hu TOF | ORNL P 1885 1 1321112 EEEE 11:25 114 LE MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS - 1963 офи - 78K CONH-660704-2 ASTS JAN 19 iyoo TIE MEASUREMENT OF NEUTRON TOTAL CROSS SECTIONS IN THE RESONANCE ENERGY REGION AND THE DETERMINATION OF RESONANCE ABSORPTION* J. A. Harvey, Oak Ridge National Laboratory ::ELEASED FOR ANNOUNCEMENT Oak Ridge, Tennessee il CLEAR SCID:"CZ ABSTRACTS I. Introduction In the design of thermal and intermediate energy reactors, the absorption cross sections of many materials for resonance energy neutrons are needed to high accuracy. It is often not possible to make direct capture cross section measurements as a function of neutron energy to the desired accuracy. For very radioactive nuclides, such as <»Pa or hot fission products, it is difficult to make even crude capture cross section measure- ments. However, on nearly all nuclides accurate total cross section measurements can be made at low energies which can be analyzed to yield accurate parameters of the resonances. From the parameters of non-fissile nuclides the resonance absorption can be computed. The resonance absorptions thus obtained are more accurate up to - 10 eV than can be obtained from direct capture measurements. They are of comparable accuracy up to 100 eV except for resonances in which the scattering is much larger than the capture and in general are less accurate above ~ 100 eV. To illustrate this method, I will refer to transmission data obtained with the ORNL fast choppert on the low energy resonances of the isotopes of tungsten. The resonance absorption of these isotopes is of interest in the design of nuclear space reactors using tungsten enriched to -93% in 10*w. - - For the 102w and 100w isotopes the low energy resonances contribute the major - - - . . . - part of the resonance absorption. LEGAL NOTICE uw This report was prepared as an account of Government sponsored work, Noithor the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makos any warranty or representation, expressed or implied, with respect to the accu- racy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, motbod, or procos, declosed in this report may not infringe privataly owned righus; or B. Assumes any liabilities with respect to the use of, or for damagos roswting from the use of any information, apparatus, mothod, or procon disclosed in this report. As used in the above, "person acting on behalf of the Commission" includes any om- ployee or contractor of the Commission, or employ of such contractor, to the extent that guch omployee or contractor of the Commission, or employme of much contractor propares, dienominatos, or provides accen to, any information pursuant to his employment or contract with the Commission, or his employment with such contractor. .. , ii II. Measurement of Total Cross Sections as a Function of Neutron Energy (a) Neutron Spectrometers 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 monoenergetic 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. Monoenergetic neutrons can be produced in a nuclear reaction by using a beam of monoenergetic protons from an electrostatic accelerator and a thin target such as lithium or tritium. This method has been used to measure neutron cross sections down to al0 keV where the energy resolution was a 0.3 keV and up to - 10 MeV where an energy resolution of ~ 5 keV has been obtained. In order to obtain monuenercetic neutrons at low energies some kind of inonochromator must be coms ined with an intende source of neutrons which has a broad energy spread such as a nuclear reactor. Although mechanical mono- chromators 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 upon a single crystal, such as beryllium, can produce monoenergetic neutrons with an energy resolution A E (in eV) 5 x 10-) E72 where E is in electron volts. This energy reso- .. lution is the Doppler broadening of low energy resonances up to - 10 eV. A crystal spectrometer at a nuclear reactor with a flux - 10+neutrons/cmY E, and resolutions (4 E/E) of ..-10°°VE have been obtained above 1 MeV. The combination of a pulsed electron linac and a booster as is used at Harwell is also excellent for high resolution transmission measurements below a few kev. Finally, a nuclear explosion provides a short, intense burst of neutrons of a 0.1 usec duration. One shot produces more neutrons than all the accelerators used for time-of-flight spectroscopy produce in many tens of years. Although the bomb has advantages over accelerators for measuring fission and capture cross sections of higlily radioactive samples, it appears to have little advantage over accelerators for transmission measurements. . (0) Total Cross Section Measurements A transmission measurement is the simplest and most accurate type of neutron cross section measurement which can be made. It consists of a measurement in "good" geometry of the neutron counting rate with a sample in the beam and one with the sample out. Backgrounds are quite 2.ow (4.10%) and a transmission measurement to statistical accuracy z 1% over a wide neutron energy range can be made in a reasonable length of time. From the observed transmission, Texo (E), the experimental neutron total cross section exp(E), in barns, is sometimes computed from the formula d'exp (E) = (1/N) 10g (1/Texn(E)) exe exp where N is the sample thickness in atoms per barn. Unless the energy resolution is much less than the width of the Doppler broadened resonance, this experimental cross section must not be used to obtain parameters of the resonances. When the resolution is poor, parameters must be derived from the experimental transmission data. Even when the resolution is good it is often preferable to determine the parameters of the resonances from the transmission data. III. Determination of the Total Width and the Neutron Width of a Resonance from Total Cross Section Measurements A. Convolution of Doppler and Resolution Broadening with the Nuclear Cross Section For osv non-fissile nuclides the total cross section can be accurately represented by the sum of single-level Breit-Wigner resonances. The single level Breit-Wigner formula consists of 3 terms, a term independent of neutron energy (potential scattering), a resonance term syrame trical about the resonant energy, i. - AC and an asymmetrical term (interference between resonance and potential scattering). For neutron energies 5. 10 keV (ignoring the (A+1)/A factor) the total cross section for the sum of single-level formulas can be written 0%(E") = 4yror? 6.52x105 5 5^(887M?" zs (E--E")(f8!Oja ad - 5 (E2-9")2+(pajaj2 - 5.725x20° G (E4-E")2+(7972)2 where on you 18 the total cross section (in barns) is the neutron energy (in electron volts) the resonant energy of resonance (in electron volts) pp is the total width of resonance 1 (in electron volts) is the reduced neutron width of resonance » (in )ev) is the neutron width of resonance » (in electron volts )and equals /OVE Ě oare is the fractional abundance of the isotope which contains resonance a to 8 is the statistical weight factor of resonance am 18 the potential scattering amplitude of the isotope and spin state which contains resonance » (in 10-12 cm) r is the effective nuclear radius for all isotopes and spin states (in 10-12 cm). However, because of the thermal motion of the atoms in the sample, this nuclear cross section must be convoluted with the Doppler broadening function. Assuming that the Doppler broadening follows a Gaussian function, the Doppler- broadened cross section, (E') is computed by convoluting the nuclear cross section, j (E"), and a Gaussian function. Then - EN ostes satis lectors onl y ) (E") exp dari where the Doppler width, A , 18 given by E", E', A and Do are measured in electron volts, Topp 18 the effective temperature of the sample in degrees Kelvin, A is the atomic weight, and Do = 0.31867 pp/293)* The theoretical transmission is obtained by the convolution of the resolution function exp[- N«' 2 (E')]. If we assume that the instrument resolution can also be represented by a Gaussian function we obtain (8,) - Peris exp - 30 % (8")] ex- (2-3)? } E', T(E) = -- exp - NOA { [ R(E) where R(E) is the resolution width at energy E,, and N is the sample thickness. From a comparison of the experimental transuission to this theoretical transuission, the paraleters 0;the resunances E?, ; and (fø10) can be obtained. The resonunt energies can be easily determined to an accuracy 0.1%. The neutron widths (assuming we know f and g) can be determined from the quantity (131) to an accuracy of 1 to 2% for strong, well-resolved resonances; however, Y weak resonances may have accuracies of only 10% - 30%. The total width of a strong resonance can be measured to v 2% accuracy if the Doppler width and the resolution are small compared to the total width!'. .:" Tie .. . 7 If they are comparable to l',!' can be measured to 5 or 10% accuracy; but if they are .1, as 18 generally true for resonances - 100 eV, ;' may be accurate to only 30. The accuracy to which the parameters can be obtained depends on the size of the resonance, the backgrounds, the counting statistics, the Doppler and resolution broadening relative to the total width of the resonance and how accurately they are known. Usually the uncertainties of the sample thickness and its uniformity are not important. The two methods which are used to obtain the resonance parameters are the shape and area methods of analysis. B. Shape Method of Analysis The shape method of analysis 18 only feasible when the Doppler width and the resolution are less than or the order of the total width of the resonance. Although the resolution can usually be made less than the Doppler width, the uncertainty of the resolution is often larger than the uncertainty of the Doppler width. The 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 energy resonances ( 10 eV) except for wide resonances (hence, predominately scattering) which may occur in nuclides where the resonance spacings are . 100 eV. The 18.8 eV resonance in coow is a good example of the quality of the data which can be obtained from the shape method of analysis. Figure 2 shows the transmission of a liquid sample of Na, wo dissolved in D20 (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 18.8 eV, the energy resolution was 0.125 eV (8.4 channels) and the Doppler width A. was 0.105 ev. Both are less than the total width of the resonance which 18 ~ 0.37 eV. The Doppler width was computed fro:n an effective temperature of 320°K determined from a Debye temperature of the metal of 400°K. 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 = 18.835 + 0.003 eV p" = 0.385 + 0.008 ev = 0.319 + 0.0033 eV and covariance (?',1m) = 0.146 x 10-4 ovar (i,in . or correlation coefficient (1', 1) = covar (1,-) = +0.55 The correlation coefficient is positive since the sample was quite thin, and are standard deviations determined from the deviations of the experimental points from 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 1/2% must be added to in due to the uncertainty in sample V ! , - * * * thickness. The estimated uncertainty in the resolution (5% ) contributes an uncertainty to m of $ 0.003 eV and an estimated uncertainty in the Doppler width (3%) contributes an uncertainty to of 0.002 eV. Adding these un- certainties to the values determined above gives the uncertainties listed in Table I. Figure 3 shows the transmission of a thin 0.3 mil tungsten metal sample. From a shape analysis the following data were obtained: . E. = 18.837 + 0.002 eV p = 0.368 + 0.004 eV Im = 0.318 $ 0.002 eV and a correlation coefficient (1, 1) = -0.10. The correlation coefficient 18 quite small and negative. For even thicker samples the correlation coefficient becomes more negative and approaches -l because the area of a transmission dip for a thick sample is a measure of the product Mm!". 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 " due to an estimated uncertainty in the sample thickness. The estimated uncertainty in resolution ( 59 ) contributes an uncertainty to of £0.00.4 eV and an uncertainty in the Doppler width (3%) contributes an uncertainty to ſ' of 0.002 eV. Adding these uncertainties to the values determined above gives the values listed in Table I. Shape analyses have also been made of the transmission of 1 and 2 mil thic's W samples. These samples were measured for the area method of analysis and are too thick to give accurate parameters from shape analysis. The para- meters obtained for these two samples are also summarized in Table I. 13 Table I also summarizes the po.rameters for the two low energy resonances at 4.16 and 7.67 eV. 'The Doppler widths of these resonances are approximately equal to their total widths and the energy resolutions for these data were some- what less than the Doppler widths: The uncertainties in the values for include uncertainties arising not only from the statistics but also those arising from a 3% uncertainty in the Doppler width and a 5% uncertainty in the resolution. The uncertainties from the various causes are approximately equal. Even though the total widths of the 4.16 and 7.67 eV resonances are measured to an accuracy of only ~10%, the resonance absorption of these resonances can be computed to the accuracy of the neutron width which is 2 to 314 as will be discussed later. Table I also summarizes the results for the 21.1 eV resonance in 182w. In order to improve the accuracy of the total widths of these resonances one would have to improve the statistical accuracy of the experiment points, improve the resolution or reduce its uncertainty, and decrease the Doppler width or reduce its uncertainty. The first two factors can be accomplished by the use of a more intense pulsed neutron source. However, the third factor, the Doppler width, is not easy to reduce. The Doppler width can be reduced (about a factor of 2) by cooling the sample down to liquid nitrogen temperature. This technique has been used at the Saclay linac for transmission measurements et ..--- m upon thorium, and additional resonances were observed due to the decreased Doppler width. However, for shape analysis this technique is of doubtful value, for although the Doppler width is less, its absolute uncertainty is about the same or may even be larger since the Doppler broadening in solid and liquid Suid- samples is very complex. A detailed study of the effects of crystalline binding on the Doppler broed- ening of a neutron resonance has been made using the BNL crystal spectrometer, . 2 11 The shape of the 4.14 eV resonance has been measured with metal samples at temperatures from 4 to 825°K and with 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 4 shows a plot of the 825°K data and the computed fit to the data Table II which were summarizes the data reported 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, and the author warns that the errors have to be increased somewhat to take care of systematic uncertainties and that the accuracy of E 1s = 0.006 eV. The author prefers the parameters from the 825°K data because the Doppler effect is known more accurately at this temperature than at the lower temperatures. It would appear that the uncertainty of ™ is as small as t 1 mV and that of ľn is only $ 0.02 mv. It should be mentioned that there are special cases where the shape method of analysis is used even when 4 and/or R are »;!'. This is the case when one is looking for the interference between resonance and potential scattering to determine if a resonance is an s or a p-wave resonance. Thick samples are measured in transmission, and the energy region of interest is many electron volts away from the resonant enerwy. 0. Area Method of Analysis Then the Doppler width and/or the resolution are greater than the width of a resonance, the shape method is not practical; so the area total me thod must be used to obtain the parameters of the resonances. This area method is useful up to a few hundred electron volts until the Doppler width, 4 , or resolution, R, is om 10 times the total width, 1. For higher energies where 12 A or R is >> 101", on?.y the neutron width can be obtained from an area measurement. The area method can often give resonance parameters more accurately than the shape method even when A and R are < ľ. The transmission data for the 18.8 eV resonance of Figures 2 and 3, as well as data for thicker samples,can be used to illustrate the area method of analysis. In the area method, values of p are determined for each sample thickness for various assumed values of 1 to produce a transmission dip which has the same area as the area of the experimental transmission dip. For a thin sample Na < 1 (where are 18 the peak cross section and N is the sample thickness) the resulting value of rm is quite independent uit the kosuned values of ™ . 'mis is to be expected since the area under a resonance is proportional to do which is a nieasure of Man For thick samples Nos » 1, the area above a transmission dip is proportional to at / or proportional to Man. The result of a single sample thickness can be represented approximately by an equation of the form pam = k. For thin samples m 18 x 0.1, for thick samples m -- 0.9, and for intermediate samples m~ 1/2. Figure 5 shows the data obtained from the five sample thicknesses for the 18.8 eV resonance in 100w using the area analysis program of Atta and Harvey. The problem 18 to determine the best values for 1"and l' and their standard deviations which are consistent with the lines in Figure 5 which have uncertainties of 0.9 to 5%. One simple method is to determine values for m and k in the vicinity of M and "' from plots of log vs log for the 5 samples. The uncertainties due to sample thickness and Doppler broadening are small and are included iri is 18 1. i L; uncertainty in the resolution is not important in the area method. The values obtained are listed in Table III. A least squares program to solve the linear equations log "n + me log” = log ke 18 used to obtain values for log i ne 13 log " and the standard deviations of these quantities. Figure 6 shows a plot is the best value for tn, and the best value for ſ' is obtained from the slope. The parameters le ", 5 (n), aft") and the correlation coefficient between in and are summarized in Table IV. The fact that the correlation coefficient 18 - 0.84 means that ľn and ™ are almost completely anti-correlated. Table IV summarizes the results of the area analysis for the other low energy resonances. The correlation coefficients between 1 and for all Before leaving this area method, I would like to mention that data from other types of cross section measurements such as capture and scattering can also be included on Figures 5 and 6. Often they give a curve in Figure 5 for a given sample thickness with quite a different value of m than can be obtained from a transmission measurement. The combination of data from several types of measurements will be discussed in detail in the next paper. IV. Determination of !, from and his The radiation widths for low energy resonances can be readily determined from the equation: I'= ;. "n, where land 'n are determined from total cross section measurements. For resonances in non-zero spin target nuclides, It must be remembered that transmission measurements yield (8!') and not ."'.. Hence, I'm = "-18112 and g must be determined from other data such as scattering. Of course, if is 10!or g is ~ 1/2, the uncertainty of the correct value of produces 14 n. little additional uncertainty in ” . The standard deviation of " can be computed from the standard deviations of ſand and the correlation coefficient between 'n and . The standard deviation of any function f( I'm!!") 16 j(s) = 635)*,-1!) + Combo 24") + 2(3) Jouem 'in cel"> p)* Hence will's) = (272,-2(!!) + (-2,3% ,20 %) + 2(1)(-1)0*(!) }() CCC", "m)! If the correlation coefficient, col",") = +1 then «I! 0) = lu(1) - :(In)! or 1f CC(1, ) = -1 (!",) = 3 (!") + 3 (! or if co(I",) = 0 .:(!",) = (12(!') +,2(",) Since the shape method, at least when N si 2.1, results in positive co:relation coefficients, the standard deviations on 1. are somewhat less than if one ignores the correlation coefficient and uses the formula Yu?(!!) + v.217m). However, for the area method, since col "no 1) - - 1, the standard deviation on 1', is nearly the sum of the 2 standard deviations of i' and 'n Values computed for the 5 low energy resonances of tungsten are listed in Table I and Table IV. For the low energy resonances I'm can be obtained from the shape and area method with comparable accuracy. However, the area method is superior for the 27.1 eV resonance where both 15 and R are > ľ. It is obvious that when I'n 18 >>", 1t is difficult to determine !", accurately, and for these resonances it is desirable to have also capture cross section measurements. Capture measurements have been made upon this 18.8 eV resonance by Block et al. After correcting for multiple capture a value for m e to a 5% accuracy was obtained. Combining this capture data with the neutron width obtained from transmission data resulted in a value of ", of 53 - 3 mv (a factor of 2 more accurate than from the transmission data alone). For the two lowest energy resonances where i capture data would not, improve the accuracy of '. An area measurement of capture cross section data for a thin sample can also be plotted in the manner of Figure 5 where the exponent m 18 (!!!)/(1 - 2"n!). Hence, when 'n is 61 the data obtained from capture is the same as can be obtained from transmission. The capture data are limited to an absolute accuracy of - 576 where the trans- mission data may be 1 to 2%. However, when > 1, capture data are needed be discussed in detail in the next paper. V. Determination of Resonance Absorption from "' and 'n From the resonance parameters E.," and "' determined from trans- mission measurements alone, it is often possible to compute the contribution of low energy resonances to the resonance absorption integral to a better accuracy can be obtained than .from capture measurements. Sometimes the resonance absorption can be de termined to a better accuracy than the accuracies of and from which it is computed. If we consider the resonance absorption for an infinitely thin sample, the contribution of a particular resonance can be computed from one of the 16 formulas 606 x DIA I Do - 2.696 x 2096 -*3. 1 2.6062008(2... & 2.606 x 200 Although the value for the resonance absorption will be the same regardless of the formula used, the accuracy computed will be different unless correlations -.-..--- . between the parameters are known (and used correctly). We have seen that there .-.. . . .. is often quite a strong correlation between 1 and (particularly in the area analysis) and hence also between !', and ', 1, and I'm: 1; and 'ye etc. The resonant energy is known to sufficient accuracy that it does not contribute any error to the resonance absorption. Agoin using the formula that the sta:dard deviation of a function f( m ) is 14) - 14 .11.2015, 2016) + 2 ): 334):{"p" (n) wat je » S aidd using the form that the resonance absorption ) we get U (RA) RA (*.2439 261.2m ) (2m): ('') ccl 'no . -. -- For resonances with !!! I we can see that the uncertainty in the resonance -- ---- - -- - - - absorption is determined mainly by the first term, the relative uncertainty of i The second term is very small because of the factor (2/1) (for a reasonable value of :( :)/:!) and the third tern 18 small and is usually negative from area analysis. For example, using parameters obtained from area analysis for the 4.16 eV resonance in 182W (/" = 0.027, 10 = 0.021, minds the meantimitowa - 17 -()/' = 0.026, and col!.?) = -0.85) the relative uncertainty of the resonance absorption, P A, 1s 0.020 or 2.0%. The contribution of this resonance to the resonance absorption is 339 $ 7 barns ev. As the ratio increases the second and third terms can be lurger than the first term. For resonances with F/?' approaching 0.5 the second term predominates and i!-(RA)/RA%;"(1')/1. For example, using the parameters obtained from area analysis for the 21.1 eV resonance in Low (W/" = 0.394, :-(!)/n = 0.027, “(?)/" = 0.059 and co(!n.") = -0.95), the relative uncertainty of the resonance absorption, g'(RA)/RA, 1s 0.030 or 3.0%. The con- tribution of this resonance to the resonance absorption is 223 $ 7 barns ev. For the isotope 102w, the two low energy resonances contribute 562 = 10 barns ev to the infinitely dilute resonance integral. Higher energy resonances contribute only 30 barns eV and, hence, need not be measured to high accuracies. For resonances where :/.is 0.5 and approaching unity, the relative uncertainty of RA is considerable larger than those of and". For example, using the parameters obtained from area analysis for the 18.8 eV resonance in 186W ( 1. = 0.887, 0-(w)/ *0.012, 0:1)/1 = 0.018, and cclI.") = -0.84), the value of (RA)/RA 1s 0.22 or 22%. For a resonance with so much scattering, the negative correlation coefficient increases the uncertainty of RA. A more accurate value for the resonance absorption of this resonance can be obtained from the parameters obtained from the shape analysis where the correlation co- efficient 18 small or even positive. From the parameters obtained from shape analysis )-(RA)/RA = -10%. Since capture cross sections measurements can be made to an accuracy of rw 5%, it can produce a factor of 2 better accuracy for the resonance absorption of this resonance. Values of the resonance absorption from she,pe and area analyses are listed in Tables I and IV. 18 annes The relative uncertainty of RA is very simply related to the relative uncertainty of the product ro p 18 m has the value (1/1)/[1 - (271/)]. It can readily be shown that + 2m her 10.m arket (9) a.m] Hence, for n = (P/13/[ - (21/72] = m. .. . . m=m . - . .. (estelar un and ostorm O (RA RA mem Thus if one is concerned with an accurate determination of the resonance absorption for a resonance where M/M <<0.3, it is not necessary to determine the parameters Mme M and their standard deviations independently. It is only necessary to determine the relative uncertainty of r'n maand to know approximately the value of Tor. Hence, for resonances where ľm/'<< 0.3 a sample thickness can be selected so that m = (PM/">/ [1 - (21//!')) and the value for in - for this m (which is equal to cs-(k)/k) can be determined to high 'n accuracy. Other sample thicknesses are selected in order to obtain approximate values for " and l as was illustrated in Figure 6. If the sample thickness 1s not selected so that m 18 exactly equal to m, the least square fitting procedure of Figure 6 permits one to compute the relative uncertainty of rm for m = m , and it will not differ much from the measured 5'(x)/x if m is nearly equal to me. For low energy well-resolved resonances for which !!**0.3 this quantity can be measured to 1 or 2% accuracy and, hence, the contribution to the resonance absorption for such a resonance can be in determined to the same accuracy. Although I have considered only the problem of resonance absorption for an infinitely dilute sample, a similar treatment should apply for a thick sample where there is self absorption. For a thick sample one would need the quantity v'(k)/k for a value of m which is somewhat greater than the value of m selected for the infinitely dilute sample. One could expect to be able to compute the resonance absorption for thick samples to a few percent accuracy. This treatment should be applicable as long as the multiple capture is not too large (1.e., for resonances with 14:"). VI. Conclusions For low energy resonances < 10 eV for which 'n 22., transmission accuracy of a few percent. For higher energy resonances up to w 100 eV for which I'm <) in the region of the 21.1 and 18.8 eV resonances. Figure 4. Total neutron cross section of Wat 825°K in the region of the 4.14 eV resonance. Curve A 1s the theoretical shape, curve B includes Doppler broadening, and curve C includes both Doppler and resolution broadening. Figure 5. Plot of ľn vs assumed ľ' for the five sample thicknesses listed in Table III for the 18.8 eV resonance in low from area analysis. Figure 6. Least squares fit to obtain the parameters for the 18.8 eV resonance Fruin area analysis. A- . 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EV-Total BE3300 10 II III EM IKIT 11ttD EN-EV 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 1600 1620 1640 1660 1680 1700 1720 1740 1760 1780 1800 DU IIIIIII TITION Hilllll MINISillifil'' NililllllllllllllllllllHNINIANIHIIHIIII|||llilan IIIIIIIIIIIIIIIIIII|II||titl|llllllllllllllllllllllllll!!!!!!!IIIIllllll !1111111tii l ilIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIINNNI IllIIIHIMUONTINUUMIIritillllllllllllllllllllllllii"nin olu l i1: 1 '11111MINIDIMINUTDU0 0 nttiilWMMTI1N111111111111111lltilin 0111 owo no 0 0 8 UUTUU tp : '... HU Ibu..' Illllll 2 I IIIIIIIIIIII Oy -BARNS U UUUUUUUU IT UNTU11 MINIM IITTAM MILINIB 01 bi1!!! !!llim lili ll tillliiii. Il null||lllIllIIlIIIIIIIIIIIIIIIIIIHIRHIMINI IL MOIHD U MIHIIIlllllllllll' TITHIN II||||1||| (Silnic !! "INITIAT 111111ITMITTITI NIIHIIIIIIIIIIIIIIIIIIIIIIIlIllimin InfilthANTAR 1 . . illiljlillliiii . UNI Hullu bo-As . 00012 :1Iliu PIN 13-N3 1400 1180 1200 1220 1240 1260 1200 1300 1320 1340 1360 1380 1140 (160 1120 1000 1020 1040 1060 1080 1100 1 mini DUREN ARAMIDA MASALAHTNINNITTINEN Lililin, thliliitrit hairul MIMONTINU WINNI Unliminimini NIIIIIIIMMMWMWINN niiNininitin tinniinIINIIN Intimin ImWm3 HIINIME DIMINUT KUMMITUT DIPINTIMIDI Wmtiminimin. WIWUMINIMUI G ILI O, BARNS Whiwulkartu n MRMwmnnullINTII AU TITANIUM WINNIWINUN Ininth in INITWIN INIMA wwmumuumilinn INNImi innnnnnn innim h ommUHNITI . 11 OOC 11 : ESTOI EV-Total 1.41X 1010 y 90 In Th 232 ORNL-DWG 65-43024 ORNL-DWG 65-13047 486W RESONANCE 1.99 x 10-5 atoms of W/ barn Eo = 18.84 1 0.04 eV r=0,385 1 0.008 eV 1,-0.319 1 0.003 eV MIL WH JUDWIDE 11 "MIOIMITIL ID: " MULTILU JINNI W TOLDI Ep IL WIKIDO IL UUUUU CULOLUIII. IUNI WILD 11 JLDU T169 1IS TIR TUIUIU MIUI 2 IM UIO LORIA ] MIT u.MINUIUIII YTUIT 11 O 1110 LILLIITTITIO I 10 ! UUUU IIIITILITIUITIUUIIIIIIIIIIIII (ILIIIIIIIIIIIIII ( ITIINIUDUDITT UITLUIDIIDIIDIINNITTITUIT IIIIIIIIIIIIIIINNOITTUUDUTOIMITIMIT NATUUTDOUTDAT OLOIIIIIIIIIIIITTTTT UITIIIIIIIIIIIIITT LIIIIIIIIIIIIITTIILITLIITUTIIL IIIIIITITITIT LIIIIIIIIIII JUDITUTIITITITO IIIIITTO LIIIIIIIIDID TITIONIN TIIIIIIIIITTITUD ITUIINID TIITTITITITUTT IIIIIOTTI IIIIIIIIIIIIIIIITTTTT IT IIIIIIIIIIIIIIIIIIIIIIIIII NTTIIIII CILIIIIIIIIIIIIIIIIDIITI,J. UTTINDIUIIIIIIIIIDI III IIIIIIIIIIIIIIIIIIIITTU DINI DICITU ILLINOILUUNIIN ODONTITION ULUULUTOIMIIT UNOT QILINU LUI DOI ILLULITI ITITUTI DIMWILI TIIUL TRANSMISSION HILUUUUL III OLUN INCLUD TIIULITUDIITTIT ILOILUUNIIDIIDII UUTIIIIIIIIIIII UIIIIIIIIIIIIIIDU LIULUIIIIIIIIIIIIII NIINIMITTIIII IIIIIIIIIIIIII IIIIIIITITUITUD IDUNIIDIIDII UITINIDOTTITOLI DIIDIIÍIIULIDUTI IIIIIIIIIIIIIIIIIIIIIIIII TII D IDUNTIINTITUL IIIIIIIIIIIIIIIIIIIIIIIIII TIITTIOITTUTTIIND DIUIN IIIIIIIIIIIIIIIIIIIIIIIIIIIT TTTTTTTTTIINTIL TIITTITOUDUTTITOITTIINT IIULIDULIITTUUNNIT LIIIIIIIIIIIIIIIIILTTILIINIT OIIIIIIIIIIIIIIILID INDI TTTTTTTTTTTTTTTT TTTIIIIIIIIIION In QUI NULIIIIIIIIIIIIIIIIIITTTT NIITTITUUT 100 MILLID IUUIITTTTTT ILLLUL IL T UDI TILL UOTIDIO III IINNUIONIIUOTINLIIMIIIIIIIIIIIIIIIAIIOINNININDINIUI DIULI TTTTTTIIIIIIIIIIIIII UUUUUUUUUUUU UULILULUIIIIIIDIITUDITUOLIDUIDIUINUUTIIL TOUJINONTTTTIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII LLLLLLULULLLILUUUTTTIIIIIIULIDIIIIIITINIIIIII TNTTITAITOTTTT OOTUDIITUIT LLLLLLLLLLS UTIIIIIIIIIIITUMIINI CINTILIIIIIIIIIIIIIINNOIUDICIUUT. DITIIIIIIIIIIIIIIIIIIDIIOONIDINULUI LIUDDIN WUDIJU TUNUIIDIDUNTIITTI TOTUOTTTTTTIILOTTI OTINUINIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII ILU DIIDIIDIIIIIIIIII IIIIIIIIIIIIIIIIIIIIMIIL NDODUINULITID 480 500 520 540 560 580 Figure 2 482W Ep=21,10 + 0.04 eV =0.100 +0.010 eV T,=0.040 + 0.002 eV Lara ORNL-DWG 65–43018 486w 5.79x10-5 atoms of w/ barn Eo= 48.84 +0.04 eV T= 0.368 + 0.005 eV T,= 0.318 + 0.003 eV UT X S 90: TRANSMISSION WOO U Hinta NINKY 430 450 470 490 510 530 CHANNEL NUMBER 550 570 ORNL-DWG 65-13020 5000 TUNGSTEN 182 (METAL) Ex=4.1422 eV TEMPERATURE 825 % 4000 * ಇರ್ತಿತ CROSS SECTION - BARNS 1000 o 4.000 4.050 4.100 4.50 4.200 4.250 4.300 NEUTRON ENERGY-LV ORNL-DWG 64-1844 340 mil 18.8 ev 5 mils 12 mils 320 0.1 mil- 310 300 290 320 340 360 380 400 Figure 5 ORNL-DWG 65-13019 186W 18.83 eV 2 k=r, rm r=361 7 mV "(FROM SLOPE) .... . .. ........ -T, = 320 4 mV - (AT m=0) - - - - - - 0 0.2 0.4 0.6 0.8 1.0 Migure 6 U I . S ...: END ***** * 13 . - DATE FILMED 2 / 21 /66 1 T