On the Calculation of the Fast Fission Factor

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AE-27

On the Calculation of the w < Fast Fission Factor

B. Almgren

AKTIEBOLAGET ATOMENERGI

STOCKHOLM • SWEDEN • I960

AE-27

ON THE CALCULATION OF THE FAST FISSION FACTOR

E. Almgren

Summary:

Definitions of the fast fission factor e ars discussed. Different methods of calculation of e are compared. Group constants for one -, two- and three- group calculations have been evaluated using the best obtainable basic data. The effects of back-scattering, coupling and (n, 2n)-reactions are discussed.

Completion of manuscript in June 1960 Printed and distributed in November 1960

LIST OF CONTENTS

Page

1. Definitions 3 2. Formulas for the fast fission factor e and the fast fission ratio R 3 3. Calculation of group constants 7 4. Collision probabilities 10 5. Numerical results 12 6. Discussion 12 7. Acknowledgements 13 Preferences 14

On the calculation of the fast fission factor.

1. Definitions

The fast fission factor e may be defined in different ways. Carlvik and Pershagen (3) have defined e as "the number of neutrons which are either slowed down below 0, 1 MeV in the fuel or leave the fuel, per primary neutron from thermal fission". This definition has been used in Sweden, since it was proposed (1956). Another commonly used definition is due to Spinrad (2) "the number of neutrons making first collision with moderator per neutron arising from thermal fission".

The choice of definition must be consistent with the definition of the resonance escape probability. The above-mentioned definitions include in e some of the capture below the fast fission threshold. However, they do not include the effects of (n, 2n)-reactions or capture of high energy neutrons in the moderator. A definition which includes these effects is the following "the number of neutrons that are slowed down past a given energy per primary fission neutron". In this case € could be called the fast multiplication factor. Different energies can be chosen as boundary energy. However, the capture below the fission threshold could, be included in the resonance escape probability. If so the definition of e is identical with that of Glas stone-Edlund (1) p. 276.

2. Formulas for the fast fission factor e and the fast fission ratio R

The following general expressions for R and s which include moderator effects above the boundary energy can be used for different definitions of e and different number of groups.

e = no + > N. a.n + ) M. b.A 0 ' iL x i0 L'-~ !•i0 i= 1 i= 1

(2) N. = n. P. + P. > a:. N. + (1 - Q.) ) b.. M. (3)

M. = n. (1 - P.) + (1 - P.) > a.. N. + Q. > b.. M. (4)

where

n. is tke probability for a fission neutron to be born in group j ' ">

N. and M. are the numbers of collisions per unit time in group i in the fuel element and the moderator respectively, if the number of ne-atrons per ".nit time emitted from thermal is taken to be one

a., and b.. are the numbers of neutrons that are transferred to group j altar a collision in group i in the fuel element and the moderator respectively • •

P. is the probability for a fission neutron that has not collided and belonging to group j to make a. collision in the fuel

P. is the probability for a neutron having collided in the fuel and belonging to grou.p j to make its next collision in the fuel

Q. is the probability for a neutron having collided in the moderator and belonging to group j to make its next collision in the moderator

v is the number of fission neutrons pex* thermal fission

cr is the fission cross-section in group j

«r. is the total cross-section in group j

The following assximptions are made, when the above formulas are derived. 1) The form of the fission spectrum is the same for fast and thermal fission

2) The fuel element and the moderator can be treated as homogeneous medias.

Additional assumptions are needed for the calculation of the group constants.

Usually moderator effects such as back-scattering and (n, 2h)~reactions are neglected. This can be done here by putting Q. = 1 andS b.. = 1, J J Then the formulas are identical with Carlvik-Pershagen's (3), if the neutron groups are:

Group 0 Neutrons with energies below 0, 1 MeV 11 1 " " " between 0, 1 - 15 49 MeV " 2 " " " above 1, 49 MeV

They are identical with Glasstone-Edlund's formula (1), if the neutron groups are:

Group 0 Neutrons .with energies below the fast fission threshold " 2 Neutrons with energies above the fast fission threshold

They are identical with Spinrad's formula (2), if n« = 0 arid a._ = 0 and the neutron groups are:

Group 1 Fission neutrons with energies above the fast fission threshold " 2, Fission neutrons with energies below the fast fission threshold " 3 Neutrons which have been scattered .out of group 1

The general multigroup formulas are difficult to handle without fast computers. Simplified formulas for one group are therefore given below. The group is called group 2 to get the nomenclature to agree with that of the group constants later in this report. Glass tone -Edlund's definition of e is -ased. 6.

P2(l - 6 = nO + n2 a20 (1 - a22 P2) (1 - b,, Q2) - a,, b22 (P2 - 1) (Q2 - 1)

(5) 1 P t ~ 2> - a22 P2 n2 b20 (1 - a?? P ) (1 - h?? Q ) - a b (P - 1) {Q - l)

Q2) - b22 (1 - P2) (QQ2 -

- b 22 Q 22

If we put P? = P_ and use Q ~ 1 and that the neutron multiplication in

the moderator has a small importance (b77 + b_n), we get

nZ P2 (a22 + a20 b22 + b?,0

1 " a22 P2 a22 P2 1 ~ b22

b (1 - P ) (1 - Q ) .

P b (1 - P) (1 - Q-) (P + 1 - a P) R = n

The first term in equation 7 comes from fast fission in the fuel element and it is the usual term found for instance by Glasctone-Sdlund (1). The second term comes from (n, 2n)-reactions and capture in the moderator. The third term represents the effect of the neutrons that are scattered back into the fuel elements after collisions in the moderator. 3. Calculation of group constants

Formulas for a., and b..

The transfer coefficients a., and b.. can be expressed in the following way

cr.. + n. v. o- + 2 m.. cr. (n, 2n) a.. -

where the cross-sections of the fuel element are used

cr.. + 2 m.. cr. (n, 2n) b..=-J yJ (10) where the cross-sections of the moderator are used.

Nomenclatur e

ij is the average transfer cross-section for a ne^^tron colliding in

group i to be transferred to group j cr. is the average capture cross-section for group i

ar.f _ » " " " (n, 2n) " " " " i i(n, 2n) v '

S. " " " 'macroscopic cross-section for group i v. " " average number of fission neutrons produced by a fission in group i m.. is the probability for a neutron from a (n, 2n)-reaktion in group i to be emitted in group j 8.

Neutron spectrum Unfortunately the energy spectrum of fast neutrons in a reactor is badly known. For high energies the fission spectrum is a rather good approximation, but for energies below 1 MeV it is certainly a bad one. It is, however, assumed that the fission spectrum can be used when the average cross-sections are calculated. The following fission spectrum given by (4) has been used:

N(E) = 0, 45270 exp ( - Q ^5) sinh J 2, 29E (11)

Number of fission neLxtrons per fast fission

According to (17) vZ38 (£) = 2, 65 + 0, 141 (E-l, 5) (12)

The average over the fissions spectrum for energies above 1, 5 MeV is v = 2, 86. This may be compared with the value recommended in (19) 2, 80 + 0, 05 if vf'35 = 2, 43.

Calculation of the transfer cross-sections The cross-sections have been taken from references 5, 6, 7, 8, 9 and 18. The cross-sections of the different references agree quite well. We have looked for newer cross-section data in Nuclear Science Abstracts 1958 and 1959, but none of importance for this work has been found. When calculating the transfer probabilities of elastically scattered neutrons, anisotropic scattering has been considered for the light elements D, C and O. The transfer function then is • •

) (2L + 1) fL (E) P 1 -± (1 -£-) (13)

where fT (IC) are the coefficients of the expansion

/._,% CO f E F cos e 14 cr(E, cos 6) = ^ \ (2 L + 1) L( ) Lj ( ) ( ) 9.

and P, are Legendre polynomials.

For inelastic scattering the transfer function is

V f.n(E5E-) = | E' e (15)

has been used for U, Zr, Al and at high energies for C. The constants a have been taken from ref. (10) and (11).

The transfer cross-sections are then of the type

fN(E){

The cross-sections have been calculated for the following energy groups: -

Group 1 Neutrons with energies between 0, 03 and 1, 5 MeV 11 2 " " - . .» M 1} 5 ii ,6j 5 Mey- "3' " " " above 6, 5 MeV • " - 2 " "• ' '' " • " -.1, 5 MeV :" •.

The small contribution to the fast-fission cross:-section for energies below 1, 5 MeV has been included in the cross-sections cr andcr^. and cr . has been considered to be 0.

The calculations of the cross-sections have been performed on AE's Mercury computer. The calculated values are found in table 2. These cross-sections cannot be used in Spinrad' s formulation as his definition of the energy groups is quite different from the one used here. 10.

4. Collision probabilities

Collision probabilities in a fuel element

In order to calculate the collision probabilities in a cylindrical fuel element it has been assumed that the source density of the uncollided fission 2 neutrons can be approximated by a function of the type a + f3 r and that the source density of the neutrons, which have collided, can be considered to be constant.

Coupling effect The collision probabilities for a lattice is somewhat greater than the collision probabilities for a fuel element owing to the possibility for a neutron emitted in a fuel element to get to another fuel element without colliding in the moderator.

This effect has sometimes practical importance. The effect is studied here for hexagonal lattices.

We consider a hexagonal lattice consisting of cylindrical fuel elements •with radius a. The distance between two neighbouring fuel elements is d. If d » a and the mean free path in the fuel » a, the formula of Carlvik- Pershagen (3) is valid. We then get the additional collision probability

AP = 6 f (d, a) + 6f (d

+ 12f (d NT13, a) + • (17)

where 2 a sf. f (x. A) x) (18) 2x 1 . mi

00 -y cos hudu Ki, (19) i (y) = I cos hu 0 1.1.

J,. = the total macroscopic cross-section of the fuel

' = " " " ' " - " " " moderator mi

The errors in e depending on approximations in the above formula are less than 0, 5 % if — > 5 for uranium ?.nd uranium oxide in heavy water. a. .

The collision densities in. the fuel then, are

(1 - P'.) A P. P.'= P". + — Si -- (20) 1 C1 1 - (1 - P'.) AP. x er i

(1 - P .) AP. P. = P . +'— ~ i 2 (21) 1 C1 • 1 - (1 - P .) A P. • x ci' i v/here P'. is the collision probability in a fuel element for uncollided fission neutrons, and'P . is the collision probability in a fuel element for neutrons •which have collided in the fuel.

Collision probabilities in the moderator

A detailed calculation of the collision probability in the moderator Q. should include space-variations of the neutron flux and anisotropy and is therefore very difficult to perform. A rough approximation can be obtained, if the neutron flux in the moderator is assumed to be constant. This is a rather good approximation, if the distance between the fuel elements is less than the mo an.'free path in the moderator. • '

Then

(1 - Q.) S. V = (1 - P.) S V, (22) v r im m s x fi f

See for instance ref. (21).

V. is the moderator volume and Y-. is .the fuel volume, m f 12.

5. Numerical resulta

e for single cylinders of uranium and uranium oxide has been calculated in different ways. One, two and three groups have been used. The effect of varying the values of cr. and P has been studied, (n, 2n)-reactions in the fuel element are included. It has been assumed that the neutrons after (n, 2n)- reactions have low energies f i. e.m... = 1 Q -f • V n i • Curves of e and R as a function of radius are found in figures 1, 2 and 3. For comparison curves of e and ?. according to Carlvik-Pershagen (3) and Spinrad (2, 15) have been drawn. Experimental values of R have also been plotted in figure 3 and can be compared with the calculated values.

The effect of (n, 2n)-rsactions in the moderator has been estimated by calculating the second term of equation 7. The same assumption as above for the energies of the neutron after (n, 2n)-reactions^has been used. Curves have been drawn in figure 4 for uranium and uranium-oxide cylinders in heavy water.

Back-scattering to a fuel element of neutrons which have collided in the moderator has been estimated by calculating the third term of equation 7. Equation 22 has been used to evaluate Q..

6. Discussion

From the curves it is seen that the use of a special energy group for neutrons with energies above 6, G has a small influence on e . This is not surprising as there are few fission neutron above 6, 5 MeV. 6, 5 MeV was 238 chosen as it is the energy of the second fission threshold of U .A lower boundary energy might give a greater difference. It is also seen that the assumptions for the source density have a small influence except for large radii.

The capture below the fission threshold is more important. Curves are found for uranium for cr =0, for cr = 0, 13 b ( which is the average over the fission spectrum) and for cr, = 0, 30 b (which is a high but not un- realistic estimate of the cross-section for the actual spectrum). The uncer- 13.

tainty depends on the bad knowledge of the neutron spectrum for low energies and on the large energy interval used. This capture could alternatively be included in the calculation of the resonance escape probability and in which case the formulas for e could be simplified,

Figure 4 shows a good agreement between, experimental and calculated values of R. Unfortunately R is weakly dependent on the method used to calculate R and the experimental values cannot be used in order to select a method for calculating R and e .

The estimate of the effect of the (n, 2n)-reactions in the moderator shows that this effect may be important. Becatise the cross-sections are very uncertain the curves should be used with caution. The cross-sections cannot even be regarded as upper and lower bounds.

It is found that 'he effect of neutrons that are scattered back into the fuel elements usually is small but it may be of importance for certain lattices. The effect of heterogeneity of the fuel elements has not been considered in this work. A method for calculating this effect is given by A Jons- son (?0).

The accuracy of the methods for calculating e and R can be improved by a better knowledge cf the space and energy variations ox the neutron density in the fuel element and in the moderator. Ivlore accurate values of v for U ° and the (n, 2n)-cross-sections of deuterium are also desirable.

7. Acknowledgements

The calculations on the Ferranti Mercury computer have been performed by C. Johansson, K. Nyman and L. Persson. 14.

References

1) GLASSTONE - EDLUND The Elements of Nuclear Reactor Theory, 1952

2) SPINRAD Nuclear Science and Engineering, 1, 455, 1956

3) CARLVIK and PERSHAGEN The fast fission effect in a cylindrical fuel element AE-21 (AEF-70)

4) Reactor Physics Constants ANL-5800

5) Neutron cross-sections BNL-325, Second edition, 1958

6) Semiempirical neutron cross-sections UCRL-5351

7) An interim report on the neutron cross-sections of oxygen NEA C86-2

8) Fact Neutron Data for Carbon NDA 12-12

9) HÖGBERG T Private communication

10) Some neutron cross-sections for multigroup calculations APZX-467

11) MANDE VILLE and KAVANAGH^ The Scattering of Neutrons by U CWR-4028

12) WEINBERG A M Lectures, Ch. 24, 1953

13) CHERNICK J Proceedings of the Geneva Conference 5, 222, 1956

14) CRITOPH E Private communication to Pershagen.

15) DZSSAUER G Proceedings of the second Geneva Conference 12, 328, 1958

16) HONE D W Proceedings of the second Geneva Conference 12, 358, 1958

17) BONDAPvENKO et. al. I I Proceedings of the second Geneva Conference 15, 355, 18) F7.ANK and GAMMZL Phys. Rev. 93, 463, 1954

19) SHER A and LZROY J ... The value ef v for fission spectrum induced and spontaneous fission of U^3C Report from Centre d'Etudes Nucleaires dc Saclay

20) JONCSON A Heterogenous calculations of e

21) ROTHENSTEIN W Collision probabilities and resonance integrals for lattices BNL-563 16.

Fission spectrum and number of fission neutrons per fast.'fission

n. V, i i Group 0' 0 - 0, 03 MeV 0, 0023 -' 1 0, 03 - 1, £ MeV 0, 469V - 2" i, 5 - 6, 5 MeV 0, 5113 2, 85 3' 6, 5 - co MeV .0, 0167

0 0 - 1, 5 MeV 0, 4720 2 1, 5 - oo MeV 0, 5280 2, 86 17. TABLE 2

Cross- sections

D C O Al Zr U D2O UO2 2,99 3, 00 4,09 3, 62 7, 05 7,44 10, 07 15,62 0, 03 - 0, 06 0, 01 0, 00 0, 01 0, 00 0, 02 0, 12 0, 02 1, 5 MeV °"lc 0, 00 o, oo 0, 00 0, 00 0, 00 0, 13 0, 00 0, 13 °"l 3, 05 3, 01 4,09 3, 63 7, 05 7,59 10, 19 15, 77 °V 0, 89 1, 52 1,66 2, 08 2,91 4,59 3,44 7,91 1, 32 0,28 0, 28 0, 50 1, 23 2, 34 2,92 2,90 1, 5-6, 5 *2l 0, 00 0, 00 0, 00 0, 00 0, 00 0, 01 0, 00 0, 01 MeV ^20 0, 00 0, 00 0, 01 0, 00 0, 03 0, 01 0, 05 'Zc 0, 00 0, 00 0, 00 0, 00 0, 00 0, 00 0, 57 0, 00 0, 57 /2f or^n, 2n) 0, 01* 0, 00 0, 00 0, 00 0, 00 0, 00 0, 02 0, 00

°"2 2, 22 1, 80 1,95 2, 58 4, 14 7, 54 6,39 11,44 0, 27 1,22 0, 62 0, 65 2, 25 3, 32 °-33 1, 16 4, 56 0, 57 2, 14 0, 51 0,69 0, 53 0,41 °-32 1, 65 1,43 6, 5- oo 0, 17 0, 20 0, 06 0, 52 0,95 °-31 1,43 0,40 1, 07 MeV 0, 00 0, 00 0, 00 0, 00 °"30 0, 00 0, 00 0, 00 0, 00 vic 0, 00 0, 00 0, 24 0, 08 0, 00 0, 00 0, 24 0,48 0, 00 0,97 /3f 0, 00 0, 00 0, 00 0, 00 0, 00 0,97 0, 00 0, 77 °"3(n, 2n) 0, 09* 0, 00 0, 00 0, 06 0, 17 0, 77 °"3 1, 10 3, 56 1,43 1,94 4, 27 6,42 3, 62 9,28 ^22 0,89 1, 58 1, 64 2, 06 2,91 4, 56 3, 42 7, 84 °"zi 1,29 0, 28 0, 28 0, 50 1, 23 2, 30 2, 86 2, 86 1, 5-00 ^20 0, 00 0, 00 0, 00 0, 00 0, 00 0, 01 0, 00 0, 01 - MeV °"2c 0, 00 0, 00 0, 02 0,01 0, 00 0, 03 0, 02 0, 07 (r2f 0, 00 0, 00 0, 00 0, 00 0, 00 0, 58 0, 00 0, 58 °"2(n, 2n) 0,01* 0, 00 0, 00 0, 00 0, 00 0, 02 0, 03 0, 02

°"2 2, 18 1, 86 1, 94 2, 57 4, 14 7, 50 6, 33 11, 38

Calculated with datas from ref. 18. With datas from ref. 6 the cross-sections are cr ', = 0, 04, cr' . = 0, 22 and 2(n cr x = 0, 05. o[n} en) 2(no/ , o2n) '

BA/EL

The fast fission factor for a single uranium cylinder as a function of Figure 1 radius. Moderator effects are not included.

1,05

i, 00

ESSELT 4446

8. According to Spinrad

Tiie tast iission factor tor a uranium oxide cylinder as a function of radius. Moderator effects are not included.

radius for single uranium and uranium Figure 3 oxide cylinders. Moderator effects are not included.

0,1 G

0, 05

ESSELTE 4446

a cm

The effect of (n, 2n)-reactions in a moderator of heavy water as a function Figure 4 of the fuel element radius (see eq. 7)

O, 015

O, 01

, 005

ESSELTE 4446

4 å cm a moderator of heavy water as a function of the fuel element radius (see eq. 7 and 22)

4 a cm

LIST OF AVAILABLE AE-REPORTS Additional copies available at the library of AB ATOMENERGI Stockholm - Sweden

Printed Price AB No Title Author. in Pages Jo Sw. er.

1 Calculation of the geometric buckling for reactors N. G. Sjöstrand 1958 23 3 of various shapes 2 The variation of the reactivity with the number, H. McCnrick 1958 24 3 diameter and length of the control rods in a heavy water natural uranium reactor. 3 Comparison of filter papers and an electrostatic R. Wiener 1958 4 4 precipitator for measurements on radioactive aero- sols. 4 A slowing-down problem. I. Carlvik, B. Pershagen 1958 14 3 5 Absolute measurements with a 4rr-counter (2nd Kerstin Martinsson 1958 20 4 rev. ed.) 6 Monte Carlo calculations of neutron rhermaliza- T. Högberg 1959 13 4 tion in a heterogeneous system 8 Metallurgical viewpoints on the brittleness of be- G. Lagerberg 1960 14 4 ryllium 9 Swedish research on aluminium reactor technology. B. Forsin 1960 13 4 10 Equipment for thermal neutron flux measurement! E. Johansson, T. Nilsson, 1960 9 6 in Reactor R2. S. Claesson 11 Cross sections and neutron yields for UtM, U*** N. G. Sjöstrand 1960 34 4 and PuM> at 2200 m/sec j. S. Story 12 Geometric buckling measurements using the pulsed N. G. Sjöstrand, J. Mednis, 1959 12 4 neutron source method T. Nilsson 13 Absorption and flux density measurements in an R. Nilsson, J. Braun 1958 24 4 iron plug in Rl. 14 GARLIC, a shielding program for GAmma Radi- M. Roos 1959 36 4 ation from Line- and Cylinder-sources. 15 On the spherical harmonic expansion of the S Depken 1959 53 4 neutron angular distribution function. 16 The Dancoff correction in various geometries 1. Carhtk, B. Pershagen 1959 23 4 17 Radioactive nudides formed by irradiation of the K. Ekberg 1959 29 4 natural elements with thermal neutrons. 18 The resonance integral of gold. K. Jtrlow, E. Johansson 1959 19 4 19 Sources of gamma radiation in a reactor core M. Roos 1959 21 4 20 Optimisation of gas-cooled reactors with the aid P. H. Margen 1959 33 4 of mathematical computers. 21 The fast fission effect in a cylindrical fuel element I. Carlvik, B. Pershagen 1959 25 4 22 The temperature coefficient of the resonance inte- P. Blomberg, E Hellstrand, 1960 25 4 gral for uranium metal and oxide S. Hörner 23 Definition of the diffusion constant in one-group N. G Sjöstrand 1960 8 4 theory. 25 A study of some temperature effects on the pho- K-E Larsson, U. Dahlborg, 1960 32 4 nons in aluminium by use of cold neutrons. S. Holmryd 26 The effect of a diagonal control rod in a cylindrical T. Nilsson, N. G. Sjöstrand 1960 4 4 reactor. 28 RESEARCH ADMINISTRATION: A selected and E. Rhenman, S. Svensson 1960 49 6 annotated bibliography of recent literature-

29 Some general requirements for irradiation experi- H. P. Myers, R. Skjöldebrand 1960 9 6 ments. 30 Metallographic Study of the Isothermal Transfor- G. Östberg 1960 47 6 mation of Beta Phase in Zircaloy-2. 32 Structure investigations of some beryllium materials. I. Fäldt, G. Lagerberg 1960 15 6 33 An Emergency Dosimeter for Neutrons J. Braun, R. Nilsson 1960 32 6 35 The Multigroup Neutron Diffusion Equations /I S. Linde 1960 41 6 Space Dimension.

Affärstryck, Stockholm 1960