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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963

ORNU-PL.2001
CONF-660303-29...
CFSTI PRICES
H.C. $ 2.70; MN 50
1966
MASTER
MAY 5
MEASUREMENTS OF LO:1 ENERGY NEUTRON CROSS SECTIONS OF
NON-FISSILE NUJLCIDES AND THEIR INTERPRETATION
J. A. Harvey, Oak Ridge National Laboratory
Oak Ridge, Tennessee

RELEASED FOR ANNOUNCEMENT
IN NUCLEAR SCIENCE ABSTRACTS
LEGAL NOTICE
This report mo propared as an account of Government sponsored work. Neither the United
States, nor the Commission, nor any person acting on behalf of the Commission:
A. Makes any warranty or representation, expressed or implied, with respect to the accu-
racy, completeness, or usefulnous of the information contained in this report, or that the use
of any information, apparitus, method, or proces disclond in this report may not infringe
privately owned ripota; or
B. Assumos nay liabiliues with respect to the use of, or for damages resulting from the
use of way Information, apparatus, method, or process disclosed in this report,
As used in the abovo, "por son acting on behalf of the Commission" includes anyom-
ployoo or contractor of the Commission, or omployee of such contractor, to the extent that
such employee or contractor of the Commission, or employee of much contractor prepares,
disseminates, or provides access to, any information pursuant to his employment or contract
with the Commission, or dio nmployment with such contractor.
MEASUREMENTS OF LOW ENERGY NEUTRON CROSS SECTIONS OF
NON-FISSILE NUCLIDES AND THEIR INTERPRETATION*
J. A. Harvey, Oak Ridge National Laboratory
Oak Ridge, Tennessee
itovi
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v a
Transmission, capture, and scattering measurements can
be made with high resolution and accuracy up to 100 eV upon
any non-fissile stable nuclide if a suitable sample (~ grams)
is available. These nieak urements can be analyzed by the
shape or area method to give accurate parameters for all but
the very small resonances. Measurements upon small samples
(u moms) with low isotopic enrichment or which are highly
radioactive can give accurate parameters up to ~ 10 eV and
for the strong resonances up to ~ 100 ev. Cross section
measurements and resonance analysis of several nuclides
which are or have been of interest in the design of thermal
and intermediate energy reactors are described.
te mira.martensiastirminnfl. incarni
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I.
INTRODUCTION
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se mi moc,
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Neutron cross section measurements have been made below 100 eV
upon several hundred non-fissile elements and enriched isotopes, and
parameters for more than a thousand resonances have been determined in
this energy region. Average level spacings for these nuclides range from
None to several hundred electron volts. If we consider a nuclide with a
level spacing of 10 eV and an s-wave neutron strength function of 1004,
we compute that the average neutron width (in eV) would be about
0.001 / E (in ev). Since neutron widths obey a Porter-Thomas distribution,
we know that we must expect many resonances with small neutron widths (i.e.,
8% have a neutron width 41% of the average value) but only a few reso-
nances with neutron widths several times the average value. Very small
p-wave resonances also occur below 100 eV, and these may have neutron widths
which are comparable to those of the very weak S-wave resonances. Since
radiation widths of nuclides range from ~ 0.5 to 0.02 electron volts, the
neutron widths of most resonances below 100 eV are smaller than their
radiation widths. Many resonances occur with same several with
In a Fe and a few with ľn >>
TS
"Research sponsored by the U. S. Atomic Energy Commission under contract
with the Union Carbide Corporation.
.
IT
The accuracy to which resonance "parameters can be obtained from
the various types of measurements (transmission, capture, and scattering)
depends upon the sample which is available and the ratio of r to . For
small resonances with rearv, capture cross section measureirents zive
essentially the same data as total cross section measurements; but scat-
tering measurements can give additional information which cannot be oo-
tained from transmission measurements. For resonances with m ar, it is
desirable to have reasurements of all three types. Finally, for resonances
with
, scattering measurements give the same information as trans-
mission data but capture measurements are needed to obtain accurate values
of
II. NEUTRON SPECTROMETERS FOR CROSS SECTION MEASUREMENTS
Two techniques are used to measure cross sections as a function of
neutron energy. One technique consists of the production of a beam of
nearly mor.oenergetic neutrons and the capability of varying its energy.
The second technique is the time-of-flight technique where intense bursts
of neutrons of short time duration are produced and the energies of the
neutrons are determined from the flight times of the neutrons to a neutron
detector many meters away.
In order to obtain monoenergetic neutrons at low energies some kind
of monochromator must be combined with an intense source of neutrons, such
as a nuclear reactor, which has a broad energy spread. Although mechanical
monochromators have been constructed to produce thermal and subthermal.
beams of neutrons, they are not practical for resonance energy neutrons.
However, a well-collimated resonance energy neutron beam incident upone
single crystal, such as beryllium, can produce monoenergetic neutrons with
an energy resolution DE (in eV) ~ 5 x 10-) 3372 where E is in elecéron
volts. This energy resolution is the Doppler broadening of low energy
resonances up to ~ 10 ev. A crystal spectrometer at a nuclear reactor with
flux w 10-) neutrons/cm/sec provides an adequate source of monoenergetic
neutrons for accurate, high-resolution cross section measurements up to
~?0 eV. The crystal spectrometer has the unique advantage over the time-
of-flight spectrometer in the measurement of activation cross sections.
Most of the measurements made with crystal spectrometers have been trans-
mission measurements and accurate parameters of many of the resonances
below 10 eV have been determined from these transmission measurements.
Considerable effort has been devoted to the problem of determining the
spins of resonances using polarized samples and polarized neutrons produced
by a magnetized, monochromating crystal.
Above 10 eV, time-of-flight techniques using intense pulsed neutron
sources are superior, except for activation measurements. A fast chopper
with, a wl usec burst and a 100 m flight path at a high flux reactor
(1044 neutrons/cm/sec) has sufficient intensity for making good transmission
measurements upon samples larger than a few grams up to ~100 eV with an
energy resolution less than the Doppler broadening (~0.4 eV at 100 eV).
Capture and scattering measurements have been made with several fast choppers
at low energies, but with considerably poorer energy resolution than that
possible for transmission measurements. Fast choppers have also been used
extensively in the measurement, with NaI spectrometers, of gamma ray
spectra from individual resonances.
Pulsed accelerator spectrometers (such as electron linacs or proton
cyclotröns), due to their narrow bursts, are far superior to fast choppers
for practically all types of measurements above 100 eV. Even below 100 eV
they are superior to choppers for partial cross section measurements, and
comparable for transmission measurements with small samples. The combination
of a pulsed electron linac and a booster as used at Harwell is excellent
for all types of measurements below 100 eV; transmission measurements have .
been made upon samples as small as 0.1 cm2 with a resolution of a 0.5 eV
at 100 eV.
Two other pulsed fission neutron sources are also used for cross
section measurements below 100 eV: a pulsed reactor and a nuclear explosion.
A pulsed reactor (such as the IBR, at Dubna) is not a particularly suitable
source for high resolution measurements even below 100 eV because of its
wide pulse width (w 40 usec). However, interesting results from all types
of cross section measurements have been obtained from this pulsed reactor.
Finally, a nuclear explosion provides a short, extremely intense burst of
neutrons of no.1 usec duration. Since the flight paths used are ~200
meters, the resolution is less than the Doppler broadening up to several
keV. Although the bomb has advantages over accelerators for measuring
fission and capture cross sections of small, very radioactive samples as
will be described later in this conference, it appears to have little over-
all advantage over pulsed accelerators for transmission measurements below
100 ev.
III. MEASUREMENTS OF LOW ENERGY NEUTRON CROSS SECTIONS
A.
Total Cross Section or Transmission Measurements
A transmission measurement is the simplest and of ten the most
accurate type of neutron cross section measurezent which can be made. It
consists of a measurement in "good" geometry of the neutron counting rate
with a sample in the neutron beam and one with the sample out. Many kinds
of neutron detectors have been used, such as BFz proportional counters,
boron-loaded liquid scintillators, Li scintillation glass and a 108 slab
with NaI. Their efficiencies range from ou I to ~ 90%, depending upon the
neutron energy, and the backgrounds may range from vi to -50%. A high
resolution transmission measurement can be made over a wide energy region
to a statistical accuracy of 71% in a reasonable length of time (sa
few days for a few cm? of sample). From the observed transmission,
Texn(E.), the experimental neutron total cross section, (E,) in barns,
can be computed from the formula
Prexp(s) - () 2016
LO:
ZIN
where N is the sample thickness in atoms per barn.
Figure 1 shows such a plot of (E) of <**Cm obtained vith the
ANL fast chopper. Two samples were used: a 46 and a 7 mgm sample
covering a beam area of only ~ 0.1 cm2.. Since the thicker sample had a
thickness of only 2.24 x 10°, atoms of 144com per barn, cross sections
$10 barns could not be measured. The best resolution using a 60-meter
flight path was 35 nsec/m which corresporads to an energy resolution of
wl eV at 100 eV.
Figure 2 shows 0(E) for 3 Pa obtained with the MTR fast chopper."
The measurements were made with less than a gram of chemically separated
Puos, which had a sample thickness at the time of the first measurements
ofw 1/300 atoms/barn. The energy-reso ution was less than the Doppler
broadening up to 10 eV. Sjgçe <5>Pa decays to 233U with a 27 day half
life, large resonances from u soon a popear in the data. Except for the
measurement on the 9-hour 1.95Xe isotope which was measured up to only I eV,
235 Pa is the hottest sarap.'.e and the shortest half life nuclide upon which
an energy dependent cross secuion has been measured. Problems associated
with the analysis of data for raäioactive samples will be described in
paper Alt in this conference.
These experimental data must not be used directly to obtain pare-
meters of the resonances. Detailed calculations to correct for Doppler and
resolution broadening must be made as. Will be discussed in Section IV of
this paper.
B. Capture Cross Section Measurements
The measurement of a capture cross section requires an absolute
measurement of the number of neutrons as a function of neutron energy
hitting the sample under investigation and an absolute measurement of the
number of neutrons which are captured as a function of neutron energy. The
problem of measuring the absolute neutron flux as a function of energy is
usually separated into two parts. The first part is finding the energy
dependence of the flux iyith a neutron detector having a well-known energy
response such as a 1/v de tector like BF a. Second, the absolute flux can
be determined at one energy by (i) the saturated resonance" technique by
using a thick sample which has zero transmission at a low energy resonance
which is predominantly capture; this resonance may be in the nuclide under
investigation, (ii) comparison of the capture data for weak resonances,
hence, predominantly capture, to parameters obtained from transmission data,
and (iii) by calibration with thermal neutron cross sections. Often the
neutron flux ® (E) and detector efficiency E (BE) are not measured
separately, but only the product is determined. This product can usually
be determined to an accuracy of 3 to 5%.
Two techniques are used to determine the number of capture neutrons
(a) a large liquid scintillation detectar and (b) a Moxon-Rae detector
The gamma ray detectors should have the following properties (i) detection
efficiencies which are essentially independent of the gamma-ray cascade
mode, (ii) low sensitivities to scattered neutrons, (iii) high efficiencies,
(iv) low backgrounds and (v) fast response times if used with time-of-flight.
The large scintillator) at General Atomic has a volume of 4000 liters to
achieve (1) to a few percent accuracy, an efficiency w 90% for a sample
in the center of the 6" tube passing through the tank, a LOB liner around
this tube to reduce the efficiency for detecting scattered neutrons to
<<1% that of capture gamma rays and a time resolution of 30 nsec which
is adequate for measurements < 100 eV. Backgrounds are important with
the large scintillator, and below 100 eV are w 1% that of a "saturated"
capture resonance.
Figure 3 shows a plot of the capture cross sections of isotopically
enriched samples of tungsten measured with the 1.25 meter O RNL tank4 at
the RPI linac. These "thin-sample" capture cross sections (in barns per
atom of w) were computed fron: the formula
1
Oxy(E) = NT
C(E)
E, (BE) (E,)
, amit kontakt a tak nak
taon. Namun
where N, is the sample thickness of sample x in atoms of W per barn, and
C (E,) is the number of counts in the gamma ray detector from the capture
of nêutrons of energy E, in sample x. Since ail three resonances shown in
Figure 3 are in 103w, the ordinates must be multiplied by 40.5, 54.4, 364
and 1.05 from top to bottom respectively. When the sainple is thick as in
the lowest curve, the shapes of the resonance are distorted and the peak
cross sections are meaningless. Monte Carlo calculations for this thick
sample give a shape in agreement with the observed data but the computed
shape is not sensitive to the parameters of the resonance.
Sektoru zarukat na
than
"Another
The Moxon-Rae detector is designed so that it achieves (i) above
by making the efficiency of the detector proportional to the energy of the
gamma ray (to ~ 10% above 1 MeV) and keeping the efficiency 109. The over-
all efficiency to detect a neutron capture event for a binding energy, BE,
of 6 MeV is only a few percent. However, it does have much lower back-
ground than the large scintillation detectors since it is small and can be
tasily shielded. Its time response is fast (~ 2 nsec), and it has a low
sensitivity to scattered neutrons. Figure 4 shows à plot of capture dataº
on the 59 and 69 eV resonances in 292 Tn obtained with a Moxon- Rae detector
at the Harwell linec and booster. The ordinate was computed from the above
"thin-sample" formula. Because of the background of hard garrma rays from
daughter products, the sample had to be chemically processed and the meas-
urements made within 48 hours.
Unless the samples are very thin (NOK), where an is the peak
total cross section) this "thin-sample" capture cross section, ou (E.), as
computed above must not be used to obtain parameters of the resonances.
Usually the samples are not thin, and detailed Monte Carlo calculations
using Doppler broadened cross sections must be made to compute the capture
of both the incident and the scattered neutrons.
C. Scattering Cross Section Measurements
For scattering measurements at low energies the neutron detector
must be quite insensitive to gamma rays since capture may greatly exceed
scattering in low energy resonances. Measurements have been made with an
energy resolution - 1%, which is less than the Doppler broadening up to
N10 eV using an assembly of BF3 counters (which are very insensitive to
gamma rays) surrounding the scattering sample. At higher energies a smaller
more-efricient detector is desirable such as a 'Li giass detector, or a
LUB loaded liquid scintillator. These two detectors are quite sensitive
to gamma rays, although pulse shape discrimination helps the problem con-
siderably.
Figure 5 is a plot of the counts per timing channel from a thin
isample of 292 Th obtained with Li glass detectors' at the Harwell linac
with an energy resolution of w1/3%. The scattering from strong resonances
was determined to an accuracy of a few percent, but the scattering of the
59 eV resonance which is a 13% scattering was determined to an accuracy of
only 50%. The ordinate in Fig. 5 is only approximately proportional to the
scattering cross section since the energy dependence of the neutron flux
did not cxactly' compensate for the variation in channel width with energy.
Usually the neutron flux and the efficiency of the neutron detector are not
determined separately, but the product is determined by measuring the
scattering from an element such as lead or graphite whuse scattering cross
section varies slowly with energy and is well-known. This product can be
determined to an accuracy of a few percent.
Unless the samples are very thin (Nasci), calculations must be
made to correct for the self absorption of both the incident and the scatt-
ered neutrons in order to obtain the primary scattering from a resonance.
IV.
ANALYSIS OF NEUTRON CROSS SECTION MEASUREENTS
A. Convolution of Doppler and Resolution Broadening with the Nuclear
Cross Sections
For most non-fissile nuclides the neutron cross sections below
100 eV can be accurately represented by the sum of single-level Breit-
Wigner resonances. The cross-section expressions for an s-wave resonance
with a resonant energy En are
41.Ton a (E"_B^)
(Er-sty2 + (%)
O (E") = 5x28-
For
(p=5432 = (
0,(E") = not
124 4902
O-{") - Page 6 not
fer-sty? (F)
hai k 1, a("-3", + kape
(evesty ? CELLS
where On (E"), 0, (E") and (E") are the total, capture and scattering
cross sections at neutron enerey E"
29X is the de Broglie wavelength in the center-of-mass system of a
neutron of energy E"
r is the neutron width
is the radiation width
ŕ is the total width, P= P +
8 is the statistical weight factor which equals a ai i i
I is the spin of the target nuclides
J is the spin of the target compound nucleus which equals I + Ź
a is the potential scattering amplitude for the spin state that
contains the resonance
4or is the potential scattering cross section for both spin states
and is equal to 4582 if the a's for the two-spin states are
the same.
If the resonance is a p-wave resonance, the potential scattering amplitude
is very small and the interference term is negligible.
However, because of the thermal motion of the atoms in the sample,
these nuclear cross sections must be convoluted with the Doppler broadening
function. Assuming that the Doppler broadening follows a Gaussian function,
the Doppler-broadened cross sections (E') are computed by convoluting
the nuclear cross sections, a (E"), and a Gaussian function. Then
E'
E
Sway it comes mo-fez
where the Doppler width, A, is given by
AW
E", E', A and D. are measured in electron volts,
Toep is the effective temperature of the sample in degrees Kelvin,
.
7
and AW is the atomic weight.
These Doppler broadened cross sections must be used in analyzing trans-
· mission, capture, and scattering data.
.
Before these Doppler broadened cross sections can be compared to
experimental data, 9 calculation musi be made to allow for the instrumental
resolution. In the case of a transmissiori measurement, if we assunie that
the instrumental resolution can also be represented by a Gaussian function,
We obtain
T(EL
exp - No
(E
)
1
exo-
Y
a
'
RE,) 75
| RED
where R(E) is the resolution wash at energy E., and N is the sample thick-
ness, From a comparison of the shape of the experimental transmission to
resonance parameters, Ei, ſand (er). Ir. the case of capture or scatter-
ing measurements, detailed Monte Carlo calculations must be made to compute
the multiple capture and scattering in addition to the resolution convo-
lution. The experimental capture data can be analyzed to give E, T,, and
8 mm m/l; the experimental scattering data can be analyzed to give E, r
and gra/r. If r«r, the capture data give the same information as the
transmission data, but the scattering data allow us to determine g. If
Max", the scattering data give the same information as the transmission
data; but the capture data allow us to determine & M. For resonances
510 eV where the Doppler broadening and resolution is less than the width
r, the above parameters can be obtained from a single thin sample measure-
ment by shape analysis. However, if the resolution and/or the Doppler
broadening is » ľ, an area method must be used to obtain which requires
measurements with several sample thicknesses for each type of measurement.
However, the thick sample measurements of capture or scattering are not as
useful as the thick transmission ones because of the problem of multiple
scattering in the thick samples. The results from all types of measure-
ments can be combined to give the best consistent set of resonance para-
meters.
B. Analysis of Transmission or Total Cross Section Measurements
(1) Shape method of analysis. The shape method of analysis is
feasible only when the resolution and A are Źr. Although the reso-
lution can usually be made less than the Doppler width below 100 eV except
for very small samples, the uncertainty resulting from the uncertainty of
the resolution is often larger than the uncertainty of the Doppler width.
Doppler width of a resonance at 10 eV for an atomic weight of 100 is 0.1 eV,
which is approximately equal to the total width of a typical low energy
resonance. Hence, this shape method is restricted to low enerey resonances
(.10 eV) except for wide resonances where ľ»rwo
The 18.84 eV resonance in toºw 16 a good example of the quality
of the data which can be obtained from the shape method of analysis.
Figure 6 shows the transmission of a liquid sample of Na2WO4 dissolved
in Do (equivalent to a thickness of about 0.1 mil of tungsten metal)
measured with the ORNL fast chopper. The individual transmission points
have a statistical accuracy of w1.5%. At the resonant energy of 18.84 eV,
the energy resolution was 0.125 e! (8.4 channels) and the Doppler width A
was 0.105 eV Both are < r which is ~ 0.37 eV. The Doppler width was con-
puted from an effective temperature of 3200K determined from a Debye temp-
erature of the metal of 4000K. The problem of determining the Doppler
broadening in solids and liquids will be discussed in some detail later.
From the shape analysis program of Atta and Harvey the following parameters
were obtained:
E = 18.335 $ 0.003 eV
m = 0.385 0.008 eV
m = 0.319 + 0.0033 ev
and a correlation coefficient (7", rn) = + 0.55.
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:
:
-
The correlation coefficient is positive since the saniple was quite thin,
and, hence, the peak cross section is a measure of M r . The uncertainties
quoted are standard deviations determined from the deviations of the exper-
imental points fron the theoretical fit and the statistical accuracy of the
points. A systematic uncertainty of # 0.02 eV must be added to the resonant
energy. An uncertainty of £ 1/2% must be added to a due to the uncertainty
in sample thickness. An estimated uncertainty of 5% in the resolution con-
tributes an uncertainty to l' of $ 0.003 eV and an estimated uncertainty of
3% in the Doppler width contributes an uncertainty to rof † 0.002 eV.
Adding these uncertainties to the values determined above gives the uncer-
tainties listed in Table 1.
Figure 7 shows the transmission of a thin 0.3 mil tungsten metal
sample. From a shape analysis the following data were obtained:
E = 18.837 1 0.002 eV
m = 0.368 + 0.004 eV
m = 0.318 + 0.002 ev
and a correlation coefficient (I', M) = -0.10. The correlation coeffi-
cient is quite small and negative. For even thicker samples the correlation
coefficient becomes more negative and approaches -] because the area of a
transmission dip for a thick saiaple is a measure of the product M.
The uncertainties listed are .again standard deviations determined from the
deviations of the points from the theoretical fit and the statistical
accuracy of the points. Again a systematic uncertainty of $ 0.02 eV must
be added to the resonant energy. · An uncertainty of £ 1/2% must be added
to ſ because of the uncertainty in the sample thickness. An estimated
uncertainty of 5 in the resolution contributes an uncertainty tor of
$ 0.0014 eV and an uncertainty of 3% in the Doppler width contributes an
uncertainty to ľ' of $ 0.002 eV. Adding these uncertainties to the values
determined above gives the values listed in Tay).- 1.
Shape analyses have also been made of the transmission of 1 and
2 mil thick w samples. The transmissions of these samples were measured
for the area method of analysis and are too thick to give accurate para.
meters from shape analysis. The parameters obtained for these two samples
are also surmarized in Table 1.
The radiation width of this resonance can be computed from the
equation" =ľ. . Its standard deviation s( rw) can be computed
from the standard deviations of r and and the correlation coefficient
between 'n and
If
or 11
or if
olry) - [P(r) + 8?(rn) - 2011) ocra) col r, nd* .
cclr, rm) = + 1, o ry) = 18() - 81 7m),
cccr , rum) = -2, 6( rv) = 8(m) + el ron] ,
ccl, 1) = 0, ory) [6217) +6317)]*.
Since the correlation coefficient is positive for the thinnest sample,
8( rwd is somewhat less than the value obtained if one ignores the cor-
relation coefficient; whereas for the thicker samples, sl ) is somewhat
larger.
For these measurements upon this 18.84-eV resonance the uncer-
tainties of the parameters ľ, and carise mainly from the statistical
accuracy of the experimental points. However, for the measurements upon
the other low energy resonances where the resolution and the Doppler
broadening are more nearly equal to the total width, uncertainties in the
parameters arising from uncertainties in the resolution and the Doppler
broadening are about equal to those from the statistical accuracy of the
data. Although it may be possible to reduce the resolution or its un-
certainty, it is not easy to reduce the uncertainty due to the Doppler
broadening. The Doppler width can be reduced (about a factor of 2) by
cooling the sample down to liquid nitrogen temperature, and this technique
he's been used at the Saclay linac for transmission measurements upon
thorium. However, although the Doppler width is less, its absolute un-
certainty is about the same or may even be larger since the Doppler broad-
ening in solid and liquid samples is very complex. Hence, it may still
produce considerable uncertainty in the determination of /' from this shape
analysis method.
20
Doppler broadening of a neutron resonance has been made using the BNL
crystel spectrometer. The shape of the 4.14 eV resonance has been
measured using metal samples at temperatures from 4 to 825°K and a heavy
water solution at room temperature. Although the purpose of this work
was a study of the Doppler broadening, the parameters of this 4.14 eV
resonance were necessarily determined. Figure 8 shows a plot of the
825°K data and the computed fit to the data. Table 2 summarizes the data
reportedlo which were determined from the effective temperature model for
the metal samples and the ideal gas model for the solution. The quoted
errors are from a least squares analysis, but the errors have to be in-
creased somewhat to take care of systematic uncertainties. The authordo
prefers the parameters from the 825°K data because he feels that the
Doppler effect 18 known more accurately at this temperature than at the
lower temperatures. It would appear that the uncertainty of risn t 1 mv
and that of ľn is only 0.02 mV.
:
.
.
.
.
. .
.
.
.
in.
a
. Finally it should be mentioned that there are special cases where
the shape method of analysis is useful even when A and/or R are >>".
In this case the interference between resonance and potential scattering
or the asymetry of the resonance may determine whether the resonance 18
an 8 or a p-wave resor.ance. Thick samples are measured in transmission,
and the energy region of interest is many electron volts away from the
resonance energy.
.
z
.
.
.
.
.
**...
...-1
-
.
(2) Area method of analysis. When the Doppler width and/or the
resolution are greater than the total width of a resonance, the shape
method is not practical for the determination of resonance parameters; so
an area method must be used to obtain the parameters of the resonances.
The area method is useful up to a 100 eV or until the Doppler width, A,
or resolution, R, is comparable to the level spacing, D. If A or R is
>> 10 r or comparable to D, it is often not possible to determine a mean-
ingful value for . The area method can often give resonance parameters
more accurately than the shape method even when A and Rare < M.
mihi...'.",*
The transmission data for the 18.84 eV resonance of Figures 6 and
7, as well as data for thicker samples, can be used to illustrate the
area method of analysis. In the area method, values of are determined
for each sample thickness for various assumed values of which produce a
transmission dip having the same area as the area of the experimental
transmission dip. For a thin sample (No <1) the resulting value of
is quite independent of the assumed values of r. This is to be expected
since the area under a resonance is proportional to a r, which is a
measure of Tn. For thick samples (NO>>1) the area above a transmission
dip is proportional to a ns, or proportional to PM. The result of
a measurement upon a single sample thickness can be represented approxi-
mately by an equation of the form of 10 = k. For thin samples m is
No.1, for thick samples 'm is ~0.9,"and for interniediate samples m is
w2/2. Figure 9 shows the data obtained from the five sample thicknesses
for the 18.84 eV resonance in 186w using the area analysis program of
Atta and Harvey.9 The problem now is to determine the best values for
r and r and their standard deviations which are consistent with the
11
lines in Figure 9, which have uncertainties of 0.9 to 5%. One simple
method is to determine values from plots of loa vs log ” for the five
samples for m and k in the vicinity of the correcť m. The uncertainties
due to sample thickness and Doppler broadening are small and are included
in ; uncertainty in the resolution 3.8 not important in the area method.
The values obtained are listed in Table 3. A least squares program to
solve the linear equations log + m, log = log k, 18 used to obtain
values for log Me logſ and the standard deviations of these quantities.
Figure 10 shows a plot of this fit to the five experimental points (m.,
log k.). The intercept at m = 0 is the best value for me and the best
value for r is obtained from the slope. The parameters pleine M, 8(ru),
s(r) and the correlation coefficient between and M are summarized in
Table 4. The fact that the correlation coefficient 18 - 0.84 means that
and r are almost completely anti-correlated as might be expected from
the nature of Figure 10.
The radiation width and its standard deviation are computed as
before. Since the correlation coefficient between m and ľapproaches
- 1, the standard deviation on equals the sum of $(M) and 8( m.).
Table 4 summarizes results of the area analysis for other low energy
resonances in tungsten.
C. Analysis of Scattering Measurements
Low energy scattering data can be analyzed by a shape metnod pro-
vided the resolution and Doppler broadenings are = ŕ and No 51; how-
ever, it is seldom done. It is somewhat more complicated than a shape
analysis of transmission data since one must include attenuation of the
scattered neutrons by the sample, allowing for the energy shift upon
scattering. In general, only an area method is used; and the result is
combined with results from transmission and capture measurements.
For very thin samples (no <<1) the area under a scattering
resonance gives the quantity gr2/. On a plot of log ľn vs log this
quantity gives two lines of slope 1/2, one line for each assumed value of
8. Figure ll shows results from scattering, transmission and capture
measurements upon the 17.4 eV resonance in 269 Im using the Harwell linac7.
It can be readily seen that it is the scattering information which is .
needed to determine the spin. The capture data for this resonance give
essentially the same information as a reasonably thin transmission
measurement.
Scattering measurements upon samples of intermediate thicknesses
may sometimes be analyzed if the energy loss by the neutron upon scattering
is sufficient so that the attenuation of the scattered neutrons is not
too large. For these intermediate thicknesses the slope of the line on
the log r vs log l' plot is somewhat less than that of the thin sample.
For thick samples the slope approaches zero.
Although an accurate scattering measurement combined with
2.2
transmission or capture measurements on a resonance will result in a
somewhat more accurate determination of m and than would have re-
sulted without the scattering data, the unique value of the scattering
data is that it enables one to determine the spin state of the resonance.
Spins of resonances can be determined by other methods, such as polarized
neutrons and polarized targets, capture gamma spectra, etc; but scattering
measurements, when possible, are very valuable in determining spins of
compound states.
D. Analysis of Capture Measurements
-
--
-
-----
--
--
---
-
-
If the resolution and Doppler broadening are < M and No. 51,
low energy capture data can be analyzed by a shape method to obtain the
parameters E^, r , and & n rull. Since low energy resonances are pre-
dominantly capture, multiple capture is not of much importance and shape
analysis of capture data is quite similar to that of transmission data.'
However, if mais comparable to me detailed Monte Carlo calculations
must be made to determine the multiple capture. Because of this com-
plexity and because many low energy reuonances have already been investi-
gated accurately by shape analysis of transmission data, little shape
analysis of capture data has been done. Almost all capture data have
been analyzed by the area method.
-
-
.
.
.
-
.
.
hod.
.
.
:o
*V
*
. For very thin samples (NO <1) the area under a capture reso-
nance gives the quantity up. For very small resonances where
Miss this quantity reduces to 8 m, which is the same quantity
deterioined from a thin sample transmision measurement. However, if only
a small amount of sample is available, accurate capture data on a small
resonance (N I <<1) can give a more accurate determination of gr than
can be obtained from a transmission measurement. If 1? 18 = "capture
measurements combined with transmission and scattering data will result
in more accurate determinations of random. When 7 18 = Pw, it is
informative to write the quantity & Paltas
On a plot of log
o vs log l' this quantity has a slope
a
1. Tarim)
When ľn is a ry, the slope is large; and when an is >> Pw, the slope
approaches +1. '.
Capture measurements upon samples of intermediate thicknesses
may also be analyzed by the area method. It is necessary to make accurate
Monte Carlo calculations for these samples since the multiple capture
13
may even exceed the primary capture if > The results can be
plotted on a log ™ vs log ^ plot, and the liries have somewhat steeper
slopce than those for thin sempl.es vaath the se: ratio of M r . If we
assure the transmission area for an interijediate semiple thickness to be
a measure of run, then the slope of the line on the log r vs log →
plot from a capture measure.rent is
* (2-4(,-)).
1 1
t un
If is <<", a plot of log vs log l'18 very similar to a
plot of 103 pm v8 log i Figure 12 shows such a plot for the 21.69 eV
resonance in 29219 from capture, scattering, eni transın18sion measure-
ments at Horwell.6 Curves labelled 1, 2 and 3 are from capture, curve 8
18 from scattering and the others are from transmission measurements.
A least squares fit to these data gives = 1.88 + 0.08 mV and
r. = 24.6 1.2 mv. When 18 « r, a plot of log ľ ve log ľn or
106 r 18 more meaningful to determine the resonance parameters
Capture measurements upon 1, 2 and 5 mil tungsten samples have
been made at RPI to determine the capturell in the 18.84 eV resonance.
About half of the capture in these samples 18 due to primary capture and
half due to multiple capture. Combining these results with the trans-
mission measurements discussed eariier, Blockii obtained a value for
of 0.053 $ 0.005 eV. About half of this error is due to the Monte Carlo
correction and the other half due to systematic and statistical errors
in the capture measurement.
From the parameters of this 18.84 eV resonance (P = 0.319 $ 3 eV,
and r = 0.053 $ 0.005 eV), we can compute that it should contribute
45 £ 4' barns to the thermal capture cross section. This 18 not in good
agreement with the recently measured values 02 37.8 + 1.2 barns. Con-
versely, if we assume that this resonance is responsible for all the
thermal capture cross section, we compute a radiation width of only
0.045 $ 0.002 eV. Since the multiple capture was so large for the samples
used for the capture measurements, it would be desirable to make capture
measurements with much thinner samples.
Before leaving capture measurements, I should mention the capture
measurements made at General Atomic upon samples of the four abundant
tungsten isotopes from 0.01 to 10 ev.12 Measurements of this type are
valuable in determining capture cross sections between resonances and the
energy dependence of capture cross sections in the thermal energy region
for nuclides where the scattering is comparable to or exceeds the capture.
E. Analysis of Other Types of Measurements
Since the self-indication ratio measure.rents will be described in
detail in paper A3, I will mention only that they they are a variation of
14
transmission measurements and are additional data to be added to the area
analysis plots.
The determination of spins of resonances from (n,a) reactions will
be described in paper El.
Very few measurements have been made of activation cross sections,
but the method of analysi8 18 similar to those aleady described.
V. Computation of Resonance Absorption from Resonance Parameters
The contribution of a particular resonance to the resonance ab-
sorption for an infinitely thin sample can be computed from one of the
following expressions
Connect
Top
(2. a) o en , (2-5)
Although the value for the resonance absorption will be the same regardless
of the expression used, the accuracy computed will be different unle88
correlations between the parameters are known (and used correctly). We
have seen that there is often quite a strong correlation between and
m (partialarly from area analysis). For resonances with a scriit
is convenient to use the expression in terms of r and to Compute the
accuracy. Assuming the uncertainty in E^ is small we get
RA
8(,) 8(r) corner)
For
For resonances with ľ«l we can see that the uncertainty in the reso-
nance absorption is determined mainly by the first term, the relative un-
certainty of in. The second term is very small because of the factor
mur (for a reasonable value of slr)/), and the third term is small
ana is usually negative from area analysis. For example, the contri-
bution to the resonance absorption of four of the low energy resonances
in tungsten listed in Table 4 can be computed to an accuracy of a few
percent even though the total widths are accurate to only 10%. However,
the contribution to the resonance absorption of the 18.84 eV resonance
15
(rear) has an uncertainty of 8% due to the 10% uncertainty of .
The relative uncertainty of the contribution of a resonance to
the resonance absorption is very simply related to the relative uncertainty
of the quantity Pro, as discussed iri Section IV, if m has the value
(r/m)/(1 - (2 r A )]. It can readily be shown that
acea) (--
) or im
RA
where
Por
O 1 - (2 PM)
Thus if one .is concerned with an accurate determination of the Infinitely
dilute absorption of a resonance where in << "", it is not necessary to
determine the parar.eters per and their standard deviations independently.
It 18 only necessary to determine the quantity Prim and its relative un-
certainty for m = m.
VI. CONCLUSION
If a suitable sample is available, resonance parameters can be
obtained for almost any desired non-fissile nuclide up to 100 eV. The
accuracy to which the parameters can be obtained depends a great deal
upon the sample (quantity, enrichment, activity), the strength of the
resonance, and the Doppler and resolution broadening relative to the total
width. Resonant energies can readily be determined to an accuracy of
~0.1% except for small resonances occurring very close to large reso-
nances. The quantity gran can be determined, depending upon the resonant
energy, to an accuracy of"to 2% for strong resonances, and from 2 to
20% for weaker resonances providing No 20.2. The total width, r, of
a strong resonance can be measured to an accuracy of ~ 2% if the Doppler
and resolution broadening are small compared to the total width; to an
accuracy of 5 to 10% if either is comparable to r ; and to an accuracy
of > 30% if either is > > r. Radiation widths can be obtained to an
accuracy of 2 to 10% for strong resonances if resolution and Doppler
broadening are <r, and to an accuracy of 2 30% if either is>>!.
The spin of the capturing state can usually be determined if rur is
20.01 to 0.1 depending on the spin of the target nucleus.
If only milligram quantities of samples are available, neutron,
total, and radiation widths of all but the weakest resonances can be
obtained with good accuracy up to ~10 eV. Above 10 eV, neutron widths
26
can be obtained providing No
for only the strong resonanceš.
0.2, but total widths can be obtained
-
-
-
REFERENCES
1.
R. E, Cote, R. F. Barnes, and H. Diamond, Total Neutron Cross Section
of 244cm, Phys. Rev. 134: B1281 (1964).
2. F. B. Simpson, J. R. Berreth, J. W. Codding, and R. P. Schuman, Total
Neutron Cross Section of 233pa, BAPS 9, 433 (1964) and MTR-ETR Tech-
nical Branches Quarterly Report, Jan. 1 - March 31, 1964, USAEC Report
IDO-16994, p. 23, Phil11p8 Petroleum Company, August 1964.
1.3.
E. Haddad, R. B. Walton, s. J. Friesenhahn, and W. M. Lopez, A High
Efficency Detector for Neutron Capture Cross Section Measurements,
Nucl. Instr. Methods 31, 125 (1964).
4. R. C. Block, J. E. Russell, R. W. Hockenbury, Phys. Div. Ann. Prog.
Rept. Dec. 31, 1964, USAEC Report ORNL-3778, pp. 53-64, Oak Ridge
National Laboratory, May 1965.
5. M. C. Moxon and E. R. Rae, A Gamma- Ray Detector for Neutron Capture
Cross-Section Measurements, Nucl. Instr. and Methods 24, 445 (1963).
6. · M. Asghar, c. Ñ. Chaffey, M. C. Moxon, N. J. Pattenden, E. R. Rae and
C. A. Uttley, The Neutron Cross Sections of Thorium and the Analysis
of Resonances up to 1 keV, British Report EANDC UK 565, 1965.
7. M. Asghar and F. D. Brooks, Neutron Resonance Scattering Measurements
with Liº Glass Detectors, British Report, AERE-NP/GEN/ 40, June 1965.
8. D. Paya, K. D. Pearce, J. A. Harvey and G. G. Slaughter, Phys. Div.
Ann. Pros. Rept., Dec. 31, 1963, USAEC Report ORNL-3582, pp. 56-60,
Oak Ridge National Laboratory, June 1964.
9.
S. E. Atta and J. A. Harvey, Numerical Analysis of Neutron Resonances,
USAEC Report ORNL-3205, Oak Ridge National Laboratory, Dec. 1961.
A. Bernabei, Effects of Crystalline Binding on the Doppler Broadening
of a Neutron Resonance, USAEC Report BNL-860, Brookhaven National
Laboratory, June 1964.
11. R. C. Block, Oak Ridge National Laboratory, personal communication,
1966, and ORNL Status and Prog. Rept., Sept. 1964, USAEC Report
ORNL-3718, pp. 17-18, Oak Ridge National Laboratory, Oct. 1964.
12.
S. J. Friesenhahn, E. Haddad, H. F. Frohner, and W. M. Lopez, The
Neutron Capture Cross Section of the Thingsten Isotopes from 0.01 to
10 Electron Volts, Report GA-6882, General Atomic, Jan. 1966.
TABLE 1
.
Parameters of 18.84 resonances in W from shape analysis
.
--.
..
.
.
..
..
Isotope E. t 8(E) Sample
... ...
Pt(m) retelro) ccernor) r, Is(r)
(mv)
Thickness
(mils)
.
(ev)
...
18.84+0.02 Solution
28.84.0.02 0.3
. 18.84£0.02
18.85+0.02
Best
Values: 18.84.0.02
38549
36846
356=30
386017
37426
31914
318+3
323–20
304–11
0.52
-0.20
w-0.9
-0.95
6648
5037
33:50
82+28
Nr
31813
5746
en
cament

TABLE 2
Parameters for the 4.14 eV resonance in tow from transmission measurements
with the BNL crystal spectrometer
Sample Material
Sample Temp. (°K)
Metal : Metal
825 296
4.142 4.141
55.5 $ 0.5 53.2
1.50 $ 0.01 1.48
Metal
77.8
4.140
52.2
1.47
Metal
4.2
4.138
52.2
1.47
Solution
296
4.140
53.0
1.47
TABLE 3
Determination of m, k and 8(k)/k from area measurements for the 18.84 eV.
resonance in 186W
Sample Thickness
(ev) Mlls Atoms of W per Barn Exponent, m
k
1 (mv) |
% Error in k
((k) x 100)
1.7

18.85 Liquid
18.81 0.3
18.83 1
18.83 2
18.81 5
1.986 x 10"
5.793 x 10-5
1.72 x 10°!
3.23 x 10-4
8.28 x 10-4
-0.012
0.21
0.75
0.82
0.96
297
1103
25800
40300
89000
5.0
0.9
2.5
The negative values for m arise due to the nature of the area analysis
program
-
TABLE 4
Parameters of low energy resonances in W from area analysis
Isetupe
182W
E ts(s)
4.17+0.02
7.68+0.03
ſ' ts(r) (unts (7) col Poem)
55.141.4 1.48+0.03 -0.85
79+3 1.74+0.03 -0.68
ts(,)
53.61.4
7713
4
183w
186
(8 = 3/4)
41:11
182w
18.83+0.04
21.0810.05
27.08+0.06
36117
10216
118£7
32014
40.101.1
43.3=1.4
(8 = 3/4)
-0.84
-0.95
-0.82
6257
183w
7518
FIGURE CAPTIONS
Figure 1. Neutron Total cross section of <**Cm up to 100 eV measured with
the ANL fast chopper. -
Figure 2. Neutron total cross section of > Pa up to 9 eV measured with
the MTR fast chopper."
Figure 3. Neutron capture cross sections (per atom of W) of isotopically
enriched samples of tungsten measured with a large liquid
scintillator.4 These cross sections must be multiplied by
factors of 40.5, 54.4, 364 and 1.05 respectiyely to obtain the
capture cross section of the resonances in 103w.
Th measured with a Moxon-Rae
Figure 4. Neutron capture cross section of “
detector at the Harwell linac.
Figure 5. Scattered neutron counts per timing channel ys neutron energy
from 232Th measured with "Li glass detectors at the Harwell
linac.
Figure 6. Transmission of a sample of Na Wos, dissolved in DOO
(N = 1.99 x 10-5 atoms of w/bafn) in the region of the 18.84 eV
resonance in 186w.
Figure 7.
Transmission of a 0.3 mil tungsten metal sample (N = 5.79 x 10°
atoms of w/barn) in the region of the 21.08 and 18.84 eV
resonances.
Figure 8.
Total neutron cross section of W at 825°K in the region of the
4.14 eV resonance. Curve A is the theoretical shape, curve B
includes Doppler broadening, and curve C includes both Doppler
and resolution broadening.
Plot of ľ vs assumed r for the five sample thicknesses. listed
in Table 3 for the 18.83 eV resonance in 100w from area analysis.
Figure 10. Least squares, fit to obtain the parameters for the 18.83 eV
resonance in low from area analysis.
Figure 11. Area analysis of 17.4 eV resonance in toyim from scattering,
(labelled S), trar.smission (T), and capture (c) measurements
to determine the spin J.
-.
..
.
..-...
.
-.,
Figure 12. Area analysis of 21.69 eV resonance in Th from capture,
scattering, and transmission measurements at Harwell.º Curves
. 1, 2 and 3 are from capture, curve 8 is from scattering, and the
others are from transmission measurements.
e
sortit
23
9 Cm 244
.
CROSS SECTION CURVES
18.1 y
IIIIIIIIIIIIIII
IHIIRIMU
UUTUULIOLI
IIIIIITTIR
00001
0008
0009
10000
8000
6000
4000
3000
2000
MIH
#lojol-AS
FAROE
T EISESFETITIE
FEFERBERREFERE
: FE
:
55
4000
3000
UNT:12
2000
IULIUUIULUIDITUUL
0001
008
009
0001
008
IDEER
600
III
400
300
S.
0,-BARNS
200
NO
1.8
PC
.o
5
10
15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

TC.T...?.
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-
::::::::::::::::::
:::
::::
::::
:;:::.1::::
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::::::::::
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4000
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2000
6000
4000
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.
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.
2000
0
0
n. 233
TOCO
800
600
1000
800
600
.
27.4 d
400
400
300
300
.
.
.
.
.
200
Oy - BARNS
SC
.
.
.
80
60
.
D
.
....
TOL
o
..0.5
.0
1.5
2.0
2.5
3.0
3.5
4.0
5.0
5.5
6.0
6.5
7.0
7.5
8
8.5
9.0

4.5
En-EV
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...........................؟
11LLLLLLL
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SD
15
NEUTRON ENERGY (OV)
omoc 65-27


1000
... n = 2.776 * 10 3 ald
+++ n = 1.418 x 10-3 ald
*** N = 0.7132 x 10°3 ald
LOGARITHMIC SCALE
BARNS
OBSERVED NEUTRON CAPTURE YIELD
• ATOMS / BARN
اللللللل
LINEAR
SCALE
55
65
75
NEUTRON ENERGY
EMOCUK Ss'. Alg a
Figure 4
..
...
-
..
.
-
-
-
-
-


TH232 SCATTERING DATA
n=1-317x10*atoms/barn
Resolution=5 nanoseconds/metre
250
200 NEUTROS ENERGI OCV) — 140
"
120
COUNTS/TIMING CHANNEL
-7570
65
24
60
NEUTRON ENERGY
Figure 5
ORNL-DWG 65-13017R
486 W RESONANCE
1.99 x 10-5 atoms of W/ barn
Eo = 48.835 $ 0.003 eV
r=0,385 + 0.008 eV
r,= 0.319 $ 0.0033 eV
co (Tn, ) = + 0.55

UDUTUL
ITIWI
SINNIT
LIITOL
LUULU LU
UK
In
NNI
KIINNIWIT
JUOOUUU
UULUUUUUUUU.
UUTUUD
UITHOUDIID
ITULUIIIILU
WDUODUOIO
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UUUUUUUUU
UDI
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IIIIDIDUNT
DUUHUDDIN
TUDI OTTI
JUDODODUODUIN
O
OD
O
C01
INH
DID
L
OD
480
500
520 540.
CHANNEL NUMBER
560
580
Figure 6
182W
=21.08 +0.004 eV
ř=0:100 +0.012 eV
T,=0.040 + 0.001 eV
CC (,,r) = 0.63
ORNL-DWG 65-43048R
186W
5.79x10-5 atorns of w/ barn
Ep = 18.837 +0.002 eV
P=0.368 + 0.004 eV
T, = 0.318 + 0.002 eV
CC (I,,)=-0.10

INTUIT
Atli
TRANSMISSION
O
001
II
MI
XIII
UrnTuin
430
450
470
550
570
490 510 530
CHANNEL NUMBER
Figure 7
ORNL-DWG 65-43020
.
... 5000
TUNGSTEN 182 (METAL)
Ep=4.1422 eV
TEMPERATURE 825 °K
4000
000
CROSS SECTION-BARNS
Ida
1000
0
4.000
4.050_
4.100
4.150
4.200
4.250
4.300
NEUTRON ENERGY-eV
Figure 8

ORNL-DWG 64-1841
340

1 mil
18.8 ev
5 mils
12 mils
330
0.3 mil
320
0.1 mil-
· 300
2904
320
340
340
360
360
380
380
400
r
Figure 99
ORNL-DWG 65-13019

186 w
18.83 eV
k=r, som
T=361 $ 7 mV
(FROM SLOPE)
-T, = 320$4 mv
(AT m=0)
0
0.2
0.4
0.6
0.8
1.0
m
Figure la
Tn
4 eV
کل 12
لا
-
13 .x2
و
6 (M۰۷)
ی )
. x2. 03
ر
IOoO
400
600

r (MeV)
AERE NP/GEN/40 FIG. 7
Figure
21.69 eV

܂
ܬ
Tn (mer)
ܣ
ܙ
-
ܝܝ ܟ
ܢ
1.5
ܗ
15
30
35
40 45
20 25
To (mev)
EANDC(UK)56's Fig. 6
Figure 12
.
END
-
-
DATE FILMED
6 / 16 /66







De
W