238 G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264
Consequently, these source neutrons are more likely to be absorbed in the fuel and cause more fis- sion chains to be initiated per source neutron. To further clarify this example, assume that the point source emits 1000 n/s. For this source strength, the equivalent fundamental-mode source will be a fictitious source that emits a total of 1700 n/s and is distributed exactly as the fundamen- tal-mode fission source.
The concept of an equivalent fundamental-mode source has received very little attention in the literature, yet, it is an important concept in reactor physics with numerous practical applica- tions. Most notably, the equivalent fundamental-mode source strength, Q, appears in the reactor point-kinetic model, 1
dn P-~, - - - n + ~ , i C i + Q , (2)
dt A
dCi ~i d--t = - ~ t l - ~ i C i f o r i=l , m, (3)
where n is the total neutron population, p is reactivity, ~ is the effective delayed neutron fraction, A is the neutron generation time, ~,l is the decay constant of the i th precursor group, Cl is the pre- cursor concentration of the i th group, J~ is the effective delayed neutron fraction of the ~h group, and m is the number of delayed neutron groups.
The equivalent fundamental-mode source also appears in many equations dealing with reactor startup. For example, Hansen 2 analyzed the problem of assembling fissionable material in the presence of a weak neutron source. Using point kinetics, he defined the weak source condition
as
2Qx + 9t~F----- ~ ,~ 1 , (4)
where Q is the equivalent fundamental-mode source strength, x + is the adjoint-weighted, neutron- removal lifetime, ~p is the average number of prompt neutrons released per fission, and 1"2 is the neutron dispersion factor (also known as the Diven factor). 3,4
The equivalent fundamental-mode source is also an important concept in reactor noise the- ory. When performing neutron noise experiments, such as Rossi-¢~ and Feynman's variance-to- mean experiments, s,6 Spriggs ~ showed that these measurements can only be performed in suberit- ical assemblies if the following condition is satisfied:
(5)
G.D. Spriggs et al./.4nnals of Nuclear Energy 26 (1999) 237-264 239
where the product gF2 is the adjoint-weighted neutron dispersion factor s'l° and Ps is the reactivity of the system in the units of $. (Note, when the system is subcritical, ps<0; hence, the minus sign in front of the lips term.) The product, g'S, in Eq. (5) is synonymous with the equivalent funda- mental-mode source strength, Q, but is expressed in terms of the actual source strength, S, and a factor, g*, that converts the actual source strength to an equivalent fundamental-mode source strength.
Furthermore, the equivalent fundamental-mode source is particularly important in reactor experimentation. For example, the equivalent fundamental-mode source appears in the Cf-source technique originally developed by Carpenter et al.II to measure the effective delayed-neutron frac- tion, [le#, in a test assembly. In this technique, a calibrated point source of strength S is placed somewhere in the system (usually in the center of the assembly), and the resulting integral fission rate, F, is measured at some known, subcritical reactivity, Ps, just below delayed critical. The effective delayed-neutron fraction is related to these quantities by
S ~e ~e:: - ipsl~, F W: ' (6)
where ~7 t is the average of the total number of neutrons released per fission, and Ws and Wf are the average importance of the source neutrons and the fission neutrons, respectively. The quantity
S ~s (7)
is approximately equal to the equivalent fundamental-mode source strength when the system is close to delayed critical.
It is essential to understand the equivalent fundamental-mode source because of its impor- tance in reactor kinetics, reactor noise theory, criticality safety, reactor start-up operations, and reactor experimentation. In this work, we describe the concept of an equivalent fundamental- mode source, and we derive an expression for the factor, g', that converts any arbitrary source dis- tribution to an equivalent fundamental-mode source. Furthermore, we discuss the dependence of g* with ke~, and the dependence of g* with the energy spectrum of the source producing the sys- tem multiplication. And finally, we present a new experimental method that can be used to mea- sure the equivalent fundamental-mode source strength in a multiplying assembly. We demonstrate the method on the zero-power, XIX-1 assembly at the Fast Critical Assembly (FCA) Facility, Japan Atomic Energy Research Institute (JAERI).
240 G.D, Spriggs et aL/Annals of Nuclear Energy 26 (1999) 237-264
II. THEORY
In terms of the angular flux, ~, the steady-state, neutron transport equation for a suberiti- cal system in equilibrium with a fixed external/intrinsic neutron source can be written as
X, = Y', (r, E) = total macroscopic cross section, X I = Y.f(r, E) = total macroscopic fission cross section, Z's = Y"s ( r ;~ ' , E' --+ ~ , E) = macroscopic scattering cross section, Zf = fission source spectrum (normalized to 1.0), X, = external/intrinsic source spectrum (normalized to 1.0), s = s (r, ~ ) = source distribution per unit volume per unit angle,-and Xs s = X, (E) s (r, f~) = energy-dependent source distribution.
Equation (8) is merely a statement of neutron conservation as applied to an infinitesimal element of direction, energy, and space. If it is integrated over all phase space, it becomes a state- ment of neutron conservation for the integral system.
I ~ . V dp df~dVdE + IY.tO df ldVdE - IZ' fP'df~'dE' d~dV dE =
I~f:tY.' fo'd~2'dE" d~dVdE 4" JZsS df~dVdE . (9)
From the definition of the total cross section, the difference between the total interaction rate and the total scattering rate is equal to the total absorption rate:
As is customary in k-eigenvalue problems, any additional neutron production reactions [e.g., (n,2n), (n,3n), etc.] occurring in the system are accounted for by altering the scattering and absorption cross sections such that
Y , = ~so+2Y-2n+ . . . . ( I I )
G.D. Spriggs et al./ Annals of Nuclear Energy 26 (1999) 237-264 241
and
Y'a = Y'ao -- 2Y'2n - - " " ' (12)
thus, preserving the total macroscopic cross section defined as ~ = Ys + Y'a. We can rewrite Exl. (9) in a simplified form as
0 = P - L + S , (13)
where P is the unweighted, neutron production rate due to fission,
P = IZ f ~ t ~ ' f p ' d ~ ' d e ' d a d V d e , (14)
L is the unweighted, neutron loss rate due to leakage and absorption,
L = I ~ 2 . V 0 d t2dVdE + f~..a{~ d t2dVdE , (15)
and S is the unweighted, extemaYintfinsic source rate,
S = Igss d t ) d V d E . (16)
We define the solution of Eq. (13) corresponding to the source distribution, S, as the fixed-source solution, Op.
For a subedtieal system, the fixed-source multiplication, Mp, is defined as the total neu- tron production rate plus the extemaYintfinsic neutron source rate divided by the neutron source rote. That is,
P + S MIs= ' S (17)
242 G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264
From Eq. (13), we can also express the fixed-source multiplication as the total loss rate divided by the source rate,
L taf, = (18)
By dividing Eq. (13) by L and using the definition of Mp from Eq. (18), we obtain
P 1 0 = ~ - l + - ~ - - (19)
If we now define the ratio of the neutron production rate due to fission, P, to the total neu-
tron loss rate, L, as theftred-source multiplication factor, kl,, Eq. (19) can be written as
1 M f s - l _ k f s (20)
We must stress that kls is not the k-eigenvalue, ks#. Unlike the k-eigenvalue, kfs is a very strong function of the source distribution. If we place an external point source in the center of a multiplying system and solve the fixed-source solution, we would find that kp > ke#. If, however, we place the same source outside of the system, we would find that k~ < k,#. In fact, in the fimit, as the source is moved an infinite distance from the system, kfs approaches zero, whereas, ke#remains the same.
In most reactor physics applications, it is more customary to write the source multiplica- tion equation, Eq. (20), in terms of the k-eigenvalue, ke~,, rather than the fixed-source multiplica- tion factor, kp. Theoretically, this is not possible because, by definition, the k-eigenvalue solution is independent of the source distribution; consequently, one cannot rearrange the steady-state transport equation to include both the source term and a rigorous definition of the k-eigenvalue. However, one can derive the source multiplication equation in terms of an adjoint-weighted source multiplication factor, k~s, that is numerically similar to the k-eigenvalue when the system is not too far suberitical. This derivation is accomplished by multiplying the neutron transport equa- tion, Eq. (9), by the adjoint angular flux, which is a direct measure of a neutron's importance in the multiplication process. The adjoint angular flux satisfies the adjoint equation,
-f~. VW + Zt~P = ~Y.'sW'df~'dE' + l ~z'f~tY./W'd~'dE' (21)
where Y~'s = Y:s (r;t2, E -o tT, E') is macroscopic scattering cross section similar to that in Eq. (8) but the primed and non-primed variables are exchanged.
G.D. Spriggs et al./Annals o f Nuclear Energy 26 (1999) 237-264 243
Integrating the adjoint-weighted transport equation over all phase space leads to an equa-
tion of the form
0 = P + - L + + S + , (22)
where P+ is the adjoint-weighted, neutron production rate due to fission,
p+ = I V ZfqtZ'f~'fsdf~'dE' df~dVdE , (23)
L + is the adjoint-weighted, neutron loss rate,
L + = I V a" V ~fs d ~ d V d E + I V Y~a~fs d ~ d V d E , (24)
and S + is the adjoint-weighted, neutron source rate,
S + = f~P ZsS d[2dVdE , (25)
where ~P(r,D,E) is the solution of Eq. (21). Dividing Eq. (22) by L +, we obtain
p+ S + 0 = - - - 1 + - - (26)
L + L +
By defining the ratio P÷/L + to be the adjoint-weighted, neutron multiplication factor, k~s, Eq. (26)
can now be rewritten as
1 M o - (27)
where Mo is defined as the adjoint-weighted, fixed-source multiplication and is identically equal to L+/S +. The quantity M o represents the multiplication that would occur if the source S were distrib- uted in space, energy, and angle identically to the fission source distribution obtained from the fixed-source solution.
244 G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264
From a calculational standpoint, Eq. (27) is more convenient than Eq. (20) because it is a function of the variable, k~s, which is more of a function of the assembly configuration rather than the source distribution causing the multiplication. However, because external/intrinsic neutron sources usually occur as point sources or as uniformly distributed sources, the actual fixed-source multiplication, MI,, obtained from the solution of Eq. (13) corresponding to these real-world source distributions can differ significantly from Mo. Therefore, we seek an expression that relates Mp to the variable k~s. This is accomplished by defining a factor, G', that is the ratio of Mp to M o. Hence, the fixed-source multiplication can he expressed as
M/s= G, Mo _ G* 1-k]s
where G* is numerically equal to
(28)
G * ~ m (29)
Alternatively, we can derive a deterministic expression for G* from Eq. (22) by dividing through by L ÷ and then multiplying and dividing the last term on the right-hand side by the unweighted quantifies S and L. These manipulations lead to
L SL + J (30)
Using the definition of Mis from Eq. (18) and the definition of Mo from Eq.(27), Eq. (30) shows that G* can also he written as
G* = Ls+ (31) SL +
Using the definitions of L and L + from Eqs. (13) and (22), G* can also be expressed as a function of the neutron production rates, P and P+,
G* = S+ (P + S) (32) S (P+ + S ÷)
From this expression we note when P and P+ ate set equal to zero, G* equals 1.0, as it should; in a non-multiplying system, every neutron has equal weight. So, the fixed-source multiplication must equal the adjoint-weighted multiplication and, therefore, G* must also equal 1.0.
G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264
When written in terms of the angular fluxes, G" corresponds to
245
G* = SW gss dt~dVdE
~gss d~dVdE
[[gcqtY.'/~'~,,di'2'dE'.,__ __,. dfldVdE + S] X (33) tf,I,., a avae + s+l
The physical meaning of G* is seen from Eq. (31) by noting that the ratio of S+/S is equal to the average importance of a source neutron, and that the ratio of L+/L is the average importance of a neutron lost (i.e., leakage plus absorption) from the system. Consequently, G* is the ratio of the average importance of a source neutron, ~ s ' to the average importance of a neutron lost from the system, ~Pt"
G* $ = ~ • ( 3 4 )
u/l
Furthermore, from Eq. (32) we note that when the fission source terms, P and P+, are much bigger than the external/intrinsic source terms, S and S +, G* can also be interpreted as the ratio of the average importance of a source neutron to the average importance of a fission neutron, ~Ff.
I~,/$
G~ ~,.~. ~ ° v: (35)
In summa. , we have just derived an expression for the fixed-source multiplication, Mrs, of a multiplying system in terms of a factor, G*, that allows us to relate Mrs to an adjoint-weighted multiplication factor, k~s. It is important to note, however, that from a theoretical standpoint k~s is not equal to ke# because the solution of the steady-state transport equation containing an inhomo- geneous fixed-source term will yield a different solution than that of the steady-state, source-free transport equation. Although the differences in these solutions are insignificant in the vicinity of delayed critical, nevertheless, we cannot strictly claim that G* is the factor that converts any arbi- trary fixed-source distribution to its equivalent fundamental-mode source strength since the fluxes used in Eq. (33) are not the fundamental-mode fluxes (i.e., the fluxes obtained from a k-eigenvalue solution at delayed critical). This leaves us with a small dilemma.
Historically, it has been customary in reactor physics to define afundaraental-mode multi- plication, M/, in terms of the k-eigenvalue, k~H, as
1 - ( 3 6 )
Mf I - keff
246 G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264
It would be most difficult at this point in time to convince the nuclear industry to express the mul- tiplication of a system in terms of the adjoint-weighted, fixed-source multiplication factor k~s rather than the k-eigenvalue. So, we have chosen to define another factor, g*, that converts an arbi- trary distribution of sources to its equivalent fundamental-mode source strength.
As with G*, g* is formally defined as the ratio of the fixed-source multiplication to the fun- damental-mode multiplication. That is,
* Mrs (37) g ffiMf '
which, by Eq. (36), corresponds to
g* .Mf,(1-k,ff) . ( 3 8 )
Hence, the fixed-source multiplication can now be written in terms of the k-eigenvalue as
g *
Mrs - 1 ----kerr (39)
We presume that when the system is in the vicinity of delayed critical and the fluxes obtained from the homogenous and the inhomogeneous transport equations are essentially identi- cal, g* will be equal to G*. Therefore, g* can be accurately calculated using Eq. (33). However, as the system becomes more subcritical, the fluxes obtained from the fixed-source solution may depart significantly from the fluxes obtained from a k-eigenvalue solution. For these situations, Eq. (38) should be used to calculate g* if the fixed-source multiplication, Mrs, is to be expressed in terms of ke#.
III. MULTIPLE SOURCES
Although we cannot derive an exact deterministic equation for g* similar to the expression for G* [see Eq. (33)], we assume that g* will exhibit the same properties as that of G* when multi- ple sources are present in a given system. Hence, by analogy, we can make the following observa- tion concerning the equivalent fundament-mode source for multiple sources.
For convenience, we define the equivalent fundamental-mode source strength, Q, as g* times the total neutron source strength, S. That is,
Q =- g* S , (40)
G.D. Spriggs et aL/Annals of Nuclear Energy 26 (1999) 237-264 247
From Eq. (31), we surmise that when multiple sources are in the system, Q is the sum of the equivalent fundamental-mode sources of each of the constituent sources.
{2 = g* S = g~ Sl + g2S2 + ... (41)
A separate solution for each g~ can be obtained for each some, Sl, using Eq. (38). The effective g* of a multiple-source distribution is simply a source-weighted average of the individual values of g~ for each of the constituent sources.
Xg~ Si
Y si i
(42)
IV. CALCULATION OF g*
In principle, the evaluation of Eq. (38) is rather straightforward when using a deterministic code. For subcritieal systems, it is only necessary to perform two calculations. The first calcula- tion is a fixed-source solution (sometimes referred to as an inhomogeneous source solution) corre- sponding to a given source distribution. From this fixed-source solution, one obtains the fixed- source multiplication of the system, Mp. The second calculation is a k-eigenvalue solution, which yields a value for keN.
We now demonstrate this method of calculating g* with the following numerical example. Consider a bare, spherical system comprised of 92.2% 235U with a uranium density of 18.6 g/co containing a 100 n/s 252Cf start-up source in the center of the assembly. Using the deterministic transport code ONEDANT ~2 and the 16-group Hansen-Roach cross section set, ~3 the effective multiplication factor, keg, of this system was calculated to be 0.9914 when the outer radius of the sphere was 8.85 cm. It is well known that both 235U and 23SU undergo spontaneous fission; 235U produces 0.01 n/s per kg and 23SU produces 13.6 n/s per kg (see Table I). For this system, the 235U produces a total of 0.5 n/s, uniformly distributed over the volume of the assembly, and the 23sU
produces a total of 57.3 n/s, also uniformly distributed over the volume of the assembly. The spon- taneous fission spectra for 235U, 238U, and 252 Cf are listed in Table II.
The equivalent fundamental-mode source for this fictitious system was determined by run- ning a k-eigenvalue solution to determine ke#, and then three fixed-source problems: 1) a 235U spontaneous fission source distributed uniformly over the volume of the assembly, 2) a 23sU spon- taneous fission source distributed uniformly over the volume of the assembly, and 3) a 252Cf point
248 G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264
Table I: Spontaneous Fiss ion Data
Spont. Fission Isotope Half-Life
(y)14-1s
232Th 7.82 x 10+ 2o
233U 2.65 x 10+ 17
~ U 1.49 x 10+ i4
~p16-21 n s
2.14:1:0.20 1.56 x 10 ~
-1.76 0.38
-1.81 6.83
Z~sU 1.00 x 10 +19 -1.86 0.01
Z~6U 2.49 x 10+ :~ 1.66:1:0.11 3.72
~mU 9.11 x 10 +!7 -1.87 0.12
~sU 8.20 x 10 +Is 2.00 5:0.02 13.6
~9U 5.55 x I0 +l~ -2.04 2.03
z36Pu 2.09 x 10 +09 2.12 + 0.14 5.70 x 10 +o7
237pu 2.05 x 10 +13 -1.88 5.09 x 10 +o3
~Spu 4.74 x 10+ l° 2.21 5:0.06 2.59 x I0 +°~
239pu 8.05 x I0 +Is 2.24 + 0.10 15.5
24°pu 1.14 x 10 +11 2.151 + 0.005 1.04 x 10 +~
241pu 5.98 x 10+ 16 -2.25 2.07
242pu 6.74 x 10+ 1° 2.141 :i: 0.006 1.74 x 10 +~
U3pu 2.00 x 10 +15 66.2 -2.43
2.29 + 0.19 6.68 x 10 +l° U4pu 1.86 x 10 ~
zszcf 8.55 x 10 +°l 3.768 + 0.012 2.31 x 10 +15
Watt Parameters a
A B
0.5934 8.030
0.8548 4.032
0.7712 4.925
0.7747 4.852
0.7352 5.358
0.6931 5.994
0.6483 6.811
0.7356 5.261
0.9883 3.104
0.9546 3.308
0.8478 4.169
0.8853 3.803
0.7949 4.689
0.8425 4.152
0.8192 4.367
0.7354 5.387
0.6947 6.004
1.0250 2.926
a. Obtained by extracting A and B from the Los Atamos Model, 22"24 where A and B are defined by
f (e) = Cexp (-~) sinh ,~-E
This yields a S. F. spectrum with an average neutron energy that is accurate to within +10% in most cases.
G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264
Table H: Spontaneous Fission Spectrum
249
Groupa Energy 235U 238U (O.,n)b 252C f Range
1 3 - ~ Mev 0.186 0.140 0.011 0.275
2 1.4 - 3 Mev 0.364 0.362 0.307 0.353
3 0.9 - 1.4 Mev 0.174 0.190 0,036 0.149
4 0.4 - 0.9 Mev 0.179 0.200 0.237 0.146
5 0.1 - 0.4 Mev 0.083 0.093 0.342 0.067
6 17 - 100 key 0.013 0.014 0.064 0.010
7 3 - 17 key 0.001 0.001 0.003 0.000
8-16 3 kev& ,l, 0.0 0.0 0.0 0.0
Average Energy = 1.89 Mev 1.69 Mev 1.01 Mev 2.31 Mev
a. Corresponding to Hansen-Roach group structure. 13 b. This spectrum was calculated specifically for the soft blanket region of the XIX-1 core using the SOURCES code. 25"26
source located at the center of the assembly. Because the 235U and ~ U spontaneous fission spec-
tra are similar, there was little in the multiplication calculated by ONEDANT; it was found to be 97.82 for ~ su and 97.48 for 23sU. Hence, g* for 235U corresponds to
g* = 97 .82(1-0 .9914) = 0.841 ,
and g* for 23sU corresponds to
g = 97 .48(1-0 .9914) = 0.838
The multiplication produced by the centrally-located ~2Cf point source was calculated to be 208.7; so, g ' for the point source corresponds to
g* = 208 .7(1-0 .9914) = 1.795
250 G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264
When combined, the equivalent fundamental-mode source for these source distributions corre-
From a physieai standpoint we can interpret these results as follows. A uniformly distrib- uted Z35U + 2-~U spontaneous fission source producing 57.8 n/s in this spherical assembly will ere-
ate the same total neutron multiplication as an equivalent fundamental-mode source of strength
48.4 n/s (i.e., 0.841 x 0.5 + 0.838 x 57.3). A z~2Cf point source emitting 100 n/s in the center of the
assembly will create the same total neutron multiplication as an equivalent fundamental-mode source of strength 179.5 n/s (i.e., 1.795 x 100). The combined sources (235U + Z3sU + ~2C0 of
157.8 n/s will produce the same total neutron multiplication as an equivalent fundamental-mode
source of strength 227.9 n/s.
V. DEPENDENCE OF g* ON k,#
The factor g* is dependent on keh,. However, depending on how the source is distributed,
this dependence may be relatively weak. For example, using the assembly described in the previ-
ous section, the radius of the uranium sphere was varied from 8.85 em to 0.85 em in 1-em decre-
ments; keg varied from 0.9914 to 0.0993, respectively. As can be seen in Fig. 1, g* for the
uniformly distributed spontaneous fission source varies from 0.84 to 0.99 as kezr decreases. In comparison, g* for a Z~2Cf point source located at the center of the assembly decreases from 1.78
to 1.03 over the same range in ke~,. As another example, we calculated g* for a reflected system with a total radius of 15 cm.
For the uranium core used in the previous example at a radius of 7 cm surrounded by an 8 cm
thick spherical graphite reflector (density = 1.65 g/co), keh' = 0.9807. When the core radius was
reduced to 1.0 em (while maintaining the outer radius of the system at 15 cm), keH decreased to 0.1296. For the case of a uniformly distributed source, g* increased from 0.949 to 0.995 as ke# decreased. However, for a point source located at the center of the assembly, g* decreased from 1.52 to 1.04 as k~,decreased (see Fig. 1).
G.D. Spriggs et aLIAnnals of Nuclear Energy 26 (1999) 237-264 251
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.0
'''''''l'l'''''''''l'''l'''''l'''''''''l''''''''.~. ~ e - U ~ o ~ m
• Bare Sptmre-Point Source A
- - *-" -Refl. S ~ r ~ n i f ~ m ~ r ~ /
- - - - - Refl. S ~ r ~ n t ~ r c e S
- F u n ~ m ~ e ~ u r ~ ~ ,,-= / l
~ i~ , ,~ ~
; l l l l l i l l i [ l l i l i l l i l [ l i i l l l l l i i l l l l l l l l l [ I I I I I l i l l " 0.2 0.4 0.6 0.8 1.0
Fig. 1. keff-dependence of g* for four different cases: 1) a bare sphere with a uniformly distributed spontaneous fission source, 2) a bare sphere with a point source at the center of the assembly, 3) a reflected sphere with a uniformly distributed spontaneous fission source, and 4) a reflected sphere with a point source at the center of the assembly.
VI. DEPENDENCE OF g* ON THE SOURCE SPECrRUM
As expected, g* is also dependent on the energy spectrum of the source. However, this dependence can be relatively weak in some systems. To illustrate this, consider the same assembly described in Section IV in which a point source is placed in the center of the system. If we assume that the energy spectrum of this point source corresponds to that of 252Cf, 235U, or 23sU (see Table II), then g* calculates to be 1.791, 1.787, and 1.788, respectively. These changes in g* are surpris- ingly small when you consider the differences in average neutron energy of the three different sources. In terms of the Watt parameters, A and B, the average energy of each spectrum is given by 15
= ,43,
252 G.D. Spriggs et aL/Annals of Nuclear Energy 26 (1999) 237-264
Using the values of A and B from Table I, zsR2f has an average energy of 2.31 Mev, whereas, Z~su and ~ U have average energies of 1.89 Mev and 1.69 Mev, respectively.
What is even more surprising is the fact that g* increases as the average energy of the source increases. [In fact, if one assumes that all source neutrons are born in group 1, which
would represent a very hard-spectrum source in which all neutron energies are greater than 3 Mev,
g* increases to 1.853.] This increase seems to contradict intuition as it implies that a hard-spec- tnun source multiplies the assembly more effectively than a soft-spectrum source. In this particu-
lar case, the increase in g* with source energy is real. As the energy of the source neutron
increases, the fission cross section decreases, but the number of neutrons released per fission increases, which produces greater multiplication.
VII. MEASUREMENT OF THE EQUIVALENT FUNDAMENTAL-MODE SOURCE
The equivalent fundamental-mode source can be easily measuredin a multiplying system by placing a neutron detector in or somewhere near the core and observing the change in the count
rate produced by a calibrated neutron source at some known location within the system. The
count rate of the detector is proportional to the unweighted, neutron loss rate and the detector effi- ciency, e, where e is defined as
C e - ( 4 4 )
In this equation, C is the detector count rate and N/~ is the total neutron loss rate in the system.
Based on the above definition of detector efficiency, Eq. (39) earl be rewritten as
eg* S C - (45) l-k,::
Assume for the moment that we wish to measure the equivalent fundamental-mode source strength of an intrinsic source present in a critical assembly containing a large amount of one or more isotopes that undergo spontaneous fission. If the system reactivity is adjusted to be just slightly subcritical, then the detector count rate produced by the intrinsic source distribution will
correspond to
~g~ Si = , ( 4 6 )
Ci 1 - kef f
G.D. Spriggs et al./Annals of Nuclear Energy 26 (1999) 237-264 253
where Cl is the detector count rate at that subcritical configuration and St is the intrinsic source strength.
If we then place a calibrated point source at some known location within that system and reestablish the same subcritical configuration, the detector count rate produced by the intrinsic source plus the point source will now correspond to
v-(g* Si * g;Sl,) Cpi = 1 -- kef f '
(47)
where Cvt is the new count rate, Sp is the strength of the point source, and gv corresponds to the g*-value at the location of the point source.
Taking the ratio of Eq. (47) to Eq. (46) and solving for g~ S i leads to
g~Si _ g;Sp (%- ) (48)
Note that we have assumed that the efficiency of the detector does not change upon insertion of the point source. In far subcritical systems, this assumption may not be valid. However, if the sys- tem is just slightly subcritical, then the multiplication of the system will be high enough to excite the fundamental mode. When this occurs, the flux distribution will be determined by the fission source distribution and will be relatively insensitive to the actual source distribution. Therefore, the detector efficiency will remain essentially constant for both source distributions used during the measurements.
Furthermore, in the above derivation, we have optimistically assumed that the observed count rates, C~ and Cpt, are produced only by neutrons. However, most neutron detectors are somewhat sensitive to gamma rays--which are always present in a multiplying system. When using fission chambers, this sensitivity can be effectively eliminated by adjusting the lower-limit discriminator to be high enough to detect only those pulses produced by the fission fragments generated within the chamber when a neutron is detected. When using SHe or BF3 detectors, it is much more difficult to discriminate out all gamma rays without simultaneously reducing the abil- ity to detect neutrons as well. Consequently, one must be able to determine the gamma ray back- ground count rate, C~/, to obtain the correct values for Ct and Cp~ to be used in Eq. (48). This can be accomplished if these measurements are repeated at several different subcritical configurations in the vicinity of delayed critical. Then, Eq. (48) can be modified as follows.
254 G.D. Spriggs et al,/Annals of Nuclear Energy 26 (1999) 237-264
When written in terms of reactivity [i.e., Ps = (ko'- 1)/[$ke#], Eq. (45) becomes
~g* S C = Eg*S- ~ (49)
where 1$ is the effective delayed neutron fraction. When p$ --> -** (i.e., ke#= 0.0), the count rate must approach eg*S, which is the count rate that would be produced by an unmultiplied source in that particular source/detector geometry. If we define eg*S as Co, we can write Eq. (49) as
where
m C = C o -p--~ (50)
Eg* S m = 13 (51)
Because Co = I~m and I$ is not larger than 1%, we expect Co to be negligible relative to -m/p$ when p$ is in the vicinity of delayed critical. Therefore, to a first approximation, we can state that
and
m i C i - Cio . . . . C i , (52)
PS
%i- %io -- _mp Ps = c~'i (53)
From these two equations, we can see that the ratio C~n/Ct is approximately equal to the ratio of rnpJrn t. Therefore, Eq. (48) can also be written as
gTs,- g;s (54) mpi F -1)
The slopes mpL and mt can be readily determined by plotting the detector count rate as a function of the inverse reactivity of the system and then performing a least-squares fit of these
G.D. Spriggs/Annals of Nuclear Energy 26 (1999) 237--264 255
data to determine the y-intercept, Co, and the slope, m, for both source configurations. If the detec-
tor system happens to be sensitive to gamma rays, then Co will be noticeably greater than 13m. When this occurs, we can infer the gamma ray background to he
cv = c o - Bm (55)
However, determining the gamma ray background is for informational purposes only. When eval- uating Eq. (54), we presume (albeit, somewhat optimistically) that the gamma ray background doesn't change much with the reactivity of the system over a small range of reactivity in the vicin- ity of delayed critical. We expect the slopes mpt and m I to be relatively independent of the back- ground count rate, whatever it may be. We now illustrate this technique on a real system.
Measurement
VIII. INTRINSIC SOURCE MEASUREMENT AND CALCULATION FOR THE XIX-1 CORE
The experimental procedure described in the previous section was performed on the zero- power XIX-1 assembly located at the Fast Critical Assembly (FCA) facility operated by the Japan Atomic Energy Research Institute (JAERI). The XIX-1 assembly is a multiregion system com- prised of an inner core fueled with highly-enriched 235U metal. The core is surrounded by an inner blanket (referred to as the 'soft blanket') containing a significant amount of depleted uranium- oxide and sodium, and an outer blanket (referred to as the 'depleted blanket') containing only depleted uranium metal. Cross-sectional views of the core are shown in Figs. 2, 3, and 4.
For this measurement, four 3He detectors were inserted into the soft blanket region of the assembly (see Fig. 2). Count rates from these detectors were obtained at three different subcritical configurations when the system was driven by just the intrinsic source. The reactivity ranged from
--0.0725 to --0.8925 with an uncertainty of:t: 0.0015 (1 sigma) as determined by a series of Rossi- a measurements. A calibrated 252Cf source was then inserted into the center of the assembly, and the count rates from the 3He detectors were obtained at eight different subcritical configurations over the same reactivity range. A least-squares fit of count rate vs. inverse reactivity was per- formed on these two sets of data.
With just the intrinsic source driving the system, it was found that m~ =-55.8 + 1.3 Us and
(C~+ C~y) = 22 + 10 cps (see Fig. 5). The y-intercept is obviously too large; from the measured slope and the measured value of the effective delayed neutron fraction, 13=0.00729, the y-intercept should have been approximately 0.4 cps if the detectors were only counting neutrons. Conse- quently, from Eq. (55), we can infer that the gamma ray background, C~t , for these 3He detectors during the intrinsic source measurements was approximately 21 cps.
256
110
G.D. Spriggs/Annals of Nuclear Energy 26 (1999) 237-264
10 15 20 25 30 35 40
115
120
125
130
135
140
Fig. 2. Cross-sectional view of FCA XIX.1 assembly. Two pairs of He detectors are located in positions 26-118 and 26-135.
G.D. Spriggs/Annals of Nuclear Energy 26 (1999) 237-264 257
Fig. 3. Effective radii of the three regions of the XIX-1 Assembly. R 1 = 32.958 cm, R2= 68.30 cm, R3= 86.36 cm.
H3
Fig. 4. Effective heights of the three regions of the XIX-1 Assembly. H 1 = 50.80 cm, H2 = 121.92 cm, Hj = 132.08 cm.
258
Q.
ILl
<
Z
o o
G.D. Spriggs/Annals of Nuclear Energy 26 (1999) 237-264
8000 . . . . , . . . . , . . . .
'"~'~ m¢=-500.6 + 3.0 ~s 7000
6000 ~- ~ / ~ + ~ ' = 39 + 25 cps
500O
400O
3000 - m,=-55.8 + 1.3 $/s
o =15 -10 -5 0
INVERSE REAGTIVITY (1/5)
Fig. 5. Plot of the count rate of the four detectors summed together vs. inverse reactivity.
A similar process was repeated for the count rates obtained with the 2s2Cf source in the center of the assembly. The least-squares fit yielded mp~=-500.6 + 3 $/s and (Cpto+ Cpt7) = 39 + 25 cps. This y-intercept is also too high; C~o should have been approximately 3.7 cps. So, the gamma ray background present during the measurements with the Cf source in the center of the assembly increased to approximately 35 cps.
The ratio of the slopes is
C3..j = mpi _ 500.6 C i m i 55.8
- 8.97 + 2.4%
Using Eq. (38), the calculated value for g; for a ~2Cf point source located in the center of the XIX-1 assembly was 1.68 for ke•slightly less than 1.0. And, as determined from a previous source calibration performed at JAERI, the strength of the ~ Cf source used to perform the exper-
G.D. Spriggs/Annals of Nuclear Energy 26 (1999) 237-264 259
intent was measured to be 97,600 + 1% n/s on the day of the measurement. Substituting these val-
ues into Eq. (48) yields an equivalent fundamental-mode intrinsic source strength of
g~S i = 1.68 x 97, 600 = 20, 600 + 3% n/s 8.97 - 1
(56)
Calculation
Using TWODANT, 12 the equivalent fundamental-mode intrinsic source strength was cal-
culated using the data listed in Tables I and II. The intrinsic source was separated into its various
constituents so that the neutron spectrum from each spontaneous fission source and the (a,n) source could be properly accounted for in each region of the assembly. The results of this analysis
are shown in Table III. Based on the given masses of materials, the total intrinsic source for this system was calculated as 343,160 n/s.
Although there is a 30% difference between the measured (20,600 n/s) and the calculated
(14, 399 n/s) equivalent fundamental-mode source strength, this difference is not particularly sig-
nificant for this system. A small increase of 2% in g* in the soft blanket (i.e., 0.0896 to 0.1096)
Table HI: Equivalent Fundamental-Mode Intrinsic Source for XIX-1 Core
Region Isotope Mass (Kg) S (n/s) g* g*S (n/s)
Core 235U 153.0 2 0.994 2
23SU 11.7 159 0.991 158
Soft Blanket 235U I 1.7 <0.1 0.0916 0
238U 5,841.2 79,400 0.0896 7,114
(a,n) a 772 0.0796 61
Depl. Blan. 235U 38.8 <0.4 0.0285 0
238U 18,713.7 255,000 0.0277 7,064
~ S = 343,160 gi Si = 14,399
a. The uranium in the soft blanket is in the form of uranium-oxide ~lates. The alpha particles from the uranium interact via (a,n) reactions with other elements in the soft blanket (primarily the lSO and 23Na) to produce an additional source of intrinsic neutrons.
260 G.D. Spriggs/Annals of Nuclear Energy 26 (1999) 237-264
and a 2% increase in g* in the depleted blanket (i.e., 0.0277 to 0.0477) would increase the calcu- lated effective intrinsic source strength to 20,800 n/s--which would compare very favorably with the measured value. A 2% increase in these two values of g* is very plausible considering the fact that the 16-group Hansen-Roach cross section set ~3 used to perform these calculations is not well suited for this particular system with significant intermediate energy reactions. This lack of suit- ability is best seen from a comparison of the measured and calculated adjoint-weighted neutron lifetimes. Near delayed critical, the adjoint-weighted neutron lifetime was calculated to be 550 ns, which is 21% lower than the measured value ofT00 ns. Because the neutron lifetime varies as the inverse of the absorption cross section, it follows that the 16-group Hansen-Roach cross section set overpredicts the absorption rate in this particular system. Hence, in accordance with the calcu- lations, source neutrons born in the soft and depleted blankets will have a much harder time pene- trating those regions then they would in reality. This shorter mean-free path, in turn, yields a calculated g* that will be lower than the actual value.
Finally, using Eq. (42), we note that the measured value of the effective g* corresponds to 0.06 [i.e., g* = 20,600/343,160], which means that thet equivalent fundamental-mode source is only 6% of the total neutron source present in this system.
IX. DISCUSSION
The XIX-1 assembly is a prime example of the importance of understanding the equiva- lent fundamental-mode source. If we evaluate the weak-source condition using the total source strength present in the system, we obtain
2S% 2 x 343, 160 x 700xlO -9 ~V 2 2.6 x 0.80
-- 0.23
which marginally satisfies the condition of << 1. However, when we use the equivalent fundamen- tal-mode source strength of 20,600 n/s, then the weak-source condition becomes 0.0138, which clearly satisfies the condition of << 1. Hence, the equivalent fundamental-mode source strength for this assembly is clearly not strong enough to ensure consistent behavior of the system during start-up despite the fact that the total source strength is in excess of 340,000 n/s.
In addition to the applications previously mentioned in this manuscript, the factor g* also has an impact on one of the most basic experimental techniques used in the field of nuclear engi- neering--the classical approach-to-cri t ical experiment using a I / M plot. In this experiment, a source is placed in an assembly, and a reference count is taken with a nearby detector. Then a small amount of fuel is added to the assembly, and a new count is obtained. The ratio of the refer- ence count to the new count (i.e., the inverse of the multiplication, Air) is pIotted as a function of the amount of fuel in the system and then extrapolated to the point where I IM = 0.0 to obtain an
estimate of the critical mass. In principle, if this process is repeated for a series of small mass
G.D. Spriggs/ Annals of Nuclear Energy 26 (1999) 237-264 261
Fig. 6. 1/M plot vs. ke# for a spherical, uranium assembly with a ~2Cf point source positioned in the center of the assembly.
additions, the 1/M plot will be a linear function of the amount of mass in the system assuming each mass addition produces the same Ake~ In general, this is not a very good assumption; if the system is loaded with fuel from the center out, then the initial mass additions will produce a much greater change in ke~,than will the later mass additions at the outer surface of the assembly.
Assume that we load a new system in such a way as to produce equal changes in ke~ Because IlM = (1 - ken')/g*, the 1/M plot will be linear with keH only if g* is constant with ke•. However, as was shown in Fig. 1, having a constant g* over a wide range of k~is highly unlikely. Using the numerical example presented in Section V, when a point source is positioned in the center of that assembly, g* increases as the radius of the assembly increases (i.e.,/~increases). As shown in Fig. 6, the 1/M plot for this example is noticeably non-lineur with k~ However, in this particular instance, the curve is considered to be conservative because an extrapolation to 1/ M = 0.0 using any two successive points underestimates the critical mass. Nevertheless, it is pos- sible to perform an approach-to-critical experiment in which g* decreases as k~, increases, thereby, producing a non-conservative estimate of the critical mass. When preparing to perform an apprcach-to-critical experiment, it would be useful to understand how g* is going to change as keffincreases.
262 G.D. Spriggs/Annals of Nuclear Energy 26 (1999) 237-264
X. CONCLUSIONS
The factor g* is the parameter that converts any arbitrary distribution of sources to an equivalent fundamental-mode source distribution and strength. The equivalent fundamental-mode source strength, in turn, is the effective source strength that is multiplied by the factor 1/(1 - ke~,) to yield the correct neutron production rate in the system (keh, in this expression corresponds to the k-eigenvalue of the system and is, by definition, independent of the source distribution).
The equivalent fundamental-mode source strength has many applications in reactor kinet- ics, particularlypoint kinetics, as well as being an important parameter in criticality safety, reactor operations, and reactor experiments. In subcritical systems, the equivalent fundamental-mode source strength is easily calculated using deterministic or Monte Carlo methods and can be readily measured in existing critical assemblies using the technique described in this work.
Acknowledgment
The authors would like to thank Dr. Bill Wilson and Dr. David Madland (Los Alamos National Laboratory) for supplying the spontaneous fission and (tz,n) data used in this analysis. We would also like to thank Dr. Murao, Mr. Mukaiyama, and Mr. Osugi (JAERI) for allowing us to use the FCA facility to test our theory and our experimental technique.
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2.
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