AE-99

W! The Effective Resonance Integral

of Thorium Oxide Rods

J. Weitman

AKTIEBOLAGET ATOMENERGISTOCKHOLM SWEDEN 1962

AE-99

THE EFFECTIVE RESONANCE INTEGRAL OF THORIUM

OXIDE RODS.

J. We it man

Summary:

The effective resonance integral of thorium oxide rods has been deter-

mined as a function of their surface to mass ratio. The range of S/M values

covered is 0. 15 - 0.65 cm /g. An experimental technique based on the

comparison of activities obtained in thermal and slowing-down neutron fluxes

was employed. The shape of the resonance neutron spectrum was determined

from measurements with a fast chopper and from calculations, permitting

deduction of a correction factor which relates the experimental values to the

ideal 1/E case. The results are summarized by the following expression:

1 5 ' 6

The main contribution to the: margin of error arises from the uncertainties

in the 1. 5 % spectral correction applied in the K 5 b "l/v" part deducted and

.in the 1520 b infinite dilution integral of gold, used as a standard.

i »

In order to compare the consistency of Dresner's first equivalence

theorem and Nordheim's numerical calculations relative to our results, the

resonance integral values for thorium metal rods obtained previously by

Hellstrand and Weitman have been recalculated, using recent cross section

and spectrum data. The new formula is

RlTh = ( 3 . i t l 6 . 1 l/ S/MTh) t 5. 0 %.

It differs from the old one mainly because of the proved non-l/v behaviour

of the thorium cross section below the first resonance.

Printed and distributed in December 1962.

CONTENTS

Page

Introduction 3

General discussion 3

Method of measurement 7

Experimental procedure 7

Special effects and corrections 10

The thorium cross section below

the first resonance 16

Results 17

Discussion 22

References 28

Tables 30

Figures 39

Addendum 41

The effective resonance integral

of thorium oxide rods.

J. Weitman

1» Introduction.

As a continuation of the work on resonance integrals, per-

formed at the Swedish Rl reactor (1, 2, 3), the effective resonance

integral of sintered ThO • rods has been measured. The method of

measurement and the experimental procedure are described in

sections 3 and 4. Important effects, including the sensitivity of

the resonance integral to the form of the neutron spectrum, are

discussed in section 5. The situation regarding the energy depen-

dence of the thorium cross section below the first resonance is

reviewed in section 6» The results of the present measurements

are evaluated in section 7, and a comparison with theoretical and

experimental values, obtained elsewhere, is undertaken in section 8.

During the last few years several papers have appeared

which facilitate a critical attitude to resonance integral measure-

ments and calculations» It is therefore felt that a short discussion

of problems of current interest should precede the description of

this experiment,

2. General discussion.

_conce_p_ts.

It is usual to define the effective resonance integral as the

lethargy integral of that cross section which, when multiplied by the

flux which would exist in the absence of the resonance, gives the true

absorption rate (4). In theoretical calculations, a spatially and

lethargy-wise uniform flux is further assumed. This definition is

satisfactory in so far as it clearly states under what conditions a

resonance integral measurement should be made in order to give

results comparable with theoretical values. Before the sample is

placed in position for irradiation, there should exist an 1/E slowing

down neutron distribution, and the perturbation caused by the sample

should, except at the resonances, be negligible. Unfortunately, in

the range of interest of reactor physics, it is very difficult to fulfil

these requirements. Even if the perturbation caused by the sample

and the experimental equipment is small, as it should be in these

experiments (section 5), the flux still departs from the dE/E form,

which has beendemonstrated theoretically by Brooks and Soodak (5),

Bigham and Pearce (6) and experimentally by Johansson and Jonsson

(7). It is possible to correct for this departure only to some extent* -

We now assume that it is feasible to perform a resonance integral

determination which is roughly in agreement with the requirements

of the definition. The question then arises as to the utility of the

quantity measured. The use of fast computers has made it possible

to show that in a real reactor environment, the fuel surface spectrum

deviates drastically from the l/E law over wide energy ranges (6).

A resonance integral obtained under idealized conditions may there-

fore give only approximate p values. Indeed, considering the type

of experiments done by Hellstrand, Critoph and Hone (8) have demon-

strated that extra factors must enter into the calculations of p , as

long as one wants RI to depend on the properties of the absorber

only.

The situation is consequently that a resonance integral

measurement, based on the theoretical definition, does not give

results which are immediately useful for reactor calculations, be-

cause one first has to take into account the variation of the flux in

space and lethargy. Nevertheless, the need of a check on theore-

tical methods and differential data justifies integral experiments of

the kind described here.

The geometric dependence of the effective resonance inte-

gral is generally summarized in formulas, with the surface to mass

ratio as single parameter. The theoretical justification for these

formulas is based on several simplifications. Their validity is

therefore restricted. As usual, a spatially and lethargy-wise uniform

- 5 -

flux is assumed. Further, the Doppler broadening of the resonances

is neglected. Making use of Wigner' s rational approximation for the

average escape probability for neutrons from the lump, the geometric

dependence deduced in the narrow resonance (NR) and infinite mass

(IM) approximations will be (4)

NR: R I ~ 2 b/V b - i.res _1

IM:

provided that b & /<y is much less than one.p o

Now the potential scattering cross section a* of the ab-pa

sorbing nucleus is at most equal to the total microscopic potentialcross section, a1 > of the lump, and the neutron width F isp nalways less than the total width T . The statistical spin factor g,

is for thorium equal to one. The quantity i is therefore less than one,

while —k— is at most equal to one. Writing i for the mean value of

i , we obtain

1 'Sn \\ / \ r~~~• ^ N R "^(1 + T "TT/V k — ^ y k "^or * a r S e k values

RIIM ~ f'- v ^ p ->

Especially for ThO^ lumps, one finds the following expressions

(1 = 4 V/S, M = p ' V, c = 12 b, a* = 19. 6 b) :pa p

RI N R = konst.x [ (0.179+ S/M)+ 1 / ' 2 + -™- (0. 179 + S/M)

_ _ _ _ _RI = konst. x\/0. 069 + S/M

For not too small S/M values one would therefore expect the

formula proposed by Vernon (9), RI = a y b + S/M , to fit experimen-

tal and calculated points well. It should be noted that none of the above

expressions is of the form proposed by Gurevich and Pomeranchouk

(10), RI = A + B yS/M . However, they make essentially the same

assumptions as those mentioned above, and the difference, which is

- 4 -

satisfactory in so far as it clearly states under what conditions a

resonance integral measurement should be made in order to give

results comparable with theoretical values. Before the sample is

placed in position for irradiation, there should exist an l/E slowing

down neutron distribution, and the perturbation caused by the sample

should, except at the resonances, be negligible. Unfortunately, in

the range of interest of reactor physics, it is very difficult to fulfil

these requirements. Even if the perturbation caused by the sample

and the experimental equipment is small, as it should be in these

experiments (section 5), the flux still departs from the dE/E form,

which has been demonstrated theoretically by Brooks and Soodak (5),

Bigharn and Pearce (6) and experimentally by Johansson and Jonsson

(7). It is possible to correct for this departure only to some extent. -

We now assume that it is feasible to perform a resonance integral

determination which is roughly in agreement with the requirements

of the definition. The question then arises as to the utility of the

quantity measured. The use of fast computers has made it possible

to show that in a real reactor environment, the fuel surface spectrum

deviates drastically from the l/E law over wide energy ranges (6).

A resonance integral obtained under idealized conditions may there-

fore give only approximate p values. Indeed, considering the type

of experiments done by Hellstrand, Critoph and Hone (8) have demon-

strated that extra factors must enter into the calculations of p , as

long as one wants RI to depend on the properties of the absorber

only.

The situation is consequently that a resonance integral

measurement, based on the theoretical definition, does not give

results which are immediately useful for reactor calculations, be-

cause one first has to take into account the variation of the flux in

space and lethargy. Nevertheless, the need of a check on theore-

tical methods and differential data justifies integral experiments of

the kind described here.

The geometric dependence of the effective resonance inte-

gral is generally summarized in formulas, with the surface to mass

ratio s.s single parameter. The theoretical justification for these

formulas is based on several simplifications. Their validity is

therefore restricted. As usual, a spatially and lethargy-wise uniform

- 5

flux is assumed. Further, the Doppler broadening of the resonances

is neglected, Making use of Wigner* s rational approximation for the

average escape probability for neutrons from the lump, the geometric

dependence deduced in the narrow resonance (MR) and infinite mass

(1M) approximations will be (4)

NR: RI ~ 2 b/l/ b - i.res V k

IM: RI ~\l b- <r J¥ pa'1pa P

provided that b 1$ is much less than one.

Now the potential scattering cross section cr of the ab-

sorbing nucleus is at most eqiial to the total microscopic potential

cross section, <r > of the lump, and the neutron width F isp n

always less than the total width F . The statistical spin factor g_

is for thorium equal to one. The quantity i is therefore less than one,

a'while —— is at most equal to one. Writing i for the mean value of

crp

i , we obtain

RI :NJR ~ ( 1 + - -rP-)y b — > \ / b for large b values

RI IM

Especially for ThO., lumps, one finds the folio-wing expressions

(T = 4 V/S, M = p ' V, o- = 12 b, o- = 19. 6 b) :Pa P

R I N R - konst, x [ (0. 179 + S/M)+ 1/f 2 + — (o. 179 + S/M)

(3)

RI = konst, x Wo. 069 + S/M

For not too small S/M values one would therefore expect the

formula proposed by Vernon (9), RI = a lib + S/M , to fit experimen-

tal and calculated points well. It should be noted that none of the above

expressions is of the form proposed, by Gurevich and Pomeranchouk

(10), R.[ = A + B \jS/M . However, they make essentially the same

assumptions as those mentioned above, and the difference, which is

not very serious, is due to another approximation in their work. The

resonance integral is divided into two parts, one corresponding to

the case l/A » 1, which is true for all low lying levels, and the

other part is obtained for l/A « 1, which is true for very highcl

energy levels. The first part is the "surface" absorption, the second

part is the "volume" absorption, which in the limit S/M > 0 should

be best approximated by the NR expression given above. - Experience

gives support to both the Vernon-type and the standard Russian formu-

las (2).

The rational approximation is sometimes as much as 20% in

error. Therefore, equivalence theorems relating resonance integrals

of absorbers with different admixture of moderating material are not

exact, though they seem to be better than the approximations leaain^

to them. More precise, but less easily handled relations have been

discussed by Wolfe (11) and Jirlow (12). We will confine our interest

to the simple expressions, obtained directly from the NR and IM

formulas for the effective resonance integral:

RI ) ( b • <r ) +xj % T V

NRo

(4)

IM J

oo

" du(b er ) +cr ~<r

The quantity (b ° ) is solely responsible for the geometric depen-

dence of these integrals, as well as for the sensitivity to admixture

of moderating materials. Within the limits of accuracy discussed

above, lumps with the same value of (b'c ) should thus possess

equal resonance integrals. Relating thorium metal and oxide lumps,

we find that the integrals for metal are transformed into the corre-

sponding integrals for oxide, if

(S/M) , = 0.079 + 1.138 (S/M)L>

This relation is used in section 8 to compare our thorium metal ana

oxide data.

- 7 -

3. Method of measurement.

In principle, the experiment consisted of measuring the

ratiores . . th

AThO9 Au

th

of four activities, which, provided the resonance and thermal fluxes

used are properly monitored, can be made to depend only on the

cross sections involved. These are the thermal cross sections of gold

and thorium oxide, and the effective resonance integrals of the same

materials. After application of necessary corrections, the resonance

integral of ThO? was calculated from these q values, assuming the

three other cross sections as known. - The resonance flux was ob~

tained inside a cadmium cylinder in the central channel of the D.O

moderated Rl reactor, while the "thermal irradiations" were per-

formed in the graphite column of the same reactor. - The main

features of the experimental procedure have been previously described

(2). In the following sections only deviations from the thorium metal

experiments will be discussed in more detail.

4. Experimental procedure.

a)

The experimental assembly, consisting of a 150 mm wide

cylindrical container with an axial tube, through which the samples

were inserted, occupied the central lattice position in the hexagonal

fuel array of the Rl reactor. Its 29 mm natural uranium fuel rods are

placed at a center-to-center distance of 145 mm, and the D7O layer

between neighbouring rods is 115 mm. Conditions are thus similar to

those in the Canadian NRX reactor, which is of some importance for

later discussions (section 5). - The thermal irradiations were per-

formed with the aid of an aluminium drawer, in which the samples were

stacked between graphite pieces. When the drawer was introduced into

the graphite column of the reactor, the sample positions were 250 mm

from the reflector. The cadmium ratio for thin gold foils in that posi-

tion is about 1500.

Four identical series of experiments were performed in

these facilities» Each series consisted of ten independent irradiations,

done within a period of about 24 hours.

For the central channel runs of each series, seven cadmium

clad holders were prepared according to the specifications in Table la.

The thoria rods were built up from cylinders with flat ground ends

and kept in place with the aid of an aluminium tube and Al spacers, as

shown in Fig. 1. The central part of the rods consisted of two 4.5 mm

thick measuring samples, their mean distance to the monitor foils

being 120 mm. The same distance3 within - 1 mm, was secured be-

tween the lead-gold foils and the monitor foils. Neglecting very small

end effects, the ThO2 measuring samples correspond to infinitely

long rods of the same material, with S/M values in the range

0.14 - 0. 65 cm /g. - The holders were individually let down into the

container in the center of the reactor. As the container was normally

air filled, a D O layer of about 50 mm separated the thorium and

nearest fuel rods. Irradiations were made at a power level of 60 kW

(corresponding to an integrated epicadmium flux of about

40 neutrons/cm , sec), for times varying between 20 and 90 minutes.

IH this way suitable activities for the (3 and y measurements were

obtained.

The conditions for the thermal column irradiations are

evident from Table Ib and Fig. 2. Activities comparable to those induced

in the central channel runs could be obtained at a power level of 600 kW

and for irradiation times between 1 and 3 hours.

b) .P ep_aring_ solutions_and

As in the thorium metal experiments, it was found con-233

venient to measure the 27. 4 d activity of the Pa isotope, formed233

in the 22. 2 m (3 -decay of Th . Two y peaks, with energies at

105 keV and about 330 keV, were studied in the scintillation spectro-

meter. In order to get another value, independent of possible drifts

in the electronic circuits of the spectrometer, a single channel unit

was also used, with a suitable "window" at the 330 keV peak.

The values reported here refer to measurements on dissolved

samples. However, before a suitable dissolving technique was

developed, a set of measurements on 4. 5 mm thick solid samples was

performed, with only small deviations in other respects. Due to the

appreciable thickness of the solid samples, large corrections had to

- 9 -

be applied for geometric effects, giving the formula obtained a low

weight. In order to avoid such difficulties, new measurements were

undertaken on solutions.

The ThO? pellets of each series, with weights ranging be-

tween 1 to 27 gram s, were put into good quality glass retorts, together

with 100 ml concentrated HNO, and 50 mg Na_SiF, <> After about one

week at a temperature of 70 C, which is low enough to prevent too fast

vaporization» even the biggest pellets (diameter 28 mm, thickness

4.5 mm) were dissolved» However, in all solutions there was a few

mg of a fine grained sediment, which proved to contain up to 20% of233

the total induced Pa activity. A possible way to avoid this difficulty

might have been extraction of protactinium according to one of the

methods described in the literature, for instance use of MnO? as

carrier or use of extracting organic solutions, such as trialkyl

phosphate or TBP. The presence of sediments and fluoride ions in the

present case makes such methods less feasible. A more direct

approach was therefore chosen, which also had the advantage of being

less mysterious from a physicist's point of view: the 100 ml amounts

of raw solution were filtered through suitable filter papers. The Pa-

activity of each pellet was thus distributed between a clear, filtered

solution and a filter paper. The solutions were diluted with HNO, to

give a constant concentration of about 50 mg ThO?/ml HNO, - the total

volume of the solutions now ranging between about 20 ml and 500 ml.

With the aid of a pipette, the volume of which was V (V almosts s

exactly 5 ml), two small identical beakers were filled, one with the

active solution, the other with pure HNO, in which the corresponding

filter paper was dispersed. The activity of the beakers, Nc and 1\L

respectively, was then measured in the gamma spectrometer and

single channel unit. If the total volume of the solution is V , the

induced Pa-activity of a pellet is given by

(Pa-233) = Ns • 2 1 + NF (7)

The only more serious difficulty in this procedure was the ability of

protactinium to adhere to glass. In order to avoid activity losses during

the preparation of solutions it was necessary to wash and check the

activity of all equipment that during the process had been in contact

- 10 -

with the active solution. The rinsing solution, consisting of HNO, +

small amounts of Na^SiF,, was added to the main solution before theC. O

determination of V and the gamma measurements.

In this way, four series of pellets, with eleven pellets in

each, were treated. Due to a couple of mishaps while making solutions,

seven of the solutions had to be disregarded, giving 37 accepted solu-

tions with corresponding filtrates. Three to five counts were made on

each of these samples. In a typical case, 20 000 pulses were registered

under the 105 keV peak, 8 000 under the 330 keV peak and about 8 000

in the single channel unit. Totally, 3 x (20000 + 8000 + 8000), or about

100 000 pulses were registered for every solution and filter paper

dispersion. The order of measurement was chosen in such a way, that

the effect of systematic drifts in the electronic equipment should be

reduced. Also, the height of the crystal above the samples was large

enough to make small differences in the activity distribution of a solu-

tion and a dispersion of little importance.

Before the lead-gold samples (containing 0.1 wt % Au) were

used, an inter calibration was undertaken in order to assure equal

gold proportions in all of them. After irradiation foil activities were

measured, using a sample changer with a printing unit and a (3 -scin-

tillation counter. As the foils had diameters ranging from 8 to 20 mm,

counting was performed with the crystal both a few millimeters away

from the foil surface and about 250 mm from it. Small geometric

effects were observed.

Finally, the monitor foils, punched from 0.2 mm thick gold

sheets, were also measured in the sample changer, when their activity

had decayed to a suitable level.

5. Special effects and corrections.

a) J?eadtime,_ J %cJ r_o_urid_ and_deca.Y cp_rrec_tions.

Because of the unfavourably large deadtime (654 - 30 fAsec) of

the gamma spectrometer available for these experiments, it was

necessary to compromise between the need to obtain good statistics

in short measuring periods, small background percentage and in

addition small dead-time corrections. The latter could generally be

kept less than 5 % for the solutions. The absolute amount of protac-

- 11 -

tinium in the dispersions varied considerably, however, giving, in

the worst cases, dead-time corrections of about 20% in N_ . For-

tunately, the ratio of N~ to N, . was not more than about 0. 1 for• •. • ' F tot

all large samples, which considerably reduces the uncertainty intro-

duced in N . Moreover, the single channel unit had a very short

dead-time, thus giving values independent of dead-time corrections.

The same is true of the |3 -measurements.

The background was generally of the order of 10% or less .

Only small relative decay corrections had to be applied, as the time

spread of irradiations and measurements was moderate. The Mercury

programs used also included corrections for activity saturation during

irradiation. When determining the irradiation time, the starting up

period of the reactor could be neglected, as during this short time

very little gold or protactinium can decay. Because of the relatively

long half -lives of Au-198 and Pa-233, small timing errors are also of

minor importance.

b) Fi_s_s_ipri_p_roducts_ ajnd_uppurity_

Spectrographic and chemical analyses were performed on the

thoria samples, but they showed no serious content of impurity ele-

ments with high thermal or epithermal cross sections. During i r ra -

diation in the central channel, however, fission products are formed.

A separate investigation was made in order to detect interference from

their activities. Samples of different sizes were irradiated both in the

thermal column and in the central channel. Measurements on these

samples were made on different occasions from 2. 5 to 32 days after

the irradiation. The time dependence of the ratio

R(t) = —kk = Konst.x [1 + X(t) ] , (8)

TK

where X(t) = N1 '™/N „„ is the quotient of fission product to central

channel induced Pa-activity, was studied. In Fig. 3, R(t) for a medium

size sample is plotted against time. The function is almost constant

from 4 to 32 days. During this time, the Fp activity has decreased at-1

least a factor 8 (assuming a t law), while the decrease in Pa activity

is only a factor 2. X(t) has therefore decreased at least a factor four.

As R(t) is nearly constant, X{t) must be approximately zero. Thus,

about 30 days after irradiation, the fission product activity is negli-

gibly small.

12 -

c) Variations_in density and gol_d percentage._

The density of the thoria pellets was determined from their

weight and accurate dimensions rather than by displacement methods,

which always give somewhat higher values, due to the porosity of the

sintered material. Because of the limited number of samples available,

even those with very low or very high densities had to be accepted, as

is evident from Table II. However, the samples were chosen to give

a mean density, nearly equal to the density of the surrounding pieces

of the rod, thus neutralizing individual deviations.

Though even a rather large radial variation in density would

be of little importance^, microscopic examination of several samples

was undertaken, and the pore distribution was studied. As usual for

sintered materials local differences in density existed. On the micro-

photographs, no systematic effects could be observed, however.

For the calibration process it is essential that the lead gold

foils compared have equal gold contents. Therefore, the foils were

irradiated in the same flux the activities measured and compared.

From these measurements correction factors of a few percent were

deduced. After a couple of weeks, the inter calibrated foils could be

used for the experiments proper, now with a known relative jjold

proportion.

effec_ts •

Possible sources of systematic, geometry dependent errors

have been considered.Such errors might arise both in irradiation and

activity measurement.

Small variations occured in the height of the monitor foils

above the samples (of the order - 1 mm), due to mechanical short-

comings of the holders. At the position of irradiation in the central

channel, however, the neutron flux can be considered as nearly gra-

dient-free and isotropic, according to earlier experience (2). Such

height variations could therefore be accepted without corrections.

To investigate whether the monitor foil activity is influenced

by neutron scattering and absorption in the thick thoria rods, cadmium

wrapped reference foils were placed more than 40 cm above the hol-

ders. The ratio of monitor foil to reference foil activity was deter-

mined, both with and without the thickest thoria rod inside the holder.

- 13 -

Three experiments were performed, showing that the monitor to re-

ference foil activity with ThO_ is 1.009 - 0.006 times the same ratio

without ThO_. As this figure certainly is sensitive to local flux per-

turbations, we may conclude that the monitoring procedure is not

seriously influenced by the presence of the thoria rods.

In the thermal column, two lead-gold foils were always

irradiated, one on each side of the 1. 7 mm thick ThO? discs. The

ratio of their activities was calculated, this being a measure of the

flux gradient. The mean value of this ratio for twelve foil pairs was

1.0095. The error introduced by neglecting this effect is therefore

less than 0.5%. - As the thorium disc is 1.7 mm thick, some self-

screening will occur. In order to improve the statistics and to find

out whether a correction for self-screening should be applied, three

samples with different diameters were irradiated in each series, which

is evident from Table Ib, As no systematic variation in the gold to

thorium activity was observed, it is concluded that the self-screening

effect is less than the experimental errors.

The self-shielding of the 4. 9 eV resonance in the 0.1 wt %

lead-gold foils and possible overlapping of cadmium and thorium re-

sonances, has been discussed in ref. (Z). In the former case a 1. 5 %

correction is applied,, In the latter case, no correction is necessary»

This conclusion is based on a perusal of a list of parameters for the

resolved resonances in thorium and cadmium.

For the calculation of the total Pa activity, according to

formula (7), one has to compare values obtained from measurements

on solutions and dispersions. In the present geometry, the solutions

contained about 20 mg Th/cm . Self-screening effects might occur.

As the 105 keV line falls below the K-edge of the photo-electric

absorption in both thorium and uranium, the cross section values for

y-absorption in uranium (13) could be extrapolated, assuming a Z -law.

With a1 - 340 b, the mean screening effect of the solution is 1 %. The

dispersion, on the other hand, does not contain thorium, and its

gamma activity is therefore not liable to screening. However, the

dispersion represents at most 20 % of the total activity, the 330 keV

line is practically not attenuated and the error introduced tends to

cancel» as the resonance integral depends on the ratio of two Pa acti-

vity measurements. It was therefore not thought worthwhile to correct

for this geometric effect.

- 14 -

The same conclusion applies to a slight geometric dependence

in the sample changer measurements of the lead-gold foils, due to

their varying diameters (8-20 mm). From the values obtained with

the crystal in the high and low positionSj it was concluded that an

error of 0. 35 % in the final result is introduced by this effect.

e) _ Choice of _ca_chpiurn_ cutoff.

As in the thorium metal experiments, a cutoff energy of

0.60 - 0.05 eV was adopted. The presence of moderating material in-

side the cadmium tube makes the cutoff somewhat less sharp than for

metal. This effect is of little importance, compared with the uncer-

tainty due to the fact that the energy dependence of thorium in the

cutoff region is not accurately known (section 6).

f) _ _ Nc_ut rpn_ s p_ e c trurru

Since the thorium metal experiments were published, our

knowledge about the shape of the neutron spectrum has improved

considerably. Firstly, the fast chopper measurements of Johansson

have been repeated with improved accuracy and extended to about

10 keV (7)o Secondly, calculations by Bigham and Pearce (6) for the

NRX-reactor (rod diam. 34 mm, pitch 173 mm), which is rather

similar to the Rl reactor (rod diam» 29 mm, pitch 145 mm) can be

used to give an approximative picture of the form of the flux up to

fission energies. In Fig» 4, the neutron spectrum in the central

channel, according to Johansson and Jonsson (7) is compared with

the NRX cell boundary flux found by Bigham and Pearce (6). The

curves, which are normalized at 100 eV, should only be compared

in the energy region above a few eV, where the effect of the therma-

lization process is negligible. At high energies, the NRX curve

should be lower than Johansson's curve, as the D O layer around

the NRX cell is about 70 mm, compared with 50 mm around our

central channel. The opposite effect is, however, displayed above

1 keV. As the curves agree very well between about 10 eV and 1 keV,

this discrepancy does not depend on the choice of normalization

energy. Neither can the tendency of the experimental curve be

changed by invoking experimental e r rors , as the uncertainty in the

relative height of two experimental points, e. g. at 1 keV ana 10 keV,

- 15 -

is only a few percent. The discrepancy might be explained by smaJl

errors in the theoretical curve. Nevertheless, the calculations of

Bigham and Pearce give enough, information to permit a tentative

extrapolation of measured values toward higher energies»

Though the neutron spectrum in the empty channel may be

regarded as approximately known, it still remains to prove that th ;

perturbation caused by the experimental arrangement (ThO? rod axd

Cd cover) is not very serious. There are two reasons to believe

this. In the first place, there is a rather large distance (approxo

70 mm) between the surface of the rod and the moderator. Conse-

quently, the negative sources produced by the absorption in the rod

will be moved far away, and their influence on the rod surface

spectrum will therefore be decreased. Secondly, measurements

with threshold detectors like sulphur and phosphor (3) show that the

flux depression caused by the Cd cover is very little, probably less

than 1% at high energies (where absorption in thorium, unlike that

in gold, is still appreciable). The curves in Fig. 4 should therefore

give an approximate representation of the flux impinging on the rod

surface.

We are now in a position to calculate a correction factor

which relates our measured integral to the ideal l/E case. This

correction factor will be denoted y . To obtain it, we regard theLO L

contributions to the resonance integral from the resolved energy

region, A (res), from the unresolved region below 30 keV, A (unres),

and from the region above 30 keV, A(> 30 keV). Due to the small

slope of the spectrum curve in the two first mentioned regions, it

is permissible to choose the same mean energy of absorption for all

rod dimensions used. Thus, mean energies of about 100 eV and

5000 eV, respectively, have been assumed. From the graph we now

determine two factors, y (res) = 1,05 and y (unres) = 1.05, giving

the weight of A (res) and A (unres) relative to the 4.9 eV level.

Finally, y ( > 30 keV) = 0. 75 is determined from numerical inte-

grations of the & (n, y) curve given in ref. (14). The (n, y) cross

section for thorium was thus integrated on the one hand in a l/E

spectrum, on the other in our actual spectrum, as read from the

extrapolated graph of Fig. 4. Writing RI' for the measured epi-

cadmium cross section of the thoria rods, we find the "ideal" effec-

- 16 -

tive resonance integral from

RI = [RI' - t

y\> iv KeV) ^J- ! Hot

or

. A(unres)

•y (res) RI Y(unres) RI

'] (9)

v(res) RI + Y(unres} RI keV) RI

K(res) K(unres) K(>30 keV)

The K values have been calculated for some different rod

sizes, using Nordheim 's data (15). The details of the calculation-1

are shown in Table III. The last line of the table gives y , cal-

culated from eq= 10. With regard to the uncertainties of the spectrum-1

curve and the partial v values, y may be considered constant in

the range of S/M-values covered by the experiment. The total spectral

correction, y , thus amounts to 1.0Z5 - 0.025, where the limits

of error take into account uncertainties in the assumptions made

and the spectrum curves used.

6« The thorium cross section below the first resonance.

There are no direct measurements of the thorium capture

cross section in the region 0,6 - 20 eV, which gives an appreciable

contribution to the activity measured in these experiments. The

usual l/v dependence assumed previously has proved not to be valid

for thorium (15, 16, 17, 18),, This can be inferred from the fact that

the "tails" of the positive energy resonances give a total contribution

of somewhat less than 1 barn to the thermal cross section, which is

7.36 b. In such cases, the remainder is usually ascribed to a single

negative energy resonance, although in principle several resonaces

could be invoked» Requiring the potential scattering cross section,

o" = or, - cr ~ <r , to be constant near zero energies for throium,p t c rs °

- 17 -

Seth et al. (17) find that this condition can be satisfied, provided that

there is a negative resonance at about -E = - 7 eV. The contribu-

tion of this resonance to & is about 6. 6 b (16). In ref. (18),o

however, the value of E is given as 4.3 eV, and its contribution as

6.2 b. We roughly assume thatE" = 5eV

r

0* — 6.6 b

= 0 .8boTh

and moreover that the whole contribution to o* is due to the nega-o

tive resonance and the two first positive energy resonances at about

22.6 eV (more exactly 21. 8 and

Breit-Wigner formula now gives

22.6 eV (more exactly 21. 8 and 23.5 eV). Thus E+ ** 22.6 eV. The

(E + E )v r

Just above the cadmium cutoff this cross section falls off much

faster than l/v.

The expression 11 can be integrated, thus giving the "l/v ! l

contribution

15

-(E) ^ ~ 1.5 b. (12)

0.6

In the thermal region, expression 11 does not deviate very much- 1/2

from the E ' law, however. Consequently, the g value.will be

nearly unity. Tirén (18) has used g = 0. 994 - 0. 003. Sjöstrand ana

Story (19) recommend g = 0.997. The value 0.995 is used in this

report.

7. Results.

§1 _ _Aksolute_ value_s_.

All data from the j3 and v measurements were treated in-

dividually, using a pair of programs developed for the Mercury

computer (20). Corrections for saturation, background and dead-time

- 18 -

were included and mean values per gram absorber calculated, thus

reducing the data to one figure for each sample. The protactinium

and lead-gold values were subsequently treated in somewhat different

manners. For protactinium, eq. 7 was used to calculate the total

Pa activity. As no monitoring was necessary in the thermal column,til 37 G S

this directly gave A , , while A T , . was obtained by norma-

lizing N (Pa) to unit monitor activity. - The lead-gold activities

had to be corrected for small variations in gold content, and a

straightforward calculation resulted in A . and A . . All factors

in eq. 6 were thus determined, allowing also q to be calculated for

different rod sizes. The mean values of q , containing all experi-

mental data, are given in Table V. The statistics behind these values

will be discussed below.

The determination of effective resonance integrals from the

dimensionless quantity q proceeded as follows. According to section

3, we have

where RI* , and RI . denote the effective epicadmiurn cross

sections of thoria and gold in the actual neutron spectrum, while

(go1 )rvu ant^- (s of ) A are the 2200 m/sec cross sections, corrected

for deviations from l/v in the thermal region» Further, eq. 9 gives

RIfThO2 = W ^ T h O ^ ^ T h ^

Similarly, the infinite dilution integral for gold, I . , is related to

RI . by the expression

RI . = 1 . - A I . + A ( l / v ) A (15)Au Au Au w / A u v /

where A I . is the 1.5% self-screening correction»

Combining eqs. 13-15, one obtains

- 19 -

RI =—^--fcq ~ A(l/v) \ (16)2 ' tot *» J

where

[I-AI-A(Vv)]Au-(g<ro)Th

is a calibration constant. Its explicit use will make renormalization

easier if changes in the accepted cross section values should occur.

Table IV gives the data from which C = 114.3 - 3,1 b has been cal-

culated. Compared with the thorium metal work (2), the only note-

worthy change is that I . . has been increased by 20 b. The reason

is that the improved chopper measurements indicate a slight de-

pression in the spectrum curve around the gold resonance energy.

Using eq. 16 the q values given in Table V could be trans-

formed into resonance integrals. These were not directly fitted to

any of the possible functions of S/M, in order not to disguise the

explicit dependence of such formulas on the auxiliary quantities C,

•y. and A(l/v)_, . Instead, the q values were fitted to ex-TOT J, iX

pressions of the form

q = & 1 + b±

q = }and the constants determined by the method of least squares. Inser

ting expressions 18 into eq. 16, we obtain

^7 {RIThO2 = 7 {C al ~

respectively

With . . .C a - A(l/v) C b

A = é i£ . B _

(21)

- 20 -

equations 19 and 20 can be written as

= A + B \ / S / M (22)

and

R IThOR IThO = a \ / b + S / M ~ c (23)

The constants in this expressions were calculated according to eqs, 21,

using the experimental values for a., a?) b . and b_ given in

Table V and the auxiliary quantities C, v and A (l /v)T , to be

found in Table IV. As our final result we have

R I ThO

orRI = 20.7y0.22 + S /M T h Q - 1 . 5 (25)'

X j l v J A -» -*- J.J.N^' ,^

When plotted as a function of \/S/M, eq. 24 gives a straight line, which

fits the experimental points extremely well. In Fig. 5, only this line

has been drawn within the range of measurements. However, eq. 25

is also a good representation, in spite of a slight curvature caused

by the large constant term under the square root sign. For small

S/M values, the contribution from narrow resonances is predominant,

and a y b + S/M-dependence can be expected (section 2). Eq. 25 has

therefore been used to extrapolate the measured values to S/M = 0,

giving a'Volume" term of 8. 2 b. This point is far outside the ranrje

of measurements, and the figure 8. 2 b is consequently only approxi-

mate. However, it compares favourably with Nordheim's calculated

value 7.3 b (22). - The formulas 24 and 25 will be further discussed

in section 8,

Tables Via and b give a survey of the statistics behind the

final results. In Table IV b, the percental limits indicate how much

individual determinations deviate from respective mean values. J£

systematic effects of the kind discussed in section 5 are disregarded

to begin with, those limits constitute a rough estimation of probable

errors. Their origin is certainly not to be found in poor counting

statistics, as each single A value was based on 3-5 measurements

of the activity from corresponding samples, for which totally at least

- 21 -

10 pulses were registered. For instance, the purely statisticalT" éi g

error for A . , being the ratio of a lead-gold and a monitorAU ' i _ r

activity, should be less than \l2 • 10 *" 0.45%, and this is also

true for all the other A values. Their average values, given in

Table VIb, should be still more accurate. Indeed, in the A .

column; all errors are less than 0.5%. It may be concluded thatA. mainly suffers from statistical uncertainties. However, in the

Au y

remaining columns of Table VIb, other errors apparently predomi-

nate. The main contribution most probably comes from the ability ofprotactinium to adhere to glass. Thus, some A and A .,

values came out a few percent too low. The uncertainties introduced

thereby have been regarded as accumalative when calculating the

errors in q . But as a matter of fact, q depends on the ratio ofT* & *? th

Am, _ and A^., _ . The former quantity has been determinedIhOp ihOo

5-8 times for each rod size, the latter totally 11 times. Therefore,

it is very likely that in the long run, errors due to Pa-losses cancel

each other. For this reason it is believed that the errors attributed

to q generally do not underestimate experimental uncertainties.

From the above discussion it follows that large limits of

error in q do not necessarily charcterize this value as bad. In-1

the absence of a better weighting factor, (d q) has been used,

however, and final q values calculated as the average of the results

from respective series. The details are again evident from Table VIb.

Possible sources of systematic errors have been previously

discussed (section 5). They are reviewed in Table VII. Only in a few

cases were corrections necessary. Most of the remaining effects

are small (of the order 0.5 %) and several counteract each other.

The limits of error inferred from the "statistical" spread of q

values should therefore be large enough to cover the contribution

to the overall uncertainty from, small uncorrected systematic effects.

Least square fits yielded the following values for the con-

stants in eq, 18:

q = (0.0577 - 0.0041) + (0.1408 - 0 . 0 0 5 3 ) \ / S / M

andf\ ^^ f\ f* ^\ \ * • / ^ ^ *~\ ^\ ^\ *q = (0.1852 - 0.0059)\J (0. 220 t 0.016) + S/M

(26)

- 22 -

From the above uncertainties in a , b , a and b^ one could cal-

culate probable errors in the related quantities A, B, a and b. The

drawback is, however, that d A and d B would not be independent

of each other. Instead, we prefer to calculate dq/q , which is used

to determine the total uncertainty in RI m, Q > a s inferred from

formulas 24 and 25. Using a procedure implied by the method of

least squares, it is found that dq/q may be put equal to 1. 5% over

the whole range of measurements, from which d(C.q)/C.q = - 3.1%

is obtained. Tentatively, the error in A ( l / v ) ^ is put equal to

-0 .3 b. Then, with the aid of eq. 16, we can write

+ rd(Cq)1 2 -}-0.09

(C q - 1.5)2

(Cq- 1.5)2

and this expression varies betwen 4, 4 - 5. 1 % as C q goes from

19.5 to 12.7 b. For convenience, the total uncertainty in RI ,

can be taken as - 5.0% over the whole range of measurements.

Because of the good fit, this error is of the same order as the one

attributed to single measuring points.

8. Discussion.

a. Cpmp_ari_son of e_xperimental values.

In Table VIII and Fig. 6, results from earlier measurements

of the effective resonance integral of ThO? are reviewed. Though un-

doubtedly the activation technique is the most straightforward for ab-

sorbers like thorium, all previous results have been obtained by

reactivity methods»

The measurements of Moore have been per-

formed with the Dimple pile oscillator. By using gold as standard,

some of the difficulties arising from uncertainties in the cadmium

cutoff energy could be avoided. The calibration constant is allocated

an error of 5 %, mainly due to an assumed 4 % uncertainty in the

- 23. -

1513 b infinite dilution integral of gold. A scattering correction of

about 1,8 - 0 . 4 b per Th atom was deducted. Possible need of a

correction for fast fission in thorium, which can be appreciable

because of the hard spectrum in the oscillator void, is not discussed.

However, the main uncertainty is most probably due to the non-

constancy of the product W (u) of the flux, $(u), and the neutron

importance, P(u), W (u) being the appropriate weighting factor in

reactivity measurements. For energies near the cadmium cutoff,

P (u) is expected to fall off sharply, because a neutron, even if not

absorbed in the sample, runs the risk of being captured during its

second pssage through the cadmium tube wall. Such a neutron has a

low importance. The effect of the cadmium tube is also extended to

resonance neutrons by the moderator surrounding the tube, because

some resonance neutrons will be slowed down and reflected back into

the tube, and subsequently captured. At high energies, P (u) is also

known to fall off, because high energy neutrons are subject to leakage

and resonance capture in the fuel. However, due to the hard spectrum

used, W (u) may exhibit a rising tendency towards higher energies.

Totally, a systematic variation may occur over the whole energy

range of interest. As an illustration, the ratio

W (E a 100 eV) . . , T , 4. J r .1. ^ r r< ( \ A

— J *- = 1. 16 was calculated from the graphs of Q> hi) andW (E = 4. 9 eV)

P (u) , given by Pettus (25) for the Lynchburg Pool Reactor. This

ratio may, in spite of the difference between the reactors, be re -

presentative for the Dimple measurements also, mainly because of

the appreciable cadmium tube thickness used (2 mm), Too high effec-

tive resonance integrals are therefore expected. Another uncertainty

is due to the " l /v" contribution in reactivity work. This problem is

discussed below. With regard to the above remarks, the difference

between the present measurements and those of Moore seem

explicable.

1959j| Pettus' thoria results are based on period measure-

ments in the Lynchburg Pool reactor. Boron has been used as stan-

dard, and a calibration error of 5 % is quoted, which may be enough

to account for the uncertainty in the cadmium cutoff energy. Scatte-

ring corrections are included. Also, mean energy importance factors

- 24 -

for boron and thorium are calculated. The "cadmiun sleeve effect"

is mentioned, but not considered in the calculations, which partly

accounts for the fact that the mean energy importance of thorium

captures, <( P , )> , came out smaller than ^ p B / * (0*91 and

0.96 respectively). The opposite behaviour is obtained from Pettus'

own recent calculations (25). Assuming the latter to be correct,

Pettus' old results should be 10-20 % too high. As in the work of

Moore et al., the true "l/v" absorption is difficult to assess. Addi-

tional uncertainties may arise from inelastic scattering and high

energy fissions.

Redm_an_ and_Thie,__l_9_58.. The measurements quoted by Redman and

Thie were made by pile oscillator technique, using indium, gold and

boron as standards. The information available about these measure-

ments is sparse, and does not permit an estimate of their value.

Dayis,__l_9_57. Like all the measurements previously discussed this

one was also performed by a reactivity method (period measurement)

inside a cadmium tube in the Hanford Test Pile. A uranium metal

rod was used as standard. Again, the energy dependence of the

weighting factor W (u) is neglected, but this is less serious when

comparing thorium and uranium in a graphite moderated reactor

than in a case where for instance boron is used for calibration in a

H,O or D?O moderated reactor. Moreover, the effect of scattering

and moderation due to oxygen is neglected, that is, the effect on the

pile is regarded as entirely due to resonance absorption. Too high

values may follow.

In Table VIII, all data include the "l/v" contribution, and

no attempt has been made to use corrections of the kind

J -75— calculated by the authors. The reason is that in

E Er / z

c

reactivity work under cadmium, the neutron importance function .

falls off steeply near the cadmium cutoff. The "l/v" contributionmust therefore be weighted by the factor wTp T" ' w ^ e r e

norm'E is a mean normalization energy,

norm b7

(^ t

The integral \ v . )£•••' <r & (E) dE is considerably smaller thanJ *• n o r m '

the first mentioned, partly because of the non-l/v behaviour of the

- 25 -

cross section <r (E) , part ly because the weighting factor is l ess

than unity (for E ?S> E ) in the region where <r (E) is still3 x norm c' b

large. The fact that the reactivity measurements all include an

uncertain "l/v!" contribution, makes comparison with the present

values and theoretical calculations difficult.

In spite of this low "l/v" contribution, all reactivity mea-

surements, except those of Redman,' give considerably higher

values than the present report, which may be partly explained by the

experimental and calibration errors of the order of 8% attributable

to previous measurements as well as our 5% uncertainty, and may be

partly due to the fact that proper account of neutron importance has

not been taken.!?•. __Comgarij3on_ with_thepretical_concep_ts_ and_ca c_ula.tio_n_s_.

The first advances in resonance integral theory were based

on several limiting simplifications. The assumptions made per-

mitted deduction of analytical relations, from which essential

physical insight could be gained. In accurate calculations, severe

simplifications must be avoided and one then has to rely on purely

numerical solutions of basic equations. Nordheim "s (15) calculations

of resonance absorption represent for the moment the ultimate in

this direction. The necessary simplifications in his work are the

assumptions of a flat flux in absorber and moderator and the narrow

resonance approximation for moderator collisions. The resulting

resonance integrals for thorium metal and oxide, including the

p wave and high energy contribution of about 2 b, together with our

experimental results, are as follows:

Rl£ s 3 . 4 + 1 7 . 6 \ - £ - (27)v Th

M H - . 3 . 3 * 1 6 . 1 ^ (2S)

R I?ho2 = 5 - 4 + 1 7 - ° \ h l ~ <*»R 1 n o 2 = 5.0 + 45.6 \ ^ — o (30)

(N = Nordheim; H + W = Hellstrand and Weitman)

- 26 -

The formulas are valid in an S/M range from about 0.15 to about

0. 7 or 0. 8 cm /g. The expression for thorium metal given in (Z) has

been recalculated, assuming a total flux correction of 1. 025 and a

"l/v" contribution of 1. 5 b (the calibration constant C is practically

unchanged). A direct comparison of the above formulas shows, I ha t

the experimental and calculated "volume" terms agree very well,

while the theoretical "surface" terms are consistently higher. In fact

a n d ^ B N ^ = 1 . 0 9 0W

T h B

which suggests that the parameters for the low lying levels in

thorium are in error. Further evidence for this assumption has been

given by Brose (28). Using a cadmium ratio technique with gold as

standard, Brose obtained for the infinite dilution integral of thorium,

I , (+ A(l/v) , ) , 82. 7 - 1. 8 b, while his calculated value came

out as 96.3 b. In Brose's experiments, I is proportional to the

same calibration constant, C = 114. 3 - 2O 7 %, as used in the present

report. His limits of error, - 2.2 % , therefore seem too small. A

more probable figure is - 4 %. To the theoretical value, Brose adds

2.9 b for the "l/v" contribution. Assuming A(l/v) , = 1.5 b , and

adding a tentative value of 2 b for the neglected p-wave contribution?

we obtain 96. 3 - 1. 4 + 2. 0 = 96. 9 b, to be compared with

82.7 - 3. 5 b. In the case of gold and uranium, Brose's calculations

give results which are in good agreement with experimental values.

The large discrepancy for thorium is therefore most probably

caused by errors in the resonance parameter data available. The

96. 9ratio Q '••* • = 1. 172 is also expected to be higher than the ratio of

the surface terms, as the infinite dilution integral is more sensitive

to the low lying levels.

The simplified treatment of resonance absorption, implying

use of the NR, IM and rational approximations, can partly be

checked by means of the equivalence theorems predicted, without

reference to calculations based on resonance parameter data.

Applying eq.5 to our metal values given by eq. 28, one obtains

- 27 -

R I T h O = 3.3 + 17. 2 y 0.069 + S/M ThO,

which can be transformed into

RI eqv.ThO,

= 5.4 + 15.4

Swithin the range 0. 14 —M ThO,

ThO,

0.65. A comparison of

equations 29 and 31

R I

RI

exp.ThO,

eqv.ThO

= 5. 0 + 15.6

= 5.4 + 15.4

Th00

ThO2 J

shows that there is excellent agreement. It seems as if our thorium

metal and oxide data are better related by Dresner's first equiva-

lence theorem, than are Hell strand's recent uranium metal and

oxide values (3). No direct theoretical motivation for such a be-

haviour can be found. Dresner's calculations indicate, however, that

the rational and flat flux approximations introduce with energy") lump-

size and scattering power widely varying, partly cancelling errors

into the NR and IM formulas, from which the equivalence theorem

is inferred. The consistency of the thorium results relative to the

theorem may therefore be fortuitous . Indirectly, however, our

thorium metal and oxide data give some support to the approximate

theory,

Acknowledg em ent s

The author is indebted to E. Hellstrand for his help and

guidance, to G. Lundgren for his investigations of the solubility of

thoria, and to B. Karmhag and L.E. Bjelvestad for skilful technical

assistance.

- 28 -

R e f e r e n c e s .

1. E. Hellstrand, J. Appi. Phys. , 28, 1493 (1957),

2. E. Hellstrand and J. Weitman, Nuclear Sci.and Eng., 9»

507 (1961).

3. E. Hellstrand and G. Lundgren, Nuclear Sci.and Eng., 12,

435 (1962),

4. L. Dresner, Resonance Absorption in Nuclear Reactors,

Pergamon Press (I960).

5. W.L. Brooks and H. Soodak, NDA 2131-19(1960).

6. C.B. Bigham andR.M. Pearce, CRRP-1006 (1961).

7. E. Johansson and E. Jonsson, Nuclear Sci.and Eng., 13,

264 (1962).

8. E Critoph and D. W. Hone, Conf. Heavy Water Lattices,

Vienna, (1959).

9. A.R. Vernon, Nuclear Sci. and Eng. , 7_> 2 5 2 (i960).

10. I.I. Gurevich and I. Y, Pome ranch ouk, Proc. 1st Intern.

Conf. Peaceful Uses of Atomic Energy, Geneva, 5_, 467 (1956).

11. B. Wolfe, Nuclear Sci. and Eng., i±, 227(1961).

12. K. Jirlow, RPI-91 (1962).

13. G.W. Grodstein, National Bureau of Standards, Circular 583.

14. S. Yiftah, D. Okrent and P. A. Moldauer, Fast Reactor Cross

Sections, Pergamon Press (I960).

15. L.W. Nordheim, GA-2734 (1961).

16. P. A. Egel staff, Phys. Rev., 115, 182(1959).

17. K.K. Seth et al. , Phys. Rev. , ll£, 692(1958).

18. L.I. Tirén and J.M. Jenkins, AEEW-R 163 (1962).

19. N.G. Sjöstrand and J. S. Story, AEEW-M 57 (I960).

20. R. Nilsson, RSA-27 and RFA~79> AB Atomenergi internal

memorandum.

- 29 -

21. K. Jirlow and E. Johansson, J, Nuclear Energy, Part A:

Reactor Sci-, i±, 101 (I960).

22. L.W. Nordheim, GA-638 (1959).

23. P.G.F. Mooreetal . , AEEW-R 57 (1961).

24. W.G. Pettus, BAW-TM-203 (1959).

25. W.G. Pettus et al. , BAW -1244 (1962).

26. W.C. Redman and J. A. Thie, Proc. 2nd Intern. Conf.

Peaceful Uses of Atomic Energy, Geneva, P/600 (1958).

See also J.A. Thie, Heavy Water Exponential Experiments

Using ThO2 and UO2> Pergamon Press (1961),

27. M.V. Davis, Nuclear Sci. and Eng., 2, 488(1957).

28. M.Brose, Thesis, Technische Hochschule Karlsruhe (1962).

- 30 -

Table 1 a.

The holder loadings for the central

channel irradiations of one series.

Holder Cadmium Loadingthickness,

mmn r

Dimensions,

mm

Number ofmeasuringsamples

Number and kindof monitor foils

6

7

1. 1

r ö d

Pb-Aufoil, 0. 1wt %Au

Diameter 6. 50 Two

Length 123

Diameter 13.00 "Length 12 3

Diameter 19.00 "Length 123

Diameter 28.00 "Length 123

Diameter 15. 00 OneThickness 0. 1

_ it tt

Two 0. 2 mm thickgold foils, diam.7 mm

Table 1 b.

Sample specifications for the thermal

column irradiations of one series.

Holder

n'r

1

2

Loading

One ThO^ sample +

2 Pb-Au foils

_ ti

DimensionSj

m m

ThO 2 : 1.7 x

Pb-Au: O . l x

ThO 2 : 1.7 x

Pb-Au: 0. 1 x

8

8

14

14

_ It _ ThO2: 1.7 x 20

Pb-Au: 0. 1 x 20

- 31 -

Table 2.

Density of thoria rods and pellets

irradiated in the central channel.

Roddiameter,mm

6.50

13.00

19.00

28. 00

Densityof end-pieces,

g/cm

9.51

9.26

9.25

9.76

Density of samples measured

SeriesI

9.25

9.24

9.24

9.80

9.79

Series

II

9.50

9.49

-

9.02

9.75

9.80

9.83

Series

. Ill

9.50

9.50

9.74

9.02

9.19

9.26

9.65

9.75

Series

IV

9.51

9.45

9.14

9.43

8.97

9.78

9.83

9.57

Mean densityof samplesmeasured,g/cm

9.49

9.32 .

9.31

9.75

- 32 -

Table 3.

Factors for the calculation of flux corrections.

Resonance S/MT ,n

integral i h U 2contributions .—.—

0.751 0.531 0.376

: resolvedenergy contribution,upper limitabout 1000 eV 14.00 10.31 8.00

A(unres): s andp wave contribu-tions below 30 keV 2 .25+1 .0 1.98+1.0 1.77 + 1.0

A(>30 keV): Highenergy contribution 1.0 1.0 1.0

RI t o t a l 18.25 14.29 11.77

K(res) = p / r e S ^- 0.767 0.721 0.680Kitot

K(unres) = (unres) 0.178 0.209 0.235RItot

K(>30 keV) =

= A(>30keV)

tot

Y t o t 1.029 1.024 0.017

- 33 -

Table 4.

Data for the calculation of the calibration constant.

Quantity- Symbol Absolute value Refer.ence(s)

Infinite dilution resonance. integral for gold 'Au

1520 t 30 b (21)

Self-screening correction for ZJilA

lead-gold foils in res. flux20 I 5 b (2)

The l/v contribution to theepicadmium cross sectionof gold 'Au

40 t 5 b

Correction factors for non-l/v behaviour in thethermal region.

For gold

For thorium

2200 m/sec cross sectionfor gold and thoriumrespectively

Au 1.005

0.995

98.4 +

7.36 t

0.

0.

5 b

10 b

(19)

(18, 19)

(19)

(19)

Calibration constantaccording to eq, 17

114.3 + 3.1 b

Total correction for devia-tions of the neutron spectrum yfrom l/E, as calculated totfrom eq. 10

1.025 t 0.025

The contribution to the epi-cadmium cross section forthorium from the energyregion below the firstresonances, calculatedfrom eqs. 11 and 12

' T h1.5 ± 0.3 b

- 34 -

Table 5.

Measured values of q and their transformation to resonance integrals,

S/MThO-

2with total experi-inental uncertain- with limits ofties error due to

uncertaintiesin C and q

with limits of error dueto uncertainties in C, q, Y.and A(l/v)Th

0.6486 t 0.0010

0.3298 ± 0.0044

0.2266 t 0.0025

0. 1465 t 0.0006

0. 1710 t 0.0036

0.1379 t 0.0021

0o1243 t 0.0017

0. 1115 t 0.0015

19.5'4 t 0.67

15.76 t 0.49

14.21 t 0.43

12.74± 0.38

17.60 t 0.84 or t 4. 8 %

13.91 t 0.66 or t 4.7 %

12.40 t 0.60 or t 4. 8 %

10.97 t 0.55 or t 5.0 %

Least square fits:

q = ax + bj p/M ;

q =

R I ThO = a V b

= 0.0577 t 0.0041;

= 0. 1852 t 0.0059;

Ca - A ( l / v ) v.»,A = - — = 4. 9 8 ; B = .... . ..= 15.62

= 0. 1401 + 0.0083;

= 0.2196 t 0.0155

Cb,

t o t to t

C a .- c ; a = -—=-=20 .65 ; b = b ? ; c =•

Ttot' T h = 1.46

to t

- 3 5 -

Series

nr

I •

II

III

IV

Total

Table 6 a,,

Schedule of measurements performed.

AthThO- A

thAu

3

2

3

3

11

3

2

3

3

11

Number of determinations of

A resAu

3

3

3

3

12

6.50

. r e s

13. 00

0 1

1 0

I I

1 I

5 5

diam.

19.00 28.00

I

2

2

2

8

2

2

2

2

8

Table 6 b.

Data for the calculation of final q values and limits of errors.

The A values can only be interrelated within the same series.

Q u a n t i t y

Series

nr

Mean values of

AthAu

r e s I* G SThO for rod diam. q =

A r e S thT h ° 2

AAu

A.thTh

resAu

thThO.

• for rod diam.

6.50 13.00 19.00 28.00 6.50 13.00 19.00 28.00

0.1224 0.3607

0. 8 % 0. 25 %

0.3967 0.3587 0.3211

1.2% 0 . 3 % 1.2 %

0.1346 0.1217

1.5 % 0 . 9 %

0.1090

1.5 %

II

0.1164 0.3376 0.5182

2.7 % 0. 19 % 2 . 4 %

0. 3751 0.3359•f

0 . 2 % 1.6%

0.1787

3.6 %

0.1293

2.7 %

0.1158

3.1 %

III

IV

0.1137 0.3425 0.5039 0.4285 0.3855 0.3448

2 . 8 % 0 . 4 8 % 2 . 2 % 0 . 6 % 0 . 5 %

1.5

0.5 %

0.1080 0.4527 0.7033 0.5832 0.5195 0.4626

0 . 2 1 % 2 . 5 % 0 . 3 % 0 . 9 % 0 . 5 %

0.1673 0.1422

3.6 %

2.9 %

2.9 %

0.1678 0.1391+ +

1.5 %

0. 1280

2.9 %

0.1239

1.8 %

0. 1145

2.9 %

0.1104

1.6%

Final q value0.1710 0.1379 0.1243 0.1115

0.0036 0.0021 0.0017 0.0015

- 37 -

Table 7.

Possible sources of systematic errors.

Kind of effect Magnitude of the effect

Irradiation

Deviations of neutron spectrumfrom l/E

Gradients and self-screeningeffects in the thermal column.

Local flux pertubation at themonitor foil position.

Activity measurement

Dead-time, background anddecay corrections.

Fission products.

Differencies in gamma rayattenuation for solutions anddispersions.

Geometric effects in the |Jmeasurements on lead-goldfoils.

A correction factor Y. . = 1. 025 ± 0. 025

has been applied.

Maximum error introduced 0. 5 %oNo correction applied.

This effect may amount to 0. 9 % for thethickest thoria rod. No correction hasbeen applied.

Large samples suffer from an extra un-certainty of less than 0. 2 % due to dead-time corrections. Negligible effect.

Interference less than 0.5 %.

Very little effect.

An error of 0. 35 % introduced into thefinal result. No correction applied.

General

Variations in thoria density.

Varying gold content in lead-gold foils.

Impurity elements.

Self-screening in the lead-goldfoils from the central channel.

Overlapping of thorium andcadmium resonances.

Uncertainty in the cadmiumcutoff energy.

Contributes to the uncertainty in theS/M-values given.

Corrections of a few percent have beenapplied.

No serious content of impurities foundin chemical analysis.

A 1.5 % correction has been applied.

This effect should be considerablysmaller than 1 % (see ref. 2).

Contributes to the limits of errorstated for A( l /v) T h .

Table 8.

Comparisons between results from different experiments,

•Author andyear ofpublication

Method StandardResults

including the actual l/v-contribution in the exp.

Limits of errors,according tothe authors

I. Moore et al. (23)1961

Pileoscillator

Gold 4. 4 + 21.2M ThO.

± 8 %

II. Pettus (24)1959

III. Redman and Thie(26). 1958

Periodmeasure-ment

Pileoscillator

Boron

Boron,gold,indium

3.7 + 26.9 SM

0.ThO

ThO2

<1.34

- 2 . 2 + 2 5 . 3 SM,ThO2

<I l . 8

t (6-8%)

Not stated

oo

IV. Davis (27)1957

Present report

Dangercoefficient

Activation

RI= 10.9 b for a1. 7 3 - cm uraniumrod. The more re-cent value 9.3 b isused for renormali-zation.

Gold

13.2 b forM

0.12ThO.

(Renormalized).

6.5 + 15.6M ThO.

0. 14<S/M_ ^0.65ThO.

t 5.2 %

t 5,0%

MONITOR FOILS

TOP OF THORIA ROD

MEASURING SAMPLES

ALUMINIUM TUBES

ALUMINIUM SPACER

CADMIUM COVER

en I"Pb-Au

ThO,

Fig. 2. Sample arrangement for thermalcolumn irradiation.

Fig. 1. Sample holder for central channelirradiations.

R ( t ) ARBITRARY UNITS

1.08"

toe-

1.04 •

1.03

lor

1.01'

1.00

0.98-

T 1 r—i 1—

Q 2 4 6 8 10 IB 20 25 30

DAYS AFTER IRRABIATION

Fig. 3. The time dependence of the ratio of central channelto thermal column induced thoria activity.

O.i-

04—-10'

AMC JOHSSCt* Wi

10"

Fig. 4. The shape of the neutron spectrumat the position for irradiation in thecentral channel.

P0W1 Of MEASUREMENT WITH EXWRIMENTAl ERRORS

PRESENT REPORT WITHIN RANGE OF MEASUREMENTS

FXTHAFOiATION ACCORDING TO SO. 25

« » « EO. 24

Tho

Fig. 5. Experimental results for the effectiveresonance integral of thoria rods as afunction of their surface-to-mass ratio.

Ri2 0 -

1S-

18-

17 -

16-

<S-

1 4 -

1 J -

12-

1 1 -

70-

9-

3-

7-

6-

5-

<-

S-

t-

1-

0

no,

PRESENT REPORT

N O R O H E I M ' S CALCULATIONS (15 !

0.6 0.7 0.8

Fig. 6. Comparison between results fromdifferent experiments.

36-

34 •

32

30-

28-

26-

24-

22-

20-

18-

16

H

12-

1 0 - -

I —H —jn—H. *i —

- ~ Moor» at— P«ltijs

— R«dman

- Oovis

— Prasant

ol 123)(24>

<26)

(J71

roport

o» 0s

Fig, 7. Comparison between Nordheim'scalculations and the presentexperiments.

Addendum

When this report was almost completed, the results of a

new set of calculations -were communicated to us by Dr. Nordheim.

These calculations are based on the resonance parameter data given

by the Harwell group in report AERE-PR/NP2. As is evident from

the tables below, excellent agreement is obtained assuming a p-wave

and high energy contribution of 2,1 b, A strong support for this value

is the fact that it suffices for both metal and oxide,,independent of S/M.

In spite of a few remaining approximations in the theoretical

treatment, for the isolated rod case satisfactory agreement now

exists between experimental and theoretical values for both uranium

and tho~ium rcetal and oxide.

Thorium metal

Nordheim 's

recent

calculations

Hell strand a»Weitman

Resolved

Unresolved

Total+ 2. 1 b

3.3 + 16.l\/s/MTh

1.240

17.04

1. 79

20.9

21.2

s/

0,623

11.92

1.59

15.6

16.0

MTh

0.311

8.45

1.38

11.9

12.3

0.155

6.16

1.21

9.5

9 . 7

0.078

4.65

1.08

7.8

7 . 8

Thorium oxide

Nordheim ""s

recent

calculations

Present report

Resolved

Unresolved

Total + 2, 1 b

5 - 0 + 1 5 - t / M - ° 2

1.125

17. 741.82

21. 7

21. 5

S/M

0.564

12.67

1.63

16,4

16.7

ThO2

0.282

9.341.45

12,9

13.3

0.141

7.25

1.31

10,7

10.9

LIST OF PUBLISHED AE-REPORTS

1—29. (See the back cover of earlier reports.)

30. Metallographic study of the isothermal transformation of beta phase inzircaloy-2. By G. Östberg. I960. 47 p. Sw. cr. 6:—.

31. Calculation of the reactivity equivalence of control rods in the secondcharge of HBWR. By P. Weissglas. 1961. 21 p. Sw. cr. 6:—.

32. Structure investigations of some beryllium materials. By I. Fäldt and G.Lagerberg. 1960. 15 p. Sw. cr. 6:—.

33. An emergency dosimeter for neutrons. By J. Braun and R. Nilsson. 1960.32 p. Sw. cr. 6:—.

34. Theoretical calculation of the effect on lattice parameters of emptyingthe coolant channels in a DjO-moderated and cooled natural uraniumreactor. By P. Weissglas. 1960. 20 p. Sw. cr. 6:—.

35. The multigroup neutron diffusion equations/1 space, dimension. By S.Linde. 1960. 41 p. Sw. cr. 6t—.

,36. Geochemical prospecting of a uraniferous bog deposit at Masugnsbyn,Northern Sweden. By G. Armands. 1961. 48 p. Sw. cr. 6:—.

37. Spectropholometric determination of thorium in low grade minerals andores. By A.-L. Arnfell and I. Edmundsson. I960. 14 p. Sw. cr. 6:—.

, 38. Kinetics of pressurized water reactors with hot or cold moderators. ByO. Norinder. 1960. 24 p. Sw. cr. 6:—.

39. The dependence of the resonance on the Doppler effect. By J. Rosén.1960. 19 p. Sw. cr. 6:—.

40. Measurements of the fast fission factor (£) in UOi-elemenls. By O. Ny-lund. 1961. Sw. cr. 6:—.

44. Hand monitor for simultaneous measurement of alpha and beta conta-mination. By I. D . Andersson, J. Braun and B. Söderlund. 2nd rev. ed.1961. 5w. cr. 6:—.

45. Measurement of radioactivity in the human body. By I. O. Anderssonand I. Nilsson. 1961. 16 p. Sw. cr. 6:—.

46. The magnetisation of MnB and its variation with temperature. By N.Lundquist and H. P. Myers. 1960. 19 p. Sw. cr. 6:—.

47. An experimental study of the scattering of slow neutrons from HjO andD2O. By K. E. Larsson, S. Holmryd and K. Otnes. 1960. 29 p. Sw. cr. 6:—.

48. The resonance integral of thorium metal rods. By E. Hellslrand and. J.Weitman. 1961. 32 p. Sw. cr. 6:—.

49. Pressure tube and pressure vessels reactors; certain comparisons. By P.H. Margen, P. E. Ahlström and B. Pershagen. 1961. 42 p. Sw. cr. 6:—.

50. Phase transformations in a uranium-zirconium alloy containing 2 weightper cent zirconium. By G. Lagerberg. 1961. 39 p. Sw. cr. 6 i—.

51. Activation analysis of aluminium. By D. Brune. 1961. 8 p. Sw. cr. 6:—.52. Thermo-technical data for D2O. By E. Axblom. 1961. 14 p. Sw. cr. 6:—.53. Neutron damage in steels containing small amounts of boron. By H. P.

Myers. 1961. 23 p. Sw. cr. 6:—.54. A chemical eight group separation method for routine use in gamma

spectrometric analysis. I. Ion exchange experiments. By K. Samsahl.1961. 13 p. Sw. cr. 6:—.

55. The Swedish zero power reactor R0. By Olof Landergärd, Kaj Cavallinand Georg Jonsson. 1961. 31 p. Sw. cr. 6:—.

56. A chemical eight group separation method for routine use in gammaspectrometric analysis. I I . Detailed analytical schema. By K. Samsahl.18 p. 1961. Sw. cr. 6:—.

57. Heterogeneous two-group diffusion theory for a finite cylindrical reactor.By Alf Jonsson and Göran Näslund. 1961. 20 p. Sw. cr. 6:—.

58. Q-values for (n, p) and (n, a ) reactions. By J. Konijn. 1961. 29 p. Sw. cr.Ol™•

59. Studies of the effective total and resonance absorption cross sections forzircaloy 2 and zirconium. By E. Hellstrand, G. Lindahl and G. Lundgren.1961. 26 p. Sw. cr. 6:—.

60. Determination of elements in normal and leukemic human whole bloodby neutron activation analysis. By D. Brune, B. Frykberg, K. Samsahl andP. O. Wester. 1961. 16 p. Sw. cr. 6:—.

61. Comparative and absolute measurements of 11 inorganic constituents of38 human tooth samples wilh gamma-ray speclrometry. By K. Samsahland R. Söremark. 19 p. 1961. Sw. cr. 6i—.

62. A Monte Carlo sampling technique for multi-phonon processes. By ThureHögberg. 10 p. 1961. Sw. cr. 6 i—.

63. Numerical integration of the transport equation for infinite homogeneousmedia. By Rune Håkansson. 1962. 15 p. Sw. cr. 6:—.

64. Modified Sucksmith balances for ferromagnetic and paramagnetic mea-surements. By N . Lundquist and H. P. Myers. 1962. 9 p. Sw. cr. 6i—.

65. Irradiation effects in strain aged pressure vessel steel. By M. Grounesand H. P. Myers. 1962. 8 p. Sw. cr. 6s—.

66. Critical and exponential experiments on 19-rod clusters (R3-fuel) in heavywater. By R. Persson, C-E. Wikdahl and Z. Zadw6rski. 1962. 34 p. Sw. cr.6:—.

67. On the calibration and accuracy of the Guinier camera for the deter-mination of interplanar spacings. By M. Möller. 1962. 21 p. Sw. cr. 6i—.

68. Quantitative determination of pole figures with a texture goniometer bythe reflection method. By M. Möller. 1962. 16 p. Sw. cr. 6:—.

69. An experimental study of pressure gradients for flow of boiling wafer ina vertical round duct. Part I. By K. M. Becker, G. Hernborg and M. Bode.1962. 46 p. Sw. cr. 6:—.

70. An experimental study of pressure gradients for flow of boiling water ina vertical round duct. Part I I . By K. M. Becker, G. Hernborg and M. Bode.1962. 32 p. Sw. cr. 6s—.

71. The space-, time- and energy-distribution of neutrons from a pulsedplane source. By A. Claesson. 1962. 16 p. Sw. cr. 6:—.

72. One-group perturbation theory applied to substitution measurements withvoid. By R. Persson. 1962. 21 p. Sw. cr. 6:—.

73. Conversion factors. By A. Ambernlson and S-E. Larsson 1962. 15 p. Sw.cr. 10:—.

74. Burnout conditions for flow of boiling water in vertical rod clusters.By Kurt M. Becker 1962. 44 p. Sw. cr. 6:—.

75. Two-group current-equivalent parameters for control rod cells. Autocodeprogramme CRCC. By O. Norinder and K. Nyman. 1962. 18 p. Sw. cr.6:—.

76. On the electronic structure of MnB. By N. Lundquist. 1962. 16 p. Sw. cr.

77.

78.

79.

80.

81.

82.

83.

84.

85.

86.

87.

88.

89.

90.

91.

92.

93.

94.

95.

96.

97.

98.

99.

The resonance absorption of uranium metal and oxide. By E. Hellstrandond G. Lundgren. 1962. 17 p. Sw. cr. 6i—.

Half-life measurements of 'He, « N , »O, *>F, »Al, "Se™ and "«Ag. By J.Konijn and S. Malmskog. 1962. 34 p. Sw. cr. 6:—.

Progress report for period ending December 1961. Department for ReactorPhysics. 1962. 53 p. Sw. cr. 6-.—.Investigation of the 800 keV peak in the gamma spectrum of SwedishLaplanders. By I. O. Andersson, I. Nilsson and K. Eckerstig. 1962. 8 p.Sw. cr. 6:—.

The resonance integral of niobium. By E. Hellstrand and G. Lundgren.1962. 14 p. Sw. cr. 6>—.

Some chemical group separations of radioactive trace elements. By K.Samsahl. 1962. 18 p. Sw. cr. 6>—.Void measurement by the [y. n) reactions. By S. Z. Rouhani. 1962. 17 p.Sw. cr. 6:—.

Investigation of the pulse height distribution of boron trifluoride pro-portional counters. By I. D. Andersson and S. Malmskog. 1962. 16 p.Sw. cr. 6:—.An experimental study of pressure gradients for flow of boiling waterin vertical round ducts. (Part 3). By K. M. Becker, G. Hernborg and M.Bode. 1962. 29 p. Sw. cr. 6:—.

An experimental study of pressure gradients for flow of boiling waterin vertical round ducts. (Part 4). By K. M. Becker, G. Hernborg and M.Bode. 1962. 19 p. Sw. cr. 6:—.Measurements of burnout conditions for flow of boiling water in verticalround ducts. By K. M. Becker. 1962. 38 p. Sw. cr. 6:—.

Cross sections for neutron inelastic scattering and (n, 2n) processes. ByM. Leimdörfer, E. Bock and L. Arkeryd. 1962. 225 p. Sw. cr. 10s—.

On the solution of the neutron transport equation. By S. Depken. 1962.43 p. Sw. cr. 6:—.Swedish studies on irradiation effects in structural materials. By M.Grounes and H. P. Myers. 1962. 11 p. Sw. cr. 6:—.The energy variation of the sensitivity of a polyethylene moderated BFjproportional counter. By R. Fräki, M. Leimdörfer and S. Malmskog. 1962.12 p. Sw. cr. 6:—.

The backscattering of gamma radiation from plane concrete walls. ByM. Leimdörfer. 1962. 20 p. Sw. cr. 6:—.The backscattering of gamma radiation from spherical concrete walls. ByM. Leimdörfer. 1962.

Multiple scattering of gamma radiation in a spherical concrete wallroom. By M. Leimdörfer. 1962.

The paramagnetism of Mn dissolved in a and B brasses. By H. P. Myers,and R. Westin. 1962.Isomorfic substitutions of calcium by strontium in calcium hydroxy-apatite. 1962. By H. Christensen.

A fast time-to-pulse height converter. By O. Aspelund. 1962.

Neutron streaming in D2O pipes. By J. Braun and K. Randen. 1962.

The effective resonance integral of thorium oxide rods. By J. Weitman.1962.

Förteckning över publicerade AES-rapporter

1. Analys medelst gamma-speklrometri. Av Dag Brune. 1961. 10 s. Kr 6:—.

2. Bestrålningsförändringar och neutronatmosfär i reaktortrycktankar —några synpunkter. Av M. Grounes. 1962. 33 s. Kr 6:—.

Additional copies available at the library of AB Atomenergi, Studsvik, Nykö-ping, Sweden. Transparent microcards of the reports are obtainable throughthe International Documentation Center, Tumba, Sweden.

EOS-tryckerierna, Stockholm 1962