Hafnium Resonance Parameter Analysis Using Neutron Capture and Transmission Experiments M. J. Trbovich,*† D. P. Barry,† R. E. Slovacek,Y. Danon, R. C. Block, N. C. Francis, and M. Lubert Rensselaer Polytechnic Institute, Mechanical, Aerospace and Nuclear Engineering Department Troy, New York 12180-3590 and J.A. Burke, N. J. Drindak, G. Leinweber, and R. Ballad Knolls Atomic Power Laboratory, P.O. Box 1072, Schenectady, New York 12301-1072 Received July 23, 2007 Accepted June 27, 2008 Abstract – The focus of this work is to determine the resonance parameters for stable hafnium isotopes in the 0.005- to 200-eV region, with special emphasis on the overlapping 176 Hf and 178 Hf resonances near 8 eV. Accurate hafnium cross sections and resonance parameters are needed in order to quantify the effects of hafnium found in zirconium, a metal commonly used in reactors. The accuracy of the cross sections and the corresponding resonance parameters used in current nuclear analysis tools are rapidly becoming the limiting factor in reducing the overall uncertainty on reactor physics calculations. Experiments measuring neutron capture and transmission are routinely performed at the Rensselaer Polytechnic Institute LINAC using the time-of-flight technique. Lithium-6 glass scintillation detectors were used for transmission experiments at flight path lengths of 15 and 25 m, respectively. Capture experiments were performed using a 16-section NaI multiplicity detector at a flight path length of 25 m. These experi- ments utilized several thicknesses of metallic and isotope-enriched liquid Hf samples. The liquid Hf samples were designed to provide information on the 176 Hf and 178 Hf contributions to the 8-eV doublet without saturation. Data analyses were performed using the R-matrix Bayesian code SAMMY. A combined capture and transmission data analysis yielded resonance parameters for all hafnium isotopes from 0.005 to 200 eV. Additionally, resonance integrals were calculated, along with errors for each hafnium isotope, using the NJOY and INTER codes. The isotopic resonance integrals calculated were significantly different from previous values. The 176 Hf resonance integral, based on this work, is ;73% higher than the ENDF/B-VI value. This is due primarily to the changes to resonance parameters in the 8-eV resonance; the neutron width presented in this work is more than twice that of the previous value. The calculated elemental hafnium resonance integral, however, changed very little. I. INTRODUCTION The majority of measurements and analyses of haf- nium cross sections in the region below 200 eV were performed prior to 1965. There were a few measure- ments by Liou et al. 1 and Moxon et al. 2 made in the mid-1970s. However, most of the ENDF 0 B-VI reso- nance parameters for hafnium in this region are based on much older experiments. These older experiments pro- vided lower-resolution data and, because of the tight level spacing of hafnium, led to many missed resonances. An example of this is best shown in the case of the reso- nance pair near 8 eV. A very strong resonance ~;25 kb! near 8 eV was attributed solely to 178 Hf up until 1974, when measure- ments by Moxon et al. 2 showed the existence of a 176 Hf resonance at nearly the same energy. Although this new resonance made no significant impact on the total neutron *E-mail: [email protected]†Current address: Knolls Atomic Power Laboratory, P.O. Box 1072, Schenectady, New York 12301-1072 NUCLEAR SCIENCE AND ENGINEERING: 161, 303–320 ~2009! 303
Transcript
Hafnium Resonance Parameter Analysis Using Neutron Captureand Transmission Experiments
M. J. Trbovich,*† D. P. Barry,† R. E. Slovacek, Y. Danon, R. C. Block,N. C. Francis, and M. Lubert
Rensselaer Polytechnic Institute, Mechanical, Aerospace and Nuclear Engineering DepartmentTroy, New York 12180-3590
and
J. A. Burke, N. J. Drindak, G. Leinweber, and R. Ballad
Knolls Atomic Power Laboratory, P.O. Box 1072, Schenectady, New York 12301-1072
Received July 23, 2007Accepted June 27, 2008
Abstract – The focus of this work is to determine the resonance parameters for stable hafnium isotopes inthe 0.005- to 200-eV region, with special emphasis on the overlapping 176Hf and 178Hf resonances near8 eV. Accurate hafnium cross sections and resonance parameters are needed in order to quantify theeffects of hafnium found in zirconium, a metal commonly used in reactors. The accuracy of the crosssections and the corresponding resonance parameters used in current nuclear analysis tools are rapidlybecoming the limiting factor in reducing the overall uncertainty on reactor physics calculations.
Experiments measuring neutron capture and transmission are routinely performed at the RensselaerPolytechnic Institute LINAC using the time-of-flight technique. Lithium-6 glass scintillation detectors wereused for transmission experiments at flight path lengths of 15 and 25 m, respectively. Capture experimentswere performed using a 16-section NaI multiplicity detector at a flight path length of 25 m. These experi-ments utilized several thicknesses of metallic and isotope-enriched liquid Hf samples. The liquid Hf sampleswere designed to provide information on the 176Hf and 178Hf contributions to the 8-eVdoublet without saturation.
Data analyses were performed using the R-matrix Bayesian code SAMMY. A combined capture andtransmission data analysis yielded resonance parameters for all hafnium isotopes from 0.005 to 200 eV.Additionally, resonance integrals were calculated, along with errors for each hafnium isotope, using theNJOY and INTER codes. The isotopic resonance integrals calculated were significantly different fromprevious values. The 176Hf resonance integral, based on this work, is ;73% higher than the ENDF/B-VIvalue. This is due primarily to the changes to resonance parameters in the 8-eV resonance; the neutronwidth presented in this work is more than twice that of the previous value. The calculated elementalhafnium resonance integral, however, changed very little.
I. INTRODUCTION
The majority of measurements and analyses of haf-nium cross sections in the region below 200 eV wereperformed prior to 1965. There were a few measure-ments by Liou et al.1 and Moxon et al.2 made in themid-1970s. However, most of the ENDF0B-VI reso-
nance parameters for hafnium in this region are based onmuch older experiments. These older experiments pro-vided lower-resolution data and, because of the tight levelspacing of hafnium, led to many missed resonances. Anexample of this is best shown in the case of the reso-nance pair near 8 eV.
A very strong resonance ~;25 kb! near 8 eV wasattributed solely to 178Hf up until 1974, when measure-ments by Moxon et al.2 showed the existence of a 176Hfresonance at nearly the same energy. Although this newresonance made no significant impact on the total neutron
*E-mail: [email protected]†Current address: Knolls Atomic Power Laboratory, P.O.
Box 1072, Schenectady, New York 12301-1072
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cross section for natural hafnium, it did affect the waythe hafnium interactions would change with exposure toa neutron flux. This is one example of the importance ofaccurate resonance parameters for analysis of nuclearsystems.
The work described in this paper was completed atthe Rensselaer Polytechnic Institute ~RPI! LINAC facil-ity and is described more thoroughly in the doctoral the-sis3 found on file at the RPI library.
II. EXPERIMENTAL SETUP
Transmission experiments at the RPI LINAC areperformed in two primary configurations, referred to as“thermal” and “epithermal.” Thermal transmission ex-periments are optimized for low energies ~0.001 to 20eV! and utilize an ;15-m flight path arrangement. Thisshort flight path provides for a higher intensity of neu-trons. The detector used at the;15-m station is a 5.08-cm~2-in.!-diam and 3-mm-thick 6Li loaded glass scintilla-tor that is optically coupled to a photomultiplier tube~PMT!. A more detailed description of the completeexperimental setup at the RPI LINAC can be found inRef. 4. The samples for the thermal transmission exper-iments are mounted on a sample changer that is located;14 m from the neutron production target. The neutronproduction target used for thermal transmission experi-ments is the enhanced thermal target.5 This target isdesigned to provide the low-energy neutrons needed forthese experiments.
Epithermal transmission experiments are done witha detector ;25 m away from the neutron productiontarget. This detector is a 12.7-cm ~5-in.!-diam and 1.27-cm~0.5-in.!-thick 6Li loaded glass scintillator that is opti-cally coupled to a PMT. A rotary sample changer is lo-cated ;14 m away from the neutron production target.
The target used to produce neutrons for the epithermaltransmission measurements is known as the bounce tar-get6 and more recently as the bare bounce target.7 Thesetargets are designed for neutron measurements from afew electron volts up to ;1 keV.
Capture experiments at the RPI LINAC are performedusing a 16-segment NaI~Tl! multiplicity-type detector.8
This detector is located;25 m from the neutron produc-tion target. Samples are inserted into the center of the de-tector and held in place by hollow aluminum tubes. Eightsamples can be mounted on a wheel that translates androtates in order to change samples. The detector is insidea 15.24-cm ~6-in.!-thick lead shield with through-holes forthe neutron beam and sample insertion and extraction.
III. HAFNIUM SAMPLES
The majority of resonances in hafnium below 200 eVwere measured using various thicknesses of metallic nat-ural hafnium. The metal samples were all of natural iso-topic abundance and were in the form of disks with;5.08-cm ~2-in.! diameters. The sample thicknesses ofthe metallic hafnium used are given in Table I along withthe experiments they were used in. This variation in sam-ple thickness enabled the analysis of widely ranging crosssections for the many resonances in hafnium below 200 eV.
The metallic samples did not allow for accurate analy-sis of the pair of resonances near 8 eV. The total crosssection due to the 8-eV resonance pair is predicted to beas high as 30 000 b. This extremely large cross section at8 eV ensures that this resonance is saturated in all but thetwo thinnest metallic samples.
Liquid samples were used in experiments specifi-cally designed for analysis of the 8-eV pair of overlap-ping 176Hf and 178Hf resonances. The large cross sectionof this resonance pair ~;30 000 b! required an extremely
TABLE I
Metallic Hafnium Sample Specifications*
Thickness Experiment~s! Used In
Nominal~cm!
Nominal~mil!a
N~atom0b!
ThermalTransmission
ThermalCapture
EpithermalTransmission
EpithermalCapture
0.00127 0.5 4.621 � 10�5 X0.00254 1 9.984 � 10�5 X0.00508 2 2.369 � 10�4 X X X0.01016 4 4.537 � 10�4 X X0.0254 10 1.139 � 10�3 X X X X0.0508 20 2.303 � 10�3 X X0.127 50 5.755 � 10�3 X X X0.254 100 1.154 � 10�2 X
a1 mil � 0.001 in.
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thin sample for adequate experimental capture and trans-mission data. Metal foils could not feasibly be fabricatedthin enough with adequate quality for these experiments.The liquid samples provide a more uniform thicknessthan metal, and the thickness can be controlled by thehafnium concentration in the solution.
The liquid samples were created using both naturalhafnium oxide and hafnium oxides enriched in 178Hf and176Hf. The isotopic content of the enriched oxides wasmeasured using a mass spectrometer. Table II shows theresults of the mass spectrometer analysis of the enrichedhafnium samples in terms of isotopic percentages.
Dissolving the hafnium into a liquid solution wasthought to be a superior alternative to solid oxide sam-ples. The solution provides a uniform distribution of haf-nium as long as the solution is not near its saturationpoint. The solvent also has to have a low and constantcross section. It was determined that hafnium could bedissolved into deuterated nitric acid ~DNO3!. The DNO3provided a low, flat cross section in the energy range ofinterest for these experiments.
Liquid sample hafnium concentrations were basedon hafnium densities that would produce transmissionvalues sufficiently above background and below satura-tion levels to allow for accurate measurements. The firstset of liquid samples was called “generation I.” Table IIIshows the properties of this generation I set of liquid
samples, indexed by a unique serial number on each cell.The generation I samples were contained in cells madefrom two;5.08-cm ~2-in.!-diam, 0.159-cm ~0.0625-in.!-thick quartz flats with a polyvinyl chloride ~PVC! spacerring glued between them with acid-resistant epoxy.
The concentration for each generation I liquid sam-ple is shown in Table III. The concentration of each so-lution was measured by inductively coupled plasmaemission spectroscopy.
After the experiments with the generation I liquidcells, two of the cells were found to have leaks. Theleaks were not significant enough to affect the experi-mental results; however, this prompted a new cell designfor the next set of experiments. The new cell design re-placed the PVC spacer ring with a quartz ring that wasfused in place. This design eliminated the glue joint inthe previous generation, which seemed to be the sourceof leaks. The new cells were referred to as “generationII” cells and were of the same nominal dimensions as thegeneration I cells. The properties of the generation IIliquid samples are given in Table IV.
IV. DATA ANALYSIS
The first hafnium data set analyzed was the epither-mal metallic transmission data. The ENDF0B-VI
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resonance parameters were used as starting parametersfor the SAMMY fit9 to the data. A combined fit wasperformed on the 0.0254-cm ~10-mil!, 0.0508-cm ~20-mil!, 0.127-cm ~50-mil!, and 0.254-cm ~100-mil! metal-lic sample data. This was done by fitting each data setsequentially and using the SAMMY parameter file alongwith the SAMMY covariance matrix file created by theprevious fit as input to the next. After a fit to the trans-mission data sets with a minimum xr
2 ~reduced chi-squared3 ! was achieved, epithermal metallic capture datasets were added to the combined analysis. The0.00127-cm ~0.5-mil!, 0.00508-cm ~2-mil!, 0.0254-cm~10-mil!, and 0.127-cm ~50-mil! sample capture datawere added to the fitting sequence of epithermal trans-mission data sets to form a combined fit. This com-bined fit sequence was run until a minimum xr
2 wasachieved.
Not all of the resonance parameters were allowedto vary during the SAMMY fit. If a radiation widthparameter Gg did not affect the overall fit and was al-lowed to vary, it was found that the value of this Ggparameter could “run away.” This meant the radiationwidth value would continuously increase or decreasewith each run of SAMMY without converging. In suchcases, the Gg parameter was fixed to an average value ofthe radiation widths for that particular isotope. Thisaverage value of Gg was found by averaging the “sensi-tive” Gg values that were fit by SAMMY within eachisotope. Barry10 developed the criteria used to deter-mine which Gg values should be varied. The methodwas based on the ratio of the radiation width to theneutron width Gg 0Gn and was used to give a sensitivityfactor for Gg . The cases where the ratio was �5 gavegood indication the fit would be sensitive to Gg . Thissolved the problem for most of the insensitive Gg valuesthat were running away. However, there were still afew Gg values that were deemed sensitive based on theBarry10 criteria that did not converge. These cases werefound in resonances that overlapped a neighboring res-onance ~or several resonances!. In some of these casesone Gg would constantly change while being compen-
sated for by a nearby resonance’s Gg changing in theopposite direction. Through trial-and-error SAMMY runs,these parameters were also fixed to an average value ofGg based on the Gg values that did converge for thatparticular isotope.
The average Gg for each isotope was calculated froma weighted average of the converged Gg values for thatisotope. All of the insensitive Gg values were fixed to thisaverage value for that isotope, and the fit was repeated inan iterative fashion. This iteration continued until thecalculated average Gg for each isotope agreed with theprevious iteration’s average Gg .
When no further improvements in the fit were ap-parent and the resonance parameters remained unchangedrelative to the previous iteration, the parameters weredeemed final. SAMMY was then used to calculate trans-mission and capture curves based on these final reso-nance parameters to compare with the experimental datafrom each sample.
After the resonance parameters were determined forthe epithermal region ~10 to 200 eV!, the thermal datawere then analyzed using SAMMY. The thermal trans-mission data set was analyzed first, and the capture datawere added to the analysis once reasonable transmis-sion fits were achieved. The combined analysis of the0.00254-cm ~1-mil!, 0.00508-cm ~2-mil!, 0.01016-cm~4-mil!, 0.0254-cm ~10-mil!, 0.0508-cm ~20-mil!, and0.127-cm ~50-mil! thermal transmission samples and the0.00508-cm ~2-mil!, 0.01016-cm ~4-mil!, and 0.0254-cm~10-mil! thermal capture samples were run in SAMMYuntil a minimum xr
2 was achieved and there were nosignificant changes in parameters between runs.
The insensitivity of the low-energy resonances tothe energy resolution and the Doppler broadening effectallows accurate simultaneous determination of all reso-nance parameters below 10 eV.
The resonance doublet at 8 eV was not analyzedusing the metallic sample data, as it was saturated orclose to saturation and provided little information. Thethinner isotope-enriched liquid sample data were used todetermine the parameters for these resonances.
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A sample of the transmission and capture fitted curvesand data is shown in Fig. 1. The 0.0254-cm ~10-mil!metallic capture sample shown in Fig. 1 shows the effectof a strong resonance with approximately equal scatter-ing and capture probabilities in a relatively thick sample.This effect produces a depression at the resonance en-ergy that is due to the high scattering cross section at theresonance energy, which scatters a large portion of neu-trons away from the sample before penetrating the sur-face. Neutrons that have energies slightly above or belowthe resonance energy have a much higher probability ofpenetrating the sample but will most likely interact in-side the sample. The neutrons that scatter once insidethe sample will most likely be captured before leavingthe sample, creating increased counts in the wings of theresonance. Table V contains the final resonance param-eters determined from this analysis, which all of thesefits are based on.
The resonance parameters for the 176Hf and 178Hf res-onances near 8 eV were determined from the isotope-enriched liquid sample data. Two generations of thesesamples were run in both capture and transmission exper-iments, as described previously. The transmission exper-iment data using the generation I and II liquid samples wereanalyzed first using SAMMY. Once the transmission data
were fitted, the capture data were added to the analysis.The combined transmission and capture analysis showedsignificant differences between the data sets. The yield val-ues from the capture data did not agree with the transmis-sion data over the energy range being analyzed. Becauseof this, SAMMY was initially unable to determine a set ofparameters that fit all of these data sets.
In order to determine if there was a problem with theflux normalization of the capture data or if there was adifference in detection efficiency between isotopes, the8-eV resonance parameters fitted to the transmission datawere used to calculate the expected yield for the captureexperiments using SAMMY. The energy region beinganalyzed was increased to include surrounding 177Hf res-onances. The parameters for these resonances had beendetermined from the previously described analyses usingnatural metallic samples. The yield curve, calculated fromthe transmission data, was then applied to the capturedata for comparison; Fig. 2 shows this comparison for a176Hf-enriched liquid sample. Figure 2 shows that theregion over the 8-eV doublet is the only area of signifi-cant disagreement. The surrounding 177Hf resonances inFig. 2 show good agreement between the yield data andthe calculated yield from the transmission fitted reso-nance parameters. The lower yield data over the 8-eV
Fig. 1. Thermal metallic hafnium transmission and capture data with SAMMY calculated transmission from fitted resonanceparameters.
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TABLE V
Fitted RPI Resonance Parameters Compared to ENDF0B-VI and Those Measured by Moxon et al.2*
Energy ~eV! Gg ~meV! Gn ~meV!
RPI ENDF0B-VI Moxon et al.2 RPI ENDF0B-VI Moxon et al.2 RPI ENDF0B-VI Moxon et al.2Isotope
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doublet was an indication of a lower detection efficiencyfor 176Hf and 178Hf neutron capture gamma rays relativeto those from neutron capture in 177Hf, which was usedfor flux normalization ~at the 1.1-eV resonance!.
The multiplicity distribution for these isotopes wasexamined to look for significant differences in averagemultiplicity. Differences in average multiplicity couldindicate differences in detection efficiency, due to differ-ences in the number of gamma rays emitted or in theenergy of the gamma rays. Figure 3 shows a plot of thefraction of total counts versus multiplicity number.The counts for each unit of multiplicity were summedover a resonance for each of the isotopes 176Hf, 177Hf,and 178Hf. The 8-eV resonance in the 178Hf-enriched liq-uid sample data was used to obtain the multiplicitydistribution for 178Hf, and the 48-eV resonance in the176Hf-enriched liquid sample data was used for 176Hf.The 1-eV resonance was used to get the 177Hf multiplic-ity distribution. This plot shows that 177Hf has a higheraverage multiplicity ~4.2! than 176Hf and 178Hf ~3.8!.This means that on average a neutron capture in 177Hfproduces;11% more gamma rays than in 176Hf or 178Hf.This higher number of capture gamma rays should in-crease the chance for detecting a capture event in 177Hf
relative to 176Hf or 178Hf. The binding energy for eachisotope can also have an effect on detection efficiencyby determining the total energy emitted by the capturegamma rays. Table VI shows the binding energy for 176Hf,177Hf, and 178Hf along with the average multiplicity atselected resonances near 8 eV. The higher binding en-ergy of 177Hf, along with the higher average multiplicity,are expected to increase the probability of detection dueto more energy and gamma rays being released on aver-age for each capture event. This effect would cause thedetection efficiency for 177Hf to be relatively larger, andthus, a lower yield would be observed for 176Hf and178Hf resonances relative to 177Hf resonances. This trendis also in agreement with detection efficiency calcula-tions based on capture gamma-ray cascades in hafniumdone using the DICEBOX code.11,12
SAMMY was used to fit a normalization factor thatwould correct for the difference in detection efficiency.This was accomplished by using the resonance param-eters fitted to the liquid transmission data as input toSAMMY. A combined fit of all capture data sets wasthen run allowing only normalization to vary. This analy-sis determined there was a 24% difference between theyield data and the SAMMY calculated yield from the
TABLE V ~Continued!
Energy ~eV! Gg ~meV! Gn ~meV!
RPI ENDF0B-VI Moxon et al.2 RPI ENDF0B-VI Moxon et al.2 RPI ENDF0B-VI Moxon et al.2Isotope
*RPI errors calculated from SAMMY are in ~ !, errors propagated from resolution function uncertainties are in @ #, and the standard deviation isshown in $ % where the average Gg was used, errors from Moxon et al.2 are in 6 6.
aThese resonances in 174Hf were fixed to the ENDF0B-VI values because of their very low cross-section values and overlapping neighboringresonances, which precluded them from analysis in this work.
bThese resonances were fitted using a narrow energy range and manually changing values. They were then not allowed to vary during the fit overthe full energy range as the values would run away due to the number of overlapping resonances. Therefore, the error due to resolution functionuncertainties was not able to be determined.
cOnly one resonance in 178Hf was found to be sensitive to Gg; therefore, this value was applied to the other two resonances.
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transmission fitted parameters. The 176Hf and 178Hfdetection efficiencies were comparable because of theirsimilar average multiplicities and binding energies. Thisnormalization factor was then used to correct the yielddata.
A combined transmission and capture data analysiswas then performed using the corrected capture data.This analysis included both first- and second-generationliquid sample data from capture and transmission exper-iments. Figure 4 shows a plot of 176Hf-enriched liquidsample capture data with calculated curves based on both
ENDF0B-VI.1 resonance parameters and those deter-mined from this analysis. The fit to the 176Hf-enrichedsamples is not as good as that to the 178Hf-enriched sam-ples. Figure 4 shows the fit slightly underpredicting theyield of the thickest sample ~Hf-1-1! and overpredictingthe two thinner samples ~Hf-2-4 and Hf-2-5!. These in-consistencies may be due to inaccuracies in the solutioncontents of the 176Hf-enriched liquid samples. The fitsfor these samples are still acceptable and are a significantimprovement over the yields calculated from the ENDF0B-VI.1 values, as shown in Fig. 4. Figure 5 shows thetransmission results for the 176Hf-enriched samples, whichshow good agreement between experiment and calcu-lated values. Figure 6 shows the capture results for 178Hf-enriched samples compared to ENDF0B-VI.1 values.Figure 6 also shows significantly better agreement be-tween experiment and calculated yields for the 178Hfparameters derived in this analysis as compared to thosebased on ENDF0B-VI.1 parameters. Figure 7 shows the
Fig. 2. 176Hf-enriched liquid capture data compared tocalculated yield based on 8-eV resonance parameters fitted totransmission data showing the normalization problem.
Fig. 3. Multiplicity distribution for 176Hf, 177Hf, and 178Hf.
TABLE VI
Binding Energy and Average Multiplicityfor 176Hf, 177Hf, and 178Hf*
Fig. 4. 176Hf-enriched liquid capture samples withSAMMY calculated yield from fitted resonance parametersand calculated yield based on ENDF0B-VI.1 parameters.
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transmission results for the 178Hf-enriched samples, whichalso show good agreement between experiment and cal-culated transmission values, based on resonance param-eters determined in this analysis.
V. RESULTS
Resonance parameters determined from the previ-ously described analyses are presented in Table V, shown
as the RPI resonance parameters. Two error values arereported for the RPI resonance parameters. The errorvalue determined by SAMMY is shown in parenthesesand is based primarily on the statistical accuracy of theexperimental data used in the fit. The error value shownin square brackets is an estimate of the error in the res-onance parameters due to uncertainties in the resolutionfunction. Reference 3 contains a detailed description ofthe methods used to calculate the uncertainties shown inTable V.
Table V also shows the ENDF0B-VI parameters,which were used as starting values for the SAMMY analy-sis, and the parameters reported from Moxon et al.2 TheGg values with errors in curly brackets $ % are those res-onances that were deemed insensitive to changes in Ggand were set to the average Gg value for that isotope. Theerror quoted for these average values of Gg is 1s.
The resonance parameters determined for the 176Hfand 178Hf resonances near 8 eV are significantly differ-ent from the few previous measurements available. Thebiggest change is in the Gn value in the 176Hf resonanceat 7.8891 eV. The value quoted by Moxon et al. of 4.71meV is approximately one-third the value determined inthis analysis of 10.15 meV. The value quoted by Moxonet al., which is also the ENDF0B-VI value, is quotedwith an extremely high error ~.100%!, and it is there-fore not surprising to see a large change in this param-eter. This analysis provides a Gn value with a significantlylower uncertainty than was previously available. Asrecommended by Moxon et al., this analysis has led tothe same conclusion that a more highly enriched 176Hfsample would allow for an even more accurate set ofresonance parameters to be determined for this resonance.
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The thermal cross sections based on the fitted RPIparameters and the ENDF0B-VI parameters are shown inTable VII. As expected, the RPI value is within the quotederror of the ENDF0B-VI value. This is due to good agree-ment between the ENDF0B-VI and fitted RPI resonanceparameters at low energies. The majority of previoushafnium measurements was done in the thermal energyregion, making the lower-energy hafnium resonance pa-rameters quite reliable, with the exception of the tworesonances at 8 eV.
V.A. Resonance Integrals
Resonance integrals for each of the hafnium iso-topes analyzed were calculated along with errors. Theresonance integrals ~shown in Table VIII! were calcu-lated based on the resonance parameters determined inthis analysis. ENDF0B-VI resonance parameters were usedoutside the energy range analyzed in this work for theresonance integral calculations. NJOY ~Ref. 13! and IN-TER ~Ref. 14! were used to calculate the resonance in-tegral for the hafnium isotopes. Table VIII shows thecalculated resonance integral for each of the hafniumisotopes analyzed compared to those based on other eval-uated hafnium resonance parameters. As is shown inTable VIII, significant changes in some of the hafniumisotopic resonance integrals were calculated based onthe resonance parameters determined in this analysis.The elemental hafnium resonance integral calculated from
the abundance weighted sum of the isotopic resonanceintegrals also differs from resonance integrals calculatedfrom other data sets. Table IX shows the resonance inte-grals calculated from ENDF0B-VI and RPI resonanceparameters with an integration region from 0.5 to 200eV. This was calculated to show the energy region thatincludes only resonances that were analyzed in this work.The errors calculated are based on resonance errors listedin Table V.
VI. CONCLUSIONS
This paper presents the results of both capture andtransmission experiments using various hafnium sam-ples. These experiments provided energy-dependent trans-mission and yield data that were analyzed using theR-matrix Bayesian fitting code SAMMY. The transmis-sion experiments were done utilizing a 6Li glass scintil-lation detector at an ;15-m flight path for low-energymeasurements ~0.005 to 10 eV! and a similar detector at;25 m for higher-energy measurements ~10 to 200 eV!.A 16-section NaI~Tl! multiplicity detector was used forthe capture experiments at a flight path of ;25 m.
The samples used in these experiments were variousthicknesses of metallic hafnium and deuterated nitric acidsolutions of isotope-enriched hafnium. The isotope-enriched samples were designed to provide experimentaldata that could be used to determine resonance param-eters for the overlapping resonances of 176Hf and 178Hfat ;8 eV. The liquid solution samples were needed toprovide sufficiently thin samples to prevent saturation orblacking out of the 8-eV resonance pair in capture andtransmission experiments. Enriched samples were usedto determine the individual contribution of 176Hf and178Hf to the resonance pair. The only previously found176Hf parameters for the 8-eV resonance were from mea-surements done by Moxon et al.,2 which have an ex-tremely high quoted error and are referred to as “not wellknown” in the report. This analysis provides a muchmore accurate set of resonance parameters for this 8-eV
TABLE VII
Hafnium Thermal Cross Section Based on ENDF0B-VIand RPI Resonance Parameters
Thermal Cross Section ~st at 0.0253 eV!
ENDF0B-VI 114.5 bRPI 115.36 0.8 b
TABLE VIII
Resonance Integrals Calculated from Resonance Parameters Determined in This Analysis ~Labeled RPI!Compared with Those from Other Evaluated Hafnium Resonance Parameters*
*All were integrated from 0.5 � 105 to 1.0 � 105 eV.
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doublet. A combined analysis of capture and transmis-sion data using SAMMY was performed to determineresonance parameters for all stable isotopes of hafniumin the energy range of 0.005 to 200 eV.
Resonance integrals for each hafnium isotope, basedon the fitted resonance parameters, were calculated usingthe NJOY and INTER codes. A method to estimate theerror on the resonance integral due to the error on reso-nance parameters is also presented. The 176Hf resonanceintegral, based on this work, is ;73% higher than thatusing ENDF0B-VI parameters. This change is primarilydue to the significant change in 176Hf resonance param-eters near 8 eV. This change is not surprising, given thesmall amount of experimental data available for this pairof resonances and the high level of uncertainty in previ-ous work. A much smaller change in the 178Hf resonanceintegral, which is ;2% lower than ENDF0B-VI, is alsoprimarily due to the changes in the 178Hf resonance pa-rameters near 8 eV.
The hafnium experimental data and resonance pa-rameters provided are a significant improvement overprevious measurements, due to improved sample design,experimental resolution, and analysis tools.
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TABLE IX
Resonance Integrals Calculated from Resonance Parameters Determined in this Analysis ~Labeled RPI!and from ENDF0B-VI Parameters, Integrated from 0.5 to 200 eV
