AE-277 Epithermal and Thermal Spectrum Indices in Heavy Water

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AE-277 UDC 621.039.512

IS IS CN ÜJ Epithermal and Thermal Spectrum Indices in Heavy Water Lattices

E. K. Sokolowski and A. Jonsson

AKTIEBOLAGET ATOMENERGI

STOCKHOLM, SWEDEN 1967

AE-277

EPITHERMAL AND THERMAL SPECTRUM INDICES IN HEAVY WATER LATTICES

Evelyn K. Sokolowski and Alf Jons son

ABSTRACT

Spectral indices have been measured by foil activation technique in a number of different D? O-moderated lattices in the Swedish zero power reactor RO and the pressurized exponential assembly TZ. In most cases the fuel was in the form of single rods, distributed uniformly in the lattice. Parameters in these cases were lattice pitch and fuel composition. A 31-rod cluster lattice was also investigated, with the moderator temperature varying up to 210 C. On the basis of these measurements, as well as measurements on cluster lattices» reported by other investigators, it has been possible to derive simple correlations for the spectral indices, which seem to be of fairly general valid- ity for D?O lattices. The experimental results have also been compared to calculations with the multigroup collision probability program FLEF.

Printed and distributed in May 1967. LIST OF CONTENTS Page

1. Introduction 3

2. Experimental conditions and techniques 4 2. 1 Lattice geometry 4 2.2 Choice of spectral indices 6

3. Theory 7 3. 1 Description of method 7 3.2 Data used 10

4. Epi thermal index 1 1 4. 1 Correlation of data 11 4. 2 Comparison with theory 22

5. Thermal index 25 5. 1 Correlation of data 25

5. 2 Comparison with theory 31

6. Summary 32

7. Acknowledgements 33

8- References 34

Figures - 3 -

1. INTRODUCTION

The prediction of neutron spectra is one of the key problems in connection with the long-time behaviour of power reactors. The resonance absorption, and hence the conversion ratio> depends upon the epithermal component of the spectrum. Although the dependence of 235 the U reaction rate on neutron temperature is small due to the almost-l/v behaviour of the fission cross section, it does have some importance e. g. for the temperature coefficient of pressure tank re- 239 actors. For Pu the spectral dependence is appreciable, primarily due to a resonance at the high energy end of the "Maxwellian" distribution of thermal neutrons. Finally the effective cross sections of some of the highly absorbing fission products are sensitive to the spectrum. Much effort is being devoted to developing theoretical multigroup methods, applicable to power reactor conditions, which entail a number of complications, such as intricate geometry, heterogeniety as well as the various consequences of burn-up. Comparisons between theoretical and experimental results, pertaining to such conditions, are the ultimate test of the theoretical methods, but their integral nature makes an interpretation difficult. It is therefore desirable to put the theories to test against experiments, performed under simplified conditions and, preferrably, as parametric investigations. A series of experiments has been carried out on uniform, single- rod lattices in the D^O moderated zero power critical assembly RO [l]. Parameters in this investigation were lattice pitch and fuel composition. (The present report deals with the determination of spectral indices for these lattices, while other aspects of the work are reported in [ 2] .) Experiments have also been performed on a lattice of 31-rod cluster assemblies, simulating the fuel assemblies of the Marviken power reactor. These experiments were made in the pressurized exponential assembly TZ (e. g. [3]) at various temperatures up to 210 C. For accurate calculations of lattice parameters, a 58-group collision probability program, FLEF, has been developed. For uniform, single-rod lattices the treatment of geometry in FLEF is nearly exact, and hence a comparison with experiments for these cases is a test primarily of the group cross sections. The ability of the program to calculate spectral indices for cluster lattices has also been investigated. In machine programs, intended for survey calculations, one needs simple expressions for the strength of the epithermal component of the spectrum and for the so-called neutron temperature. The present measurements offer the opportunity of deriving such correlations for both these quantities. The correlations will, of course, contain the main independent variables of the cell, and hence may also serve as a useful basis for comparisons between experiments and more detailed calculations -e.g. of the FLEF type - since any observed discrepancies may then be more easily translated into errors in the main variables of the system.

2. EXPERIMENTAL CONDITIONS AND TECHNIQUES

2. 1 Lattice geometry

Four different types of fuel rods were available (see table 1):

Table 1 ; Fuel composition

Fuel Enrichment Fuel Fuel Canning mate rial diameter density notation (cm) (g/cm ) Material |OD(cm) ID(cm)

nat. isotopic UO (nat) 1. 35 10.45 Al 1. 52 1.37 ? composition

U-235 enrich- UO- (1. 2%) ment 1.25 10.5 AlSiNi.x 1.47 1.29 (1 . 21+. 01) %

nat. 2 containing 2. 04+. 04)% PuO? UO (Pu)3 89. 0"% Pu-239, 1.25 10.5 Zraloy-2 1.39 1.26 9. 6 % Pu-240, 1.4 % Pu-24J)

ThO2 ** 1.26 9.06 AlSiNi* 1.47 1.29

90 % Al, 9% Si and 1 % Ni xx The UO2 (Pu) was obtained from the UKAEA and the ThO2 from the AEE Trombay, India. - 5 -

The rods were arranged uniformly in a square array of variable pitch in the test lattice, which was composed of either one type of rod ("pure lattice") or of a regular pattern of two rod types ("mixed lattice").

The moderator was D?O of (99. 54 -. 04) w/o purity and at a temperature of 20 C. Of each type of rod, a few 0. 75 - 1.0 meter sections (in the case of UOT(PU), only one), were openable to allow the accom- modation of detector foils between the fuel pellets. For the plutonium- enriched fuel the loading of foils between the pellets posed severe problems of contamination. To avoid these, the pellets of the openable rod section were sealed into machined brass containers of thickness 0. 01 cm, and the inner diameter of the canning for this section was increased slightly. The test lattice region had to be surrounded by several driver zones, but could generally be made large enough for asymptotic spectrum conditions to be established at the center. The pure lattices of plutonium-enriched fuel were exceptions in this respect, due to the fact that only 68 rods of that type were available. To study the influence of the driver lattice for these configurations, macroscopic calculations were performed (see section 4). In general, the assumption of an asymptotic spectrum was checked by studying the activity ratio Cu /in across the test region. It was usually possible to adapt the driver zone adjacent to the test zone with regard to the spectrum conditions.

In the case of the cluster lattices investigated, the UO?(1 . 2%) rods were gathered into 31-rod clusters (fig. 1). Nine such clusters were arranged in a square lattice of pitch 24. 0 cm. The test region was surrounded by a buffer zone, consisting of 19-rod natural UO? fuel clusters of the type used in the Âgesta power reactor [3]. The buffer zone was fairly well matched to the test zone with regard to the spectrum, and the conditions in the central test cell were deemed to be close to the asymptotic ones. In order to retain a sufficient margin to criticality, each peripheral test assembly was poisoned with five stainless steel rods of 0. 8 cm diameter. On the basis of the correlations derived in the subsequent sections it was judged that the influence of these on the spectral indices of the unpoisoned, central assembly would be small. - 6 -

2. 2 Choice of spectral indices

To determine an epithermal spectrum index, the sandwich foil method, with In as the resonance detector, was employed. In has a dominating resonance at 1.457 eV, which is shown in relation to relevant fuel resonances in figure 2. The sandwich foil method makes use of s elf-shielding around the main detector resonance - approximately between 1. 2 and 1. 7 eV for the indium foils used - and, as demonstrated in [4], leads to a spectral index, q, which is the ratio . between the unperturbed neutron density in the detector resonance interval and the total neutron density. The normalization to total neutron density is achieved by means of the reaction rate in the nearly - ]/v detector Mn , appropriately corrected for resonance absorption. Since the energy limits of the resonance interval depend on the particular foils used and are difficult to determine accurately, the spectrum index q is normalized to the corresponding quantity for a known calibration spectrum, q . It is assumed that the spectrum shape within the detector resonance interval is always the same. This assumption is a very general one, and, as shown by multigroup calculations, valid for most uranium -fuelled lattices, where there is no significant inter- ference from fuel resonances. For lattices containing appreciable 240 quantities of Pu , the assumption is more doubtful (see figure 2). For the present experiments, the calibration site was the central channel of the RT reactor, for which the spectrum is accurately known from chopper measurements, and hence the q-values for the test lattices were obtained in an absolute sense. The ratio q/q , although having a simple interpretation, independent of the spectrum assumptions in- herent in the Westcott formalism, may also be expressed in terms of the familiar Westcott parameters T and r L4} : ? • I? • 0) c nc where the subscript 'V indicates calibration quantities. The sandwich foil method, as described in [ 4] and used here, is not based on - 7 -

quantitative knowledge of the detector cross section function nor on any limiting assumptions about the neutron spectrum. The shape of the thermal spectrum component was studied by means of the reaction rate ratio Lu /Mn relative to the same ratio in the Rl graphite thermal column, where the spectrum was assumed to be Maxwellian with the characteristic temperature equal to the physical temperature. The manganese reaction rates were corrected for resonance absorption, so that the final spectrum index was the relative lutetium-to- l/v reaction rate ratio, which could be calculated directly with the FLEF program. To facilitate the systematization of the data, the reaction rate ratios were converted to neutron temperatures.

3. THEORY

3. 1 Description of method

The calculations of cell and region averaged constants, fundamental mode and infinite lattice criticality, reaction rates etc. were done with the IBM 7044 program FLEF. This paragraph gives a brief account of the program. A more detailed description will be published elsewhere L5J. FLEF is a multi-energy-group program, solving the cell eigen- value problem by means of integral transport theory. The geometry of the fuel can be either of the cluster type or circular cylindrical. The outer boundary of the cell is assumed to be circular, and here the boundary condition of isotropic return with specified albedos is used. The program consists of four major parts as shown in table 2. The different parts correspond roughly to the chain structure of the Fortran program and are ordered in the table in the sequence in which they are used. - 8 -

Table 2: Structure of the FLEF program

Number of space Number of energy Function regions, typically groups, typically

Part 1 1 0 materials 60 Calculate macroscopic cross-sections

Part 2 4-5 60 Calculate rough space dependence and "micro- group" spectra

Part 3 up to 25 space- Less than 25, Calculate detailed points normally 10-15 space dependence in "macro-group" structure

Part 4 All physically dif- Any number less Edit results: calcu- ferent regions as than or equal to late group constants required by problem the number of and reaction rates groups in the library

In part 1, microscopic cross sections are fetched from the library or calculated as necessary (resonance region). They are then compound- ed to form macroscopic cross sections. The data used will be described in the next paragraph. Effective group cross sections in the resonance region are calculated, using equivalence theorems described by Leslie et al.L6, 7]. The procedure for establishing group data and forming effective cross sections, using the calculated equivalence, follows that described by Askew [8]. Part 2 of the program contains a collision probability calculation, using the full library group structure. Because of computer restrictions, the full geometry of the cell cannot be used at this stage. Such a calculation would also be unnecessarily time-consuming. Typically, four space regions (flat flux regions) are used. In cluster geometry these regions are chosen as fuel, canning, coolant and moderator, including guide tubes. In regular lattice geometry the division of the cell for the pur- pose of calculating micro-group spectra can be determined by the user. Collision probabilities are calculated as described by Leslie and Jonsson [9] in cluster geometry. In regular lattices, and also for the elementary cells of the cluster, collision probabilities are worked out with the method of Carlvik [lO]. The end products of this part of the program are micro-group spectra (and in cluster geometry the internal fine structure necessary for homogenization) in typically 4-5 space regions. Before entry into part 3 , this information is used to con- dense the group structure to a maximum of 25 energy groups. Normal- ly 1 0 to 15 are sufficient- The geometry is now circular cylindrical and the one-dimensional transport problem is solved using the so-called Discrete Integral Transport theory (in contrast to Collision Probability theory) described by Carlvik [ 11 ]. In problems of the present type this choice makes for a faster computation and a more efficient use of the core memory, since, with a given fine structure, Discrete Integral Transport theory needs fewer space points than Collision Probability theory needs space regions. For a more detailed discussion of these points the reader might wish to consult the cited paper by Carlvik and also a survey paper by Ahlin et al. [l2]. ' Finally in part 4 the geometry and the energy group structure are both returned to the original ones, so that, for comparison with experiments, the group constants and reaction rates may be edited using any desired group structure. Reaction rates are calculated for each physically different region of the cell and for any detector chosen from the library. In addition reaction rates for isotopes contained in the fuel may be calculated. The group constants are given for the cell or for two distinct parts of the cell (for use in so-called heterogeneous calculations). A fundamental mode calculation is carried out in the "macro- group" structure, and the group constants refer to the spectrum accordingly modified by leakage. The reaction rates are given both with and without leakage. In the calculations reported here, the fundamental mode calculations were made using diffusion theory with the diagonal transport correction to allow for anisotropic scattering. The diagonal transport correction is also used in the transport calculations of parts 2 and 3. The program is also equipped with a link for solving depletion equa- tions in up to 10 fuel regions. A typical running time on the 7044 is 6 minutes, including a quite extensive output. - 10 -

3.2 Data used

The present calculations were made not only as part of the test- ing of FLEF but also as a check of the cross-section data contained in its library. The group structure, as indeed much of the data, coincides with that used earlier in the Winfrith program WIMS [l3]. In the thermal region, below 4 eV, there are 42 groups with concentrations around the plutonium resonances (fig. 2). The U-238 resonance region, 4 eV to 0. 1 keV, is covered by 10 and the fast energy region by 6 groups. Group boundaries may be found in a report by Barclay [14]. About fifty "isotopes" are represented in the library. Most of them are fission products or aggregates of fission products, for which group cross sections were calculated by Rumpold [15]. He also computed the cross sections for various detectors, of which Lu is of special interest in this paper. The resonance parameters used for Lu were from measurements by Roberge and Sailor [l6], and are identical to those employed by Westcott [17]. Data for fuel and moderator isotopes are, with the exceptions mentioned below, the same as those used in WIMS. The 2200 m/s normalizations of the most important isotopes used in the present calculations are given in Table 3.

Table 3: 2200 m/s normalization in FLEF

238 Absorption u 2.718 b i o tr Absorption u235 680 b 235 Fission u 580 b 235 V u 2.438 u235 2.078 Absorption Pu239 1025 b T-. 239 Fission Pu 739 b 9 V Pu" 2.091 Pu239 2.093 Absorption D 0.46 mb Absorption Al 0.23 b Absorption Zraloy 0.212 b - 17 -

For all isotopes, except deuterium and hydrogen, present in trace amounts, scattering matrices were calculated from the free gas model. For deuterium, Egelstaffs effective width model [18] was used. The parameter q in this model (q = 0 corresponds to free gas) was taken to be 2.7. 238 In the resonance region the tabulated cross sections for U are reduced by 10 % with respect to those derived from current differential data [74, 79]. The fission cross sections of U are adjusted to give a fission integral of 275 barns above 0. 5 eV, in accordance with integral experiments-

4. EPITHERMAL INDEX

4. 7 Correlation of data

As pointed out in section 2. 2, the indium spectrum index is a measure of the neutron density in a spectrum range, characterized by small variations in the important neutron parameters, and hence one might hope to derive a correlation for the spectrum index from simple neutron balance considerations. Consider two energy groups, "e" (for "epi"-) and "t" (for "ther- 9 •J Q mal"), which both lie below the U resonance region. The epithermal group is fed by neutrons, slowing down in the moderator. For not-too- wide rod spacings it is thus reasonable to assume the epithermal neutron source density, Q, to be constant throughout the moderator and zero in the fuel. Let us further make the simplifying assumption that all scattering takes place in the moderator and all absorption in the fuel and in the thermal group only. With subscripts "f" and "m" referring to the fuel and moderator, respectively, E denoting the macroscopic epithermal removal cross section of the moderator and

Sf the group averaged macroscopic thermal absorption cross section of the fuel, we have

The bar indicates the volume average.

It can be shown that the effect of neglecting the epithermal fuel absorption is small in the cases of interest. - 12 -

Disregarding leakage, neutron balance requires

S 0 = Q (3) rr/m, e v and

Combining (2) - (4) we obtain

* = (Vl)/(V S ) • (5) , t v if" x m m' x '

The quantity determined in the experiments is not the flux ratio but the ratio between epithermal and total neutron density. Within the frame of the two-group treatment we then get

v t h e n V e e t K (6) n + n, v t e e t 7 + hi V e K t

As long as q is « 1 j it is, in view of (5) and (6) > a linear function of (V\.vJL)/(V Sj> passing through the origin. If the fuel is a l/v absorber in the thermal region, the product v.S- is a constant, independent of neutron temperature. If, on the other hand, the fuel cross section deviates from a 1/v function, the variation of vS, will be described by 2200 the Westcott effective cross section g(T )S [20]. In the following, we shall let Sf denote this quantity. The assumption of constant source density in the moderator, which underlies equ:n (5), starts breaking down when the rod spacing exceeds the slowing-down length from fission to group "e"j in the moderator. For such cases the effective moderator volume is less than the actual volume, and the curve q vs (V\2,.)/(V S ) becomes concave upwards, intersecting the ordinate axis at a finite value, which corresponds to the spectrum of an isolated rod in infinite moderator. For values of q, not negligible compared to unity, the curve, ac- cording to (6), tends to flatten out with increasing (VJE-)/(V E ). - 13 -

The above discussion, as indeed most multigroup calculation s > refer to a cell of an infinite lattice without leakage, whereas the experiments are performed in lattices of finite buckling. The leakage probability down to indium resonance energy will differ from the average for the neutron population, which is concentrated around thermal energy. To correct for this differential leakage effect in a simple manner, the experimental ratio q/q should be multiplied by the ratio,

Pr between the non-leakage probabilities for thermal neutrons and indium resonance neutrons in the test lattice, which can be written approximately as

where B is the buckling, T 1 ., and T is the age from fission to indium resonance and to thermal energy, respectively, and L is the

diffusion length for thermal neutrons. In D2O, T , ., v~ 0.9 T , (see e.g. [2l], ch. 16). This has been used in the evaluation of P. It should be noted that the leakage correction is mainly due to the ther- 2 mal diffusion. L and T , were obtained from few-group lattice calculations, and the bucklings were taken from measurements. A condition for the correction (7) to be adequate is that the site of measurements is far enough from the test lattice boundaries for asymptotic conditions to prevail. In view of the above, a natural presentation of the data is in the form of a plot of Pq/q versus (VXr)/(V £ ). However, since we shall only deal with D^O moderated lattices, we prefer to eliminate the moderator cross section, retaining only its variation with temperature.

Thus we choose for our independent variable the quantity (p (293°) V J!f)/ /(p(T )V E ), p being the moderator density and S the effective thermal macroscopic absorption cross section of natural UO9 of density 10. 5 g/cm . For natural uranium dioxide at 20 C, this variable re- duces to the fuel-to-moderator volume ratio. For the fuels not containing plutonium, the thermal cross section is nearly l/v-dependent, the variation in the Westcott g-factor between T =50 and 150 °C being for 235 238 n U about 2 % and for U much less than 1 %. Hence for these fuels - 14 -

as well as for thorium, the quantity Zf may be regarded as a characteristic constant for any one type of fuel rod, independent of rod spacing. In calculating E, for the low temperature cases, the Westcott g-values for T =100 C were used for uranium, in combination with the 2200 m/s n cross sections given in ref. [22] for fissile and in [23] for non-fissile nuclides. This leads to the value 0. 172 cm for S . For thorium, g was taken to be equal to unity. For the cases of moderator temperatures above 200 C the g-values corresponding to T = 330 C were used for uranium, changing 2f by about 2 %. For mixed lattices, containing two types of fuel rods, the following expression was used:

V (8) 1 + 3

Here $^ and $? is the thermal flux in rods of type 1 and 2, respectively, and j is the number of rods of type 2 in the lattice per rod of type 1. The flux split between the different types of rods in the mixed lattices was determined experimentally (see table 4).

Table 4: Flux split in mixed lattices

Fuel Fuel Ratio Lattice ni ii "2" iti ii.n^'i pitch cm '1 / *z

3.2V2 .958 i. 006 UO2(1.2%) ThO2 1:1 6.4 .952 ±.015

3.2V2 .947 ±.005 UO2(1.2%) ThO2 3:1 6.4 .959 ±. 006 3.2VT .750 t. 005 UO2(Pu) ThO2 1:1 6.4 .715 ±.005

UO2(Pu) UO2(nat) 1:1 6.4 . 744 ±. 015

For the UO2(1. 2%)/ThO2 lattices, 0. /02 refers to the Cu reaction rates on the can surfaces; for the plutonium lattices, to the Mn" reaction rates inside the fuel. - 15 -

Expression (8) implies a certain homogenization of the lattice. Since the epithermal index may be expected to vary approximately in- versely with the thermal neutron flux level, also locally, the measured epithermal indices for the two rod types should differ by a factor equal to the thermal flux split. Their mean value should correspond to that value of the independent variable (VX2/{V 2 ), obtained by expression (8). Accordingly, the values (Pq/q ), and (Pq/q )2 were multiplied by [] i (tf. - $J)/{$, + $2) ^' respectively. With this correction the epithermal indices of two constituent rod types were found to be within experimental error of each other and of the volume-weighted mean of the un corrected values (see table 6). For plutonium-bearing lattices the situation is complicated by the fact that the effective thermal fuel absorption cross section is sensitive to the neutron temperature and hence dependent on rod spacing. To calculate the independent variable for the different plutonium lattices, the 2200 m/s cross sections for the fuel constituents were multiplied by the Westcott g-factors, corresponding to the experimental Lu/Mn raac- tion rate ratios. However, this must be expected to result in too high values for the effective fuel absorption cross sections (see section 5. l). The uncertainty in the experimental results is greater for the plutonium lattices than for the uranium and thorium lattices, due to the fact that only one measuring position in the lattice was available for the former. Since the fuel rods were not very straight, the results might be affected by local non-uniformities in the lattice» which, in the case of the plutonium lattices, could not be eliminated by averaging over many positions. The experimental data for uniform rod lattices are given in tables 5a and 6. Of the two sets of values for P, given in table 5, one was calculated by expression (7), and the other is the ratio between q/q as c calculated by FLEF for infinite and for critical lattices. The two sets differ by less than 2 %. The different experimental values of Pq/q c (for mixed lattices, the flux split corrected values) are plotted against (V, E,)/(V E .) in figure 3. All the points are well gathered around a X I XTX II ell common straight line through the origin, although the plot tends to be concave upwards for very wide and downwards for very tight lattices, as predicted above. - 16 -

The points for the uniform, natural UO2 rod lattices are re- plotted in figure 4a, and a straight line through the origin is fitted through the points, giving

Pq/qc =18-6 ^i-| (9) m nat

As indicated by figure 3, the correlation also fits the other types of uniform rod lattices. This is brought out further in tables 5a and 6, which include the values Pq/q obtained by (9). It is seen that the agreement between these and the experimental values is mostly much better than 10 %. Figure 4b, which gives a more detailed representa- tion of the data for the pure UO,>(?. 2 %) rod lattices, shows a very slight but systematic trend for the points to lie above those for natural UO,, indicating a slight dependence on enrichment over and above that entailed in the simple correlation (9). The latter may therefore break down for enrichments higher than a few percent. The agreement between the correlation and the experimental points is good also for the plutonium-enriched lattices. For the pure plutonium-enriched lattices, the values used for Sf are clearly too high (see section 5. 1).

In fig. 4d the uncertainty in Sf is indicated by a lower limit of the 239 independent variable, corresponding to g(Pu ) = 1. The correlation (9) can be expressed in terms of the Westcott parameters r and T by using (1) together with the numerical values of r and T . r has been determined by chopper measurements to c H c c be 0. 0400 ± 0. 0015 [24]. (Measurements of the thin-foil activity ratio In /Mn relative to the same ratio in a thermal column could be brought into agreement with this value only by assuming that the Westcott s.-factors for In are about 7 % too high [25], which is in accordance with the findings of [26]. The neutron temperature in the calibration position, T , was determined by different methods [27, nc 28] to be(T + (22±4))°K. For the present calibrations the moderator temperature was 301 °K. Expression (9 ) can then be rewritten as V 2 =13.4^4f_ 00) m nat - 17 -

This expression may be compared to the approximate formula for a homogeneous medium, given by Westcott et al. [20]:

TiT

which, by extension to the heterogeneous case, gives

r: m nat

It should be noted that this expression, like expression (10), refers to the mean value in the fuel: the transition to the heterogenous case in- troduces a factor ~$. / ^ ,, in the cell-averaged spectrum index, which again is cancelled when taking the fuel-averaged value. Substituting numerical values, equ:n (11) gives

m nat which is within 1 5 % of equ:n (10) Table 5: Fuel epithermal index for pure lattices 5a: Uniform rod lattices. T = 20 "oC. m—

—•—••'-—•••" • ' 1 1 Fuel Pitch (Pq/q ) V E P q/qr k/^exp Pc f f 1 (cm) (q/q } (Pq/q ) - Vm £nat two-grouf FLEF exp. FLEF c FLEF exp. correl. (9) 6.4 0.0366 0. 897 0.903 0. 81 4±. 0100.822 O.73±.O1 0.68 +7 3.8V2" 0.0529 - 0.931 - 1. 13 - 0.98 -

UO2(nat) 3.2V2" 0.0766 0.952 0.955 1.47 ±.08 1.59 -8 1.40±.08 1.42 -1 4. Ox 0. 104 1. 000 - 1.9 ±.2 - - 1.9 ±.2 1.93 -2 3.2 0. 169 1. 013 1.022 2.98 ±. 06 3. 15 -5 3. 02±. 06 3. 14 -4

00 6.4 0. 0448 0.814 JO. 830 1. 13 ±.03 1. 12 + 1 0.92±.03 0. 83 + 11 1 3.8^ 0. 0648 0.851 | 0. 864 1.48 ±. 03 1.54 -4 1.26±. 03 1.20 + 5 UO2(1.2%) 3.2V2" 0. 0938 0. 883 ! 0. 896 2. 05 ±. 03 2. 12 _3 1.81±. 03 1.74 +4 3.2 0.206 0.957 JO. 971 3.93 ±.07 4. 14 -5 3.76±. 07 3. 83 -2

6.4V2 0.0585 0. 746 0. 740 1. 65 ±.20 1.65 ±0 1.22±. 20 1. 09 + 12

UO2(Pu) 6.4 0. 121 0. 841 0. 825 2.46 ±.20 3. 16 -22 î.07±.20 2.26 -8 3.2V2" 0.269 0.906 (0. 909)XS5. 09 ±.20 6. 29XX -19 i. 76±. 2(f3 [ 5.00

Moderator temperature = 90 C. The independent variable has been increased accordingly. Non-asymptotic correction gives P = 0.936, which has been used in the FLEF calculation and in calculating Pq/q . For the other lattices an asymtotic spectrum can be assumed. Table 5b: Cluster lattices

t V S P (Pq/q ) Fuel CoolantPitch T p(20°) f f q/< fa/*c)exp Pq /qc c exp p(r vT ) V E . m m nat two- FLEF (q/qc)FLEF (Pq/q ) ] i i(cm) group ; exp. FLEF exp. correl (%) (9)1 (%) 20 0. 104 0.91 2. 19±.O5 2.31 c I.99±.05 1.93 + 3

UO2(1.2%) 89 0. 107 0.92 2. 14±.05 (2. 35)** -9 1.97±.05 1.99 -1 _ 31 -rod clust. D2O 24. 0 140 0. 112 0.92 — 2.38 - 2.08 -

Marviken 208 0. 119 0.93 _ 2.07±. 15 (2. 53)** -19 1.93±. 15 2.2] -13

220 0. 122 0.93 - - 2. 60 - - 2.27 -

ÙO- (nat), 35 0. 0640 0.912 - 1.40 - - 1.28 1. 19 + 7 19-rod clust. DOO i 27. 0 x ù Âgesta [31 ] \ k212 0. 0734 0.925 _ 1.60 _ _ 1.48 1.37 + 8 36 20 0.0291 0.92 0.89 0. 63±. 02 0.75 -16 0. 58±.02 0. 54 + 7 UO2(nat) 28 0. 0508 0.91 1. 05±. 03 - - 0. 95±.O4 0.94 + 1 19-rod clust. Air 24 0.0730 0.93 - 1.4O±.O4 - 1.30±. 05 1.36 -4 AECL [32]* 21 H 0. 1007 0.97 1.9Ü.06 - - 1.85±.O8 1.87 -1 18 • • 0. 1522 0.99 £.69±.08 2.66±.10 2. 83 -6 ... i 36 H 0. 0286 0.93 0.89 0. 59±. 02 0.73 -19 0. 55±.02 0.53 +4

H UO2(nat) 28 0. 0493 0.91 - - - - 0. 87±.04 0.92 -5

19-rod clust. D?O 24 0. 0698 0.93 - - - - 1. 18±.O5 1.30 -9 AECL [32]* 21 H 0. 0947 0.96 - - - - 1.68±.07 1.76 -5 18 it 0. 1389 0.99 0.99 2.39±. 07 2.71 -12 2.37±. 09 2.58 -8 The experimental data given here were deduced from the rVT values of ref. :s [31 ] and [32],using relation (1).

These values have been interpolated from the plot of q/q vs p(D->O). Table 6: Fuel epithermal index for mixed lattices. T = 20 C £ ___—, .—__ m

Fuel Pitch V S Site of Pq/qc (Pq/q ) f f measure- x n/ q/qc v(Pq/ H/ q^c'ex ) p ,' (cm) v s <. ment exp. corr. for correlation ^c' correl. m nat flux split (9)

UO2 rod 0.697±.030 0. 78±.O3 0. 76±.03 +12 , j UO2(1.2% 6.4 0. 0367 0. 68 ThO? " 0. 650±. 030 0.73±. 03 0. 75±.O3 +10 1 ThO2 uo2 " 1.41 ±.09 1.52±.O9 1.47±. 09 +3 | 1:1 3.2^ 0.0768 1.43 ThO2 " 1.32 ±. 07 1.42±. 07 1.46±.O7 +2 j UO2 rod 0.84 ±.03 0. 77±.03 0. 75±.O3 -1 1 UO2{1. 2%) 6.4 0. 0407 0.76 ThO2 " 0.74 ±. 03 0.68±. 03 0. 70±.03 -8 j ThO2 + 1 | uo2 " 1. 70 ±.06 1.64±.O6 1. 59±.06 3.2/2" 0.0850 1.58 o 3:1 ThO2 " 1.48 ±. 04 1. 43±.04 1.47±. 04

UO2(Pu)rod 2. 06 ±.20 1. 73±.20 1. 51±.2O + 14 | 1.32 UO2(Pu) 6.4 0. 0712 UO2(nat) " 1.47 ±.20 1.24±. 20 ).40±. 20 +6 UO2(nat) UO2(Pu) " 3.72 ±.20 3.36±. 20 2.92±.2O + 1 1:1 3.2V2" 0. 1557 2.89 UO2(nat) " 2.94 ±.20 2.66±.2O 3.00±.20 +4

UO2(Pu) " 1. 60 ±.20 1.44±.20 1.25±.2O ±0 UO2(Pu) 6.4 0.0670 1.25 ThO2 1.25 ±.20 1. 13±.2O 1.28±.20 +2 ThO2 UO2(Pu)rod 3.90 ±.20 3.70±. 20 3.21±.2O + 17 3.2V2" 0. 1473 2.74 1:1 TI1O2 " 2. 85 ±.20 2. 70±. 20 3.05±.20 + 11 It can be seen from fig. 3 that even for moderately wide lattices the epithermal index is determined almost entirely by the lattice sur- rounding the rod in which the measurement is made: the "eigenspec- trum" of an isolated rod is very soft. This confirms that the slowing- down density is largely a property of the homogenized lattice and leads one to expect the following: a) the epithermal index varies over the lattice cell as the inverse of the neutron density, or, approximately, the thermal flux density (this has already been assumed in the analysis of the mixed lattice data), and b) the correlation (9) might also be valid for cluster lattices. Assumption a) can be applied to the measurements of cell boundary

(cb) spectrum indices for natural uranium metal rod lattices in D?O (RO reference lattices), reported in [4]. Table 7 gives a comparison between the measured epithermal indices and those of correlation (9) , multiplied by the ratio between the neutron density in the fuel and at the cell boundary. This fine structure correction factor was obtained by interpolation from the experimental data of [29]» which was in fair agreement with diffusion theory calculations, using an empirical fuel diffusion length [30].

Table 7. Epithermal indices at the cell boundary in natural uranium metal rod lattices

Pitch VfSf (Pq/qc) cb V S , (cm) m nat exp. [4] correl. (9)

19 0. 0420 0.418±.010 0.40 17 0.0527 0.530±.015 0.51 14 0.0792 0. 822±. 016 0.79 11 0. 1323 1. 397±.016 1.37

Table 5b gives the average epithermal index in the fuel for various cluster lattices. The 31-rod cluster lattice is that described in section Z. Measurements were made in the pressurized exponential assembly - 22 -

at 20, 89 and 208 C. As seen from the table and from figure 3, the agreement between the correlation and the measurements is within experimental error for the lower temperatures. For the highest temperature there is a discrepancy» which is probably due to an erroneous experimental value. Preliminary high temperature results are also available from low power activation experiments in the Âgesta reactor [31 ], in which the Westcott two-foil method was employed to determine r and T . Because of the long time required to remove the foils from n 115 the reactor, In could not be used, and a number of other detectors with higher energy resonances were substituted. Hence the leakage correction factor P as calculated for In must be expected to give slightly too high an experimental value for the infinite lattice epithermal index. The indices q/q and Pq/q were obtained from the values r 1 C C of rVT given in [31 ] by means of expression (1). Looking only at the tempe rature dependence of the infinite lattice index, it is in fair agreement with the correlation. A large number of experimental spectrum indices for natural UO? 19-rod clusters in D?O have been reported by Green and Bigham [32]. Their results were based on the thin-foil reaction rate ratios In /bAn in the fuel, normalized to thermal-column values. The analysis was made in terms of the Westcott formalism, the 115 s .-factors for In u having been lowered by about 6 %, which is consis- tent with equ:n (10). The r- and T -values quoted are those of the critical lattices. However, the leakage correction factor P can be calculated from lattice data given in [32] and [33], using expression (7). In calculating the independent variable (VXf /(V S ), the moderating effect of the large quantities of aluminium in the lattices were taken into ac- v count by setting Vm = V^Q + Al(&s)A1/(&s) D, Q- The correction is, however, at most of the order of 2%. It is seen from table 5b that the correlation (9) reproduces the experimental data, also for cluster lattices, to well within 10 %. 4. 2 Comparison with theory Due to the simple interpretation of the quantity q given in section 2.2, the spectrum index q/q (or Pq/q ) lends itself readily to comparisons with multigroup calculations. In this particular case such comparisons could be made in an absolute sense, since the detailed form of the - 23 -

calibration spectrum was known from chopper measurements [27J. When normalizing the theoretical spectrum to the chopper-measured one, it is of course essential that the latter be treated in the same way as the former with regard to group structure and end points. In order to make the present data available for comparison with other multigroup programs, the chopper-measured calibration spectrum is given in table 8. The following relations hold between the spectrometer channel number i, the flight time t, and the energy E:

t = (i - |) • 200 |j,s,

The neutron beam was extracted from the RJ central channel by means of a graphite scatterer, which was found to cause a minimum of spectral distortion. The chopper measurements were performed at a moderator temperature of 37 C. while the foil calibrations were made at 28 C. However, this discrepancy does not influence the q- values. The normalized theoretical values may contain a systematic error of about 2 %, due to the error in the chopper-measured spectrum.

Table 8. (E) from a graphite scatterer in the R? central channel T = 37 °C [27]. m

i 0 1 2 3 4

00 149.4 146.4 147. 7 05 142.4 145.4 147.6 150.8 751. 7 10 160.3 164.7 174.6 183.4 202.2 15 222.7 253.9 310.9 393.3 492. 0 20 616.5 746.9 913.8 1045. 1 7213.8 25 I 323.7 1440.2 1562.3 I 620.6 7720.9 30 1753.2 1796.5 1814.7 1875.5 1778^2 35 1736.2 1737.0 1677.0 7627. 0 7606. 1 40 1557.2 1494. 5 1440.1 1412.2 1347.5 45 1300.9 1222.5 1166.5 17 49.4 1094.3 50 1035.9 985.7 949.7 888.4 837.4 55 800. 7 770.5 725.9 688.3 670. 6 60 638.9 600.2 559.9 542.4 574.6 65 497.9 474.8 449.4 436.4 410. 7 70 408.5 394. 7 361. Î 343.5 328.2 75 310. 1 301.8 300. 7 287.6 287.2 80 270.2 254.5 243. 1 227.8 230.4 85 203. 1 212. 5 199-5 781.9 7 74.3 90 155. 1 160.2 146.5 144.0 139.4 95 135. 0 138.6 134.2 7 24.4 125.4 - 24 -

Comparisons between the experimental spectrum, indices and those calculated by FLEF are given in tables 5 and 6 and in figures 4a, b, c» and d. A criterion for satisfactory agreement is obtained by con- sidering an error in the relative epithermal level equivalent to an error in the effective resonance integral. The uncertainty in measurements of the latter is about 4 % [34]. These errors, naturally, become more important the harder the spectrum. If, for instance, the conversion ratio (CR) for uranium fuel is to be calculated from given spectrum and resonance data, we have

d(CR)/(CR) - [ resonance absorption, . ^ v " v ' total absorption 238 ' dE/E being the relative error in either the epithermal level or the resonance integral. For the tightest uranium lattices investigated here, the bracketed factor is about 0. 6. It should, of course, be remember- ed that agreement with regard to the indium spectral index does not necessarily imply a corresponding accuracy for all epithermal absorption. It is seen from table 5a and figures 4a and b that, for the uniform

UO?(nat) and UO?(l.2%) lattices, the FLEF values generally lie within 5 % of the experimental ones, the latter being consistently lower. In view of the above, this agreement must be considered quite satisfactory. For the two tightest plutonium-enriched lattices the discrepancy between FIJEF and the experimental results is much greater (table 5a and figure 4d). The possibility of an explanation in terms of perturbations due to 240 n • 4.- * J * n i. -A 197 JT 115 Pu was investigated experimentally by comparing Au and In epithermal indices in fuel rods with and without plutonium, the assump- 240 tion being that, whereas the Pu resonance might interfere with the In " resonance, the flux at the main Au resonance would remain unperturbed. The outcome of these measurements, as well as studies of 240 the theoretical flux shapes» showed that the Pu effect was not sufficient to explain the discrepancy. The comparatively good agreement between the experimental data, for the pure as well as the mixed plutonium lattices, (tables 5a and 6) and the correlation (9) gives increased confidence in the experimental results. - For the cluster lattices the discrepancies are in the same sense as for the uniform rod lattices - 25 -

(table 5b and figure 4c). Disregarding the case of the 31-rod cluster lattice at 208 C, where the experimental value is apparently too low, the discrepancy between FLEF and experiments is between 5 and JO % for the 31-rod cluster lattice and somewhat larger for the AECL lattices. It may be noted that if the multigroup program overestimates 238 the epithermal flux level throughout the U resonance region, then this might, in the calculation of resonance absorption, be offset by an error in the FLEF resonance cross sections, which have purposely been chosen 10 % lower than current differential data in order to give agreement with integral results (see section 3.2).

5. THERMAL INDEX

5. 1 Correlation of data

The shape of the thermal spectrum component was studied by means of the reaction rate ratio Lu /Mn relative to this ratio in the Rl graphite thermal column, where the spectrum was assumed to be Maxwellian with the characteristic temperature equal to the physical temperature, 25 C. The resonance component of the manganese absorption was separated from the l/v component by using the measured epithermal spectrum indices together with the effective resonance integral for the foils. The calibration was checked by measuring the Lu/Mn ratio in the central channel of the Rl reactor, for which the Maxwellian shape of the thermal spectrum component has been established by chopper measurements [27], and the neutron temperature has been determined from chopper as well as pile oscillator measurements with Cd, Sm and Gd [28]. Using the Westcott parameters for lutetium in conjunction with the r -value given in section 4. 1, the following result was obtained: c

T (Lu/Mn) = (52 ± 2) °C as compared to

Tn[27, 28] = (51.5 ± 3.5) °C - 26 -

The thermal spectrum indices (RT /R, / )/(RT /R, / ) are given, for the pure lattices in tables 9a and b,and for mixed lattices in table 10. Table 9b comprises the results for the 31-rod cluster lattice investigated here, as well as preliminary results from the Agesta reactor [31] and the results of Green and Bigham [32], Although its physical significance is doubtful, the neutron temperature is a useful concept for systematizing the data. As shown by Cohen [35], the neutron temperature for homogenous absorber - moderator mixtures should be given by the expression

We note (cf. section 4. 1) that for homogenous media S /% T. is propor 3* S tional to the epithermal index rVT . We shall assume that, in a heterogenous lattice , AT has an "over-all lattice " component, depending in the same way on the epithermal index, and, in addition, a "local" component, depending on the thermal absorption in the rod in which the measurement is made. A similar approach has been taken in [26]. We thus write for the fuel averaged temperature increment

T / Vf i mhr4-J 1t / 2W „ \ m nat/ lattice rod In the case of cluster assemblies, the last term is taken to refer to a single rod in the cluster. Expression (12) has been fitted by a least squares procedure to all the experimental low temperature data of tables 9 and 10, with the exception of those pertaining to the plutonium-enriched rods. With V, expressed in cm per cm height and H. in cm , the fit gave the result

k] = 1.42 ± .09

k2 = 1 04 ±10

In analogy with the homogeneous case, the temperature dependence of the first term in expression (12) is obtained by taking into ac-

Introducing the disadvantage factor in the variable does not improve the correlation. - 27 -

Table 9: Thermal spectrum index in fuel for pure lattices 9a: Uniform rod lattices

Fuel Pitch T (RL>]/v) exp. . AT (°c) 111 (RLu/R FLEF

(cm) (°0 exp. FLEF (%) exp. correl. (1 3) 6.4 19.1 1.14±.01 1. 12 +2 38±2 41

UO2(nat) 3. 2V2 20.4 1.20±.01 1. 18 +2 50±4 57 3.2 21. 1 1.31±.01 1.29 +2 83±3 96 isol, rod 20.0 - - - 28±2 33 6.4 21.2 1.19±.01 1. 15 + 3 51±4 50

UO2(1.2%) 3.2/2 19.9 Î.26±.O1 1.23 +2 70±3 70 3.2 21. I 1.42±.O1 1.34 +6 124±3 117

6.4 20.5 1.22±. 02 1.23 -1 (55±6) 134 UO2(Pu) 3.2/2" 20.5 1.36±. 02 1.31 +4 (117±6) 196 - 28 •-

Table 9b: Cluster lattices

(RLu/R]/v) Fuel Coolant Pitch T exp. , r m 1 "D / *D A r(°c) 1 XVT / Xv . /Vc FLEF" correl. (cm) exp. FLEF exp. (13)

UO2(1.2 %), 20 1.30±. 01 1.30 ±0 82±6 75 x 31 -rod clust. D2° 24.0 89 1. 54±.O1 (1.54) ±0 90±5 94 Marviken 140 - 1.70 - - 110 208 1.94±.O2 (1.90)K +2 125±20 133 220 - 1.93 - - 138

UO2(nat), 35 69 67 1 9-rod clust. D2O 27. 0 212 - - - 142 117 Âgesta [31]

36 20 1.T8±.07 1.22 -3 49±3 41

UO2(nat), 28 " - - - 58±3 50 19-rod clust. Air 24 it - - - 63±3 59 KK AECL [32] 21 " - - - 72±3 70 18 H - ' - - 86±3 92

36 20 1. 1 6±. 01 1.20 -3 44±3 40

UO2(nat), 28 - - - 51±3 50 H 19-rod clust. D2O 24 - - - 59±3 57 HK AECL [32] 21 " - — - 72±3 68 18 1.31±.01 1.31 ±0 86±3 86

These values have.been interpolated from a plot of the reaction rate ratio vs temperature. The reaction rate ratios have been deduced from the T and r values of ref. [32], using the Westcott parameters for lutetium. - 29 -

Table 10; Thermal spectrum index in fuel for mixed lattices

Site of Fuel Pitch R R AT (°C) measurement ( Lu/ l/v)

(cm) c exp. correl. (1 3)

UO2 rod 1.13±.01 33±3 47 UO2(1.2? 6.4 20.5 ThO2 " 1. 10±. 01 26±3 36

ThO2 UO2 rod 1.22±. 01 57±2 63 3.2/2" 21. 0 1:1 ThO? " 1.23±. 01 59±2 52

UO2 rod 1. 17±.01 43±3 48 6.4 21. 0

ThO2 UO rod 1.28±.O1 70±4 67 3:1 2 3. 2/2" 20. 0 ThO2 » 1.27±.O1 67±4 56

UO2(Pu) rod 1.23±. 02 (60±5) 114 UO2(Pu) 6.4 18.2 UO2(nat) " 1. 19±.01 50±4 55 UO2(nat) UO2(Pu) rod ].36±. 02 (ÏO8±8) 149 1:1 3.2/2 20.0 UO2(nat) " 1.33±.O1 93±3 90

UO2(Pu) rod ?.20±. 02 (50±6) 112 UO2(Pu) 6.4 19.8 ThO2 1. 16±. 01 39±3 48 ThO2

1:1 UO2(Pu) rod 1.33±.02 (89±6) 145 3.2/2" 20. 0 ThO2 1.29±. 01 78±4 82 - 30 -

count the variation in the moderator density. There is no obvious reason for introducing a temperature dependence in the second term. However, better agreement with the experimental results is obtained T V S We thuS get f r if the second term is rewritten 1 04 -jsj * ( f f) rod' ° the temperature dependent fuel spectrum index correlation:

AT = 1.42 + 0. 355 v(V,S.) , (13) p(T ) f frod m K v vT . lattice m' m nat A comparison between this correlation and the measured neutron temperature values is given in figure 5 as well as in the tables.

The coefficient k1 of equ:n (12) can be identified with k of the homogeneous case, multiplied-by the factor S /§ E . For D7O at room temperature this latter factor is approximately unity, and hence the value 1. 42 ± .09 should be compared to the value of k given in [.35], i. e. k = 1.8 (from Monte Carlo calculations) and 1. 2 ^ k 41 (from analytical approximations). V* For wide lattices it may be expected that, for the cell boundary, k~ = 0, since AT for that position should extrapolate to zero as V\./V £t i m goes to zero, k., on the other hand, should remain unchanged, since the first term in (12) is an over-all lattice component. Table 1 1, giving the cell boundary data of ref. [4], indicates that this is true to a good approximation, at least for rod lattices. (The T values of [4] were slightly incorrect and have been adjusted.) Table 1 1. Cell boundary neutron temperatures in natural uranium metal rod lattices. k1 = 1.42 ±.09 % ko = 0. Square V S lattice f f AT (°C) pitch Vm 2 na .t (cm) exp. correl. [4] (12) 19 0. 0420 18± 4 17 ± 1.5 17 0. 0527 19 ± 4 22 ± 2 11 0. 1323 46 ± 4 54 ±6 In the case of the ]9-rod cluster lattices studied by Green and Bigham [32], measurements at the cell boundary give lower values of AT than would be obtained by setting k_ s 0. The plot of AT vs V /V r m intersects the ordinate axis at a negative value, which may be due to the assumption of too low a neutron temperature for the AECL reference spectrum [36]. A correction for this would raise the fuel- averaged neutron temperature as well and give poorer agreement with correlation (13). This might be expected for tight clusters, where the "rod" component of (13) would tend to give an underestimate of the local hardening in the fuel. As seen from tables 9a and 10, the correlation breaks down for plutonium-enriched fuel, for which the experimentally determined neutron temperature is much lower than that of the correlation. This is not surprising in view of the considerable overlapping of the Lu and the 239 239 Pu resonances (see fig. 2). If Pu had been used as a spectrum indicator, this would obviously have led to even lower neutron temperatures, which illustrates the ambiguity of the neutron temperature concept. Conversely, the neutron temperatures, deduced from lutetium measurements, when used in conjunction with the Westcott parameters for plutonium, will give too high effective cross sections for plutonium- enriched fuel, as stated in section 4. 1.

5. 2 Comparison with theory

For the pure lattices the spectrum index (R. /R, / )/(Ry /Rj /) was calculated with FLEF, and these results are also shown in tables 9a and b. In all cases, including those with plutonium enrichment, the calculated reaction rate ratios agree within about 3 % with the experimental ones. It may be observed that whereas the experimental values lie systematically above the calculated ones for the Swedish lattices, the converse is true for the Canadian ones. This speaks for the possibility, mentioned in section 5. 1, that the neutron temperatures of ref. [32] are slightly too low. The lutetium reaction rate is generally very sensitive to the shape of the thermal spectrum - about three times as sensitive to 239 a shift in the Maxwellian as the Pu fission rate. Since for plutonium-enriched fuel, where the shape of the thermal spectrum is of - 32 -

particular importance, the uncertainty in the plutonium content is usually of the order of a few percent (see table 1), the agreement obtained must be considered satisfactory. Both the experiments and the calculations on pure plutonium-enriched lattices show that the lutetium reaction rate - and this would be even more true for the fission rate in a 239 Pu detector - is fairly insensitive to the amount of plutonium present in the lattice (or to the cross section normalization for the plutonium), since the general shift in the neutron population with increasing absorption towards the detector resonance is offset by the increased shielding of the latter by the fuel resonance. 6. SUMMARY

Fuel-averaged spectral indices have been examined for a large number of heavy water lattices, differing with regard to geometry, fuel composition and moderator temperature. It has been found that practically all of the experimental material can be summarized by means of simple correlations, which, when expressed in terms of the Westcott parameters, take the form:

rrr m nat lattice

^TTt 1(t. m HX m' m nat lattice

In these expressions Sf is the effective thermal macroscopic absorption cross section of the fuel, i.e. gS, » and S = 0. 172 cm . p is the x nat density of D?O at a certain temperature. The factor P corrects for differential leakage in a finite lattice and depends somewhat on the detector 1 1 5 used. For In it takes the form p 1+0.9 T B2 (1 + T B2) (1 + L2BZ)

The volume ratios in the "lattice" terms refer to a lattice cell. The volume V, in the "rod" term is that of a single fuel rod, in cm per cm height, also for cluster lattices. As might be expected, the concept of neutron temperature is not applicable to plutonium-enriched lattices. - 33 -

Calculations of the spectral indices have been performed with the 58-group collision probability program FLEF. Good agreement with experiments is obtained for the thermal index. For the epithermal index the calculations give values that are systematically higher than the experimental ones. The discrepancy is small for uniform rod lattices containing only uranium, but larger for tight, plutonium- enriched lattices and cluster lattices.

7. ACKNOWLEDGEMENTS

We should like to thank the RO operating crew under Mr E. Wendler for precise and flexible work, and Mr Rolf Bladh for thor- oughness in handling the foil activations. We are also indebted to many colleagues for fruitful discussions, and in this context would particularly like to mention Dr Rolf Persson and Mr C -E. Wikdahl. The thorium oxide rods, used in part of this investigation, were put at our disposal by the Atomic Energy Establishment Trombay, which is gratefully acknowledged. - 34 -

8. REFERENCES

1. LANDEGÂRD O, CAVALLIN K, and JONSSON G, The Swedish Zero Power Reactor RO. 1961. (AE-55)

2. WIKDAHL, C.-E. étal., Paper to be presented at the British Nuclear Energy Society Conference on the Physics Problems of Thermal Reactor Design. 1967.

3. PERSSON R et al. , Buckling Measurements up to 250 C on Lattices of Âgesta Clus- ters and on D?O Alone in the Pressurized Exponential Assembly TZ. 1966. (AE-254)

4. SOKOLOWSKI E K, J. Nucl. Energy_2J (1967) 35.

5. JONSSON A, AE report to be published.

6. LESLIE D C, HILL J G, and JONSSON A, Nucl. Sei. Eng. ^(1965) 78.

7. LESLIE D Cand JONSSON A, Nucl. Sei. Eng. JÎ3_(1965.) 82.

8. ASKEW J R, The Calculation of Resonance Captures in a Few-Group Approxima- tion. 1966. (AEEW-R489)

9. LESLIE D Cand JONSSON A, Nucl. Sei. Eng. ji3_(1965) 272.

1 0. CARLVIK I, Internat. Conf. on the Peaceful Uses of Atomic Energy, 3. 1964, Geneva. Proc. Vol. 2. New York 1965. p. 225.

11. CARLVIK I, Integral Transport Theory in One-Dimensional Geometries. 1966. (AE-227)

12. CARLVIK I, AHLIN A, and JONSSON A, AB Atomenergi, Sweden. 1966. (Internal report RFR-548. )

13. ASKEW J R, FAYERS F J, and KEMSHELL P B, J. of B.N.E.S. , _5 (1966) 564.

14. BARCLAY F R, An Analysis of Uranium Metal- Graphite Systems Using the Multi- group Code WIMS. 1966. (AEEW-R473) - 35 -

15. RUMPOLD A, Calculation of Group Cross Sections for the Neutron Capture in Fission Products and Detector Materials. 1965. (internal Report HÏR-50.)

16. ROBERGE J P,and SAILOR V L, Nucl. Sei. Eng. 2(1960) 502.

17,. WESTCOTT C H, Effective Cross Section Values for Well-Moderated Thermal Reactor Spectra. 3 ed. corr. 1962. (AECL-1101)

1 8. EGELSTAFF P A, The Physics of the Thermal Neutron Scattering Law. 1962. (AERE-NP/GEN 29.)

19. ASKEW J R, Some Problems in the Calculation of Resonance Capture in Lattices. ANS, San Diego, 1966. Proc. Nat. Topical Meeting. Vol. 2, Resonance Absorption. MIT Press. 1966. p. 395.

20. WESTCOTT C H, WALKER W H, and ALEXANDER IK, Internat. Conf. on The Peaceful Uses of Atomic Energy, 2. 1958, Geneva. Proc. Vol. 16. Geneva 1958. p. 70.

21. BECKURTS K H, and WIRTZ K, Neutron Physics. Springer Verlag, Berlin, 1964.

22. WESTCOTT C H et al. , Atomic Energy Review^ (1965) 2, 1.

23. HUGHES D J, and SCHWARTZ R B, Neutron Cross Sections. 2 ed. 1958. (BNL-325)

24. JOHANSSON E et al. , Ark. Fys. ]_8 (I960) 513.

25. SOKOLOWSKIE, AB Atomenergi, Sweden. 1965. (Internal Report FFR-38.)

26. BIGHAM C B, CHIDLEY B G, and TURNER R B, Experimental Effective Fission Cross Sections and Neutron Spectra in a Uranium Fuel Rod. Part 2. CANDU-type Ura- nium Oxide Clusters. 1961. (AECL-1350)

27. JOHANSSON E, Some Studies of Thermal and Epithermal Neutron Spectra in Heterogenous Systems. Thesis. Chalmers Technical University, Gothenburg. 1966.

28. SOKOLOWSKIE, PEKAREK H , and JONSSON E, Nukleonik £ (1964) 245. - 36 -

29. COHEN E R, Internat. Conf. on the Peaceful Uses of Atomic Energy, 1. 1955, Geneva. Proc. Vol. 5. New York 1956. p. 268.

30. HOROWITZ J, Internat. Conf. on the Peaceful Uses of Atomic Energy, 1. 1955, Geneva. Proc. Vol. 5. New York 1956. p. 256.

31. JOHANSSON H,and SUND L, AB Atomenergi, Sweden. 1964. (Internal report RFX-341.)

32. GREEN R E.and BIGHAM C B, Lattice Parameter Measurements in ZED-2. Exponential and Critical Experiments. Proc. of the Symp. Vol. 2. IAEA, Vienna. 1964. p. 457.

33. CHIDLEY B G, TURNER R P, and BIGHAM C B, Experimental Effective Fission Cross Sections and Neutron Spectra in a Uranium Fuel Rod. Part 3. CANDU-Type Fuel with H?O-D2O Mixtures as Coolants. 1961. (AECL-1419)

34. HELLSTRAND E, and LUNDGREN G, The Resonance Absorption of Uranium Metal and Oxide. 1962 . (AE-77)

35. COHEN E R, Internat. Conf. on the Peaceful Uses of Atomic Energy, 1. 1955, Geneva. Proc. Vol. 5. New York 1956. p. 405, 53 7.

36. JONG R A, and SERDULA K J, Neutron Spectrum Measurements in-Non-multiplying and Multi- plying Media. 1966. (AECL-2626) Geometry of the 31-rod cluster. The hatched areas indicate the foil positions. Energy dependenoe of the tetal cross Pig. 2 section for relevant detector and fuel nuelides (different scales are uaed for different nuclides). At the top of the fig« the low energy group structure of FLEF ie indicated. I The infinite lattice, epithermal spectrum Pig. 3 2 index Pq/qc in the fuel, vs p(@93)Vf f/p(T for D20 lattices with different fuels, pitches and moderator temperatures.

1 5.0-

4.0-

3.0-

Pure lattices!

• U02 (1.2 %)

U02(1.2 %), ThO2, 111

U02(Pu), U02(nat),lil O,B U02(Pu), Th02 Correlation {<$) l.o-

^06 io8 .10 .12 .16 .18 .20 .22 .24 .26 .28 Comparison between experimental data, Fig. 4a FLEP calculations and correlation ( 9), t«r the fuel epithermal index Pq/q of U0o(nat) c 2 ' uniform rod lattices In Dg0.

O r-t

o * PS oin

T oî Comparison between experimental data, FLEF Fig. 4b calculations and correlation (3)tov the fuel epithermal index Pq/qc of U0„ (1.2 %) uniform rod lattices in D~0.

tf-.

o r-H •p w a) o r-t GO •H V

(3 ft) Pc] C) OJ tÉ. *—' O oc1 -H ö

-o

CX3 "O

-oin

-8

o o o CM • r-4 Comparison b*ttw«n ABCL «xpcriaantal data, Pig. 4c FLEP calculation and correlation (9-), for the average fu*l «plthanaal indbx Pq/q of 19-rod U#2(nat) cluster lattice« in DgÔ.

-8

? Jt I •.-o- in

O r-i

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LIST OF PUBLISHED AE-REPORTS 243. On shear stress distributions for flow in smooth or partially rough annuli. By B. Kjellström and S. Hedberg. 1966. 66 p. Sw. cr. 8:-. 1-200. (See the back cover earlier reports.) 244. Physics experiments at the Agesta power station. By G. Apelqvist, P.-A. Bliselius, P. E. Blomberg, E. Jonsson and F. Akerhielm. 1966. 30 p. Sw. 201. Heat transfer analogies. By A. Bhattacharyya. 1965. 55 p. Sw. cr. 8:-. cr. 8:-. 202. A study of the "384" KeV complex gamma emission from plutonium-239. 245. Intercrystalline stress corrosion cracking of inconel 600 inspection tubes in By R. S. Forsyth and N. Ronqvist. 1965. 14 p. Sw. cr. 8:-. the Agesta reactor. By B. Grönwall, L. Ljungberg, W. HUbner and W. 203. A scintillomeler assembly for geological survey. By E. Dissing and O. Stuart. 1966. 26 p. Sw. cr. 8:-. Landström. 1965. 16 p. Sw. cr. 8:-. 246. Operating experience at the Agesta nuclear power station. By S. Sand- 204. Neutron-activation analysis of natural water applied to hydrogeology. By ström. 1966. 113 p. Sw. cr. 8:-. O. Landström and C. G. Wenner. 1965. 28 p. Sw. cr. 8:-. 247. Neutron-activation analysis of biological material with high radiation levels. 205. Systematics of absolute gamma ray transition probabilities in deformed By K. Samsahl. 1966. 15 p. Sw. cr. 8:-. odd-A nuclei. By S. G. Malmskog. 1965. 60 p. Sw. cr. 8:-. 248. One-group perturbation theory applied to measurements with void. By R. 206. Radiation induced removal of stacking faults in quenched aluminium. By Persson. 1966. 19 p. Sw. cr. 8:-. U. Bergenlid. 1965. 11 p. Sw. cr. 8:-. 249. Optimal linear filters. 2. Pulse time measurements in the presence o< 207. Experimental studies on assemblies 1 and 2 of the fast reactor FRO. Part 2. noise. By K. Nygaard. 1966. 9 p. Sw. cr. 8:-. By E. Hellstrand, T. Andersson, B. Brunfelter, J. Kockum, S-O. Londen 250. The interaction between control rods as estimated by second-order one- and L. I. Tirén. 1965. SO p. Sw. cr. 8:-. group perturbation theory. By R. Persson. 1966. 42 p. Sw. cr. 8:—. 208. Measurement of the neutron slowing-down time distribution at 1.46 eV 251. Absolute transition probabilities from the 453.1 keV level in 183W. By S. G. and its space dependence in water. By E. Möller. 1965. 29 p. Sw. cr. 8:-. Malmskog. 1966. 12 p. Sw. cr. 8:-. 209. Incompressible steady flow with tensor conductivity leaving a transverse 252. Nomogram for determining shield thickness for point and line sources of magnetic field. By E. A. Witalis. 1965. 17 p. Sw. cr. 8:-. gamma rays. By C. Jönemalm and K. Malén. 1966. 33 p. Sw. cr. 8:-. 210. Methods for the determination of currents and fields In steady two- 253. Report on the personnel dosimetry at AB Atomenergi during 1965. By K. A. dimensional MHD flow with tensor conductivity. By E. A. Witalis. 1965. Edwardsson. 1966. 13 p. Sw. cr. 8:-. 13 p. Sw. cr. 8:-. 254 Buckling measurements up to 250°C on lattices of Agesta clusters and on 211. Report on the personnel dosimetry at AB Atomenergi during 1964. By DiO alone in the pressurized exponential assembly TZ. By R. Persson, K. A. Edvardsson. 1966. 15 p. Sw. cr. 8:-. A. J. W. Andersson and C.-E. Wikdahl. 1966. 56 p. Sw. cr. 8:-. 212. Central reactivity measurements on assemblies 1 and 3 of the fast reactor 255. Decontamination experiments on intact pig skin contaminated with beta- FRO. By S-O. Londen. 1966. 58 p. Sw. cr. 8:-. gamma-emitting nuclides. By K. A. Edwardsson, S. Hagsgârd and A. Swens- 213. Low temperature irradiation applied to neutron activation analysis of son. 1966. 35 p. Sw. cr. 8:-. mercury in human whole blood. By D. Brune. 1966. 7 p. Sw. cr. 8:—. 256. Perturbation method of analysis applied to substitution measurements of 214. Characteristics of linear MHD generators with one or a tew loads. By buckling. By R. Persson. 1966. 57 p. Sw. cr. 8:-. E. A. Witalis. 1966. 16 p. Sw. cr. 8:-. 257. The Dancoff correction in square and hexagonal lattices. By I. Carlvik. 1966. 215. An automated anion-exchange method for the selective Sorption of five 35 p. Sw. cr. 8:-. groups of trace elements in neutron-irradiated biological material. By 258. Hall effect influence on a highly conducting fluid. By E. A. Witalis. 1966. K. Samsahl. 1966. 14 p. Sw. cr. 8:-. 13 p. Sw. cr. 8:-. 216. Measurement of the time dependence of neutron slowing-down and therma- 259. Analysis of the quasi-elastic scattering of neutrons in hydrogenous liquids. lization in heavy water. By E. Möller. 1966. 34 p. Sw. cr. 8:-. By S. N. Purohit. 1966. 26 p. Sw. cr. 8:-. 217. Electrodeposition of actinide and lanthanide elements. By N-E. Bärring. 260. High temperature tensile properties of unirradiated and neutron irradiated 1966. 21 p. Sw. cr. 8:-. 20Cr-35Ni austenitic steel By R B Roy and B Solly. 1966. 25 p. Sw. 218. Measurement of the electrical conductivity of He1 plasma induced by cr. 8:-. neutron irradiation. By J. Braun and K. Nygaard. 1966. 37 p. Sw. cr. 8:-. 261. On the attenuation of neutrons and photons in a duct filled with a helical 219. Phytoplankton from Lake Magelungen, Central Sweden 1960-1963. By T. plug. By E. Aalto and A. Krell. 1966. 24 p. Sw. cr. 8:-. Willen. 1966. 44 p. Sw. cr. 8:-. 262. Design and analysis of the power control system of the fast zero energy 220. Measured and predicted neutron flux distributions in a material surround- reactor FR-O. By N. J. H. Schuch. 1966. 70 p. Sw. cr. 8:-. ing av cylindrical duct. By J. Nilsson and R. Sandlin. 1966. 37 p. Sw. 263. Possible deformed states in ":ln and "'In. By A. Bäcklin, B. Fogelberg and cr. 8:-. S. G. Malmskog. 1967. 39 p. Sw. cr. 10:-. 221. Swedish work on brittle-fracture problems in nuclear reactor pressure 264. Decay of the 16.3 min. '"Ta isomer. By M. Höjeberg and S. G. Malmskog. vessels. By M. Grounes. 1966. 34 p. Sw. cr. 8:-. 1967. 13 p. Sw. cr. 10:-. 222. Total cross-sections of U, UOi and ThOj for thermal and subthermal 265. Decay properties of "'Nd. By A. Bäcklin and S. G. Malmskog. 1967. 15 p. neutrons. By S. F. Beshai. 1966. 14 p. Sw. cr. 8:-. Sw. cr. 10: -. 223. Neutron scattering in hydrogenous moderators, studied by the time de- 266. The half life of the 53 keV level in '"Pt. By S. G. Malmskog. 1967. 10 p. pendent reaction rate method. By L. G. Larsson, E, Möller and S. N. Sw. cr. 10:-. Purohit. 1966. 26 p. Sw. cr. 8:-. 267. Burn-up determination by hight resolution gamma spectrometry: Axial and 224. Calcium and strontium in Swedish waters and fish, and accumulation of diametral scanning experiments. By R. S. Forsyth, W. H. Blackadder and strontium-90. By P-O. Agnedal. 1966. 34 p. Sw. cr. 8:-, N. Ronqvist. 1967. 18 p., Sw. cr. 10:-. 225. The radioactive waste management at Studsvik. By R. Hedlund and A. 268. On the properties of the s,/ >- d / transition in "'Au. By A. Bäcklin Lindskog. 1966. 14 p. Sw. cr. 8:-. 2 3| 2 and S. G. Malmskog. 1967. 23 p. Sw. cr. 10:-. 226. Theoretical time dependent thermal neutron spectra and reaction rates in HiO and D2O. S. N. Purohit. 1966. 62 p. Sw. cr. 8:-. 269. Experimental equipment for physics studies in the Agesta reactor. By G. Bernander, P. E. Blomberg and P.-O. Dubois. 1967. 35 p. Sw. cr. 10:-. 227. Integral transport theory in one-dimensional geometries. By I. Carlvik. 1966. 65 p. Sw. cr. 8:-. 270. An optical model study of neutrons elastically scattered by iron, nickel, cobalt, copper, and indium in the energy region 1.5 to 7.0 MeV. By B. 228. Integral parameters of the generalized frequency spectra of moderators. Holmqvist and T. Wiedling. 1967. 20 p. Sw. cr. 10:—. By S. N. Purohit. 1966. 27 p. Sw. cr. 8:-. 271. Improvement of reactor fuel element heat transfer by surface roughness. 229. Reaction rate distributions and ratios in FRO assemblies 1, 2 and 3. By By B. Kjellström and A. E. Larsson. 1967. 94 p. Sw. cr. 10:-. T. L. Andersson. 1966. 50 p. Sw. cr. 8:-. 272. Burn-up determination by high resolution gamma spectormetry Fission pro- 230. Different activation techniques for the study of epithermal spectra, app- duct migration studies. By R. S. Forsyth, W. H. Blackadder and N. Ron- lied to heavy water lattices of varying fuel-to-moderator ratio. By E. K. qvist. 1967. 19 p. Sw. cr. 10:-. Sokolowski. 1966. 34 p. Sw. cr. 8:-. 273. Monoenergetic critical parameters and decay constants for small spheres 231. Calibration of the failed-fuel-element detection systems in the Agesta and thin slabs. By I. Carlvik. 24 p. Sw. cr. 10:-. reactor. By O. Strindehag. 1966. 52 p. Sw. cr. 8:-. 274. Scattering of neutrons by an anharmonic crystal. By T. Högberg, L. Bohlin 232. Progress report 1965. Nuclear chemistry. Ed. by G. Carleson. 1966. 26 p. and I. Efabsjö. 1967. 38 p. Sw. cr. 10:-. Sw. cr. 8:-. 275. ThelAKI = 1, E1 transitions in odd-A isotopes of Tb and Eu. By S. G. Malm- 233. A summary report on assembly 3 of FRO. By T. L. Andersson, B. Brun- skog, A. Marelius and S. Wahlborn. 1967. 24 p. Sw. cr. 10:-. felter, P. F. Cecchi, E. Hellstrand, J. Kockum, S-O. Londen and L. I. Tirén. 1966. 34 p. Sw. cr. 8:—. 276. A burnout correlation for flow of boiling water in vertical rod bundles. By Kurt M. Becker. 1967. 102 p. Sw. cr. 10:-. 234. Recipient capacity of Tvären, a Baltic Bay. By P.-O. Agnedal and S. O. W. Bergström. 1966. 21 p. Sw. cr. 8:-. 277. Epithermal and thermal spectrum indices in heavy water lattices. By E. K. Sokolowski and A. Jonsson. 1967. 44 p. Sw. cr. 10:-. 235. Optimal linear filters for pulse height measurements in the presence of noise. By K. Nygaard. 1966. 16 p. Sw. cr. 8:-. 236. DETEC, a subprogram for simulation of the fast-neutron detection pro- cess in a hydro-carbonous plastic scintillator. By B. Gustafsson and O. Förteckning over publicerade AES-rapporter Aspelund. 1966. 26 p. Sw. cr. 8:-. 237. Microanalys of fluorine contamination and its depth distribution In zircaloy 1. Analys medelst gamma-spektrometri. Av D. Brune. 1961. 10 s. Kr 6:—. by the use of a charged particle nuclear reaction. By E. Möller and N. 2. Bestralningsförändringar och neutronatmosfär i reaktortrycktankar — nägra Starfelt. 1966. 15 p. Sw. cr. 8:-. synpunkter. Av M. Grounes. 1962. 33 s. Kr 6:—. 238. Void measurements in the regions of sub-cooled and low-quality boiling. 3. Studium av sträckgränsen i miukt stâl. Av G. Ostberg och R. Attermo. P. 1. By S. Z. Rouhani. 1966. 47 p. Sw. cr. 8:-. 1963. 17 s. Kr 6:-. 239. Void measurements in the regions of sub-cooled and low-quality boiling. 4. Teknisk upphandling inom reaktoromrâdet. Av Erik Jonson. 1963. 64 s. P. 2. By S. Z. Rouhani. 1966. 60 p. Sw. cr. 8:-. Kr 8:-. 240. Possible odd parity in ">Xe. By L. Broman and S. G. Malmskog. 1966. 5. Agesta Kraftvärmeverk. Sammanställning av tekniska data, beskrivningar 10 p. Sw. cr. 8:-. m. m. för reaktordelen. Av B. Lilliehöök. 1964. 336 s. Kr 15:-. 241. Burn-up determination by high resolution gamma spectrometry: spectra 6. Atomdagen 1965. Sammanställning av föredrag och diskussioner. Av S. from slightly-irradiated uranium and plutonium between 400-830 keV. By Sandström. 1966. 321 s. Kr 15:-. R. S. Forsyth and N. Ronqvist. 1966. 22 p. Sw. cr. 8:-. Additional copies available at the library of AB Atomenergi, Studsvik, Ny- 242. Half life measurements in "!Gd. By S. G. Malmskog. 1966. 10 p. Sw. köping, Sweden. Micronegatives of the reports are obtainable through Film- cr. 8:—. produkter, Gamla landsvägen 4, Ektorp, Sweden.

EOS-tryckerierna, Stockholm 1967