach. Further properties of the time moments and their connection with the experiment are pointed out. EUR 3275.e THE CONNECTION BETWEEN THE TIME-MOMENTS AND THE FUNDAMENTAL DECAY CONSTANT OF A THERMALIZED NEUTRON PULSE by H. HEMBD European Atomic Energy Community — EURATOM Joint Nuclear Research Center — Ispra Establishment (Italy) Reactor Physics Department — Reactor Theory and Analysis Brussels, February 1967 - 14 Pages - FB 40 The decay of a thermalized neutron pulse in a large moderating sample is studied by the time moments method applied to the Boltzmann equation. The time moments define the mean emission time or lifetime for neutrons before escaping or being absorbed at a certain energy. The decay constant of the neutron pulse is derived as a mean value of the inverse emission time. The first three coefficient in the expansion of the decay constant in terms of the geometrical buckling are identical with those obtained by Nelkin in his eigenvalue approach. Further properties of the time moments and their connection with the experiment are pointed out. EUR 3275.e EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM THE CONNECTION BETWEEN THE TIME-MOMENTS AND THE FUNDAMENTAL DECAY CONSTANT OF A THERMALIZED NEUTRON PULSE by H. HEMBD 1967 Joint Nuclear Research Center Ispra Establishment - Italy Reactor Physics Department Reactor Theory and Analysis SUMMARY The decay of a thermalized neutron pulse in a large moderating sample is studied by the time moments method applied to the Boltzmann equation. The time moments define the mean emission time or lifetime for neutrons before escaping or being absorbed at a certain energy. The decay constant of the neutron pulse is derived as a mean value of the inverse emission time. The first three coefficients in the expansion of the decay constant in terms of the geometrical buckling are identical with those obtained by Nelkin in his eigenvalue approach. Further properties of the time moments and their connection with the experiment are pointed out. THE CONNECTION BETWEEN THE TIME­MOMENTS AND THE FUNDAMENTAL DECAY CONSTANT OF A THERMALIZED NEUTRON PULSE (*) The time behaviour of a neutron pu lse in a l a r g e the rmal iz ing medium can be descr ibed by the time dependent Boltzmann equation in t h e d i f ­fusion approximation J- · JJ + ( 2^+ DB*) ψ = t f f +S(E,B)^(t) (1) where ^(E,B,t) is the neutron flux, Β the buckling, S(E,B)*0(t) the source distribution induced as a pulse at t = O, D(E) the diffusion coefficient, y^ (E) the absorption cross section, and % the thermalization operator defined in the usual way by o? % ψ s f :ξ(Ε·­*Ε) ^(E')dE' ­ ¿á(E) ^>(E) (la) with 2"s (E) = ƒ Z,(E-» E»)dE The problem of solving eq. (1) becomes more simple by assuming /v - ab­sorption ν·2ΰι(Ε) = a = const, (lb) a condition generally fulfilled in practical situations. The substitution <^(E,B,t) = e"at ψ (Ε,B,t) (IO reduces eq. (1) to the absorption free case (*) Manuscript received on December 13, 1966. - * -ν τ τ + r ø if = £f + s*^(t) (2) which will be the basis of the following investigation. Nelkin £l] has studied the eigensolutions of the source-free eq. (1) by the ansatz <^(E,B,t) = e~°Ct . (f> (E.B). (3a) This is equivalent to applying the ansatz ^(E,B,t) = e"(0C_a)t. ^ (E,B) (3b) to eq. (2). For the fundamental eigenvalue in terms of the buckling he finds 2. t s OC- a = DnB - C B + ... (4) with J D(E) M(E)dE D0 - τ*- (4a) J - M(E)dE jf(D(E) -£s_ ) ^0(E)dE (4b) °Ύ 1 ƒ ~ M(E)dE Í V where M(E) is a Maxwellian and ψλ ^ (E) the solution of D , (­ ­ D(E) ) M(E) ·= % %D<-E>· < 4 c ) - 5 -At it is wellknown, the solution of eq. (1) by a superposition of eigen-functions cannot be done in the classical way since the eigenfunctions do riot form a complete set [2] . However, instead of solving eq. (1) ex­plicitly in time and energy we can obtain the same information by con­sidering the time moments which satisfy a simpler equation. The time moments of order k are defined as Μ^Ε,Β) =r it" <p(E,B,t) dt = J e~at t* ^(E,B, t)dt (5) or m (Ε,Β) Ξ ƒ t ^(E,B,t) dt (5a) in the absorption free case of eq. (2). The time moments are of interest for three reasons: (i) They can be drawn out from pulsed experiments with great preci­sion since they are time-integrated quantities. Moreover, the time- and energy-integrated reaction rates measured by a /v -detector E max rK(B) = J J- mK (E,B) dE (6) E min are connected with the fundamental eigenvalue Λ in the follo­wing way r0 C B ) τ 2 2 ,D, = λ (Β), (for Β2 « ΣΖ ) (7) r (Β) 8 a result which will be shown afterwards. ­ 6 ­(ii) They satisfy a Boltzmann equation which is stationary but with a modified source term. Hence they may be calculated by a rou­tine program like GATHER f3] . The latter provides an option for time moments. (iii) The ratio mY (E,B) T(E) = (8) mc (E,B) has a physical meaning. Called "emission time" it is the mean lifetime of a neutron before escaping with an energy E. , v k To derive the equations for the time moments we multiply eq. (2) by t and integrate from zero to infinity: DB Z mo(E,B) = if mo(E,B) + S(E,B); (k = 0) (9,0) k DB mk(E,B) = ¡ζ m^E.B) + ­ m^íE.B). (k* 1) (9,k) To solve eqs. (9) we expand all quantities in a power series of B and 2 compare terms of the same order in B : m0 (E,B) = moe(E) + B^m^ (E) + B*n^(E) + ... (10,0) mi (E'B) = Bx fmio (E) + B* m « <E) + Β*».,/Β> + ­­'J (10,1) mz (E,B) = 5­#[mi0(E) + B^m^ (E) + Β^πι^(Ε) + ...J (10,2) The expansions (10,k) observe the fact that the k­th moment is an order of magnitude (in Β ) larger than the (k­l)­th. This can be seen from the con­dition of neutron conservation - 7 -B2(D,ii i) ■ J % mK dE + k ­ ( ^ , m ^ ) , (11) Û since for a l l d i s t r ibut ions Jf(E) ] tf fdE s 0 (11a) ρ which is a consequence of the definition of the operatori la). The brackets mean QO (u,v) s J u(E) v(E) dE. The source in (9,0) must be of the same order as the leakage term on the left, hence S(E,B) = B2 Sx (E) + Β S (E) + ... (12) Eq. (9,0) splits up into 0 = Ζ m (13,0) D**, = £ "ey + sy (13'2> We infer the solution m = M(E) » ­ ­ exp (­E/kT) (14,0) and,observing eq. ( l i a ) , the conservation conditions (D,M) = ƒ S¿ dE (14,1) où (D,mei) = ƒ S^ dE (14,2) - 8 -Eq. (9,1) splits up into 0 Dm>« 40 = = ^m1o * " » + V m o o V mo¿ (15,0) (15,1) (15,2) with the conservation conditions m1o = c M(E) (16,0) c1 (D,M) = (- , M) or c, = Ό~ (see eq. (4a) ) (16,1) (D,me) = (^  , m M ) . (16,2) For the determination of the ratio (7) of reaction rates (6) we have for small Β r*(B) = ( Γ , " ) + B Z ( ; , m 0 1) + ... or with (16,2) = ( J- , M) + BZ (D, m<a ) + ν and ■< C B > = B* f D c ( v ' M ) + ß Z ( v ' m « ) + '··] Hence De —7­T­ = D Β ­ Β · τ + 0(B' ) . (17) r, (Β) ° < è . M) ­ 9 ­This is the same result as given by Nelkin (see eqs. (4) ) for the funda­mental eigenvalue. The solution m (E) of eq. (15,1) satisfies la D ^(­D om^) = ( — ­ D ) MCE) which is exactly Nelkin's equation for his φ . 2 The reciprocal emission time in terms of Β is given by m CE) . meCE,B) MCE) [ l + B - ^ r + . . ] MCE) T ( E ) m^E.B) r 2 "VL<E) Τ (B2Do) 1 M C E ) [ l + B \ - ^ i T + . .] 2 = D c ß 2 ' f 1 - MT?) C D c m , z ( E ) - m c x ( E > J + 0 ( B 4 ) J . C18) Averaging this quantity over all energies with a Maxwellian density as weight function fi MÍEJ. /L> = *Έ(Ε) ν dE OÇ Β- dE f Míj Κ ν (19) yields same right hand side as eq. (17) so that we have the agreement ik -M 0(B6) C20) 2 for small Β . From a physical point of view it may be more justified not to use the Maxwellian density in the averaging prescription Cl9) but the actual neutron density connected with the solution of eq.C9,0). Operating with m0CE,B) we get the same asymptotic relation C20), now 6 with another coefficient of Β . The behaviour of the emission time for small values of the energy is of interest. For the functions m and m which enter eq. Cl8) we make the ansatz m02CE) = MCE) [a0 + a^ + .. ] C21.0) m nCE) = MCE) [bo + bxE + .. ] . C21,l) - 10 -These expansions are coupled by the condition of neutron conservation written in the form of eq. Cl6,2). With the help of eq. Cl6,l) we find CD b - a ) (- , M) = a. C~ , EM) - b CD,EM). C22) Ο Ο Ο γ T V 7 The desired expression to be inserted into eq. (18) is MTU K m « ( E ) - mo*(E)J - [(D.bo - V + E <Dob, - a 1 > ] · Do or , again with Cl6, l ) and C22) Do Φ » M> J a1 [ (J· , EM) - EC ·^ , M)] + ^ [CD,M)E - CD,EM)] ƒ . C23) Averaging in the sense of definition C19) cancels the cofactor of a., whereas the second member in (23) yields a diffusion coefficient Cf which is an approximation to the true diffusion coefficient C due to the fact that we have retained only two terms in the expressions C21,0) and (21,1): c* " °Λ°. * <L > M ) " 1 < — - D> EM>· <24> 1 Ù i ν ν The coefficients a^  , b can be determined from eqs. Cl3,l) and Cl5,l). The equation for &. is DCE) MCE) = a,, % [EMJ+ SZCE) . The common way to solve for a^ is to multiply by E and to integrate. But then the mean source energy would dominate in the above equation. This is not desirable since the source has not been specified, except that it is concentrated near a mean source energy Esin the high energy range. Instead, we weigh by E so that I o» | s z C E ) d E = ¿ ­ | s z d E = J ­ · CD,M)^C~ , DM) ­ 11 ­can be neglected. CUse has been made of eq. C14.1)). Now é , DM) a ­ ­f ­= ■£·<*. DM) C25) 1 φ^ΕΜ) f? E where the energy transfer moment A*= JJ^—^MCE») ^ CE'­>E) dE'dE = 2 (^EM)^O C25a) σ has been introduced CFor the heavy gas operator U - 2*^2^ » where %Σ^ is the constant slowing down power). In the same manner we get from f^- MCE) = b, ¡C ÍEM] + ~ MCE) the coefficient b = JL. . (P­ _ Ì M ) (26) In the limit of zero energy the emission time Τ is given by ^ 3 7 ­ Daß* ­ C^B* + OCB). (27) According to eq. (23) = Do C^ , M)" 1 [a, C~ , EM) ­ b, CD,EM)] . C28) The difference of the C­values turns out to be Co C ­ C = Λ - . D0 . (v>EM)(v ' E} > 0 (29) o i U* ï > Γ (r- , M) ν ­ 12 ­so that on the basis of the linear Ctwo terms) approximation (21,0) and C21,l) the relation | è c o T " λ | l i n = ­ (Cc ­ 0Λ ) Β 4 C30) holds. Since the difference C20) is only of order Β we state that the correct substitute for the fundamental eigenvalue up to corrections of order Β is ^  _ ,„. ^  instead of τ CE) ~ * — — TT CO)* So far we have studied the absorption free case with time moments m^CE.B). What is really measured or calculated by GATHER is the time moment M^CE,B) over the true flux established under the influence of absorption. Assuming the absorption to be small we may expand in C5) with respect to the small parameter a: 2 M^CE.B) = mxCE,B) ­ a m^fCE,B) + J rn^CE.B) ­+ C31) Then the inverse of the emission time comprising absorption I^CE) is MÖCE,B) m CE.Β) m CE,Β) m CE,Β) C „ ,„ D,l ­ D + 0U¿) . C32) 1*^ CE) Μ/Ε,Β) m., CE,Β) rn^CE.B)3 We find m CE,Β) by introducing the ansatz Cl0,2) into eq. C9,2) and com­z. pa r ing q u a n t i t i e s of t h e same order in Β : O = % m¿0 C33.0) Dm2£> =£mzl+-m40 C33,l) Dm2.i " t *¿y­ + 7 m 4 2 . « S , 2 ) I t fo l lows m_ = c„ MCE) (34 ,0) 2.0 2 ­ 1 ­1 (v" ' M ) 2 c2 - D0 - O M O - - ¿J <".« (D,m^ = 2 (J­ , m^) . (34,2) Further 2 2. m . _ Β D m ■••■t ο L1 + B T T Q Ä ­ J ­ t i * 2 M(EV 2 m ι m Γ 2 "fi Τ 1 " J Γ t »βΡ> D0 mJE> *ΛΕ> 1 4 = 2 L 1 + B ( ­ 5 ( E 5 F ­ÏÏCE) 2 D " ­ M 1 Ì ) > J + 0(B > thus the average value of (32) in the sense of (19) becomes ¿hç- > = a +<-^-> + <T£(E)>. C35) We will show that the error term <i£(E)> vanishes so that 1 2. ¥ <-£r-> = a + <7P> = a + D B ­ C B C36) agrees ­ up to terms of higher order ­ completely with Nelkin's result C4) for the fundamental eigenvalue <X = a + ^A . The error term is <£(E>> « aß2" (¿ , M)"1^ 2(¿ , . . , ) + τξξ , m^i ~ 4 De<J" / ­ , , > J o r , us ing (16,2) and (34 ,2) - c o Π oO I} = a B^cJ­ , M)_ 1 · [~2 J CD­ ^­ ) m4ZdE ­ D0 ƒ CD - 7 ° ) » ^ J # (37) Equation (15 ,1) can be w r i t t e n - 1* -Ζ m,z= (^ - 7 ) · MCE), C38) and equation (33,1) , because of (34,1) and (16,0), ^ = D 7 ' ( D ; - 7 > · Μ ( Ε > . ( 3 9> Consequently the difference 2 m - D^ m is a solution of the homogeneous equation if [2 « „ - D m z J = 0, that means a Maxwellian: 2 m.,, - D m„ = const. MCE). C40) Due to the definition C4a) of D0 the error term C37·) vanishes if we insert expression C40). References : fl] M. NELKIN: Nucl. Sc. Eng. 7, 210-216, C1960) [2] M. NELKIN: Physica 29, 261-273 Q963) [3] G.D. JOANOU et al.I GATHER-II, GA-4132, General Dynamics 1963, San Diego. ri¡r¡ NOTICE TO THE READER All Euratom reports are announced, as and when they are issued, in the monthly periodical EURATOM INFORMATION, edited by the Centre for Information and Documentation (CID). For subscription (1 year : US$ 15, £ 5.7) or free specimen copies please write to : Handelsblatt GmbH "Euratom Information" Postfach 1102 D-4 Düsseldorf (Germany) Office central de vente des publications des Communautés européennes 2, Place de Metz Luxembourg taß « χ w : · · · . .;... · . ::.. I To disseminate knowledge is to disseminate prosperity — I mean ! general prosperity and not individual riches — and with prosperity | disappears the greater part of the evil which is our heritage from ! darker times. -• Alfred Nobel Su Λ ι. 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The Connection Between the Time-Moments and the Fundamental Decay Constant of a Thermalized Neutron Pulse. EUR 3275.

EUR 3275.e 
EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM 
THE CONNECTION BETWEEN THE 
TIME-MOMENTS AND THE FUNDAMENTAL 
DECAY CONSTANT OF A 
THERMALIZED NEUTRON PULSE 
by 
H. HEMBD 
. 
1967 
-
•uhi · Joint Nuclear Research Center 
Ispra Establishment - Italy 
Reactor Physics Department 
Reactor Theory and Analysis 
LEGAL NOTICE 
This document was prepared under the sponsorship of the 
Commission of the European Atomic Energy Community 
(EURATOM). 
Neither the EURATOM Commission, 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 rights ; or 
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. 
This report is on sale at the addresses listed on cover page 4 
at the price of FF 4.—- FB 40. DM 3.20 Lit. 500 Fl. 3.— 
When ordering, please quote the EUR number and the title, 
which are indicated on the cover of each report. 
Printed by Vanmelle 
Brussels, February 1967 
This document was reproduced on the basis of the best available copy. 
EUR 3275.e 
THE CONNECTION BETWEEN THE TIME-MOMENTS AND THE 
FUNDAMENTAL DECAY CONSTANT OF A THERMALIZED 
NEUTRON PULSE by H. HEMBD 
European Atomic Energy Community — EURATOM 
Joint Nuclear Research Center — Ispra Establishment (Italy) 
Reactor Physics Department — Reactor Theory and Analysis 
Brussels, February 1967 — 14 Pages — FB 40 
The decay of a thermalized neutron pulse in a large moderating 
sample is studied by the time moments method applied to the Boltzmann 
equation. The time moments define the mean emission time or lifetime 
for neutrons before escaping or being absorbed at a certain energy. 
The decay constant of the neutron pulse is derived as a mean value of 
the inverse emission time. The first three coefficient^ in the expansion 
of the decay constant in terms of the geometrical buckling are identical 
with those obtained by Nelkin in hi« eigenvalue approach. Further 
properties of the time moments and their connection with the experiment 
are pointed out. 
EUR 3275.e 
THE CONNECTION BETWEEN THE TIME-MOMENTS AND THE 
FUNDAMENTAL DECAY CONSTANT OF A THERMALIZED 
NEUTRON PULSE by H. HEMBD 
European Atomic Energy Community — EURATOM 
Joint Nuclear Research Center — Ispra Establishment (Italy) 
Reactor Physics Department — Reactor Theory and Analysis 
Brussels, February 1967 - 14 Pages - FB 40 
The decay of a thermalized neutron pulse in a large moderating 
sample is studied by the time moments method applied to the Boltzmann 
equation. The time moments define the mean emission time or lifetime 
for neutrons before escaping or being absorbed at a certain energy. 
The decay constant of the neutron pulse is derived as a mean value of 
the inverse emission time. The first three coefficient in the expansion 
of the decay constant in terms of the geometrical buckling are identical 
with those obtained by Nelkin in his eigenvalue approach. Further 
properties of the time moments and their connection with the experiment 
are pointed out. 

EUR 3275.e 
EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM 
THE CONNECTION BETWEEN THE 
TIME-MOMENTS AND THE FUNDAMENTAL 
DECAY CONSTANT OF A 
THERMALIZED NEUTRON PULSE 
by 
H. HEMBD 
1967 
Joint Nuclear Research Center 
Ispra Establishment - Italy 
Reactor Physics Department 
Reactor Theory and Analysis 
SUMMARY 
The decay of a thermalized neutron pulse in a large moderating 
sample is studied by the time moments method applied to the Boltzmann 
equation. The time moments define the mean emission time or lifetime 
for neutrons before escaping or being absorbed at a certain energy. 
The decay constant of the neutron pulse is derived as a mean value of 
the inverse emission time. The first three coefficients in the expansion 
of the decay constant in terms of the geometrical buckling are identical 
with those obtained by Nelkin in his eigenvalue approach. Further 
properties of the time moments and their connection with the experiment 
are pointed out. 
THE CONNECTION BETWEEN THE TIME­MOMENTS AND THE FUNDAMENTAL DECAY CONSTANT 
OF A THERMALIZED NEUTRON PULSE (*) 
The time behaviour of a neutron pu lse in a l a r g e the rmal iz ing medium 
can be descr ibed by the time dependent Boltzmann equation in t h e d i f ­
fusion approximation 
J- · JJ + ( 2^+ DB*) ψ = t f f +S(E,B)^(t) (1) 
where 
^(E,B,t) is the neutron flux, 
Β the buckling, 
S(E,B)*0(t) the source distribution induced as a pulse at t = O, 
D(E) the diffusion coefficient, 
y^ (E) the absorption cross section, and 
% the thermalization operator defined in the usual way by 
o? 
% ψ s f :ξ(Ε·­*Ε) ^(E')dE' ­ ¿á(E) ^>(E) (la) 
with 
2"s (E) = ƒ Z,(E-» E»)dE 
The problem of solving eq. (1) becomes more simple by assuming /v - ab­
sorption 
ν·2ΰι(Ε) = a = const, (lb) 
a condition generally fulfilled in practical situations. The substitution 
<^(E,B,t) = e"at ψ (Ε,B,t) (IO 
reduces eq. (1) to the absorption free case 
(*) Manuscript received on December 13, 1966. 
- * -
ν τ τ + r ø if = £f + s*^(t) (2) 
which will be the basis of the following investigation. Nelkin £l] has 
studied the eigensolutions of the source-free eq. (1) by the ansatz 
<^(E,B,t) = e~°Ct . (f> (E.B). (3a) 
This is equivalent to applying the ansatz 
^(E,B,t) = e"(0C_a)t. ^ (E,B) (3b) 
to eq. (2). For the fundamental eigenvalue in terms of the buckling he 
finds 
2. t 
s OC- a = DnB - C B + ... (4) 
with 
J D(E) M(E)dE 
D0 - τ*- (4a) 
J - M(E)dE 
jf(D(E) -£s_ ) ^0(E)dE 
(4b) 
°Ύ 1 
ƒ ~ M(E)dE Í V 
where M(E) is a Maxwellian and ψλ ^ (E) the solution of 
D , 
(­ ­ D(E) ) M(E) ·= % %D<-E>· < 4 c ) 
- 5 -
At it is wellknown, the solution of eq. (1) by a superposition of eigen-
functions cannot be done in the classical way since the eigenfunctions 
do riot form a complete set [2] . However, instead of solving eq. (1) ex­
plicitly in time and energy we can obtain the same information by con­
sidering the time moments which satisfy a simpler equation. The time 
moments of order k are defined as 
Μ^Ε,Β) =r it" <p(E,B,t) dt = J e~at t* ^(E,B, t)dt (5) 
or 
m (Ε,Β) Ξ ƒ t ^(E,B,t) dt (5a) 
in the absorption free case of eq. (2). 
The time moments are of interest for three reasons: 
(i) They can be drawn out from pulsed experiments with great preci­
sion since they are time-integrated quantities. Moreover, the 
time- and energy-integrated reaction rates measured by a /v -
detector 
E max 
rK(B) = J J- mK (E,B) dE (6) 
E min 
are connected with the fundamental eigenvalue Λ in the follo­
wing way 
r0 C B ) τ 2 2 ,D, = λ (Β), (for Β2 « ΣΖ ) (7) r (Β) 8 
a result which will be shown afterwards. 
­ 6 ­
(ii) They satisfy a Boltzmann equation which is stationary but with 
a modified source term. Hence they may be calculated by a rou­
tine program like GATHER f3] . The latter provides an option for 
time moments. 
(iii) The ratio 
mY (E,B) T(E) = (8) 
mc (E,B) 
has a physical meaning. Called "emission time" it is the mean 
lifetime of a neutron before escaping with an energy E. 
, v k To derive the equations for the time moments we multiply eq. (2) by t 
and integrate from zero to infinity: 
DB Z mo(E,B) = if mo(E,B) + S(E,B); (k = 0) (9,0) 
k DB mk(E,B) = ¡ζ m^E.B) + ­ m^íE.B). (k* 1) (9,k) 
To solve eqs. (9) we expand all quantities in a power series of B and 
2 compare terms of the same order in B : 
m0 (E,B) = moe(E) + B^m^ (E) + B*n^(E) + ... (10,0) 
mi (E'B) = Bx fmio (E) + B* m « <E) + Β*».,/Β> + ­­'J (10,1) 
mz (E,B) = 5­#[mi0(E) + B^m^ (E) + Β^πι^(Ε) + ...J (10,2) 
The expansions (10,k) observe the fact that the k­th moment is an order of 
magnitude (in Β ) larger than the (k­l)­th. This can be seen from the con­
dition of neutron conservation 
- 7 -
B2(D,ii i) ■ J % mK dE + k ­ ( ^ , m ^ ) , (11) 
Û 
since for a l l d i s t r ibut ions Jf(E) 
] tf fdE s 0 (11a) 
ρ 
which is a consequence of the definition of the operatori la). The brackets 
mean 
QO 
(u,v) s J u(E) v(E) dE. 
The source in (9,0) must be of the same order as the leakage term on the 
left, hence 
S(E,B) = B2 Sx (E) + Β S (E) + ... (12) 
Eq. (9,0) splits up into 
0 = Ζ m (13,0) 
D**, = £ "ey + sy (13'2> 
We infer the solution 
m = M(E) » ­ ­ exp (­E/kT) (14,0) 
and,observing eq. ( l i a ) , the conservation conditions 
(D,M) = ƒ S¿ dE (14,1) 
où 
(D,mei) = ƒ S^ dE (14,2) 
- 8 -
Eq. (9,1) splits up into 
0 
Dm>« 40 
= 
= 
^m1o 
* " » + 
V 
m o o 
V 
mo¿ 
(15,0) 
(15,1) 
(15,2) 
with the conservation conditions 
m1o = c M(E) (16,0) 
c1 (D,M) = (- , M) or c, = Ό~ (see eq. (4a) ) (16,1) 
(D,me) = (^  , m M ) . (16,2) 
For the determination of the ratio (7) of reaction rates (6) we have for 
small Β 
r*(B) = ( Γ , " ) + B Z ( ; , m 0 1) + ... 
or with (16,2) 
= ( J- , M) + BZ (D, m<a ) + ν 
and 
■< C B > = B* f D c ( v ' M ) + ß Z ( v ' m « ) + '··] 
Hence 
De 
—7­T­ = D Β ­ Β · τ + 0(B' ) . (17) 
r, (Β) ° < è . M) 
­ 9 ­
This is the same result as given by Nelkin (see eqs. (4) ) for the funda­
mental eigenvalue. The solution m (E) of eq. (15,1) satisfies 
la 
D 
^(­D om^) = ( — ­ D ) MCE) 
which is exactly Nelkin's equation for his φ . 
2 The reciprocal emission time in terms of Β is given by 
m CE) 
. meCE,B) MCE) [ l + B - ^ r + . . ] MCE) 
T ( E ) m^E.B) r 2 "VL<E) Τ 
(B2Do) 1 M C E ) [ l + B \ - ^ i T + . .] 
2 
= D c ß 2 ' f 1 - MT?) C D c m , z ( E ) - m c x ( E > J + 0 ( B 4 ) J . C18) 
Averaging this quantity over all energies with a Maxwellian density as 
weight function 
fi MÍEJ. 
/L> = *Έ(Ε) ν dE OÇ 
Β- dE f Míj Κ ν 
(19) 
yields same right hand side as eq. (17) so that we have the agreement 
ik -M 0(B6) C20) 
2 
for small Β . From a physical point of view it may be more justified 
not to use the Maxwellian density in the averaging prescription Cl9) 
but the actual neutron density connected with the solution of eq.C9,0). 
Operating with m0CE,B) we get the same asymptotic relation C20), now 6 with another coefficient of Β . 
The behaviour of the emission time for small values of the energy is of 
interest. For the functions m and m which enter eq. Cl8) we make 
the ansatz 
m02CE) = MCE) [a0 + a^ + .. ] C21.0) 
m nCE) = MCE) [bo + bxE + .. ] . C21,l) 
- 10 -
These expansions are coupled by the condition of neutron conservation 
written in the form of eq. Cl6,2). With the help of eq. Cl6,l) we find 
CD b - a ) (- , M) = a. C~ , EM) - b CD,EM). C22) 
Ο Ο Ο γ T V 7 
The desired expression to be inserted into eq. (18) is 
MTU K m « ( E ) - mo*(E)J - [(D.bo - V + E <Dob, - a 1 > ] · Do 
or , again with Cl6, l ) and C22) 
Do 
Φ » M> 
J a1 [ (J· , EM) - EC ·^ , M)] + ^ [CD,M)E - CD,EM)] ƒ . C23) 
Averaging in the sense of definition C19) cancels the cofactor of a., 
whereas the second member in (23) yields a diffusion coefficient Cf which 
is an approximation to the true diffusion coefficient C due to the fact 
that we have retained only two terms in the expressions C21,0) and (21,1): 
c* " °Λ°. * <L > M ) " 1 < — - D> EM>· <24> 
1 Ù i ν ν 
The coefficients a^  , b can be determined from eqs. Cl3,l) and Cl5,l). 
The equation for &. is 
DCE) MCE) = a,, % [EMJ+ SZCE) . 
The common way to solve for a^ is to multiply by E and to integrate. But 
then the mean source energy would dominate in the above equation. This is 
not desirable since the source has not been specified, except that it is 
concentrated near a mean source energy Esin the high energy range. Instead, 
we weigh by E so that 
I o» | s z C E ) d E = ¿ ­ | s z d E = J ­ · CD,M)^C~ , DM) 
­ 11 ­
can be neglected. CUse has been made of eq. C14.1)). Now 
é , DM) 
a ­ ­f ­= ■£·<*. DM) C25) 
1 φ^ΕΜ) f? E 
where the energy transfer moment 
A*= JJ^—^MCE») ^ CE'­>E) dE'dE = 2 (^EM)^O C25a) 
σ 
has been introduced CFor the heavy gas operator U - 2*^2^ » where %Σ^ 
is the constant slowing down power). In the same manner we get from 
f^- MCE) = b, ¡C ÍEM] + ~ MCE) 
the coefficient 
b = JL. . (P­ _ Ì M ) (26) 
In the limit of zero energy the emission time Τ is given by 
^ 3 7 ­ Daß* ­ C^B* + OCB). (27) 
According to eq. (23) 
= Do C^ , M)" 1 [a, C~ , EM) ­ b, CD,EM)] . C28) 
The difference of the C­values turns out to be 
Co 
C ­ C = Λ - . D0 . (v>EM)(v ' E} > 0 (29) 
o i U* ï > 
Γ (r- , M) 
ν 
­ 12 ­
so that on the basis of the linear Ctwo terms) approximation (21,0) 
and C21,l) the relation 
| è c o T " λ | 
l i n 
= ­ (Cc ­ 0Λ ) Β 4 C30) 
holds. Since the difference C20) is only of order Β we state that the 
correct substitute for the fundamental eigenvalue up to corrections of 
order Β is ^  _ ,„. ^  instead of τ CE) ~ * — — TT CO)* 
So far we have studied the absorption free case with time moments m^CE.B). 
What is really measured or calculated by GATHER is the time moment M^CE,B) 
over the true flux established under the influence of absorption. Assuming 
the absorption to be small we may expand in C5) with respect to the small 
parameter a: 
2 
M^CE.B) = mxCE,B) ­ a m^fCE,B) + J rn^CE.B) ­+ C31) 
Then the inverse of the emission time comprising absorption I^CE) is 
MÖCE,B) m CE.Β) m CE,Β) m CE,Β) 
C „ ,„ D,l ­ D + 0U¿) . C32) 1*^ CE) Μ/Ε,Β) m., CE,Β) rn^CE.B)3 
We find m CE,Β) by introducing the ansatz Cl0,2) into eq. C9,2) and com­
z. 
pa r ing q u a n t i t i e s of t h e same order in Β : 
O = % m¿0 C33.0) 
Dm2£> =£mzl+-m40 C33,l) 
Dm2.i " t *¿y­ + 7 m 4 2 . « S , 2 ) 
I t fo l lows 
m_ = c„ MCE) (34 ,0) 
2.0 2 
­ 1 ­
1 (v" ' M ) 2 
c2 - D0 - O M O - - ¿J <".« 
(D,m^ = 2 (J­ , m^) . (34,2) 
Further 
2 2. 
m . _ Β D m 
■••■t ο L1 + B T T Q Ä ­ J ­ t i * 2 M(EV 
2 m ι m Γ 2 "fi Τ 1 
" J 
Γ t »βΡ> D0 mJE> *ΛΕ> 1 4 
= 2 L 1 + B ( ­ 5 ( E 5 F ­ÏÏCE) 2 D " ­ M 1 Ì ) > J + 0(B > 
thus the average value of (32) in the sense of (19) becomes 
¿hç- > = a +<-^-> + <T£(E)>. C35) 
We will show that the error term <i£(E)> vanishes so that 
1 2. ¥ 
<-£r-> = a + <7P> = a + D B ­ C B C36) 
agrees ­ up to terms of higher order ­ completely with Nelkin's result C4) 
for the fundamental eigenvalue <X = a + ^A . 
The error term is 
<£(E>> « aß2" (¿ , M)"1^ 2(¿ , . . , ) + τξξ , m^i ~ 4 De<J" / ­ , , > J 
o r , us ing (16,2) and (34 ,2) 
- c o Π oO I} 
= a B^cJ­ , M)_ 1 · [~2 J CD­ ^­ ) m4ZdE ­ D0 ƒ CD - 7 ° ) » ^ J # (37) 
Equation (15 ,1) can be w r i t t e n 
- 1* -
Ζ m,z= (^ - 7 ) · MCE), C38) 
and equation (33,1) , because of (34,1) and (16,0), 
^ = D 7 ' ( D ; - 7 > · Μ ( Ε > . ( 3 9> 
Consequently the difference 2 m - D^ m is a solution of the homogeneous 
equation 
if [2 « „ - D m z J = 0, 
that means a Maxwellian: 
2 m.,, - D m„ = const. MCE). C40) 
Due to the definition C4a) of D0 the error term C37·) vanishes if we insert 
expression C40). 
References : 
fl] M. NELKIN: Nucl. Sc. Eng. 7, 210-216, C1960) 
[2] M. NELKIN: Physica 29, 261-273 Q963) 
[3] G.D. JOANOU et al.I GATHER-II, GA-4132, General Dynamics 1963, 
San Diego. 
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The Connection Between the Time-Moments and the Fundamental Decay Constant of a Thermalized Neutron Pulse. EUR 3275.

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