PHYSICAL REVIEW A VOLUME 35, NUMBER 7 APRIL 1, 1987

Precision measurement of the bound-coherent-neutron scattering length of 235U

M. Arif, H. Kaiser, and S. A. WernerDepartment of Physics and Research Reactor, Uniuersity ofMissouri, Columbia, Missouri 65211

J. O. WillisPhysics Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

(Received 10 November 1986)

With use of the neutron interferometric technique, the real parts of the bound-coherent scatteringlength, b, of 'U have been precisely measured for five different incident neutron energies in therange of 30 to 92 meV. The value of b at each incident neutron energy was obtained from the aver-

age of two independent measurements using two samples of different thicknesses. The results areb =10.47+0.04 fm (E =30.34 meV) b =10.47+0.03 fm (E =41.02 meV), b =10.51+0.03 fm

(E =51.43 meV), b =10.39+0.03 fm (E =74.90 meV), b =10.41+0.03 fm (E =91.39 meV). Theuncertainty in these results is about an order of magnitude smaller than the present literature value.

The small energy dependence of the observed scattering lengths has been used to establish the pa-rameters of the negative energy resonance level ( —1.4 eV) using the Breit-Wigner formalism.

I. INTRODUCTION

The total and fission neutron cross sections of Uhave been extensively studied with several different typesof spectrometers. ' But, in comparison, a relativelysmall effort has been made to measure the low-energycoherent scattering length b by direct means. The diffi-culty in carrying out precision measurements of b arises(i) from the presence of two nearby energy resonances (at0.29 and 1.124 eV) which produces large sample absorp-tion at thermal energy, and (ii) from the necessity of em-

ploying special handling techniques for uranium samples.The current literature b value of U in the thermal ener-

gy region has a 2% uncertainty which is about an order ofmagnitude higher than the uncertainty in the measure-ments of b of many isotopes through modern neutron op-tical techniques, such as neutron interferometry.

One of the most difficult problems in interpreting Ucross-section data arises from the shape of the curvebelow 2 eV. In this region, taking into consideration thecontribution of all known positive energy resonances, thecalculated cross section using Breit-Wigner formalismshows appreciable discrepancy with the experimentalvalues. This disagreement can be explained to a great ex-tent by assuming the existence of so-called "negative ener-gy" resonance levels, which can contribute significantly tothe scattering length (or cross-section) values at thermalenergies. ' '" These levels cannot be accessed directly, butthe existence of such levels can be felt from the magnitudeof coherent and incoherent scattering amplitudes and theshape of cross-section curves.

The neutron interferometric technique provides us withthe opportunity to measure the energy-dependent b of

U at thermal energies more accurately than has beenpreviously possible. Realizing the extensive use of U inmany critical nuclear technology areas, an improvementin the accuracy of b would appear to be important. Since

our measurements were taken in the low-energy range of-30 to 92 meV, the effect of negative energy resonancelevels was pronounced. The accurately measured valuesof b allowed us then to compare the results with thetheoretical predictions from Breit-Wigner formalism, tak-ing into consideration the contribution from both positiveand negative energy resonance levels of U.

II. DESCRIPTION OF THE SAMPLES

Two metallic uranium foils with 0.498+0.002- and0.598+0.002-mm thicknesses furnished by the LosAlamos National Laboratory were used for the experi-ment. The total concentration of uranium isotopes was99.996% in both samples. Each foil had 93.5% U iso-topic purity. The impurities consisted mainly of otherisotopes of uranium, with U being the largest withabout a 5% concentration. The concentrations of Uand U were about 1.1% and 0.3%, respectively. Therewere also various trace elements whose concentration wastypically a few parts per million and were considered in-significant. The foils were identical in composition to+0.02% variation in U and U concentration.

Since U is radioactive and fissionable, the samplefoils required a special handling technique. To minimizehandling hazards and eventual oxidation, each foil wasencapsulated in a specially designed aluminum capsuleand sealed with electron-beam welding to make the assem-bly airtight and leakproof. A photograph of such a cap-sule is shown in Fig. 1. The cross-sectional view of thealuminum capsule with the uranium sample inside isshown in Fig. 2.

III. EXPERIMENT

The measurements were carried out at the neutron in-terferometer setup at beam port B at the University of

35 2810 1987 The American Physical Society

35 PRECISION MEASUREMENT OF THE BOUND-COHERENT-. . . 2811

R @JRIN

FIG. 1. Photograph of an assembled aluminum capsule. The uranium foil has been sealed inside.

Missouri Research Reactor using 30.34-, 41.02-, 51.43-,74.90-, and 91.39-meV incident neutron energies. ' Theschematics of the experimental arrangements are shown inFigs. 3 and 5. The incident neutron wavelength wasselected by a double crystal monochromator assembly andthe collimated incident beam was 5 mm in diameter. Thesamples were driven in and out of the direct beam in thefirst leg of the interferometer (Fig. 3) causing a relativephase shift between the direct and diffracted beams. The

/0

A l Capsu le

advantage of the double crystal monochromator is evidentfrom this experimental geometry. Since for this arrange-ment the incident-beam direction remained the same at allincident neutron energies, the beam transversed the sam-ples through the same spot and experienced identicalthickness and composition at all energies, thereby elim-inating potential unreliable sample thickness and composi-tion corrections for different incident neutron directions.

In the measurement a very flat 1-cm-thick aluminumphase shifter was rotated in steps in the downstream legof the interferometer. For each angular position of thephase shifter, the counts in detectors Cz and C3 with orwithout the sample in the direct beam were recorded.Figure 4 shows a typical intensity variation of the in-terfering beams entering detectors C2 and C3 as a func-tion of the angle 6 of the phase shifter. If we label thephases of the interferograms with sample in and out of

g~~~ Uraniumfoil

Si(220) planes sam

2. 5g 5 from

monochromtorphaseshifter

He

etector

FIG. 2. Cross-sectional view of the aluminum capsule show-ing the placement of the uranium foil.

FIG. 3. Experimental geometry to measure the phase shift5N.

2812 M. ARIF, H. KAISER, S. A. WERNER, AND J. O. WILLIS 35

5000A = 1.261 A

4000 SAMPLE OUT

Q)

CQ

CJ

I

3000

2000

1000

0-3 -2 -1 0 1 2 3

(deg}FIG. 4. Typical data for experiments carried out for sample

in (out) geometry as illustrated in Fig. 3.

the beam as N'" and 4""', respectively, then the phaseshift h4 caused by the sample is given by

b, N =4'"—4 "'= —(2mm +5N),

If the sample contains more than one isotope, b is givenby

b=gx;b;,

where x; and b; are the fractional abundance and scatter-ing length of the individual isotope labeled by i.

4'" and N'"' in Eq. (1) are obtained by nonlinear least-squares fitting of the measured data points (Fig. 4) with afunction of the form

where m is an integer and 5@& 2m. . In terms of b,N, theatom density N of the sample, the incident wavelength k,and sample thickness T, the scattering length b of thesample is given by the relation'

shifter. From the least-squares fit of the data from thisexperimental geometry, it is not possible to identify the in-tegral multiple of 2m of the total phase shift in Eq. (1),and as a result the actual value of the phase shift obtainedfrom the fit gives only 6N.

To determine the integer number m, a separate experi-ment was performed as shown in Fig. 5. The phaseshifter was replaced with the sample plate. Thus the for-mulas (4), (5), and (6) were now applied to the sample.Again as a function of rotation angle 5 of the phase shift-ing sample, interferograms were obtained for the twodetectors (Fig. 6). When fitted to the function given byEq. (4), the value of the coefficient c corresponding to thesample was obtained. From this fitted value of c thephase shift b,N (=c/2) due to the passage of neutronsthrough the sample along its normal was obtained. Therewere five runs made for each sample, and for the datagathered at the two detectors, a total of ten values of cwere obtained which in turn provided a measured value ofAN to within an error e. With 6@ already known, the in-teger number m could now easily be determined uniquelyfrom the relation

AN —e(2~m +6&~ AN+g .

Using the values of m and M&, Eq. (1) now yielded a pre-cise value for the sample phase shift AN. The advantageof this measuring technique can now be seen. It is ap-parent that once the integer number m is known, the errorin AN lies with the measuring error in 6N. This meansthat for a sufficiently large m the relative error in AN be-comes very small. In our case, a relative error of 1 part in5000 was easily obtained.

Since the uranium foils were encapsulated, there was nodirect way to measure the phase shifts caused by theuranium foils alone. For our experiment we determinedthe phase shifts due to the empty aluminum capsules andthe capsules with enclosed uranium foils separately. Thuswe were able to subtract the contribution of the aluminumcapsule to determine the phase shift due to uranium only.The phase shift due to the empty capsule was about 75%%uo

of the total phase shift due to the sample plus the capsule.The experimental procedure described above was identicalin both cases.

I(5)=A +Bcos[cf(5)+40],where

(4)

and

c = —2TpkXpbp

sin5 sinOgf(5)=cos Oz —sin 6

(6)

Si(220) planes

He

In the above equations 3 and B are the mean and ampli-tude of the interferograms. @0 is the initial phase of eachinterferogram, 2T~f (5)=b,T(5) is the path difference ofthe two interfering beams traversing the phase shifter as afunction of its angle 5, Tz is the thickness of the phaseshifter, Oz is the interferometer Bragg angle correspond-ing to the neutron wavelength k, Nz and b~ are the atomdensity and scattering length, respectively, of the phase

etector

frommonochromator

C::

uraniums amp le

FICx. 5. Experimental geometry to determine the phase shift

35 PRECISION MEASUREMENT OF THE BOUND-COHERENT-. . . 2813

3000

2000-

p, =0.946AI 1

11.0

10.8 .

E xperimental data

Breit-wigner f it

EDLD

1000—

O

C310.6 .

10.4 .

0—12 0

(deg )

12

10.2 .FIG. 6. Typical intensity patterns recorded in detectors Cz

and C3 corresponding to the experimental geometry of Fig. 5.10.0

0 20I

40 60J

80 100

IV. RESULTS

Table I shows our experimental results. The b valueswere obtained by using Eqs. (2) and (3). To determineatom density N we used the average of the density ofnatural uranium obtained from lattice parameter measure-ments and more recently directly by Archimedes balance.This gave the value of N to be (0.4816+0.0005)X10atoms/cm . The measured accuracy of A, was -0.001 Aand this was taken to be the error for all wavelengthsmeasured. The thicknesses of the samples were measuredat Los Alamos National Laboratory using a comparatordevice and the measuring error for each sample was +2tMm. First, the scattering lengths of the two uranium sam-ples were obtained, and then the values of b for U wereobtained by correcting for the uranium isotopes by usingEq. (3). This correction was about 0.8%. The scatteringlengths of the other uranium isotopes are only slightly en-

ergy dependent. The literature values at the thermal ener-

gy region were used for correction at all incident neutronenergies. The values were 8.55, 10.0, and 12.4 fm for

U, U, and U, respectively. At each incident neu-tron energy the final value of b for U was obtained byaveraging two independently measured b values of the

Energy ( ~ev i

FIG. 7. Comparison of the measured scattering length values(in fermis) with the calculated values from the Breit-Wigner for-malism. The solid line represents the calculated values.

two sample foils. Calculation of the scattering lengthswas also carried out using the Breit-Wigner formalism.About 230 positive and negative energy resonances forwhich resonance parameter values are known were used inthe calculation. The results of this calculation were thencompared to the experimental values. Figure 7 shows thecalculated Breit-Wigner curve normalized to the best linefit value of the experimentally determined scatteringlengths at 60 meV. In order to do so, the 2gl „value ofthe negative energy resonance at —1.4 eV was adjusted to1.89 meV from the literature value of 2.6 meV. ' Theslope of the linear fit to the experimentally determinedvalues was —1.39X 10 fm/meV and the slope obtainedfrom the Breit-Wigner line (assuming a straight line forthis small experimental energy range) was —1.18X 10fm/meV. Within the limits of error the values of theslopes agreed.

TABLE I ~ Experimental scattering lengths.

(A)

1.642

1.412

1.261

1.045

0.946

E(meV)

30.335

41.022

51.434

74.895

91.391

T(mm)

0.4980.5980.4980.5980.4980.5980.4980.5980.4980.598

bU sample

(fm)

10.376+0.06110.404+ 0.05310.375+0.04610.401+0.04010.419+0.04610.438 +0.03910.307+0.04710.331+0.03910.329+0.04710.342+0.040

b235U

(fm)

10.452+ 0.06510.482+ 0.05810.451+0.05010.479+0.04410.499+0.05010.519+0.04210.379+0.05010.404+0.04310.402+0.04910.416+0.044

b235U

(fm)

10.47+0.04

10.47+0.03

10.51+0.03

10.39+0.03

10.41+0.03

2814 M. ARIF, H. KAISER, S. A. WERNER, AND J. O. WILLIS 35

V. DISCUSSION AND CONCLUSIONS

These results, to our knowledge, are the first directlymeasured energy-dependent scattering lengths of U.The scattering length values of 'U obtained from thisexperiment agree very well with the mean literature bvalue. However, the error limit of +0.03 fm obtainedfrom this experiment is about 1 order of magnitude small-er than that in the presently known best value of 10.5+0.2fm. ' The independently measured b values of U fromtwo different samples agree within 0.3%%uo in all cases andthis agreement demonstrates the consistency of our mea-surements.

As expected from Breit-Wigner formalism, the best fitof the experimentally measured b values has a negativeslope, i.e., the scattering length increases at lower energiesaway from the first positive energy resonance at 290 meV.The contributions of the negative energy resonance levels( —10%) were significant in the calculation of the b valueto obtain a good fit to the experimental data. This sug-gests the possibility that for isotopes with negative reso-nances and resonances in the thermal energy range,analysis of accurate scattering length or cross-section dataat thermal energies may be a reliable and accurate way toestablish the parameter values for negative energy reso-nances.

The accuracy of these measurements depended primari-ly on our knowledge of the sample dimensions and com-position rather than the technique itself. The largest con-tribution to the error in our results came from the thick-

ness uncertainties of the uranium samples. The contribu-tion of this uncertainty could be made smaller by choos-ing a larger sample thickness T. The foil thicknesseschosen in this experiment were the results of compromisebetween a reasonable data collection rate and acceptablerelative uncertainty in the thickness measurements. Theenergy range for this experiment was limited by the avail-able flux from a steady-state reactor. Extending this ener-

gy range would require a higher epithermal flux source,such as a pulsed neutron source. Our aim is to extendthese interferometric measurements to -2 eV at a pulsedneutron source, encompassing the first two positive energyresonances of U at 0.290 and 1.124 eV. This will en-able us to obtain a precise set of b values for this extendedenergy range and allow a better and more complete com-parison of theory and experiment. Initial exploratory ex-periments have already been carried out in this respect.

ACKNOWLEDGMENTS

This work was supported by the Physics Division of theNational Science Foundation through Grant No. PHY-841063. Work at Los Alamos was performed under theauspices of the U.S. Department of Energy. We wish tothank Dr. G. H. Lander for initiating this collaborationbetween Missouri and Los Alamos. We also wish tothank Dr. L. Passell of Brookhaven National Laboratoryfor calling helpful references to our attention. The expertphysics machine shop services of Mr. C. Holmes were im-portant in apparatus construction.

i J. A. Harvey, Proceedings of the International Conference onthe Peaceful Uses of Atomic Energy, Geneva, 1955 (UnitedNations, New York, 1956), Vol. 4, p. 147.

F. J. Shore and V. L. Sailor, Phys. Rev. 112, 191 (1958).E. E. Maslin, J. A. Moore, J. M. Reichelt, and J. B. Crowden,

Phys. Rev. 139, B852 (1965).4H. Palevsky, R. S. Carter, R. M. Eisberg, and D. J. Hughes,

Phys. Rev. 94, 1088 (1954).5W. Bauspiess, U. Bonse, and H. Rauch, Nucl. Instrum.

Methods 157, 495 (1978).6H. Kaiser, H. Rauch, G. Badurek, W. Bauspiess, and U. Bonse,

Z. Phys. A 291, 231 (1979).7U. Bonse and U. Kischko, Z. Phys. A 305, 171 (1982).A. Boeuf, R. Caciuffo, R. Rebonato, F. Rustichelli, J. M. Four-

nier, U. Kischko, and L. Manes, Phys. Rev. Lett. 49, 1086

(1982).9R. E. Word and S. A. Werner, Phys. Rev. B 26, 4190 (1982).

H. L. Foote, Jr., Phys. Rev. 109, 1641 (1958).J. A. Harvey and J. E. Sanders, Progress in Nuclear Energy,Series I (Pergamon, London, 1956), Vol. 1, pp. 55—89.

~ H. Kaiser, M. Arif, S. A. Warner, and J. O. Willis, Interna-tional Conference on Neutron Scattering (Santa Fe, August1985) [Physica 136B, 131 (1986)].

A. G. Klein and S. A. Werner, Rep. Frog. Phys. 46, 259(1983).S. F. Mughaghab, M. Divadeenam, and N. E. Holden, Neu-tron Cross Sections (Academic, New York, 1981), Vol. 1, PartA; Neutron Cross Sections (Academic, New York, 1984), Vol.1, Part B.