Neutron optics?

Rep. Prog. Phys., Vol. 46, 1983. Printed in Great Britain

Neutron optics? A G Klein4 and S A Werner§ $ School of Physics, University of Melbourne, Parkville 3052, Australia

0 Department of Physics, University of Missouri-Columbia, Columbia, Missouri 6521 1, USA

Abstract A range of phenomena similar or analogous to those of classical optics is exhibited by slow neutrons. The aim of this review is to discuss this similarity and to display the results obtained in a wide range of experimental work on reflection, refraction, diffraction and interference of neutrons. The motivation of such experiments is discussed in the introductory section. The review is divided into three main sections: geometrical optics, wave optics and crystal optics; with subsections that parallel the typical textbook development of classical optics. Special emphasis is placed on the recent developments in neutron interferometry and on the fundamental tests of quantum-mechanical principles made possible by these novel techniques. This review was received in its present form in July 1982.

t This review commemorates the 50th anniversary of the discovery of the neutron,

0034-4885/83/030259

+ 77$09.00

@ 1983 The Institute of Physics

259

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A G Klein and S A Werner

Contents 1. Introduction 1.1. The analogy between neutron and classical optics 1.2. Motivation of neutron-optical experiments 1.3. The experimental challenge 2. Geometrical optics 2.1. The Hamiltonian analogy 2.2. The wave equations 2.3. The eikonal equation 2.4. Neutron radiography 2.5. Beam deflection 2.6. Refraction 2.7. Reflection 2.8. Polarisation 3. Wave optics 3.1. Wave theory of refraction 3.2. Diffraction 3.3. Diffraction from macroscopic objects 3.4. Interference and interferometry 4. Crystal optics 4.1. Kinematical theory of nuclear Bragg scattering 4.2. Magnetic diffraction by a single crystal 4.3. Single-crystal monochromators 4.4. Beam filters 4.5. Dynamical diffraction of neutrons by perfect crystals 5. Conclusion 5.1. Future prospects 5.2. Retrospect Acknowledgments References

Page 26 1 26 1 262 265 266 266 267 267 269 270 27 1 280 285 289 289 290 290 296 314 3 14 317 319 319 320 329 329 331 331 331

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1. Introduction 1.1. The analogy between neutron and classical optics Not long after their discovery 50 years ago (Chadwick 1932a, b) neutrons were recognised as nuclear constituents and were pressed into service as tools for probing the atomic nucleus. In the course of experiments on artificial radioactivity, in which neutrons were used as bombarding particles, Fermi (1934) discovered the inordinate effectiveness of slow neutrons. The measured cross sections were greatly in excess of the geometrical cross sections that might have been expected in short-range interactions between particles, suggesting that wave-like properties of the neutron were involved (Amaldi 1977). Elsasser (1936) was the first to suggest that, if the motion of neutrons is determined by quantum mechanics, they should be diffracted by crystalline materials. Such properties were later confirmed by diffraction experiments from crystals (Halban and Preiswerk 1936, Mitchell and Powers 1936) which showed that thermal neutrons behaved in accord with de Broglie's hypothesis and had wavelengths comparable to those of x-rays. Theories of neutron scattering and diffraction were soon worked out by Fermi (1936), Wick (1937) and others, based largely on the analogy with x-ray scattering. The analogy between x-ray scattering and elastic neutron scattering is not surprising; it reflects the behaviour of waves of comparable wavelength, though the nature of the individual interactions is very different: purely electromagnetic, in the case of x-rays, and mainly nuclear, in the case of neutrons. In the early experiments, the neutrons came from the nuclear reaction responsible for their discovery, namely 'Be(a, n)12C. Thus, radioactive alpha-sources, such as polonium, were mixed with powdered beryllium and the neutrons were thermalised in hydrogenous moderators. With the advent of fission reactors providing a more copious flux of neutrons, beams of thermal neutrons became available for experiments in nuclear physics as well as for elucidating further the nature and properties of the neutron itself. At the same time, beams of thermal neutrons became recognised and established as important probes of the structure and dynamics of condensed matter in experiments which were seen as being adjuncts, as well as analogous, to x-ray work. A large and important branch of experimental physics thus came into being. Elastic and inelastic neutron scattering are currently burgeoning fields of endeavour, well served by an extensive literature of textbooks, monographs and reviews (see list of references). In the course of such experiments, slow neutrons, as de Broglie waves, have been shown to exhibit not only Bragg diffraction but also many of the phenomena of classical optics, such as refraction. For example, total reflection from plane surfaces has been known and utilised for almost 40 years (Hughes 1954). However, in the last decade or so, a growing body of experimental work can be identified, forming a new strand in the experimental uses of neutrons. These experiments have in common a striking analogy with the experiments of classical optics, involving not only reflection and refraction but also diffraction by macroscopic objects and, particularly, interference on a macroscopic scale, The aim of this review is to gather such experiments and

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their theoretical foundations, scattered throughout the literature, under the obvious heading of neutron optics, and to discuss them more or less in the order in which their analogues occur in texts on classical optics. We shall be dealing, therefore, with the optics of massive de Broglie waves, pointing out the similarities of principle and the differences in technique compared with classical optics. The similarities of principle are easy to understand. The time-independent Schrodinger equation is formally equivalent to the Helmholtz scalar wave equation which, in classical optics, accounts for the behaviour of light waves (aside from polarisation-dependent effects which may be considered separately). The differences in technique will emerge in due course, but the results are, assuredly, similar to those of classical optics. What, then, is the motivation for carrying out such experiments? Before embarking on the elaboration of our scheme we shall attempt to answer this pertinent question. In the first instance, one should not ignore their pedagogical value; indeed, some have called them ‘textbook experiments’. As such they are pleasant and instructive exercises. There are, however, at least three other aspects of greater importance to the progress of physics, one or more of which have motivated most of the experiments in neutron optics. These will be dealt with in the next subsections of the introduction.

1.2. Motivation of neutron-optical experiments 1.2.1. Tests of fundamental propositions in quantum mechanics. From its inception, quantum mechanics has dealt with the dynamics of electrons in atomic and molecular systems, governed by the electromagnetic interaction. Indeed, the most rigorous tests of quantum mechanics have nearly all employed electrons as their test objects. The applicability of quantum mechanics to other objects and their interactions is much less securely founded. In the case of the nuclear interactions, rigorous tests have been excluded by the very complexity of the interactions as well as by the lack of rigorous theories. In the case of the gravitational interaction, the very small mass of the electron has precluded precise measurements of all but the limiting classical effects. It has also been found that the gravitational interaction is difficult to study with electrons because of the long-range Coulomb interaction between the electron and the walls of the vacuum chamber (Witteborn and Fairbank 1967). The behaviour of thermal neutrons as de Broglie waves as seen in Bragg diffraction from crystals was, of course, an early verification of quantum-mechanical principles. It was made possible by the particular properties of the neutron, namely its large mass (1836 m J , its lack of electric charge (which simplifies its interactions with matter) and the relative ease of detection by means of highly specific nuclear reactions. These very same properties have made it possible, in more recent neutron-optical experiments, to probe certain aspects of quantum mechanics which have hitherto remained untested or only indirectly inferred. For example, the observability of relative phase shifts by means of neutron interferometry has allowed the direct verification of several important principles. The role of a purely gravitational potential on the phase of a Schrodinger wave has been explicitly exhibited (Colella et a1 1975, Staudenmann et a1 1980) as will be discussed in § 3.4.4. In a follow-up experiment (Werner et a1 1979) the inertial effects due to the Earth’s rotation were also demonstrated in a neutron interferometer, thus verifying the principle of equivalence in the quantum limit. Other experiments, to be described in § 3.4.6, have directly demonstrated the spinorial nature of fermions by showing that the neutron wavefunction reverses its

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sign after rotation by 27r, and is only restored to its initial phase by a rotation of 47r (Werner et a1 1975, Rauch et a1 1975, Klein and Opat 1976). This effect, while in accord with the accepted formalisms of quantum mechanics, had nevertheless remained without direct verification for over 40 years. Another interesting test of quantum mechanics concerns the linearity of the Schrodinger equation. Various non-linear variants had been put forward (Bialynicki-Birula and Mycielski 1976, 1979) by adding an extra term to the equation, proportional to In lq+lz, whose role is to counteract the indefinite spreading of wave packets implied by Schrodinger's linear quantum mechanics. In this way it would be possible for macroscopic particles, such as large organic molecules, to move classically, along localised trajectories, while allowing smaller particles to behave in accordance with the established laws of quantum mechanics. The remarkable agreement of the Lamb-shift measurements with ordinary linear quantum mechanics has put an upper limit of the order of 4 X 10-l' eV on the strength of such a hypothetical non-linear term. Recent experiments in neutron optics, to be described in § 3.4.7, have reduced this upper limit by an impressive five orders of magnitude (Shull et a1 1980a, Gahler et a1 1981). The neutron-optical analogue of the Fizeau experiment was proposed in 1978 and performed by Klein et a1 in 1980. It concerns the propagation of de Broglie waves in a moving medium. Though not comparable in significance with the 1853 original experiment, it nevertheless is a direct test on the transformation laws of the ( U , k) four-vector in a moving frame of reference and thus constitutes another verification of basic quantum-mechanical principles. Its description will be given in § 3.4.8. A search for a neutron Aharonov-Bohm effect has recently been carried out by Greenberger et a1 (1981) as described in § 3.4.9. Further fundamental experiments in quantum mechanics have already been proposed or await developments in technique. 1.2.2. Measurements of the fundamental properties and interactions. The currently known values of fundamental properties of the neutron are shown in table 1. In recent years neutron-optical techniques have contributed significantly to the accuracy of several of these quantities in ways which will be elaborated in later sections. There is, undoubtedly, plenty of scope for improvement, particularly in the measurement of the lifetime (Byrne et a1 1980) which turns out to be a particularly difficult task, and in the measurement of the electric-dipole moment. This latter quantity is of great theoretical interest since a non-vanishing value is linked to CP violation in elementary particle physics (Dress and Miller 1978, Ramsey 1981). Neutron-optical techniques are also responsible for the most accurate measurements of the interactions of neutrons with matter. Of the greatest practical importance are accurate data on the neutron-nucleus interaction for an exhaustive set of nuclear species. From the point of view of optics, such measurements are strictly analogous to measurements of refractive indices, a connection which will be elaborated in § § 2.6.1 and 3.1. Accurate values are of crucial importance to the investigation of condensed matter by means of neutrons, yet with a few exceptions such data are not calculable from nuclear theory. The exceptions concern the simple two-body n-p and three-body n-d interactions and to some extent the four-body n-3H and n-3He interactions for which some useful theoretical results are available (Kharchenkov and Levashev 1976, Sears and Khanna 1975). The newer measurements, by means of neutron interferometry (Rauch 1979), are therefore of great importance. Of similar importance are experimental data on the neutron-electron interaction. This very feeble interaction is made up of two parts; the first is the quantum electrodynamic interaction between

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A G Klein and S A Werner Table 1. Fundamental properties of the neutron. Mass: m = 1.6747 x g. Spin: s = 1/2 Charge: q,.

(4.36)

The first term is the current carried by the a -branch wavefunction $ a ( r ) , the second term is the current carried by the @-branch wavefunction iLp(r)and the third term

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arises from interference effects between the a and @ branch wavefunctions. Although we will not establish it here, it is a straightforward matter to show that (&) is normal to the a-branch dispersion surface and ( J p ) is normal to the @-branch dispersion surface, and the interference current (Jap)is a vector directed along G, being positive at certain depths, x , in the crystals and negative at others. Thus, the total current will oscillate between the incident beam direction and the diffracted beam direction. Keeping in mind that the region of the hyperbolae in figure 36 is vastly expanded in scale relative to the length of ko, we note that KO is essentially parallel to k o = ko&(within a second of arc) and KG is essentially parallel to ko + G = kOsG (the diffracted wave direction as defined by the kinematical theory). As the mis-set angle A 8 is varied from positive values to negative values over a range of only a second of and arc or so, the currents (.la) and ( J p )sweep over the angular range between encompassing the entire 'Borrmann triangle' ABC in figure 37. There is an enormous angle amplification effect here. For a given mis-set angle A$, the current corresponding to the a branch will propagate at an angle R with respect to the lattice planes as shown in figure 37, while the current corresponding to the @ branch will propagate

so

Figure 37. This diagram shows the Borrmann triangle ABC and the directions of current flow J, and J p corresponding to the a and p branches of the dispersion surfaces of figure 36. The angle 2fl between these two vectors depends upon the mis-set angle, AB, from the exact Bragg condition (see equations (4.35) and (4.36)).

across the crystal at an angle -a.Since these currents are vectors normal to the dispersion surfaces it is a matter of geometry to relate A 8 to R. Defining p =tan R/tan

8B

(4.37)

one finds AB =

* (Eo sin 2&

VG v - G ) ' l 2

P (1-p2)1/2'

(4.38)

This is a highly non-linear relation, but for small R one obtains (4.39) For silicon, the amplification factor in brackets is typically 105-106.


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This angle amplification effect was exploited by Kikuta et a1 (1975) to measure small directional changes of a neutron beam resulting from prism refraction (figure 38). Neutron-absorbing slits, one in front and one on the back face of the crystal plate, act as a ‘crystal collimator’, selecting only those neutrons which travel through the crystal parallel to the lattice planes. These neutrons travel in the second crystal plate, again in the same direction. Now, if the beam is deflected in the space between the two crystals, its path in the second crystal plate splits according to equation (4.38).

c

c Scan

-1

0

1

-1 0 1 Scanning slit position [mm)

i’/

Figure 38. Demonstration of the angle amplification effect (Kikuta et al 1975). The wedge shifts the angle of the neutron beam off the exact Bragg condition, and the currents corresponding to the (Y and j3 branch dispersion surfaces are split apart in the second crystal. This splitting is observed by scanning a slit across the diffracted beam leaving the second crystal as shown by the data on the right-hand side of this figure.

This same splitting phenomenon occurs if the wavelength, rather than the direction of the beam, is altered between the two crystals. This feature was utilised by Zeilinger and Shull(l979) in a measurement of the wavelength change of a neutron in a magnetic field ( S h / h =fpn8/2Eo). The results of their experiments are shown in figure 39. The energy shift due to the magnetic field was only 10-8eV, demonstrating the extreme sensitivity of this angle amplification effect in perfect single crystals of silicon. At the exact Bragg condition (A6 = 0), the currents (Ja)and ( J p )oscillate with a period A corresponding to the difference in the wavevectors IK; -Kg I = D,thus A=

.rrkoh2COS 6 B m ( vGv-G)1’2‘

(4.40)

For silicon this is typically 105-106A. At the exit face of the crystal ( x = T ) , the neutron wavefunction splits up into a single incident wave (0 beam) and a single diffracted wave (G beam). The relative intensity of these two beams depends upon the phase of the interference current at the exit boundary. As the thickness of the crystal is varied, the neutron flux will be ‘swapped’ back and forth between the 0

Neutron optics

\

327

pole face

I

5

l 0 o - 4 -2 0 2 4w

I

ib)

Exit slit position i m m l

Figure 39. Experimental arrangement used by Zeilinger and Shull (1979) to observe the small shift in the neutron wavelength due to the application of a magnetic field. Spin-up neutrons are shifted down in wavelength, while spin-down neutrons are shifted up in wavelength. This splitting of the wavelength results in an angular separation of the neutron currents propagating through the second crystal, which can be observed by scanning a slit across the exit face of this second crystal. The splitting is proportional to the magnetic field as shown by the data on the right-hand panel of the figure.

beam and the G beam. Alternatively, since the period depends upon the neutron wavelength, these oscillations can be observed experimentally by varying A for a crystal of a fixed thickness T. An experiment of this type was carried out by Shull (1968, 1973) and the results are shown in figure 40. These data show directly the Pendellosung interference effects. There are a number of important subtleties involved in properly describing these Pendellosung interference effects, which fundamentally arise from a ‘beating’ of the a and p branch wavefunctions. This beat period is macroscopic in real space because ]Kg --KE I, the difference in the wavevectors corresponding to the a and p branches, is so small. In general, the incident beam is not sufficiently well collimated so as to generate separated plane waves inside the crystal. Instead, the entire dispersion surface is simultaneously ‘illuminated’ by the various divergent rays in the incident beam. Thus, in order to describe these interference effects properly, it is necessary to specify the precise phase relationships of the various plane-wave components in the incident beam. Scans across the diffracted beam (figure 41) carried out by Shull (1968,1973) are correctly described if one assumes that the incident beam is a coherent spherical wave (Shull and Oberteuffer 1972), for which the phase relations and amplitudes of a plane-wave decomposition are, of course, known. Calculations based upon this assumption were first carried out by Kat0 (1961, 1968) for the x-ray case. The time of flight of neutrons through a perfect single crystal under Bragg-reflecting conditions is of special interest. It is easily shown that the expectation value of the momentum for either the a or p branch wavefunctions is

[

( p >= h[ KO+ 1+ (fi)2]-1G]. *G

(4.41)


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Incident neutrons

I 1-

10

I_

Detector

Neutron

Crystal

absorber

_---------- 1 Reflecting __---planes

11 Wavelength (A1

1 2

Figure 40. ( a ) Arrangement used for observing Pendellosung interference fringes in silicon (Shull 1968); (6) interference oscillations observed by Shull for crystals of various thickness: (A) 10 mm, (B) 5.94 mm, (C) 3.32 mm.

Thus, the momentum component perpendicular to the lattice planes is 2 -1

(py)=h(Ko,-[l+(~)

]

G]

(4.42)

but the momentum component pertinent to transport of neutrons across the crystal is simply ( p x )= mOx = hko COS eB.

(4.43)

Consequently, the time of flight of neutrons across the crystal of thickness T is

Tm T =

hko cos OB '

(4.44)

For large Bragg angles, this time of flight is considerably different than the time of flight of a neutron in free space. Experimental verification of equation (4.44)by ShulI et a1 (1980a, b) using a 20.4 cm thick Si crystal, and neutrons of wavelengths A = 2.7 A and 3.7 A shows that on a microscopic scale the neutron is taking a 'zig-zag' trajectory within the crystal. Recently, Werner (1980a) has worked out the theory of dynamical diffraction of neutrons under the influence of an external gravitational or magnetic field. It is found that the average trajectories corresponding to the cy and p branch wavefunctions are no longer straight lines, but are hyperbolic. Experiments in several laboratories are currently under way to observe these effects.

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Neutron optics

160 80

.-c c1 + c

80

a 40

U

0

-20

0

0 20

- 20

0

20

Scanning slit position (mils 1

Figure 41. Scans across the diffracted beam of figure 40 (Shull 1968, 1973). These results required that the incident beam be described by a spherical wave. Left: 1.020 A; right: 1.034 A.

5. Conclusion

5.1. Future prospects It is, of course, very difficult to predict the progress of any field of scientific research. This is particularly true in neutron optics where experience has shown that major advances have depended very heavily upon new technologies, such as the availability of large single crystals of silicon. In the short term, neutron interferometry is likely to be applied to a continuing programme of interesting experiments. For example, another fundamental test of quantum mechanics is the following. Quantum theory rests on the principle of superposition, which states that if JI1and $2 are two possible states of a system and C1 and C2 are arbitrary numbers, then CI$I+C~IJI~ is also a possible state of the system. It is usually taken for granted that the coefficients C1 and Cz are complex numbers. However, it is possible to imagine a quantum theory based on quaternions (numbers of the form a +ib + j c +kd). Peres (1979) has proposed a simple commutativity test to see if the neutron-nuclear scattering lengths are strictly complex numbers or whether they have additional features allowed by a quaternion quantum theory. We have already mentioned the fact that the trajectories of Bragg-reflecting neutrons in perfect crystals are substantially affected by an external force, such as the gravitational force or a force arising from a magnetic-field gradient. The calculations of Werner (1980a) suggest that the neutron-nuclear absorption of Bragg-reflecting neutrons may also be affected by an external force. Another proposition, to which we have alluded earlier, is to explore the temporal coherence of neutron beams by means of an unequal-arm interferometer. Such an interferometer, of the Michelson type, could also be used to repeat the MichelsonMorley experiment with neutrons, as proposed by R Deslattes (1978 private communication). The importance of an experiment of this type on the empirical foundations of special relativity has been discussed earlier by Breitenberger (1971).

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A Michelson interferometer using ultracold neutrons has already been proposed (Steyerl et a1 1980). Its concrete realisation may well lead to surprising results. For example, it has been suggested that a gravitationally induced quantum interference experiment with such slow neutrons would be of interest since the neutron trajectory would no longer be uniquely defined, and the usual WKB technique for calculating phase shifts fails in the limit of zero velocity. It is also conceivable that such an interferometer could be used to perform a gravitational experiment of the Cavendish type, i.e. a measurement of the universal constant G. Higher-order general relativistic effects, however, seem to be somewhat beyond currently imaginable technology, although recently several experiments have been proposed by Anandan (1982) to test possible parity and time-reversal non-invariance effects in the interaction of the neutron with electromagnetic and gravitational fields, including detection of its electricand gravitational-dipole moments. This is also the case regarding higher-order coherence effects such as the ones exhibited in the Hanbury-Brown-Twiss experiment in optics. Further along these speculative lines, there has been discussion for some time of the possibility of using two coherent beams available in a neutron interferometer to help solve the ‘phase problem’ in certain crystallographic studies. It appears to us that this would require perfect sample crystals and extreme positional stability of the sample. It is important to note that most neutron experiments are limited at some stage by the intensity of the source. To take a far-reaching example, the feasibility of a neutron microscope would require a considerably higher intensity than is currently available. So far, nuclear reactors have provided the highest intensity neutron beams, but recently, another kind of source is showing promise of overcoming, to some extent, the flux limitations of reactors imposed by heat removal technology. In these sources, currently under development in the United States, Great Britain and Japan, highintensity proton beams are made to collide with heavy nuclei and to generate intense bursts of neutrons by spallation (see Carpenter et a1 1979). It appears that peak thermal neutron fluxes of about 10I6 neutrons cm2 s-’ will be available from these sources by the end of this decade. The moderators in these sources can be tailored to produce very intense beams of either ‘cold’ or ‘hot’ neutrons. The prototype source at Argonne National Laboratory has already been utilised to generate ultracold neutrons using Doppler-shifted Bragg scattering (Dombeck et a1 1979, Brun et a1 1980). This new type of pulsed source will undoubtedly play an important role in neutron-scattering investigations of the structure of condensed matter. Experiments utilising the unique time structure of these beams to investigate temporal coherence questions may also prove fruitful. With the advent of these pulsed-neutron spallation sources on the horizon, it is clear that the energy dependence of the neutron-scattering length through the epithermal Breit-Wigner resonances of certain isotopes will need to be measured. An experiment on the resonance at 0.096 eV in 149Smis currently in progress at Missouri. It will also be of interest to carry out interferometer experiments in which both polarised neutrons and polarised targets (probably dynamically polarised) are used, so that the scattering lengths of both spin states of the compound nucleus can be measured directly. Finally, further progress may be expected in the more precise measurement of the fundamental properties of the neutron (Ramsey 198l ) , such as the electric-dipole moment (Dress and Miller 1978) and the lifetime (Byrne et a1 1982).

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5.2. Retrospect In this review we have concentrated on exploring and elucidating the analogies between classical optics and the rapidly expanding field of neutron optics. Vast areas of great practical importance have had to be omitted or only briefly mentioned, particularly in the field of elastic and inelastic neutron scattering for structural and dynamic investigations of condensed matter. Many excellent texts, monographs and reviews are listed in the bibliography. Other fields of great interest and importance, such as ultracold neutrons, parity-violating weak interaction experiments, neutron lifetime measurements and several others, were also rather skimped; more specialised reviews are readily accessible. Various aspects of the dynamical theory of diffraction could only be touched upon lightly, in spite of the growing trend of experimental investigation of subtle effects in perfect crystals. The rather elementary treatment of dispersion theory given here is adequate for the description of practically all of the experimental work carried out so far, but the field is on the threshold of exploring the higher-order effects neglected here. A more thorough treatment, which is, in many ways, complementary to this work, is the review by Sears (1982a). Our emphasis has been on the exposition, in one place, of the continuing series of optical experiments aimed at probing various aspects of quantum mechanics, measuring the properties of the neutron and its interactions, and developing new experimental techniques. We make no claim regarding completeness and apologise to those whose work was inadvertently overlooked. However, we hope to have communicated the flavour of an interesting and exciting area of physics in which almost every experiment has aspects which are totally novel or unique. Acknowledgments We would like to express our appreciation to Varley Sears (Atomic Energy of Canada, Ltd), Anton Zeilinger (Massachusetts Institute of Technology) and Geoffrey Opat (University of Melbourne) for their helpful advice and criticism of the manuscript. We are indebted to Bonnie Beckett for her careful and patient typing of the various drafts of this review paper. This work was supported by the Australian Research Grants Committee and the US National Science Foundation (grant no PHY 792079).

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