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Precision measurement of the scattering amplitude of 235U
Physica 136B (1986) 134-136 North-Holland, A m s t e r d a m
PRECISION MEASUREMENT OF THE SCATTERING AMPLITUDE OF 2aSu H. KAISER, M. ARIF and S.A. WERNER Research Reactor and Department of Physics, University of Missouri, Columbia, MO 65211, USA
and J.O. WILLIS Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
T h e thermal neutron coherent scattering amplitude b e for b o u n d atoms of the isotope Z3~U has been m e a s u r e d with the neutron interferometer at two different wavelengths. The results, an average of two independent m e a s u r e m e n t s using two samples with different thicknesses and corrected for impurities, are: b c = (10.50 +_0.03) fm b c = (10.39 -+ 0.03) fm
for A = 1.261/~ (E = 0.051 e V ) , for A = 1.045 A (E = 0 . 0 7 5 eV).
Comparison with the B r e i t - W i g n e r formalism shows that, by adjusting the n e u t r o n width F, for the first resonance at resonance energy E 0 = - 1 . 4 eV, reasonable a g r e e m e n t between calculated and m e a s u r e d b c values can be obtained.
I. Introduction With the successful operation of pulsed neutron sources, much higher neutron energies are available for scattering experiments. In the case of resonances in this energy range, it is not only important to know the neutron nuclear scattering amplitude at a particular energy, but also this amplitude as a function of the neutron energy through the resonances. Neutron interferometry is one of the most precise measuring techniques to determine directly the bound coherent scattering amplitude of an isotope [1-5]. Though we cannot cover the desired energy range, we have begun some preliminary measurements using our neutron interferometer spectrometer at MURR. Our ultimate goal, however, is to build an interferometer instrument at a pulsed source [6] and to do, among other experiments, energy dependent scattering amplitude measurements of various isotopes.
2. Experiment and results The measurements were performed on two
235Umetal foils furnished by Los Alamos National Laboratory (-93% enrichment) with 0.06 and 0.05 cm thicknesses. An accurate analysis of the impurities ( - 5 % 238U, the rest other uraniumisotopes) was done at Los Alamos. Each foil is enclosed in an Al-capsule, which is sealed by e-beam welding. A scheme of the experimental arrangement is shown in fig. 1. The neutron interferometric measuring technique can be described briefly as follows (for more detailed description, see [5,7]): the Al-phase rotator is rotated in steps and each time the intensities behind the interferometer are measured. The intensity oscillation as a function of the phase rotator angle 6 is equal to
1(3) --
1011 + c cos(at
(
) -
i)l,
(1)
where A f t ( 6 ) = K A t ( 6 ) = - A N b c At(3),
where I 0 is the mean value, C the contrast, A f t ( 6 ) the phase difference for the neutron wave traversing the path ABD relative to ACD, ~bi is the initial phase, K is the oscillation period, At(8 ) the
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
H. Kaiser et al. / Scattering amplitude o f Z35U
135
2asU: t=0.06cm, A=1.261A 5OOO
Si(220) planes
A ~=--(2W. 22+A~)=--142.586(radians)
, 235U s a m p l e ~ 4000
&q~=4.3~6
( SAMPLE OUT
O . ~ .om
rnonochromator
~" ~ " " ~ 1 I N
cl~
u \
~
detector
t I
3000 z O 2000
I ~ \Lphas e
'-- SAMPLE IN
rotator
1000 Fig. 1. Sketch of the experimental set-up. The Z35U-sample can be driven in and out of one of the two neutron paths in the first section of the interferometer.
geometrical path difference, A the wavelength of the incident neutron, N the atom density and b~ the coherent scattering length of the phase rotator material. A nonlinear least squares fit of our data to eq. (1) yields values of the four parameters (I0, C, K , q~i).
At each step of the phase rotator, the neutron intensity was measured with and without the encapsulated 235U-sample. Thus, two intensity oscillation patterns are obtained (fig. 2). The phase shift, Aq0, due to the 235U-sample plus capsule, is given by the change in the initial phase of sample in, ~biin, versus sample out, ~b,o u t , Aq~ = -(27rm + ~bln - i#io u t )= -(27rm + A~b'), (2) where m is an integer and A 4 / < 27r. In this type of interferometric measurement, only A~b' can be determined directly. In order to determine m, the ~hase rotator was replaced by the encapsulated U-sample and rotated. In this case A qo can be determined via the oscillation period K (eq. (1)). Though it is a less accurate measurement, the value for m can be obtained very easily. The combination of m and A~b', however, yields a very precise value for A qO. Previously, the same two measurement steps were performed for each capsule alone and thus the phase shift AqOcap due to the empty capsule was determined. The phase shift A qb ~ u due to the 235U-sample alone is then expressed simply as
0
-3
i
-2
L
i
L
i
-1 0 1 2 ROTATOR ANGLE b (DEG)
3
Fig. 2. Neutron intensity of the forward beam as a function of the phase rotator angle 6. The × represents the data points for sample out, the Q for sample in. The loss of contrast for sample in is due to absorption of 23SU. A ~ z 3 5 u = A¢~ - Aq~ca p =
27r/A(1 - n ) t = - N t A b c ,
(3) where n is the real part of the refractive index, t the 235U-sample thickness, N the atom density, and b c the real part of the coherent scattering amplitude for bound atoms of the the 235Usample. It is clear that, when Aq0 and m~cap are measured and N, A and t are known, b c can easily be determined. The results of this experiment thus far are shown in table I. The values for A q0 and m~cap are the average of five runs. The b c values (uncorrected), b~°rr (corrections made due to impurities) and the final results are summarized in this table.
3. D i s c u s s i o n
Looking at the error bars of all the physical quantities involved, the accuracy of this measurement is primarily limited by the error of the two 235U-foil thicknesses A t / t - - O . O 0 4 (see table I). Aside from this, the accuracy of the experiment is comparable to earlier intefferometric experiments [3,4]. Comparison of our results with the literature v a l u e b~'t = (10.5 - 0.2) fm [8], without
H. Kaiser el al. / Scattering amplitude of 233U
136 Table I Experimental results
bc
b ....
Final result bc
( 1 0 -13 c m )
( 1 0 -13 c m )
( 1 0 -13 c m )
t (cm)
Aq~ (radians)
Aq~ap (radians)
A4~.,35U (radians)
1.261 +-0.1301
0.0498+-0.0002 0.0598 -+0.0002
-136.253 -+0.032 - 142.588 -+0.008
104.754 -+0.013 - 104.695 -+0.028
-31.499 ---0.009 -37.893 ---0.010
10.41-+0.04 10.43+- 0.04
10.49 -+0.05 10.51 -+0.04
10.50 -+ 0.03
1.045 -+0.001
0.0498-+0.0002 0.0598 -+ 0.0002
- 112.633 ~- 0.033 - 117.841 -+0.017
-86.809 +-0.011 -86.760 -+0.023
-25.824 -+0.008 -31.081 -+0.009
10.30-+0.04 10.32-+0.04
10.37 +-0.05 10.40 -+0.04
10:39 -+0.03
(/~)
-
been calculated and plotted (fig. 3). It was necesr, lit sary to adjust the neutron width 2 g l , = 3.076 (meV) to 2gFn~ew = 1.620 (meV) for the first resonance ( E 0 = - l . 4 e V ) in order to have reasonable agreement between calculated and measured values. Obviously, more experimental values for b c at higher energies are necessary in order to accomplish a reasonable fit and to test the quality of calculating b c in terms of BrietWigner resonances using the current literature values.
11.00 11
'~
,
10
91
10.75
,
,
0
.5
1.124eV 1~ 24eV 1.0
1.5
ENERGY (eV) 10.50
10.25
Acknowledgments 10.00
t
0
i .025
i .050
L
i .075
i
.100
ENERGY (eV)
Fig. 3. Comparison of our data x and the literature value Q with calculated values (solid line) using the Breit-Wigner formula. The insert shows these calculated be-values over a wider energy range; the first two positive resonances are at E 0 = 0.290 eV and E 0 = 1.124 eV.
This work is supported by the Physics Division of the National Science Foundation through Grant No. PHY-841063. Work at Los Alamos is performed under the auspices of the U.S. Department of Energy.
References taking the energy dependence of b c into account, nevertheless shows excellent agreement. In the case of slow s-wave neutron scattering, the real part of the energy dependent scattering amplitude b~ can be written in terms of the Breit-Wigner formalism. By using the resonance parameters of the first 40 resonances and including them in the summation of the Breit-Wigner formula [8], and by correcting for the neutronelectron interaction and the Foldy term, the real part of the energy dependent scattering amplitude as a function of the incident neutron energy E has
[1] W. Bauspiess, U. Bonse and H. Rauch, Nucl. Instr. and Meth. 157 (1978) 495. [2] H. Kaiser, H. Rauch, G. Badurek, W. Bauspiess and U. Bonse, Z. Phys. A291 (1979) 231. [3] U. Bonse and U. Kischko, Z. Phys. A305 (1982) 171. [4] A. Boeuf, R. Caciuffo, R. Rebonato, F. Rustichelli, J.M. Fournier, U. Kischko and L. Manes, Phys. Rev. Lett. 49 (1982) 1086. [5] R.E. Word and S.A. Werner, Phys. Rev. B26 (1982) 4190. [6] IPNS Progress Report 1983-1985, p. 66. [7] J.-L. Staudenmann, S.A. Werner, R. Colella and A.W. Overhauser, Phys. Rev. A21 (1980) 1419. [8] Neutron Cross Sections, S.F. Mughabghab, ed., vol. 1, part B (Academic Press, New York, 1984) pp. 92-12.
PRECISION MEASUREMENT OF THE SCATTERING AMPLITUDE OF 2aSu H. KAISER, M. ARIF and S.A. WERNER Research Reactor and Department of Physics, University of Missouri, Columbia, MO 65211, USA
and J.O. WILLIS Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
T h e thermal neutron coherent scattering amplitude b e for b o u n d atoms of the isotope Z3~U has been m e a s u r e d with the neutron interferometer at two different wavelengths. The results, an average of two independent m e a s u r e m e n t s using two samples with different thicknesses and corrected for impurities, are: b c = (10.50 +_0.03) fm b c = (10.39 -+ 0.03) fm
for A = 1.261/~ (E = 0.051 e V ) , for A = 1.045 A (E = 0 . 0 7 5 eV).
Comparison with the B r e i t - W i g n e r formalism shows that, by adjusting the n e u t r o n width F, for the first resonance at resonance energy E 0 = - 1 . 4 eV, reasonable a g r e e m e n t between calculated and m e a s u r e d b c values can be obtained.
I. Introduction With the successful operation of pulsed neutron sources, much higher neutron energies are available for scattering experiments. In the case of resonances in this energy range, it is not only important to know the neutron nuclear scattering amplitude at a particular energy, but also this amplitude as a function of the neutron energy through the resonances. Neutron interferometry is one of the most precise measuring techniques to determine directly the bound coherent scattering amplitude of an isotope [1-5]. Though we cannot cover the desired energy range, we have begun some preliminary measurements using our neutron interferometer spectrometer at MURR. Our ultimate goal, however, is to build an interferometer instrument at a pulsed source [6] and to do, among other experiments, energy dependent scattering amplitude measurements of various isotopes.
2. Experiment and results The measurements were performed on two
235Umetal foils furnished by Los Alamos National Laboratory (-93% enrichment) with 0.06 and 0.05 cm thicknesses. An accurate analysis of the impurities ( - 5 % 238U, the rest other uraniumisotopes) was done at Los Alamos. Each foil is enclosed in an Al-capsule, which is sealed by e-beam welding. A scheme of the experimental arrangement is shown in fig. 1. The neutron interferometric measuring technique can be described briefly as follows (for more detailed description, see [5,7]): the Al-phase rotator is rotated in steps and each time the intensities behind the interferometer are measured. The intensity oscillation as a function of the phase rotator angle 6 is equal to
1(3) --
1011 + c cos(at
(
) -
i)l,
(1)
where A f t ( 6 ) = K A t ( 6 ) = - A N b c At(3),
where I 0 is the mean value, C the contrast, A f t ( 6 ) the phase difference for the neutron wave traversing the path ABD relative to ACD, ~bi is the initial phase, K is the oscillation period, At(8 ) the
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
H. Kaiser et al. / Scattering amplitude o f Z35U
135
2asU: t=0.06cm, A=1.261A 5OOO
Si(220) planes
A ~=--(2W. 22+A~)=--142.586(radians)
, 235U s a m p l e ~ 4000
&q~=4.3~6
( SAMPLE OUT
O . ~ .om
rnonochromator
~" ~ " " ~ 1 I N
cl~
u \
~
detector
t I
3000 z O 2000
I ~ \Lphas e
'-- SAMPLE IN
rotator
1000 Fig. 1. Sketch of the experimental set-up. The Z35U-sample can be driven in and out of one of the two neutron paths in the first section of the interferometer.
geometrical path difference, A the wavelength of the incident neutron, N the atom density and b~ the coherent scattering length of the phase rotator material. A nonlinear least squares fit of our data to eq. (1) yields values of the four parameters (I0, C, K , q~i).
At each step of the phase rotator, the neutron intensity was measured with and without the encapsulated 235U-sample. Thus, two intensity oscillation patterns are obtained (fig. 2). The phase shift, Aq0, due to the 235U-sample plus capsule, is given by the change in the initial phase of sample in, ~biin, versus sample out, ~b,o u t , Aq~ = -(27rm + ~bln - i#io u t )= -(27rm + A~b'), (2) where m is an integer and A 4 / < 27r. In this type of interferometric measurement, only A~b' can be determined directly. In order to determine m, the ~hase rotator was replaced by the encapsulated U-sample and rotated. In this case A qo can be determined via the oscillation period K (eq. (1)). Though it is a less accurate measurement, the value for m can be obtained very easily. The combination of m and A~b', however, yields a very precise value for A qO. Previously, the same two measurement steps were performed for each capsule alone and thus the phase shift AqOcap due to the empty capsule was determined. The phase shift A qb ~ u due to the 235U-sample alone is then expressed simply as
0
-3
i
-2
L
i
L
i
-1 0 1 2 ROTATOR ANGLE b (DEG)
3
Fig. 2. Neutron intensity of the forward beam as a function of the phase rotator angle 6. The × represents the data points for sample out, the Q for sample in. The loss of contrast for sample in is due to absorption of 23SU. A ~ z 3 5 u = A¢~ - Aq~ca p =
27r/A(1 - n ) t = - N t A b c ,
(3) where n is the real part of the refractive index, t the 235U-sample thickness, N the atom density, and b c the real part of the coherent scattering amplitude for bound atoms of the the 235Usample. It is clear that, when Aq0 and m~cap are measured and N, A and t are known, b c can easily be determined. The results of this experiment thus far are shown in table I. The values for A q0 and m~cap are the average of five runs. The b c values (uncorrected), b~°rr (corrections made due to impurities) and the final results are summarized in this table.
3. D i s c u s s i o n
Looking at the error bars of all the physical quantities involved, the accuracy of this measurement is primarily limited by the error of the two 235U-foil thicknesses A t / t - - O . O 0 4 (see table I). Aside from this, the accuracy of the experiment is comparable to earlier intefferometric experiments [3,4]. Comparison of our results with the literature v a l u e b~'t = (10.5 - 0.2) fm [8], without
H. Kaiser el al. / Scattering amplitude of 233U
136 Table I Experimental results
bc
b ....
Final result bc
( 1 0 -13 c m )
( 1 0 -13 c m )
( 1 0 -13 c m )
t (cm)
Aq~ (radians)
Aq~ap (radians)
A4~.,35U (radians)
1.261 +-0.1301
0.0498+-0.0002 0.0598 -+0.0002
-136.253 -+0.032 - 142.588 -+0.008
104.754 -+0.013 - 104.695 -+0.028
-31.499 ---0.009 -37.893 ---0.010
10.41-+0.04 10.43+- 0.04
10.49 -+0.05 10.51 -+0.04
10.50 -+ 0.03
1.045 -+0.001
0.0498-+0.0002 0.0598 -+ 0.0002
- 112.633 ~- 0.033 - 117.841 -+0.017
-86.809 +-0.011 -86.760 -+0.023
-25.824 -+0.008 -31.081 -+0.009
10.30-+0.04 10.32-+0.04
10.37 +-0.05 10.40 -+0.04
10:39 -+0.03
(/~)
-
been calculated and plotted (fig. 3). It was necesr, lit sary to adjust the neutron width 2 g l , = 3.076 (meV) to 2gFn~ew = 1.620 (meV) for the first resonance ( E 0 = - l . 4 e V ) in order to have reasonable agreement between calculated and measured values. Obviously, more experimental values for b c at higher energies are necessary in order to accomplish a reasonable fit and to test the quality of calculating b c in terms of BrietWigner resonances using the current literature values.
11.00 11
'~
,
10
91
10.75
,
,
0
.5
1.124eV 1~ 24eV 1.0
1.5
ENERGY (eV) 10.50
10.25
Acknowledgments 10.00
t
0
i .025
i .050
L
i .075
i
.100
ENERGY (eV)
Fig. 3. Comparison of our data x and the literature value Q with calculated values (solid line) using the Breit-Wigner formula. The insert shows these calculated be-values over a wider energy range; the first two positive resonances are at E 0 = 0.290 eV and E 0 = 1.124 eV.
This work is supported by the Physics Division of the National Science Foundation through Grant No. PHY-841063. Work at Los Alamos is performed under the auspices of the U.S. Department of Energy.
References taking the energy dependence of b c into account, nevertheless shows excellent agreement. In the case of slow s-wave neutron scattering, the real part of the energy dependent scattering amplitude b~ can be written in terms of the Breit-Wigner formalism. By using the resonance parameters of the first 40 resonances and including them in the summation of the Breit-Wigner formula [8], and by correcting for the neutronelectron interaction and the Foldy term, the real part of the energy dependent scattering amplitude as a function of the incident neutron energy E has
[1] W. Bauspiess, U. Bonse and H. Rauch, Nucl. Instr. and Meth. 157 (1978) 495. [2] H. Kaiser, H. Rauch, G. Badurek, W. Bauspiess and U. Bonse, Z. Phys. A291 (1979) 231. [3] U. Bonse and U. Kischko, Z. Phys. A305 (1982) 171. [4] A. Boeuf, R. Caciuffo, R. Rebonato, F. Rustichelli, J.M. Fournier, U. Kischko and L. Manes, Phys. Rev. Lett. 49 (1982) 1086. [5] R.E. Word and S.A. Werner, Phys. Rev. B26 (1982) 4190. [6] IPNS Progress Report 1983-1985, p. 66. [7] J.-L. Staudenmann, S.A. Werner, R. Colella and A.W. Overhauser, Phys. Rev. A21 (1980) 1419. [8] Neutron Cross Sections, S.F. Mughabghab, ed., vol. 1, part B (Academic Press, New York, 1984) pp. 92-12.