New developments in cold and ultracold neutron research

Physica 137B (1986) 270-281 North-Holland, Amsterdam

NEW D E V E L O P M E N T S IN C O L D A N D U L T R A C O L D N E U T R O N RESEARCH A. S T E Y E R L Fakultiit ffir Phvsik. J'echnische UnicersttiJt Mi'mchen, 1)-8046 Garching. I'~'d. Rep. (iermanv

S.S. M A L I K Department of Pl~v.wc.~, Unwersit.v of Rhode Island, Kingston. RI 02881, USA

The paper contains recent contributions to cold and ultracold neutron physics by groups at Garching and Grenoble, and, in a second part. by a group at the University,of Rhode Island. We describe first results in neutron microscopy obtained at the ILL and (}arching reactors. Two versions of a two-mirror neutron microscope were used: one employing uhracold neutrons and total-reflecting mirrors, and the other utilizing very cold neutrons in conjunction with muhilayer mirrors. We then describe the essential features of the new source of ultracold and cold neutrons which ls presently being set up :it the HFR. Grenoble. and discuss the novel all-metallic neutron mirrors and guides which were developed in connection with this project. Finally a proposal by the University of Rhode Island for application of high-resolution neutron optical techniques to the stud~, of critical phenomena in liquids is presented.

!. Recent developments at Garching and the Institute Laue-Langevin, Grenoble (by A. Steyerl, in collaboration with H. Nagel, K.-A. Steinhauser, P. H e r r m a n n , R. G~ihler, and H. F r y d m a n of Technische Universit~it, Mi]nchen, a n d with P. Ageron, P. Astruc, W. Mampe, a n d W. Drexel of Institut Laue --Langevin, G r e n o b l e ) 1.1. Introduction

The long wavelength of very slow n e u t r o n s and special focussing and high-resolution techniques developed for these n e u t r o n s have e n a b l e d us to study n e u t r o n wave diffraction a n d resonance phen o m e n a in macroscopic structures like a ruled grating or composite films of different materials [1-5]. A c o m m o n feature of the spectroscopic methods applied in these studies is the ample use of specific, energy-selective features of the parabolic flight path described by slow n e u t r o n s in the earth's gravitational field [6]. G r a v i t y plays an i m p o r t a n t role also in attempts to use very slow n e u t r o n s for image formation and microscopy, since it can give rise to chromatic aberrations due to the wavelength d e p e n d e n c e of flight-path curva0378-4363/86/$03.50 '!~ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

ture. The consequences of this p h e n o m e n o n and first experiments in n e u t r o n microscopy are described in the following sections. The properties of a concave mirror as a system for ultracold n e u t r o n imaging were first discussed by Frank in 1972 [7]. In the first n e u t r o n imaging system designed for c o m p e n s a t i o n of chromatic effects we used, in 1980, a Fresnel-zone-type mirror [8,9] a n d o b t a i n e d achromatic images at magnifications up to six. Successful operation of a four-mirror system at magnifications 1.375 and 0.725 was reported in 1984 by F r a n k and his collaborators [ 101. In the present paper we describe two versions of a n e u t r o n microscope: the 50 times magnifying microscope for ultracold n e u t r o n s ( U C N ) [11,12] tested by us at the ILL reactor, a n d a Schwarzschild system e m p l o y i n g very cold n e u t r o n s (VCN) which was tested at the F R M reactor [13]. i. 2. Ultracold neutron microscope

A schematic view of the U C N microscope setup is shown in fig. 1. A large, concave, parabolic mirror (radius of curvature R = 585 mm, diameter 200 mm, 5~Ni coated) and a smaller, convex,

A. Steyerl and S.S. Malik / CoM and ultracold neutron research

7 rror

bject Stit

ic Mirror Fig. 1. Scheme of the utracold neutron microscope. Neutron trajectories are shown for two different velocities. The instrument is free of chromatic aberrations to first order, and it is designed for utilization of a wide entrance ~tperture (1 : 1.5) and for a magnification of 52.

spherical mirror ( R ' = - 1 2 1 0 mm, diameter 45 mm, Ni coated) are arranged one above the other on their common optical axis. A special feature of the design is the following: on their way between the mirrors the neutrons pass the highest point of their flight trajectory. This peculiarity, in conjunction with the choice of suitable dimensions, is a necessary requirement for the possibility of compensating the two lowest-order chromatic aberrations. Specifically, both the image distance (663 mm above the second mirror) and the magnification ( M = 52) were rendered stationary in the neighbourhood of the design value v = 5.7 m / s (A = 700 ,~) of neutron velocity at the primary mirror. Higher-order chromatic aberrations cannot be eliminated at the same time, but they can be strongly reduced by coarse monochromatization. Geometrical aberrations are dominated, at the large geometrical apertures required for intensity reasons, by spherical aberration and coma, and these were drastically reduced by the choice of a parabolic shape for the primary mirror. Thus a very large f-factor ( f = 1.5) was realized. (The parabolic shape of the primary mirror was so essential because most of the magnification - a factor of 20

271

- takes place at the first mirror. The second mirror was made spherical without any significant loss in imaging quality.) The image may be scanned in two dimensions in the image plane. However, in the present experiments a single, fixed image mirror of 17 x 17 mm 2 (which corresponds to an object field width of 0.35 x 0.35 mm 2) was used. It reflects the neutrons into an inclined guide tube which conducts them to the UCN detector (BF3 with reduced 1°B content and a thin aluminium window). The experiments were performed on the guide tube PN5 of the ILL reactor. The measurement consisted in a simple scan of a cadmium slit serving as the object. Simple as this proposition might appear, it turned out to be quite demanding. The problems encountered were essentially twofold: (a) the weak intensity on PN5 ( < 20 UCN cm-2s -1 during much of the experiment), and (b) the requirement of very precise adjustment of the optical components. After many time-consuming futile attempts - and at least as many speculations on whether or not we saw a peak, and if not, why not - both problems were finally solved. The intensity on PN5, which had dropped considerably over the years due to contaminations not yet identified, was boosted by a factor of six by new liners inserted into the in-pile part and partial replacement of the external guide. The novel technique applied by us to produce these purely metallic liners, neutron guides and mirrors was, after initiative tests at the ILL and in close cooperation with the ILL, developed by one of us (H. Nagel). It will be described below in section 1.4. The second problem, precise adjustment of the optical components, was, of course, connected with the first, since an adjustment by use of the neutron beam was practically ruled out by the weak intensity. Light-optical techniques suffer from the absence of gravitationally induced beam curvature. But in the end the following scheme developed by K.-A. Steinhauser was successful: a light-emitting diode and a photodiode were installed in the object plane symmetrically on both sides of the object slit. If a glass plate is inserted horizontally between the object and the parabolic mirror and the object center is positioned precisely in the light-optical focal point (corrected for refraction in

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A. Steverl and S.S. M a l i k / Cold and ultracold neutron research

the intervening glass plate) then the LED light is exactly focussed onto the photodiode. This scheme also ensures that the optical axis is precisely vertical. A similar arrangement with a small glass plate placed onto the center of the large mirror enabled us to check the optical adjustment also after pumping the system and during data collection. As a curiosity we may remark that the absence of a clearly detected image for over two years gave rise to weird speculations. For example, we started to wonder what might happen to the neutrons near the reversal point of their flight trajectory when they are brought nearly to a full stop. Their lateral velocity component then is only of the order of a few c m / s (~---0.01 mm), and to our knowledge neutrons in this wavelength range had never been observed before. Of course, wave mechanics provides a full description of the wave function near the reversal point, and we expect no strange phenomena to happen apart from the phase change by rr/2 known to occur upon touching the caustic surface. Such a phase change, however, was unobservable in the present experiment. In spite of the low neutron velocity we should not be conflicted with the limit to the simple wave description which had been pointed out by Shull [14]. He had defined "snail's-pace" neutrons (SPN) to be so slow that, during the neutron lifetime for radioactive decay, they could advance only a distance comparable to their wavelength. Thus ~,spN/t, spN -- r, and hence Osp N ----- 1 0 - 3 c m / s . Naturally, this consideration implies that the neutron spend a good deal of its life as an SPN, and not only a short instant during reversal of its flight direction. After this fruitless excursion into the realm of speculation we were quite content to observe that indeed nothing serious happened to the neutrons at their reversal point, and that they formed an image of the object slit exhibiting the expected brightness and quality - if only the system was properly adjusted. The result of an object scan is shown in fig. 2. The observed edge width of --- 0.3 mm is explained by the "coarse-grained" image detection technique described above. The edge width corresponds to the width of 17 mm for the image mirror. Aberrations would tend to smooth the sharp corners. However, the counting statistics

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was not sufficient to derive from the data a reliable value for the aberrations. Monte Carlo simulations yielded a theoretical edge blurring by --- 20 p,m for the velocity band width (5.5 < ~, < 6.7 m / s at the primary mirror) and apertures used. This value is c~)nsistent with a direct calculation of differential aberrations, and it corresponds to a resolution of -- 20 p,m. In the future we hope to be able to make full use of the UCN microscope when it is installed at the new " N e u t r o n Turbine" at the ILL reactor. The source will be described in section 1.4. In addition to the gain factor of -- 100 expected for the new source, as compared to PN5, further significant gains can be achieved by (a) the use of a different spectral region with a nominal velocity at the first mirror of --- 7.0 m / s (instead of 5.7 m / s ; thc estimated gain factor is = 2). and (b) provision of a two-dimensional UCN detector in the image plane. Work on a detector using a ~Li converter on high-quality semiconductor material is in progress at Garching. Step (a) will require a replacement of the second mirror and an extension of the microscope height to --- 3.5 m. We will aim at a magnification of 100 and hope to obtain, as a first direct application of the microscope, a two-dimensional image of the domains in a thin ferromagnetic film. The microscope design allows for both bright-field and dark-field illumination.

A. Steverl and S.S. Malik / CoM and ultracoM neutron research

Let us pause for a while and ask for specific applications of this novel instrument: a neutron microscope. This question has recently been discussed in detail by Steinbach [15]. It is possible, on the one hand, to exploit the "neutron contrast" where especially the protons can be marked by p - d exchange. On the other hand we can expect that neutron studies an organic matter, specifically prepared with respect to its isotopic composition, will be associated with less radiation damage than photon or electron microscopy. At the presently available low source strengths the problem of radiation damage is irrelevant, of course, but we may expect that the intense (~,, n)-sources proposed by Kumakhov [16] and Eremeev [17] ("beryllium focus") and by Steinbach [18] ((~,, n)-reaction on the deuteron) will be capable of boosting the thermal neutron source brightness available for microscopy beyond the present standards by several orders of magnitude. The scheme of an ultracold neutron microscope described above represents, of course, only one design possibility. The four-mirror arrangement of Frank [10] has been mentioned before. The use of 20,4, neutrons, in conjunction with zone plates as the focussing elements, has been proposed [19], but it is not obvious that the application of shorter wavelengths would necessarily imply an advantage, for two reasons: a) Imaging systems, and in particular: aberration-free systems of sufficient speed, seem to be unavailable. The geometrical aperture of the very chromatic - Fresnel zone plate used with 20 ,~ neutrons by Klein et al. [19] was f = 500, and the wavelength and divergency limitations imposed by the more futuristic proposals of thermal neutron focussing by perfect crystals [20] seem to be still much more severe. b) It follows from simple phase-space considerations for a Maxwellian source spectrum that the gain in useful beam intensity, from a guide tube, within a fixed velocity bandwidth to be used for imaging, is proportional to v. But for a (1/v)-absorbing object this factor is compensated by the reduced contrast, and for a coherently scattering object the contrast factor ( - 1 / 0 2 ) over-compensates the lower intensity at long wavelengths. The wavelength limit of resolution will probably

273

not be reached in the near future, unless significantly higher source strengths will become available. It will be shown in the following section that a microscope using somewhat faster than ultracold neutrons can be built using multilayer mirrors. Such a system has recently been developed and tested by us at Garching [13]. 1.3. Cold neutron microscope

The design of the VCN microscope is depicted in fig. 3. It is based on the principle of the Schwarzschild telescope, using two spherical mirrors with a common (vertical) optical axis. By use of a specific geometry the Schwarzschild system is made aplanatic, i.e., spherical aberration and coma can be corrected. Thus a very wide geometrical

prim( mirr( (conca

2 Fig. 3. Scheme of the Schwarzschild microscope using two spherical mirrors. Spherical aberration and coma are corrected by the choice of a suitable geometry. Therefore it was possible to use a very wide entrance aperture of 1 : 1.2. A magnification of 6 was chosen.

274

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aperture ( f = 1.2) was realized. Unfortunately, it is not possible in this scheme (where the neutrons do not pass the highest point of their flight trajectories between the two mirrors) to compensate firstorder chromatic aberrations. Due to the higher neutron velocity these are less severe than they would be in a U C N microscope, but they are sill considerable, as shown below. The system was designed for a magnification of six. The primary mirror is concave (R = 100 ram, diameter 100 mm) and the second mirror is convex ( R = - 3 0 mm, diameter 20 mm), and they arc separated by a distance of 72.29 mm. We selected a neutron velocity band 10 m / s < t, < 13 m / s (300 < )~ < 400 ,~) by appropriate mirror preparation. This was achieved by a ( N i - T i ) multibilayer coating kindly performed by 0. Schfirpf at the ILL, Grenoble. The reflectivity of these mirrors for 4-A neutrons, measured by use of the scheme described in [21], showed the expected broad plateau (with reflectivity above 90%) within the calculated range of velocity c o m p o n e n t normal to the mirror. We made use of the neutron turbine [22] at the F R M reactor, Garching, as a source of neutrons in the required wavelength region. A 90" arc of neutron guide channelled the beam to the microscope and turned its axis into the vertical direction. The guide tube was designed to be conically convergent in both lateral dimensions in order to ensure a sufficiently wide incident beam divergence. The reflecting walls were purely metallic, and they were produced in the manner described below in section 1.4. As the object (123.15 m m below the primary, large, mirror) we used a c a d m i u m coated razor blade edge, and the image was scanned by a 3 m m wide exit slit at a distance of 39.8 mm above the large mirror. A nickel coated foil served as a barrier preventing the ultracold neutrons below 6.7 m / s from reaching the detector. These neutrons which make up = 10% of the beam are also reflected by the imaging mirrors. They were removed from the beam in order to avoid increased chromatic aberrations. Fig. 4 shows the result of an image scan for two settings of the object edge, 1.0 m m apart from one another. From the data the magnification was determined to be 6.0 +_ 0.2. in agreement with the

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Fig. 4. l ) a t a of image scans in the Schwarzschild microscope for two settings of an edge which served as the object. For an aberration-free system the intensity would increase linearly within a range of 3 m m (the width of the s c a n n i n g slit). From the observed b l u r r i n g of the edges a total a b e r r a t i o n of 1.4 -.- 0.2 m m was derived. This value corresponds to a resolution of 230 p. m.

design value of 6.0. l'he observed edge blurring by + (1.4 + 0.2) mm corresponds to a measured resolution in the object plane of 230 p.m. The observed total aberration is somewhat higher than the calculated value of 1.0 mm, 80go of which is due to the uncompensated first-order chromatic aberrations. Trying to draw a comparison between the two types of neutron microscopes described above we point out two essential features: a) The Schwarzschild scheme offers a significant intensity advantage over the U C N microscope. Direct conversion of the measured intensities to identical U C N / V C N source strengths and magnifications suggests a gain factor ot' = 20. This result is not in contradiction with the previous statement that the intensity should increase only like - v. That comparison had been based on the assumption of a given width in m o m e n t u m space for the source neutrons, as determined by a guide tube. However, in the U C N microscope only one tenth of the m o m e n t u m region conducted by the PN5 guide could be used. In the Schwarzschild system, on the other hand, it was - just barely possible to fully illuminate the geometrical aperture (2 × 24") by use of the convergent feeding

275

A. Steyerl and S.S. Mafik / Cold and ultracold neutron research

guide. At higher velocities this would be impossible - but then no imaging optics admitting large apertures would seem to be available either. Use of a two-dimensional detector will, of course, be equally beneficial for both types of microscopes. b) The uncompensated first-order chromatic aberrations seem to constitute a significant handicap for the Schwarzschiid system. We could think of reducing the wavelength bandwidth by a suitable modification of the multibilayer coating, but limits are imposed by the property of Bragg-type reflection to be selective not to the magnitude I k I of the wavevector but to its normal component, k.. The Schwarzschild scheme requires a fairly wide range of angles of incidence and - ideally selectivity of the reflection process to I kl. This problem can be relaxed by a suitable variation of layer thicknesses over the mirror surfaces, and we can also think of other methods of monochromatization selective to I k [. Of course, we have to keep in mind that any narrowing of the bandwidth will be associated with a proportional drop in speed. This situation is to be compared to the case of the U C N microscope where further monochromatization - if necessary - may be achieved very easily by an adjustment of beam stops and where a bandwidth limitation would be accompanied by a quadratic reduction of chromatic effects. 1.4. The new U C N - V C N source at the HFR, Grenoble, and development o f novel neutron mirrors and guides

In 1978 Ageron proposed [23] installation at the ILL reactor of a new source of ultracold and very cold neutrons based on the Garching-type total-reflection turbine. This source has now been jointly constructed at Garching and the ILL. Fig. 5 shows the curved guide tube (with radius of curvature 13 m, length 12.8 m, cross section 7 0 × 7 0 mm 2) connecting a vertical straight guide (of 5 m length, placed inside the D 2 Cold Source neck) to the turbine. The straight guide is cylindrical with a diameter of 70 mm. The turbine is positioned on the platform of reactor operations. A unique feature of the vertical design is the possibility of avoiding thick windows anywhere along the beam. (The tube nose dipping into the

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vertical straight guide dipping into the liquid deuterium of the "Vertical" Cold Source. One half of the beam is decelerated from h ~ 80 ,~, to the UCN regime by reflection on the receding nickel blades in the turbine. We expect to obtain a UCN current of =6×103 c m - 2 s - t and a total current exceeding 1× 106 UCN/s. The second half of the beam bypasses the turbine wheel and may be u~d for cold neutron experiments. The guides and blades were produced on an all-metal basis using the novel technique described in the text.

liquid D z consists of a 1 mm thick aluminium dome.) Primarily cold neutrons with wavelengths centered around 80 A are conducted by the guide where they undergo only a moderate number of wall reflections. A part of the beam is then decelerated, with = 50% efficiency, to the UCN region by reflection on the receding turbine mirror blades. 690 blades are mounted on the periphery of the turbine wheel with a radius of 0.85 m which rotates at 260 rpm. For this scheme the overall efficiency for UCN production ( = 10%) is expected to be significantly higher than with direct UCN beam extraction from a primary source. We

276

A. Stel'erl and S.S. Malik / ( "old and ultracoM neutron re~'ear~'h

expect that the turbine will supply a current density of 6 × 10 ~ U C N / c m 2 s in a beam with a cross section of 20 × 10 cm -~. In addition to UCN the source will also supply a strong VCN beam bypassing the turbine. The cryogenic section of the guide was installed in early 1985, and the curved guide followed in July, 1985. The turbine is scheduled to be set up in October, 1985. The new source will be used for a number of different experiments, partly on a beam sharing basis and partly in time sharing. The kind of research envisaged includes neutron microscopy, highest-resolution spectroscopy (e.g. in surface studies as described in part It of this paper), UCN storage experiments in magnetic and material bottles, the search for the electric dipole moment and the charge of the neutron, and demonstration experiments in neutron optics. Rather than going into details of these planned activities I should like to discuss the new technological development mentioned before, which was stimulated by the specific requirements of the new source: for safety reasons glass had to be avoided for the vertical guide and thus all-metal, high-quality neutron guides and mirrors had to be developed. Plane or cylindrical mirrors are produced as replicas of high-quality glass surfaces in the following way: about 1000 ,~ of nickel and copper are consecutively evaporated on the glass substrates. Nickel or copper (or cadmium) and, in some cases, an aluminium coating, are then electroplated on these films up to a thickness of 0.15 to 0.30 mm. This composite metal sheet can subsequently be removed from the glass substrate by gentle mechanical deformation. This procedure was applied, e.g., in the fabrication of the wall plates for the curved, rectangular guide. These plates were then glued onto thicker aluminium supports and curved. The circular cylindrical tubes used for the cryogenic section of the guide were cut along their full length of 70 cm prior to removal from the glass tube serving as the substrate. The cut edges were later reconnected by spot welding. An analogous technique was used for the production of the semi-circular turbine blades. In this case a fairly smooth electroplated nickel coating was de-

posited on the external surface. (This was necessary because a small number of neutron reflections takes place on the convex surface near the blade ends.) Reflection measurements on flat and cylindrical neutron mirrors produced in this manner, using 80-A neutrons on the ILL guide PN5, yielded very favourable results. In the region of total reflection, the reflectivity showed a linear decrease with angle of incidence, as previously observed in ref. 22 and expected from ref. 24 where the scattering of neutrons on slightly rough surfaces had been analyzed. The mean reflectivity in the region of total reflection was determined to be about 99%. This value is considerably in excess of the reflectivities (98 to 98.5%) measured for nickel-coated glass mirrors using the same setup. Uncoated glass plates showed about the same reflectivities (within a narrower region of total reflection) as the metallic mirrors. These results can be explained by our observation that in electron micrographs the vapour deposited nickel films appear to display a larger microroughness than the glass used as the substrate. In the replication process, on the other hand. the quality of the original glass surface seems to be conserved and directly transferred to the metal surface. We add the remark that comparative tests with the best polished bulk metal plates available (stainless steel and also nickel), which were carried out by P. Ageron and W. Mampe, yielded significantly lower reflectivities (not for light but for neutrons!) than for the replica mirrors. In addition to the high neutron-optical quality the new metal guides and mirrors offer further obvious advantages: a) They can be used in high irradiation fields, e.g., as the initial sections of neutron guide tubes. In this way full illumination of the guides, up to the critical angle for total reflection, can be ensured even for long neutron wavelengths, say, beyond 10 Jk. This is not the case for most existing guide facilities based on nickel-coated glass. b) The limited mechanical stability of glass, especially under irradiation, seems to rule out its use in vertical or inclined guides like the new V C N / U C N source. c) The replica technique offers the advantage over glass mirrors of much higher flexibility in

A. Steyerl and S.S. Malik / Cold and ultracold neutron research

geometrical design. Thus, rather exotic guides with strong curvature and variable cross section have been made (see, e.g., section 1.3). The thin metal plates with high reflectivity on both sides can also be used in lamellated guides and "super-benders". d) The present technique of combined evaporation on glass and electroplating can also be applied to produce special multilayer mirrors and polarizers.

Acknowledgements For active help in carrying out the work described above we are very gratefyl to F.-X. Schreiber and A. Beynet, to the FRM and ILL reactor crews, and to the workshops in Garching headed by J. Schiller and B. SchriScker. O. Sch~irpf (ILL) kindly performed the multilayer coating of the mirrors for the VCN microscope. Very useful discussions with W. Gl~iser are acknowledged.

Addendum After resumption of operation of the ILL reactor in September, 1985, we measured the intensity spectrum on the new vertical U C N / V C N guide by gold-foil activation, UCN bottle experiments, and by the time-of-flight method. The data show a gain factor of 70 to 100 over the ILL guide PN5. Thus we expected to achieve, after installation of the turbine, a UCN current density of --- 1 × 10 4 cm 2s-1 and a UCN density of = 90 cm -3. These values have now been confirmed [25]. They are in excess of the previous estimates by a factor of about 2.

!1. Very slow neutrons and liquid-vapour phase transition (by S.S. Malik, in collaboration with F. Ludecke and F. Khan, University of Rhode Island) I1. I. Introduction

Reflection and transmission of long wavelength neutrons from material media can be treated in

277

terms of transfer matrix formulation by solving a 1-d Schi~dinger equation involving only the neutron wave vector component k z, perpendicular to the potential barrier. The material medium may be represented by optical potential U: U=

2~rh 2 Nb; m

(1)

N and b, respectively, are the number density (in c m - 3 ) and the coherent scattering length of the medium; m is the neutron mass and h is the Planck constant divided by 2~t. The shape of the potential barrier is determined by the nature of the medium. For instance, for a uniform thin film, with constant density, the height of the potential is determined by the scattering length density ( N b ) and the width corresponds to the thickness of the film. For a bi-layer system (e.g. those employed in fabricating multilayer polarizing monochromators) each subunit consists of a two-potential region with a step at the common boundary of the media making up the bi-layer. A similar optical potential distribution also attains for a liquid-vapour system in the two-phase region. The transfer matrices, M n, in generalized form for multiple boundaries connect the incident wave (amplitude assumed to be unity), reflected and transmitted waves with amplitudes r and t respectively, according to the equation

M l, M2, etc., are transfer matrices at the boundary

identified by the subscript of M. A variety of experiments using long wavelength neutrons have found satisfactory explanation in terms of the transfer matrix [1,26-30]. The interference patterns calculated and observed display characteristic features of the system under investigation. We have recently examined the possible investigations of the phase transition of classical fluid systems using very cold neutrons (VCN). The specific feature explored is the shape of the coexistence curve in the immediate vicinity of the critical point. The effects arising from the temperature dependent width of the liquid vapour inter-

27F.

A. Steverl and S.S. Mahk / ('oM and uhracoht neutron rcsear
face were also examined. We add that wetting. penetration depth for superconductors and interdiffusion between two dissimilar material media constitute the same general type of problems. 11.2. l,iquid- capour phase transition The discussion presented here does not dwell upon the details of the phase transition itself. There are several extensive articles which discuss the topic in considerable details [31 .34]. For present purposes it suffices to say that the liquid -vapour phase transition at the critical point belongs to the universality' class of 3D Ising-like systems with one c o m p o n e n t order parameter. The order parameter, in the case of classical fluid for this transition, is the reduced density, d - - ( p & ) / O , where p is the bulk density' and the subscript c designates its critical value. The coexistence curve is, generally, a plot of k vs t where t = ( T - T.)/T~. T is the temperature of the fluid with T as its critical value. The relationship between ,.1 in the immediate neighbourhood of the critical point for this universality class is governed by the critical exponent fl through the equation [35,36]

A= _+Bl-tl/~:

(3)

B is a system dependent constant. The value of 3, calculated using renormalization group (RG) technique [37], is 0.325 _+ 0.02 (the High Temperature Series (ttTS) techniques [38] values arc somewhat different but appear to be converging toward the R G values). Experimental values for /9 display it considerable range not only a m o n g different fluids but also for different experinaental techniques for the same fluid [39]. The primary source of the difficulty is attributed to inhomogeneities caused by gravity in a finite size sample, typically --=- 1 mm in height. For details on the effects due to gravity, we refer to other articles (these also contain references to most literature on the subject) [40-42]. The required averaging of experimental quantities for this effect and for sample sizes :- 1 m m leads to the consequence of limiting approach to the critical point to It[ -- 10 4. The relationship embodied in eq. (3) assumes an asymptotic critical behaviour which is supposedly reached only when It[ < 10 4

In a recent paper we (along with F. Ludecke and K.-A. Steinhauser) proposed a new method to explore the asymptotic behaviour [43]. The method is based on the reflection of very slow neutron,,, from a thin film of a fluid. Calculations show that the reflection pattern for thin fihns ( < 0.1 ram) is sensitive to very small changes in the bulk density. Two variants of a reflection experiment were disct.ssed: one employing a thin fihn of 5000 ,A thickness, for which the excluded temperature region due to gravity-induced effects anaounts to It I < 10 ~'. In the other, technologically' much less d e m a n d i n g variant, a liquid film of - 0 . 0 5 turn thickness could be investigated. In that case. I tl < 10 s. These techniques can provide a precise mapping of the coexistence curve. The specific example discussed here is that of "~I-te. although the method, in principle, can be applied to study, the critical behaviour of any fluid. In the two-phase region the film consists of two regions with different densities: A~ and A l for the vapour and liquid phases respectively. One can then express (assuming an infinitely sharp interface) the average bulk density, d, in terms of Z~ (the scaled coordinate for the bottom of the film), .k\. and k~ its = -Z,k,(-t)+(l

+ Z 1)..Iv(--;).

(4)

If it is assumed (as is normally the case) that the coexistence curve is symmetrical, then k , . ( - t ) = •..1~ ( - t ) . Furthermore. if it is assumed that the coexistence curve is describable by a linear model. then AI ( - I)~= B ( - - I ) [~.

15)

I:or '*He we employed the values of the parameters B and /9 respectively as 1.435 and 0.359. These values are the result of the analysis of existing data by 1,evelt Sengers and Sengers. A calculation of .k~ and ..1~ (together with the scattering density, Nh, for "~lle) enabled us to generate the neutron optical potential for the 4tie fihn for different values of ( - t ) . It consists of one value for the vapour phase with a width equal to (1 + Z ~ ) and another value for the liquid phase with a width - Z I. The discontinuity from A v to A i. is customarily taken to correspond to Z = 0. The interference pattern for a thin film (of

279

A. Steyerl and S.S. M a l i k / Cold a n d uhracold neutron research

0.8-

I

I

I

O:

t 1-1

1: t = 2: t -

I

• 1041

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l

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.

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.<2_

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Energy (neV) Fig. 6. The calculated reflectivity patterns for several values of ( - t ) for an average bulk density ,~ = 0.05 are shown as a function of neutron energy for normal incidence. The reflectivity maxima and minima beyond the characteristic cut-off are typical of the thin films. The curve labelled 0,1 is for two values of t indicating that the fluid for these two temperatures consists of a single phase. Curves labelled 2 and 3 are for different temperatures and the film consists of two phases. A change in the temperature of the fluid brings about a change in the fractional volume occupied by each of the two phases, thus presenting a different potential distribution to the incident neutrons.

thickness 5000 ,~,) was calculated for several values of t (assuming that the optical potential of the film boundaries is zero). Fig. 6 shows the values of the reflection coefficient as a function of the energy corresponding to the wave vector, k:, for normal incidence. An examination of the figure shows that it is characterized by a number of maxima and minima. If we examine the first minimum, its position is identical for ( - t ) = 1 x 10 -5 and 8 × 10 -5 . The corresponding positions for ( - t) = 2 × 10 -4 and 3 × 10 -4 are different not only from each other, but also from that of ( - t ) = 1 × 10 -5 and 8 × 10-5. This behaviour is directly related to whether the fluid is in the one- or two-phase region• For the one-phase region, the reflectivity pattern will remain unchanged irrespective of the temperature (due to the constancy of the density, the height and the width of the optical potential remain unchanged). In the two-phase region, the positions of the minima, as well as the overall pattern, change with temperature due to the combined effect of the shift in the location of the interface

0

"6 9./-. >,

~, tp

,. '~

.

~10

,

z

tm

"'

; I'~ ~ 2's Reduced Ternperoture -t, 10~

o:s

Fig. 7. The figure shows the variation in the neutron energy of normal incidence, E l, corresponding to the first reflectivity minimum as a function of the fluid temperature. For values of t between 0 and the point marked, t o, E I is constant (the fluid is one phase). For ( - t) values in the vicinity of tp but beyond it,, E~ varies logarithmically (the variation is due to the change in the distribution of the two phases in the fluid of the film)• Insert in the figure is a semi-logarithmic plot of E I vs. In( - t). The intersection of the two straight lines yields directly a value

of t o .

(between the two phases) and the variation in the density of each phase. From the data in fig. 6, it is thus evident that for ( - t ) < 8 × 10 -5 the film I

.

I

, I~I,I

I

,

I

J [¢l,[

I

•

I

i

I

il

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1! Ill t'-

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2-

0'1 U

=

Reduced

Temperature tp

Fig. 8. Values of tp as a function of corresponding ,~ values are plotted on a l o g - l o g scale. The resulting straight line is a consequence of the power law dependence of ,~ on ( - t). The slope of the line determines ft. B is determined by the functional relationship between ,~ and ( - t).

281.t

.4. Steverl and S.S. Mahl, . / ( "oM and ultra
consists of one phase. For ( - t ) > 2 x I 0 "~ the fluid in the film has a two-phase composition. A precise value of the temperature, tp, at which the system reaches the coexistence curve (for a given ~ ) can be obtained by plotting 1"I. the energy, for normal incidence corresponding to the first minimum in the interference pattern. Fig. 7 shows E I as a function of ( - t ) for ~ = 0.05. l:'l is seen to be constant for all values of ( - t ) up to tp (the temperature at which the two-phase composition ensues) and then increases, with further increase in ( - t). A more precise determination of tp is obtained by plotting E~ versus ( - t ) in the vicinity of lp on a semi-log plot as shown in the insert in fig. 7. Intersection of the two straight lines thus enables a unique determination of tp. In this manner, one is able to obtain for different values of ~ corresponding values of %. The pairs of numbers (,.~, tp) then can be employed to map out the coexistence curve. Fig. 8 shows a plot of ,..1 versus tp on a l o g - l o g scale. The resulting straight line then enables a determination of B and /9. The analysis of the 4He data yielded B = 1.412 _+ 0.042 and /9 = 0.358 _+ 0.003 in good agreement with the initial values of B = 1.435 and /9 = 0.359. Although the method, in principle, works very well, the technical difficulties relating to the preparation of large and extremely thin films which must be flat (to - 100 A) are indeed formidable. M a n y of these difficulties can be avoided if we consider the reflection of very cold neutrons (X 80 ,k) at a glancing angle ( - 1 °) on the l i q u i d - v a p o u r interface within a bulk sample of several cm in height and only a few cm-' in extension. The vertical energy, can be defined with a precision - 0 . 0 1 mm by the height of free fall in the gravitational field (1 cm corresponding to 1.025 neV) by use of a "gravity refractometer," designed for ~. ~ 80 ,~ and small fall heights ( < 50 cm). An intense beam of 80 ,~ neutrons exists at ILL. Grenoble. This technique permits use of thick ( - 5 mm) vertical quartz glass windows encasing the sample. Due to their vertical arrangement, the vertical c o m p o n e n t of the neutron wave vector remains unchanged upon entrance into the sample fluid. Thus a reflection measurement is sensitive not to the value of the helium density but only to its variation near the interface. If thin horizontal

entrance and exit slits of - 0 . 0 5 m m width are used. only a narrow region near the interface will be scanned. The neutrons can be allowed to fall on the interface either from above (as in the usual refractometer design) or from below if an intermediary, horizontal mirror is used to reflect the neutrons back up. Thus both the wave transition from the optically thicker (vapour) to the thinner medium (liquid) and the reverse can be utilized to obtain information on the interface formation. The excluded temperature due to gravity effects for this arrangement is reduced to the level It I -< 10 5. similarly as in the proposed space laboratory experiments [41 ]. We now adress the question: "'Do the conclusions of the above analysis aher in any significant way' when the liquid vapour interface has a temperature dependent width". Further calculations were performed by F. Khan. F. Ludecke and S. Malik to examine tills aspect [44 I. Details of these calculations will appear elsewhere. Here we note the two main consequences arising from that work which are independent of the specific profile assumed for the description of the interface. 1) The position of the first minimunl in reflectivily displays a finite but small shift toward higher neutron energy. This shift is somewhat temperat u r e d e p e n d e n t but for I - t l > 3 × 10 " the shift is < 0.03 neV. A more relevant number, howcver, is the dispersion about an average shift. This dispersion was found to be < 0.01 neV. It is thus clear that a precise determination of tp for a given ..1 is unaffected by inclusion of an interface with finite width. 21 The value of the reflectivity at the first minimum is always higher than that calculated using an infinitely sharp interface. This increase is again temperature dependent and to a lesser degree dependent on the profile used for the description of the interface. In any case identification of the reflectivity minimum remains clearly definable. It is thus apparent that the specific features of the slow neutron interference pattern from thin films are capable of precise mapping of the coexistence curve. The method, for all practical purposes, does not suffer from the serious limitations imposed by gravity and introduces an entirely new technique for such studies.

A. Steyerl and S.S. Malik / Cold and ultracoM neutron research

References For Part 1 [1] A. Steyerl, Z. Physik 252 (1972) 371. [2] H. Scheckenhofer and A. Steyerl, Phys. Rev. Left. 39 (1977) 1310. [3] K.-A. Steinhauser, A. Steyerl, H. Scheckenhofer and S.S. Malik, Phys. Rev. Left. 44 (1980) 1306. [4] A. Steyerl, T. Ebisawa, K.-A. Steinhauser and M. Utsuro, Z. Physik B41 (1981) 283. [5] A. Steyerl, J. de Physique 45, Coll. C3, Suppl. No. 3 (1984) C3-255. [6] A. Steyerl, B. Gmal, K.-A. Steinhauser, N. Achiwa and D. Richter, Z. Physik B50 (1983) 281. [7] I.M. Frank, Priroda 9 (1972) 24. [8] A. Steyerl and G. Schiitz, Appl. Phys. 17 (1978) 45. [9] G. Schbtz, A. Steyerl and W. Mampe, Phys. Rev. Lett. 44 (1980) 1400. [10] S.S. Arzumanov, S.V. Masalovich, A.N. Strepetov and A.I. Frank, Pis'ma Zh. Eksp. Teor. Fiz. 39 (1984) 486 [JETP Lett. 49 (1984) 590]. [11] P. Herrmann, Ein Neutronenmikroskop ft~r ultrakalte Neutronen: Entwicklung und erste Messungen, Dipl. thesis, Technische Universit~lt Mbnchen (1982). [12] P. Herrmann, K.-A. Steinhauser, R. Gahler, A. Steyerl and W. Mampe. Phys. Rev. Lett. 54 (1985) 1969. [13] H. Frydman, Ein Schwarzschild-Mikroskop f't~r sehr langsame Neutronen, Dipl. thesis, Technische Universit~it Mianchen (1985). [14] C.G. Shull, in: The Neutron and Its Applications - 1982, P. Schofield, ed., lOP Conference Proceedings No. 64 (Institute of Physics, Bristol, London, 1983) p. 157. [15] A. Steinbach, Cell Biophysics 7 (1985), in press. [16] M.A. Kumakhov, Phys. Lett. 57A (1976) 17; Phys. Stat. Sol. (b) 84 (1977) 41. [17] I.P. Eremeev, Pis'ma Zh. Eksp. Teor. Fiz. 27 (1978) 13 [JETP Lett. 27 (1978) 10l. [18] A. Steinbach, A Photoneutron Source Based upon an Electromagnetic Undulator, in: Proc. Particle Accelerator Conference, Vancouver (May 1985), IEEE Trans. Nucl. Sci. (Oct. 1985) in press. [19] A.G. Klein, P.D. Kearney, G.I. Opat and R. G~thler, Phys. Lett. 83A (1981) 71. [20] V.L. Indenbom, JETP Lett. 29 (1979) 5; A. Zeilinger and C.G. Shull, Phys. Rev. B19 (1979) 3957; A. Zeilinger, Nukleonika 25 (1980) 871. [21] A. Steyerl, K.-A. Steinhauser, S.S. Malik and N. Achiwa, J. Phys. DI8 (1985) 9.

281

[22] A. Steyerl, Nucl. Instr. and Methods 125 (1975) 461. [23] P. Ageron, Cold and Ultracold Neutrons Source Projects, Progress Report of the Studies in October 1978, lnstitut Laue-Langevin, report 78AG224T (1978). [24] A. Steyerl. Z. Physik 254 (1972) 169. [25] A. Steyerl, H. Nagel, F.oX. Schreiber, K.-A. Steinhauser, R. Glhler, W. Gl~er, P. Ageron, P. Astruc, W. Drexel, R. Gervais and W. Mampe, Phys. Lett. A (1986), submitted.

For Part II [26] K.-A. Steinhauser, A. Steyerl, H. Scheckenhofer and S.S. Malik, Phys. Rev. Lett. 44 (1980) 1306. [27] K.-A. Steinhauser, Dissertation, Technische Universit~tt Mbnchen (1981). [28] K. Scheckenhofer and A. Steyerl, Phys. Rev. Lett. 39 (1977) 1310. [29] A. Steyerl, T. Ebisawa, K.-A. Steinhauser and M. Utsuro, Z. Phys. B41 (1981) 283. [301 A. Steyerl, K.-A. Steinhauser, S.S. Malik and N. Achiwa, J. Phys. DI8 (1985) 9. [311 A. Levelt-Sengers, R.J. Hocken and J.V. Sengers. Physics Today 30(12) (1977) 42. [32] J.V. Sengers and J.M.H. Levelt-Sengers, Progress in Liquid Physics, C.A. Croxton, ed. (Wiley, New York, 1978) p. 103. [33] M.R. Moldover, Phase Transitions, Carg~se (Plenum, New York, 1980) p. 63. [34] J.V. Sengers, Phase Transitions, Carg~se (Plenum, New York, 1980) p. 95. [351 B.D. Josephson, J. Phys. C2 (1969) 1113. [36] P. Schofield, Phys. Rev. Lett. 22 (1969) 606. [37] J.C. Le Guillon and J. Zinn-Justin, Phys. Rev. B21 (1980) 3976. [38] B.G. Nickel, Phase Transitions, Cargd:se, M. Levy, J.C. Le Guillon and J. Zinn-Justin, eds. (Plenum, New York, 1980) p. 291. [391 D. Beysens, Phase Transitions, Car#:se (Plenum, New York. 1980) p. 47. [401 F. Ludecke, M.S. Thesis, Univ. of Rhode Island (1982). [41] P.C. Hohenberg and M. Barmatz, Phys. Rev. A6 (1972) 289. [42} M.R. Moldover, J.V. Sengers, R.W. Gammon and R.J. Hocken, Rev. Mod. Phys. 51 (1979) 79. [43] F. Ludecke, S.S. Malik, K.-A. Steinhauser and A. Steyerl, Physica 132B (1985) 1. [44] F. Khan, F. Ludecke and S.S. Malik, to be published.