Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 315–317www.elsevier.com/ locate/elspec

¨N eutron–electron scattering length deduced from Pendellosunginterferometry in neutron Bragg reflections on silicon

a,b , a*J.-M. Sparenberg , H. Leeba ¨ ¨ ¨Atominstitut der Osterreichischen Universitaten, Technische Universitat Wien, Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria

b ´ ´ ´ ´Physique Nucleaire Theorique et Physique Mathematique, Universite Libre de Bruxelles, CP 229, Campus de la Plaine,B-1050 Brussels, Belgium

Abstract

A new method to deduce the neutron–electron scattering length from high-precision thermal-neutron measurements of thenuclear scattering length and of the scattering amplitudes of Bragg reflections is proposed. It consists of an extension of a

¨Pendellosung interferometric technique proposed by Shull in 1968 to higher-order Bragg reflections in silicon. Error-barestimates show that it could solve the ambiguity between disagreeing experimental values of the neutron–electron scatteringlength and provide a very precise temperature factor of silicon. 2003 Elsevier Science B.V. All rights reserved.

Keywords: Neutron charge radius; Neutron–electron scattering length; Neutron interferometry; Silicon; Temperature factor; Bragg reflection

1 . Introduction: neutron material science implies a high penetration in condensed matter, (ii)their spin and magnetic dipole moment, which allow

In the next 10 years, very intense accelerator- their polarization and their use for the study ofdriven neutron sources will be available, for instance magnetic materials, (iii) their wavelength, similar toin the United States of America (Spallation Neutron atomic spacing in condensed matter, which allowsSource) and in the European Union (European structure studies, (iv) their energy, similar to elemen-Spallation Source). Such machines will boost the tary-excitation energies in solids, which allows thedevelopment of the neutron material science, i.e. the study of atomic motion, and (v) their sensitivity tostudy of materials through thermal-neutron scatter- nuclei (rather than to electrons), which allows theing. These techniques are complementary to electron, study of materials rich in light atoms such asphoton or ion material sciences, the main advantages biological molecules[1].of thermal neutrons being (i) their neutrality, which

2 . The neutron–atom interaction

*Corresponding author. TRIUMF, 4004 Westbrook Mall, Van-The interaction of thermal neutrons with atoms iscouver, BC V6T 2A3, Canada. Tel.:11-604-222-1047; fax:

mainly due to the strong nuclear interaction with11-604-222-1074.E-mail address: jmspar@triumf.ca(J.-M. Sparenberg). nuclei, which can be characterized by the nuclear

0368-2048/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0368-2048(03)00084-7

316 J.-M. Sparenberg, H. Leeb / Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 315–317

scattering lengthb . The knowledge of this have recently proposed a new method to measurenuclear

quantity for each type of nucleus is thus in principle b , which could help solving this ambiguity[9]. Wene

sufficient for the analysis of neutron-material-science summarize this method below.scattering experiments. However, neutrons also inter-act with electric charges (as well with nuclei as withelectrons), a phenomenon characterized by the so-called neutron–electron scattering lengthb . The 3 . Proposed new experimentne

total neutron–atom ‘scattering length’ thus reads[2]Eq. (1) shows that, for neutrons scattered off

b(Q)5 b 2Zb 1Zf(Q)b , (1) atoms in the forward direction (Q 5 0), the electro-nuclear ne ne

static effect of the electronic cloud exactly compen-whereZ is the atomic number,Q is the transferred sates the electrostatic effect of the nucleus. Thismomentum of the scattered neutron andf(Q) is the property is exploited to precisely measurebnuclearatomic form factor normalized tof(0)5 1 with through interferometric methods on perfect crystalsf(`)5 0. The second and third terms of this equation [10]. For instance, a measurement on silicon[11]are due to the electrostatic interaction and are smaller provides b 5 4.1507(2) fm. SinceZ 514 fornuclearthan the nuclear term; the second term comes from silicon, the maximum electrostatic effect is aboutthe nucleus punctual charge while the third term 0.02 fm for Q 5`, as shown by Eq. (1). The valuecomes from the electronic cloud. This electrostatic of b can thus in principle be deduced frombne nuclearinteraction is due to the fact that, despite its neu- and b(Q) for Q ± 0, but the precision on thistrality, a neutron has a non-vanishing charge density, measurement has to be very high to get a reasonabler(r), where r is the distance to the neutron center. precision onb . Such a measurement was performedneThis charge density can be characterized by the by Shull in 1968[12,13], with the help of Pendel-2neutron mean square radiuskr l, which can ben ¨losung fringes in the (111) Bragg reflection on ashown to be proportional to the neutron–electron perfect silicon crystal. The typical error bar is 0.001scattering length[2]: fm, which implies a 10% error onb . In order tone

maximize the electrostatic effect in Eq. (1), we1 "c2 2 propose to extend this experiment to higher-order] ]]kr l5 E r(r)r dV5 3 b ,n 2 nee am cn Bragg reflections (largerQ). Moreover, in order toneutron

reduce the error bar onb , we suggest to measurenewhere " is the Planck constant,c is the speed of several Bragg reflections instead of one. However,light, a is the fine-structure constant andm is the this project raises several difficulties.n

neutron mass. First, this experiment is performed in Laue trans-In order to precisely estimate the electrostatic mission geometry with a thermal-neutron white beam

effect in neutron-material-science scattering experi- (typical spectrum range 80#l#250 pm). Thisments, a good knowledge ofb is required. More- imposes geometrical constraints (through the Braggne

2over, kr l is an important structure constant for the condition) which prevent one from measuring reflec-n

test of models describing the internal structure of the tions higher than (642). Second, since a white beamneutron. Despite this importance, available ex- is used, contamination between Bragg reflections hasperimental values forb disagree with one another: to be taken into account. These considerations leavene

the Argonne[3] and Garching[4] groups provide us with nine reflections[9] which can in principle be23b 5 2 1.31(3)310 fm, in agreement with the measured by Shull’s technique: (111), (422), (511),ne

Vienna–Oak Ridge collaboration[5], while the (531), (620), (533), (551), (711) and (642). Another23Dubna[6,7] group providesb 5 2 1.59(4)310 difficulty arises from the temperature effect[14],ne

fm. Moreover, the Argonne–Garching value is small- which is absent forQ 5 0 but which increases2er and the Dubna value is larger than the theoretical exponentially withQ , the actually measured quanti-

23estimate of Foldy[8]: b 5 2 1.4683 10 fm. We ty beingne

J.-M. Sparenberg, H. Leeb / Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 315–317 317

R eferences2Q]F S D Gb (Q)5 b(Q) exp 2B .meas 4p

[1] h ttp: / /www.sns.gov,Spallation Neutron Source Internet Site,Oak Ridge National Laboratory, TN, 2002.To get a good precision onb(Q), one needs a good

[2] V .F. Sears, Phys. Rep. 141 (1986) 281.precision on bothb (Q) and the so-called Debye–meas[3] V .E. Khron, G.R. Ringo, Phys. Rev. D 8 (1973) 1305.Waller temperature factorB. The present-day value[4] L . Koester, W. Waschkowski, L.V. Mitsyna, G.S. Samosvat, P.2for B is 4613(27) pm [15], which is not precise

Prokofjevs, J. Lambergs, Phys. Rev. C 51 (1995) 3363.enough for our purpose. The solution to this problem [5] S . Kopecky, J.A. Harvey, N.W. Hill, M. Krenn, M. Pernicka,is to deduce bothb andB from the measurement of P. Riehs, S. Steiner, Phys. Rev. C 56 (1997) 2229.ne

[6] Y .A. Alexandrov, T.A. Machekhina, L.N. Sedlakova, L.E.the nine Bragg reflections.Fykin, Sov. J. Nucl. Phys. 20 (1975) 623.

[7] Y .A. Alexandrov, M. Vrana, G.J. Manrique, T.A.Machekhina, L.N. Sedlakova, Sov. J. Nucl. Phys. 44 (1986)

4 . Conclusions: simulated results 1384.[8] L .L. Foldy, Rev. Mod. Phys. 30 (1958) 471.[9] J .-M. Sparenberg, H. Leeb, Phys. Rev. C 66 (2002) 055210.In Ref. [9], we estimate the error bars which could

[10] H . Rauch, S.A. Werner, Neutron Interferometry. Lessons inbe obtained on bothb andB by measuring the nineneExperimental Quantum Mechanics, Oxford Series on Neu-reflections mentioned above, assuming the sametron Scattering in Condensed Matter, Vol. No. 12, Oxford

precision as that obtained by Shull in 1968. We get University Press, New York, 2000.2an error onB of 2.7 pm , which is 10 times smaller [11] A . Ioffe, D.L. Jacobson, M. Arif, M. Vrana, S.A. Werner, P.than the error on the present-day value, and an error Fischer, G.L. Greene, F. Mezei, Phys. Rev. A 58 (1998)

23 1475.on b of 0.06310 fm, which is comparable to thene[12] C .G. Shull, Phys. Rev. Lett. 21 (1968) 1585.errors on the disagreeing Argonne–Garching and[13] C .G. Shull, J.A. Oberteuffer, Phys. Rev. Lett. 29 (1972) 871.Dubna values. Since the Dubna value is partly[14] V .F. Sears, Neutron Optics. An introduction to the Theory of

deduced from a Bragg-reflection experiment, while Neutron Optical Phenomena and their Applications, Oxfordthe other values are obtained with gas and liquid Series on Neutron Scattering in Condensed Matter, Vol. No.targets, the experiment proposed here would comple- 3, Oxford University Press, New York, 1989.

[15] P .J.E. Aldred, M. Hart, Proc. R. Soc. London A 332 (1973)ment existing ones and could shed new light on the223.discrepancy between experimental values.