ÅE-39
The Dependence
0\
of the Resonance Integral
on the Doppler Effect
J. Rosen
AKTIEBOLAGET ATOMENERGI
STOCKHOLM • SWEDEN • I960
AE-39.
THE DEPENDENCE OF THE RESONANCE INTEGRAL ON THE DOPPLER
.
EFFECT.
J. Rosén
Summary:
The Doppler sensitive contributions to the resonance integral for
metal and oxide cylinders have been calculated using tables compiled
by Adler, Hinman and Nordheim.
The temperatures 20, 200, 350, 500 and 650
C have been investi-
gated for the pure metal and 20, 300, 600, 900 and 1200 °C for the oxide.
Contributions from the separate resonances in the resolved region
and for certain energies in the unresolved region are accounted for in
detail. Integration over adequate statistical distributions has been carried
out for the resonance parameters in the unresolved region.
The increase in the resonance integral at elevated temperatures due
to the Doppler effect is given separately in tables and diagrams.
Printed December I960.
LIST OF CONTENTS
Page
I.
Introduction
3
II.
Cross sections
3
III.
Resonance absorption in the heterogeneous case
7
IV.
The Calculations
9
V.
Results
15
VI.
Summary of the results
16
VII.
Accuracies
17
VIII. The results in the form of diagrams
18
IX.
18
Concluding remarks
References
20
The Dependence of the Resonance Integral on the Doppler Effect.
I.
Introduction
An investigation of the dependence of the resonance integral on the
Doppler effect has been made, Pure metal and oxide cylinders of different radii have been studied at five different temperatures.
II. Cross sections.
The following types of cross sections are encountered in the reso
nance region:
1.
ir, / = konst
the background absorption. It is assumed to consist of two parts, a true
l / v - c r o s s section and a part which approximately corrects for the skirts
in many resonances at higher energies. This component is in the reactor
theory not treated in connection with the resonance escape probability
and does not enter into the resonance integral. It is not dependent on the
Doppler effect.
2.
o-a (E, T) =.
the resonance absorption for a line at energy E~. I* and P are the
radiative capture and neutron widths at exact resonance. P = P + P is
y
n
the width of the resonance peak at half-maximum,
<rQ= 2, 60 1 0 *
P
and
0
%' (x-y)
4
00
x,
-Ju
2 */
-.CO
1+y2
where
-2
x'= — {'E'- E n )
a= '
4kTE
E = the neutron energy, A = the ratio between the masses of the uranium
nucleus and the neutron, k = Boltzmann'' s constant and T = the absolute
temperature.
ijj (x, £) is the shape function obtained if one assumes that the resonance can be described by a simple Breit-Wigner expression and with
the Doppler effect included assuming maxwellian velocity distribution of
the nuclei.
i
This expression is derived assuming that
kT « EQ
r
«E0
2 -3 Q
which is approximately true for U
3.
P
<r r g = - £ - crQ q; ( X) £ ) , '
P
the resonance scattering cross section.
4.
c
= <r + <r ,
p
h
m
the potential scattering cross section (energy independent). <r is the cross
section for uranium and <r for any other element in the fuel such as oxygen
in the case of the oxide.
5.
a scattering cross section which describes the interference between the
potential scattering and the resonance scattering.
Here
?
7
dy
This cross section is asymmetric about E_. Its presence tends to increase
the slowing down properties of the uranium itself just above the resonance
and decrease it just below. The neutrons are concentrated to the. parts of
the resonance where the absorption cross section is the largest and consequently tends to increase the absorption.
6.
The necessary quantities for a satisfactory description of the cross
sections under 2, 3 and 5, are evidently F , P and E_. Much work has
been done during the last couple of years to obtain information about these.
Information below E» = 540 eV is also rather complete for the more important resonances. The number, of resonances in this resolved energy
region amounts to 22. Above 540 eV - the unresolved or statistical region certain assumptions can be made on a theoretical or semi-empirical basis.
One assumes, that the resonances up to energies around 30 keV are sufficiently far apart so that no interference between the lines will exist; that
they are uniformly distributed along the energy scale; that they have a
constant radiative ,-capture width P j that they posses neutron widths having
"V>
the property that
P°
w
n
n
is normally distributed,
•6.
where F = P
n
n
7.
v^E and where F
n
is the average value for P .
°
n
Above 30 keV the picture becomes different. A levelling otit of the
resonance structure has taken place. The width of the lines is dominated
by the neutron width and increases while the top value at the same time
decreases. Processes involving neutrons with spin numbers I > 0 become
important. Chernick and Vernon (l) have discussed the problem of the best
values for the parameters for neutrons with t - 1 and have also estimated
additional contributions from this specific energy region. The dependence
on the Doppler effect can be neglected.
8.
In the present investigation the necessary data according to point 6
have been those recommended by Harvey and Schwartz. Values for the 22
resolved resonance peaks are given in table 1. They have been separated
into two groups and numbered consecutively according to increasing energy
in each group. The first group includes peaks 1 through 6 and the second
group 7 throLigh 22. Regarding the principles governing the separation into
groups, see section 9 below.
The unresolved region is characterized by the average values for the
quantities mentioned in 6. They have been obtained by statistical analysis
of the parameters in the resolved region. Harvey and Schwartz have here
used available data only in the energy region below 300 eV. It has been found
that weak resonances though of negligible importance to the value of the
resonance integral have a major influence on the average values of the
resonance parameters. Åt energies between 300 and 540 eV these weak
resonances have probably not been resolved in most cases.
The values obtained by Harvey and Schwartz are:
P = 0. 024 - 0. 001 ev;
y
P° = 0. 0022 - 0. 004 x(ev)
n
'
l
'
c
and D = 16 - 3 ev.
The values <r, = 10 b and <r = 8. 4 b for the oxide have been used for the
h
m
scattering cross sections.
•7.
III. Resonance absorption in the heterogeneous case
9.
The model used for the resonance absorption is the model originally
introduced by Wigner (2). The resonances are separated into two classes.
One of them contains the broad resonances and the other those which are
strongly concentrated around the resonance energy. In order to determine
if a resonance is broad or not, Wigner introduced the concept of the practical width A. The peak has the width Awhen the condition <r + <r = <r
J
a
rs
p
is satisfied. If ZK »
E o (1 - a) the peak is broad and if A «
E^ (1 - a) it
is not. E» (1 - a) is the maximum possible energy loss experienced in a
collision between a uranium nucleus and a neutron of energy E~. The two
groups are designated; IA-infinite absorber, due to reasons given below and NR - narrow resonance. In the IA-group it is assumed that an elastic
collision between a neutron and a nucleus leaves the neutron at the same place
in the resonance. This would be exactly true if the nucleus had an infinite
weight, thus the name. For the other group it is assumed that such a collision completely removes the neutron from the resonance. The lines 1
through 6 belong to the IA-group and the rest to the NR-group.
The expressions for the resonance absorption are based on these
rather crude models. It has been found that boundary cases between the
IA- and NR-approximations are not very sensitive to which of the groups
they are referred. It is obvious that subtle effects such as the influence of
the interference structure according to point 5 cannot be expressed with
these models.
10.
The resonance integral is" thus composed of the following components:
The contribution from the IA-linfes, the resolved NR-lines, the unresolved
NR-lines in the statistical region up to 30 keV and finally the region above
this energy. In the last part is also included the contribution from neutrons
with spin numbers I > 0. Only the Doppler dependent line contributions are
considered below. Wigner has deduced expressions for these according to
the following reasoning. He separates the neutrons in the fuel into two groups,
those having experienced their last collision in the fuel and those having
experienced it in the moderator. The contribution from the distinct lines
is thus separated into volume and surface terms.
8.
The number of neutrons from the first group slowing down to an energy
interval between E and E + dE is approximately cr • ,-, . This should be
,. ,
. •
P -k
equal to the number scattered out of the interval when no resonance structure exists, and the presence of resonances does not make too much difference, since the width of the resonances is insignificant compared to the
energy interval from which the neutrons being slowed down originate. The
number scattered out of the interval is now, however, a product of the volume flux and cr . By equating these two quantities an expression for the
volume flux can be obtained and the corresponding component be computed.
In the other neutron group one considers the excess number of neutrons
from the moderator, which enter through a surface element, and proceeds
to correct this number by multiplying with the probability for a collision in
the fuel and also the probability that such a collision would lead to capture.
This is then the surface term.
In thé NR-approximation the above expressions will look like:
p
cr
A,
E
It should be noticed chat the interference term is not included in cr .
4=
V
In the above expression, r- = —
— , i.e. the average chord for an isotropically
incident flux, r = 2a in the cylindrical case if the radius is a.
238
3
N = the number of U
nuclei per cm in the fuel, F.= the probability that
an incident neutron will experience a collision in the fuel.
F = N <r r Pn(N <r r), where P_ = the average escape probability computed
and tabulated by Case, de Hoffmann and Placzek (3).
At an earlier stage in the development of the theory F was put equal
to 1 and one obtained Ao -» l / r
o.
~ S/M and the resonance integral became
equal to
RI = A + B ' -|j
where A and B are constants.
The surface term will now become a more complicated function of the
geometry but the designation is maintained unchanged,
11.
By letting <r —» cr
and o* —JO in the expressions in 10, cor-
responding expressions are obtained in the IA-approximation for the oxide
case. For a pure metal it is assumed that all volume neutrons which are
slowed down to the resonance are absorbed, i . e . , the volume contributions
are all of the same size and equal to the slowing down density
A-. = 2/A • or = 0. 084 b
i
(Doppler independent)
By introducing this modification the corresponding surface terms for
the metal become
Ag = 1/Nr
IV. The Calculations
12.
As mentioned in the introduction the calculations are intended to
estimate the increase in the resonance integral at elevated temperatures
due to the Doppler effect. They have been carried out by calculating the
Doppler sensitive contributions first at room temperature and then at
higher temperatures.
20 .
13.
By inserting the cross sections given in II into the expressions in 10
and 11, the contributions can be expressed in a few standard formulae. The
interference term and the variation in N E-/E in the expression for <r . have
been excluded. The following expressions have then been obtained:
14.
The IA-lines
The volume terms for the metal are obtained according to 11, and for the
oxide one gets
where
dx
The surface terms for the metal are
i
i
with
Th 2 - *Z
Z = No-, a
h
dx
11.
and for the oxide
cr
= -£2
A
E.
1
Z = No- a
m
o
15. The NR-lines
Volume terms
i
i
cr
Surface terms
A
=
S.
o-F PY
^
L (Z, g, P)
ÄI.
i i
B
r
Z = No- a
P
For the metal <r = <r and for the oxide tr = cr, + cr
p
h
p
h
m
16.
The unresolved region
The expressions in 15 will hold in general, but integration over the distributions according to 6 must be carried out.
12.
The expressions become;
y
_
00
E
CO
_
—
c
2
CO
E
y
CO
c
Here is
2
y
n
=
n
F
(T
8=
i
E
With the expression
2
00
j -
~
J^ e
\
J(|,P)dy
0
and correspondingly for L,/p one gets
C
—
13.
It has been found desirable and also possible with respect to the
accuracy to express the variable as E = E~ • .2
and evaluate J (jfe)
at points, n = 0 , 1, 2 . . . .
One then obtains the contributions in the form
00
In 2 \
J (n) dn
0
and correspondingly for A« .
Only a few n-values have to be used,
18.
The basis for the calculations,
.
Tables for the functions L / , J and JL have been compiled by Adler, Hinman
and Nordheim (4). Their tables and instructions for carrying out the calculations have been followed in the present investigation. Tables for J are
also available from other sources (5).
The J- and L ' -tables are two-dimensional while the L-table is in
principle three-dimensional. One has, however, proceeded by constructing
two-dimensional L-tables for a series of Z-values. These Z-values are:
0.05
0.10
0.20
0.40
0.80
1.60
3.20
and
6.40
The boundaries for the variation of the geometrical quantities are
thus fixed. Wigner succeeded by approximating P~ with a discontinuous
function and assuming tr «
<r- in taking out a penetration factor
from the surface term in which the total geometry dependence is included.
According to Bakshi (5) the resulting error in the estimation of the contribution is about 10 %.
.14.
Besides the cylinder case also the slab and spherical cases have been
tabulated. The authors, however, point out that for a constant value of the
average chord the difference between the different cases becomes negligible.
19.
The tables are evaluated for the following ranges of the arguments:
£ = 0.02,
(0.01) 0.05,
6 = 2 k • 10" 5 ,
k = 4,
Z'=2"j-104;
(0.05) 0, 5 and oo
(0, 5), 20
j = -2,
-1,5,
(0.5),
...9
These arguments are1 rather far apart and a 4-point interpolation
method has therefore been developed and used. It has furthermore been
necessary to increase the table with values for £ = 0, in which case one sets
00
lim
J (£, P) = \ 7^ dx =
J
oe
g-o
lim
L (t, g, p) s 0
An accurate estimate of the contribution from the statistical region
cannot be obtained otherwise.
20. The calculations in detail
The calculations have been carried out for the following temperatures:
Metal:
Oxide:
20
20
200
300
350
600
500
900
and
and
650 °C
1200 °C
These temperatures together with values for the parameters in Table 1
have been used to calculate the input values £ and 5 . The Z'-values for
the L ' -table corresponding to the Z-values have then been computed for
15.
each line. The integration over the variable y in the unresolved region has
been carried out for proper values of T and E. E is chosen from the eeries
of increasing n according to 16. (3 is very large for small values of y and
the calculations start by determining the minimum value of y for which one
can enter the table. Values for the functions outside the table are considered negligible. The value of the integrand has been evaluated at 50 points
with a density between the points which has been varied in four steps. The
integration has been carried out using Simpson' s rule. The same formula
has been used in the integration over n.
21.
All calculations have been made on Facit EDB. The programming has
been performed by Å. Norden, L. Persson and B. Tollander of the Atomic
Energy Company. A calculation
of the contribution from the unresolved
region for a T-Z combination takes about 16 min.
V. The Results
22.
The direct results, i.e. the output data from the computer are found
in Tables 2 - 6 . The tables 2, 3 and 4 refer to the metal while tables 5 and
6 refer to the oxide.
23.
Table 2
refers to the IA-lines, the surface terms. The Z-value is
given at the head of each sub-table, and then follows in five columns the
contributions from the separate lines. Each temperature has its own column
with its value heading the column. They end with an asterisk and the sum
of the contributions from the IA-lines. All numbers are presented in the form
of a decimal expression and a 10-exponent. The value 293. 2 for instance is
written as 0. 2932 03.
The tables start with Z = 0. 20 instead of the minimum value 0. 05.
The reason for this is that the Z-values 0. 05 and 0. 10 give Z x -values for
certain lines which fall outside the L ' -table.
24.
Table 3 contains the contributions from the 16 NR-lines. The con-
struction is analogous to table 2. The Z-value 0.00 in the first sub-table
16.
indicates that the content refers to the volume term. This procedure is
used throughout the rest of the tables.
25.
Table 4 shows the contribution from the unresolved region. The table
is divided according to the 9 different Z-values with sub-tables for the 5
different temperatures. The Z-value is preceded by 3 asterisks. The value
of the temperature heads each of the sub-tables. The sub-table itselt lists
the values of J (E) and L. ,^
as a function of E = E _ 2 . The sub-table
ends by an A followed by the numerical values of A,r and A~.
1
V
b
26.
Table 5 corresponds to tables 2 and 3 for the metal case. The 6 IA-
lines and their sum head the table and are followed by corresponding data
for the NR-lines.
27.
Table 6 corresponds directly to table 4
VI. Summary of the results
28.
It is now possible to summarize the Doppler sensitive contributions
to the resonance integral from tables 2 through 6,
29.
The metal. From the relationship Z = Ncr a = 0.473 a (the NR-case)
using the Z-series in the tables one obtains the following radii:
0.423
0.845
1.69
3.38
6.76
and
13.5 cm
The IA-contributions are missing as pointed out earlier for the two
smallest Z-values and the minimum radius is therefore 0.423 cm.
30.
The oxide. In this case Z-TT, = N (<r + cr ) a while Z T . = Ncr a.
•
.
.
NR
h
m
IA
m
If one lets Z ^
run through the sequence of values given in the tables and
solves for the radii in the corresponding expression, one obtains the corresponding Z T --values from the relationship
Z TA /Z, TO =<r /(a-,
+0- ) = 8.4/18.4= 0.457
IA' NR
rrr v h
m
. 17,
Using these ZT. -values it is then possible to obtain the IA-contributions
through interpolation in table 5. For the two smallest Z-values
ZTA will fall outside the sequence of values and the radii are therefore
0.50
31.
1.00
2.00
4.00
8.00
16.0 cm
The results of the above outlined summaries are found in tables 7 and
8. A separate presentation of the increase in the resonance integral above
the value at 20
C due to the Doppler effect is given in tables 8 (metal) and
9 (oxide). These two tables can be said to comprise the essential results
of the present investigation.
VII. Accuracies
32.
A complete calculation of the absolute value for the resonance integral
contains a large number of approximations. Spinrad, Chernick and Corngold (6) have discussed rather throughly the validity of assuming an undisturbed l/E-apectrum, other assumptions for the NR- and IA-approximations,
the validity of describing the resonances by a one-level Breit-Wigner expression etc., and only a reference to their discussion will therefore be
offered below. The above approximations can be considered to have a relatively minor influence on the differential quantities such as the Doppler
contributions.
33.
The disregarding of the interference structure will, however, intro-
duce an error which is especially noticeable in the Doppler contributions.
Nordheim (7) concludes that this effect is of subordinate importance in a
calculation of che total resonance integral. At low temperatures the structure is v/ell developed and increases the absorption as mentioned above.
At higher temperatures, however, a certain smoothening out takes place
due to the Doppler broadening. A leaving out of the interference therefore
leads to an apparent increase in the temperature effects.
For the IA-resonances the uranium scattering cross section is of very
little importance to the absorption and consequently the interference plays
18.
a minor
role in this case. Furthermore, the contribution to the scattering
cross section from the oxygen in the oxide case makes the interference become even less important. The maximum influence is obtained in metal
cylinders with large radii. The magnitude of this error might be possible
to determine by a Monte-Carlo technique (8).
34.
Another major source of error in this connection is the one introduced
due to the uncertainty in the estimation of statistical data. This is particularly true for the line density l / B . It is possible to estimate the error in
the Doppler contribution due to this uncertainty in D to between 6 and 12 %
for the metal and between 3 and 8 % for the oxide depending on the radius
considered.
VIII. The results in the form cf diagrams.
35.
The Doppler contributions according
rdin to tables 9 and 10 have been drawn
in
diagrams 1
1 and
and 2
2 as
as functions
functions of
in diagrams
of T ' and with the radius as parameter.
A remarkable linearity is obtained.
36.
An attempt to interprete the results in the form
ARI= ( ^ T - NfTQ)f (A)
has been made in diagrams 3 and 4.
For practical use it seems possible to describe the Doppler contributions at elevated temperatures by a single curve having the above shape
and in the metal case by just a straight line.
IX. Concluding remarks
37.
The results of the present investigation as presented in tables 7
through 10 have a limited direct application in practical problems in reactor
physics. For fuel elements in the form of clusters one uses for instance
19.
somewhat
different probability functions P_, which are very closely related
to the fundamental functions. It may, however, be possible to extend the
range of applicability by performing some sort of artificial interpretation
of the results using penetration factors and Dancoff-Ginsburg corrections.
38.
The calculations have been carried out under the assumption of a
uniform temperature throughout the fuel body. The quantitative effect of a
strongly varying temperature is difficult to estimate. This problem has
been discussed by for instance Pearce (9) and Keane (10) but it is far from
satisfactorily explained.
39.
The tables 2 through 6 indicate certain possibilities for some useful
applications. After having included the Doppilcr independent contribution
they permit one to obtain a detailed picture of the variation of cr .- in the
ett
resonance region and can in a multi-group calculation be used to compute
the resonance capture taking into account each separate resonance and
assuming an arbitrary neutron spectrum. The road to a more sophisticated
treatment ?f che resonance escape probability should then be open without
having to elaborate on the rather artificial concepts which make up the
resonance integral at che present.
20.
References:
1.'
2.
CHERNICK J, VERNON R
Some refinements in the calculation of resonance integrals
Nucl.. Sci. Eng. 4, 649-672 (2950)
CREUTZ E, JUPNIK H, WIGNER E P
Effect of temperature on the total resonance absorption of neutrons
by spheres of uranium oxide.
. J. Appl. Phys. 26, 276 (1955)
3.
"CASE KM, de HOFFMANN F, PLACZEK G
Introduction to the theory of neutron diffusion.
Vol. 1, Los Alamos Scientific Laboratory (1953)
4.
ADLER F T, HINMAN G W, NORDHEIM L W
The quantitative evaluation of resonance integrals
GA-350 (1958)
5.
DRESNER L
The effective resonance integrals of U-238 and Th-232
Nucl. Sci. Eng. 1_, 68-79 (1956)
6.
SPINRAD B I, CHERNICK J, CORNGOLD N
. X Resonance capture in uranium and thorium lump's.
Second International Conference on the Peaceful Uses of Atomic
Energy, P / l 847 (1958)
7.
NORDHEIM L W
The theory of resonance absorption
GA-630 (1959)
8.
RICHTMEYER R D, van NORTON R, WOLFE A
The Monte-Carlo calculation of resonance capture in reactor lattices.
Second International Conference on the Peaceful Uses of Atomic
Energy, P/2489 (1958)
9.
PEARCE R M
Radial dependence of the Doppler effect in bars of uranium and
thorium.
AERE R/R 2806 (1959)
10.
KEANE A
Resonance absorption in a slab with a parabolic temperature distribution
AERE R/M 198 (1958)
JR/SL
Diagram 1
- metal
0,423
0,5
ARI
1,0
o
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Table 1.
The values' of "the -resonance parameters Sor the resolVtfd U-2-3-8 resonances
r Y = o, 024
i
E.i
r
6,7
n
0, 00148
12
i
E.
i
r
13
278
297
n
0,025
0, 020
0, 033
14
346
0, 047.
66,3
0, 023
15
368
0, 0027
5
103
0, 072
395
6
191
8.1, 1
117
0, 140
1.6
17
418
0, 008
0, 012
0, 00.21
18
0, 017
431
460
0, 00.9
0, 007
475
0, 005
0, 039
0,058
1
2
21,0
0, 0090
3
36,8
4
7
8
9
10
166
211
0, 003
19
20
0, 045
21
515
ii
239
0, 030
22
531
In the unresolved region the following v&lues have been used.
= 0, 024
r
= o, 0022
n
D
=16
Taljell 2
0.20
0.2932 03
0.4732 03
0.6232 03
0.7732 03
0.9232 03
0.4508 01
0.2006 01
0.1671 01
0.6166 00
0.5606 00
0.3217 00
0.4536
0.2027
0.1685
0.6429
0.5776
0.3362
0.4561 01
0.2046 01
0.1697 01
0.6644 00
0.5919 00
0.3475 00
0.4586
0.2066
0.1710
0.6848
0.6060
0.3582
0.4612
0.2087
0.1724
0.7060
0.6198
0.368 3
0.9683 01
01
01
01
00
00
00
01
01
01
00
00
00
01
01
01
00
00
00
0.9805 01
0.9908 01
0.1001 02
0.1012 02
0.4732 03
0.6232 03
0.7732 03
0.9232 03
0.40
0.2932 03
0.3083 01
0.1364 01
0.1144 01
0.4140 00
0.3825 00
0.2185 00
*
0.6606 01
0.3092 01
0.1371 01
0.1149 01
0.4233 00
0.3885 00
0.2240 00
0.6648 01
0.3100 01
0.1377 01
0.1153 01
0.4314 00
0.3937 00
0,2*88 00
0.6684 01
0.3109 01
0.1383 01
0.1157 01
0.4393 00
0.3991 00
0.2335 00
0.6721 01
0.3117 01
0.1390 01
0.1162 01
0.4481 00
0.4046 00
0.2381 00
0.6760 01
0.80
*
0.2932 03
0.4732 03
0.6232 03
0.7732 03
0.9232 03
0.2086 01
0.9183 00
0.7769 00
0.2802 00
0.2624 00
0.1499 00
0.2089 01
0.9207 00
0.7783 00
0.2832 00
0.2644 00
0.1519 00
0.2092 01
0.9228 00
0.7798 00
0.2858 00
0.2661 00
0.1536 00
0.2095 01
0.9247 00
0.7812 00
0.2885 00
0.2679 00
0.1554 00
0.2098 01
0.9267 00
0.7826 00
0.2917 00
0o2699 00
0.1572 00
0.4474 01
0.4488 01
0.4500 01
0.4512 01
0.4526 01
1,60
0.2932 03
0.4732 03
0.6232 03
0.7732 03
0.9232 03
0.1386 01
0.6069 00
0.5194 00
0.1889 00
0.1795 00
0.1033 00
0.1388 01
0*6077 00
0 o 52G0 00
0.1839 00
0.1802 00
0,1040 00
0.1389 01
0,6084 00
0*5206 00
0.1907 00
0.180« 00
0.1046 00
0 o 1389 01
0.6C90 00
0.5210 00
0 a 1915 00
0o1814 00
0,1052 00
0 e 1391 01
0.6096 00
Q.52J.5 00
0.1925 00
0,1820 00
0 o 1058 00
0.2984 01
0.2989 01
0.2994 01
0.2997 01
0.3002 01
3,20
*
0.2932 03
0.4732 03
0.6232 03
0 0 7732 03
0.9232 03
0.8968 00
0.3897 00
0.3390 00
0.1255 00
0.1217 00
0.7105-01
0.8972 00
0.3898 00
0.3393 00
0.1259 00
0.1220 00
0.7128-01
0.8976 00
0.3000 00
0.3394 00
0.1261 00
0.1221 00
0.7148-01
0.8978 00
0.3902 00
0o3396 00
0.1264 00
0.1223 00
0.7168-01
0,8982 00
0.3904 00
0.3399 00
0.1267 00
0.1226 00
0o7188-01
0.1944 01
0.1945 01
0.1947 01
091948 01
0.1950 01
6.40
0.2932 03
0.4732 03
0.6232 03
0.7732 03
0.9232 03
0.5576 00
0.2397 00
0,2137 00
0.8140-01
0.8117-01
0.4850-01
0.5576 00
0.2398 00
0.2137 00
0.8150-01
0.8127-01
0.4858-01
0.5577 00
0,2398 00
0.2138 00
0.8157-01
0,8134-01
0,4864-01
0,5579 00
0.2399 00
0.2138 00
0,8167-01
0.8140-01
0.4871-01
0 c 5579 00
0.2400 00
0.2139 00
0.8178-01
0.8148-01
0.4878-01
0.1222 01
0.1222 01
0.1223 01
0.1223 01
0.1224 01
Tabell 3
0.00
2932
03
4732
03
6232
03
7732
03
9232
03
1025
1085
5232
5391
4453
3674
3391
2787
2158
2447
2359
2259
2072
1907
1747
1612
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
1145
1115
5840
5584
4737
4016
3805
2980
2493
2854
2721
2639
2428
2223
1958
1747
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
1230
1148
6350
5746
4972
4289
4015
3106
2708
3137
2991
2901
266 0
2421
2083
1851
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
1287
1188
6825
5879
5236
4456
4249
3247
2874
3376
3225
3120
2845
2574
2187
1956
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
00
00
-01
-01
-01
-01
-01
1348
1234
7255
6000
5402
4S44
4485
3400
3011
3574
3428
3301
3000
2704
2297
2043
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
6238
00
6862
00
7301
00
7680
00
8036
00
-01
-01
0.05
2932
03
4732
03
6232
03
7732
03
9232
03
4443
4174
1936
1968
1892
1666
1575
1167
3953
8108
8974
7442
5662
4098
7832
6951
00
00
00
00
00
00
00
00
-01
-01
-01
-01
-01
-01
-01
-01
4944
4814
2009
2242
2182
1911
1776
1341
3835
8365
9406
7679
5735
4057
8788
7911
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
-01
-01
-01
-01
-01
5325
5611
2083
2597
2539
2137
1938
1546
3723
8689
9911
7983
5847
4014
9513
3958
00
00
00
00
00
00
00
00
-01
-01
-01
-01
-01
-01
-01
-01
5444
5971
2X12
2739
2645
2215
2008
1630
3650
8817
1014
8090
5837
3955
9779
9294
00
00
00
00
00
00
00
00
-01
-01
-01
-01
-01
-01
-01
-01
5188
5231
2050
2434
2372
2054
1861
1453
3784
8525
9683
7825
5801
4058
9218
8497
00
-01
-01
-01
-01
-01
2412
01
2680
01
2838
01
2964
01
3072
01
-01
-01
-01
-01
-01
0.10
2932
03
4732
03
6232
03
7732
03
9232
03
3350
3033
1515
1432
1380
1225
1170
8533
3174
6421
7007
5896
4533
3294
5861
5138
00
00
00
00
00
00
,00
-01
-01
-01
-01
-01
-01
-01
-01
-01
3770
3498
1586
1630
1596
1414
1332
9848
3061
6671
7417
6J26
4604
3254
6647
5891
00
00
00
00
00
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
3983
3802
1625
1771
1740
1528
1403
1069
3034
6823
7677
6265
4683
3271
7014
6360
00
00
00
00
00
00
00
00
-01
-01
-01
-01
-01
-01
-01
-01
4106
4084
1658
1890
1870
1595
1469
1142
2959
6995
7890
6432
4724
3206
7273
6739
00
00
00
00
00
00
00
00
-01
-01
-01
-01
-01
-01
-01
-01
4215
4356
1685
1994
1952
1660
1529
1208
2869
7117
8117
6 530
4691
3133
7509
7020
00
00
00
00
00
00
00
-01
-01
-01
-01
-01
-01
-01
-01
1809
01
2018
01
2143
01
2244
01
2330
01
do
0.20
2932
03
4732
03
6232
03
7732
03
9232
03
1534
1308
7889
6305
5923
5290
5139
3688
2314
3574
3608
3292
2783
2253
2627
2258
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
1778
1482
8648
7014
6804
6161
6005
4245
2380
3970
4049
3660
3035
2386
3062
2610
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
1922
1599
9187
7553
7435
6746
6407
4607
2435
4232
4359
3901
3205
2483
3288
2847
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
2012
1716
96 60
8011
8052
7034
6813
4950
2457
4463
4621
4117
3335
2523
34S2
3059
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
2100
1834
1009
8419
8442
7457
7219
5285
2452
4G53
4861
4286
3412
2536
3S35
3224
00
00
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
8537
00
9663
00
1039
01
1099
01
1153
01
0.40
9232
2932
03
4732
03
6232
03
7732
03
8189
7062
4313
3436
3163
28 OS
2725
1973
1430
1988
1967
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
»01
1034
8474
5140
4035
3930
3575
3414
2439
1589
2444
2444
2256
1926
1569
1760
1513
00
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
-01
1086
9064
5464
4260
4255
3763
3644
2619
1628
2603
2615
2404
2030
1619
185S
Ib28
00
-01
-01
-01
»01
-01
-01
-01
1352 - 0 1
1397 - 0 1
1203 - 0 1
9522
7903
4783
3772
3605
3261
3192
2254
1522
2260
2244
2086
1795
1481
1634
1386
4644
5270
00
5685
00
6031
CO
ö 345
30
9232
03
-01
-01
-01
-01
»01
-01
-01
»01
-01
-01
1833 - 0 1
1599 - 0 1
00
-01
03
113© CO
9670 - 0 1
57:li
4-3^
4-430 - 0 1
Sto-A - 0 1
\ih '£ \s - 0 1
-01
-01
•** Q JL
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—ill.
r c
-01
-01
'l l i1
-01.
?7 90 - 01
!tc 4v -01
-01
18.5«?
-01
»4. • v* *. *
-01
-01
0.80
2932
03
4732
03
6232
03
7732
03
4228
3700
2241
1811
1645
1451
1404
1026
7825
1039
1022
9589
8463
7296
7206
6232
-01
-01
-01
-01
-01
-01
-01
-01
-02
-01
-01
-02
-02
-02
-02
-02
4915
4108
2496
1972
1862
1679
1643
1166
8510
1192
1172
1101
9620
8129
8423
7151
-01
-01
-01
-01
-01
-01
-01
-01
-02
-01
-01
-Cl
-Q2
-02
-02
-'•02
5345
4386
2S94
2100
2024
1839
17 57
125Ö
8&S'"5
12B6
12S2
13 97
1039
-01
-01
-01
-01
-01
-01
-öl
»01
-02
"01
-01
-01
-01
5620
46 7 K
2874
23'5f*
-01
-01
-01
— 05
2423
00
2749
00
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15*89 - 0 1
i.S-?6 - 0 1
J.3W - 0 1
VY 03 - 0 2
134£
9302
lit SS
1373
12**0
?100
9041
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3150
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bChb - 0 2
59CO - 0 1
4981 - ö l
3036 - 0 1
2308 - 0 1
229'i - 0 1
2037 «01
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1450 "-'CR
i4^a -oi
1S&7
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3317
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1.60
2932
03
4732
03
6232
03
7732
03
9232
03
2140
1888
1135
9258
8360
7349
7101
5218
4026
5273
5171
4865
4309
3737
3647
3162
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2486
2088
1265
1005
9430
8479
8302
5908
4408
6060
5941
5597
4915
4184
4258
3620
-01
-01
-01
-01
-02
-02
-02
-02
-02
-02
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-02
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2703
2226
1367
1068
1023
9281
8876
6361
4673
8598
8 503
6094
5319
4483
4589
3940
-01
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-02
-02
-02
-02
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-02
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2841
2371
1460
1122
1105
9765
9481
6811
4849
7062
6985
6524
5639
4675
4851
4240
-01
-01
-01
-01
-01
-02
-02
-02
-02
-02
-02
-02
-02
-02
-02
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2983
2523
1544
1171
1157
1028
1008
7268
4957
7446
7413
6873
5874
4808
5118
4477
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-02
-02
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-02
-02
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1231
00
1395
00
1506
00
1599
00
1684
00
3.20
2932
03
4732
03
6232
03
7732
03
9232
03
1073
9492
5694
4658
4200
3688
3561
2621
2027
2646
2594
2442
2164
1879
1829
1587
-01
-02
-02
-02
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-02
-02
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1247
1049
6350
5053
4736
4256
4163
2965
2223
3043
2982
2811
2470
2107
2135
1815
-01
-01
-02
-02
1355
1117
6862
5368
5139
4658
4453
3193
2358
3314
3264
3061
2675
2259
2301
1976
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6182 - 0 1
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7007 - 0 1
7561 - 0 1
-02
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8027 »01
36 48
2505
3741
3721
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2245
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6.40
2932
03
5371
4752
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£332
2102
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1298
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£42 0
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Tabell 4
*** 0. 00
293.2
473.2
540
32.901708
540
36.498147
1080
29.078910
1080
32.470422
2160
24.805040
2160
27.575155
4320
19.916759
4320
21.869395
8640
14.823091
8640
16,002913
17280
10.172112
17280
1 C.783373
34560
6.464Ö28
34560
6»737267
09120
3.840561
69120
3,945518
138240
2.162870
138240
2.198218
27Ö480
1.170511
276480
1.181178
552960
0.615397
552960
G.618S69
1105920
0,316852
1105920
0.317644
A
1.346877
A
623.2
1.473049
773.2
i
540
38»92902C
540
41.014756
1080
34.681799
2 080
36.552228
2160
29.324749
2160
HO.778953
4320
23.007525
4320
24.033613
8Ö40
16.704718
8640
17.255812
17280
11.134221
17280
11.404448
34560
6.888210
34560
7.002334
69120
4.001855
69120
4.043570
138240
2.216747
138240
2.230216
276480
1.186717
276480
1.190642
552960
0.619924
552960
0.620979
1105920
0.318048
1105920
0.318328
A
1.552644
A
1.618369
***
923.2
0.05
293.2
540
42.871424
540
108.171770
1080
38.173697
1080
78.405648
2160
32.022647
2160
49.835983
4320
24.852879
4320
27,521262
8640
17.715552
864C
13.242055
17280
11.625719
17280
5.607483
34560
7.094014
34560
2.093114
69120
4.076575
69120
0,668783
138240
2.240752
138240
0.369222
276480
1.193705
276480
0.062834
5529G0
0.621819
552960
0.017139
1105920
0.318552
1105920
0.004503
A
1.674732
A
473.2
2.385705
623.2
540
117.044968
540
122.336710
1080
82.635157
1080
84.877810
2160
50.792652
2160
51.114223
4320
27.072460
4320
26.684078
8640
12.575430
8640
12.145412
17280
j . 153749
17280
4.8H^030
34560
1.865052
31560
1.7354."*)
69120
0.576998
öi»!21»
0.51908(3
J38240
0*357742
138240
0,338739
276480
0.053716
276480
O.feUK'-l.*.
552960
0.014-.&?
552H60
0,012154
1105920
0.003827
1105920
0.003144
A
2.461509
A
2.499114
773. 2
623. 2
A
540
93.569540
540
96,981779
1080
6 6.052191
1080
GV-5S7783
2160
40.432375
2160
40,706894
4320
21.276036
4320
£3 , 064712
8640
9.637646
8640
1?. 333101
17280
3.792968
17230
3.636 057
34560
1-316891
34560
1.24 5145
69120
0.411943
69120
0.385482
138240
0.118980
138240
0.110561
276480
0.03247 4
276-IS')
0.030049
05296 0
0,008547
55&980
0,007890
1105920
0.002447
1105*20
0.002271
1 .950761
1 .977252
A
***
923,?;
0.20
293.2
540
99.84.4627
540
10EÖ
6i?.r-J;28E6
1080
tl'-iU
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0.104026
A
1J8 •;'.-* 0
0.175U3
2764S0
0.077 465
552: »SO
0.007386
552960
0,012791
S20
0,002154
1105920
0.003774
1,998529
A
1.128875
923.2
773.2
540
126.528515
540
130.011329
1080
86.507888
1080
87.754347
2160
51.250012
2160
51.276504
4320
26.314672
4320
25.970787
8640
11.789304
8640
11.483686
17280
4.673615
17280
4.497990
34560
1.637535
34560
1.558566
69120
0.481166
69120
0.451687
138240
0.333338
138240
0.327121
276480
0.042121
276480
0.039318
552960
0.011156
552960
0.010387
1105920
0.002880
1105920
0.002678
2. 525345
A
***
A
2.544512
0.10
293.2
473.2
540
82.167028
540
89.282579
1080
60.605675
1080
64.117804
2160
39.068972
2160
40.002601
4320
21.7*9017
4320
21.499568
8640
10,441968
8640
9.948555
17230
4.346630
17280
3.995905
34560
1.586477
34560
1.414583
69120
0.520921
69120
0.455823
138240
0.157186
138240
0.136042
276480
0.044408
, 276480
0.038227
552960
0.011946
552960
0.010257
1105920
0.004874
1105920
0.004457
A
1.849962
A
1.915699
623.2
473.2
540
46.241745
540
49.230564
1080
36.830347
1080
38.648240
2160
25.870053
2160
26,609994
4320
15.718241
4320
15.806817
8640
8.193461
8640
8.045059
17280
3.666120
17Z80
3.517066
34560
1.419869
34560
1.333188
69120
0.484572
69120
0.445833
138240
0.151601
138240
0.134922
276480
0.070154
276480
0.063145
552960
0.010589
552960
0.008656
1105920
0.003207
1105920
0.002643
A
1.194374
A
1.231824
923.2
773.2
540
51.706430
540
53.828108
1080
40.078214
1080
41.250803
2160
27.149454
2160
27.554398
4320
15.829789
4320
15.821824
8640
7.907469
8640
7.780420
17280
3.394548
17280
3.290281
34560
1.265595
34560
1.210301
69120
0.417452
69120
0.394803
138240
0.125112
138240
0.117440
276480
0.060348
276480
0.057888
552960
0.007763
552960
0.007119
1105920
0.002422
1105920
0.002253
A
1.260656
A
1.283843
***
0.40
293.2
473.2
540
23.221588
540
25.959240
1080
19.385034
1080
21.380233
2160
14.636021
2160
15.689441
4320
9.705598
4320
10.039809
8640
5.568392
8640
5.535719
17280
2.741472
17280
2.615649
34560
1.159352
34560
1.061976
69120
0.424886
69120
0.374922
138240
0.138016
138240
0,119412
276480
0.041261
276480
0.035318
552960
0.011533
552960
0.009824
1105920
0.003075
1105920
0.002613
A
0.677390
A
623.2
0.722905
773.2
540
27.767660
540
29.271771
1080
22.587303
1080
23.552920
2160
16.273758
2160
16.717935
4320
10.190434
4320
10.280266
8640
5.485461
8640
5.430393
17280
2.530275
17280
2.457759
34560
1.003861
34560
0.957743
69120
0.347376
69120
0.326337
138240
0.107455
138240
0.099824
276480
0.030656
276480
0.028282
552960
0.008280
552960
0.007608
1105920
0.002161
1105920
0.001981
A
0.749563
A
0.770395
.***
923.2
0.80
293.2
540
30.567156
540
12.267012
1080
24.356147
1080
10.385985
2160
17.065060
2160
8.014898
4320
10.338319
4320
5.474061
8640
5.374945
8640
3.256663
17280
2.394613
17280
1.670989
34560
0.919563
34560
0.737602
69120
0.309436
69120
0.281299
138240
0.093825
138240
0.094050
276480
Ö.026437
276480
0.028654
552960
0.007089
552960
0.008125
1105920
0.001842
1105920
0.002186
A
0.787399
A
473.2
0.373823
623,2
540
13.751029
540
14.733675
1080
11.514172
1080
12.203250
2160
8.658626
2160
9.025353
4320
5.718472
4320
5.839822
8640
3.274469
8640
3.267135
17280
1.613756
17280
1.572153
34560
0.683720
34560
0.650608
69120
0.250782
69120
0.233551
138240
0.081555
138240
0.074201
276480
0.024557
276480
0.021534
552960
0.006923
552960
0.005879
1105920
0.001858
1105920
0.001544
A
0.400833
A
0.416815
923.2
773.2
540
15.552367
540
16.259777
1080
12.760040
1080
13.227340
2160
9.309219
2160
9.535140
4320
5.920252
4320
5.978559
8640
3.252006
8640
3.233592
17280
1.535660
17280
1.503118
34560
0.623889
34560
0.601495
69120
0.220291
69120
0.209552
138240
0.069086
138240
0.065057
276480
0.019893
276480
0.018615
552960
0.005405
552960
0.005039
1105920
0.001416
1105920
0.001317
A
0.429411
***
A
0.439783
1.60
293.2
473.2
540
6.259340
540
7.019297
1080
5.324511
1080
5.911305
2160
4.142950
2160
4.488366
4320
2.863583
4320
3.004267
8640
1.730251
8640
1.749024
17280
0.905781
17280
0.880337
34560
0.409359
34560
0.382185
69120
0.160059
69120
0,143725
138240
0.054738
138240
0.047631
276480
0.016940
276480
0.014542
552960
0.004862
552960
0.004144
1105920
0.001320
1105920
0.001121
A
0.194194
A
0.208579
773.2
623.2
540
7.522675
540
7.942946
1080
6.271552
1080
6.564229
2160
4.687233
2160
4.842283
4320
3.076171
4320
3.125243
8640
1.750915
8640
1.747485
17280
0.860930
17280
0.843555
34560
0.365195
34560
0.351340
69120
0.134363
69120
0.127114
138240
0.043606
138240
0.040692
276480
0.012849
276480
0.011885
552960
0.003543
552960
0.003260
1105920
0.000936
1105920
0.000859
A
0.217131
A
0.223902
*** 3.20
293.2
923.2
540
8.307.045
54.0
3.146 007
1080
6.810940
1080
2.679302
2160
4.966607
2160
2.089085
4320
3.161792
4320
1.448483
8640
1.741560
8640
0.879172
17280
0.827828
17280
0.46 2845
34560
0.339646
34560
0.210653
69120
0.121212
69120
0.083058
138240
0,038378
138240
0.028650
276480
0.011133
276480
0.008927
552960
0.003040
552960
0.002576
1105920
0.000799
1105920
0.000702
A
0.229502
A
0.098079
623.2
473.2
540
3.528345
540
3.781597
1080
2.975665
1080
3.157860
2160
2.265026
2160
2.366536,
4320
1.521254
4320
1.558745
8640
0.890062
8640
0.891864
17280
0.450693
17280
0.441272
34560
0.197127
34560
0.188625
69120
0.074773
69120
0.070004
138240
0.024986
138240
0.022913
276480
0.007672
276480
0.006798
55296 0
0.002197
552960
0.001883
1105920
0.000597
1105920
0.000499
A
0.105392
å
0.109746
923.2
773,2
540
3.993101
540
4.176359
1080
3.305968
1080
3.430947
2160
2.445765
216 0
2.509361
4320
1.584520
4320
1,603852
8640
0.890795
8640
0.888354
17280
0.432776
17280
0.425047
34560
0.181665
34560
0.175779
69120
0.066302
69120
0.063282
138240
0.021403
138240
0.020202
276480
0.006292
276480
0.005897
552960
0.001733
552960
0.001617
1105920
0.000458
1105920
0.000426
A
0.113195
A
0.116049
***
6.40
293.2
473.2
540
1.574793
540
1.766223
1080
1.341440
1080
1.489936
2160
1.046469
2160
1.134835
4320
0.726139
4320
0.762807
8640
0.441155
8640
0.446776
17280
0.232542
17280
0.226510
34560
0.105892
34560
0.099239
69120
0.041876
69120
0.037719
138240
0.014471
138240
0.012627
276480
0.004516
276480
0.003882
552960
0.001305
552960
0.001113
1105920
0.000356
1105920
0.000303
A
0.049147
A
623.2
0.052818
773.2
540
1.893078
540
1,999123
1080
1.581485
1080
1.655929
2160
1.185981
2160
1.225989
4320
0.781728
4320
0.794743
8640
0.447778
8640
0.447319
17280
0.221825
17280
0.217619
3456 0
0.095015
34560
0.091432
69120
0.035324
69120
0.033464
138240
0.011584
138240
0.010823
276480
0.003442
276480
0.003186
552960
0.000955
552960
0.000879
1105920
0.000253
1105920
0.000232
A
0.055008
A
0.056743
923.2
540
2.091004
1080
1.718639
2160
1.258112
4320
0.804511
8640
0.446160
17280
0.213776
3450 0
0.088469
69120
0.031947
138240
0.010218
276480
0.002987
55296 0
0.000820
1105920
0.000216
0 . 058179
Tafcell 5
0.00
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Tabell 6
0.00
293.2
A
573.2
540
25.900827
540
30.096672
1080
22.444431
1080
26.056065
2160
18.465834
2160
21.119904
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14.128002
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15.793933
8640
9.957749
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10.850013
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6.475765
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6.883952
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3.920475
34560
4.080748
69120
2.238724
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2.293598
138240
1.224126
138240
1.240934
276480
0.649585
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0.654316
552960
0.335782
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0.337029
1105920
i.773521
0.164825
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0.165133
i. 994340
873.2
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540
33.309516
540
35.835742
1080
28.677834
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30.674358
2160
22.956328
2160
24.318638
4320
16.890677
4320
17.678849
8640
11.407055
8640
11.795048
17280
7.125873
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7.289230
34560
4.171280
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4.230815
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2.323368
69120
2.342591
138240
1.249810
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1.255473
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0.656777
276480
0.658347
552960
0.337683
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0.338100
1105920
0.165305
2.148480
1105920
A
0.165404
2.263480
*** 0.05
293.2
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540
37.945837
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73.566857
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32.313140
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27.901525
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18.289266
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13.894229
8640
12,087651
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6.06344'.1
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2.317208
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0.758264
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0,474082
138240
1.259420
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0.072543
276480
0.659410
276480
0.019967
552960
0.338387
552960
0.005263
1105920
0.001427
1105920
0.165480
2.355919
A
573.2
A
2.578989
873.2
540
79.536978
540
82.968109
1080
50.254829
1080
50,817484
2160
27.444468
2160
26.864583
4320
13.000958
4320
12.328918
8640
5.416285
8640
4.988387
17280
1.976718
17280
1.770809
34560
0.615029
34560
0,536242
69120
0.449171
69120
0.439992
138240
0.054805
138240
0.046687
276480
0.015957
276480
0.012421
552960
0.004189
552960
0.003214
1105920
0.001055
1105920
0.000816
2.610733
A
2.610267
•
1473. 2
1173.2
540
85.159515
540
86.693278
1080
50.974416
1080
50.957717
2160
26.329889
2160
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4320
11.815097
4320
11.398908
8640
4.683901
8640
4,447053
17280
1.630834
17280
1.526084
34560
0.484788
34560
0,447552
69120
0.419220
69120
0.409979
138240
0.041164
138240
0.037529
276480
0.011020
276480
0.009870
552960
0.002845
552960
0,002538
1105920
0.000720
1105920
0.000643
2.599560
***
2.585643
0.10
293.2
540
56.693885
540
61.6(17230
1080
38.056983
1080
39.491951
2160
22.010400
2160
21.709917
4320
10.932369
4320
10.245091
8640
4.688820
8640
4.180002
17280
1.755220
17280
1.494951
34560
0.586842
34560
0.478309
69120
0.179506
69120
0.140448
138240
0.051230
138240
0,038771
276480
0.013870
276480
0.011215
552960
0.004334
552960
0.003646
0.000477
2.009575
1105920
0.000235
1105920
A
573. 2
A
2,.040779
1173.2
873.2
A
540
64.540345
540
66.440671
1080
40.092067
1080
40.406323
2160
21.349267
2160
21.029799
4320
9.753056
4320
9.39016C
8640
3.858674
8640
3.634956
17280
1.342908
17280
1.241618
34560
0.420876
34560
0.382782
60120
0.121824
69120
0,109312
138240
0,033331
138240
0.029609
276480
0.008773
276480
0.007861
552960
0.002359
552960
0.002124
1105920
0.000390
1105920
0.000176
2.047930
A
2.048412
***
293.2
1473.2
540
67.880293
540
30.853507
1080
40.593157
1080
23.1986G9
2160
20.748993
2160
15,267593
4320
9.100074
4320
8.622331
8640
3.466503
8640
4.159290
17280
1.167580
17280
1.722576
34560
0.356084
34560
0.620581
0.100945
69120
0.200240
138240
0.027222
138240
0.076138
276480
0.007102
276480
0.015195
552960
0.001848
552960
0.004408
1105920
0.000163
1105920
0.001193
69120
A
0.20
,
2.046732
A
i.314611
***
0.40
293.2
573.2
540
17.548529
540
20.101198
1080
13,717435
1080
15.218767
2160
9,493747
2160
10.034283
4320
5.675676
4320
5.672991
8640
2.898573
8640
2.731557
17280
1,263932
17230
1.123603
34560
0.474856
34560
0.400234
69120
0.156832
69120
0.126459
138240
0.047474
138240
0.036597
276480
0.013404
276480
0.010713
552960
0.003589
552960
0,002856
1105920
0.000979
1105920
0.000726
A
0.807949
A
873.2
A
0.861688
1173.2
540
21.919235
540
23.246597
1080
16.143130
1080
16.758640
2160
10.283722
2160
10.411009
4320
5.599406
4320
5.515606
8640
2.595444
8640
2.488262
17280
1.029824
17280
0.962732
34560
0.355537
34560
0.325363
69120
0.109769
69120
0.098730
138240
0.031303
138240
0.027758
276480
0,008463
276480
0.007517
552960
0.002209
552960
0.001956
1105920
0.000563
1105920
0.000498
0.892903
A
0.912735
*** 0.80
1473.2
A
293.2
540
24.288476
540
9.350660
1080
17.212360
1080
7.457551
2160
10.476204
2160
5.311739
4320
5.432287
4320
3,293305
8640
2.399729
8640
1.754046
17280
0.910884
17280
0.799102
34560
0.302965
34560
0.312800
: 69120
0.090878
69120
0.106652
138240
0.025360
138240
0.032906
276480
0.006756
276480
0.009429
552960
0.001749
552960
0.002550
1105920
0.000445
1105920
0.000699
0.926667
A
573.2
A
0.451779
873.2
540
10.787326
540
11.824252
1080
8.363259
1080
8o939446
2160
5.692309
2160
5.890157
4320
3.344926
4320
3.337437
8640
1.681676
8640
1.615675
17280
0.722498
17280
0.668994
34560
0.267471
34560
0.239548
69120
0.086997
69120
0.075873
138240
0.025652
138240
0.022001
276480
0.007539
276480
0.006010
552960
0.002029
552960
0.001578
1105920
0.000519
1105920
0.000404
0.486106
A
0.506807
1473.2
1173.2
A
540
12.588031
540
13.194431
1080
9.333929
1080
9.631510
2160
6.005734
2160
6.077836
4320
3.313467
4320
3.283649
8640
1.561215
8640
1.514824
17280
0.629803
17280
0.599034
34560
0.220409
34560
0.206052
69120
0.068551
69120
0.063250
138240
0.019573
138240
0.017908
276480
0.005342
276480
0.004812
552960
0.001398
552960
0.001252
1105920
0.000357
1105920
0.000319
0.520426
A
0.530311
*** 1.60
573.2
293.2
A
540
4.785142
540
5.532198
1080
3.844271
1080
4.328057
2160
2.768676
2160
2.984607
4320
1.742891
4320
1.783506
8640
0.947051
8640
0.916143
17280
0.441792
17280
0.403492
34560
0.177357
34560
0.153189
69120
0.061935
69120
0.050957
138240
0.019419
138240
0.015273
276480
0.005634
276480
0.004508
552960
0.001538
552960
0.001224
1105920
0.000423
1105920
0.000315
0.235953
A
0.254800
1173.2
373.2
A
540
6.072991
540
6.473351
1080
4.639135
1080
4.S54525
2160
3,101257
2160
3.171950
4320
1.780863
4320
1.782970
8640
0.885529
8640
0.853477
17280
0.376025
17280
0.355629
34560
0.138022
34560
0.127518
69120
0.044643
60120
0.040467
138240
0.013135
138240
0.011718
2764S0
0*00362'^
276480
0.003223
552960
0.000957
552960
0.000848
1105920
0.000246
1105920
0.000217
0.266307
A
0.273983
•** 3.20
1473.2
A
293.2
540
6.793242
540
2.406974
1080
5.018552
1080
1.937107
2160
3.218012
2160
1.399230
4320
1.772432
4320
0.884604
8640
0.836842
8640
0.483329
17280
0.339442
17280
0.227067
34560
0.119579
34560
0.091917
69120
0.037422
69120
0.032384
138240
0 o010736
138240
0.010226
276480
0.002908
276480
0.002984
552960
0.000760
552960
0.000818
1105920
0.000194
1105920
0.000226
0.279623
A
0.119348
873.2
573.2
A
540
2.783970
540
3.057290
1080
2.183035
1080
2.341708
2160
1.510454
2160
1.571192
4320
0.907138
4320
0.911211
8640
0.468799
8640
0.453966
17280
0.208055
17280
0.194304
34560
0.079678
34560
0.071950
69120
0.026737
69120
0.023471
138240
0.008072
138240
0.006951
276480
0.002389
276480
0.001926
552960
0.000651
552960
0.000510
1105920
0.000168
1105920
0.000131
0.129002
A
1173.2
A
0.134920
1473.2
540
3.259932
540
3.422002
1080
2.451826
1080
2.535799
2160
1.608384
2160
1,632880
4320
0.909223
4320
0.904652
8640
0.441213
8640
0,430061
17280
0.184054
17280
0.175891
34560
0.066579
34560
0.062511
69120
0.021305
69120
0.019721
138240
0.006209
138240
0.005693
276480
0.001714
276480
0.001548
552960
0.000452
552960
0.000406
1105920
0.000116
1105920
0.000104
0.138881
A
0.141798
**•
6.40
293.2
573.2
540
1.205033
540
1.393824
1080
0.970206
1080
1.093597
2160
0.701324
2160
• 0.757235
4320
0,443765
4320
0,455290
8640
0.242768
8640
0.235582
17280
0.114097
17280
0,104756
34580
0 c 046331
34560
0.040193
69120
0.016354
6Ö120
0.013513
138240
0,005173
138240
0.004086
276480
0.001511
276480
0.001210
552960
0.000415
552960
0.000330
1105920
0.000114
1105020
0.000085
A
0.0059824
A
373.2
A
0.054679
1173.2
54.0
1,530820
540
1.632676
1080
1,173348
1080
1.228924
2160
0.787919
2160
0.806704
4320
0.457492
4320
0.456612
8640
0,228235
8640
0.221830
17280
0.097702
17280
o,002726
34560
' 0.036312
34560
0,033613
69120
0.011868
69120
0 r 010776
138240
0,003520
138240
0.003145
276480
0,000976
276480
0.000869
552960
0 o 000259
552960
0,000230
1105020
0.000067
1105920
0.000059
0.067654
A
0.069660
1473,2
A
b40
1 .724146
1030
1 .271387
2260
0 .819098
4S20
0 .454407
8640
0 02l6i>25
17230
0,.088633
34560
0, ,031568
69120
0 , ,00SS 77
138240
o 4,002884
276480
0 . 00078Ö
552960
0 . 000206
11059^0
0« 000053
0.071 137
Table, 7.
The Doppler sensitive contributions to the resonance Integral for -metal
cylinders with radius a.cm
Contributions in barns
1 temperature '20 C
Volume
term
Surface term
a=0, 423
a=0, 845
IA-lines •
a=l, 69
a=3, 38
a=6, 76 I a=13, 5
4,474
2,984
1,944
1, 222
NR-4in.e-sj- v
0, 624
0, 242
0, 123
0, 062
0,031
Unresolved
region
1,347
0,374
0, 194
0, 098
0, 049
Total
i, 971
5, 090
3,301
i 2, 104
1,302
11,666
7,747
Table 7.
The Doppler sensitive contributions to the .resonance integral for
metal cylinders with radius a cm
Contributions in barns
2. Temperature 200 °C
Volume
a=0, 423
term
IA- lines
a=0, 845
Surface term
a=6, 76
a=l,69 a=3,38
a=13, 5
9,805
6, 648
4,488
2,989
1,945
1,222
NR-lines
0, 686
0,966
0,527
0,275
0, 140
0, 070
0, 035
Unresolved
region
1,473
1, 194
0,723
0,401
0,209
0, 105
0,053
Total
2, 159
11,965
7,898
5, 164
3,338
2, 120
1,310
•
Table 7.
The Doppler sensitive contributions to the resonance integral'for metal
cylinders with radius a cm
Contributions in barns
3. Temperature 350 C.
Volume
term
IA-lines
a=0, 423
Surfc .ce term
a=0, 845 a=i, 69 a=3,38
a=6, 76
a=13, 5
9,908
6,684
4, 500
2,994
1,947
1, 223
NR-lines
0,730
1,039
0,569
0,297
0, 151
0, 076
0, 038
Unresolved
region
1,553
1,232
0, 750
0,417
0,217
0, 110
0, 055
Total
2, 283
12, 179
8, 003
5,214
3,362
2, 133
1,316
Table 7.
The Doppler sensitive contributions to the resonance integ-ral for
metal cylinders with radius a cm
Contributions in barns
4. Temperature 500 C.
j Volume
term
a= 0,423
Surface term
a=0, 845 a=l, 69 a=3,38
a= 6, 76
a= 13,5
i, 948
1, 223
._ .
10, 01
IA-lines
721
4 , 512
2, 997
NR-lines
0,768
1, 099
o, 603
o, 315
o,
160
080
o, 040
Unresolved
region
1, 618
1, 261
o, 770
o, 429
o, 224
113
057
-Total--
2,386
12, 370
8, 094
5, 256
3, 381
2, 141
1, 320
Table 7.
The Doppler sensitive contributions to the resonance integral for
metal cylinders with radius a cm
Contributions in barns
5. Temperature 650 °C
Volume ;
term
a=0,423
10, 12
I A-lines
Surface term
a=0, 845 i=l,69
a=3, 38
a=6, 76
a=13, 5
6,760
4, 526
3, 002
1,950
1,224
NR-lines
0, 804
1, 153
0,635
0,332
o, 168
o, 085
o, 042
Unresolved
region-..
1, 675
1, 284
0,787
0,440
o, 230
o, 116
o, 058
Total
2,479
i 12, 557
8,182
5,298
3,400
2, 151
i,324
Table
8.
The Doppler sensitive contributions to the resonance integral for oxide
cylinders with radius a cm
Contributions in barns
1. Temperature 20
C
Surf cice term.
Volume
a=l, 00
a=2, 00
a=4, 00
a=8, 00
a=16, 0
•
6, 075
3, 265
1,784
0,921
0,464
1, 233
0, 682
0,358
0, 182
0, 091
0, 046
1,315
0,808
0,452
0, 236
0, 119
0, 060
7, 565
4,075
2, 202
1, 131
0, 570
term
a=0, 50
IA-lines
5, 385
•
11,78
NR-lines
0,921
Unresolved
region
1,774
Total
8, 080
14, 33
Table 8.
The Doppler sensitive contributions to the resonance integral for oxide
cylinders with radius a era
Contributions in barns
2. - Temperature 300 o,C
Surface term
Volume I
a=l, 00
a=2, 00
a=4, 00
a=8, 00 i a=l6, 0
12, 67
6,3 65
3,372
1,838
0, 948
0,477
1,443
0,806
0,424
0, 216
0, 109
0, 054
1,382
0,862
0,486
0, 255
0, 129
0, 055
i 15, 50
8,033
4,282
2,309
1,186
0,586
term
a=0, 50
IA- lines
5,440
NR-lines
1,071
Unresolved
region
1, 994
Total
8,505
Table 8.
The Doppler sensitive contributions to the rcsonanccintegral for oxide
cylinders with radius a cm
Contributions in barns
3. Temperature 600 C
Volume
Surface term
a=l, 00
a=2, 00
a=4, 00
a=8, 00
a=l6, 0
6,636
3,446
1,884
0,973
0,490
1, 585
0,892
0,472
0, 240
0, 121
0,060
1,418
0,893
0, 507
0,266
0, 135
0, 068
8,421
4,425
2,390
1,229
0, 618
term
a=0, 50
IA-lines
5,492
13,46
NR-lines
1, 188
Unresolved
region
2, 148
Total
8,828
16,46
Table 8.
The Doppler sensitive contributions to the resonance integral for oxide
cylinders with radius a cm
Contributions in barns
4, Temperature 900 °C
Volume-
Surfe.ce term
term
a=0, 50
a=l,00
a=2, 00
a=4, 00
a=8, 00
a=l6, 0
IA- lines
5, 541
14, 20
6, 890
3, 570
1,937
o, 996
0, 502
NR-lines
1, 283
1, 684
o, 956
0, 507
0 , 259
0, 130
0, 065
Unresolved
region
2, 263
i , 439
o, 913
0, 520
0, 274
0, 139
0, 070
Total
9,087
8, 759
4,597
2 ,470
1, 265
0, 637
17, 32
Table 8.
The Doppler sensitive contributions to the resonance integral for oxide
cylinders with radius a cm
Contributions in barns
5.. Temperature 1200- C
Volume
term
Surf a cc term
a=0, 50
14,90
a=l, 00
a=2, 00
a=4, 00
a=8, 00
a=l6, 0
7, 141
3, 668
1,986
1, 020
0, 514
IA- lines
5,592
NR-lines
1,368
1, 767
1, 010
0, 538
0, 274
0, 138
0, 069
Unresolved
region
2,356
1,452
0, 927
0, 530
0, 280
0, 142
0, 071
Total
9,316
9,078
4,736
2, 540
1,300
0, 654
18, 12
Table 9.
The increase in the resonance integral for the metal above its value
at 20 C due to the Doppler effect.
a=0, 423 cm
a=0, 845 cm a=i, 69 cm a=3, 38 cm la=6, 76 cm a=13, 5cm
200 C
0,487
0, 340
0, 263
0, 226
0, 205
0, 197
350 C
0,825
0, 568
0,436
0, 373
0,341
0,326
500 C
i, 119
0,763
0, 582
0,496
0,453
0,434
650°C
1,398
0,943
0,716
0,607
0, 555
0, 530
Table 10.
The increase in the resonance integral for the oxide above its value
at 20 C due to the Doppler effect.
a=0 , 50 cm
a = l , 00 cm
a=2, 00 c m a=4, 00 cm
a=8, 00 cm
a= 16, 0 cm
300°C
1, 595
o, 893
0, 632
0,532
0,479
0,442
600°C
2, 898
1, 604
1,098
0,936
0,845
0,797
900°C
4, 007
2, 201
1,529
1, 275
1, 140
1, 074
1200°C
5, 036
2, 749
1,897
1, 574
1,405
1, 320
LIST OF AVAILABLE AE-REPORTS
Additional copies available at the library of
AB ATOMENERGI
Stockholm - Sweden
AB No
Title
Author
Price
Printed
in
Pages
in
Sw. er.
1
Calculation of the geometric buckling for reactors
of various shapes.
N. G Sjöstrand
1958
23
3
2
Tile variation of the reactivity with the number,
diameter and length of the control rods in a heavy
H. McCrirtck
1958
24
3
3
Comparison of filter papers and an electrostatic
precipitator for measurements on radioactive aerosols.
R. Wiener
1958
4
4
4
A slowing-down problem.
1 Carlvik, B. Persbagen
1958
14
3
5
Absolute measurements with a 4«r-countcr
rev. ed.)
Kerstin Martinsson
1958
20
4
6
Monte Carlo calculations of neutron thermalization in a heterogeneous system.
T
1959
13
4
8
Metallurgical viewpoints on the brittleness of beryllium.
G Lagerberg
1960
14
4
1960
13
4
1960
9
6
(2nd
Högberg
9
Swedish research on aluminium reactor technology.
S. Forsen
10
Equipment for thermal neutron flux measurements
in Reactor R2.
E Johansson, T
S. Claesson
11
Cross sections and neutron yields for U m , U i M
and P u " ' at 2200 m/sec
N. G. Sjöstrand
J S Story
1960
34
4
12
Geometric buckling measurements using the pulsed
neutron source method
N. G Sjöstrand, J. Mednis,
T Nilsson
1959
12
4
13
Absorption and flux density measurements in an
iron plug in Rl
R. Nilsson, / . Brann
1958
24
4
14
GARLIC, a shielding program for GAmma Radiation from Line- and Cylinder-sources.
M. Roos
1959
36
4
15
On the spherical harmonic expansion
neutron angular distribution function.
S
of
the
Nilsson,
Depken
1959
53
4
1959
23
4
29
4
16
The Dancoff correction in various geometries
I. Carlvik, B. Persbagen
17
Radioactive nu elides formed by irradiation of the
natural elements with thermal neutrons
K Ekberg
1959
The resonance integral of gold
K. Jtrlow, E Johansson
1959
19
4
M. Roos
1959
21
4
P H. Margen
1959
33
4
18
19
20
Optimisation of gas-cooled reactors with the aid
of mathematical computers
21
The fast fission effect in a cylindrical fuel element.
1. Carlvik, B. Persbagen
1959
25
4
22
The temperature coefficient of the resonance integral for uranium metal and oxide.
P. Blomberg, E. Hellslrand,
S. Hornet
1960
25
4
23
Definition of the diffusion constant in one-group
theory.
N. G. Sjöstrand
1960
8
4
25
A study of some temperature effects on the phonons in aluminium by use of cold neutrons.
K-E Ursson, U. Dahlborg,
S. Holmryd
1960
32
4
The effect of a diagonal control rod in a cylindrical
reactor.
7
1960
4
4
28
RESEARCH ADMINISTRATION: A selected and
annotated bibliography of recent literature..
E. Rhenman, S. Svensson
1960
49
6
29
Some general requirements for irradiation experi-
H. P. Myers, R
1960
9
6
30
Metallographic Study of the Isothermal Transformation of Beta Phase in ZircaIoy-2.
G. Östberg
1960
47
6
32
Structure investigations of some beryllium materials
I. Faldt, G. Lagerberg
1960
15
6
33
An Emergency Dosimeter for Neutrons.
/. Brann, R. Nilsson
1960
32
6
35
The Multigroup Neutron Diffusion Equations /I
Space Dimension
S. Linde
1960
41
6
26
Nilsson, N G. Sjöstrand
Afförslryck, Stockholm \960
Skjoldebrand