Dispersion free measurements in neutron interferometry

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<ul><li><p>*Corresponding author. Fax: #420-2-20941130.E-mail address: vrana@ujf.cas.cz (M. VraH na)</p><p>Physica B 283 (2000) 400}402</p><p>Dispersion free measurements in neutron interferometry</p><p>M. VraH na!,*, P. Mikula!, P. LukaH s\ !, A. Io!e", W. Nistler#!Nuclear Physics Institute, 250 68 Rez near Prague, Czech Republic</p><p>"Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, MO 65211, USA#Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany</p><p>Abstract</p><p>Neutron interferometry is a method of high-accuracy determination of neutron scattering length b#. The phase shift</p><p>acquired by a neutron wave that propagates through a sample inserted in one of the beams of an interferometer isproportional to the neutron wavelength. A knowledge of it precisely de"nes the accuracy of the determination of b</p><p>#. As</p><p>a solution of this problem, the non-dispersive sample geometry was proposed. In this case the phase shift is j-independentbut the very precise orientation of the sample relative to the crystal planes of the interferometer is required. To overcomethis obstacle a new nondispersive phase di!erence method was developed which achieves an accuracy in the determina-tion of b</p><p>#of even better than 10~4, i.e. at the level of 10~5. This method was used for the high-accuracy measurement of</p><p>the coherent scattering length of Si. Experiments carried out in HMI Berlin and NIST Gaithersburg and NPI Rez ondi!erent interferometric facilities at di!erent neutron wavelengths proved that the coherent scattering lengths can bemeasured with a relative uncertainty of 0.002%. ( 2000 Elsevier Science B.V. All rights reserved.</p><p>Keywords: Neutron interferometry; Scattering length</p><p>It has been commonly accepted that neutroninterferometry is a method of high-accuracy deter-mination of neutron scattering length b</p><p>#. The phase</p><p>shift acquired by a neutron wave that propagatesthrough a sample of thickness t</p><p>0and atomic den-</p><p>sity N, inserted in one of the beams of an inter-ferometer, is proportional to the neutronwavelength j: U"Nb</p><p>#jt</p><p>0. Therefore, one must</p><p>know the wavelength j, which precisely de"nes theaccuracy of the determination of b</p><p>#from the mea-</p><p>sured value of U. As a solution of this problem,the non dispersive sample geometry was proposed[1], where the sample was installed parallel to the</p><p>lattice planes of the interferometer crystal. In thiscase, the e!ective thickness is t</p><p>0/sin h</p><p>Band using</p><p>the Bragg law j"2dsin hB</p><p>(hB</p><p>is the Bragg angle,d the lattice constant), one immediately obtainsU"2dNb</p><p>#t0, so that the phase shift is j-indepen-</p><p>dent. However, now the very precise orientation ofthe sample relative to the crystal planes of theinterferometer is required. To overcome this ob-stacle a new nondispersive phase di!erence methodwas developed [2] which achieves an accuracy inthe determination of b</p><p>#of even better than 10~4, i.e.</p><p>at the level of 10~5. This method was used for thehigh-accuracy measurement of the coherent scat-tering length of Si [2]</p><p>Let us consider a sample that is placed in one ofthe beams of the interferometer, so that it is nearparallel to crystal planes of the interferometer</p><p>0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 3 5 3 - 7</p></li><li><p>crystal. For a small declination angle e of the sur-face of the sample from the Bragg planes of thecrystal. one can obtain acquired phases for thesample placed in beams I and II,</p><p>UI"2dNb</p><p>#t0G1!e cot (hB)#</p><p>e22</p><p>[1#2cot2(hB)]H,</p><p>UII"!2dNb</p><p>#t0G1#e cot (hB)</p><p>#e2</p><p>2[1#2cot2(h</p><p>B)]H. (1)</p><p>The basic idea of the proposed method is not toadjust the sample relative to the interferometertrying to get e+0, but to record the interferencepatterns for both positions of the sample in theinterferometer for a set of positive and negativevalues of e. Fitting the dependence of the phasedi!erence U</p><p>II!U</p><p>Ito a parabolic function of e</p><p>(Eq. (1)), one can de"ne the minimum of this parab-ola, which corresponds to the parallel position ofthe sample and crystal planes. Then the neutronscattering length, b</p><p>#, can be calculated by the value</p><p>of UI!U</p><p>IIand does not depend on the wave-</p><p>length j.Thus, an aim of the experiment is to determine</p><p>phase shifts UI</p><p>and UII, introduced by a sample</p><p>placed in beams I and II of the interferometer, fordi!erent angles e between the sample and crystalplanes of the interferometer. However, the neutroninterferometry method does not allow the directdetection of the total phases U</p><p>I(U</p><p>II), but only their</p><p>fractions uI(u</p><p>II) with 0(u</p><p>I,II(2p, so that</p><p>UI"u</p><p>I#2pn</p><p>1, U</p><p>II"u</p><p>II#2pn</p><p>2</p><p>with n1, n</p><p>2integers. Fitting the experimental data</p><p>to a sinus-like theoretical dependence one can de-termine phase shifts u</p><p>Iand u</p><p>II, measured for di!er-</p><p>ent angles d of the surface of the sample withrespect to the interferometer crystal (here d isa physically measured value in contrast to e whichis a model one). Carrying out such measurementswith a small step for d, which allows to follow 2pmultiple variations of U</p><p>I(U</p><p>II), one can de"ne</p><p>UI!U</p><p>II"f (d). Then "tting this function to d2</p><p>(Eq. (1)), we can determine the value of d.*/</p><p>(e"0),which corresponds to the parallel alignment of the</p><p>surface of the sample and the crystal planes of theinterferometer. Because in this case phase shiftsintroduced in both beams of the interferometer areequal (so that U</p><p>I"DU</p><p>IID and n</p><p>1"n</p><p>2), we obtain</p><p>b#"(uI#uII).*/#4nn1</p><p>4dNt0</p><p>. (2)</p><p>Therefore, in order to determine b#from Eq. (2) one</p><p>should de"ne the number of integer oscillationsn1</p><p>for the parallel position of the sample and crys-tal planes. This can be done using a knowledge ofb#</p><p>from preliminary low-accuracy experiments bya method described in Ref. [5].</p><p>The experiment has been done on the neutroninterferometer installed at the beam line HC8 at10 MW reactor in Rez. The instrument arrange-ment is a non dispersive setting of a double mono-chromator and an interferometer crystal placed ona vibroisolated foundation. The interferometercrystal (conventional LLL-type) manufactured atthe Atominstitut Vienna features 70% interferencepattern visibility for the 220 re#ection (neutronswavewlength of 0.1 nm). A new sample holder per-mits to position, rotate in a horizontal plane andtilt the sample in both beams inside the inter-ferometer. The upgraded interferometer facility wastested by measuring the coherent scattering lengthof Si performed by a nondispersive phase-di!erencemethod [2,3]. In this method interference patternsare measured by rotating the phase shifter for threepositions of the sample } outside the interferometerand in beams I and II (Fig. 1) in dependence on thedeclination of the surface of the sample from theBragg planes of the interferometer crystal. As canbe seen from Fig. 2. the phase shift produced by thesample inserted in one of the beams of the inter-ferometer depends nearly linearly on the angle ofmisaligment. The di!erence between the phaseshifts produced by successively inserting the samplein beams I and II, respectively, only weakly de-pends on the declination angle in the vicinity of thevertex of the parabola (Fig. 3) From a quadratic "t,one can determine the phase shift di!erencemodulo 2p corresponding to the minimum of thiscurve. The total phase shift can be determined bythe procedure described in Ref. [5]. In our caseof an Si plate of a thickness of 3.0055(1) mm we</p><p>M. Vra&amp; na et al. / Physica B 283 (2000) 400}402 401</p></li><li><p>Fig. 1. Interferograms taken at the three positions of the sample} outside the interferometer, in beams I and II.</p><p>Fig. 2. Phases UI</p><p>and UII</p><p>and sum of the phases UI#U</p><p>IIfor</p><p>di!erent angular positions of the sample.</p><p>Fig. 3. Phase di!erences UI!U</p><p>IIfor di!erent angular positions</p><p>of the sample.</p><p>obtained the value of 477.3392(75) rad. Using thecrystal lattice spacing of silicon a"0.543102 nmand a correction factor due to the displacement of</p><p>air by the sample [4] we obtain the value forb#"4.1511(2) fm. The result of this experiment is</p><p>in very good agreement with the results obtainedfrom the measurements carried out at the HMIBerlin and at NIST Gaithersburg [3,4]. It provesthat phase shifts can be measured on our instru-ment with a relative uncertainty of 0.002%.</p><p>Acknowledgements</p><p>This research is supported by the Grant Agencyof the Czech Republic under contract No.202/97/K038 and the TMR-Network ERB FMRXCT 96-0057.</p><p>References</p><p>[1] H. Rauch, E. Seidl, D. Tuppinger, D. Petrascheck,R. Scherm, Z. Phys. B 69 (1987) 313.</p><p>[2] A. Io!e, M. VraH na, Phys. Lett. A 231 (1997) 319.[3] A. Io!e et al., Physica B 241}243 (1998) 130.[4] A. Io!e et al., Phys. Rev. A 58 (1998) 1475.[5] A. Io!e, M. VraH na, V. Zabiyakin, J. Phys. Soc. Japan. A 65</p><p>(Suppl.) (1996) 82.</p><p>402 M. Vra&amp; na et al. / Physica B 283 (2000) 400}402</p></li></ul>