Wave-particle properties measured by neutron interferometry

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  • 156

    Nuclear Instruments and Methods in Physics Research A284 (1989) 156-160North-Holland, Amsterdam

    Section V. Neutron optics and quantum mechanics


    H. RAUCHAtominstitut der sterreichischen Uniuersaten, Schttelstr. 115, A-1020 Wien, Austria

    Coherence phenomena of neutron waves have been investigated by means of perfect crystal interferometry . A perfect siliconphase shifter was used to obtain information on the longitudinal and transverse coherence lengths and to observe the strong variationof the phase shift near a Bragg direction. Interference experiments with pulsed beams displayed properties of the relative wavepacket, and the self-interference character of experiments of this land . Experiments with an absorber-detector in one coherent beamgave information about the unsharp particle-wave behaviour of neutrons .

    1. Introduction

    Interferometry with massive particles very clearlyshows the conceptual difficulties of the particle-wavedualism of quantum mechanics. Although the neutronhas well known particle properties, a wave picture ex-plains all interference experiments more easily. In thisframework, the relevant quantities are: Compton wave-length, De Broglie wavelength, coherence length, packetlength, decay length and relative phases .

    Neutron interferometry has been developed on thebasis of slit diffraction [1,2], perfect crystal diffraction[3,4] and ruled grating diffraction [5] . In this article, wewill focus on perfect crystal interferometry becausemost experiments performed up till now have used thistechnique. It provides a wide beam separation andrather high intensities due to its nondispersive responseon the neutron beam . High contrast and high-orderinterferences have been observed and it has developedinto a standard method for fundamental [6,7], nuclear[8,9] and solid state physics [10] applications .

    Up till now, only monolithic silicon crystals havebeen used for neutron interferometry, whereas for X-rays, separated crystals have also been successfully used[11,12] . A rather versatile monolithic interferometer isshown in fig. 1, where, due to the skew symmetric cut,the space for the sample is larger than in the standardsymmetric version.A phase difference between the two coherent beams

    can be produced by nuclear, gravitational, magnetic orelectromagnetic interaction . The nuclear phase shift canbe written as X = k(n - 1)D = -NbcXD, where n isthe index of refraction, n = K/k = 1 - VNbc/21T, N isthe particle density, b,, is the coherent scattering length,X = 2m/k is the wavelength and D is the optical pathdifference between the two beams. A significant gravita-tional phase shift exists for nonhorizontally orientedinterferometers, due to the Earth's gravitation and rota-

    0168-9002/89/$03.50 Elsevier Science Publishers B.V .(North-Holland Physics Publishing Division)

    tional effects [12,13] . The magnetic interaction (H=-,uB) causes a phase shift a = yBD/v, where y isthe gyromagnetic ratio of the neutron, B is the mag-netic field strength and o the velocity of the neutrons . Itcan be interpreted in terms of a 41T-symmetry of spinorwave functions [13,14] and permits the realization ofmany typical spin-interference experiments [15-17].Most electromagnetic contributions to the phase shiftcancel out in the forward direction, due to the electricalneutrality of the atoms.

    It follows from the dynamical diffraction theory, andit can be understood from general symmetry considera-tions, that the wave function of the beam in the forwarddirection behind the interferometer is composed of equalwave functions ( = %P 11 ) coming from the beam paths1 and 11 (I0a 0 + 0 2 ) and, therefore, an appliedphase shift causes a complete modulation of the beam,



    Fig. 1 . Sketch of the skew symmetrically cut perfect crystalinterferometer with the indication of various sample arrangements. A: standard rotation, D = Do /[cos(B B + e) -cos(BB -

    e)] ; B: sweep, D = Do ; C - nondispersive, D = Do/sin BB .

  • 2 . Coherence aspects

    ,P(r, t) = (21T)-3 fF(k, t) e, ( k- ") dk,

    +10 3










    H. Rauch / Wave- particle properties measured by neutron interferometry

    to cc (1 + cos X) . Imperfections of the whole setup (smalllattice distortions, variations of the geometry, vibra-tions, absorption) and of the neutron beam itself (wave-length spread, divergency) cause a reduction of thecontrast and the appearance of an internal phase Xogiving Io a A + B cos(X + Xo) . In certain cases, theimaginary part of the scattering length and of the phaseshift must also be accounted for (X = X' + iX", whereX" = a,ND/2 and v, is a kind of removal cross sectionincluding absorption and scattering processes) . In thisarticle, some more recent experiments using perfectcrystal interferometers are discussed which have notappeared in previous review articles on this topic[7,18-20] .

    Although there is a closed analogy to optical phe-nomena, some differences due to the dispersivity of freeparticle movement appear . All neutron interferometerexperiments belong to the extreme self-interference do-main, because the phase space density is very low,generally below 10-16 [21] . Coherence is an ensembleproperty of similarly prepared particles . The relatedwave function is able to describe a particle in such anensemble :

    where w = hkz/2m describes the dispersive motion . Inany stationary experiment, the shape function F(k, t)does not depend on time and, therefore, a detectorregisters :

    I= f JiP(r, t)1 2 drdt=(2ar) -3 f IF(k) 1 2 dk.


    F(k) depends on the preparation procedure and be-


    comes very anisotropic due to different collimation anddue to crystal reflection, which only influences thecomponent parallel to the reflecting lattice vector . Theaction of a phase shifter depends on the orientation ofits surface relative to the incident wave [22] .A phase shifter oriented with its surface perpendicu-

    lar to the neutron beam scans the coherence of thebeam along the interferometer axis (z-axis) . If the inci-dent beam in this direction has a Gaussian width, Skz ,centered around kzo , the intensity is given by :

    I ac eX1cosh Xo + e-cx)2csk : /kai2/z cos X, .


    This formula shows that the visibility of the interferencepattern decreases drastically if I X0Skz/ko I >_ vf2- , or atinterference orders m >_ kv/(rTr8kz ) . This phenom-enon has been observed experimentally [23,24] and de-fines the coherence length of the beam in this directionas A, = mX o = r2 /8k., which is closely related to theuncertainty relation . Coherence even persists for largerphase shifts, but the interference pattern disappears, butit can be recovered by proper phase echo methods [25] .

    When the phase shifter is oriented with its surfaceperpendicular to the lattice vector of the reflectionplane, it acts on the highly reflection-modulated beamcomponent and the phase shift becomes independent ofthe wavelength to first order, because D = Do/sin BBand X = 2d,, k , sin BB, which gives X'= -2dhk,NbcDo[26] . In this so-called nondispersive arrangement theintensity is obtained as follows [22] :

    I ac e- Xo

    cosh Xo + I 1 +dhkIX0 )


    \\ TrA

    X e-dek1X0/mA cos X 't.


    The coherence length in this direction (x) is much


    0 2 4 6 8

    0 - O B

    (min arc)Fig. 2 . Interference pattern near a Bragg diffraction position of a perfect Si crystal [28] .



  • 158


    dx =A sin BB, where A

    is the characteristiclength for this reflection ; A (Si) - (2dhk,b~N)-t , whichgives for the (220) reflection 13 ~tm.

    The method of nondispersive sample arrangementhas been used in a recent measurement with a perfectsilicon crystal as a phase shifting material . If the inci-dent beam approaches a Bragg position of crystal dif-fraction, the internal K-vector strongly deviates fromthe value given by the index of refraction and variesalong the dispersion surface, which affects the inter-ference pattern in a nonstandard way [27,28] (fig . 2) .The surprisingly high contrast near to the exact Braggdirection should be mentioned. The disappearance ofthe interference pattern near the exact Bragg directionis due to the reduction of the transmitted beam (X" -*oo). The steeper interferometer response near the Braggdiffraction corresponds to an enhanced sensitivity ofthe system.

    These experiments and the preceding discussion haveagain shown that coherence is determined by the vari-ous preparation tools applied to the beam . The experi-mental conditions determine the parameters of F(k)and, therefore, the wave function describes a particle ina defined ensemble, or, equivalently, a particle underthe experimentally defined conditions . Coherencelengths can also be defined by autocorrelation functionsof the related wave functions [29] and they are intrin-sically connected to the (incoherent) momentum distri-butions [30] .

    3. Pulsed neutron interferometry

    H. Rauch / Waoe-particle properties measured by neutron Tnterferometry

    in this case, the neutron wave or the interaction hasa marked time dependence . Pulsed beams can be pro-duced by pulsed neutron sources or by various neutronbeam choppers. Neutron bursts are produced with burstlengths 8x(0), which are usually much larger than thecoherence lengths discussed in the previous section . Thespreading occurs due to the dispersivity of free particlemotion according to the velocity spread 8v of thebeam :

    [8x(t)]2 = [8x(0)]2 + +[w.t]2,

    (5)which can be understood classically as the spreading ofa swarm of particles. If a chopper varies the velocityspread and produces a minimum uncertainty wavepacket (8k8x(0) = 1/2), one can rewrite eq . (5) :

    [8x(t)]2=[8x(0)]2 + [ht/2m8x(0)]2 . (6)

    In this limiting case, the packet length equals thecoherence length. Related experiments are in prepara-tion [31,321 .

    Here we will describe an experiment related to eq .(5), where the burst length (8x(0) = 1 .5 cm) is muchlarger than the coherence length (d z = 400 f1) but








    Fig . 3 . Sketch of the arrangement for pulsed neutron inter-ferometry [33,34] .

    smaller than the dimensions of the interferometer (10cm, fig . 3) . In this case, the wave functions in both partsof the beam are completely disconnected, because evenall the plane wave components (eq. (1)) are cut by theaction of the chopper. The mean wavelength of theneutron registered in a time channel t around the meantime-of-flight to = L/v, follows from eq . (5) as :

    (X(t)) =,\o11 +to(t - to)(o

    ) 2/(A

    t) 2J

    > (7)

    where A t represents the time resolution of the wholesystem. Thus a time-resolved measurement correspondsto a higher resolution experiment and, therefore, to ahigher visibility of the interference pattern according toeq . (3) . Such an experiment has been performed at aburst repetition rate of 40 kHz, at a wavelength X =1.900 0.007 f1, with an initial burst length of 1 .3 cmand a mean occupation number of 0.0024 neutrons perburst (fig. 4) [33,34] . The recovery of the interferencecontrast in the time-resolved case and the dispersivephase shift according to eq . (7) have been demonstrated .Experiments of this kind can be seen as a first steptowards delayed choice experiments with matter waves[35,36] .

    4. Unsharp particle-wave behaviour

    The detection of a particle in a quantum mechanicaltwo-slit experiment corresponds to a beam path detec-tion, which is complementary to the detection of waveproperties by means of the interference pattern. It hasbeen shown theoretically that the beam trajectory andinterference can be measured, at least approximately[37,38] . A partially absorbing material in one beamcauses a distinct degree of beam path detection, becauseany absorption process causes a collapse of the wavefunction and represents a measuring process where thefollowing capture gamma radiation can be registered .

  • O



    Fig. 4 . Interference pattern observed with a pulsed beam without (left) and with time resolution (right) and the dispersitivity of thephase shift measured with phase shifters of different thicknesses [33,34] .

    " 10

    1 84


    1 76

    1 72

    1 68

    1 64


    The related beam attenuation has to be described by animaginary part of the phase shift X" = arND/2. Re-lated measurements were performed at a low inter-


    01 02 03 04 05D (mm)

    H. Rauch / Wave- particle properties measured by neutron interferometry

    1 56I

    10 01 02 03 04 05


    (m m)


    Fig. 5. Interference pattern without (above) and with a 99 .5`efficient absorber-detector in one beam path [40,41] .

    ference order (XOSk,/ko 1) . Now eq . (3) can berewritten as :

    I oc [(1+a)+2~_a cos XO] ,

    where a is the usual beam attenuation factor a = I/Io= exp(-arND). In contrast to this stochastic measur-ing process with a partially absorbing detector, a de-terministic method can be applied by varying the slitwidth of the beam or by moving the chopper disk with adefined open-to-closed ratio which defines the beamattenuation factor a . In this case we obtain :Icc [(l+a)+2a cos X ],where the amplitude of the interference pattern variesproportional to a, in contrast to the 4 behaviour in






    Fig. 6 . Normalized amplitude of the interference pattern vs thetransmission probability of the absorber-detector at very low

    transmmssion probabilities [41].


  • 160 H. Rauch / Wave-particle properties measured by neutron interferometry

    the statistical case . Related experiments have verifiedthis behaviour [39,40] and have recently been extendedto the region of extremc-ly low transmission probabilityand corresponding high beam path detection probabil-ity (a


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