AECL-5236

by

J.D. CHEN.. D.G. BOASE and R.B. LYPKA

Whiteshel l Nuclear Research Establishment

Pinawa, Manitoba

January 1976

Atomic Energy of Canada Limited

NON-JESTRUCTIVE DETERMINATION OF BURN-UP BY GAMMA-SCANNING:

AN ASSESSMENT OF Ce/Pr AS A FISSION MONITOR

IN CANDU FUELS

by

J.D. Chen, D.C. Boase and R.B. Lypka

Analytical Science Branch

Whiteshell Nuclear Research Establishment

Pinawa, Manitoba ROE 1L0

January 1976

AECL-

Determination non-destructive du taux de combustion par balayage gamma:144

evaluation de Ce/Pr comme moniteur de f i ss ion dans les combustibles CANDU

p a r

J . D . Chen , D .G. Boase e t R . B . Lypka

Resume

144On presente une evaluation theorique de Ce/Pr comme moniteur de determinationdu taux de combustion par balayage gamma pour les combustibles CANDU (CanadaDeuterium Uranium) provenant du rgacteu.r WR-1 et de la centrale Pickering.Les donnees relatives aux changements isotopiques se produisant dans le combustibleen cours d1irradiation sont obtenues au moyen des codes d'ordinateur LATREPet ISQ6EN. Elles permettent de suivre les diverses etapes du calcul du tauxde combustion S partir des mesures experimentales. On decrit, pour uneirradiation continue, 1'effet de la croissance et de la decroissance de144

Ce/Pr ainsi que le calcul de la correction de decroissance sur le tauxde combustion calcule. On examine d'autres facteurs, comme les changementssurvenus dans le niveau des flux neutroniques, la concentration des isotopesfissiles, les rendements de fission et Venergie liberee par la fission. Leo'.lcul du taux de combustion a partir d'une concentration mesuree de 144Ce/Prpour une irradiation intermittente fait l'objet d'un commentaire. Bien qu'enprincipe le taux de combustion puisse etre determine de cette fagon, selon lesconditions de 1'irradiation, Vincertitude globale peut aller de +30% a -20%par rapport aux meilleures valeurs calculees. La principale incertitudeaux niveaux des taux de combustion calcules pour les combustibles de Pickeringet WR-1 est due a la courte demie-vie de Ce/Pr. De recents travaux experimentauxont montre que le rapport des activites gamma des isotopes de cesium peutconstituer une meilleure solution pour determiner le taux de combustion.

L1Energie Atomique du Canada, LimiteeEtablissement de Recherches Nucleaires de Whiteshell

Pinawa, Manitoba ROE 1L0

Janvier 1976AECL-5236

NON-DESTRUCTIVE DETERMINATION OF BURN-UP BY GAMHA-SCANNINC:

AN ASSESSMENT OF U 4 C e / P r AS A FISSION MONITOR IN CANDU FUELS

by

J . D . Chen, D.G. Boase and R .B . Lypka

ABSTRACT

144A theoretical assessment of Ce/Pr as a monitor for the

determination of burn-up by y s c anning is described for CANDU (Canada

Deuterium Uranium) fuels from the WR-1 Reactor and Pickering Generating

Station. Data on the isotopic changes occurring in the fuel during

irradiation are obtained using the computer codes LATREP and ISOGEN to

permit discussion of the various steps in the calculation of burn-up

from the experimental measurements. For e. continuous irradiation, the144

effect of the growth and decay of Ce/Pr and the computation of the

decay correction on the calculated burn-up are described. Other factors

such as changes in neutron flux level, fissile isotope concentration,

fission yields, and energy released per fission are examined. The144

calculation of burn-up from a measured Ce/Pr concentration for an

intermittent irradiation is also discussed. While in principle burn-up

can be determined in this way. depending upon the irradiation conditions,

the overall uncertainty can be as high as +30% to -20% relative to the

'best' calculated values. The major uncertainty at design burn-up levels144

for Pickering and WR-1 fuels is due to the short half-life of Ce/Pr.

Recent experimental work suggests that the ratio of y-activities of cesium

isotopes may offer a better alternative as a means of determining burn-up.

Atomic Energy of Canada Limited

Analytical Science Branch

Whiteshell Nuclear Research Establishment

Pinawa, Manitoba ROE 1L0

January 1976

AECL-3236

TABLE OF CONTENTS

1. INTRODUCTION 1

2. CHOICE OF FISSION PRODUCT NUCLIDE 4

3. EXPERIMENTAL PROCEDURE ' 6

4. ISOTOPIC CHANGES DURING IRRADIATION 8

4.1 Ce/Pr Formation During a Continuous Irradiation 94.2. Correction of a Measured l^Ce/Pr Activity for Der;iy 10

4.2.1 Effect of Flux Level on ~ 13N

4.3 Effects of Changes in the Fissile Nucludes and Fission -|»Yields

4.3.1 Energy Per Fission 16

4.4 Overall Errc-s in Burn-Up Determination 174.5 Intermittent Irradiation 18

N14.5.1 Determination of — For an Intermittent ,„

Irradiation

5. CONCLUSION 24

6. REFERENCES 26

FIGURES 29144

APPENDIX A - Calculation of the Fraction of Ce/Pr AtomsRemaining at the end of a Continuous Irradiation 46from the Fissile Isotopes 235u and 2 ^ P u

APPENDIX B - Calculation of the Fraction of 1 Ce/Pr AtomsPresent at the end of an Intermittent Irradiation 51for the Fissile Isotopes 235u ancj 239pu

- 1 -

1. INTRODUCTION

The accurate determination of the burn-up of irradiated

nuclear fuel is of cun.-|amentai interest to the fuel designer and the

reactor operator to allow correlation of the performance of the fuel

with the chemical, physical and metallurgical changes which take place

during irradiation.

The methods available for the determination ol burn-up fall

into three broad categories:

1) Measurement by destructive analysis of the amount

of one or more fission products produced

2) Destructive analysis of the fuel before and afte/(2)

irradiation to determine the heavy element isotopic

composition.

3) Non-destructive measurements using flux monitors,(3)

calorimetry or y-spectrometry

Each of these methods has advantages and disadvantages.

Methods in categories 1 and 2 are theoretically the most accurate but

they are time consuming and require care in sampling because of the(A)

asymmetric distribution of fission within fuel rods and fuel assemblies

Each destructive analysis yields a burn-up value only for the area of

the fuel sampled, which is frequently small, and additional Information,

not always available, is needed to relate a point burn-up value to the

spatial neutron flux distribution along the fuel rod and at the ends

of the assembly (Figure 1).

- 2 -

Calorimetric methods which determine the heat output of a

number of fuel assemblies in the reactor through measurement of the

temperature and flow of the reactor coolant are frequently used but

require a detailed knowledge of many reactor physics parameters, such

as the distribution of neutron energy and neutron flux, to obtain the

burn-up of individual assemblies. In a large reactor it may not be

possible to obtain the necessary data for each fuel channel assembly

and only integrated values for the whole core may be available. To

compute the burn-up of individual fuel rods, reactor lattice-parameter

codes must be used and these require detailed data on neutron spectrum,

flux, fuel geometry and other parameters.

Methods which use flux monitoring foils or wires external

to the fuel tend to be the least accurate of the experimental techniques.

Corrections must be applied for self-shielding effects and the neutron

spectrum must be known to allow an estimate of the effective cross

sections. The method is, however, attractive in some experimental

situations because of its relative ease of application.

In principle, the method of Y-scanning after irradiation

allows a rapid non-destructive measurement of the amount of selected

fission products present in a fuel with subsequent calculation of

burn-up in the same way as in the category 1 methods. Experimentally

rapid, it allows the fission product profile, and therefore the

burn-up distribution, along a whole fuel rod to be recorded. The(4-7)

technique has been widely used for relative burn-up measurement

where the specific fission product content of a number of fuel rods is

assayed relative one to the other. In this application, an absolute

calibration of the technique may not be required but it can be achieved,

if desired, by destructive analysis of samples from one or two rods when

Y-scanning is completed. In this way, the burn-up distribution in CANDU*

fuel element assemblies has been determined together with data on end-

flux peaking and radial and longitudinal burn-up gradients.

•'Canada Dfcjterium Uranium

- 3 -

This report describes a theoretical assessment of a

y- scanning method, which utilizes the fission product pair

as the fission monitor, for the absolute determination of burr-up in

CANDU fuels from WR-1 and the Pickering Generating Station (design

burn-up 192 MWh/kg). The study was requested in order to exploit

the potential advantages of speed and non-destructive measurement

inherent in the y~&canning approach.

The discussion is concerned primarily with procedures by

which burn-up may be cal"ulated from an experimentally determined144

value of the concentration of Ce/Pr in a fuel rod at some time after

discharge from the reactor. The method of experimental measurement(4)

has been described previously and is discussed only briefly here.

At the outset it was apparent that the proposed method should

meet the following criteria:

1) The method should be independent of other experimental

methods of burn-up determination, except for calibration

purposes.

2) It should offer improved accuracy and precision over

reactor lattice-parameter codes currently used for

burn-up calculation. For practical purposes a target

accuracy of 95% or better, and a precision (2a) of

± 10% or better are required.

3) It should be applicable to any fuel discharged from a

reactor of a given design, without recourse to the

detailed data which are used in lattice-parameter codes.

This criterion was adopted to attempt to obtain J

degree of independence for the yscanning method since

experimental burn-up values are often sought as a means

of checking the accuracy of lattice-parameter code

burn-jp values. For safeguards applications it is also

desirable to obtain an independent check of the

operational history of the reactor.

- 4 -

As discussed in Section 4.5, complete attainment of the144

last criterion is not possible using Ce/Pr since an accurate

knowledge of the on-power and shutdown history of the reactor is144

required to permit corrections for the decay of the Ce/Pr formed.

Several other basic requirements of a burn-up method based(12 5 9}

on the assay of a fission product isotope have been identified ' ' '

They include: accurate values of neutron cross sections, fuel enrich-

ment, fissior. yield, y-ray abundance, fission product half-life and

others. The desirable properties of a fission product monitor are

discussed below, together with an examination of changes which occur in

the heavy element isotopic composition of fuel as irradiation progresses

and the relationship between burn-up, fission product formation and

decay, time, and neutron flux.

2. CHOICE OF FISSION PRODUCT NUCLIDE

The required characteristics of a fission product nuclide

for use as a measure of burn-up in power-reactor fuels have been discussed

in detail by Fudge et al . In summary the nuclide should have:

1) a half-life which is long compared to the fuel

irradiation time,

2) a high fission yield,

3) a low neutron absorption cross section,

4) ct high y-energy and y-ray abundance.

These criteria limit the possible choices to Zr, Ce/Pr,1 %Ru/Rh, and 1 3 7Cs.

- 5 -

95

Zr (half-life 65.5 days) has been mainly used for burn-up

determination for short irradiations and for relative burn-up assays in

groups of fuel elements where the irradiation time is identical.

The disadvantage of Ru/Rh (effective half-life 369 days)239 235

is that its fission yield from Pu is much higher than that from U.

It is useful when one type of fission is predominant, for example in

highly enriched U or plutonium fuels. Forsyth and Blackadder

and others have explored the value of the difference in fission yields035 939

to determine the relative number of fissions of " U and "' Pu in lowenrichment fuels.

Cs exhibits many of the required characteristics. It lias

a half-life of 30.1 years, which is long compared to any fuel irradiation239 235

time, and its fission yield from both Pu and U is almost identical.

However, the volatility of cesium and its compounds leads to its

migration in U0_ fuels and its deposition in cooler regions of the fuel

Thus, y~scanning measurements of this isotope way not record the rrue-

distribution, or number, of fissions.

The usefulness of the measurement of ratios of fission product(12-15') 134

nuclides has been discussed elsewhere . The ratio of Cs (a133 137

product of neutron capture by the fission product Cs) to Cs appearsto be the most promising and offers a possible means of overcoming the

137errors caused by the migration of Cs.

144In this work the isotopic pair Ce/Pr was selected for

examination since it exhibits a number of the required properties of a( 9 )good fission monitor . The parent nuclide has a half-life of 284.4

days and the daughter (half-life 17.3 min ) emits a v-ray of 2.185 MeV

which undergoes minimal attenuation in uranium dioxide fuel. The

abundance of this y-ray is low but with Ge/Li detectors it is recorded

in a low background region of the y-spectrum and its intensity can be

determined with a high degree of precision. These isotopes have not

- 6 -

been found to migrate in irradiated U0 7 , but the fission yieldsnc 239 *"

from u U and Pu are somewhat different and the importance of this

to the determination of the total number of fissions in natural U0_

fuel requires further examination.

3. EXPERIMENTAL PROCEDURE

The general apparatus and experimental procedure for y-scanning(4)

has been described elsewhere . For the present work a precision scanner144

was fabricated which allowed a precise determination of Ce/Pr activity

at a specific point on a single fuel pin. Tt consists of a fuel carriage

(Figure 2A), mounted integrally with a lead collimator to prevent unwanted

relative movement between the two. The fuel pin is rotated during

measurement to effectively smooth out the asymmetric radial distribution144

of Ce/Pr arising from the asymmetric fission profile, and to provide

an average value of the activity for the whole fuel cross section examined.3

A 30 cm Ge/Li detector is aligned by a detector mount attached directly

to the collimator to minimize errors in repositioning the detector. The

remaining components of the apparatus are shown in Figure 2B.

Repetitive measurements of a single fuel specimen using this

apparatus yielded a precision (2o) of 1.5% for the determination of the144

intensity of the 2.185 MeV y-ray of Pr. The spectra were recorded

on magnetic tape and the photopeak intensities determined using the

computer program GAMAN

The determination of burn-up requires an assay of the144

Ce/Pr activity per unit mass of fuel. For the purposes of the following

discussion it is assumed that the experimental system is calibrated by

destructive analysis of a number of fuels after their y-scanning is

complete, and by repetitive measurement of a selected fuel pin to check

the instrument for long-term stability and reproducibility.

- 7 -

The activity determined in this way requires correction for

decay, during and after the irradiation, to allow the total number of, 14'

atoms of

equation 1.

144atoms of Ce formed during the irradiation to be calculated from

N. = Ac .... 1A -r

144whpre N = number nf at-nms of C.p ppr gr.im. A is the. corrected activity

in disintegrations per minute per gram, and X is the decay constant.

The number of fissions per unit mass of fuel may then be

calculated by equation 2 using the appropriate value of the fission

yield for the fissioning nuclides.

where N = number of fissions per gram and Y is the fractional fissionr

yield.

The number of fissions is then converted to burn-up in

MWh/kg by equation 3.

—?nMWh/kg = Np x E x 4.45 x 10 3

where E is the energy released per fission, in MeV.

The validity of these calculations depends upon the accuracy144

of the assessment of the radioactive decay of Ce during irradiation

and on the number of cerium atoms formed per fission. The latter number

changes during the course of the irradiation as plutonium isotopes are144 235

formed and fissioned. Since the fission yield of Ce from U is about

239a factor of 1.4 greater than the yield from Pu, a quantitative knowledge

is required of the changes in the heavy element and fission product

content of fuel as irradiation proceeds, fo»- typical conditions lor the

reactor fuels of interest. With this infomation available the uncertainties

in the application of equations 1 to 3 can b<5 assessed.

4. ISOTOPIC CHANGES DURING IRRADIATION

o T c 9 ̂flIn principle the changes in the concentration of U, U,

'39 144

Fu and Ce/Pr as irradiation proceeds may be determined by the

destructive analysis of fuel specimens. The quantity of experimental

work required is, however, prohibitive and it is more convenient to

calculate the changes from a knowledge of the thermal neutron flux and

the neutron cross sections of the fissile and fertile isotopes. In

practice, accurate values require detailed reactor lattice-cell computations.

For the present work, the LATREP code was used to determine the

heavy element isotopic composition as a function of burn-up, in Pickering

and 2.4% enriched U0 ? WR-1 fuels of average design power rating. Input

data for the fuel assembly geometry and enrichment, coolant and

moderator temperatures, and neutron flux were taken from design

documents . A second computer program, ISOGEN , which uses neutron

flux, spectrum, and cross-section data derived from LATREP, was used to

calculate the activities of Ce/Pr and other fission product nuclides.

The data produced are presented in the following sections as reference

cases to permit discussion of the steps in the calculation of burn-up.

The absolute accuracy and precision of the LATREP and ISOGEN values

cannot, in general, be given, but Griffiths has reported data which

indicate accuracies of 2 to 3% for the heavy element content of specimens

of NPD* fuel. For the specific cases given here, where defined values are

chosen for the neutron flux, the accuracy is probably about 1%.

* Nuclear Power Demonstration reactor.

- 9 -

4.1. Ce/Pr FORMATION DURING A CONTINUOUS IRRADIATION

144Ce/Pr is formed from fission by:

144 144 144Fission * Xe(9.5 s) -• Cs(l.l s) •• Ba(11.9s) '

144La(41 s) > 144Ce(284.4 d) > 144Pr(17.3 min).

Its growth during a continuous irradiation at constant flux

in 2.4% enriched WR-1 and natural uranium Pickering fuels is shown in

Figures 3, 4 and 4A. For comparison, a number of other fission product

growth curves are also given.

95The plot of Zr activity shows clearly why this nuclide is

not an effective measure of absolute burnup above 100 MWh/kg (U) where

the saturation activity is approached. Above ^140 MWh/kg, the concentration95

of Zr decreases as the fissile material is depleted and the fission

rate decreases.

144For the longer-lived Ce, the saturation activity is not

137reached before ^300 MWh/kg and for Cs the- slope of the growth curve

is positive up to and beyond 300 MWh/kg. Since the fission yields of137 239 235 (21)

Cs from Pu and U fission are almost identical , the growth

curves are almost the same for both WR-1 and Pickering fuels. In contrast,

for Ru the fission yield fror Pu is a factor of 12 higher than for235

U and this results in a much higher rate of production compared to

the other isotopes during the later stages of the fuel irradiation.

The effect is more apparent in natural (Pickering) fuel, where a higher

fraction of the total fissions is due to plutonium, than in enriche 1

fuel (see Section 4.3 and Figures 8 and 9).

- 10 -

144The Ce activity is the second largest of the five isotopes

plotted but the growth curve begins to depart significantly from

linearity (Figure 4A) at about 150 MWh/kg and thus the sensitivity of

Ce as a burn-up monitor decreases above this level, which is about(18)

80% of the design burn-up of Pickering and WR-1 fuel (192 MWh/kg).

4.2 CORRECTION OF A MEASURED 14^Ce/Pr ACTIVITY FOR DECAY

To determine experimentally the total number of fission

product atoms formed during an irradiation, the measured y-activii:y

must be corrected for the decay which has occurred during and after the

irradiation. For decay following irradiation, the correction is simply

applied, when the date of discharge of the fuel from the reactor is

known, from:

A_ = A e .... 4t o

where A is the measured activity at time t and A is the activity at

the end of the irradiation.

Corrections for decay during irradiation are more complex.

In the simplest case where the irradiation is at constant neutron flux

and the concentration of fissioning atoms is essentially constant, the

total number of atoms, N, of a fission product nuclide produced in time t

is given by:

N = N (fi IT Ytr I

- 11 -

and the number of atoms, N 1, present at the end of the irradiation is:

N,, i> oc Y (1 - e~At) 6

N, _ F r f

where N is the number of initial fissile atoms, «f> is the thermal

n

yield of the nuclide.

neutron flux, o is the fission cross section, and Y is the fission

Thus the fraction of the total number of atoms produced which

remain at the end of the irradiation is:

For this simple case the correction is independent of the

parameters of the irradiation such as neutron flux and effective neutron

cross sections. In the practical case of long-irradiated power reactor

fuel, N is not constant, and the equations must allow for the consumption235 238 239

of U and ' U and the formation of Pu and other fissionable isotopes.235 239

To illustrate, when the changes in the two major isotopes U and Pu

only are considered, the equation for N' becomes:N

N1

N

M o f AVN a _ 4>Y2 5 25 u

"25 "25 *„a

°25

a—a,, _<pt

(e Z i

a

(1-e 2 5

- e ~ U +

N28} 4 '

*t N28°28 (

) ( a .2

c°28

V(X-t

°49

J49 '

Y i>P

( O 49 (

[X(l-e+ e - 1)

... 8

- 12 -

where N.1L, and N^_ are the initial numbers of ̂-lGll and U atoms per

unit mass respectively,

f f 23i> 239c and c/g are the fission cross sections of U and Pu

a a 235 239;* and c,_ are the total absorption cross sections of "~ U and Pu

C. 7TU

ana u is the capture cross section of " u. The derivation of this

equation is given in Appendix A.

A rigorous solution of even this simplified equation is not

possible in practice since, for any fuel undergoing measurement, precise

values for the effective cross sections are not available without recourse

to reactor lattice-cell computations specific to that fuel. Approximate

values of — may however be computed using published 2200 ms cross

s<ictions(22). Xo obtain a measure of the degree of approximation incurred,N1

values of — were calculated by three methods. These values are plotted

in Figure 5 for a Pickering fuel of average design rating irradiated at

an average Westcott neutron flux in the fuel of 5.29 x 10 n m s

Curve A was obtained from equation 8 using 2200 ms cross sections;

curve B was derived from equation 8 using 'effective' cross sections

obtained using the LATREP code; and curve C gives the results of complete

calculations using LATREP and ISOGEN only, which incorporate the changes

in all the fissionable nuclide concentrations. The values of the cross

sections used in each case are given in Table 1.

Figure 6 shows the percent deviation of curve A from curve C

in Figure 5. Thus the error relative to LATREP/ISOGEN values in —N

caused by using equation 8 and readily available cross-section data is

-3.4/i for a 467 day irradiation in Pickering, equivalent to a burn-up

of 216 MWh/kg.

- 13 -

TABLE i

NEUTRON CROSS SECTIONS (Barns)

Effective for PickeringNeutrons +

2 3 5u

2 3 8U

239Pu

r0

100

3.98

440

480

Reactor

fn

540

0.24*

950

1240

For 2200

r

98.3

2.72

271.3

368.1

m/s neutrons

f0

580.2

-

741.6

1007.3

TOO

* Only fast neutron fissions occur in U. This cross section has beennormalized to the fuel thermal flux.

+ The effective neutron cross section multiplied by the average Westcottneutron flux gives the total reaction rate. This rate consists ofcotitributions from the thermal, epitViermal, resonance and iast neutronenergy range.

4.2.1 EFFECT OF FLUX LEVEL ON j-J-N

N'In equation 8, TT— is also a function of the neutron flux

and since for any fuel undergoing y-scanning the irradiation flux may

not be known with accuracy, it is necessary to examine the variationN' 144

of rj- with flux. In Figure 7 the numbers of ' Ce atoms produced from235 239

fission in U and Pu in Pickering fuel are given as a function of

neutron flux for a constant irradiation time (467 days) . The range oi

- 14 -

flux values given approximates that for the lowest to the highest rated(18)

elements in the Pickering core ' .

144Also given are the numbers of Ce atoms present at the

end of the irradiation period for each neutron flux. Values of the

fraction of Ce remaining,— , are given in Figure 7A where it isN1

seen that — changes by about 0.035 over the flux range1 7 - 2 - 1 N1

2 to 9 x 10"' n m *" s . A similar range of values of — is found17 _9 N_i

tor WR-1 fuel for a flux range of 0.6 to 3 x 10 n m s .

N'Thus if equation 8 is used to calculate --- by insertion of

17 — 2 - 1a nominal flux of 5 x 10 n m s and 2200 m/s cross sections for

17 -2 -1all Pickering fuels irradiated in the range 2 to 9 x 10 n m s ,

the total error in — (including the error given in the previous section)

will be approximately -3.4 + 3% relative to the 'best' calculatedH

values. For WR-1 fuels the corresponding value is -2.8 * 3%.

4.3 EFFECTS OF CHANGES IN THE FISSILE NUCLIDES AND FISSION YIELDS

The principal fissionable nuclides present in irradiated fuelO T C I?*^Q 0*^0 J/1 "I / /

are U, U, Pu and Pu. The corresponding Ce fission yields

are given in Table 2.

* A negative error in •— leads to a positive error in the derived burn-up.

- 15 -

FissioningNuclide

239Pu

241Pu

2 3 8u

NUCLEAR

Fission Yield

5.39

3.80

A.17

A.55

TABLE

DATA

(%)

2

FOR

(21)

144Ce

Energy/Fission (

200.8

209.3

212.5

204.6

,(24,25)

Since the values differ for each nuclide and the fission

rate of each isotope changes during the irradiation, a method of

determining an overall average yield is required. This can then be

inserted into equation 2.

Figure 8 shows the changes in fissile isotope concentrations

versus burn-up in WR-1 and Pickering fuel and Figures 9 and 10 show the'

fractional contribution of U, Pu and Pu to the total fissions.238

Fast fission in U is essentially constant at 4.5% for WR-1 and r).7%

for Pickering fuel throughout the range of burn-up examined. The

predominant changes occur in the concentrations, and fission fractions235 239 239

of U and Pu. For the enriched WR-1 fuel the Pu cumulative

fission fraction reached 0.165 at 288 MWh/kg while for Pickering fuel

the corresponding value is 0.486 and additionally 4.6% of the total

fissions occurs in Pu.

In principle, the average fission yield required for the

application of equation 2 is defined as Y, where:

144_ No. of atoms of Ce producedY = Total fissions in all isotopes

- 16 -

It can be calculated as a weighted average yield from

7 = F25Y25 + F49Y49 + F41Y41 + F28Y28

235Where Fo is the cumulative fractional contribution of I)

?35 144fissions and \' is the "' U fission yield for Ce. The remaining

^39 ^h\ 238^39 h\ 238subscripts indicate "" 'Pu, ~ Pu and U.

For a given fuel specimen undergoing measurement by yscanning,

the values of F to Foo cannot be determined since they change through-

out the irradiation (Figures 9 and 10). An approximate solution is

possible by selecting values of F?_ to F calculated for a generalized

case at the median of the burn-up range of interest (192 MWh/kg) and

applying this value to all experimental cases. The corresponding Y

is calculated and used throughout the burn-up range. The percentage

error in the derived burn-up value using this approximation is shown in

Figure 11. The largest errors occur where the true burn-up values are

in the lowest part of the range.

4.3.1 ENERGY PER FISSION

Table 2 lists the energy released per fission for each of

the fissionable nuclides. Again, for any given fuel specimen, in the

absence of additional information, a weighted average energy release

per fission must be used. A similar approach may be taken as in the

fission yield approximation. The maximum error involved is ±0.5%.

- 17 -

4.4. OVERALL ERRORS IN BURN-UP DETERMINATION

Combination of the errors quoted in the previous section

leads to burn-up errors of +10% to -5% for Pickering fuels and +8 to -2%

for WR-1 fuels for the case of an uninterrupted radiation. These errors

are relative to the 'best' values which may be calculated from LATREP

in each case and are therefore an indication of the additional uncertainty

arising from the approximate calculation methods described. In particular

cases, the errors may be minimized if the actual operating flux of the

fuel is known from reactor operating data, but for the general case

where the irradiation information is not available these errors apply.

Additional errors must now be considered which are independent

of the calculational procedures adopted. These arise from the

experimental measurements and from the non-linear relationship of144

Ce/Pr activity to burn-up. Figures 4 and 4A illustrate the growth144

of Ce/Pr activity uitli burn-up and Figure 12 shows the number of144

Ce atoms present in a Pickering fuel versus time for a constant17 -2 -1

irradiation of 5.29 x 10 n m s Clearly, for irradiation times144

greater than about 350 days, the sensitivity of Ce as a burn-up

monitor is markedly decreased and thus the uncertainty in the derived144

burn-up caused by a given error in the measurement of Ce increases

with increasing irradiation time. Thus from this cause alona a144

precision of ±5% in the Ce measurement becomes ±7% in the derived

burn-up value at the 96 MWh/kg level and -12.8% to +19.8% at the

240 MWh/kg level. The overall measurement error of ±5% taken here is

an estimate, based on the precision of the components of the y-scanning

measurement which include calibration by independent destructive radio-

chemical analysis, which in turn is dependent on the accuracy of

radiocheraical standard sources and y-detector calibration.

- 18 -

Thus, the combined errors of the ^-scanning measurement

and the subsequent calculation of burn-up for a Pickering fuel

irradiated in an undefined but uninterrupted flux for a period of

vJiOO days would be in the range +30 to -13%. If an accurate value

of the flux were available, chis error would be less by approximately

•3%. Smaller errors O12 to 15Z) apply for fuels of short irradiation

time (<300 davs).

4.5 INTERMITTENT IRRADIATION

For power isactor fuels, a continuous irradiation at

constant neutron flux is the exception rather than the rule. UnderN'

these jonditions, the calculation of — requires an accurate knowledge

of I.he fuel irradiation history to permit computation of the decay144

of Ce during the off-power periods. A simplified expression( 7f\ 1

similar in form to equation 7, has been derived by Koch et al for

the calculation of the fraction — of Cs for an intermittent

irradiation.

-XT -Xt(1 - e ) e

9N m

XZ1=1

where T. is the irradiation time of the period i, and t. is the time

elapsed from period i to the end of the irradiation.

This expression applies to the case of a constant fissile137

isotope concentration and is satisfactory for Cs since the

production of this isotope is independent of the neutron flux level

- 19 -

and the fissioning isotopes. It is not satisfactory for the shorter*144 235

lived Ce because of the difference in fission yields in U and239Pu.

In the previous discussion it was shown that where only235 239

the major fissile isotopes U and Pu are considered together

with the 2200 m/s cross sections, an acceptable calculation of —

is possible for a continuous irradiation if the neutron flux is

approximately known. The intermittent radiation case is similarly

treated below.

4.5.1 DETERMINATION OF ^- FOR AN INTERMITTENT IRRADIATIONN

For this discussion a somewhat simplified Pickering fuel

irradiation history is taken as shown in Figure 13 and Table 3. The

latter gives reactor power levels, on-power periods and shutdowns

and closely approximates an actual irradiation history of Pickering

fuel for the period July 1971 to May 1972. P detailed derivation of

the equations given below is presented in Appendix B.

144The number of Ce atoms present, n , at the end of'the

whole irradiation from the fission of235

consumption of U) is given by

235.U alone (allowing for

= N° a* Y a* E25 25 u 25 . ,

ILit). ,T. .25 j-1 l-l, 25*1 i

-AT.

- e 10

- 20 -

where '.!• is the number of irradiation periods, T. is the length of the

itii irradiation period, t. is the time elapsed from the end of the ith1

pe144

eriod to the end of the mth period and T = 0. The number of Ce0 239239

atomy present at the end of an irradiation from the fission of ~ Pu

alone is

C I

:s JSV IT 1

y:= J

- a c . ,\v i - .

, 1 : ,I ~ I

- A I .1

-- 0

a" liV'"' i

. 1 . - A-'t .

..11

The total number of Ce atoms produced from U fission is

m -J, 5*.T.1 - IT e 12

239and the total number of atoms produced from Pu fission is

c f

a ~>- 1

The fraction of Ce atoms present at the end of an intermittent

irradiation is then

illn + ni

N Nu + N ... 13

- 21 -

TABLE 3

IRRADIATION SCHEDULE FOR A PICKERING FUEL FROM

JULY 1 1971 TO MAY 31 1972

MONTH DAYS POWER LEVEL

July

August

September

October

November

December

January

February

March

April

May

31

125

14

30

12676

30

3325

68413

15311

31010

30

1021

100%

045100

100

1000

10090

100

1000

100

100075100

1000

100

010080100

100

0100

I

- 22 -

TABLE 4

COMPARISON OF N'/N FOR AN INTERMITTENT PICKERING FUELIRRADIATION USING EQUATION 13 AND 2200 m/s

CROSS SECTIONS (o) WITH LATREP/ISOGEN CALCULATED VALUES

(n

' 4k

;' 5

9

FLUX

x l O 1 7

x 1017

.29 x 10 1 7

x l O 1 7

N'/NLATREP/ISOGEN

0.6824

0.6796

0.6774

0.6714

N'/NEQUATION

0

0

0

0

TABLE 5

13A'22OOm/s

.6753

.6682

.6640

.6534

% ERROR

-1.0

-1.7

-2.0

-2.7

ERROR IN N'/N IF A STANDARD FLUX OF 5.29 x 10 1 7 n m"2 s 1 ISUSED IN EQUATION 13 RELATIVE TO LATREP/ISOGEN CALCULATED VALUES

N'/N

(n m-2s-l)

2 x 10 1 7

4 x 10 1 7

5.29 x 10 1 7

9 x 10 1 7

N'/NLATREP/ISOGEN

.6824

.6796

.6774

.6714

17 -2 -1(5.29x10 n m s )

.6640

.6640

.6640

.6640

% ERROR

-6.8

-2.3

-2.0

-1.1

- 23 -

The solution of these equations requires values for each

of the t)>. , T. and t.. As before, to avoid a detailed dependence on

the reactor operating data, the flux is assumed to be proportional to17 -2 -1

the reactor power and the full power flux is taken as 5.29 x 10 n m s

for the average rated fuel. T. and t. values are assumed to be1 1 N1

available from the reactor record. Values of —• were calculated forN

the irradiation scheme given, for fuels in a number of different

positions in the core (i.e. fuels at the ends of the channel, and the

centre bundle in the core corresponding to full-power fluxes of17 -2 -1

2 to 9 x 10 n m s ) using equation 13. They are compared in

Table 4 with values calculated using LATREP/ISOGEN and detailed data

available for the Pickering core. The Table also gives the errorN'

in — relative to the LATREP/ISOGEN values. As in the continuousN

irradiation case the approximate equation underestimates the valuesN1

of — but the largest error recorded in this example is -2.7% and

applies to the case of the highest rated fuel. In Table 5,values of17 -2 -1

the errors are given where the standard flux of 5.29 x 10 n m s

for all the fuels on the assumption that the true irradiation flux is

not known. The maximum relative error is then -6.8%, and applies to

the lowest rated fuel.

N'Using the values of — from Tables 4 and 5 together with

weighted fission yields and an average value of energy released per

fission for a burn-up of 192 MWh/kg, the fuel burn-up was calculated

and the cumulative errors relative to LATREP were obtained. The combined

errors are shown in Figure 14 as a function of burn-up. The maximum

error given is +10.7% and occurs at the lowest burn-up.

144Again errors due to the non-linear relation of Ce

activity to burn-up, and to the experimental measurement, must be

included. For an intermittent irradiation these errors are difficult

to quantify, but they will be at least as large as for the continuous

- 24 -

irradiation case. Thus the estimated combined error in the

experimentally determined burn-up value is 15 to '20%. Again, the

largest component of this error at the Pick?ring design burn-up levels144

arises from the insensitivity of the Ce activity to burn-up at the

long irradiation times.

144The weakness of Ce as a burn-up indicator is further

144illustrated in Figure 15. The growth and decay of Ce is plotted

lor a continuous irradiation to 216 MWh/kg and for a hypothetical

intermittent irradiation to the same burn-up. The difference in the

activities generated; 9% at 216 MWh/kg, and the corresponding

uncertainty in burn-up demonstrates the need for data additional to144

the measurement of the Ce activity and therefore the impracticality

of an independent burn-up assay based on the y-scanning measurement144

of Ce alone. In contrast the relative ir.sensitivity of the137

growth of Cs activity can be seen from curve 3 in Figure 15. In

this case, the half life of the nuclide is long relative to the

irradiation period (467 days) and the activity shows essentially the

same relationship to burn-up in both the irradiation cycles.

5. CONCLUSION

The discussion has shown that burn-up can, in principle,144

be calculated from a determination of the Ce activity present in

irradiated fuel provided that data on the irradiation history and

cooling time are available. The resulting burn-up value may have an

uncertainty of as much as +30 to -20% depending on the conditions of

the irradiation.

In general, no valid calculation of burn-up can be

performed without a previous generalized analysis of the reactor and

fuel system to give data to permit approximate calculations of the

- 25 -

144'rnction of Ce remaining after the irradiation, the weighted

average fission yield and the energy release per fission. Such

preliminary calculations are satisfactorily performed using lattice-

cell parameter codes but their use then compromises the independence

of the yscanning method. The method does not yield results which

meet the criteria of accuracy and precision adopted in Section 1 and

in general does not afford a method of checking the accuracy of the

reactor physics methods of burn-up calculation.

144The major weakness of Ce/Pr as a fission monitor lies

in its relatively short half-life which leads to large decay correct inns

and insensitivity to changes in burn-up. To overcome these difficulties,

a nuclide of longer half-life must be used. Only " Cs and the neutron

capture product Cs (half-life, 2.1 years) are suitable. Recent

experimental work at WNRE indicates that differential migration of

these nuclides in Pickering fuels is much smaller than the migration of

either species alone and measurements of the ratio of the y-activities

of these isotopes appear to offer the highest potential for a

non-destructive burn-up assay method.

6. REFERENCES

1. Rider, B.F., Ruiz, C.P., Peterson Jr., J.P. and Smith, F.R.,"Determination of Neodymium - 148 In Irradiated Uranium andPlutonium As A Measure of Burnup", General Electric Company,GFAP-5354, October, 1967.

2. Rider, B.F., Russell Jr., J.L., Harris, D.W. and Peterson Jr.,J.P., "The Determination of Burnup In MWd/ton", General ElectricCompany, GEAP-3373, 1960.

3. Edwards, R.R., "A Review of Recent Studies of Non-DestructiveAssay Methods For Irradiated Nuclear Fuels", Nucl. Appl., _4_ (4)245-259 (1968).

4. Boase, D.G., Chen, J.D. and Felawka, L.T., "Gamma Spectrometryof Irradiated Reactor Fuels, Experience At the Whiteshell NuclearResearch Establishment", Atomic Fnergy of Canada Limited Report,AECL-3952, 1971.

5. Fudge, A.J., Foster, K. and Murphy, I.., "The Non-DestructiveExamination of Irradiated Nuclear Fuel For Burnup By y-SpeetrometryWith Mechanical Scanning", International Atomic Fnergv Agency,SM 67/50 (1965).

6. Murphy, E.S., Mancia, G. and Christiansen, D.E., "Non-DestructiveAnalysis Of Fuel Irradiated In the EBR By y-Scanning", Battelle-Northwest, BNWL-10n5, 1 % Q .

7. Christiansen, D.E. and Murphy, E.S., "Determination of RelativeBurnup By y-Scanning EBWR Fuel Rods", Battelle-Northwest, BNWL-653,1968.

8. Robertson, J.A.L., Internal Memorandum to P.J. Dyne, 12 March, 1973.

9. Forsyth, R.S. and Blackadder, W.H., "The Non-Destructive DeterminationOf Burnup By Means Of The lkl*?v 2.18 MeV Gamma Activity", InternationalAtomic Energy Agency, ST1/PUB/105 Page 399, 1965.

10. Forsyth, R.S. and Blackadder, W.H., "Use of the Fission ProductRu-106 Gamma Activity As A Method For Estimating The RelativeNumber of Fission Events In 2 3 5U and 239Pu In Low Enriched FuelElements", International Atomic Energy Agency, SM 133/4, 1970.

11. Forsyth, R.S., Blackadder, W.H. and Ronquist, M., "Burnup Determina-tion By High Resolution Gamma Spectrometry: Fission Product MigrationStudies", Aktiebolaget Atomen^rgi, AE-272, 1967.

12. Rasmussen, N.C., Sovka, J.A. and Mayman, S.A., "The N"n-DestructiveMeasurement of Burnup by y-Ray Spectroscopy", International AtomicEnergy Agency, SM 67/45, 1965.

- 27 -

13. Hick, 11. and hammer, M., "Interpretation of y-Spectrometric Measure-ments On Burnt Fuel Elements", International Atomic Energy Agency,SM 133/5, 1970.

14. Oden, D.R, and Christiansen, D.F., "Application of y-Ray SpectrometryAs A Supplementary Mist Technique", Battelle-Northwest, BNWL-SA-4059,1971.

15. Heath, R.I.., "The Potential of High Resolution Gamma Ray SpectrometryFor the Assay of Irradiated Reactor Fuel", Atomic F.nergy Commission,WASH-1076 Page 115,

16. Felawka, L.T., Molnar, J.G., Chen, J.D. and Boase, D.C., "CAMAN - AComputer Program For The Qualitative And Quantitative Fvaluation OfGe (Li) Gamma-Ray Spectra", Atomic Fnergy of Canada Limited Report,AECL-4217, 1973.

17. Gibson, I.H., "The Physics Of LATRPP", Atomic Energy of CanadaLimited Report, AECL-2548, 1966.

IS. Whiteshell Reactor Ho. 1 Design Manual, Volume 7, Section T70.Pickering Generating Design Manual, Volume 5, Section 3700.

19. Van Tuyl, II.H., "ISOGEN - A Computer Code For Radioisotope GenerationCalculation", General Electric Company, IIW-83785, 1964.

20. Griffiths, J., "The Effectiveness of LATRF.P Calculations: A Surveyand Detailed Comparison With Experiment", Atomic Energy of CanadaLimited Report, AECL-3739, 1971.

21. Walker, W.H., "Status of Fission Product Yield Data For ThermalReactors", Atomic Energy of Canada Limited Report, AECI-4704, 1974.

22. Hanna, G.C., Westcott, C.H., Lemmel, H.D., Leonard Jr., B.R., Story,J.S. and Attru, P.M., "Revision Of Values For The 2200 m/s NeutronConstants For Four Fissile Nuclides", Atomic F.nergy of CanadaLimited Report, AECL-3436, 1969.

23. Westcott, C.H., Walker, W.ll., and Alexander T.K., "EffectiveCross-Sections and Cadmium Ratios For The Neutron Spectra ofThermal Reactors", Atomic Energy of Canada Limited Report, AECI.-612,1958.

24. Walker, W.H., "Mass Balance Estimates of The Energy Released PerFission In A Reactor", Atomic Energy of Canada Limited Report,AECL-3109, 1968.

25. Phillips, G.J. and Griffith, J., "LATREP Users Guide", Atomic Erior/vof Canada Limited Report, AECL-3857, 1971.

Ariemira, A., Bramati, I.., Calliani, M. , Cu;ilard:i, f-M'., Zaffiro, B.,Cricchio, A. and Koch, L., "Experimental and Theoretical Determina-tion of Burnup and Heavy Isotope Content In A I'uel Assembly IrradiatedIn The Carip.liano Doilinf Water Reactor", European Atonic EnergyCommunity, EUR 4(538, 1971.

12

10

J_

10 20 30

' f 'NO ALOT, THE TUEL (cm)

4 0 5 0

FIGURE 1: A TYPICAL AXIAL BURN-UP DISTRIBUTION IN A NATUPAL U0 2FUEL ROD SHOWING AN AXIAL FLUX GRADIENI ANLcENL'-PEAKING

LATA OBTAI ' iFT BY • SCANMNf, OF THE 0 . 7 ? 4 NeV • ~>r a 5 7 r

- It) -

Ge(Li) DETECTORCOLLIMATOR BLOCK ' FUEL ELEMENT

FUEL CARRIAGE

FIGURE 2A: /-SCANNING APPARATUS

DETECTOR HIGH VOLTAGE

PRE-AMPLIFIER

AMPLIFIER

MULTICHANNELANALYZER

MAGNETIC TAPE TYPEWRITER

FIGURE 2B: f- SPECTROMETRY SYSTEM

- 31 -

96 144 192 240

BURN-UP (MWh/kg U)

FIGURE 3 ; FISSION PRODUCT ACTIVIT IES IN WR-1 FUEL vsCONSTANT NEUTRON FLUX, 0 - i .e x io 1 7 n m-

286

-UP AT A

r 10 -

240

FIGURE

96 144 192BURN-UP (MWh/kg U)

FISSION PRODUCT ACTIVITIES IN PICKERING FUEL vsBURN-UP AT A CONSTANT NEUTRON FU:X, 0 = 5.29 x lo" n

288

- i'i -

CD00CVJ

cu

oCM

toCM

CVJ

00

(n 6 > | / i 3 )

N' (FRACTION OF 144Ce ATOMS REMAINING)

p oCJI

"T

5o

o

Mloho

°lo

o

o

- :35 -

33N3M3JJIQ

16

14

~ 10

TOTAL H X e PRODUCED

144TOTAL Ce PRESENT

Ce ATOMS PRESENT FROM 235U 239Pu

NEUTRON FLUX (n m"2 s"1 x 1O17;

FIGURE 7: l¥lCe ATOMS/kg U PRESENT M D DROPUCED vs NEUTRON FLUXFOR A tt7 DAY PICKERING FUEL IRRADIATION

oI—

CD

8

NEUTRON FLUX (n m"2 s"' x 101' )

"FIGURE 7A: FRACTION OF ̂ X e ATOMS REMAINING AFTER A 467 DAY PICKERINGFUEL IRRADIATION AS A FUNCTION OF NEUTRON FLUX (CALCULATEDFROM EQUATION 8 USING 2200 a/s CROSS SECTIONS)

18

16 -

14

-- 12

10

239Pu

96 144 192

BURN-UP (MWh/kg U)

240 288

FIGURE 8: CHANGES IN ISOTOPIC CONTENT OF PICKERING AND WR-1FUEL vs BURN UP

- 39 -

600

6r-6

<o6

Nouav

in

6iJ NOISSIJ

dro6 b

2£ QXA. 3K

z: ca

" 5cc

0 3

MOUDtfMJ NOISJId

PICKERING

96 120 144 168 192 216

BURN-UP (MWh/kg U)

2 4 0 2 6 4 288

FIGURE 11: ERROR IN BURN-UP USING WEIGHTED FISSION YIELD FROM192 MWh/kg u FOR BURN-UPS FROM 96 TO 288 MWhAg U

I -

100 £ 0 0 3 0 0 4 0 0 5 0 0 6 0 0 700 8 00 9 0 0 1000 1100

TIME (DAYS)

FIGURE 12: THE NUMBER OF ATOMS/Mg U vs TIME FOR A PICKERINGELEMENT AT CONSTANT NEUTRON FLUX (0 .29 x i o 1

I R R A D I A T I O N T I M E

LU

OQ.

FIGURE 13: INTERMITTENT FUEL IRRADIATION

O

UJ

I—•ZL

LLJCJo:UJ

4 -

2 -

72 96 120 144 168

BURN-UP (MWh/kg U)

192 216 240

FIGURE ERROR IN CALCULATING BURN-UP FOR AN INTERMITTENT PICKERINGFUEL IRRADIATION RELATIVE TO LATREP CALCULATED BURN-UP

4 8 _

4.2 -

3 6

3.0

2.4

1-8

0 = 5.29 x 1 0 "

J_

CONTINUOUSI

INTERMITTENT

= 5 . 2 9 x T O 1 7

Cs CONTINUOUS ANDINTERMITTENT

72 96 120BURN-UP (HWh/kg U)

144 168 192 216

FIGURE 15: PRODUCTION OF 14LlCe S 1 3 7Cs IN PICKERING FUEL FOR ACONTINUOUS AND AN INTERMITTENT IRRADIATION

- 46 -

APPENDIX A

CALCULATION OF THE FRACTION OF 1 4 4Ce ATOMS REMAINING AT THE END OF ACONTINUOUS IRRADIATION FROM THE FISSILE ISOTOPES 2 3 5U AND 239Pu

144The fraction of Ce atoms remaining at the end of an

Irradiation is the totaL number finally present, divided by the total

number formed.

A.I PRODUCTION FROM 235U FISSION

144A.I.I Ce ATOMS PRESENT

144 ?35The rate of accumulation of Ce atoms from u fission

ir» given by:

dn

-dT " N25J25 + Yu " V

144 235where n = number or Ce atoms from the fission of U

235N? = number of U atoms

f c. . . 235a = fission cross section of U

<j> = neutron flux

235 144Y = JJU fission yield for Ceu J

t = irradiation time

A = decay constant of Ce

- 47 -

235To solve equation Al the number of U atoms as a function

of time is required. This is represented by

dN25 N a* <p (A2)

dt 25 "25

r. 35where o is the absorption cross section of " U. The solution toequation A2 is

- aa 4, t

N25 = N25 e 2 5 ( A 3 )

where N is the initial number of U atoms at t = 0. Substituting144

A3 into equation Al and solving gives the number of Ce atoms, n (t),235 u

present after an irradiation of t seconds from U.

, . N° o -t Y -o^ ttn (t) = 25 25 u , 25 -At, . .u (e - e ) (A4)

(X -4 •)

144A.1.2 Ce ATOMS PRODUCED

The total number of atoms produced is given by equation Al144

without the decay term n X. The rate of accumulation of Ce atoms

is then

dNu o -°25*t fd~r= N25 e °25

144The to ta l number ot Ce atoms produced a l t e r an

i r r a d i a t i o n time of t seconds i s the re fore

N° - f Y - a htN ( t ) = ~ ^ _ r 2 -^ (1 - e - 3 ) (A5)

u a°25

A. 2 PRODUCTION FROM 239Pu FISSION

A. 2.1 U 4 C e ATOMS PRESENT

144 219The rate of accumulation of Ce from ' Fu is given by

= N49 it' Yp - % A (A6)

144 9'39where n = number of Ce atoms produced from ~ Pu

239N,^ = number of Pu atoms49

:/c, = fission cross section for Pu

144 239Y = fission yield of Ce from Pu

239 238Pu is produced from U by:

2 3 8u

I "•' - J ^ L 9 4 0 N p

239Pu

- 49 -

239 ?40Tin-1 fraction of Np converted to " Np is n

T 19N

q 19small ( 0.1%) and is ignored. The half-lives of " U and Np are

239also negligi'^y small and thus the rate of accumulation of Pu may

be written

d N,-C ... - N -a >

28 28 ' 49 49'

c 2 3 8 awhere o i s t h e c a p t u r e c r o s s s e c t i o n of " U and • / 0 i s t h e a b s o r p t i o n

z ° 239c r o s s s e c t i o n of Pu. S o l v i n g t h i s e q u a t i o n g i v e s t h e number of239

Pu atoms as a function of time

,It - •'!,,- -t

49 - e Kc ^Sl

49

•1IP:V K r o p r e n o n t n t h r i n i t i n l muiber o\239

, I ( C - •-, ;>nd K - n nt t -1 44

Substituting A7 into A6 and solving gives the number ul" Ce atoms ,is

a function of time.

n (t)P

Yp

0% X(X-a;g •)X(l-e a

49I - c

-At(AS |

A.2.2l.'.A

Ce ATOMS PRODUCED

Equation A6 without the decay term gives the rate of144 239

accumulation of Ce atoms produced from Pu

- "50

Jti i I 1 - e

144Solving t h i s equat ion gives the number of Ce atoms produced as a

function of time

N ( t ) =

N_\S JS '4+ e

49- 1J (A91

144 \'The fraction of Ce atoms present, '—- , at the end of an irradiation ol t

seconds is thus obtained from equations A4, A5, A8 and \'J:

it)(A10)

•N25 25Q u (e "J - X(l-e G49*-At

(All)

M0 f -2s 25 u

(1 -

a 'I

- 51 -

APPENDIX B

CALCULATION OF THE FRACTION OF IH4Ce ATOMS PRESENT AT THE END OF ANINTERMITTENT IRRADIATION FOR THE FISSILE ISOTOPES 2 3 5U AND 239Pu

144The derivation of the fraction of Ce atoms present at

the end of an intermittent fuel irradiation is similar to the continuous

irradiation case described in Appendix A. Each irradiation period

shown in the simplified irradiation scheme in Figure 13 is considered

individually.

B.I PRODUCTION FROM 2 3 5U FISSION

144B.I.I Ce ATOMS PRESENT

The number of a toms p r e s e n t a f t e r t he f i r s t i r r a d i a t i o n

p e r i o d (T ) i s g i v e n by e q u a t i o n A4 of Appendix. A.

fA-l

For convenience le t

- 4°25 Yu

144The number of t h e s e Ce atoms r e m a i n i n g a t t h e end o f t h e f i n a l i r r a d i a t i o n

is

V l . - ° 2 5 ^ 1 iu ^ P = ^e - e ) e

a

- 52 -

whore t is t !'•: time from the end of T to the end oi the last

irradiation. The number of atoms present at the end oi the

irradiation as a result of the first two neutron exposure periods is

1 I + I ,) - n1

(e''"• 5 XT

where the term e'•>5̂ 1 ^ 1 235" accounts for the depletion of U during

the first period. Additional irradiation periods are treated in a

similar fashion. Thus after m periods, the number of

235present from the tission ot U is given by

I 44

a !i: -%c<t>.T. -AT.'25 1 l _ i.- c

l X " i 5 * i '

Ce atoms

At .1

(B3J

M.2 Ce ATOMS PRODUCED

The number of atoms produced for any irradiation period

T. was previously derived in Appendix A.

u \ - e

25

(AS)

- 53 -

144 235The total number of Ce atoms produced from U depends upon the

flux, |;. , and the length of irradiation, T., and is not affected by1 1 144

reactor shutdowns. Therefore the total number of Ce atoms producedis

'25

B.2 PRODUCTION FROM 2 3 9 P u FISSION

5.2.1 14ACe ATOMS PRESENT

144 r139The r a t e of accumulation of Ce from ~ Pu during any

i r r a d i a t i o n period T. i s given by

dn f

where N.q i s given by equation A7

M1Q = N28 a28 n "°49 * i T i , , ~°4<AT i49 (1 - c ) + K.e

3 1a49

?39where K. is the number of *" Pu atoms present at th<.- be^innint;irradiation time T..

I

- 54 -

For the first irradiation period, K1 in equation A7 is239

equal to zero. Therefore the number of Pu atoms present after

Tl is

-74O V I IBC>)

•49

144Substituting B6 into B5 and solving yields the number of Ce atoms

239present at the end of the first period arising from fission of Pu.

a .

Rearrangement of tu^ terns anil rho aH-.lition of n dccav tern for t,

gives

r -A!

>• - °49 *,

'. B7 )

N28°28°49 \where C. and t is the time from the end of T..

to the end of the final irradiation.

- 55 -

For the second irradiation period, the initial number of

239,Pu atoms, K , in equation A7 is equal to N,q(T ) given by equation B6.

239The concentration of ' Pu atoms present at the end of T , N,- (T ), is

then given by

'49

(1 - e ) cN a

+ 28 28 (1

J49

(B8)

239Pu from T in T and thewhere the first term denotes the depletion of

239Pu atoms formed during TQ. Substituting

i / /N,_(T0) into equation B5 and solving for n' (T ) yields the number of Ce

second term is the number of

49' Vatoms present at the end of period

irradiation period.

F 239resulting from Pu fissions in this

C2*2Co47 (1 - "°49*1T- C +

2 2

-AT

r ,149'2

I - (B9)

144The number of Ce atoms present at the end of the last irradiation

period is then

n (TJp 2

-XT-(e - e

-Xt.

RIOX -

Av.Kiiiion.il irradiation periods are treated in a similar manner to144

obtain the ,i ̂ tribution trom each period to the total number of Co

atoms present at the end of the whole irradiation. For m irradiation

periods, the total number of ^ C e atoms present from the fission of

~- Pu is

n = i. , i . 4 9 r i i -AT. 1 -''t.

HI 1

uhciv T - 0

B.2.2144

Ce ATOMS PRODUCED

144The number of Ce atoms produced during any irradiation

period T. is iviven bv equation A9

N (T. i

P r

r A r4l> I i

49

- 1. (A9)

For n i r r a d i a t i o n period? the t o t a l number of Ce-144 atoms produced from

2 3 9Pu f i s s ions i s

C, f m

a49 L

( a , , , $ . T . )m - o . qC>. T .

(B12)

The fraction of 141*Ce atoms present , : - , at the end of an intermit tentN

i r r a d i a t i o n i s obta ined from equa t ions B3, B4, Bl1 and B12.

N

n + n_LI £

N + Nu p

(B13)

The International Standard Serial Number

ISSN Q067-Q367

has been assigned to this series of reports.

To identify individual documents in the serieswe have assigned an AECL- number.

Please refer to the AECL- number whenrequesting additional copies of this document

from

Scientific Document Distribution OfficeAtomic Energy of Canada Limited

Chalk River, Ontario, Canada

KOJ 1J0

Price - S4.00 per copy

2515-75