pa<

»57

Heterogeneous Two-group Bifuslon

Theory for a Finite Cylindrical

By Alf Jonsson and Göran Näslund

AKTIEBOLAGET ATOMENERGI

STOCKHOLM SWEDEN 1961

AE-57

HETEROGENEOUS TWO-GROUP DIFFUSION THEORY

FOR A FINITE CYLINDRICAL REACTOR

Alf Jonsson, Göran Näslund

Summary:

The source and sink method given by Feinberg and Galanin

is extended to a finite cylindrical reactor. The two-group diffusion

theory formulation is chosen primarily because of the relatively

simple formulae emerging.

A machine programme, calculating the criticality constant

thermal utilization and the relative number of thermal absorptions

in fuel rods, has been developed for the Ferranti-Mercury Computer.

Printed and distributed in June 1961

LIST OF CONTENTS

Page

1. Introduction 1

2. Assumptions 1

3. The critical condition 2

4. The thermal utilization 6

5. The resonance escape probability 9

6. Conclusion 10

7. Description of the programme 11

8. A sample calculation 18

9. References 19

Heterogeneous two-group diffusion theory for ,a finite cylindrical reactor

1. Introduction

The individual source and sink method of treating a heterogeneous reactor

was given by Feinberg and Galanin at the Geneva Conference 1955 (ref. 1

and ref. 2). They considered a reactor consisting of fuel elements embedded

in an infinite moderator. At the Geneva Conference 1958 this method was

extended to finite reactors of rectangular shape by Meetz and Barden et al

(ref. 3 and ref. 4). Auerbach (ref. 5) has considered a cylindrical reactor

of finite extent but the heterogeneous part of the reactor is limited to a small

central region.

This report contains a formal derivation of the critical condition for a finite

cylindrical reactor. The treatment given is easily applied to the combination

of Fermi age theory and diffusion theory used by Feinberg and Galanin, but

two-group diffusion theory has been chosen because the formulae are more

easily applied to machine calculation.

2. Assumptions

The assumptions repeated below are identical with those of Feinberg-Galanin.

(i) The problem is a two dimensional one, that is the reactor is infinite in

the axial direction or the well-known simplification of an axial cosine

flux distribution is made.

(ii) The rods are line sources of fast neutrons and line sinks of thermal

neutrons. The number of fast neutrons emitted by the fuel element for

every thermal neutron absorbed is r\ .

(iii) The number of thermal neutrons absorbed by the element is proportional

to the flux at its surface. Tha proportionality constant -y is assumed to

be dependent only on the content of the element and on the moderator

material. This implies that the neutron density at the surface is a

constant independent of the azimuthal angle.

Additional assumptions are

(iv) The reflector which may have a finite size must have the same physical

properties as the moderator.

(v) Two-group theory is valid for the neutron flux in the moderator at least

at some distance from the fuel elements.

For a discussion of (i) - (iii) the reader is referred to the papers of Feinberg

and Galanin or to a paper by D. C. Leslie (ref. 6). The other two assumptions

2.

will be discussed below.

3. The critical condition

The restriction of an infinite reflector is removed by working with finite

Fourier-Bessel transforms when solving the diffusion equation in the

moderator. For the fast flux in two-group theory this equation is

N

V2* ( r ) - i * ( F ) + * T n n V n 4 > n 6 ( F - F n ) = 0 (1)

Mf n=l

Here r"is a plane-polar vector with the components p and cp .

T is given by

M x

where T is the age to thermal in the moderator and H is the critical

extrapolated height of the reactor. Fast absorption in the fuel elements is

neglected but an approximate correction may be obtained by using the age in

the actual lattice instead of T . Actually the fast absorption in the rods

should be accounted for by a delta-function sink term analogous to the last

term in eq (1) which gives the source of fast neutrons as a sum of contributions

of all fuel elements in the reactor. The neglect of fast sinks means that this

method will be unable to deal with for example reactors containing control

rods with considerable epithermal absorption where the spatial distribution

of the fast flux is important. Likewise lattices with appreciable epithermal

fissions cannot be handled.

D. xx. is the fast diffusion coefficient of the moderator. r> • i is the number ofMf i n n _fast neutrons emitted by fuel element number n with posit ion vector r .

v cb = i is the number of the rmal neutrons absorbed by the n ' t h rod,

3.

extrapolated radius of the reactor., , / d dr \ 2I d / ^m \ m i / \ 1 , / \p d ( p ) - ib { P )- _ p —, _ — _ dip dp y p dp / 2 v m

N 2ir

+ _ 1 ) t] 7 $ •• 5 ( 7 - 7 ) e dcp' = 0 (3)2 T D . , . / • n n n j x nMf " . ,;n= 1 0

The finite Fourier-Bessel transform of ijj (r)> where r = -~- , is

defined as (ref. 7):

i

r4* (PÄ ) = ^ ( r ) r J (p« r ) dr

where pfl are the zeros of z J (z). The inversion is given by

CO

^ (r) = ) ? I|J (p. ) J (p. r) (5)^mv ' /y i- -|2 rn v r£m' m v r£m ' * '

2=1 ' J ' (ftm

Applying this to eq. (3) one obtains summing over m

V V im(cp-cp)— / TI 7 H> N

3 ^ 'n n Knn=l m=-oo

e

oo

^

According to Buchholz (ref. 8) the last series may be summed to give:

N « . /

D . , , ;/_/ n n n

N

/ / n n n i_jn= 1 m=-oo

Im

m v \n / m

(7)if r > rn

4.

Using the addition theorem for K(,(W) (ref. 7 p. 361) and introducing

p = rR one obtains:

17-7 I

oo K ' R

V eim(9- p . In the case ofn

infinite R eq. (8) reduces to the line diffusion kernel in an infinite medium,

To solve the equation for the slow flux and to obtain the critical

condition we need IJJ ( r ) , the fast flux from the n ' th rod.

N

n=l

Defining p as the probability that a neutron emitted by the n ' th

fuel rod will escape resonance capture in the lattice the equation for

the thermal flux may be written:

D NV " i / ~ " ~ \ ^ j . / * ~ " \ t Nil. \ i / ~ " \

5.

The second term of eq. (9) gives the absorption in the moderator

and the last term the absorption in fuel elements. If we substitute

\\J ("r ) from eq. (6) into eq. (9) and solve in the same way as for the fast

flux we find:

7n ^n { V F n ^' L' T > " fnn=l

Where

oo K ( •$-m V L.

V J \ Rm=-°o I -—-

m V L

1 Ö Ms

ir

and

^ ' Tri = p • TI'n ^n n T . .

M

f ("r, L.) and F ( r, L,T ) are the correctly normalized (that is giving unit

thermal moderator absorption in an infinite moderator) thermal sink and

fission-to-thermal source kernels.

When r = r, , where k may take the values k = 1,2, . . . N, eq. (10) constitutes

a set of N linear homogeneous equations for the unknown quantities -S

The condition of vanishing determinant gives the critical condition with

for example one of the r\ as critical parameter. The corresponding

eigenvector gives the relative number of thermal absorptions in the fuel

elements, when multiplied by 7 . When r —> r the diffusion kernel

f ( r, L) is replaced by 9 _• Kn ( —=— J , a being the radius of the n ' thn c. n L) u \̂ J_J j nrod.

6.

4. The thermal utilization

As mentioned above 7 should be defined so that it gives the true

number of absorptions in the element when multiplied by the surface flux

$ calculated by diffusion theory. Methods for calculating 7 have been

given by Kushneriuk - McKay (ref. 9) and Stuart (ref. 10). In terms of

the extrapolation distance X. of ref. 9 one has:

27T a D. .7 = n Msn A-

nand in terms of the rod blackness (3 of ref. 10:

y __ or - , **•n " n 2 - (3.

Zink (ref. Il) has described a method of determining 7 by measuring the

flux distribution around a single fuel rod.

The thermal utilization of an infinite square lattice with spacing s

is related to 7 by the following formula given by Galanin (ref. 12):

2 - it a2)

M

2s

4JTL?.M

r In -

1

2s

7T a 2

27ra

2~s

- ! •

° ' O 2 8 1 (11)S . TTB. -

S

Together with the t|-value determined from the critical condition

and p . the resonance escape probability of the infinite lattice, f̂ may be

used to obtain the infinite multiplication factor

koo= 1 Poofoofco

7.

In the case of a finite lattice consisting of an arbitrary number

of different rod types (fuel rods of different properties, control rods etc. )

the thermal utilization should be defined as:

LJ nf = 2zi (13)

1n=l

where i = 7 4> and M is the number of thermal neutrons absorbed inn n n v̂ ,

the moderator. ) in the numerator of (13) indicates that the summation

shall be carried over fuel rods only. An expression for the moderator

absorption M can be derived as follows.

Rewrite eq. (10) as

fn = l

Here we have introduced the reactivity k determined as the

largest eigenvalue of the set of equations giving . M is given by:

M = \\ S $ (7) d7 (15)

where S is the thermal absorption cross section of the moderator and

the integration is carried out over the moderator. It is assumed that the

volume of the rods can be neglected so that the integration can be extended

over the reactor volume. 2 being a constant we have to evaluate integralsa

of the type

— ! < —in the region r j - R. f ( r , L) satisfies the differential equation

divgrad f ( r, L) - 4 y f ( r , L) +n

with the boundary condition f (R, L) = 0

Integrating this equation one obtains:

( I f ( 7 , L ) d 7 = -~— + L 2 C g r a d f ( 7 , L ) d ? (16)

The last integral is taken over the surface of the reactor. The

result of the integration is:

M s \ \ t i— T \ -,— £— K — i \ f r , L d r = K =2 JJ n v ' nr ,Ln v ' n

l0 V L

and for F ( r, L, r ) :

( 7 , L , r ) d 7 = K F

n v ' n

Ri i -

T 1 "? T f

72 -1" " 1 o v TJLJ

\ ^

K,^ V \rz: J I 1

(18)

Introducing K and K eq. (13) r eads :

N ̂

f= 2zi T . (19)M —7~ K

2 T n

IT i IAK

i f l - — -̂- K + rtn \ . 2 n 'nn= 1 M

9.

In the case of an infinite one-component lattice, where i = i,

7 n = 7 > ^n = l i p cc ' c q > ^1 4^ g i v C S :

— 4- T, {7 + *kn

n (20)S F.

knn

where f, = f ( r, , L) and F, = F ( r, , L, T ), and becausekn n v k ' kn n v k '

f FK = K = 1 for an infinite lattice, eq. (19) becomes:

S F kn2

This result is given by Feinberg (ref. 1) and Galanin (ref. 12) has

shown that in the case of a square lattice eq. (21) reduces to eq. ( l l ) , if

the finite rod size is taken into account.

5. The resonance escape probability

In the machine programme p is calculated by a method developed

by Pershagen and Carlvik (ref. 13). Accordingly p is calculated from

the formula:

V Nm m

* s

s M

(Rl)m

>m

PM

P M

2

i=3

V

e

cell m

cell

! n4

4 T T

- Vm

m

m

2

T .lm

i m

N

- l n p n = ^ "-1 n i ' m / a. (22)

m=

10.

i m

k = 1 -m

• km

D20Mf

20TMI

In

U

E 2 8E I

s '20 -'M

20C P M

2 Ycell m

cell m Vm

V = volume/unit length of m/th rod

N = number of uranium atoms/unit volume in m ' th rodm '

(Rl) = resonance integral of m ' th rod

Jj S ) = slowing down power of moderator at 20 C

p, , = moderator density

V ,.. = cell volume/unit length of m ' t h rod

T ^ . and a. = constants in the slowing down kernel toIndium resonance energy

20 o

T _ = age to Indium resonance energy in the moderator at 20 C

E_p = equivalent resonance energy of U

ET = r e sonance energy of Indium

6. Conclusion

The machine programme based on the formulae given above and

described in section 7 is now available at the Ferranti-Mercury computer

of AB Atomenergi. It is to be regarded as a preliminary version and

the experience gained by the users of this programme should serve as

a foundation for future improvements. Further work may be required

on the points listed below.

A. The treatment of the resonance and epithermal absorption

is unsatisfactory because it gives a wrong picture of the spatial

distribution of fast neutrons slowing down to thermal energies. To

take care of the resonance absorption a third energy group containing

sinks for resonance neutrons may be introduced.

11.

B. The effect of the finite rod size should be investigated. It affects

the eigenvalue and the flux distribution in two ways. Firstly it may

be expected that the approximation of constant surface flux is a poor

one, when the fuel consists of large elements for example natural

uranium rod clusters. Meetz (ref. 3) has suggested a method of

dealing with a variable surface flux. Secondly the replacement of

the fuel volume by moderator results in an overestimate of the

slowing down. This might be accounted for by a simple density

correction.

C. The possibility of anisotropic neutron diffusion should be included.

D. Methods of calculation of the rod parameter 7 should be investigated.

Measurements (ref. 11) of 7 and the corresponding quantity for the

epithermal flux would be of great value.

E. The theory should be extended to the important case, where the

reflector and moderator materials are different. In connection

with this the possibility of introducing more realistic slowing-

down kernels should be investigated.

F. Application of the theory to exponential experiments might be useful

in evaluating effects of core configuration on the axial flux

distribution.

7. Description of the programme

Method

The problem of solving the critical determinant is equivalent with the

eigenvalue problem

(A + XB) x = 0

where only the smallest eigenvalue and corresponding eigenvector is to

be Calculated. After rewriting the equation as

(A" 1 B + -jL I ) x = 0

12.

the classical iteration method (ref. 14) can be used. To start the iteration

an approximation to the eigenvector is calculated from

2 . 4 p= J '

R

To speed up the convergence a simple over relaxation process is used.

Limitations

At present the parameters are limited as follows

N

Input list

13.

Symbol

3.

(title)

DMs

Mf

M

7

a

s

Poo

1

2

M

thermal diffusioncoefficient of moderator

fast diffusioncoefficient of moderator

age to thermal inmoderator

H = extrapolated heightof reactor

R = extrapolated radiusof reactor

k = reactivity

= thermal coefficient

= rod radius

= square lattice pitch

= infinite lattice resonanceescape probability

= age in lattice

2 _

4. M =

K

thermal diffusion lengthin lattice

number of rod groupswith unknown rj

total number of rodgroups

Remarks

The title must begin with a letter-shift and close with at least two figure-shifts

When the reactivity is to be calculateda zero should be entered in this position.When T] of some group of rods is to becalculated k is set equal to unity

This group of data is inserted onlywhen k = 1 above and is used in formulae(11) and (12) to compute a k and amaterial buckling from the rj -valuedetermined. The buckling is calculatedfrom the two group formula.

A rod group consists of rods with thesame surface flux

14.

10.

a k

x

x .

y

multiplication of rodsin group 1

, If k = 1 T]. = 1 when i < M

thermal constant of rodsin group 1

- radius of rods in group 1

resonance escape proba-bility of neutron emittedby an element belongingto group 1

Rod coordinates

i 0 enter constants\ for data group 11.! 1 fixed constantsv are used.

If * is inserted after r\ then r\ .,t O ^

If * is inserted after 7 then 7 4,n n+1

n+2' 7. are set equal to 7k ^ n

In analogy with above

In analogy with above

If p. is to be calculated by theprogramme a * should be insertedinstead of p.

Must be written in the orderdetermined by 4

Data groups 10, 11, and 12 shall beentered only if the p. are to becalculated

15.

11.20MI20Ml20M2

1

c

D

220Mf

E 28

*20'M

Confer section 5.

This group of data is omitted if q = 1

12.

13.

14.

Pj20

= density ratio ofmoderator

,1 cell 1 , 1

V V NM cell M M

V V NVK cell K iNK

11 x l

(RDK

hy l y v l

If * i s inse r ted after (Rl) then

V n + 1V c e l l

n+1

= Vn+2

n + l = "= . . . . N K

. . . =V c e l l= Nn

V K

K =

= Vn

cell n

The fast and the rma l flux will becalculated in the points (x.+y< + X.h ) where X. takes the' 1 v

' r v - vv 1values 0, 1, . . . . =-!- hx

If h = h =0 the flux will be calculatedx yin (x , y.) only.

The data group 13 can be repeated anarbitrary number of times .

The tape must close with a *.

If no fluxes are wanted data group 13is omitted.

16.

Punching instructions

Every number must be followed by sp, sp orCR, LF.

Output list

Remarks

1.

2.

Title

DMs

DMf

T ML LH

R

k

6.

If all T) r\ > i are equal onlyTJ - r\. are printed

In analogy with above

In analogy with above

In analogy with above

17.

7. t a,

8. 7

9.

10.

1 ' 1

m oo oo

y i

i is the number of iteration and a.is the corresponding approximationto the eigenvalue sought.

o r kM

These numbers are printed only whenkey 0 is set (see operating instructions).

7. . is normalized so thati i

(7 . .) = 1a = T| or kf is according to formula (19)

This group is omitted if k hasbeen calculated

Operating instructions

The programme named HETERO 1 is taken in as a normal Pig-programme.

The data tape is taken in with a prepulse. If key 0 is set the successive

iterations to the eigenvalue are printed as soon as they are calculated. By

setting key 9 the machine will stop and take the last calculated eigenvalue as

the true value. These two facilities are to be used only on problems with

slow convergence or when erroneous data have been inserted. After the

printing of the eigenvalue the machine starts reading the coordinates of

points where the flux is to be calculated. If some error occurs on this part

of the data tape the machine comes to a hoot stop. If the error can be

corrected at once the machine can be made to take in the corrected part of the

data tape by tapping key 8. Restart is made in the normal way.

18.

8. A sample calculation

To illustrate the accuracy of the

flux calculations a sample calculation has been performed on an actual

configuration of the zero-energy reactor RO. At present this reactor is

a bare, D-O-moderated reactor with a radius of 113 cm. The reference

fuel consists of natural uranium metal rods with a diameter of 3. 05 cm

and a 0. 15 cm thick aluminum sheath. The lattice pitch was 18 cm

square and the configuration was that shown on the graph. The measured

critical extrapolated height was 202 . 31 cm. The radial flux distribution

was measured along a diameter with a BF,-counter and the experimental

results corrected according to TPM-RFN-11 (ref. 15) are compared

to the calculated thermal flux on the graph. Experimental points shown

are the mean values ^ I ^(p) + ( - P) )•

19.

9. References

1. S. M. Feinberg, Heterogeneous Methods for Calculating Reactors:Survey of Results and Comparison with Experiment, 1955 GenevaConference, Vol. 5, P/669

2. A. D. Galanin, Critical Size of Heterogeneous Reactor with SmallNumber of Rods, 1955 Geneva Conference, Vol. 5, P/663

3. K. Meetz, Exact Treatment of Heterogeneous Core Structures,1958 Geneva Conference, Vol. 16, P/968

4. S. E. Barden, D.Blackburn, A.Z.Keller, J. M. R. Watson, SomeMethods of Calculating Nuclear Parameters for HeterogeneousSystems, 1958 Geneva Conference, Vol. 16, P/272

5. Auerbach, The Effect of Lattice Perturbations on Criticality,Reaktor A. G. WUrenlingen, ST-PH-29, 1959

6. D. C. Leslie, The method of Feinberg Galanin, AEEW-M18 (1959)

7. G. N. Watson, Theory of Bessel functions, Second Edition,1944 p. 576

8. H. Buchholz, Bemerkungen zu einer Entwicklungsformel aus derTheorie der Zylinderfunktionen, Zeitschrift fur angewandte Mathematikund Mechanik, Band 25/27, 1947 p. 245

9. S. A. Kushneriuk, C. McKay, Neutron Density in an Infinite Non-Capturing Medium Surrounding a Long Cylindrical Body whichScatters and Captures Neutrons, AECL-137, (1954)

10. G. W. Stuart, Neutron Multiple Scattering Methods, 1958 GenevaConference, Vol. 16, P/624

11. J. W. Zink, Use of small source theory in the determination ofthe critical size of heterogeneous thermal reactors, NAA-SR-3222,1959

12. A. D. Galanin, Theorie der thermischen Kernreaktoren, Leipzig 1959

13. B. Pershagen, I. Carlvik, The resonance escape probability ina lattice, AB Atomenergi, AEF-71 (1957)

14. R. Zurmilhl, Praktische Mathematik, Springer Verlag 1957, p. 159

15. G. Näslund, Minsta kvadratanpassning till radiell flödesfördelning,TPM-RFN-11 (1960)

AJ-GN/BMS5.4. 61

M 0 '>-, 0)

2. 0

0,5 calculated the rmal flax r;orn-.t>lizec toexper imenta l at :: exper iment

One uuarfra.'-.t -j- tho core v/2t'ire la t : \ e numoers of abiorotion1 ;

620 565 4ol 318 ] 50

•ja su rec ar.c calculated :•;-":-ja! fljx in the IH)-; eaci or.i't: i.:ei - ? fw souarc lattice pitch.

16 24 '52 40 48 56 64 80 104 112 120

9.

10.

11.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24,

25.

26.

27.

28,

List of published AE-reports.

Calculation of the geometric buckling for reactors ofvarious shapes. By N. G. Sjöstrand. 1958, 23 p. Sw. cr, 3:-.

The variation of the reactivity with the number, diameterand length of the control rods in a heavy water naturaluranium reactor. By H. McCririck. 1958. 24 p. Sw. cr. 3:-.

Comparison of filter papers and an electrostatic precipi-tator for measurements on radioactive aerosols. By R. Wiener.1958. 4 p. Sw. cr. 4:-.

A slowing-down problem. By I. Carlvik and B. Pershagen.1958. 14 p. Sw. cr. 3:-.

Absolute measurements with a 4 TT-counter. By Kerstin Martinsson.2nd rev. ed. 1958. 20 p. Sw. cr. 4:-.

Monte Carlo calculations of ncuti c " - n—•*--- ir_ a hete-rogeneous system. By T. Högberg. 1959. 13 p. Sw. cr. 4:-.

Spatial variations of the isotopic concentrations in a fuel rod.By P. -E. Ahlström and P. Weissglas. 1960. 9 p. Sw. cr. 4:-.

Metallurgical viewpoints on the brittleness of beryllium..By G. Lagerberg. 1960, 14 p. Sw. cr. 4:-.

Swedish research on aluminium reactor technology. By B. Forsen.1960. 13 p. Sw. cr. 4:-.

Equipment for thermal neutron flux measurements in reactor R2.ByE. Johansson, T. Nilsson and S. Claeson. 1960. 9 p. Sw. cr. 6:-

Cross sections and neutron yields for U , U and Pu at2200 m/sec. By N. G. Sjöstrand and J. S. Story, I960. 34 p.Sw. cr. 4:-.

Geometric buckling measurements using the pulsed neutron sourcemethod. By N. G. Sjöstrand, M. Mednis and T. Nilsson. 1959.12 p. Sw. cr. 4:-. (Arkiv för fysik 15 (1959) nr 35 pp 471-82).

Absorption and flux density measurements in an iron plug in Rl,By R. Nilsson and J. Braun. 1958. 24 p. Sw. cr. 4:-.

GARLIC, a shielding program for GAmma Radiation from Llne-and Cylinder-sources. By M. Roos. 1959. 36 p. Sw. cr. 4:-.

On the spherical harmonic expansion of the neutron angulardistribution function. By S. Depken, 1959. 53 p. Sw. cr. 4:-,

The Dancoff correction in various geometries. By I. Carlvikand B. Pershagen. 1959. 23 p. Sw. cr. 4:-,

Radioactive nuclides formed by irradiation of the naturalelements with thermal neutrons. By K. Ekberg. 1959. 29 p.Sw, cr, 4:-.

The resonance integral of gold. By K. Jirlow and E. Johansson.1959. 19 p. Sw. cr. 4:-.

Sources of gamma radiation in a reactor core. By M. Roos.1959. 21 p. Sw. cr. 4:-.

Optimisation of gas-cooled reactors with the aid of mathema-tical computers. By P. H. Margen. 1959. 33 p. Sw. cr, 4;-,

The fast fission effect in a cylindrical fuel element. By I. Carl-vik and B. Pershagen. 1959. 25 p, Sw. cr. 4:-.

The temperature coefficient of the resonance integral for ura-nium metal and oxide. By P. Blomberg, E. Hellstrand andS. HÖrner. I960. 14 p. Sw. cr. 4:-.

Definition of the diffusion constant in one-group theory. ByN. G. Sjöstrand. 1960, 8 p. Sw. cr. 4:-.

Transmission of thermal neutrons through boral. By F. Åkerhielm.2nd rev. ed. 1960. 15 p, Sw, cr. 4:-.

A study of some temperature effects on the phonons in aluminiumby use of cold neutrons. By K. -E. Larsson, U. Dahlborg andS. Holmryd. 1960. 21 p, Sw. cr. 4:-.

The effect of a diagonal control rod in a cylindrical reactor.By T. Nilsson and N. G. Sjöstrand. I960. 4 p. Sw. cr, 4:-,

On the calculation of the fast fission factor. By B. Almgren.1960. 22 p. Sw. cr. 6:-.

Research administration. A selected and annotated bibliographyof recent literature. By E. Rhenman and S. Svensson. 2nd rev. ed.1961. 57 p. Sw. cr. 6:-.

29. Some general requirements for irradiation experiments. ByH. P. Myera and R. Skjöldebrand. 1960. 9 p. Sw. cr. 6:-,

30. Me tal lo graphic study of the isothermal transformation of betaphase in zircaloy-2. By G. Östberg. I960. 47 p. Sw. cr. 6:-.

31. Calculation of the reactivity equivalence of control rods m thesecond charge of HBWR. By P. Weissglas. 1961. 21 p. Sw. cr. 6:-

32. Structure investigations of some beryllium materials. By LFaldtand G. Lagerberg. 1960. 15 p. Sw. cr. 6:-.

i5. An emergency dosimeter for neutrons. By J. Braun andR. Nilsson. I960. 32 p. Sw. cr. 6;-.

34. Theoretical calculation of the effect on lattice parametersof emptying the coolant channels in a D^O-moderated andcooled natural uranium reactor. By P. Weissglas. I960.20 p. Sw, cr. 6:- .

3-.. " ' o-uup r.outron diffusion equations/l space-dimension.By S. Linde. I960. 41 p. Sw. cr. 6;-.

ib. Geochemical prospecting of a uramferous bog deposit atMasugnsbyn, Northern Sweden. By G. Armands. 1961. 48 p.Sw. cr. 6:-.

37. Spectrophotometric determination of thorium in low grademinerals and ores. By A. -L. Arnfelt and I. Edmundsson.I960. 14 p. Sw, cr. 6:-.

38. Kinetics of pressurized water reactors with hot or cold mode-rators. By O. Norinder. I960. 24 p. Sw. cr. 6:-.

39. The dependence of the resonance on the Doppler effect.By J. Rosén, I960. 19 p. Sw. cr. 6:-.

40. Measurements of the fast fission factor (e ) m UO2-elements.By O. Nylund. 1961. Sw. cr. 6:-.

41. Calculation of the flux in a square lattice cell and a comparisonwith measurements. By G. Apelqvist. 1961. Sw. cr. 6:-.

42. Heterogeneous calculation of 6 . By A. Jonsson. 1961.Sw. cr. 6:-.

43. Comparison between calculated and measured crossection-changes in natural uranium irradiated in NRX. By P. E. Ahl-ström. 1961. Sw.cr. 6:-.

44. Hand monitor for simultaneous measurement of alpha and betacontamination. By I. Ö, Andersson, J. Braun and B. Söderlund.2nd rev. ed. 1961. Sw. cr. 6;-.

45. Measurement of radioactivity in the human body. By I. Ö. Anders-son and 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. I960, 19 p. Sw. cr, 6;-.

47. An experimental study of the scattering of slow neutrons fromH2O and D2O, 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. Hellstrandand J, Weitman. 1961. Sw, cr. 6;-.

49. Pressure tube and pressure vessels reactors; certain compari-sons. By P.H, Margen, P. E. Ahlström and B, Pershagen.1961. Sw.cr. 6:-.

50. Phase transformations in a uranium-zirconium alloy containing2 weight per cent zirconium. By G. Lagerberg. 1961. Sw. cr. 6;-.

51. Activation analysis of aluminium. By D. Brune. 1961. 8 p.Sw. cr. 6:-.

52. Thermo-technical data for D^O. By E. Axblom. 1961. 14 p.Sw. cr. 6:-.

53. Neutron damage in steels containing small amounts of boron,ByH.P. Myers. 1961. Sw. cr. 6:-.

54. A chemical eight group separation method for routine use ingamma spectrometric analysis. By K. Samsahl. 1961. Sw. cr. 6:-,

55. The Swedish zero power reactor R0. By Olof Landergård,Kaj Cavallm and Georg Jonsson. 1961. Sw. cr. 6:-.

Additional copies available at the library of AB Atomenergi, Studsvik,Tystberga, Sweden, Transparent microcards of the reports are ob-tainable through the International Documentation Center, Tumba, Sweden.

Affärstryck, Stockholm 1961