AE-145 UDC 681.177 621.039.538

If) ^r LLJ A User’s Manual for the NRN Shield < Design Method

L. Hjdrne

AKTIEBOLAGET ATOMENERGI

STOCKHOLM, SWEDEN 1964

AE-145

A USER'S MANUAL FOR THE NRN SHIELD DESIGN METHOD

Editor: Leif Hjarne Authors: E Aalto, RFraki, L Hjarne, M Leimdorfer, K Lindblom S Linde, K Malen, K Nyman

Summary:

This report describes a code system for bulk shield design written for a Ferranti Mercury computer and is intended as a ma­ nual for those using the programme. The idea of an "almost direct" flux, as in the removal theory serves as a basis for further deve ­ lopment of the theory. An important aspiration has been to mini ­ mize the manual work of administering the codes. The codes con ­ cerned are: NECO, computing necessary group constants from pri­ mary data, REFUSE and RE BOX (infinite plane or cylindrical, and box geometry, respectively), computing removal flux, NEDI a one ­ dimensional (plane, spherical, cylindrical) diffusion multigroup code, and SALOME a Monte Carlo code computing the gamma flux. Output tapes are constructed for direct use as input tapes, when required, for a following code.

Printed and distributed in June 1964 Contents Page

DESCRIPTION OF PRINCIPLES 1. Introduction 1 2. Cross-sections 1 3. The deep penetration of neutrons 5 4. Neutron diffusion and slowing down 9 5. Gamma penetration and energy deposition 11

DATA PREPARATION 6 . General instructions 13 7. Data for NECO 14 8. a. Data for REFUSE 26 8. b. Data for REBOX 31 9. Data for NEDI 33 10. Data for SALOME 44

OPERATION INSTRUCTIONS 1 1. Operation of NECO 47 12. a. Operation of REFUSE 49 12. b. Operation of REBOX 49 13. Operation of NEDI 50 14. Operation of SALOME 50 15. Error prints 52

LAY-OUTS OF DATA 16. Lay-out of data for NECO 56 17. a. Lay-out of data for REFUSE 60 17. b. Lay-out of data for REBOX 60 18. Lay-out of data for NEDI 64 19. Lay-out of data for SALOME 69

APPENDICES Appendix A 72 Appendix B. Tables 1 03 References 107 DESCRIPTION OF PRINCIPLES

K Introduction

The diffusion theory, based on the so called diffusion approxima­ tion, is accurate only far away from sources and material surfaces. It has,, however , certain advantages over the strict transport theory, 1. e. its relative simplicity and its applicability to geometrically more complicated problems. For deep penetration problems, however , diffusion theory requires some complementary information about the neutrons coming directly from the sources at a comparably high energy (about 1 Mev) . This is because of the decreasing cross-sections and the increasing scattering anisotropy with higher energies.

The methods used hitherto for neutron penetration computa­ tions in large geometries are impaired by some serious defects, essentially inherent in the semi-empirical character of the removal theory. The method which will be described here is a further develop ­ ment of the course suggested by the removal theory, but more strictly based on theoretical elements.

In this report we shall describe a code system for shielding purposes.An important aspiration has been to minimize the manual work of administering the codes. The codes concerned are: NECO, computing necessary group constants from primary data, REFUSE and RE BOX (infinite plane or cylindrical, and box geometry, respec­ tively), computing removal flux, NEDI a one -dimensional (plane, spherical, cylindrical)diffusion multigroup code, and SALOME, a Monte Carlo code computing the gamma flux. Output tapes are con ­ structed for direct use as input tapes, when required, for a following code .

2. Cross-sections

In solving neutron attenuation problems one needs a rather detailed knowledge of elementary data, such as cross-sections as functions of energy, from which the constants occurring in the transport formalism can be computed. This work can be accurately executed by NECO, a code which reads raw data from a tape library, containing experimental and theoretical information which can easily be kept up to date. The output tapes from NECO can be used directly as data tapes for the following codes. - 2 -

In the data library for NECO the cross-sections are organized from the pqint of view of neutron transport. Therefore all reactions of the type (n, y) (n, p) (n,Cf) must be looked upon as absorptions. The angular distribution of elastic scattering is very important owing to its effect on deep penetration and the energy degradation. In the data library these distributions are given in the form of Legendre poly­ nomial expansions:

cr .(E) r—, e(E;n) = ) (2 % + l) % (E) P (n) (2. 1)

with the first coefficient normalized

f E 1 o

Taking data from various compilations one must keep in mind that the Legendre coefficients, f& , can be defined in a number of ways, al­ though lately this form (eq. 2. l) has predominated. Directly from the experimental data often some kind of fitting by least squares must be accomplished. For this purpose the code COFFEE (ref. 1) for the IBM 7090 has been developed in cooperation with FOA (Res. Inst. Natl. Def.).

Inelastic scattering is treated in different ways in different energy regions. In the energy region where the energy levels are known (generally below 4-5 Mev) we give cross-sections for excita­ tion of each level. At higher energies the level density grows very fast, and we use an evaporation model to describe the secondary energy distribution. Let E be the primary energy E1 the secondary energy. The distribution is then described by:

gn 2n E' (2.2) *n 2n +V' T^

The "temperatures" T are defined according to

2 E _2 E - B , = a, • L = a2 - 3 -

B is the (n, 2n) threshold. The constants and can be deter ­ mined from experiments, but very roughly they are one tenth of the atomic weight of the residual nucleus (ref. 2), Inelastic scattering is assumed to be isotropic.

The group constants are defined as usual as weighted averages over the groups. The weight function must be the expected form of the neutron flux spectrum within the group. This spectrum can be de - scribed in an approximate way as varying as E' 6 for the diffusion group constants. If the a>g are not given, they are all set equal to - 1, making collision density constant. For the removal group con ­ stants the weight function is the source spectrum, a Watt type fission spectrum:

-PE eee sin h y E (2.3)

Ct being a normalization constant. The two constants f3 and y are for U22^ 1.036 and 2.29, respectively, for Pu22<^ 1.0 and 2.0 and for Pu241 1.0 and 2. 2 (see table II in app. B).

Thermal cross-sections are treated in a special way. For the library we need elastic and capture cross-sections at 2200 m/sec (0.0253 ev) . The elastic cross-section is assumed to be constant -i/z the capture cross-section to vary as (Energy) . Now the thermal neutron spectrum is assumed to be a Maxwellian spectrum with temperature T , which usually can be set to some fifty degrees above the temperature of the medium:

E E kT (2.4) (kT)

Observe that T is given in centrigrades for NECO, not as the absolute temperature as in (2.4).

In the data for the programme a maximum step length in lethargy (Au)^ for the group averaging integrations must be speci­ fied for each group g . The programme selects the ’’basic step length” (Au)^ slightly below the given value so that it goes an inte ­ gral number of times into the group width. Close to the group boun ­ dary certain functions can vary quickly with the energy: - 4 -

r Z (E-»E») dE' (2.5) E

E s-1 Xa,g(E) £(E) - Z S(E-* E 1) dE' (2.6) E •

This applies also, for example, to

E„

(E) = Z (E-4 E ') dE' (2.7) ’s, g+1 Jg+1

The variation is particularly rapid in the event of strong an ­ isotropy in heavy nuclei. To obtain reasonable accuracy with a simple integration formula, the basic step lowest in energy in each group is divided into seven "whole integration steps" (Au)^ .These are, in order from the group boundary,

& ^ (Au)^ ^ (Aii)^ rjL (Au)^ 1 (Au)^ ^-(Au)g and

In the remainder of the energy group a "whole integration step" is equal to the "basic step length". In the integration the flux per lethargy unit is regarded as constant over a "whole integration step" and cross­ sections and Legendre coefficients over a "basic step length".

The functions are integrated as step functions or with Simpson's formula except as regards the distribution of the secondary neutrons from the inelastic scattering process amongst discrete levels, where an analytical approach is used. The remarks above concerning step length etc. apply likewise to the integration over primary groups.

The distribution functions are integrated analytically over the secondary groups. 3. The deep penetration of neutrons

The treatment of neutron transport is divided into two stages. The first stage is the direct flux attenuation which is of a simple ex­ ponential type, with its sources situated within the core. Our "direct flux" is not exactly uncollided, as a small energy loss and small angle scattering do not necessarily remove a neutron from the direct flux, for which reason we shall retain the expression "removal flux". The theory underlying this method is presented in ref. 3. The relation to the old removal theory can be illustrated by fig. 1 which demonstrates the organization of the rectangular removal matrix of cross-sections.

R R D

------~ Fixed energy boundary, ------Boundary of 'free choice. R* Removal groups. D* Diffusion groups

a b

Fig. 1. Illustration to the inferences in the treatment of the slowing down of neutrons between (a) : the older method (represented by JtAMH-B) and (b): the one presented here (NRN). — 6 —

The removal neutrons are divided into G energy groups with boundaries Eq, E^ , ... E^,, . . *Eq • The removal flux in the energy group at a distance x from an infinite , plane fission source of thickness dx is

J_ d F„S E, dx (3.1) 2 v i (x >b) where S is the fission source density, the spectral factor for group

G and Vq the removal cross-section in group G for the material be­ tween the dose point and the source. The total removal flux in group G at a dose point in the shield is then

T ■ S(x) E1 (x Pq + Y^) dx 3 2 VG ( . ) 0 where T is the thickness of the core, S(x) the fission source density in the core at a distance x from the core-shield interface, y^, the cross-section of the core material and y~ the number of removal Cj mean free paths between the dose point and the core. If h is the thick­ ness of the i** 1 region between the dose point and the core and y.,^. the corresponding removal cross-section, then y^_, = E 1. y.^.

S(x) can be written

-3 -1 S(x) v E. cm sec (3.3) where v is the number of neutrons per fission, Z^ the average fission cross-section and <^(x) the h armal flux.

In data the values of are given at points throughout the core from x = 0 to x = T. A fit by orthogonal polynomials up to 6 ^ degree is made and the values of ^(x) neclded for the integration are obtained from these polynomials. When y^. is small or equal to zero, a new source function S* (x) = S(x) - S(0) is used and the expression (3.2) takes the form - 7 -

T *G = i"FG | S* (x) E1 (x l»G + Y q) dx + 0 Fe S(0) _E2^YG^ " E2^T yG + Y gL (3.4) 2 VG

This is done to avoid numerical difficulties in the evaluation of the integral. The programme employs a variant of the trapeziodal rule with self-modifying step length to evaluate the integral in (3. 2) or (3.4), so that the error in each integration is less than 1 %.

If the value of ^^(x) is given at one point only there is a drastic reduction of computing times because then the flux values are obtained directly from the second term in (3.4).

The values £, (G = 1, 2, ...» G) of the removal flux in a dose point form a vector sj> . To obtain the removal sources Qg (g = 1, 2, . .., g) this vector is multiplied by a gxG matrix, whose elements are determined by the material at the dose point.

Consider one removal source group Qg and a region be­ tween x = a and x = b in the shield. For simplicity we omit the index g and make the transformation

t - 2x - (b + a) r ~ b - 1 (3.5) which moves the region to (- 1, 1).

Then we make the series expansion

- N Log Q(t) = F(t) = Z 11 a T (t) (3.6 ) • r=0 where the T (t) are Chcbyshev polynomials and Sj denotes that the first and the last term in the sum should be halved. The a are . r then determined by

N • 2_ a F(cOS ~~r) COS "rrh— r (3.?) N 8=0 — 8 ~

F(t) is approximated by polynomials of successively 1 S\ 2nt * and 4^ degree until half the absolute value of the last coefficient is less than 0.01 (for REFUSE, or less than the ACCURACY value in REBOX). If this is not attained by the approximation of fourth degree, the com­ puter will proceed, with the next region but writes ACC.MAT. followed by-the region number and the magnitude of — a^ . If desired, the region can then be divided into two or more parts before the next run.

When all the shield regions have been treated, the coefficients of the Che by she v polynomials for each removal source group and shield region are punched on a tape in a form that is accepted by the input rou­ tine of the diffusion programme NEDI. This tape is headed by SOURCE POLYNOMIALS and the title of the problem (see page 42).

In the case of a small reactor the infinite plane approximation used in REFUSER is not valid and therefore a modified version, called REBOX, has been written. In this case the core is assumed to be a box with the origin of coordinates at the centre of the core face, the z-axis perpendicular to the face, positive inwards.

By divisions in the x , y and z directions the core is divided into cells dV = dxdydz . The removal flu.. in group G at a dose point P is then .

= FGl^;7 7 :rJ- exp(-^'i *iG> dV <3-8>

where the sum Is taken over all source cells dV and F is the distance from P to the centre of dV , r. being the intersection of r and the .th . 1 l region.

The removal flux and re.u-.val source distributions are calcu­ lated on an axis perpendicular to the core face. The coordinates of the intersection point P1 of this shield line and the core face are speci­ fied in data. All calculations except the evaluation of the removal flux are made in the same way as in REFUSE and the output is also similar. In input data there is one obvious difference, namely the specification of the SOURCE data. Neutron diffusion and slowing down

NEDI is a programme designed to calculate the neutron flux distribu ­ tion in a multi-layer reactor shield. Multigroup diffusion theory is used, and the equations for the neutron fluxes are written

v (D v Y - E + Sg(r) = 1,2, g (4.1a)

S,(r) = Q,(r) g_i }(4‘ lb) Sg(r) = Qg(r) + Y, ’g1'’ f°r g = 2' 3j-"-

G 1 =1

Thus the neutron spectrum is divided into g" intervals, and (4. 1) is the th diffusion equation for the flux qp^ in the g interval, (p ^ being the flux of highest energy. D is the diffusion coefficient and £ the group g g > . absorption cross-section (including transfer as well as absorption) in the gtb energy group; £ , is the cross-section for transfer of g g neutrons from group g1 to group g . These parameters are constant within a shield region. The source term Q^(r) is a function which is specified by the user of the programme.

We restrict the problem to one space dimension and rewrite the first term in equation (4. la) for interior points of a shield region:

V(DgV»g) D_ a (4.2) where r is the space variable and p a geometry parameter which takes the value 0, 1, or 2 in plane , cylindrical and spherical geometry, respec­ tively.

2 . The buckling terr \ - D^o; has been included to allow for a lateral leakage and can be used c represent, for instance, a lateral flux distri ­ bution of the form cos(a z) or J^(a r) .

The following boundary conditions are to be satisfied: dcp a) at an interface between dissimilar regions m and D ■■■:are B v g g dr continuous; b) at an outer boundary numerical values are given for the fluxes dq> ® , or the currents J = - D L'~ , or the flux to current g g g dr ’ ratios cd J . 'g g - to

NEDI solves equations (4. 1) by a finite difference method (ref. 4). We choose a^set of points r^

< < rM which divide the range of integration (r^ » r^) into M intervals and pro­ vide a suitably fine mesh for the differences.

It is not necessary to space the mesh points at equal intervals. As an example we consider a cylindrical iron-water -concrete shield with six regions, Let us space the mesh points as follows: region medium range (cm) mesh size (cm)

1 water 215.0 - 245.0 2.00 o 2 iron 245.0 - 250.0 o 3 water 250.0 - 261.0 2.20 4 iron 261.0 - 267.5 0.93 5 air gap 267.5 - 282.5 15.00 262.5 - 292.5 1.00 6 concrete 292.5 - 412.5 » 3.33 412.5 - 612.5 v 3. 33

At interfaces between media of considerably different scattering and absorbing properties the flux values can be expected to vary rapidly and a small mesh size should be chosen. Therefore an artificial inter ­ face has been introduced at r = 292. 5 , the mesh size in the left sub­ region being roughly the same as in the preceding iron layer. The reason for introducing another artificial interface at r =412.5 is the form of the source input available (see under "8. Data for NEDI", page 34).

Thus the range of integration is split into 8 divisions , a divi ­ sion being a homogeneous sub-region of the shield divided into meshes of equal size.

The choice of mesh size depends on the geometry, the physical properties of the media and the accuracy aimed at. No universal rules can be indicated, btit some rough estimates will be given in the data description, page 36. Before the calculations or a new kind of shield are started, a typical configuration should be tried with a couple of different mesh sizes in regions where rapid flux variations are expected. The numerical method used to solve the equations is a standard procedure as described, for instance, by Wachspress in ref. 4, chap­ ter 10.

The total neutron flux is the sum of the removal (from REFUSE or RE BOX) and the diffusion (from NEDI) fluxes, the removal flux, however, being negligible except in a small part of the high-energy region.

5.Gamma penetration and energy deposition

Most standard methods presently used for theoretical analysis of the migration of gamma rays in reactor shields rely on the buildup concept to account for the collided radiation component. The main shortcoming of these methods is that they may be rather inaccurate in dealing with multilayer shields, as buildup factors cannot be applied with confidence to general heterogeneous media. The method to be outlined here is designed for arbitrarily laminated shields and employs the Monte Carlo technique with various departures from direct nume ­ rical simulation to accelerate the convergence of the calculation pro­ cess. One-dimensional plane geometry is assumed in the version under consideration. The processing times, on the Ferranti Mercury computer, required for analysing a typical power reactor shield with respect to capture and fission gamma, are of the order of an hour.

The main features of the procedure are the following: Photon absorption is accounted for by analytical averaging above 0.50 MeV thus preventing undesirable break-off of photon histories. The score of transmitter" radiation is obtained by statistical estima­ tion of the escape probability at each collision. The exponential transformation of the flux is employed to increase the path lengths of photons headed toward the external shield boundary, and to de ­ crease the path lengths of photons headed oppositely, as well as to modify the source distribution. All of these "tricks" have been de ­ scribed previously (ref. 5 and 6 ) but the last one had to be elabo­ rated to permit, formally, negative cross-sections. The alterations in the basic sampling processes caused by the exponential trans ­ formation are: 1) the change of the total photon cross-section from gi to p. -y c, y being the cosine of the angle between the photon path and the aid s}

2) the multiplication of the collision kernel C(E 1 , y 1; E, y , z) by the factor —. (primed quantities refer to the pre­ collision value s), ' '

3) the multiplication of the sou/e, -J." stribution S(E,y , z ,) by e^Z.

The last substitution has the effect'of counteracting the, gene­ rally quite strong, spatidl decrease of the gamma source distribution to avoid under sampling of Source photons from parts which may give important contributions to the dose rate as well as energy deposition results. Values of c suitable to promote the last mentioned effect may sometimes turn the total cross-section, now gt-yc, negative. In such cases we use importance sampling to select a path length, d , from the probability density function

P = (7 c - pi) exp (- (7 c - F ) d ‘> instead' of

P = (p -7 c) exp {- Gt - 'j o) d} "■»

(For simplicity, the relations given here refer to the case of a homo­ geneous medium). This operation requires a multiplication of the photon weight by the" factor - exp { 2d(yc -gi)} • The negative sign of this factor is always annihilated in the modified collision kernel, which is negative simultaneously. This procedure has proved to be superior, with re­ spect to rapid convergence, to the possibly more direct approach of adding a constant quantity to the cross-section to turn it positive and compensate this by an artificial source term on the right-hand side of the transport equation.

The quantities calculated are the dose rate (D) at the external boundary of the shield and the energy deposited (Qj) in each of the arbitrarily defined shield regions (j) . Scoring is performed according to the relations 1 3 -

D = V W. exp { - ~ (d-z.) - cd }xR(E), y.> 0 aTl i 1

D = 0 , 7i<0 ,

(with trivial generalization to the multimedium problem), wne re .th w. photon weight after the i 1 d = total shield thickness R(E) dose response function and

*i _CZi l (E. W, - EJ Qj = i ie j where «til , E^ = photon energy after the i collision El = photon energy before the i^ collision Wl = photon weight before the i**1 collision

The dose rate is defined on the basis of the current instead of from the flux, as this is more favourable with respect to Monte Carlo technique s.

The optimum value of the parameter c is selected internally according to a criterion discussed in ref. 7 where a thorough account of the theory underlying this method is presented for further re­ ference.

DATA PREPARATION

6 . General instructions

In the neutron codes (NECO, NEDI, REFUSE, REBOX) the data are grouped in a logical way, each group beginning with a directive , describing the data in the group. For certain reasons, the routine for interpreting these directives in NECO is somewhat different from that of the other codes. In NECO the directive has to be correctly spelt and punched alone on one line, in the others only the first two characters have to be correct, and the data may follow on the same line. Two directives are common to the codes NEBI, REFUSE and REBOX: WAIT and START. The directive WAIT may be punched anywhere, to stop the reading - in and for continuing on the ‘same or on another tape. The .directive START is punched when all the data for one problem are complete, to initiate the calculation.

The data tapes for all the codes (including SALOME) must contain a-ctitle, which is reprinted in the output. Thus the title will also be included in some tapes used as data tapes for a following code. An exception is the tapes from NECO beginning with the title, the input list, and then the data for the following codes.

The differences between SALOME (a Monte Carlo code) •id Idle other codes with an analytical character should be noted. Thus SALOME can be run as long a time as wanted and is stopped by a setting of key. The same run can be continued later, if only a continuing tape is punched, until satisfactory accuracy is reached. It must be noted also that the last output print should not be consi ­ dered uncritically as the final re suit. See also the data example in the appendix.

7. Data for NECO

The data for NECO consist of two parts.

Data 1 gives information of group boundaries, step lengths, spectrum parameters, weight parameters, possible use of age theory, temperature of the thermal neutrons, and the composition of the various material".

Data 2 gives the microscopic cross-section data for the ele­ ments. The object is that, us microscopic data are prepared, they shall be punched and incorporated in a ranched version of a data library to be. stored with the programme. - 15

A) General

The following directive s are given:

In Data 1

TITLE REMOVAL GROUPS AND SPECTRUM PARAMETERS DIFFUSION GROUPS or DIFFUSION GROUPS WITH WEIGHT PARAMETERS TEMPERATURE (only if thermal group included) MATERIALS

In Data 2

ELEMENT and/or RAW DATA

A directive shall be correctly spelt and shall be punched alone on one line . Space and erase are ignored.

Identification of elements occurs in both in Data 1 and Data 2. An identification has the same form as the chemical designation for an element, i.e. one or two letters followed by a positive integer. No. character (except erase) may occur between the alphabetical characters. The integer shall be < 511. Example: Iron - FE 26.

Numbers shall be punched on the conventional principles, being followed by two spaces or CRLF.

B) Detailed description of data

Data 1

TITLE

Data. 1 shall commence with a title, which starts with the directive TITLE, and ends with the character / which shall not occur anywhere else in the title. The title may be of any length and is directly reproduced at the beginning of the output tape. -16-

REMOVAL GROUPS AND SPECTRUM PARAMETERS

Under this directive are specified group boundaries in MeV for the removal group scheme and integration step lengths in units of lethargy. For each group the upper energy limit E^ ^ and the maximum step length (Au)^, within the group is punched on one line. The groups are recorded in order of decreasing energy. After the line containing the last group with group number G = G(<_ 49) its lower limit E^ is punched, followed by the character ^ •

Thereafter are punched the values of the parameters p and y in the following expressions for the source spectrum:

Q (E) - exp (- pE) sinh if

DIFFUSION GROUPS or DIFFUSION GROUPS WITH WEIGHT PARAMETERS

Under either of these directives are specified group boundaries in MeV, step lengths in units of lethargy and possibly weight para­ meters for the diffusion groups. For every group g the data are punched in the following order - the upper limit Eg ^ , maximum step length (Au)^ within the group, and weight parameter • The latter parameter is indicated only if the heading specifies "WITH WEIGHT PARAMETERS". The groups are recorded in order of decreasing energy. After the last group with group num ­ ber g = g (< 49) its lower limit E^ is punched followed by the character ^ .

The flux is assumed to be proportional to the weight function mg . = E .If weight parameters are not specified in Data 1, they are assumed = - 1 . For age theory

If age theory is to be applied in th computation of the slowing- down cross-sections below some energy group boundary, an asterisk (*) is punched in the last-mentioned table before the upper limit Eg_^ for the first (in energy highest) of these age groups. TEMPERATURE

■ If it is desired to extend the diffusion group scheme by a thermal group, the directive TEMPERATURE is punched and, under this, the temperature t of the thermal neutrons . The tempe­ rature must be given in degrees Centigrade .

The absolute temperature T = 273 + t is used as parameter in a Maxwell spectrum which in the programme represents the flux spectrum in the thermal group.

Thus the code can compute the thermal constants , assuming a Maxwellian distribution in material consisting of monatomic mole­ cules, l/v absorption and constant elastic cross-sections. This is however a rather bad assumption in some cases, A better result can be reached if measured thermal constants, if available, are used to replace those calculated by NECO. In some cases the input can be modified for effective values for temperature and for the cross-sections at 2.2 km/sec, but often it is simpler to revise the values on the output tape from NECO. For values see table IV in appendix B.

MATERIALS

Under this directive the number of materials is indicated by an integer (<_ 31). By a material is meant a (homogeneous) mixture of elements.

Thereafter the identification of every material m is given in the form of a positive integer (<, 127). After the identification of the material follows its composition.The latter consists of a list of all Jm (<•. 31) elements in the material. The list includes for each ele­ ment j its identification (see below), followed by the concentration cm’J of the element j in the material m . The concentration is - 3 expressed in £ g cm -] . The end of the list is indicated with the character ^ .

The materials may be recorded in any order. There may be a maximum of 1 27 different elements, but not more than 31 per material. -■ 18

Sample problem

Thirteen direct groups = removal groups Nine diffusion groups, of which three age groups and one ther mal group Two homogeneous mixtures of elements

Age theory is employed for computation of Zg_^7 , Ey^g, and . Also Dg , Dy , Dg , Sag , Za7 and Z^g are computed using formulas which are consistent with age theory.

TITLE

CASE I 1 JULY -63 4

REMOVAL GROUPS AND SPECTRUM PARAMETERS

18.0 0.05 16.0 0.05 14.0 0.05 12.0 0.05 10.0 0.05 8.0 0.05 6 . 0 0.05 4.0 0.05 3.0 0.05 2.0 0.05 1.0 0.05 0.5 0.05 0. 1 0.05 0/01 - 4

1.03 2.29 - 19 -

DIFFUSION GRUPS WITH WEIGHT PARAMETERS

14.0 0.05 - 2 5.0 0.05 - 1 1.0 0.05 0 0. 2 0.05 - 0.5 0.05 0.05 - 1 * 0.01 0. 10 - 1 0.001 0.10 - 1 0.0001 0. 10 - 1 0.00000025 4

TEMPERATURE

250

MATERIALS

2 100 H 1 0.1 1 O 8 0.89 4 101 CA 20 0.70 FE 26 0.40 SI 14 0.40 H 1 0.02 O 8 0. 38 4

Data 2

Microscopic cross-sections in the data library.

Since the data tables are fairly extensive - probably com­ prising one to two thousand numbers per element - the collection of all cross-sections on one tape would make it unmanageable. Each tape shall therefore contain complete data only for one element. - 20 -

All cross - sections will be expressed in barns.

a) ELEMENT

This directive introduces the data for each element.

Under the directive are punched the identification of the ele­ ment and its atomic weight A , followed by Lhe character 4 ,

Example

ELEMENT

FE 26 55. SIT ^

Between the atomic weight and the character 4 there may be inserted a text of maximum 80 characters (shift characters included) indicating the data reference or the like. The text shall not include the character =/ .

Example ELEMENT . FE 26 55.85 E.S TROUBETZKOY NDA 2111-3 4s .

b) Thereafter follows a table of"sigma total”, ”sigma scattering” ”sigma nn 1" and ”sigma n2n ” as functions of the energy E .

For each tabulation point n , there is punched in order:

En ° VEJ °nn'

The tabulation shall be made with monotonously decreasing E^ . If cfn 2n is everywhere equal to 0 , this quantity need not be punched, cr f can also be eliminated if the cross-section is every ­ where equal to 0 , provided that cfn 2n * s lacking.

Number 2, ,3 or 4 is punched before the table to indicate the number of cross-sections in the table, which must be concluded with the character 4 -

Below the table should be punched the numerical values of "sigma elastic” and "sigma capture", at 2200 m/s <(0.0253 ev). ~ 21

c) After the cross-sections follows a table of Legendre coefficients. An integer L (<_ 10) indicating the number of coeffi­ cients is punched first. Thereafter follows for each tabulation point n , the Legendre coefficients in the centre-of-mass system:

E„ MEn> L...... fJEn>

The energies shall be monotonously decreasing. The table concludes with the character ^ .

If all Legendre coefficients are equal to 0 , the table as a whole can be eliminated and 0 is punched instead of L .

d) The Legendre coefficients are followed by four quantities in the order a. , B , , a0 , B . B , and B are threshold 1 nn 1 2 n 2n nn ' n 2n values in the laboratory system for nn 1 and n2n cross-sections. If the data for the element are thereby complete, the character ^ is punched after B^^,^ . e) If cross-sections for separate energy levels for nn' scattering are known, these energies are punched after B^^ in order of increasing energy. The sequence of values ^ (with t = 1, 2, . . . , T) is followed by the character . T must not exceed 1 1 .

For each tabulation point n there is then punched in order:

En 'cJEn> °eT

Tabulation shall take place with monotonously decreasing and the table concludes with the character ^ .

RAW DATA

With the programme the microscopic data can be converted into binary form (see Operation of NECO, page 48). The advantage of this is that the input time is reduced and the reliability of input increased. - 22

The binary data tapes produced by the programme start with the directive RAW DATA followed by the identification of the ele­ ment in plain text.

Note 1 . The tabulation points need not to be the same in the tables referred to in the tables described under 6 , c, and e.

Note 2. a) The cross-sections ” sigma total”, ” sigma scattering”, etc. must be tabulated up to an energy > 1 . 000001 x the upper limit of the group scheme and down to < 0. 999999 x the lower limit of the group scheme (excluding thermal group if present).

b) The Legendre coefficients must be tabulated up to 1.000001 x the upper limit of the group scheme.

c) The cross -sections for separate levels must be tabu­ lated down to 0. 999999 x B , . nn'

Note 3. Observe that the programme computes nn 1 scattering only for primary energies > the specified value of , irrespec­ tive of whether 0 , differs from zero at lower energies, nn' The same applies to n2n scattering, which is calculated for primary energies > •

OUTPUT

A title is first printed, followed by the Input List (see page 47).

The group constants are punched in the order in which they are to be read into the programmes REFUSE or REBOX and NEDI. If the directive REMOVAL GROUPS AND SPECTRUM PARAMETERS does not occur in DATA 1, data are obtained only for the NEDI pro­ gramme. The two data groups are 4separated by a long portion of blank tape and a row of erase characters. - 23

Data for REFUSE and REBOX

1) A rectangular "pattern" described under 5)

2) The directive CONSTANTS and the source spectrum para­ meters P and y and for programming reasons, an integer indicating the number of materials

Thereafter for each material

3) Identification of the material

4) The’Removal" cross-sections Z for the direct groups arranged in order of increasing G = decreasing energies

5) A table of ^ G-> g ’ arran §ec* primarily in order of increasing consecutive number of g of the diffusion groups and secondarily of increasing consecutive number of G of the direct groups, i. e .

S . ZL...... Z ZTI~N ...... "G'A

y y Z - ITv2 ...... G,2

Z T Z - Hg H-2g ...... G-»g

To reduce the volume of output and the space requirement in transport programmes the matrix elements Z^-. which are

negligible for all materials are not printed. And Zqis regarded as negligible when the corresponding microscopic

cross-section o Tq^ is zero or less than a given small quantity for all elements j . To indicate which quantities have been included and excluded, a pattern with the title PATTERN is printed before the aforementioned tables. In this pattern every Z-Q^g (for any material) that is not negligible is represented by a one and a "zero" cross-section by a zero. The indicators are printed in the same order as the corresponding In the table s.

6 ) Directives WAIT and START - 24 -

Data for NEDI

11) A triangular pattern under the directive PATTERN, analogous to the preceding pattern.

12) The directive CONSTANTS and an integer indicating the number of printed quantities per material according to 14) and 15)

Thereafter for each material

1 3) Identification of the material ■

14) The diffusion constants and the group absorption cross­ sections Z for the diffusion groups, both arranged in order of decreasing energy

15) A table of ^, arranged primarily in order of increasing consecutive number g1 = 2,3,4... g of secondary groups and, secondarily, of increasing consecutive number g = 1 , 2, . . . , g 1 - 1 of primary groups, i.e.

Z l->2

Z 1 —>3 J2->3

(7.2) t 1 ' f Z Z Z l-*g 2->g g-l-> g

The values of Z which are negligible for all materials are not printed. Ones and zeroes in the pattern matrix indi ­ cate which Z , have been and which have not been printed g-»g' in subsequent tables.

1 6 ) Directives WAIT and START

Running time etc.

It is difficult to give a genera] formula for the running time since it is dependent on many factors. Certain guiding lines will be indicated, however . ' - 25

Read-in of the programme takes about 2 minutes . The times for dealing with the data for the various elements are added. Input of microscopic data takes about 1.5 min. per 1000 binary numbers, about a couple of minutes per element . The computation time varies greatly, being dependent on such factors as the number of groups, the number in which age theory is employed (giving a shorter compu­ ting time), number of integration steps, maximum slowing -down length versus group width, threshold for nn 1 absorption, threshold for n2n absorption. The maximum output time - 49 groups and only figures "1" in pattern - is about 40 seconds per material using a high speed punch.

Data shall be read-in in the order : Data 1 , Data 2. As already noted, the latter will form part of a data library to be stored with the programme. They are then in the form of a number of tapes, one for each element. .

Data for the various elements shall be read-in in the order in which they occur in the material specifications . The input sequence is shown by an INPUT LIST which is printed during the run as soon as Data 1 have been read-in. When removal, data are requested the input list is divided into two par! s, each part containing all the ele­ ments concerned. The list may sometimes show that data for cer­ tain elements shall be read -in more than onc.e in each part. This is because the drum capacity is insufficient for storing group constants as well as certain intermediate results for all materials simultaneously. Owing to the space requirements for these intermediate results, and certain other circumstances, this may occur even if the final data can well be stored by the codes in which the group constants are to be used.

It is advisable to enclose a list of notations of the elements in question so that corresponding Data 2 lists can be selected before the run. - 26 -

8, a Data for REFUSE

Every group of data is preceded by a directive . A description of the possible directives and the corresponding data is given.

1 . TITLE The characters (maximum 254) following this direc ­ tive are interpreted as the title of the problem and are reproduced on the output. The title is terminated by at least two consecutive figure shifts, and care should be taken that these do not appear between the directive TITLE and the beginning of the title.

Ex: TITLE REFUSE TEST PROBLEM 9.8.63 (fig. sh. , fig. sh.)

2. ENERGY This directive is followed by G = the number of flux energy groups and then the energy boundaries E^(G =0, 1 ,. . . , G) in Mev beginning with the highest energy.

Ex: ENERGY 18 18 17 16 15 14 13 12 11 10 9876543210

3. GEOMETRY PLANE or GEOMETRY C ILINDRIC AL is a direc ­ tive with obvious meaning.

4. MATERIALS This precedes the data for the shield configura­ tion. First comes an integer which indicates the number of regions (at most 28). After that the coordinates of the region boundaries follow (with arbitrary origin), starting from the core-shield interface and between these boundaries identification number corresponding to the materials in the respective regions. An air-gap is denoted by 0 .

Ex: MATERIALS 7 21 5(103)245(192)250(103)261(102)267. 5(0)282.5(101)412. 5(101)61 2.5 - C l

5. SOURCE Following the directive we have V and as defined in eq. (3. 3). Then an integer giving the number of values of 6 . ^(x) , followed by a table with x to the left and

Note. If (he geometry is cyl ndricaJ, T should be set equal to the radius of the core, and the flux values at a distance x from the core face are multiplied by T/(xfT) during calculations.

Ex: SOURCE 2.5 3.84, -3

8 0 3.6, 1 3 20 6 .0, 1 3 40 7.4, 1 3 60 9.0, 1 3 80 13.6, 1 3 100 14.5, 1 3 1 50 15.0, 1 3 215 15.0, 1 3 103 215

6. PRINT If this directive appears on the data tape, the removal flux and removal source vectors for each dose point will be printed after the calculations. If the words

DOSE FACTORS are included, followed by the dose factors d^(G = 1 , 2, ... G), the sum £ *G in every dose point will be G printed in connection with the PRINT output.

8. WAIT causes the computer to' come to a hoot stop during in ­ put. Reading is resumed when key 9 is pressed and released. If the data are divided among several tapes, each tape should end with WAIT so that the next tape can be placed in the reader.

9. The removal cross-section and the matrix, elements for the computation of the removal source vector in each material are given in a distinct group of data, the Material Constants Library. The library tape is provided by t e programme NECO (see page 23). - 28 -

10. The last directive of the data is always START, which ends the data reading and initiates the calculations.

Some of the directives must be given in a certain order, as set out below:

Directive Must be preceded by

1 . TITLE 2. ENERGY 1 3. MATERIALS 1 4. SOURCE 1 5. PRINT 1 6 . DOSE FACTORS 1, 2, 5 7. PATTERN 1, 2, 3, 4

Certain faults in input data can be detected, and the computer then comes to a hoot stop, printing the word FAULT followed by a fault number (see p. 54).

If any of the directives ENERGY, MATERIALS or SOURCE are missing on the data tape, fault 4 will be printed. There is, how­ ever no such test on the directive PATTERN. This is to make it possible to make several runs in succession with the same materials without having to read the library tape each time.

Data example

For clarity, a complete set of data for a run (the same as quoted in the description of directives) is included. The reactor core is of 215 cm radius and we have 7 shield regions of -which number 5 is an air-gap. The materials are: 101 concrete , 1 02 iron and 103 water. To get more accurate results, the 330 cm concrete layer is divided in two parts by an artificial region boundary introduced at 412.5 cm. We have 18 removal flux groups, all of which go into one source group. - 29 -

TITLE REFUSE TEST PROBLEM

ENERGY 18 18 17 16 15 14 13 12 11 10 9876543210

GEOMETRY CYLINDRICAL,

PRINT SOURCE 00 2.5 VO - 3

8 0 3.6, 1 3 20 6 .0, 1 3 40 7.4, 1 3 60 9.0, 1 3 80 1 3. 6 , 1 3 1 00 14.5, 1 3 1 50 15, 1 3 215 15, 1 3 103 215 MATERIALS 7 215(103)245(1 2)250(1 03)261 (102)267. 5(0)282. 5(101)412.5(101)612.5

WAIT

The a we have the Material Constants Library tape as follows:

PATTERN 111111111111111111

CONSTANTS 1.0363 2.2900 3

101 .076 . 076 .075 . 078 . 080 . 078 . 074 .070 .068 .073 .074 . 076 .089 . 102 . 1 36 . 108 . 172 .273 . 076 . 076 . 075 . 078 . 080 . 078 . 074 .070 .068 .073 . 074 .076 .089 . 1 02 . 1 36 . 108 . 1 72 .273 - 30 -

102 .110 . 114 .119 . 118 . 125 . 134 . 141 . 147 . 154 . 164 . 172 . 180 . 182 ;177 . 166 . 153 . 142 . 166 .no . 114 .119 . 118 . 125 . 134 . 141 . 147 . 154 . 164 . 172 . 180 . 182 . 177 .166 . 153 . 142 .166

103 .054 .055 . 055 . 059 .061 .061 .060 .060 . 060 . 070 .071 .074 .086 .098 . 125 .113 . 163 . 222 .054 .055 .055 . 059 ' .061 .061 .060 .060 . 060 . 070 .071 .074 .086 .098 .125 . 113 . 163 . 222

START

The beginning of the print-out is shown here with the ACC. MAT. specifications for regions 1 and 6 and the flux and source values for the first dose points in region 1.

The SOURCE POLYNOMIALS - output is shown and described in connection with NEDI - input (page 41}.

REFUSE . REFUSE TEST PROBLEM 9. 8. 63 ACC. MAT 1 = 4. 16, - 2 ACC. MAT 6 = 1.01, - 2

L FLUX SOURCE 215. 0 8. 086, + 6 1. 856, + 7 4. 327, + 7 9. 239, + 7 2. 045, + 8 4. 671 , + 8 1. 078, + 9 2. 471 , + 9 5. 357, + 9 9. 804, + 9 2. 078, + 10 4. 187, + 10 7. 280, + 10 1. 256, + 11 1. 713, 4 11 3. 688, + 11 3. 617 , + 11 2. 487, + 11

SOURCE SUM 2. 017 , +11 - 31

L FLUX SOURCE 219.4 4.441, + 6 5.438, + 10 1.010, + 7 ' 2. 355, + 7 4. 853, + 7 1. 056, + 7 etc

8. b Data for REBOX

The new set of data introduced under the directive SOURCE is illustrated by an example in general form. Here V and B^ have the same meaning as in REFUSE, i is the identification number of the core material. The specifications under SHIELD LINE are the coordinates of P* (see also the code description page 8). The figure following ACCURACY is the relative accuracy for the polynomial approximation of the removal source, and it is used according to the description on page 8 , where REFUSE uses the fix value 0.01.

Due to the usually rather long running times for REBOX, it has proved necessary to control the number of dose points treated in each shield region, and this can be done by specifying a suitable ACCURACY value. If a large value, say >10, is given, only the region boundaries are treated, and the corresponding polynomials fit to the logarithm of the removal source values is linear. If zero is given, five dose points are treated in each shield region and the fit will be of 4^ degree.

The accuracy of the integration for the flux values method is wholly dependent on a suitable division of the core. In no direction can the core be divided into more than 50 parts, which gives a maximum number of cells = 125, 000 (note that the running time is proportional to the number of cells). These cells should be small in the vicinity of P1 and large in the more remote parts of the core. It is recommended that each problem be run with different core divisions to obtain a check on the results. - 32 -

SOURCE Example of SOURCE data in general form for REBOX. The f, g and h V Sf ' i functions are chosen so that, for every X source cell with centre x y z , we have f(-^(x0 + Xj)) x0 . f(x) g(y) h(z) =

f(|(xl_l +xL xl-l

X1

Y

g(i(Yo + y^)) yo

h({(%Q + %i))

z n

SHIELD LINE

X = x Y = y

ACCURACY F - 33

9. Data for NEDI

Data for NEDI consist normally of three parts, called here DATA 1, 2, and 3.

DATA 1 gives the relevant geometrical data and output directives for the problem. It is the only data that is not prepared by preceding codes in normal production runs.

DATA 2 gives material constants (diffusion constants and cross­ sections). It is normally prepared by programme NECO, and can in many cases be considered as a library tape for a number of problems having the same materials.

DATA 3, finally, gives the neutron sources. Normally this data is a direct output of the removal part, but may also be prepared manually.

To illustrate the use of the programme we consider its applica­ tion to a six group problem to be solved in the geometry described in section 4. The source Q^(r) is a given function not identically zero, Q2(r) = Q3(r) = ... = Q^(r) = 0.

There is slowing down from one group to the next only. The materials are numbered

concrete 101 iron 102 water 103

The boundary conditions are

the fluxes given at r^ the flux-to - current ratios given at r^

Data for this problem can be set out as follows.

TITLE . SAMPLE PROBLEM FOR NEDI REPORT 9. 8. 63. (CR LF fig. shift fig. shift)

GROUPS 6 GEOMETRY CYLINDRICAL MESH 215.0(2. 00)245. 0(1.00)250. 0(2. 20)261. 0(0. 93)267. 5(15) 282. 5(1. 00)292. 5(3. 33)412. 5(3. 33)612, 5 34 -

DIVISIONS 103 102 103 102 0 101 101 101

BOUNDARY CONDITIONS

flux RATIO 0. 287, 13 2. 13 0. 3 24, 13 2. 13 0.159, 13 2. 13 0. 669, 13 2. 13 0.961 , 13 2. 13 3.600, 13 2. 13

PRINT FLUX SPACING 5 2 2 3 1 2 6 6 GROUPS 1 2 6

PATTERN 1 01 001 0001 00001

CONSTANTS 17 101 1. 634 1. 370 1.428 1.343 1. 321 1. 250 0.0306 0. 0529 0. 1499 0. 0390 0. 0294 0. 1951 0.0302 0.0529 0.1499 0. 0389 0.0381

102 2. 057 1. 115 1.429 0.482 0. 354 0. 354 0. 0240 0. 0048 . 0.0047 0. 0066 0.0331 0. 1608 0. 037 0.0041 0.0039 0. 0050 0.0051

103 1.707 1. 589 1. 633 1. 639 1. 639 1. 637 0. 0699 0. 0651 0. 1384 0.0331 0. 0231 0. 1221 0. 0699 0. 0651 0. 1384 0. 0331 0. 0230

SOURCE POLYNOMIALS REFUSE TEST PROBLEM 1 215.0 - 245.0 4 1 45. 970 - 2. 618 0.277 - 0. 108 0.083

245. 0 - 250. 0

2 1 41,180 - 0.516 0.005

250. 0 - 261. 0 - 35 -

2 1 37. 770 - 0. 677 0.017

261.0 - 267.5

2 1 36. 330 - 0. 638 0. 005

267. 5 - 282. 5

0 1 - 350

282. 5 - 412. 5

4 1 21. 320 - 5. 952 0. 260 _ 0. 056 0.020

412. 5 - 612. 5

4

1 - 5. 748 - 7. 711 0. 074 - OlOOS 0.003

START

Data 1

The word TITLE will cause the machine to read and print out the characters following the directive up to two consecutive figure shift symbols.

The word GROUPS is followed by an integer specifying the number of energy groups.

The geometry of the problem is indicated by the directive GEOMETRY followed by one of the words PLANE, CYLINDRICAL and SPHERICAL.

Data after MESH is self-explanatory. All division interfaces in REFUSE or REBOX input must appear in the sequence even if the mesh points are equally spaced in two adjacent divisions. The maxi­ mum number of mesh intervals in a division is 1023. If the division width is not a multiple of the specified mesh size, the programme will adjpst the mesh size appropriately. Then, in computing the - 36 -

number of intervals in a division, the computer may get a result different from that intended by the problem originator. If it is essen tial to avoid this, the relative error in the specified mesh size must not exceed l/(2 k + 1), where k is the number of intervals in the division.

As mentioned before, no universal rules can be given for the choice of mesh size in a region, but as a first trial the smallest value of y D/Z^ of the non-thermal groups may be used. Near an interface special care must be taken, and special spatial regions should be introduced here with smaller mesh sizes than elsewhere. Within a few mean free paths from an interface the mesh size de ­ pends on the value y D/Z^ of the material on the other side of the interface, whence the smallest one of the mesh sizes should be used on both sides of the interface. The rules given here serve only as an orientation, however, and they can be shown to be too severe in a great many cases, but a couple of trials for a problem will indicate a suitable mesh size.

The integers after DIVISIONS specify the media of which the divisions are composed.

BOUNDARY CONDITIONS is followed by a table of boundary values arranged in two columns, one for each boundary. Each column has a heading indicating which type of boundary condition is used, FLUX for flux values, CURRENT for current values and RATIO for flux to current ratios. The column headings may be combined arbitrarily, but at one boundary the same type of condi ­ tion must be used through all groups.

Usually in a bulk shield the flux is given at the inner and the flux-to-current ratio at the outer boundary. In most cases, however the flux at the core face is primarily divided into one thermal group and only a few non-thermal groups (perhaps only one). In order to convert this into a multigroup boundary condition we have to make some assumption about the shape of the neutron spectrum. For epithermal groups (up to around 1 MeV) we can 37 -

usually assume a l/E shape. If a primarily given group is to be divided into subgroups, the flux in subgroup g will be proportional to Dg(Au)g , where (Au)^ is the lethargy width of the g*"* 1 group.

For a fast group the spectrum connects more to a fission spectrum, but as this part is provided for by means of a removal flux, it is usually satisfactory to extrapolate the l/e spectrum up to about 2 3 - 4 MeV and to vary perhaps as E” or E above this value.

For the outer boundary, the conventional extrapolated distance is used in most cases. This makes all the RATIO values = 2. 13 as it makes q>./ = - 2. 13 D , which is the usual extra- r dr polated distance.

The result from the diffusion calculation can be obtained in tabular form or as a tape to be fed into a data plotter. The directives to be used are PRINT FLUX and PLOT FLUX respectively. After the directives the spacing of the print-out is specified by one integer s. for each division. The fluxes will be listed at every s^1 point i ' th. ^ in division 1, at every s^ point in division 2 etc. The integers s. must not exceed 511. Boundary and interface values are always printed. The energy groups for which the fluxes are required are specified on the next line after the word GROUPS.

If the tabular alternative is chosen, the output consists of space variable and flux values at the mesh points required.

The layout of the output to the plotter is similar, but the flux values are replaced by their common logarithms* A few directives not used in the sample problem are available. After LATERAL 2 LEAKAGE the buckling values a may be listed, one for each 2 division. If the directive is omitted, the programme sets a = 0 .

The directive DOSE RATES n is to compute dose rates d. = C. cp,i=l, ...,n. 1 g 1> g g

The number of dose rates n must not exceed 5 in the present programme. The factors C. are listed after the directe as follows. - 38 -

DOSE 1 cm Cl, 2 * * * Ci.T

ba,i DOSE 2 C2, 2 ’ * * °2,Tg etc.

The flux output is assumed to be in tabular form, and the dose rates are printed after the fluxes and with the same spacing. A gamma source is defined as

g F

,(a) = £ for g = 1, 2, a § /, g->g' 'g-1 g '=g+1

'(a) _ ag If a print-out of the gamma source is required, the directive PRINT GAMMA SOURCE is given. It is followed by SPACING specifi­ cations in the same manner as PRINT FLUX (the GROUP line omitted, of course)

Ex: PRINT GAMMA SOURCE SPACING 3 11112 3 3

The word WAIT on the input tape will cause a hoot stop which is passed by depressing and releasing handswitch 9 on the control desk.

When the complete set of data is entered, the word START initiates the calculation. Thus all information relevant to the problem must precede this directive. During the first few seconds the data are checked and arranged in a more convenient order. Then, for each energy group, the source Sg is computed and the group equa­ tion is solved. The output required is produced and the machine will read and interpret the next directive, which —ill probably be a WAIT or the title of a new problem. - 39 ~

Data 2

The constants, i.« e, diffusion coefficients and cross sections, for the components of the shield are listed after a directive CONSTANTS followed by an integer specifying the number of constants for each component. Suppose that the values for concrete (medium No. 101) are:

g °g

1 1. 634 0. 0306 2 1. 370 0.0529 3 N 00 0. 1500 4 1. 343 0. 0390 5 1.321 0. 0294 6 1. 250 0. 1951

Zg'-*g g*1 2 3 4 5 6 g 1 2 0.0302 3 0 . 0. 0529 4 0 0 0.1499 5 0 0 0 0.0389 6 0 0 0 0 0.0281

These constants can be arranged as follows.

CONSTANTS 27 101 1.634 1. 370 1.428 1. 343 1.321 1. 250 0.0306 0.0529 0..1500 0. 0390 0.0294 0.1951

0.0302 0 0. 0529 000.1499 0 0 0 0. 0389 0 0 0 0 0. 0281 - 40 -

The order is number of the medium m (1 < m < 511) diffusion coefficients Dg absorption cross-sections Zag transfer cross-sections 2g'-»g by rows

0 as a material number is used to indicate an air gap. No con ­ stants are given, the programme automatically sets ^ = Z = for all gl and g. ^ g^g

In the slowing down model chosen for the sample problem ten of the fifteen transfer cross-sections Z , are zero in all media. -- g'-7g To reduce the input volume there is a facility which permits these constants to be omitted if the positions of the non-zero constants are indicated by a pattern symbol. Using this device the constants above can be written

PATTERN 1 01 001 0001 00001

CONSTANTS 17 101 1. 634 1.370 . 1.428 1. 343 1. 321 1. 250 0.0306 0.0529 0.1500 0.0390 0. 0294 0. 1951 0.0302 0. 0529 0.1499 0.0389 0. 0.281

The triangular pattern symbol has (g - 1) rows and columns and is related to the table of transfer cross-section in an obvious way. It is common to all media, and only constants which are zero in all media can be represented by zeros in the pattern. If no PATTERN directive is given, it is assumed that all cross-sections are in the lists, i. e. the programme will use a pattern symbol with 1 's in all positions. - 41

This part of the data is normally prepared by NECO, and the only thing to note is that the thermal constants (the last ones of the diffusion constants and the absorption cross-sections) sometimes have to be modified (see page 17 and table IV in appendix B).

Data 3. Source functions

The source functions Q^(r) can be specified in several ways.

1) A directive SOURCE ZERO is used to set Q^(r) = 0 for all g .

2) A source can be entered in tabular form with the directive SOURCE TABLES which is followed by lists of source values, one list for each of the groups in which Q (r) ^ 0 . A list . ■ § consists of the group number g and the source values Og(rk) » k = 0 , 1 , . . . At division interfaces, where Q^(r) may be discontinuous, two values shall be given, one left- hand value and one right-hand value. If 0^(r) = 0 for some g , the list for that group is omitted. Thus for our sample problem we can specify the source in the following way.

SOURCE TABLES Group number 1 'i 2.02, 11 1. 05, 11 6 . 00, 10 3.73, 10 2.46, 10 | Source values 1.70, 10 1. 22, 10 8.85, 9 6 . 52, 9 4.82, 9 j^ in division 1 3.58, 9 2. 65, 9 1.97, 9 1.48, 9 1.12, V 8. 64, 8 1.47, 9 1. 19, 9 9.68, 8 7.88, 8 6.42, , Source values 5. 24, 8 j in division 2 3. 16 , 8 2. 38, 8 1. 18, 8 1. 38, 8 1.06, 8i1 Source values 8. 15, 7 jj in division 3

etc etc

The machine will automatically set Q%(r) E Q3(r) = . * * E Q^(r) = 0.

3) The source functions can also be expressed in Chebyshev polynomials.

0g(r) = exp (J" ^ T* (r))

where S" denotes a finite sum whose first and last terms are to be halved, and T^(r) , k = 0, 1, . . . , are Chebyshev polynomials shifted to a convenient interval ranging over one or several divisions. It is not expected that source specifica- 42 -

tions of this kind will be prepared manually, but they are convenient as links between programmes like REFUSE and NEDI. The directive used in the1 REFUSE output is SOURCE POLYNOMIALS.

It is followed by a tape identification, which is copied on the NEDI output tape, and an integer specifying the number of groups with Qg(r) = 0 (i. e. the number of polynomials to be given in each range; this number must be the same in all sub-ranges of (r^, r^.)).

SOURCE POLYNOMIALS Directive REFUSE TEST PROBLEM Tape identification 1 ' • Number of polynomials 215. 0 - 245. 0 Sub-range 1 4 Degree in the sub-range 1 45.970 - 2. 618 0.277 - 0. 108 0. 083 Group number and coefficients 245. 0 - 250. 0 Sub-range 2 etc. etc.

For each sub-range is specified, as is shown above, the end ­ points, the degree of the polynomials in the sub-range (the same degree for all groups), then a group number and the polynomial coefficients a^ , for that group, another group number and the corresponding coefficients, etc. until aL group sources in the sub­ range have been defined. Group sources not included in the list will be set = 0 .

In order to allow for polynomial approximations of low degree the source in the concrete shield is defined by two polynomials ranging over the intervals (282, 5, 412. 5) and (412. 5, 612. 5) respectively.

However, the range of a source polynomial must be an integral number of divisions; that is the reason for introducing an artificial interface at r = 412. 5. on page 33. - 43 -

General value_s

A few rules as to the order of the directives must be known by the user of the programme. They are summarized in the follow­ ing table. '

Directive Must be preceded by Must precede

1 . TITLE 2. GROUPS 1 3. GEOMETRY 1 ■ 4. MESH 2 5. DIVISIONS 4 6 . BOUNDARY CONDITIONS 2 7. DOSE RATES 2 1 3 8. LATERAL LEAKAGE 4 1 3 9 a PRINT FLUX 4 9 b PLOT FLUX 4 9 c PRINT GAMMA SOURCE 4 10. PATTERN 2 1 1 11. CONSTANTS 5 12. SOURCE 3, 4 13. START 1 6 , 9, 11, 12 14. WAIT • May come anywhere

Directives 7, 8, 10 and 14 are optional. At least one of the output directives 9 a, b and c must be included in the data.

Limit a ti on s_

Let g be the number of energy groups, d the number of divisions, e the number of ones in the pattern and K the number th n of mesh intervals in the n division. Define an integer t as the integral part of the fraction

K + d + 3l)/32 . n " - 44 -

The following conditions must be satisfied

3 < 51

d < 31

K < 1023 for all n n — t(g + 2) + 3 % + e < 648

The computing time, input and output excluded, varies from a few minutes for the six group sample problem to 20 - 30 minutes for problems which use the full capacity of the backing store. The compu­ ting time is approximately proportional to the number of groups and the number of mesh intervals.

10.Data for SALOME

The input information is divided into three parts representing

1) cross- section and dose response data, common to many problems,

2) geometrical and energy data for the specific problem, and

3) source distribution data.

Each part may be entered into the computer on a separate tape. Only Data 2 is manually prepared for every problem under normal condi ­ tions.

Data 1. Material constants

This tape may be common to many problems and contains information on cross-sections and the dose response function. It may contain data for more materials than are actually used in the specific problem dealt with a) A title of not more than 1 28 characters, starting with a letter shift and ending with two or more consecutive figure shifts (blank tape). b) 25 input energies in Mev given in decreasing order of magni­ tude. c) 25 values of the dose response function, in arbitrary units per Mev /cm « s, at the energies determined in b). - 45 -

d) Number of materials, of which data are given. The fast storage restrictions of the Mercury do not allow for more than six different materials in a shield.

e) For each material the following information should be given: m = material number (m < 512) "X(E) = total mean free path at the 25 energies and in the same order as under b) in centimetres

gCompton * °pair prod. ,, atotal

pair prod. at the nine first of the energies gpair prod. + aCompton

given under b) (the last value is usually zero).

Data 2. Geometrical data

a) A title in the same way as in Data 1 a)

b) N = total number of regions in the shield (less than 3 2)

c) d Q > dj , . . . , d^ +^ = coordinates of boundaries of shield regions (in centimetres). These regions and their coordinates are of course independent of the regions used in NEDI.

d) mg , m-| , ... . , i = material identification numbers referring to the values given in Data 1 .

e) fg , , . . . , f j = conversion factors from neutron reac­ tion rate (in reactions per cm^ • s) to photon source density (in photons per cm3 • s) of the specific energy group, say 6-8 Mev, being analysed.

f) Eg = source energy in Mev

g) 'k'min ~ ener §y cutoff, below which the photons are no longer followed but are considered as having been absorbed. This energy whould be set equal to about 0, 25 Mev in most shielding problems. A new run has to be made for each source energy. - 46 -

Data 3. Source distribution data

a) A title which should always commence with the words "Gamma Source" after which an arbitrary text may follow, ended as .in Data 1 a). b) A table of the source distribution;

. Xg(cm) . S(xq) (reactions per cm^ • s)

X1 S(xl> .

...... (at most 256 values)

As mentioned before, the source distribution data tape is normally supplied by the NEDI-3 diffusion programme.

Output

The output gives a specification of input data used and is suffi­ ciently descriptive to make it self-explanatory. The information given is 1) the average energy deposition rate in each of the shield regions in watts per cm^> s and the total amount of gamma energy deposited in the shield in watts per s,

2) the total dose rate, and contributions to it from each of the shield regions in the units chosen in Data 1 , c) and

3) the dose rate contributed by uncollided quanta in the same units as the total dose rate.

An output print is provided for each accumulated group of 96 source photons started, to be enable following of the convergence.

A photon history can last from less than one second up to about 3 seconds, a typical average, however, being about 1.3 seconds. For one energy a typical shield requires about one hour. - 47

OPERATION INSTRUCTIONS 1 1.Operation of NECO

Part I, Calculation oi group constants

1) Deisolate columns 2-25 and 30. Read-in binary programme NECO. A wait signal is issued.

2) Data 1 is put in the reader and key 9 is thrown up and down. Wait signal. Write-out the data punched hitherto up to erase characters. The output is titled INPUT LIST, being a list of the notations on the required Data 2 tapes (forming part of the data library associated with the programme) and also indicating the order in which the tapes shall be taken. The list often indicates that the tapes shall be read-in twice. For rare major problems some tapes must be read several times. Manually punched Data 2 tapes or binary versions car be used according to choice.

Example of INPUT LIST

H 1 O 8 CA 20 FE 26 SI 14 xxxxxxxxxxxxxxxxxxxx

The tapes, shall be marked in analogy with the following example: "FE 26 (iron data to NECO)".

3) The read-in of the Data II tapes starts when key 9 is thrown up and down. When a tape has been read-in, computation starts and lasts some minutes. In the meantime the next tape on the INPUT LIST should be placed in the reader, is then read auto­ matically, and so on.

If the wrong tape is placed in the reader, a wait signal (fault 23) is returned when the notation is read-in immediately after the introductory directive. Then insert the correct tape and continue with key 9. — 48 —

When all data on the INPUT LIST have been read-in and the result has been punched, a wait signal is issued indicating that the computation is finished. Then one may start again from point 2 with a new problem.

Part II. Preparation of binary version of manually punched data tape belonging to data library

1) as in Part I.

2) Insert the Data 2 tape and operate key 9. The data are read- in and checked for certain faults on the fault list. The machine then punches the data in binary form and issues a wait signal.

3) The punched tape - carrying a number of characters CR at the beginning and end - is placed in the reader for checking. Key 9 is thrown up and down and the read-in starts. A check is made for faults 23, 40, 42 and 43. In the event of stoppage see "action on stoppage".

If the tape is found to be correct, a new wait signal is issued and operation can continue from point 3 of Part I or II.

Action on stoppage l) ERROR write-out

The data tape which is being read or, if computation is in progress, the last tape to be read, is replaced in the reader, provided that the data are correct. Key 9 is then thrown up and down, whereupon one of the following two events occurs.

a) Normal case The tape enters and operation continues in accordance with the Operation Instruction.

b) Exceptional case The machine does not read any character but immediately makes a new ERROR write-out. It is impossible to con ­ tinue the run. - 49 -

2) 990 stoppage on fault 40, 42 or 43 in binary data tape. The fault number in index register B 7.

The tape is moved back in the reader to the start of the sector at which the stoppage occurred. The sectors are separated by 21 figure shifts and each sector starts with CR, LF, CR, LF. Press the PREPULSE button.

The sectors have consecutive numbers (but not sector numbers) and on read-in the machine accepts only the sector with the next consecutive number.

First, sector is 800.

1 2. a_____ Operation of REFUSE '

The programme is written for a Ferranti Mercury computer with 4 drums. It occupies sectors 2 - 1 1, 128 -186 and its working space is from 250 to 940, which should be deisolated. The programme tape in binary form is entered with key 2. At the hoot stop the data tape is placed in the reader, whereafter key 9 is pressed and released. If there are more than one data tape, each one will end with WAIT, allowing the operator to change tapes.

If something goes wrong, the programme can be restarted in the usual way with first sector 128.

The computing times are a bit difficult to estimate, owing to the special method of integration, but an average of 2. 5 seconds per energy group and dose point should be normal, not counting the in- and output.

On the output tape, the PRINT output and the SOURCE POLY­ NOMIALS output are separated by 50" of blank tape.

12. b Operation of REBOX

For a problem with G energy groups and N source cells the time per field point is approximately

N(8 G + 16) milliseconds 50 -

The possible faults that can be detected during input are the same as for REFUSE.

The programme occupies sectors 2-11 and 128 - 178; the working space is the same as for REFUSE, as is the running of the programme.

13,_____ Operation of NEDI ' •

NEDI is written for a Ferranti Mercury computer with four drums (32 768 words backing store). It is located to sectors 736 - 831 and sectors 2-5, which hold part of the Mercury Input Routine. First sector is 736. Working space is sector 64 and onwards.

The programme is in binary form and is entered in the usual way by setting key 2, When the programme is in, the machine comes to a hoot stop which is passed by depressing and releasing key 9.

Some obvious errors in the data are detected by the programme. After printing an error indication the machine hoots and comes to a stop with an error number displayed in B 7. On prepulse the machine enters the WAIT routine, the usual intermittent signal is heard and the machine is ready to accept a new set of data.

When a problem is completed, the first data tape for the next problem, or a directive WAIT (last word in data 3), should be placed in the reader, as new data are read in automatically.

Data 2 ends with WAIT START, data 3 with WAIT START WAIT. The word START is punched to make possible an optional order of the tapes (eg. when SOURCE POLYNOMIALS is not used), but START may not be entered until all the data are complete.

14.Operation of SALOME

When a programme tape in binary form 1 has been entered with key 2, the computer comes to a 990 stop (PF = 0).

When starting on a new problem xor new source energy), the following procedure should be followed; - 51

Key 0 up

Enter data of type 1, "Materials data" into the tape reader, prepulse " " " " 2, "Geometrical data" " " " " " " " " " 3, "Source distribution data" " " "

Key 0 down

During computation, key 3 should be in "up" position. If the problem exceeds the capacity of the core memory, the com­ puter comes to a 991 (PF = 1) stop when reading data.

When the calculation is to be terminated:

K ey 3 down

Key 2 up, which stops the computer after a few minutes, whereupon a signal is given

Key 2 down

Key 9 is pressed, whereupon the binary continuation tape will be punched.

This tape is placed in the reader and checked by pressing key 9.

If the checking is unsuccessful, the computer comes to a hoot stop, and there are the following possibilities:

Key 6 - no further checking takes place

Key 7 - checking resumed from stopping point (to find all faults)

Key 8 - checking again from the beginning. Try this first

Key 9 - a new binary tape is punched. This should be tried at least once.

If continuing on a problem:

Key 2 up

Enter binary continuation tape into the tapereader, prepulse

Key 2 down

Key 3 up - 52 -

In case of restart from the preceding output;

Key 8 up, prepulse

Key 8 down

Key 3 up

Restart of the programme is made with first sector 128.

If keys are set in a wrong way, the computer hoots until a correct combination is found.

Error prints

Error list for NECO

Certain data errors are discovered by the programme.

1) The majority of errors result in ERROR PRINT followed by a persistent signal

Form of ERROR PRINT

ERROR ERROR

On On or B1 B1 = . ..

n is here the fault number.

Z) If Data 2 for the wrong element have been inserted in the reader - fault 23 - a wait signal is issued. See "Operation at NECO".

* 3) In the case of certain faults on a binary tape - faults 40 , * 42 and 43 - 990 stoppage P obtained with the fault number in B 7.

F ault Nature of fault number 1 The programme tries to invert a number with power of 2 < - 253 or > + 259.

4 The programme seeks the exponential function of a number > 255 log 2 - 176.

12 The programme seeks the root of a negative number. 53 -

F ault Nature of fault numbe r 14 The programme seeks the logarithm of a number < 0 .

18 The programme finds irrelevant characters when it attempts to read an integer.

19 The programme finds irrelevant characters when it attempts to read a "long number".

20 Irrelevant characters in directives.

21 Unidentifiable directive or directive in wrong place.

23 Microscopic data for wrong element.

25 Energy limits in the group scheme not monotonously decreasing.

27 One of columns 2 - 24 is isolated.

28 Intermediate result requires too much space. Data for four materials cannot be stored simultaneously. The main reason is that the number of integration steps exceeds 2200 (1100 if age theory is applied)/

29 Microscopic data for one element exceeds the space available for the purpose (about 3500 numbers. Note that, a zero cross-section often does not occupy any space. ).

31 Too many groups in the group scheme ("> 49),

32 Too many materials (> 31).

33 Too many elements in total (> 1 28).

34 Too many elements in a material (> 31).

35 The energies for the tabulation points not monotonous ’.'/ decreasing.

36 Too many separate (inelastic) energy levels (> 1 l).

37 The energy levels not recorded in monotonously increasing sequence.

38 The integer indicating the numbe, of cross-sections in the cross-section table < 2 or > 4. 54 -

Fault Nature of fault number

39 The number of Legendre coefficients > 10. * 40 Consecutive number of read-in binary sector does not agree, or sector does not start with CRLFCRLF.

42 Check-total for binary sector does not agree. * 43 Binary sector not equal to content of corresponding drum sector (on checking of afore-mentioned binary data,tape).

45 Binary data stored in wrong sectors.

48 Too many characters in the specification of microscopic data.

50 The cross-sections ("sigma total" etc.) not tabulated up to a 1.000001 x upper limit or down to « 0. 999999 x lower limit of groupe scheme (excluding thermal group if present).

51 Legendre coefficients (if given) not tabulated up to a 1. 000001 x upper limit of group scheme.

52 Cross-sections of separate energy levels (if given) not tabulated down to =» 0. 999999 x B . nn

Error list for REFUSE and REBOX

Fault Nature of fault number 1 Word incorrectly spelled (only the first two characters are essential)

2 Figure instead of word

3 • Word instead of figure

4 Directives missing or in wrong order

10 Title too long

11 Too many groups G > 32 or J > 28)

14 Too many regions (> 28)

17 Spurious character in Pattern matrix

21 Too many different materials - 55

Error list for NEDI

Fault Nature of fault number

1 Word not recognised

2 Number on the data tape when word was expected

3 Word on the data tape when number was expected

4 Directives in wrong order

5 Data incomplete

10 Title too long

1 1 Too many groups

12 Mesh incomplete

1 3 Some mesh size = 0

14 Too many divisions

15 More than 1023 steps in some divisions

16 Material index < 0 or > 511

1 7 Spurious character in Pattern

18 The number of elements on some row in pattern is wrong

19 The number of constants inconsistent with the pattern symbol

21 Capacity exceeded

22 No constants are given for some medium referred to under DIVISIONS

23 Some group number under SOURCE is > g or < 1 (generally an indication that a number has been left out or added, so that a table entry is interpreted as the group number)

24 A sub-range endpoint specified under SOURCE POLY - NOMIAES does not coincide with a division boundary

25 A group number under an output directive is > g or < 1 .

26 Spacing > 511

27 Number of dose rates > 5

If a spurious character is read, there is and indication "ERROR Q 1 00" and the contents of the accumulator and the index registers arc printed out. " 56 —

Data 1 for NECO Materials TITLE The title may be of any size, but may not con ­ tain the character /

REMOVAL GROUPS AND SPECTRUM PARAMETERS This chapter can be excluded, when removal groups are not needed. Energies E^_, are in Mey in decreasing order. Page 16. Au repre sent s 'the maximum integration mesh interval in lethargy units . Pages 3, 16.

G < 49

Spectrum parameters, see pages 3, 16.

DIFFUSION GROUPS WITH WEIGHT PARAMETERS This directive can also be just "DIFFUSION GROUPS" and in that case the third column must be excluded; all the m being assumed = - 1 . Page ®1 6 .

For E and (Au)g see preceding chapter.

g ^ 49

TEMPERATURE This chapter can be ex­ cluded when thermal group is not needed . Page 17. t Temperature t should be given in degrees centr:grade of the thermal neutrqn spect­ rum, often some 25 - 50 beyond that of the surrounding I medium. - 57 -

Data. 1 for NECO (cont. ) MATERIALS

Number of materials in the following table < 31 . Page 17.

Identification number of the fir st material < 127. Don't use 0, as it is used for air gaps in the following codes.' cm]j Element identifications (according to the data library) in the first column. In the second column the concentra ­ tion is written in g/cm3 Qf each element on the same line. The number of elements in one material must be < 31 .

m- Identification number of the second material.

Etc.

Etc. until all the materials are included. Data 2 for NECO Element microscopic data element

Ident. Element identification followed by its atomic weight. Page 20. Some text of maximum 80 characters including any ­ thing but / can be inferred before the character / .

2, 3 or 4 giving the number of cross-section columns in the following table.

Energies are in Mev, in de ­ creasing order Cross-sections in barns. Page 20. a- can be excluded only w&n un 2n is excluded too. Note.1 The absorption cross-section is the difference between the total cross-section and the sum of the others. More figure s than relevant may oft n be punched if a correct absorption is important.

a 2.2 km/sec. (0.0253 ev) c values. Pages 3, 20 Number of Legendre coeffi­ L cients in the following table. L < 10. If L = 0 the table is excluded. Pages 2, 21 Legendre coefficients in centre-of-mass system.

E VE„: f2

B . a2 B ( and B _ are thresholds nn 1 irPftie laboratory system for the (nn 1) and (n 2n) reactions. For the constants ai and a^ , see pages 2, 3, 21.

The first few (T < 1 1) nuclear levels for which cross-sections are known. €1 < e2 < • • • < V If T = 0 the following table as well is excluded. Page 21. - 59 -

Data 2 for NECO (cont. )

Energies are in Me v in decreasing order and cross­ sections in barns for the E a (E ) <7 (E ) <7 (E ) n e| n' % n separate levels. Note: Check that the sum of these cross­ sections is equal to o>nn i at each point, as the difference between the two quantities will have the effect of an absorption (or creation) cross section. Data 1 for REFUSE Geometry

TITLE The title may include at most 254 characters, with no double fig. sh. between the directive and the end of the title. The title is ended by at least two fig. sh.

ENERGY p5­ Number of energy groups

______^G__ Energy boundaries in Mev , decreasing. GEOMETRY f PLANE or CYLINDRICAL

MATERIALS Number of regions (<, 28)

Coordinates of region bounda ­ ries from the core face and outwards, alternating with identification number of the re­ spective material within each region (in parentheses). Any coordinate can be given at the core face according to what is needed in the NEDI run. See page 26. SOURCE V V and (thermal values) See page 27. Number of coordinates in the following table.

First column: coordinate measured inwards with zero X. 1 Vxi> on the core face. Second column: Thermal flux in the respective point. First and last x-value ought to be 0 and T respectively. Page 27.

m T Identification number of the o core material, and core thickness. For a cylinder, T is the core radius.

PRINT Optional. Page 27. Causes printing of removal flux and source in all dose points .

DOSE FACTORS Optional, but requires the PRINT directive. One dose fc -tm for each group.

WAIT Data 1 for REBOX Geometry TITLE The title may include at most 254 characters, with no double fig. sh. between the directive and the end of the title. The title is ended by at least two fig. sh.

ENERGY G Number of energy groups

Energy boundaries in Mev , decreasing. MATERIALS Number of regions. 28).

Coordinates of region bounda ­ ries from the core face and outwards, alternating with identification number of the respective material within each region (in parentheses). Any coordinate can be given at the core face, according to what is needed in the NEDI run. See page 26. V -y , IL and identification number of the core material. See page 27. X

X. X(x.) x-coordinate and source. 1 Page 31 .

Y

y-coordinate and source. Vi Y(y.)

Z

Z. Z(z.) a -coordinate and source 1 This is the coordinate mea­ sured inwards with zero on the core face.

SHIELD LINE

X = Y = Coordinates for the line along which the calculation is executed.

ACCURACY oee page 8, 31, Data 1 for REBOX (cont. ) Geometry

PRINT Optional. Page 27.

DOSE FACTORS Causes printing of removal flux and source in all dose points.

Optional, but requires the PRINT directive . One dose factor for each group.

WAIT -63 -

Data 2 for REFUSE and RE BOX Constants

PATTERN See page 23.

CONSTANTS P nWattM -spectrum constants , and number of materials. Pages 3, 16, 56. Identification number of the fir st material.

Removal cross -sections for the first material.

Transfer cross-sections for the first material, one value for each 1 in the pattern matrix.

WAIT (Repeated for each material.) Included by NECO only to faci­ START litate handling the tape.

This tape is normally produced by NECO. TITLE A title of at most 1 28 characters ending with at least two fig. sh. Double fig. sh. may not occur elsewhere in the title

GROUPS Number of energy groups g ^ 51 GEOMETRY {______J The word GEOMETRY ie .followed by one of the words PLANE, CY LIN DR I CAL or SPHERICAL.

MESH Coordinates for division interfaces, alternating EZZ with me sh intervals in each division. See pages IL-L i_and_4jL DIVISIONS Integers specifying the materials. The numbers refer to those given in NECO. Divisions are given in the same order as in the preceding chapter. For air, set material 0 .

BOUNDARY CONDITIONS Any one of the words FLUX, CURRENT or RATIO can head each J column. Pages 36, 37.

Inner and outer boundary values according to the heading of the column. One line for each group.

LATERAL LEAKAGE Optional . Page 3 7.

I Z Buckling values in each division.

DOSE RATES Optional n a-, 5. ... Pag. 8 ' Dose rates d. = Y C. .... i g Lg DOSE C ! i = 1, . . . , n , are calcul Lose facto) s for dose I, r»OSE 2 C [ z,8 beginning with the factor for the highest energy group, etc.

DOSE u C n, g 65 -

Data 1 for NEDI (coni.)

PRINT FLUX Integers s. giving the spacing of fete output. Page 37. \Optional SPACING Integers specifying j each group to be GROUPS printed. y

X PLOT FLUX See above Optional SPACING > See above GROUPS z X PRINT GAMMA SOURCE Page 38.

SPACING | 5; Integers s. giving Optional the spacing of the gamma source 51 1 / WAIT - 66 -

Data 2 for NEDI Constants.

PATTERN (g -1) rows and columns . The directive is optional; if omitted, the programme assumes the pattern filled with 1 1 s.

See pages 24, 39, 40

CONSTANTS Number of constants for each material. Material number <511 and > 1 .

Diffusion coefficients.

Group absorption cross­ sections (absorption + transfer) .

Transfer cross-sections one for each 1 in the g—> g'« pattern. Page 39, 40.

Next material number etc,

i t i

WAIT This tape is normally pro­ duced by NECO. Important: the word START is printed after WAIT, to make possible an optional order of data 2 and data 3. The word START should not be read until all the data is complet

START - 67 -

Data 3 for NEDI Source

SOURCE POLYNOMIALS Optional, see below, page 42,

Tape identification, a title ending with at least two fig. sh. Double fig. sh. may not occur elsewhere in the title .

Number of groups where the source is / 0 . (Counted coherently from tie first {highest} group)

Coordinates for first sub­ range .

Degree in the sub-range,

First column: group number . Second and following columns: C he by she v coefficients.

Coordinates for second sub-range.

etc. i i i WAIT The fir st WAIT make s an optional order of data 1 and START data 2 possible. Remember WAIT that START may not be read until data are complete. The second WAIT is printed to stop the computer if the data tape has not been changed for a new problem in the reader.

This tape is normally pro­ duced by REFUSE or RE BOX. - 68

Data 3 for NEDI (cont. ) Source

SOURCE ZERO Optional see below > Page 41 . SOURCE TABLES Optional see below

Group number .

Source values in division 1 , one for each mesh point, that is one for each interval + 1 .

Source values in division 2.

etc. i i Group number,

Source values in division 1 for the second group.

etc . i i WAIT Of course, it is not necessary to make a START special tape for the source when REBOX or WAIT REFUSE has not been used for the purpose. It can be printed to­ gether with the data 1 tape. The directive SOURCE POLYNOMIALS is intended for use to­ gether with REFUSE or RE BOX. In other cases one of the directives SOURCE ZERO or SOURCE TABLES should be used. - 69 -

Data 1 for SALOME Material constants . See pages 44-45. A title of at most 1 28 characters beginning with i L S and ending with at least two fig, sh. Double fig. sh. may not occur elsewhere in the title.

25 decreasing energies in Mcv

25 dose factors for the given energies.

I M The number of materials in the following tabic. M < 6 . m Identifica- S >■ (D tion number ■X (E,) f 25 values f I X (E24)

M such data °c + °p groups, one for every material. M < 6 . 25 values

a + a 9 value s c p - 70 -

Data 2 for SALOME Geometry See page 45. A title of at most 1 28 characters beginning with LS and en^ig with at least two fig. sh Double fig. sh. may not occur elsewhere in the title <

N The number of regions „ N < 32

6N 4 59M 4 54 < 448

N + I boundary coordinates

Material number for each one of the N regions in accordance with the identi ­ fication numbers specified for the materials in Data 1 . .rtn air-gap is denoted by aero (0) and no constants for air shall be given in Data 1 .

Conversion factors for each one of the N regions.

Starting energy in Mev .

The lowest energy in Mev mm to which the particle is followed. - 71

Data 3 for SALOME Gamma source

GAMMA SOURCE Only single fig. sh. between the words GAMMA and SOURCE and the beginning of the title. A title of at most 1 28 characters beginnings with LS and ending with at least two fig. sh.

X S(xJ First column: coordinates, o (cm) second column: gamma source, (reaction

X, spc) For internal boundaries 1 two values of the source X. s&.) can occur in the NEDJ- output.For boundaries between different media, two values must occur. X S(x max ' max

This tape is normally prepared by NEDI. - 72 -

APPENDICES

Appendix A

An example of an actual problem

A complete calculation with an actual configuration is presented on pages 79-102.

The configuration is presented in fig A1 . It is one of the con ­ figurations in a greater bulk shielding study (ref. 10). The neutron source is the R2-0 reactor (100 kW) at Studsvik research centre . The core, approx. 60 x60x32 cm, is transformed for the NEDI- calculations into a sphere of radius = 30 cm. The actual data are presented with some short comments. The runs form a consistent set and can thus be used for a re-run to check the code or computer.

Input data for NECO

Data 1

There are 30 removal groups. This is a recommended optimum number.

The energies for the diffusion groups (24) form two distinc ­ tive series with broader groups below 0. s Mev, where the spectrum is l/E. .

The materials with identification numbers are (see figAl):

110: reactor core 111: water 112: aluminium 113: ordinary concrete , p = 2.43 g cm ^ (not used in this ex.)

117: magnetite concrete, p = 3.74 . 1 19: Incite ( r " ~ )

Data _2

An example (Fe data) is given on p. 81 . The INPUT LIST for this run is reproduced on page 82. The running time for NECO was st 1 h total. 73

Input data for RE BOX

Data 1

This (t he geometrical data) is reproduced on p, 83. After the 30 energy groups come the material specifications . Two extra points were used (z = 87 and 1 67 cm) in the concrete , see fig. Ad.

The division of the source (reactor) needs some remarks. A study of the effect of the number of divisions x, y, and z showed in our case that no improvement in the accuracy was to be expected when x = y > 1 6 and z > 26. The 16x16x26 divisions give a running time of tt 3.5 h and are thus not suitable for test runs . The 6x6 x12 divisions used require a running time of a 0.6 h. The results using this divi ­ sion agree quite well with the more accurate re suits and were there­ fore used in test runs . It is recommended that a test is made with varying numbers of divisions to find the minimum number that gives the required accuracy (p. 31).

The results are needed only on the z -axis (x = y = 0), and with an accuracy = 10, see p. 31 , 4 extra points will be calculated, see results on pp. 89 and 94.

Data 2

An example giving the PATTERN for the six materials on p. 80 and the removal and transfer cross-sections for the concrete (117) are reproduced on pp. 84 - 88. For identification of a certain group of transfer cross-sections one has to note that the total number of erases and numbers is in our case thirty for each line .

The beginning of the first part of the output is given on p. 89 . The second part which forms data 3 for NEDI is given on p. 94.

Data 1 for NEDI (p. 90)

The principles for MESH and BOUNDARY CONDITIONS are discussed on pp. 35-36, 43. The example is in spherical geometry with core radius = 30 cm. The dose factors listed give 1) S(n, p) , 2) P(n, p) reaction rates, 3) average flux per lethargy unit in the interval 1 -1 000 ev 4) total flux 2.0-0. 3 Mev and 5) flux > 1 Mev. All of the groups are printed. - 74 -

Data 2 for NEDI (output of NECO). (pp. 91 -93) is self-explanatory. The "erase" characters in the transfer cross­ sections correspond to zero characters in PATTERN.

Data 3 for NEDI is the latter part of the output of REBOX or REFUSE. The beginning of.this data is reproduced on p. 94.

NEDI output . .

The print-out of the first group, DOSERATES and GAMMA SOURCE are reproduced on pp. 95-96, 100. The running time was 0.45 h.

As an example of the spectrum obtained, Fig. A2 gives the results at z = 65. The fluxes are given per lethargy unit and are manually corrected for the direct (removal) flux. A good agreement with the spectrum obtained by NIOBE (ref. 1 i) is seen. No normalization of the absolute values has taken place. - 75 -

Input data for SALOME

Data_2_ for SALOME (p. 97) contains, as a title, a list of the materials in question, 25 energies in Mev, 25 conversion factors, 5 as the number of materials. As an example the data for magnetite concrete is given.

Data Z_ shows that in order to get a picture of the change in energy- deposition rate throughout the shield, 20 cm thick regions have been used (fig.Al). In this way a decent statistic is arrived at in a reason ­ able time, see discussion about the results. The lowest energy has been set to 250 kev, and this can be taken as a recommended value for a typical run.

Data_3 for the source or reaction rate consists in this case of two parts:

1) the reaction rate in core, manually punched 2) "GAMMA SOURCE" in the shield, produced by NEDI

As the code works with plane geometry, the fission rate is the averaged one for the core along the z-axis. Note that the fine division for water is useless in this case, because water has no source in the 8 Mev group.

Output starts with the regional energy deposition rate. The problem was run for 1 h, during which time 28*96 = 2688 photons were started. The last output is reproduced on pp. 1 01 - 1 02. While the main interest is normally focused, on the dose on the outer shield surface, the total dose and uncollided dose with their standard deviations are given in table A! as a function of the number of photons started. - 76 -

Table A1

No. of Total dose Deviation Uncolli^ed Deviation photons Imr/h] Imr/h] dose Imr/h] Imr/h] started 8 Mev

I lx 96] 2.718, -3 1.06, -3 2.074, -3 0.868, -3 2 4.397, -3 1.81, -3 3.494, -3 1.71, -3 3 4.924, -3 1.51, -3 3.954, -3 1.45, -3 4 1.087, -2 0.662 , -2 3.274, -3 1.09, -3 5 1.535, -2 0.681, -2 3.181, -3 0.913, -3 6 2.893, -2 1.62, -2 1.818, -2 1.51, -2 7 2.515, -2 1.39, -2 1.532, -2 1.29, -2 8 2.252, -2 1.22, -2 1.430, -2 1.13, -2 9 2.051, -2 1.03, -2 1.317, -2 1.00, -2 10 1.926, -2 0.975, -2 1.203, -2 0.90, -2 1 1 1.764, -2 0.886, -2 1.107, -2 0.82, -2 12 1.663, -2 0.813, -2 1.028, -2 0.75, -2 1 3 1.761, -2 0.778, -2 9.701, -3 6.95, -3 14 1.679, -2 0.723, -2 9.316, -3 6.46, -3 15 1.609, -2 0.675, -2 8.776, -3 6.03, -3 16 1.529, -2 0.633, -2 8.379, -3 5.65, -3 17 1.472, -2 0.536, -2 7.972, -2 5.32, -3 18 1.426, -2 0.563, -2 7.654, -3 5.03, -3 19 1.367, -2 0.534, -2 7.324, -3 4.76, -3 20 1.304, -2 0.507, -2 7.004, -3 4.52, -3 21 1.258, -2 0.483, -2 6.807, -2 4.31, -3 22 1.208, -2 0.461, -2 6.555, -3 4.11, -3 23 1.167, -2 0.441, -2 6.347, -3 3.93, -3 24 1.127, -2 0.423, -2 6.160, -3 3.77, -3 25 1.088, -2 0.406, -2 5.963, -3 3.62, -3 26 1.063, -2 0.390, -2 5.795, -3 3.48, -3 27 1.043, -2 0.376, -2 5.637, -3 3.35, -3 28 1.016 , -2 0.363, -2 5.548, -3 3.23, -3 29 0.987, -2 0.350, -2 5.401, -3 3.12, -3

6 Mev

18 lx 96] 0.297, -2 0.097, -2 1.458, -3 0.687, -3 - 77 -

One sees that the accuracy of the uncoJlided dose increases with the number of photons started, as expected. In the total dose column a change with a factor of 2 is seen both between groups 3 and 4 and 5 and 6 . On a closer examination of the output this is shown to be caused by photons with a great weight originating from the core. This has an effect on the uncollided dose between groups 5 and 6 , too.

The results have not yet converged to a certain value , but the convergence after the 20^ group is so slow that one may write as an answer: (9 ™ 3), -3 mr/h. The running time for this problem was 1 h.

For control, the 6 Mev-source group has been calculated with 1 8x96 photons . The result is 1 /3 of the dose from the 8 Mev group. In this problem it was known that the dose from the highest energy group dominates.

Generally it can be recommended that for the first run of an actual problem, the assumed main energy is run for 0.7- 1.0 hr s and the other energies for y 0.2-0. 4 hr s. This gives the magnitude of the minor energies, and if the assumptions have been right, the problem is already solved. If they have been wrong or a better statistic is needed, a continuation run ( see p. 51) is easily done .

This example shows too, that one shall not automati­ cally read only the last result but must take into consideration the whole output. In this Mercury version of tue code splitting can not be used, and photons with a great weight may appear. There exists a FORTRAN version in which splitting is used to avoid the type of jumps observed in this example. - 78 -

Configuration Core H20b-Al Magnetite concrete

Z( cm) 231

Material nr 110 111 112 117 32.4 0 20""s21 231 Boundaries 3.2.4** —0 source coordinates used in: ' i N REBOX REFUSE i / 30 50 L17 157 197 /z229 26 extra boundaries produced by the code

NEDI 30 50 51 117 157 197 229 26l

SALOME 30 5(f 51 67 87 107 127 157 182 202 222 242 261

Fig. Al. Configuration for the data example in app. A.

10'* 10"s 10"1 10* 1 10° —>E(MeV) 101 Fig. A2. Neutron spectra by NRN and NIOBE methods after penetrating 20 cm water and 15 cm heavy concrete Data 1 for NECO

TITLE

CONCRETE EXPERIMENT. 4

REMOVAL GROUPS AND SPECTRUM ARAMETERS

1.800.1 0.05 1.430.1 0.05 1.136.1 0.05 9.021 0.05 7.166 0.05 5.692 0.05 4.521 0.05 3.591 0.05 2.85 3 0.05 2.267 0.05 1.800 0.05 1.430 0.05 1.136 0.05 9.021, -1 0.05 7.166, -1 0.05 5.692, -1 0.05 4.521, -1 0.05 3.591, -1 0.05 2.853, -1 0.05 2.267, -1 0.05 1.800, -1 0.05 1.430,-1 0.05 1.136,-1 0.05 9.021, -2 0.05 7.166, -2 0.05 5.692, -2 0.05 4.521, -2 0.05 3.591, -2 0.05 2.853, -2 0.05 2.267, -2 0.05 ' 1.800, -2 K 1.0363 2.29

DIFFUSION GROUPS WITH WEIGHT PARAMETERS

1.8,1 0.1 -4 1.35,1 0.1 -4 1.0,1 0.1 -4 7.8 0.1 -2.5 5.9 0.1 -2.5 4.4 0.1 -2.5 3.4 0.1 -2.5 2.6 0.1 -2.5 2.0 0.1 0 1.5 0.1 0 1.2 0.1 -1.5 0.9 0.1 -1.5 0.7 0.1 -1.5 0.51 0.1 -1.5 0.38 0.1 -1.5 80

Data 1 for NECO (cont.)

0.3 0.08 -1 0.1, 0.08 -1 3.0,-2 0.08 -1 1.0,-2 ' 0.08 -1 1.0,-3 0.08 -1 1.0,-4 0.08 -1 1.0,-5 0.08 -1 1.0,-6 0.08 -1 1.05,-7 ¥

TEMPERATURE

75

MATERIALS

6

110 HI 0.065 08 0.52 AL13 1.12 ¥

111 HI 0.111 08 0.889 ¥

112 . AL13 2.7 ¥

113 HI 0.0187 08 1.19 A LI 3 0.236 5114 0.722 CA20 0.19 FE26 0.073 ¥

117 HI 0.015 08 1.272 AL13 0.03 5114 0.18 CA20 0.254 FE26 1.99 ¥ 119

119 HI 0.17 C6 1.03 ¥ Example of a data .2 - tape for NECO

ELEMENT FE .26 55.85 E.S. TROUBETZKOY NDA 2111-3 AND GOLDSTEIN

4 ■ ' 18.001 2.35 1.05 0.28 0.960 17.1 2.40 1.08 0.37 0.878 16.3 2.45 1.12 0.48 0.775 15.5 2.50 1.15 0.61 0.650 14.75 2.54 1.18 0.68 0.580 14.0 2.58 1.21 0.76 0.500 13.3 2.62 1.23 0.84 0.420 12.7 2.67 1.29 0.92 0.345 12.1 2.74 1.35 1.02 0.270 11.5 2.81 1.41 1.11 0.191 10.9 2.91 1.51 1.23 0.080 10.4 3.00 1.60 1.32 0 Cut

4.5359,,-8 1.3363, 1 1.1400, 1 1 4.1042.,-8 1.3466, 1 1.1400, 1 i 3.7137, ,-8 1.3560, 1 1.1400, 1 l

8

22.0 .803 .686 .592 .505 .424 .350 .235 .1300 18.0 .807 .690 .580 .487 .384 .291 .195 .1135 17.1 .808 .691 .577 .480 .374 -278 .187 .1065 16.3 .810 .692 .573 .473 .364 .265 .179 .1005 15.5 .812 .693 .571 .467 .354 .253 .171 .0945 14.75 .814 .695 .569 .458 .344 .241 .163 .0880 14.0 .815 .700 .566 .450 .334 .228 .155 .0820 Cut

.0233 .00615 .00007 0 0 0 0 0 0 .0221 .00585 .00006 0 0 0 0 0 0 .0201 .0000 .00000 0 0 0 0 0 0 ¥

4.3 0.85 2.8 10.8

8.45,-1 2.09, 0 2.66, 0 2.95, 0 3.01, 0

4.9103 5.4740,-1 2.4330,-1 9.2450,-2 2.5790,-1 8.5150,-2 1.3380,-1 4.4430 5.4410,-1 2.7740,-1 1.6220,-1 2.0800,-1 5.2280,-2 9.6020,-2 ' Cut 1.095 6 4.6000,-1 0 0 0 0 0 9.9137,-1 4.1000,-1 0 0 0 0 0 8.9703,-1 2.0000,-1 0 0 0 0 0

8.1167,-1 0 0 0 0 0 0 ¥ ' - 82

Inputlist (first output) for NECO

NECO 111 SEP.63

CONCRETE EXPERIMENT.

INPUTLIST

HI 08 AL13 5114 CA20 FE26 C6

HI 08 AL13 5114 CA20 FE26 C6 - 83

Data I for REBOX

TITLE CONCRETE EXPERIMENT. CONFIG. 2.

ENERGY 30 1.8,1 1.43,1 1.136,1 9.021 7.166 5.692 4.521 3.591 2.853 2.26 7 7.8 1.43 1.136 9.021,-1 7.166,-1 5.692,-1 4.521,-1 3.591,-1 2.853,-1 2.267,-1 1.8,-1 1.43,-1 1.136,-1 9.021,-2 7.166,-2 5.692,-2 4.521,-2 3.591,-2 2.853,-2 2.267,-2 1.8,-2

GEOMETRY PLANE PRINT MATERIALS

5 30(111 )50(U2)51(117)117(117)197(117)261

SOURCE 2.5 0.0438 110

X -30.85 7.0,11 -14 1.07,12 -4 1.13,12 0 1.13,12 4 1.07,12 14 7.0,11 30.85

Y -30 .644 -14 .924 -4 0 4 .924 14 .644 30

Z 0 .86 .787 2 .768 5 .81 6.5 .855 8 .922 11 .99 18 .925 22 .81 26 .715 28.5 .678 31 .73 32.4

SHIELD LINE X = 0 Y.-0

ACCURACY 10

WAIT - 84 -

D^c 2 for REBOX or REFUSE

PATTERN

1 1 000000000 00000000 00 00000 0000 This is or output I 1 lOO'OOUOOOOOOOOOOOOOOOOOOOOOO tope from NECO 1111000000000000000000000001)0 ------111110000000000000000000000000 1 1 1 1 1 1 1 00000000000000000000000 1 1 1 111 11 0000000000000000000000 liiinmooooooooooooooooooooo 111111111100000000000000000000 immimooooo oooooooooooooo immmiioooooooooooooooooo lmiimm Vo oooooooooooooooo 111111", vnimooooooooooooooo 111111111111111100000000000000 mmmnininooooooooooooo 111111111111111111000000000000 111111111111111111111110000000 II m m 11 m i m n m m m oo ii i m i m m i i i m i i 11 m m i • i m l n 111 n n 111 n 11 i l i n 1111 11 u i i i n i i n i 111 n 11 i n i i 1111 i n on i i n m i i i i l i 11 n i i 11 i i i n n om i i ooi oi i 11 i i i i n i i i i i i 111110011000001000000000111111 111000001000000000000000000000

CONSTANTS 1.0363 2.2900 6

110

.070541 .073155 .073626 .079334 .089675 .098684 .120702 .136394 .138693 .160493 .191421 .206314 .250268 .260309 .271545 .326596 .414760 .373628 .382731 .445487 .538721 .497112 .573816 .616621 .538154 .581949 .718886 .785992 .607512 .627401

.006647 .000063 XXX XXXXX XXXXX XXXXX XXXXX XXXXX .006664 .008865 .000900 XX XXXXX XXXXX XXXXX XXXXX XXXXX .005251 .006530 .008745 .001460 XXXXX XXXXX XXXXX Cut - 85

Data 2 for REBOX (cont.)

117

.102298 .097103 .086565 .086454 .090365 This is the complete .094424 .128975 .141814 .113246 .133709 set of data for magne. .180383 .171523 .260705 .206243 .195581 tite concrete (No. 117) .283719 .432776 .275195 .279519 .296414 .294101 .302784 .306416 .333757 .316087 .345527 .375215 .723053 .596724 .318162

.009953 .000149 XXX XXXXX XXXXX XXXXX XXXXX XXXXX .005037 .012790 .001469 XX XXXXX XXXXX XXXXX XXXXX XXXXX .002545 .003558 .011567 .002563 X XXXXX XXXXX XXXXX XXXXX XXXXX .004294 .003703 .004409 .018162 .009074 XXXXX XXXXX XXXXX XXXXX XXXXX .006202 .005168 .005082 .005698 .026115 .024913 .000126 XXX XXXXX XXXXX XXXXX XXXXX .006350 .005283 .005240 .005455 .006303 .025090 .050586 .001410 XX XXXXX XXXXX XXXXX XXXXX .006688 .005652 .005625 .006056 .006383 .007460 .043054 .079836 .006469 X XXXXX XXXXX XXXXX XXXXX .005930 .005166 .005080 .005673 .006142 .007081 .005836 .031053 .058817 .01^690 XXXXX XXXXX XXXXX XXXXX 86

i 2 for RE BOX (ceilit.)

.055335 .004860 .004644 .005360 .006005 Magnetite .007025 .007732 .005529 .024333 .077206 concrete (cont.) .043766 ’1 xxxx XXXXX XXXXX xxxxx .003210 .003075 .002835 .003353 .003873 .003979 .004743 .003979 .008541 .014145 .094621 .043249 XXX XXXXX xxxxx xxxxx .003047 .003075 .002732 .003288 .003980 .003 953 .005284 .004427 .002826 .015526 .011321 .092756 .074662 XX XXXXX xxxxx xxxxx .001837 .001959 .001680 .002046 .002515 .002331 .002716 .003358 .002305 .002930 .011682 .006019 .151822 .113181 .000277 XXXXX XXXXX XXXXX .001513 .001699 .001417 .001737 .002167 .002098 .001r54 .002926 .002845 .002436 .009559 .006631 .007095 .065143 .142789 .025317 XXXX XXXXX XXXXX .000858 .001012 .000828 .001018 .001285 .001306 .000837 .002346 .001772 .001727 .002403 .008768 .004855 .006943 .023530 .205636 .137070 XXX XXXXX xxxxx .030439 .000537 .000436 .000538 .000683 .000727 .000505 .001441 .000844 .001198 .001471 .005206 .002987 .004273 .006075 .020413 .237586 .067118 XX XXXXX xxxxx .000730 .000939 .000775 .000963 .001256 .001438 .001147 .003147 .002100 .002892 .003668 .006520 .014228 .010682 .015187 .021483 .042742 .186471 .249342 ‘ .254394 .235971 ,204591 .036335 XX XXXXX .000103 .000150 .000139 .000181 .000249 .000338 .000318 .000673 .000974 .000953 .001284 .001829 .004270 .003945 .005315 .007519 .010675 .015034 .021028 .029301 .040587 .073917 .236812 .283507 .255247 .264465 .262694 .172629 XX .000015 .0.;022 .000025 .000034 .000050 .000092 .000090 .000190 .000268 .000262 .000367 .000522 .000747 .001458 .001519 - 87

Data 2 for REBOX (cent.)

.002148 .003050 .004296 .006008 .008374 Magnetite .011596 .016085 .022046 .030053 .040440 concrete (cant.) .053889 .076768 .503482 .535662 .239059 .000004 .000006 .000009 .000013 .000019 .000040 .000054 .000126 .000363 .000127 .000165 .000235 .000337 .000646 .000683 .000967 .001372 .001933 .002704 .003768 .005218 .007238 .009920 .013524 .018198 .024250 .032024 .042077 .054784 .071020 .000000 .000000 .000001 .000001 .000002 .000004 .000005 .000020 .000106 .000013 .000017 .000024 .000034 .000050 .000063 .000097 .000137 .000193 .000270 .000377 .000522 .000724 .000992 .001352 .001820 .002425 .003202 .004208 .005478 .007102 0 0 0 X0 .000000 .000000 .000002 .000010 .000001 ,000001 .000002 .000003 .000005 .000007 .000010 .000014 .000019 .000027 .000038 .000052 .000072 .000099 .000135 .000182 .000243 .000320 .000421 .000548 .000710 0 0 0 0 X 0 0 .000000 .000001 0 X X 0 X 0 .000000 .000001 .000001 .000002 .000003 .000004 .000006 .000009 .000013 .000018 .000024 .000032 .000042 .000055 .000071 0 0 0 0 0 X X 0 .00COO0 x X X X X 0 X X X X X X X X X .000000 .000001 ,000001 .000002 .000004 .000005 0 0 0 X X X X X X X X X X X X X X X X X X X X X X

119

.085593 .098634 .109064 .139181 .151127 .171933 .219191 .247486 • .268718 .300483 .344159 .392512 .446777 .504230 .564991 .629413 .702013 .777035 .854088 .935067 1.017311 1.108246 1.194617 1.281725 1.359880 1.430930 1.494773 1.554740 1.602742 1.644486

.003096 0 XXX X X X X X X X X X X X X X X X Cut 88

Data 2 for REBOX (cant.)

.000001 .000001 .000001 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .002999 .011693 .022388 .035562 .051803 .072101 .397279 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 X0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 X 0 0.0 0 0 X X 0 X 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 X X 0 0 X X X X X 0 X X X X X X X X X 0 0 0 0 0 0 0 0 0 X X X X X 0 X. X X X X X X X X X X X X X X X X X X X X

WAIT

START 89

Output from RE BOX

REBOX 15.10.63

CONCRETE EXPERIMENT. CONFIG. 2.

L FLUX SOURCE

7.489, +6 2.498, +4 This is the complete 7.911, •1-7 7.282, 4 5 output for the first point. 4.851, +8 4.905, +6 Further points computed: 1.763, + 9 4.300, + 7 40.0, 50.0, 50.0, 51.0, 51.0, 4.257, + 9 1.930, + 8 84.0, 117.0, 117.0, 157.0, 7.863, 49 4.891, +8 197.0, 197.0, 229.0, 261.0 1.041, 4-10 1.108, + 9 1.234, •i 10 1.186, + 9 1.413, + 10 1.814, +9 1.271, -no 1.791, -19 1.001, + 10 2.185, + 9 8.250, ->-9 2.141, +9 5.565, + 9 2.103, 49 4.329, + 9 1.796, + 9 3.254, 19 1.233, 19 1.954, 49 3.500, + 9 1.027, + 9 1.492, -.9 9.085, + 8 4.716, + 8 6.474, -8 2.162, 48 3.707, 1-8 2.161, *-7 CO o CO + 8 1.974, + 6 1.602, +8 8.212, 44 8.740, 4 7 9.322, 42 5.387, 4-7 1.000, -75 5.099, + 7 3.112, + 7 1.384, + 7 7.878, 4 6 1.025, *7 6.814, r6

SOURCE SUM = 2.179, + 10

L FLUX SOURCE

40.0 1.587, +6 5.290, +3 1.611, +7 1.485, +5 Cut 90

Dota 1 for HEDI

TITLE CONCRETE EXPERIMENT. CONFIG. 2. SPHERICAL. GROUPS 24 GEOMETRY SPHERICAL ■ MESH 30(0.5)50(0.1)51 (0.5) 117(1 )157(2) 197(2)227( 1)261 DIVISIONS 111 112 117 117 117 117 117 BOUNDARY CONDITIONS FLUX RATIO 1.2,7 2.13 3.0,8 2.13 1.7.9 2.13 6.7.9 2.13 1.67.10 2.13 2.24.10 2.13 3.77.10 2.13 4.34.10 2.13 4.47.10 2.13 3.41.10 2.13 4.13.10 2.13 3.53.10 2.13 3.61.10 2.13 2.73.10 2.13 1.96.10 2.13 6.4.10 2.13 4.6.10 2.13 3.67.10 2.13 6.58.10 2.13 6.42.10 2.13 6.33.10 2.13 6.21.10 2.13 4.85.10 . 2.13 5.5,11 2.13

DOSE RATES 5 DOSE 1 4.7,-3 6.39,-3 6.5,-3 6.4,-3 5.65,-3 4.45,-3 2.26,-3 1.17,-3 1.2,-4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 DOSE 2 2.82,-3 2.82,-3 2.82,-3 2.82,-3 2.53, -3 1.88,-3 1.48,-3 7.33,-4 8.5,-5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 DOSE 300 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .1449 J449 .1449 0 0 DOSE 4 00 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 DOSE 5 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

PRINT FLUX SPACING 4 10 14 10 18 10 10 GROUPS l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

PRINT GAMMA SOURCE SPACING 4 10 14 10 18 10 10 WAIT 91

Doto 2 for HEDl

PATTERN

1 This is an output 1 1 tape from NECO 111 • mi inn linn liniii linnii liiiiim 1111111111 I 1 U 111 1 n 1 liiiininii liiiinimn 1)11111113)1)1 111111111111111 1111111111111111 11111111111111111 m 11 m m ii n 111 n i 11 n n i 111111 i 11 II i i i i i i i 11 n i i i i i i i l n 11 n n oi m m i 111 1111111000011000111111 11111 1 1 000011 00000 1 1 1 1 1

CONSTANTS 307

no

5.544267 5.361239 5.234705 4.332417 3.830900 2.958233 2.765369 2.552449 2.050170 1.852382 1.505174 1.451282 1.308911 .992368 1.091517 .853474 .657753 .690595 .712440 .705093 .705060 .704260 .630984 .424273

.068032 .071527 .074499 .083299 .090194 .114944 .114722 .128305 .143123 .168700 .206514 .213758 .228605 .304406 .334727 .263218 .362989 .441280 .306458 .308735 .308803 .310009 .375110 .015180

.010656 .005278 .011783 .005847 .007894 .016147 .005876 .007699 .010163 .023762 .004630 .006093 .007922 .010678 .028143 .004050 .005433 .007103 .009519 .013254 .048971 Cut 92

Ds.'j 2 for MED! (cont.)

117

3.689195 3.936324 4-338008 3.710003 3.513436 This is the complete 2.5 22572 2.996902 3.253676 2.164052 2.116112 set of data for 1,516692 1.850608 1.658788 1.040002 1.633668 magnetite concrete 1.136917 .912954 .912844 .813994 .684592 .663750 .664061 .641912 .517851

.096278 .087429 .076577 .079555 .075853 .109701 .083080 .082093 .102330 .108624 .173855 .114993 .107852 .215198 .186906 .080041 .113754 . .124495 .082419 .084064 .085778 .039349 .116353 .048211

.098689 .002567 .009563 .004322 .003692 .012329 .006223 .005110 .005148 .021830 .006356 .005223 .005359 .005643 .032615 .006.685 .005561 .005868 .006362 .007086 .072990 .005926 .005021 .005406 .006099 .007344 .006862 048999 ,005333 .004639 .005029 .005903 .006972 .007004 .012418 .048546 .003211 .002883 .003107 .003767 .004208 .005237 .002772 .013899 .065976 .003050 .002844 .003015 .003755 .003629 .004600 .004159 .007325 .012664 .0746 89 .001841 .001796 .001859 .002372 .002046 .003619 .0027f 2 .002223 .009643 .004814 .140642 .001518 .001555 .001566 .002040 .001824 .002310 .002873 .002479 .005556 .009434 .006899 .086520 .000861 .000930 .000911 .001210 .001139 .001366 .001885 .001794 .002115 .008452 .004721 .00/157 .075276 .000441 .000497 .000478 .000645 .000634 .000585 .001252 .001116 .001295 .003926 .003766 .004404 .006831 .173782 .000734 .000835 .000849 .001175 .001271 .001172 .003136 .002531 .003266 .004828 ,013429 .011011 .017077 .027750 .166703 .000109 .000143 .000156 .000228 .000301 .000393 .000931 .000765 .001175 .001685 .004153 .003955 ,005977 .009476 .014053 .062631 .000015 .000023 .000029 .000045 .000108 .000111 .000260 .000216 .000336 .000481 .000727 .001330 .001708 .002707 .004015 .011509 .086869 .000004 .000007 .000011 .000017 90055 .000098 .000617 .000099 .000164 .,00217 .000328 .000625 .030768 .001218 .001807 .005179 .024031 ,114343 .000000 .000001 .000001 .000002 .000005 - 93 -

Data 2 for NEDI (cont.)

.000013 .000191 .000011 .000016 .000022 Magnetite .000033 .000373 .000077 .000122 .000181 concrete (cont.) .000518 .002403 .008958 .075170 0 0 0 0 .000000 .000001 .000019 .000001 .000001 .000002 .000003 .000037 .000008 .000012 .000018 .000052 .000240 .000896 .006260 .076420 0 0 0 0 0 .000000 .000002 0 0 X 0 .000003 .000000 .000000 .000001 .000004 .000024 .000090 .000626 .006303 .076396 0 0 0 0 0 0 .000000 XKX X .000000 oxx X .000001 .000007 .000062 .000627 .006266 .076122 0 0 0 0 0 0 0 XXX X .000000 OXX XXX .000005 .000071 .000732 .007357 .098171

119

4.223874 3.666697 3.07776} 2.75827.' 2.217310 1.709291 1.576715 1.403132 1.222917 1.061507 .909551 .793328 .686946 .596767 .532324 .440133 .319807 .265953 .283560 .279635 .279621 .279297 .250607 .168211

.091837 .107686 .135044 .153395 .178878 .235012 .267276 .298234 .324539 .395836 .440757 .522528 .574483 .677325 .799244 .684188 .939420 1.162865 .805362 .811316 .811495 .814647 .984628 .027649

.021719 .009480 .025377 .008495 .015262 .043033

0 .076464 0 0 0 0 0 0 0 XXX X 0 oxx XXX 0 0 0 0 .078119

WAIT

START 94

Data 3 for MED!

SOURCE POLYNOMIALS This is on output tope from REBOX CONCRETE EXPERIMENT, CONFIG. 2

24

30.00 - 50.00

2 1 1.731, +1 -1.389, H O 1.633, -1 2 2.398, < 1 -1.423, HO 1.668, -1 3 2.773, +1 -1.453, +0 1.711, -1 4 3.185, +1 -1.560, +0 1.850, -1 5 3.458, H i -1.687, +0 1.989, -1 6 3.598, +1 -1.901, +0 2.387, -1 7 3.707, +1 -2.152, +0 2.757, -1 8 3.704, +1 -2.234, +0 2.808, -1 9 3.751, +1 -2.406, +0 3.179, -1 10 3.701, +1 -2.606, +0 3.875, -1 n ' 3.717, +1 -2.702, 40 4.385, -1 12 3.662, +1 -2.893, +0 5.626, -1 13 3.655, +1 -2.908, 40 5.689, -1 14 3.592, +1 -3.018, 40 6.575, -1 15 3.501, +1 -3.073, 40 7.107, -1 16 3.691, +1 -3.136, 40 7.683, -1 17 3.491, +1 -3.235, 40 8.635, -1 18 3.246, '!•! -3.285, 40 9.139, -1 19 3.087, +1 -3.294, 40 9.230, -1 20 2.626, +1 -3.297, 40 9.225, -1 21 2.081; +1 -3.615, 40 9.537, -1 22 8.515, +0 -6.597, 40 9.233, -1 23 -8.214, +0 -1.071, 41 4.699, -1 24 -3.454, +2 2.384, -7 4.768, -7

50.00 - 51.00

1 1.728, +1 -1.146, -1 1 1.728, +1 -1.146, -1 2 2.302, +1 -1.159, -1 3 2.640, -M -1.158, 1 4 2.964, +1 -1.181, -1

16 -1.243, 41 -3.190, 40 2.997, -2 17 -1.558, 41 -3.209, 40 3.319, -2 13 -1.860, 41 -3.229, 40 3.443, -2 19 -2.029, 41 -3.278, 40 4.687, -2 20 -2.462, 41 -3.376, 40 6.712, -2 21 -3.246, 41 -3.919, 40 2.161, -2 22 -3.707, 41 -3.917, 40 2.090, -2 23 -1.556, 42 -1.050, 41 4.204, -2 24 -3.454, 42 2.384, -7 4.768, -7 WAIT

START

WAIT 95

Output from NEDI

NEDI-4. 3.3.64. 4 DRUMS.

CONCRETE EXPERIMENT. CONFIG. 2. SPHERICAL

SOURCE POLYNOMIALS

CONCRETE EXPERIMENT. CONFIG. 2.

GROUP 1

30.0 1.20, +7 32.0 8.96, +6 34.0 6.71, +6 36.0 5.05, +6 38.0 3.81, +6 40.0 2.89, +6 42.0 2.20, +6 44.0 1.68, +6 46.0 1.29, f6 48.0 1.00, +6 50.0 7.83, +5

50.0 7.83, +5 51.0 6.99, +5

51.0 6.99, +5 58.0 2.09, 45 65.0 6.40, +4 72.0 2.01, +4 79.0 6.51, +3 86.0 2.17, 43 93.0 7.43, +2 ' 100.0 2.63, +2 107.0 9.63, +1 114.0 3.65, +1 117.0 2.43, +1

117.0 2.43, +1 127.0 6.56, +0 137.0 1.86, +0 147.0 5.46, -1 157.0 1.65, -1

157.0 1.65, -1 193.0 2.59, -3 197.0 1.65, - 3

197.0 1.65, - 3 " 217.0 1.76, -4 227.0 5.82, -5

227.0 5.82, -5 . 237.0 1.92, -5 247.0 6.39, -6 257.0 2.09, -6 267.0 1.30, -6

GROUP 2 Cut Output from MEDl (cent.)

227.0 1.49, + 1

227.0 , 1.49, +1 237.0 5.30, +0 247.0 1.85, +0 257.0 4.67, -1 261.0 9.83, .2

DS. RATE 1 DS. RATE 2 DS, RATE 3 DS. RATE 4 DS. RATE 5

30.0 3.91, +8 2.00, 2.75, 4 10 2.38, +11 2.49, +11 32.0 2.58, +8 •. 1.31, + 8 2.01, + 10 1.44, + 11 1.54, + 11 34.0 1.71, +8 8.68, +7 1.33, +10 8.65, +10 9.71, +10 36,0 1.14, +8 5.78, +7 8.31, + 9 5.23, + 10 6.18, + 10 38.0 7.70, +7 3.87, +7 5.10, +9 3.21, + 10 3.98, +10 40.0 5.23, +7 2.62, + 7 3.13, + 9 2.00, + 10 2.59, + 10 42.0 3.59, +7 1.79, +7 1.95, +9 1.26, + 10 1.71, +10 44.0 2.49, +7 1.24, +7 1.25, +9 8.19, +9 1.15, + 10 46.0 1.76, +7 8.70, +6 8.28, +8 5.49, +9 7.87, +9 48.0 1.27, +7 6.26, + 6 5.89, +8 3.91, +9 5.61, +9 50.0 9.50, +6 4.69, +6 4.57, +8 3.14, +9 4.28, +9

50.0 9.50, +6 4.69, +6 4.57, +8 3.14, +9 4.28, + 9 51.0 8.63, +6 4.26, +6 4.52, +8 3.02, +9 3.96, +9

51.0 8.63, +6 4.26, + 6 4.52, +8 3.02, +9 3.96, +9 58.0 3.52, +6 1.77, +6 2.86, +8 1.75, +9 1.86, +9 65.0 1.43, +6 7.31, +5 1.74, +8 3.92, +8 8.37, +8 72.0 5.84, +5 3.01, +5 9.66, +7 4.24, +8 3.66, +8 79.0 2.39, +5 1.24, +5 4.99, +7 1.94, +8 1.58, +8 86.0 9.89, +4 5.13, +4 2.44, +7 8.61, +7 6.74, +7 93.0 4.12, +4 2.14, +4 1.14, + 7 3.77, +7 2.87, +7 100.0 1.74, +4 9.02, +3 5.22, + 6 1.64, +7 1.22, +7 107.0 7.43, +3 3.85, +3 2.34, + 6 7.07, +6 5.22, + 6 114.0 3.22, +3 1.67, +3 1.04, +6 3.06, +6 2.25, +6 117.0 2.26, +3 1.17, +3 7.31, +5 2.14, +6 1.57, +6

117.0 2.26, +3 1.17, +3 7.31, +5 2.14, +6 1.57, +6 127.0 7.10, +2 3.65, +2 2.26, +5 6.53, +5 4.83, +5 137.0 2.29, +2 1.17, +2 7.00, +4 2.02. +5 1.52, +5 147.0 7.52, +1 3.84, + 1 2.19, +4 6.39. +4 4.86, +4 157.0 2.52, +1 1.28, + 1 6.91, +3 2.05, +4 1.59, +4

157.0 2.52, +1 1.28, + 1 6.91, +3 2.05, +4 1.59, +4 193.0 5.67, . 1 2.86, -1 1.20, +2 4.07, +2 3.32, +2 197.0 3.76, -1 1.89, -1 8.34, +1 2.67, +2 2.18, +2

197.0 3.76, -1 1.89, -1 8.34, + 1 2.67, +2 2.18, +2 217.0 4.92, -2 2.47, -2 1.01, + 1 3.34, + 1 2.79, + 1 227.0 1.80, -2 9.04, -3 3.61, +0 1.20, + 1 1.01. + 1

227.0 1.80, -2 9.04, 3.61, +0 1.20, + 1 1.01, + 1 237.0 6.64, -3 3.32, -3 1.29, +0 4.35, +0 3.69, +0 247.0 2.44, -3 1.22, -3 4.56, -1 1.57, +0 1.34, +0 257.0 8.58, -4 4.27, -4 1.20, -1 4.93, -1 4.52, -1 261.0 5.18, -4 2.57, - A 3.08, -2 2.30, -1 2.51, -1 97

Date 1 for SALOME

DATA 1 TILL SALOME MTR 110 111 117 113 121

10 8 6 5 4 3 2 1.5 1 0.8 0.6 0.5 0.4 0.3 0.2 0.15 0.1 0.08 0.06 0.05 0.04 0.03 0.02 0.015 0.01 .00105 .00111 .00119 .00125 .00133 . .00146 .00164 .00177 .00192 .00199 .00123 .00124 .00123 .00199 .00184 .00175 .00162 .00164 ' .00202 .00270 .00435 . .0100 .0314 .0714 .286

5

no

24.8 23.4 21.1 Cut - 98 -

Poto 1 for SALOME (cont.)

117

10.3 This is the complete 9.98 data for magnetite 9.35 concrete (Ho. 117) 8.91 8.25 7.38 6.13 5.33 4.34 3.89 3.40 3.18 2.70 2.50 2.03 1.65 1.02 .694 .371 .237 .129 .0565 .0177 .00782 .00336

1 1 1 1 1 1 1 1 .998 .996 .991 .969 .982 .963 .902 .796 .548 .392 .221 .144 .0808 .0368 .0119 .00535 .00235 99

Data 1 for SALOME (cent.)

,431 Magnetite .349 j concrete (cent.) .253 ~ .158 .137 .0760 .0216 .00494 0

113

18.0 16,9 15.3 14.2 ' Cut

Data 2 for SALOME

CONCRETE EXPERIMENT. CONFIG. 2. 8-0.25 MEV 13 0 30 50 51 67 87 107 127 154 182 202 222 242 261

no 111 113 117 117 117 117 117 117 117 117 117 117

.03 0 .19 .37 .37 .37 .37 .37 ..37 .37 .37 .37 .37

8.0 .25 100

Data 3 for SALOME

GAMMA SOURCE

CONCRETE EXPERIMENT. CONFIG. 2. SPHERICAL. <------30.0 1.061. +10 32.0 1.502,+10 34.0 1.387,+ 10 36.0 1.093,+10 38.0 7.934, +9 40.0 5.490, +9 42.0 3.091, +9 The reaction 0 3.32, 10 44,0 2.428, +9 (fission) rate 2 3.48, 10 46.0 1.545, +9 in core is 4 3.66,10 48.0 8.880, +8 manually 6 4.0, 10 50.0 3.047, +8 punched and 8 4.31, 10 edited to 10 4.55, 10 50.0 1.982, +8 precede the 12 4.65, 10 51.0 1.890, +8 table produced 14 4.95, 10 by NEDI 16 4.95, 10 51.0 8.345, +8 18 4.85, 10 58.0 1.395, +8 20 4.65, 10 65.0 4.850, +7 22 4.36, 10 72.0 2.427, +7 24 4.0, 10 79.0 1.260, +7 26 3.81, 10 86.0 6.312, +6 28 3.76, 10 93.0 3.037, +6 30 4.45, 10 100.0 1.416, +6 107.0 6.44?, +5 114.0 2.391, +5 117.0 2.043, +5

117.0 2.043, 4-5 127.0 6.378, +4 137.0 1.984, M 147.0 6.200, +3 157.0 1.956, +3

157.0 1.956, +3 193.0 3.565, +1 197.0 2.317, +1

197.0 2.317, +1 217.0 2.791, +0 227.0 9.903, -1

227.0 9.903, -1 237.0 3.535, -1 247.0 1.239, -1 257.0 3.178, -2 261.0 7.450, -3 101

An example of one output group from SALOME

SALOME - 23.3.64.

DATA 1 TILL SALOME MTR 110 111 117 113 121

CONCRETE EXPERIMENT. CONFIG. 2. 8-0.25 MEV

ENERGY= 8.00 - 0.25 MEV D= 261.0 CM

STARTED = 2688

ENERGY DEPOSITION RATES, REGIONAL

1.136, -3 4.17, -4

1.635, -4 8.90, -5

1.575, -5 1.11, -5

1.899, -3 1.44, -3

3.227, -4 1.95 -4

1.940, -6 5.91, -7

2.148, -7 4.59, -8

2.746, -8 7.19. -9

1.855, -9 6.15, -10

1.189, -10 2.80, -11

6.206, -12 9.41, -13

1.403, -12 2.88,-13

2.998,-13 1.16, -13

ENERGY DEPOSITION RATE,AVERAGE

2.845, -4 1.13, -4 102

Output group from SALOME (cont.)

PARTIAL DOSE RATES

3.261, -3 3.26, -3

0.0 0.0

0.0 0.0

2.860, -4 1.49, -4

4.854, -4 2.37, -4

8.403, -4 6.66, >4

1.300, -3 9.62, -4

1.669, •3 1.02, -3

7.946, -4 2.33, -4

2.619, -4 6.05, -5

4.273, -4 1.22, -4

3.945, .4 5.36, -5

4.391, -4 3.63, -5

TOTAL DOSE RATE

1.016, -2 3.63, -3

UNCOLLIDED DOSE RATE

5.548, -3 3.23, -3 103 - T able s BI. Recommended data for REFUSE and REBOX

Dose factors Activation "dose factors" in per „ 8 • sec cm^sec Horn ('Jicrgv gro biological physical lnI15(n,n') boundary Ai27(n,e) Na24 P31(n,p)Si3:l S32(n,p)P32 Ni59(n,p)Co58 num mrcrn neutr. neutr. in Mrv mrad , a h ** cm^sec InUSm n cm^sec o 18.00 i 14. 30 0.. 1 46 0.0235 0.00259 0.00282 0.00432 0. 00383 0 2 11 . 36 0.149 0.0248 0.00236 0.00282 0.00601 0.00546 0 3 9.021 0. 150 0.0253 0.00175 0.00282 0.00657 0.00655 0.00019 4 7.166 0.149 0.0255 0.00075 0.00282 0.00648 0.00662 0.00058 5 5.692 0.144 0.0239 0.00017 0.00282 0.00620 0.00623 0.00090 6 4.521 0.137 0.0212 0.000008 0.00248 0.00533 0.00523 0.00124 7 3.591 0. 1 31 0.0188 0 0.00213 0.00483 0.00379 0.00156 8 2.853 0.126 0.0168 0 0.00149 0.00280 0.00203 0.00175 9 2.267 0.124 0.0156 0 0.00111 0.00158 0.00120 0.00181 10 1.800 0. 126 0.0153 0 0.00031 0.00062 0.00062 0.00139 11 1.430 0.129 0.0151 0 0.000059 0.000016 0.000156 0.00103 12 1.136 0.132 0.0147 0 0 0 0 0.00052 1 3 0,9021 0.130 0.0134 0 0 0 0 0.00029 14 0.7166 0.118 0.0118 0 0 0 0 0.000162 15 0.5692 0.100 0.0100 0 0 0 0 0.000083 16 0.4521 0.0885 0.00890 0 0 0 0 0.000023

17 0.3591 0.0740 ' 0.00795 0 0 0 0 0

18 0.2853 0.0645 0.00705 0 0 0 0 0

19 0.2267 0.0545 0.00630 0 0 0 0 0

20 0.1800 0.0463 0.00560 0 0 0 0 0

21 0. 1430 0.0405 0.00495 0 0 0 0 0

22 0.1136 0.0353 0.00442 0 0 0 0 0

23 0.09021 0.0305 0.00400 0 0 0 0 0

24 0.07166 0.0258 0.00355 0 0 0 0 0

25 0.05692 0.0220 0.00321 0 0 0 0 0

26 0.04521 0.0181 0.00289 0 0 0 0 0 27 0.03591 0.0151 0.00261 0 0 0 0 0 28 0.02853 0.0124 0.00239 0 0 0 0 0 29 0.02267 0.0103 0.00220 0 0 0 0 0 30 0.01800 0.0089 0.00208 0 0 0 0 0

groups have equal lethargy width = 0. ..303 units. 104

Table BII. Recommended data for REFUSE and RE BOX (cont. ) Isotope p 7 v 233 1.05 2. 3 Constants in Watt-spectrum u235 1.036 2.29 e-SE .inkfE- Pu23? 1.0 2.2 which is used as a weight function ,0 ; Pu241 1.0 for removal groups.

Table Bill. Recommended data for NEDI

Dose factors Activation "dose factors’1 in ------per — Lower Lethargy Diffusion Internal Biological Physical energy width group spectrum mrad boundary Lethargy U27 (n,K)Na 24 P31(n,p)Si 31 S32(n,p)P 32 Ni58(n,p)Co 58 ln 115(n,n)ln 1I5m number exponent h T Per Mcv units neutr. neutr. cm^sec cm^sec2

0 18.0 1 1 3. 5 0. 288 -10 0.146 0.0236 0.00259 0.00282 0.00470 0.00410 0- 2 10.0 0. 300 -5 0.149 0.0250 0.00212 0.00282 0,00640 0.00616 0.000037 3 7.8 0.248 -4 0. 1 50 0.0256 0.00109 0.00282 0.00652 0.00669 0.00044 4 5.9 0.279 -4 0.146 0.0242 0.00047 0.00282 0.00639 0.00644 0.00083 5 4.4 0.293 -2. 5 0. 1 37 0.0212 0.00001 1 0.00246 0.00565 0.00519 0.00124 6 3.4 0.258 -2.5 0.129 0.0182 0 0.00198 0.00445 0.00315 ' 0.00159 7 2.6 0.268 -2. 5 0.125 0. 0163 0 0.00142 0.00226 0.00180 0.00182 8 2.0 0.262 -2.5 0.126 0.0154 0 0.00073 0.00117 0.00086 0.00161 9 1.5 0.288 0 0.128 0.0152 0 0.00011 0.00012 0.00032 0.00116 1 0 1.2 0.233 0 0.131 0.0148 0 0 0 0 0.00064 1 1 0.9 0.288 -1.5 0.131 0.0137 0 0 0 0 0.00033 1 2 0. 7 0.251 -1 . 5 0.114 0.0114 0 0 0 0 0. 000152 1 3 0.51 0. 317 -1.5 0.0950 0.00960 0 0 0 0 0.000075 1 4 0.38 0.294 -1.5 0.0780 0.00815 0 0 0 0 0.000010 15 0. 3 0.236 -1.5 0.0655 0.00725 0 0 0 0 0 16 0. 1 1.099 -1 0.0420 0.00495 0 0 0 0 0 1 7 0.03 1 . 204 -1 0.0175 0.00290 0 0 0 0 0 18 1,-2 1.099 -] 0.00740 0.00210 0 0 0 0 0 19 1,-3 2. 303 -1 0.00475 0.00210 0 0 0 0 0 20 1, -4 2.303 -1 0.00500 0.00235 0 0 0 0 0 21 1,-5 2.303 -1 0.00500 0.00250 0 0 0 0 0 22 1, -6 2.303 -1 0.00500 0.00250 0 0 0 0 0 23 1.05,-7 2.254 -1 0.00500 0.00250 0 0 0 0 0 Ther - — 0.00375 0.001151 0 0 0 0 0 24 mal - - i K>5 -

Table B IV. Thermal data.

Aiomic number Element Density Diffusion Macr. absorption constant cross section

g,A'm3 cm cm~^

1 H (2.. 2 km/s! cr =0* 332 r a ~ yb,. ?g.= 38 b) - H^O 1.00 0.164 0.022 * 1 D (2_. 2 km/s;• 4 Be 1.85 0.60 0.00124 5 B 2.45 1.28 1 03 6 C 1. 60 0.778 2. 6, -4 7 N (2. 2 km/s

15 P 1.82 1.90 0. 007 ( * ) 1 6 S 2. 07 7.24 0. 019 ( * ) 19 K 0.87 13. 1 0. 026 * 20 Ca 1.55 4.70 0. 010 ( * ) 24 Cr 7. 1 1.13 0.238

* 25 Mn 7. 2 0.839 1.04 * 26 Fe 7.86 0.345 0.215 27 Co 8.9 0. 251 3.37

( * ) 28 Ni 8.90 0.199 0.420 29 Cu 8.94 0.497 0.313 * 40 Zr 6.4 0.99 0.008 48 | Cd ' 8.65 0.011 ‘ 1 54 56 Ba. | 3. 5 2. 64 0. 01 8 * 82 Pb 11.35 0.918 0. 006 90 Th 11.3 1.49 0.205

92 U 18.9 0.683 0.364

uo2 10. 5 0. 615 0.169

* Data in library tape form summer 1964 (*) In work Data from ref. 1 2

Note that the densities in this table and ref. 12 are not always equal to those given in table B V. Table B V* Recommended data for SALOME m3) A-l; p = 2,70 Fe; p 7. 80 Pb; p = 11.40 Watc r (p = . o 8 A - 1 7 “ ■ . „ 0- MeV + +

10 1 . 353 16.2 1 .515 4. 36 .984 .734 1, 80 1 0,2202 45.66 ft 27 6 15.4 . 431 4. 35 .979 .682 1.91 0. 1657 41.67 6 . 194 14. 0 . 326 4,22 .966 .585 2. 01 0.1106 36. 36 5 . 148 13.1 . 259 4.10 .959 .517 2.06 0.08102 33,22 4 . 102 12. 0 . 183 3.88 .944 .412 2.09 0.05456 29.50 3 . 0544 10. 5 . 105 3.56 .923 ,293 2.08 0.02233 25.25 2 .0155 8, 58 . 0305 3.02 .873 . 125 1.92 0,007458 20,28 1.5 . 00.3 39 7.41 .00713 2.64 . 830 .0375 1.71 0.001687 17. 39 1 0 6.03 .995 2. 15 . 736 0 1.28 0 14. 16 0.8 5.42 .992 1,93 .671 1.05 12.72 0.6 4.77 .986 1.68 . 558 .769 11. 16 0.5 4,41 .979 1. 55 .476 . 606 10.35 0.4 .998 4.02 .966 1.39 , 364 .422 9.434 0. 3 .997 3.60 ,936 1.21 .237 .246 1.000 8.475 0,2 .984 3.09 . 826 .929 .108 .0980 0.9975 7. 353 0. 15 .962 2.77 .678 . 701 .576 . 0476 0.9955 6.711 0. 1 839 2.30 .401 .373 . 222 .0166 0. 9860 5.988 0.08 . 807 1.99 .261 .231 .0876 . 0621 0. 9718 5,618 0. 06 .639 1.49 . 201 . 113 . 0415 . 0279 0.9333 5.102 0. 05 .499 1. 14 .139 . 0701 . 0257 .0169 0.8891 4. 739 0.04 .331 . 731 . 0466 . 0368 . 0141 .00901 0. 7961 4. 115 0.03 167 . 356 . 0208 . 0160 .00638 .00394 0. 5964 2.985 0.02 . 0538 . 111 .00671 ,00497 .00211 .00125 0.2932 1.418 0.015 . 0233 .0473 .00225 .00220 .00586 ,00342 0.1414 0.6711 0. 01 .00706 .0141 .00101 .000720 .00191 .00110 0.04313 0.2016

Ev ' Air; p S lOx 1. 293 • 10"J Magnetite concrete; p = 3.74 Ordinary concrete; p ~ 2,43 R2-0- core P - . 74 (75. 5 % N, 23.2 ? O, !. 3 % A) (34.0 % O, 6 8 % Ca. 4. 8 % Si, 0. 8 % A1 (49. 0 % O, . 8 % Ca, 29, 7% Si,9. 7 % A1 (Voi-%: 58.4 % H2C, 4 1.4 % Al, 0, 2 % U) 53.2 % Fe 0,4 % H) 3.0 % Fe. o. 80 % 10 1 .247 3. 82,3 1 .431 10. 3 1 .326 18.0 .998 . 330 24, 8 8 . 182 3. 52,3 .349 9.98 .255 16.9 .998 . 257 23.4 6 . 121 3.09.3 .253 9.35 . 1 15. 3 .998 . 160 21. 1 5 . 0905 2.82, 3 . 198 8,91 . 134 14.2 .998 . 136 19.7 4 . 0593 2. 52,3 . 137 8.25 . 0912 12.9 .998 . 0948 17.8 3 .0317 2. 17,3 .0760 7. 38 .0485 11.3 . 998 . 0511 15,6 2 .00840 1.74,3 .0216 6. 13 .0137 9. 19 .995 . 0151 12*7 1.5 .00191 1.50,3 .00494 5. 33 .00301 7. 93 . 995 ' .00351 11.8 1 0 1.22, 3 .998 0 4, 34 0 6.46 ,991 ) 8. 85 0.8 1.10,3 .996 3. 89 5.80 .976 7.94 0. 6 9.62,2 .991 3.40 5.10 .966 6.90 0.5 8.91,2 ,989 3. 18 .999 4.71 .962 6.25 ’ 0.4 8.12,2 .982 2,70 .996 4. 30 .943 5.71 0,3 7.30,2 .963 2.50 .991 3. 81 .915 4.98 0.2 6. 29, 2 .902 2.03 .994 3.29 . 825 3. 89 0. 15 5.77,2 .796 1.65 ,957 2.94 . 706 3. 06 0. 1 . 980 5. 12,2 . 548 1.02 . 860 2.39 .811 3.15 0.08 .969 4. 80, 2 . 392 .694 .768 2.03 . 703 2.60 0. 06 .921 4.34,2 .221 . 371 . 587 ■ 1.46 . 516 1.81 0.05 .871 3.99,2 . 144 . 237 .452 1. 10 . 385 1,32 0.04 .770 3.42, 2 . 0808 . 129 .294 .696 .249 , 826 0.03 . 568 2.44,2 .0368 .0565 . 148 . 337 .124 .400 0.02 .267 1.11,2 .0119 . 0177 . 0491 , 109 .0732 .227 0.0(5 . 128 5.23, 1 .00535 .00782 . 021 7 . 0470 .0297 . 0910 0. 01 . 0395 1.58.1 .00235 .00336 .00682 .0145 .00922 . 0277

Light water moderated research reactor, MTR fuel elements Data from ref. 9 and 1 0 References

1. F ORS B E RG/H J A rne L and PALS SON S Bestamning av Legendrekoefficienter , RFA-483 (1963). Internal report of AB Atomenergi

2. NEWTON T D Shell effects on the spacing of nuclear levels Can. J. Phys. 34, 804-829 (1956)

3. HJARNE L and LEIMDORFER M A new method for predicting the penetration and s lowing - down in reactor shields To be published

4. RALSTON A and WILF H S Mathematical methods for digital computers New York, Wiley, 1960

5. GOERTZEL G and KALOS M H Progress in Nuclear Energy, Ser. I, Vol. 2, p. 315 New York, Pergamon Press, 1958

6 . FANO U, SPENCER L V and BERGER M J Encyclopedia of Physics, Vol. 38/2, p. 660 Berlin, Springer-Verlag, 1959

7. LEIMDORFER M A Monte Carlo method for the analysis of gamma radiation transport from distributed sources in laminated shields AE-135, Nukleonik 6 , 58-65 (1964)

8. RASO D J Monte Carlo calculations on the reflection and transmission of scattered gamma rays Nucl. Sc. and Eng. JT7, 41 1-418 (1963)

9. GRODSTEIN G W X-ray attenuation coefficients from 10 kev to 100 Mev. Wash. D. C. 1957 NBS Circular 583 and Me Ginnies, R T, Suppl. (i960)

10. AALTO E and NILSSON R Measurements of radiation attenuation in massive laminated shields and a study of the accuracy of some design methods To be published

1 1. PREISER S, RABINOWITZ G and de DUFOUR E A program for the numerical integration of the Boltzmann transport equation - NIOBE ARL-TR-60-314 (Dec. I960)

12. ' DAVIS M V and HAUSER D T Thermal-neutron data for the elements Nucleonics 16 , 87-89 (1958) Data Sheet No 23

LIST OF PUBLISHED AE-REPORTS 111. The paramagnetism of small amounts of Mn dissolved in Cu-AI and Cu-Ge alloys. By H. P. Myers and R. Westin. 1963. 7 p. Sw. cr. 8:—. 1—70. (See the back cover earlier reports.} 112. Determination of the absolute disintegration rate of Cs137 -sources by the tracer method. S. Hellstrom and D. Brune. 1963. 17 p. Sw. cr. 8:—. 71. The space-, time- and energy-distribution of neutrons from a pulsed 113. An analysis of burnout conditions for flow of boiling water in vertical plane source. By A. Claesson. 1962. 16 p. Sw. cr. 6 :—. round ducts. By K. M. Becker and P. Persson. 1963. 28 p. Sw. cr 8:—. 72. One-group perturbation theory applied to substitution measurements 114. Measurements of burnout conditions for flow of boiling water in vertical with void. By R. Persson. 1962. 21 p. Sw. cr. 6 :—. round ducts (Part 2). By K. M. Becker, et al. 1963. 29 p. Sw. cr. 8:—. 73. Conversion factors. By A. Amberntson and S-E. Larsson. 1962. 15 p. Sw. 115. Cross section measurements of the 58Ni(n, p)58Co and 29Si(n, a )26 Mg reac­ cr. 10:—. tions in the energy range 22 to 3.8 MeV. By J. Konijn and A. uauber 74. Burnout conditions for flow of boiling water in vertical rod clusters. 1963. 30 p. Sw. cr. 8;—. By Kurt M. Becker. 1962. 44 p. Sw. cr. 6 :—. 116. Calculations of total and differential solid angles for a proton recoil 75. Two-group current-equivalent parameters for control rod cells. Autocode solid state detector. By J. Konijn, A. Lauber and B. Tollander. 1963. 31 p. programme CRCC. By O. Norinder and K. Nyman. 1962. 18 p. Sw. cr. Sw. cr. 8:—. 6 :—. 117. Neutron cross sections for aluminium. By L. Forsberg. 1963. 32 p. 76. On the electronic structure of MnB. By N. Lundquist. 1962. 16 p. Sw. cr. Sw. cr. 8:—. 6: —. 118. Measurements of small exposures of gamma radiation with CaSO^Mn 77. The resonance absorption of uranium metal and oxide. By E. Hellstrand radiothermoluminescence. By B. Bjarngard. 1963. 18 p. Sw. cr. 8:—. and G. Lundgren. 1962. 17 p. Sw. cr. 6 :—. 119. Measurement of gamma radioactivity in a group of control subjects from 78. Half-life measurements of 6 He, UN, lsO, 2CF, 23A1, 77 Se" and no Ag. By J. the Stockholm area during 1959—1963. By I. O. Andersson, I. Nilsson Konijn and S. Malmskog. 1962. 34 p. Sw. cr. 6 :—. and Eckerstig. 1963. 19 p. Sw. cr. 8:—. 79. Progress report for period ending December 1961. Department for Reac­ 120. The thermox process. By O. Tjalldin. 1963. 38 p. Sw. cr. 8:—. tor Physics. 1962. 53 p. Sw. cr 6 :—. 121. The transistor as low level switch. By A. Lyden. 1963. 47 p. Sw. cr. 8:—. 80. Investigation of the 800 keV peak in the gamma spectrum of Swedish 122. The planning of a smoll pilot plant for development work on oqueous Laplanders. By 1. O. Andersson, I. Nilsson and K. Eckerstig. 1962. 8 p. reprocessing of nuclear fuels. By T. U. Sjoborg, E. Haeffner and Hult- Sw. cr. 6,i—. gren. 1963. 20 p. Sw. cr. 8:—. 81. The resonance integral of niobium. By E. Hellstrand and G. Lundgren. 123. The neutron spectrum in a uranium tube. By E. Johansson, E. Jonsson, 1962. 14 p. Sw. cr. 6 :—. M. Lindberg and J. Mednis. 1963. 36 p. Sw. cr. 8:—. 82. Some chemical group separations of rodioactive trace elements. By K. 124. Simultaneous determination of 30 trace elements in cancerous and non- Samsahl. 1962. 18 p. Sw. cr. 6 :—. cancerous human tissue samples with gamma-ray spectrometry. K. Sam­ 83. Void measurement by the (y, n) reactions. By S. Z. Rouhani. 1962. 17 p. sahl, D. Brune and P. O. Wester. 1963. 23 p. Sw. cr. 8:—. Sw. cr. 6 ,:—. 125. Measurement of the slowing-down and thermalization time of neutrons 84. Investigation of the pulse height distribution of boron trifluoride pro­ in water. By E. Moller and N. G. Sjostrand. 1963. 42 p. Sw. cr. 8:—. portional counters. By I. O. Andersson and S. Malmskog. 1962. 16 p. 126. Report on the personnel dosimetry at AB Atomenergi during 1962. By Sw. cr. 6,t—. K-A. Edvardsson and S. Hagsgdrd. 1963. 12 p. Sw. cr. 8:—. 85. An experimental study of pressure gradients for flow of boiling water in vertical round ducts. (Part 3). By K. M. Becker, G. Hernborg and M. 127. A gas target with a tritium gas handling system. By B. Holmqvist and Bode. 1962. 29 p. Sw. cr. 6 :-—. T. Wiedling. 1963. 12 p. Sw. cr. 8:—. 86 . An experimental study of pressure gradients for flow of boiling water 128. Optimization in activation analysis by means of epithermal neutrons. in vertical round ducts. (Part 4). By K. M. Becker, G. Hernborg and M. Determination of molybdenum in steel. By D. Brune and K. Jirlow. 1963. Bode. 1962. 19 p. Sw. cr 6 :—. 11 p. Sw. cr. 8:—. 87. Measurements of burnout conditions for flow of boiling water in vertical 129. The Pt-approximation for the distribution of neutrons from a pulsed round ducts. By K. M. Becker. 1962. 38 p. Sw. cr. 6 :—. source in hydrogen. By A. Claesson. 1963. 18 p. Sw. cr. 8:—. 88. Cross sections for neutron inelastic scattering and (n, 2n) processes. By 130. Dislocation arrangements in deformed and neutron irradiated zirconium M. Leimdorfer, E. Bock and L. Arkeryd. 1962. 225 p. Sw. cr. 10:—. and zircaloy-2. By R. B. Roy. 1963 18 p. Sw. cr. 8:—. 131. Measurements of hydrodynamic instabilities, flow oscillations and bur­ 89. On the solution of the neutron transport equation. By S. Depken. 1962. nout in a natural circulation loop. By K. M. Becker, R. P. Mathisen, O. 43 p. Sw. cr. 6 r—. Eklind and B. Norman. 1964 . 21 p. Sw. cr. 8:—. 90. Swedish studies on irradiation effects in structural materials. By M. Grounes and H. P. Myers. 1962. 11 p. Sw. cr. 6 :—. 132. A neutron rem counter. By I. O. Andersson and J. Braun. 1964. 14 p. Sw. cr. 8:—. 91. The energy variation of the sensitivity of a polyethylene moderated BF3 proportional counter. By R. Fraki, M. Leimdorfer and S. Malmskog. 1962. 133. Studies of water by scattering of slow neutrons. By K. Skold, E. Pilcher 12. Sw. cr. 6 :—. and K. E. Larsson. 1964. 17 p. Sw. cr. 8:—. 134. The amounts of As, Au, Br, Cu, Fe, Mo, Se, and Zn In normal and urae­ 92. The backscattering of gamma radiation from plane concrete walls. By mic human whole blood. A comparison by means of neutron activation M. Leimdorfer. 1962. 20 p. Sw. cr. 6 :—. analysis. By D. Brune, K. Samsahl and P. O. Wester. 1964. 10 p. Sw. cr. 93. The backscattering of gamma radiation from spherical concrete walls. 8:—. By M. Leimdorfer. 1962. 16 p. Sw. cr. 6: —. 135. A Monte Carlo method for the analysis of gamma radiation transport 94. Multiple scattering of gamma radiation In a spherical concrete wall from distributed sources in laminated shields. By M. Leimdorfer. 1964. room. By m. Leimdorfer. 1962. 18 p. Sw. cr. 6 :—. 28 p. Sw. cr. 8:—. 95. The paramagnetism of Mn dissolved in n and ft brasses. By H. P. Myers 136. Election of uranium atoms from UO2 by fission fragments. BMG. Nilsson. and R. Westin. 1962. 13 p. Sw. cr. 6 :—. ^ 1964. 38 p. Sw. cr. 8:—. 96. Isomorphic substitutions of calcium by strontium in calcium hydroxy ­ 137. Personnel neutron monitoring at AB Atomenergi. By S. Hagsgdrd and apatite. By H. Christensen. 1962. 9 p. Sw. cr. 6 :—. C-O. Widell. 1964. 11 p. Sw. cr. 8:—. 97. A fast time-t'o-pulse height converter. By O. Aspelund. 1962. 21 p. Sw. cr. 138. Radiation induced precipitation in iron. By B. Solly. 1964. 8 p. Sw. cr.

98. Neutron streaming in DaO pipes. By J. Braun and K. Randen. 1962 139. Angular distributions of neutrons from (p, n)-reactions in some mirror 41 p. Sw. cr. 6: —. nuclei. By L. G. Stromberg, T. Wiedling and B. Holmqvist. 1964. 28 p. Sw. cr. 8:. 99. The effective resonance integral of thorium oxide rods. By J. Weitman. 1962. 41 p. Sw. cr. 6: —. 140. An extended Greuling-Goertzel approximation with a Pn -approximation in the angular dependence. By R. Hakansson. 1964. 21 p. Sw. cr. 8:—. 100. Measurements of burnout conditions for flow of boiling water in vertical annuli. By K. M. Becker and G. Hernborg. 1962. 41 p. Sw. cr. 6: —. 141. Heat transfer and pressure drop with rough surfaces, a literature survey. By A. Bhattacharyya. 1964. Sw. cr. 8:—. 101. Solid angle computations for a circular radiator and a circular detector. By J. Konijn and B. Hollander. 1963. 6 p. Sw. cr. 8:—. 142. Radiolysis of aqueous benzene solutions. By H. Christensen. 1964. Sw. cr. 102. A selective neutron detector in the keV region utilizing the 19F(n, y) 2°F reaction. By J. Konijn. 1963. 21 p. Sw. cr. 8:—. 1 2 3 4 143. Cross section measurements for some elements suited as thermal spect­ rum indicators: Cd, Sm, Gd and Lu. E. Sokolowski, H. Pekarek and 103. Anion-exchange studies of radioactive trace elements in sulphuric acid E. Jonsson. 1964. Sw. cr. 8:—. solutions. By K. Samsahl. 1963. 12 p. Sw. cr. 8:—. 144. A direction sensitive fast neutron monitor. By B. Antoikovic, B. Holm­ 104. Problems in pressure vessel design and manufacture. By O. Hellstrom qvist and T. Wiedling. 1964. Sw. cr. 8:—. and R. Nilson. 1963. 44 p. Sw. cr. 8;—. 145. A User's Manual for the NRN Shield, by L. Hjarne. 1964. 105. Flame photometric determination of lithium contents down to 10-3 ppm in water samples. By G. Jonsson. 1963. 9 p. Sw. cr. 8:—. Forteckning over publicerade AES-rapporter 106. Measurements of void fractions for flow of boiling heavy water in a vertical round duct. By S. Z. Rouhani and K. M. Becker. 1963. 2nd rev. 1. Analys medelst gamma-spektrometri. Av D. Brune. 1961. 10 s. Kr 6: —. ed. 32 p. Sw. cr. 8r—. 2. Bestrdlningsforandringar och neutronatmosfar i reaktortrycktankar — 107. Measurements of convective heal transfer from a horizontal cylinder ndgra synpunkter. Av M. Grounes. 1962. 33 s. Kr 6: —. rotating in a pool of water. K. M. Becker. 1963. 20 p. Sw. cr. 8:—. 3. Studium av strackgransen I mjukt stdl. G. Ostberg, R. Attermo. 1963. 17 s. 108. Two-group analysis of xenon stability In slab geometry by modal expan ­ Kr 6: —. sion. O. Norinder. 1963. 50 p. Sw. cr. 8r—. 4. Teknisk upphandling inom reaktoromrddet. Erik Jonson. 1963.64 s. Kr. 8:—. 109. The properties of CaSO^Mn thermoluminescence dosimeters. B. Bjarn- gard. 1963. 27 p. Sw. cr. 8:—. Additional copies available at the library of AB Atomenergi, Studsvik, Nyko- 110. Semianalytical and seminumerica! calculations of optimum material ping, Sweden. Transparent microcards of the reports are obtainable through distributions. By C. I. G. Andersson. 1963. 26 p. Sw. cr. 8:—. the International Documentation Center, Tumba, Sweden.

EOS-tryckerierna, Stockholm 1964