The Wave-Particle Dualism || Neutron Wave Optics Studied with Ultracold Neutrons

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    A. Steyerl

    Technische Universitat MUnchen, D-8046 Garching, Germany

    The paper discusses experiments demonstrating or utilizing the wave properties of neutrons with wavelengths of about 100 nm.


    The preparation of the present paper was difficult for me mainly because I tried at first to relate our experimental data on ultracold-neutron optics, which are simple, to that difficult subject: "non-linear wave mechanics", which has been created by de Broglie over 50 years ago (see /1/) and which seems to be re-sumed nowadays at a time of rapidly growing interest in non-linear phenomena (e.g. /2/). I must confess that for a simple-minded ex-perimental physicist like me it turned out that this programme was too difficult. Therefore, I finally decided not to annoy you with, perhaps, unsound interpretations but present the plain data. I should add that this restraint was facilitated by the fact that, within experimental precision, we did not observe any deviation in the behaviour of ultracold-neutron waves from the predictions made within the framework of standard, linear, quantum mechanics, and therefore an urgent need of possible nonlinearities has not arisen.


    The designation "ultracold" is commonly used for extremely slow neutr"Ons with ~elocitie~ below "'10 m/s. Their energies are of the order of 10- eV (10- K) or below and their d~ ~roglie \'Iavelengths reach nearly macroscopic demensions, ,v 10 A. Ultra-cold neutrons (UCNs) cannot penetrate matter as easily as faster


    S. Diner et al. (eds.). The WaveParticle Dualism, 85-99. is) 1984 by D. Reidel Publishing Company.

  • 86 A.STEYERL

    neutrons, and they suffer total reflection at the walls of suit-able substances, at any angle of incidence. This unique property makes them suited for storage in closed cavities, the so-called "neutron bottles". The neutrons are contained in carefully pre-pared vessels for hundreds of seconds, travelling to and fro be-tween the walls from which they rebound many thousands of times before loss processes due to reactions with the wall nuclei and the finite lifetime for beta-decay become significant /3 - 6/. Storage lifetimes reaching the limit of the fl-decay lifetime have been obtained in a "magnetic storage ring" /7/, and an improve-ment on the lifetime value seems to be possible.

    The main applications of UCNs envisaged until now are high-precision investigations of the properties of the neutron itself, like its lifetime for fl-decay and searches for a possible elec-tric charge and dipole moment. The low energy of UCNs seems to make them suitable for high-resolution spectroscopy, and a third branch of attractive physics with UCNs seems to be neutron optics which is the subject of the present paper.


    Dealing with neutrons of extremely low energy we must take into account a number of interactions which are usually negli-gible in thermal or fast neutron research because of their weak-ness, but play an important role at very low energies. These in-teractions include the gravitational potential (mg ~ 10-7 eV per m of height), the interaction of the neutron magnetic dipole mo-ment ~ with a magnetic field B (~ ~ 6 x 10-8 eV/tesla) , and the interaction with the "scattering potential" U which arises as an effect of multiple coherent scattering in the forward direction and gives rise to refraction and total reflection, as in light optics. For most substances, U is of the order of 10-7 eV. These various interactions may be incorporated in a spatially variable index of refraction, n{r) = [1 - (U(r) + ~B(r) + mgz)/E ]1/2. The sign of the magnetic term + ~B is negative for neutPon spin antiparallel to B and positive for ~ liB. (The magnetic inter-action can be inCluded in this sjmpTe Torm whenever the "adiabat-ic condition" is satisfied, i.e., changes of magnetic field direction "sensed" by the neutron along its trajectory occur slowly compared with the frequency of Larmor precession in the local magnetic field.)

    As a consequence of the gravitational interaction even empty space in the absence of magnetic fields may be considered as a refracting medium with a spatially variable index of refraction. Therefore, the optics of ultracold neutrons deals with curvilin-ear rays, and this introduces new features into the usual frame-work of optics.


    UCN Source (From "Neutron Turbile")

    Fi g. 1:


    Ruled Grating or Mirror


    Scheme of the "gravity diffractometer" for ultracold neutrons. The entrance slit is imaged onto the exit slit.



    The gravitational interaction has been utilized for UCN beam focusing in high-resolution experiments studying interference, diffraction and inelastic scattering phenomena.

    Fig. 1 shows the scheme of the "UCN gravity diffractometer" /8,9/ installed at the Research Reactor at Garching near Munich. In this instrument, the vertical component of neutron velocity is precisely determined by the height of fall from the entrance slit to the horizontal mirror which may be replaced by a reflection sample or a combination of transmission sample and mirror. The beam spreading associated with the free fall from the entrance slit to the sample is undone in the phase of ascent from the sample to the exit slit. Thus the entrance slit is imaged onto the exit slit. The instrument admits of a resolution of 2 neV for the energy corresponding to the vertical neutron motion.

    Before proceeding to the neutron-optical studies made possi-ble with this high resolution, let me mention another scheme of

  • 88 A.S1EYElU.

    UCN focusing which we applied in a "gravity spectrometer" /10/. With this instrument quasi-elastic neutron scattering can be in-vestigated with a resolution of a few neV in energy transfer. This resolution which is much higher than in conventional neutron scattering spectrometers, is achieved by exploiting the simple fact that the maximum reach of the flight parabola described by a very slow neutron in the earth's gravitational field at a launching angle of 45, is a very sensitive measure of its ini-tial kinetic energy. An instrument working according to this prin-ciple has been installed at Garching and a scheme of its design is shown on Fig. 2. It seems worth mentioning that this scheme admits of focusing both in energy and in space. In the monochro-mator, the neutron source is line-to-line focused on the sample, and in the analyser the sample is point-to-point focused onto the detector, using the focusing properties of the slightly ellipti-cal, cylindrical analyser mirrors which divide the analyser flight path into five segments. Energy scanning is accomplished by displacement of the monochromator mirror, thus effectively changing the reach of the monochromator flight path.

    Fig. 2: Scheme of the "gravity spectrometer" NESSIE (for "NEutronen-Schwerkraft-SpektrometrIE"), in whiCh the maxTmum reach oT the flighr-para-bola described by the ultracold neutrons is used for precise energy selection


    The spectrometer is being used for measuring quasi-elastic line broadening (by only a few neV) in polymer solutions. Fig. 3 shows. as a first example. the quasi-elastic scattering from a polydimethylsiloxane (0.05 g/cm3 ) solution in deuterated benzene (C6D6) at 50 C (above the critical "8"-point). and this data is compared to the resolution curve measured with a purely elastic scattering line (electro-graphite). The reason for showing this - preliminary - data in a talk on neutron optics is that it seems to demonstrate the precise functioning of the spectrometer in the way it was designed. The calculations for the fairly complicated neutron paths had all been based on classical mechanics and (curved-) ray optics. assuming an ideal geometry.



    ~~ / \ !l7.3 t O.8) neV M Graphite

    ...... 10 /il 4)\

    _0/ 0 ~~-----~~------------~Q~ Ul -C :::::I o


    8 0 >.

    -.~ Polymer $ 10 in Solution

    ~. -~ 6 -~?t? '~~~-~-----

    ? 4 (20*3.5) neV 2

    o 3~80--~--40~0--~-4~~------4~4-0~~-4~~----~48~0--~-S~oo~--

    Energy [neVJ Fig. 3: The energy resolution curve of NESSIE measured

    with an elastically scattering graphite sample is compared to the slightly broadened scattering curve for a polydimethylsiloxane solution in deuterated benzene. The curves are plotted vs the incident neutron energy and are centred on the analyser energy of 412 neV.

  • 90 A. STEYERL


    The experiments on UCN diffraction and interference to be discussed now. had been initiated at a time when the early UCN storage experiments had yielded surprisingly short containment lifetimes and a number of suggestions and speculations had been put forward to account for the apparent anomaly. The most drastic idea was that the observed imperfection of neutron reflection from the bottle wall was perhaps an indication of a fundamental fault in the simple analysis in terms of standard wave mechanics (Ignatovich /11.12/). 5.1 Diffraction from a Ruled Grating

    In a first direct experiment testing the interference pro-perties of ultracold-neutron waves. we measured the diffraction from a ruled reflection grating /8.9/ with the high resolution provided by the "gravity diffractometer". The grating had 1200 mechanically ruled grooves per mm and the groove profile was appropriate for the "blazing" condition in first-order reflection. The grating was coated with nickel (a good neutron reflector) and arranged vertically in the "gravity diffractometer" at the posi-tion indicated on Fig. 1. Since the action of the grating can be understood as the transfer of momentum. hn/d. parallel to the sur-face (d: groove spacing. n: diffraction order). the neutron re-ceives a well-defined push up or down. for n F O. This momentum transfer changes the height of the ascending flight parabola, and this can be sensitively analysed by a vertical displacement of the scanning exit slit. Fig. 4 shows intensity profiles measured in this way. The observed orders of diffraction are clearly sepa-rated. and the linewidths may be fully explained by the instru-mental resolution.

    We have, very tentatively. interpreted the absence of a detectable line broadening in terms of a lower limit for an "in-trinsic coherence length for the neutron wave train". Such a hypo-thetical limit - beyond those determined by the instrumental re-solution and by the finite neutron lifetime - does not exist in ordinary wave mechanics. and I don't have any idea what kind of theoretical modifications would be required to account for a speculative finite "coherence length". Nonlinearities, as proposed by de Broglie /1/ or Bialynicki-Birula and Mycielski /2/ would certainly require drastic revisions in our present view of quantuM phenomena. Thus, in the absence of a clear model even our inter-pretation of the absence of line broadening in terms of a lower limit for the "coherence length" seems doubtful, because even this interpretation may depend on the model. Therefore, the value of NO.1 mm derived from the minimum number of coherently illumina-ted grooves necessary for the observed linewidth, on the basis of the linear theory, should be taken only as a guide number.


    (For the specific, logarithmic, non-linearity, -bf~ al~12, proposed by Bialynicki-Birula and Mycielski /2/ this would corre-spond to a limit for the constant b of 10-15 eV, similar to the limiting value recently reported by Gahler, Klein and Zeilinger /13/ and 100 times lower than the limit given by Shull et ale /14/. However, the proposed modification of theoSchrodinger equation must be understood to be on the classical level, since this equation has, e.g., non-quantized oscillatory "breather" mode solutions, and it would, therefore, seem to require quanti-zation. In the given form for the non-linear term,~(lvI2) (logarithmic or other), the diffraction process, o/partial reflection or any scattering from fixed objects, would require energy for wave dispersion in space, which poses problems regard-ing the proposed probabilistic interpretation of the wave function. At any rate, a non-stationary solution of the non-linear equation for reflection from a potential wall, nothing to say about diffraction from a ruled grating, has apparently not yet been found. For all these reasons, I would rather hesitate to interpret our simple experiments in terms of something as complicated as non-linearitiesl)

    50r n= 1 n=O n=-1

    ~ *

    ~ .r.

    -40 1/1

    -c:: ~q+~ ':) l~\# 0 Col 30 q .+++. 9 ?: 9 t~

    "iii +

    9 9 q + .+ ... .+ +tf~ + H 20 + ~9~ c:: ~ c::


    t + ,-\ I \ ,

    -\ I \ : \ I \ I \ I \ I

    I \ I \ I \ \

    -30 -25 - 20 -15 -10 -5 5 10 15 Relative Slit Position in em

    Fig. 4: Several orders of diffraction from a ruled grating (1200 grooves per mm), measured by vertical scanning of the exit slit in the "gravity diffractometer". Resolution curves are included for comparison.


  • 92 A. STEYERL

    5.2 Mirror Reflection

    The main reason for mentioning the next, even simpler, experiment is that it seems to have provided one of the first direct clues as to the reason for the short UCN containment life-times in "bottles". We placed a float glass plate at the horizon-tal sample position in the "gravity diffractometer" (exchanging the grating by a vertical mirror). The reflection curve measured as a function of the neutrons' height of fall to the sample is shown in Fig. 5 /8/. It exhibits the typical steep edge at the critical height of fall which is determined by the limit for total reflection. However, the measured slope is considerably steeper than that calculated for reflection from a potential step function, i.e., a sharp transition from the vacuum to the bulk glass. The most plausible interpretation seems to be in terms of surface contamination.




    ., .c;

    80 -.. c: e 'S 60 Z


    ~ 40 ! c:


    0 74 78 82 86

    Variation of primary intensity ... T-- _ _ 1 ___ _ I I

    , I I I I

    ~ \ \

    \+ \+ \ ,


    Assumed wall potentials

    d= 73' J& b) +'i.t.. rL '~j

    hcr =f93.6 t o.l)cm - .. _--f-----

    90 94 98 102 106 110 "eight of fall (mJ

    Fig. 5: Reflectivity vs neutron fall height in the "gravity diffractometer" for a float glass sample. The data points are compared to calcu-lations for a) a step-function potential distribution, and, b) a potential distribution smoothed due to a hydrogenous surface contamin-ation with a gradual change in composition.


    Hydrogenous substances had for a long time been considered a prime candidate for UCN losses at the bottle walls, because of the extraordinarily high cross section for inelastic neutron scattering from protons, but some authors were reluctant to accept the large quantities of hydrogen necessary to explain the data (e.g., /15/). The reflection data in Fig. 5 can be explained by a smoothed wall potential due to an inhomogeneous contamination layer, and for a specific analytical form for the potential distribution near the surface /8/ an effective thickness of (73 3) A for the transition is required to obtain a good fit to the data (dashed curve in Fig. 5). The corresponding amount of hydrogen is sufficient to account for the containment lifetimes reported in most of the experiments.

    This conclusion was corroborated by independent experiments using different approaches /16, 17/. Recently, Mampe, Ageron and Gahler succeeded in directly demonstrating long containment life-times (several hundred seconds) in bottles after glow-discharge cleaning of the surfaces /18/.

    5.3 Interferences at a Single Thin Film

    As another approach to testing the interference 'properties of ultracold neutrons we examined the thin-film interference pattern for the reflection from a glass substrate with a 250 nm thick, homogeneous, evaporation film of gold /8/. The data taken at the "gravity diffractometer" is shown on Fig. 6. The interferen~e pattern is clearly established and agrees with the simple plane-wave theory, allowing for resolution effects. The fitted numerical value of (268 6) nm for the film thickness agrees with the quartz data taken during evaporation.

    5.4 Neutron Resonances in More Complex Stratified Systems

    An extension of the thin-film experiments to more complex stratified systems was undertaken with a view of possible applications as narrow-band interference filters. In addition, such systems offer the opportunity to study elementary quantum phenomena in synthetic, one-dimensional, potential distributions with almost macroscopic dimensions.

    A simple example is a sequence of three thin films such that a layer of a material with a low scattering potential for neutrons (e.g. aluminium, for which U = 53 neV) is sandwiched between two identical layers of a medium with a high scattering potential (e.g. copper, for which U = 165 neV). Such a composite film con-stitutes a potential well in which neutrons with an appropriate energy of the motion perpendicular to the film plane may form quasi-bound states. These states can be studied by observing the resonance behaviour of reflection from, or transmission through, such samples, as in an optical Fabry-Perot interferometer. Simi-larly as in light optics, resonances are expected for neutron

  • 94




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    ~Xl '" c ..

    'E 20


    0 1!6 90



    --. --

    her =(113.0~0.2)em

    9S 100 lOS 110 11S 120 12S 130 135 Height of fall [em]

    Fig. 6: High-resolution interference pattern for UCN reflection from a thin, homogeneous, gold film evaporated on glass. The solid curve 2 is a calculation with an adjusted film thickness.

    wavelengths "matching" the width d of the potential well, and sharp resonances require, in addition, a high reflectivity of the "mirrors" (which are represented by the high-barrier copper films). The quasi-classical approximation yields the resonance condition

    kz -g (7r'/d) (n + 1/2) [n = O,1, J for the perpendicular wavevector component kz within the well, and k~ should lie within the region of total reflection for copper. Even ln conditions of "total reflection" a slight transparency of the "mirrors" is ensured by the finite probability of tunneling through the thin copper films.

    The samples were prepared by successive evaporation of the thin layers (-20 to 200 nm) on glass or silicon substrates, and experiments were performed both with col d neutrons (A ~ 10 A) at grazing incidence and with ultracold neutrons (~ - 103 A) incident at large angles in the "gravity diffractometer" /19,20/.





    " ~l - 30~ c:: ::J 0 U 0 -~


    --~ n-I - -- -- -_ _ r t 3.9 neV Reference

    + -----1- Intensity '+\J. -------1. __

    T t -----~ +/+- f-:-2r-:( ~ I Cu Cu / 240A 240A

    \ ~ ~1t-V ~t. t::::! 5mm ~ IOOA 860A Glass Al Al

    1~4~~----------------~------------~----------------~----------------r-90 100 110 120 130

    Fall height [cm)

    Fig. 7: Intensity reflected from a sample with the indicated layer sequence, which corresponds to a two-hump potential. The narrow minimum


    is due to the (n = 1) - state in the artificial potential well.

    Fig. 7 shows the reflected intensity measured in the "gravity diffractometer" for a target with the following layers succes-sively evaporated on glass: Cu (24 nm), Al (86 nm), Cu (24 nm), and, as a protection from oxidation, Al (10 nm). The potential function representing this sequence is shown in the inset. The observed pronounced reflectivity minimum at a fall height of 108.5 cm corresponds to n = 1 (the second stationary state). The data is well represented by the exact solution of the one-dimensional Schrodinger equation for the multistep potential, including the small instrumental resolution broadening (solid curve). The measured linewidjh of 2r = 7.7 neV corresponds to a lifetime of ~ = ~/2r ~ 10- s for the quasi-bound state. The most plausible explanation for the noticeable enhancement of minimum reflectivity at resonance seems to be a slight roughness of the film interfaces /21/. Similar deviations were observed with all the other samples.

  • 96 A.STEYERL

    ,.., '\ 1\ ~ 1.0 I, n -_ 0 '\ 220 I, I , C , , I , 5 0.8 Q) , , I,

    170A 200A 170t.. Cu Cu Cu

    -.. .... --t ~ --I S--

    u 16 u , , I, L...I c: I, I ~ I Si ~ 12 0.6 g :', '/ I 'I U __ l. l. __ I J L VI .- Q. I I \'-1"'~ ~~ ..... ~ E 'Ii' ' I ~ 1 \1 '" AI:600A /0.25 "E 8 0.4 g I 19\ ... ~~ I 9, AI: 100A mm

    \oJ ~1'9 I~' ,!::: V I- -I 9.

    4 0.2 9 Ll=6.3 neV ~ o ?-+-+j!- ------------~:::~::-y +

    65 70 75 80 85 90 95 E [neVJ

    Fig. 8: Level splitting observed in the UCN trans-mission through a coupled-resonator potential created by the layer sequence indicated in the inset

    As an extension of the above Fabry-Perot-type neutron-wave resonator we also studied a system of two coupled resonators, created by two identical potential wells arranged in series and weakly coupled by a common, sligthly transparent, potential barrier. Such a potential distribution may be generated by a layer sequence Cu-Al-Cu-Al-Cu, and we expect a "tunneling" splitting of the single-resonator states into two each. These states are characterized by a symmetric and an anti symmetric wavefunction, respectively. This is similar to the inversion splitting in the ammonia molecule, or, simpler, to the mechanical example of two coupled pendulums.

    Fig. 8 shows the results for the UCN transmission through a target with the indicated structure, plotted vs the energy of vertical neutron motion in the "gravity diffractometer". The data clearly shows a splitting of the (n = 0) - state by only 6.3 neV. The measured points are compared to the calculation for mono-energetic neutrons on the basis of the multistep potential function, for the nominal film thicknesses and potential heights (dashed curve). An improved representation (the solid curve) is


    obtained by taking into account, mainly, the instrumental reso-resolution and a roughness scattering whose strength was adjusted to the data but appears to be plausible /20,21/. Again, we do not have to invoke a fundamental deviation from the elementary analysis in terms of plane waves.


    As a last example of UCN optics I should like to discuss, briefly, the possibility of image formation of an object and the prospects of a possible neutron microscope. The main incen-tive for work on the development of a neutron microscope arises from the interest in exploring the information to be obtained from the contrast in matter, as "seen" by neutrons - which is quite different from the contrast for light or electrons, and which is adjustable by proton-deuteron exchange. For thermal or cold neutrons, however, no satisfactory focusing systems are available, and it is only with ultracold neutrons, owing to their property of being reflected from mirrors even at large angles of incidence, that high magnifications at tolerable aberrations appear to be attainable. The wavelength limit to the resolution is of the order of 100 nm, i.e. about one order of magnitude below that for visible light, but with presently existing UCN sources a much more severe limitation is imposed by the small available intensity.

    In designing systems for magnified image formation with UCNs we are faced with the problem of gravity and, hence, must deal with the optics of curvilinear rays. For example, a concave mirror, which is an achromatic system in light optics, would exhibit a chromatic aberration if used with UCNs because their ray curvature strongly depends on the wavelength.

    A possibility of compensating this effect in a single-component magnifying system consists in "crossing" a concave mirror with a Fresnel-zone plate to obtain a reflecting, concave, "zone mirror" /22/, as sketched in Fig. 9. By choosing an appropriate geometry such a device can be made to achromatically diffract the UCN intensity incident from a point object into one image point. The zone mirror then exhibits much the same proper-ties as a lens or a concave mirror in light optics. This has been demonstrated in an experiment where we obtained sharp images of an object slit with a magnification of up to six /23/.

  • 98

    Glass Substrate ~r,x n 1, N . , +'2


    I I 1 Layer rrx,n- 12 I

    Optical Axis



    Fig. 9: Scheme of the "zone mirror", which provides achromatic image formation with ultracold neutrons.

    At present we are testing, in an experiment at the Institut 'Laue-Langevin at Grenoble, a different scheme of achromatic imag-ing with UCNs, which should admit of much higher magnifications /24/. In this instrument the chromatic aberration of one, parabo-lic, mirror is compensated by that of a second, spherical, mirror. This scheme includes the possibility of using multilayer mirrors which reflect 2 to 3 times faster neutrons than ordinary mirrors, and this could help to increase the available beam intensity.



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    Dubna, E4-8039, unpublished 12 Ignatovich, V.K.: 1977, Preprint Joint Inst. Nucl. Research,

    Dubna, R4-10650, unpublished 13 Gahler, R., Klein, A.G. and Zeilinger, A.: 1981, Phys. Rev.

    A 23, p. 1611 14 Shull, C.G., Atwood, D.K., Arthur, J. and Horne, M.A.: 1980,

    Phys. Rev. Lett. 44, p. 765 15 Luschikov, V.I.: June 1977, Phys. Today 42, p. 42 16 Stoika, A.D., Strelkov, A.V. and Hetzelt, M.: 1978, Z. Physik

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