When sodium chloride crystallises in the presence of protein the usual cube shape derived from aqueous solutions is converted to an extensively branched or dendritic form. A study was made of factors affecting the transformation, cube-dendritic structure. The following were observed to affect the process: type and structure of protein, substrate on which the crystallisation occurred, events concurrent with the crystallising process but occurring from .05-30 m distant in space such as crystallising of salt-protein solutions where the state of the protein had been altered, crystallisation in the presence of a lead mass and in the presence of certain chemical reactions possibly associated with the release of ions. Some of the action at a distance was shown to be associated with ambient oxygen. The effects were noted irrespective of whether the corresponding crystals were electrically shielded. Some of the effects found could be regarded as warranting a re-examination of the plenary concept of space in eclipse since the early part of this century. 

INTRODUCTION 

Growth of biological matter remains as a formidable intellectual problem for which presently there are no clear concepts. The application of solid state physics in the field of energy flow and dielectric properties of biological matter has been of such profit as to suggest that there may be other aspects of biology including growth, where the physicist may provide insight of equal value. An approach to the topic of growth in terms which are more familiar to the physicist than those usually afforded within complex biological systems, is provided by the growth of a crystal. This approach is not as alien to the biological system as could be thought in as far as much biological material is, in effect, crystalline in both structure and function. Many illustrations of ultrastructure attest to the repetitive nature of biological structure, while the success of the Frohlich-Davydov approach to energy distribution properties of this material rest heavily on coherence, in other words on one of its prime functional properties. 

There exists a very simple system wherein biological material can be allowed to perturb the growth of a crystal of a simple inorganic salt. Some of the properties of this system reveal that the energy available to the crystal in its growth is highly manipulable experimentally and that the results of this manipulation generates a new pattern in the crystal array not found in the absence of the bioloqical additive. lt is with the results of crystallising aqueous solutions of sodium chloride in the presence of protein in the solvent that this paper is concerned. It is to be emphasised that such results are intended to be introductory only, in the sense that they carry no quantitative data even though they do expose fields in which even minimal quantitative study will accelerate progress. 

From the aqueous environment, sodium chloride crystallises as a cube. In the presence of certain types of protein in this environment, the simple cube is broken up or subdivided into a branched structure. It is with this transition, from cube to branched structure, that this study is primarily concerned, keeping in mind, in the design of experiments, the query: whence comes the energy for the vast increase in surface area of the crystal evident in this transition? A term used to express the subdivided state is 'fern' and the phenomenon of sodium chloride 'ferning' (alternatively termed dendrite-formation or arborisation) first became apparent in clinical gynecological literature When mucus from the cervix uteri is removed during clinical examination and allowed to dry on a glass slide, fern patterns appear when the dried material is examined under the microscope. 

The elements of the fern structure are crystals which by x-ray diffraction studies can be shown to be composed of nearly 100 per cent sodium chloride (1). The clinician uses this as a test for ovulation in the female because it so happens that the endocrine cycle is associated with a cycle of proteins of several types. One of these proteins, that indicative of ovulation, permits vicinal sodium chloride to crystallise in the so-called fern pattern. It is now known that other proteins and other substances than those of cervical mucus permit 'ferning' of sodfum chloride (2) and it is with some of these proteins that the present study is concerned. 

MATERIALS AND METHODS 

We have standardised our test solution to bovine serum albumen (Armour Pharmaceuticals) dissolved in 0.15 m sodium chloride at a concentratfon of 10 mg ml ^-1. A drop of this solution (0.1 ml) was allowed to dry at ambient room temperature and pressure on a microscope slide for subsequent microscopic study and for photomicrography. In later studies the drying temperature was raised to 70° with no obvious change in the fern appearance. Variations on this standard are described in the results. 

RESULTS 

I. Observations on Normal Solutions 

The subdivision parameter 

If a sufficient number of preparations is studied under the microscope, a series of structures can be tentatively assembled into a sequence suggesting stages in the transition from single cubic crystal to dendritic pattern or 'ferning'. Each corner of the cube (Fig la) is first projected by a sequence of chevron-like titles which give rise to shorter or longer arms 30 µm in diameter (Fig lb). Next, these arms subdivide always at 90° into shorter arms each with a stem of rhomboidal crystals which give off side branches also at 90° (Fig 1c). The process continues until the whole drop area is occupied by a labyrinth of geometrical aspect where there is precise interdigitation without overlap when different preparations are examined (Fig ld). The subdivision process varies in its extent. It is often much finer branches have stems no greater than 10 µm in diameter with finer side arms (Fig le). 

Figure 1. Selection of photomicrographs to illustrate grades of subdivision of the original sodium chloride cubic crystal structure in providing a parameter whose variation is the subject of this paper. The crystals have been air dried from a solution with the following specifications: 0.15 M sodium chloride in distilled water containing 10 mg ml ^-1 bovine serum albumen. Bar = 0.1 mm. 

la.   Three cubes, the right hand pair showing changes along the upper edge indicative of prospective branch formation. The left cube upper surface branching has progressed to the issue of a brench with commencing subdivision.	1b.  Two cubes in a process of growth. The left is more advanced stage of branch origin than is the right crystal.
1c.  Three of the branches of this cube have progressed to increasingly fine subdivision. The fourth branch (facing left) has not branched at this stage.	1d.  A higher magnification view of a typical subdivision pattern in the side branches. Primary and secondary branching occurs at 90° to the next bigger branch.
	1e.  The finest calibre branches which come to compose most of the area of the dried drop in most preparations often have next largest branch at an angle other than 90°. This is the typical fern pattern that has given the phenomenon its clinical name of 'ferning'.

The branches in this finer subdivision state diverge at less than 90°. In short, there is a sequence in grade of subdivision of crystal growth sufficiently obvious as to allow its subsequent use in this study as a parameter: degree of subdivision. 
When preparations are examined on a daily basis there is a demonstrable variation in the degree to which this sequence is realised. While on most days the sequence progresses to the stage of finest subdivision, on other days, at a frequency of 3 or 4 days per year, there is a sudden arrest to the subdivision process and the dried drop is composed mainly of cubes with short coarse side arms. 

II. Experiments Varying Certain Conditions 

At least 5 observations were made of each phenomenon to be discussed and in many cases over 70 have been made at various times over a two year period. 

1. Ambient temperature. The crystal pattern does not change when the preparation is dried at 4°, 22°, 37°, 70°C. When the solution from which the slide preparation is made is heated to 80°C for 10 m, no crystal pattern appears. The preparation is amorphous (Fig 2). The crystal pattern does not change over a range of atmospheric humidity occurring at the following wet and dry bulb thermometer readings: 3°-9°C. 

Figure 2.  Sometimes the substance added to the solvent will not sustain a branched pattern. This amorphous pattern is the result of polyarginine in the solvent. Bar = 0.1 mm.

2. Effect of pH. The same crystal pattern is sustained over the range pH 3 - 10. At extremes of pH, 1 and 14, the subdivision into ferns disappears and only cubes are formed. 

3. Concentration. When the protein concentration is lowered tenfold, the characteristic fern pattern disappears to be replaced by a new pattern wherein concentric laminae of salt crystals surround a central cube. We term this a whorled pattern (Fig 3). Its chief difference from the pattern produced at the higher concentration is the absence of linear branched crystals. 

Figure 3.  Photomicrograph of the "whorled" pattern typical of reducing the albumen concentration in the solvent tenfold. The crystal pattern is less ornate. Bar = 0.1 mm.

4. Protein type. Although albumen from bovine serum was most frequently used, the following proteins produced the typical fern patterns at concentrations of 10 mg ml ^-1: serum globulin, human chorionic gonadotrophin, follicle stimulating hormone, lysozyme, trypsin as well as the polymer polylysine. The protein protamine as well as polyarginine exhibited only an amorphous picture, although the latter polymer produced a fern pattern when a separate preparation was dried concurrently alongside a preparation made with polylysine (see later). 

5. Substrate. The substrate used most frequently was glass. In some experiments the substrate was glass overlayered by a thin polymer film of undetermined thickness prepared as follows: 

(a) Polyvinylchloride. A 1% solution of polyvinylchloride resin was prepared in tetrahydrofuran. Cleaned microscope slides were immersed in and quickly withdrawn from the solution. The film on the reverse side of the glass was wiped off before drying. That on the obverse side was air dried, then released from the glass by flotation on a water bath at room temperature. The film was picked up from the surface of the bath in one of two ways, either on the surface of a fresh slide or very rapidly on the surface of a small ladle fashioned from aluminium foil containing 3 g of crystalline sodium chloride. Both preparations were brought to the glass temperature of the polymer (120°C) for 5 m and then allowed to cool at room temperature. The film on the salt was recovered by reimmersion in a water bath at room temperature followed by mounting on a clean slide. The two slides with their adherent films, the one brought to glass temperature but otherwise untreated, the other brought to the glass temperature in contact with salt, were then used as the substrate for drying drops of albumen-salt solution as used previously. 

(b) Polystyrene. The resin was dissolved (1% w/v) in acetone from which solution, films were prepared and subsequently heated to the glass temperature with or without contact with sodium chloride crystals in the manner described for polyvinyl chloride. The films were used as substrates for dried drops of albumen-saline solution. In experiments with both polymer films, control drops were dried on glass or on unheated films dried either on glass or on sodium chloride crystals. Observations were also made on slides dried in an enclosing earthed metal screen (Faraday Cage). 

(c) Observations on Experimental Substrates There was no difference in the microscopic appearance of the crystals in drops dried on unheated polymer films when compared with those drying on glass. With both polymers, differences were noted in those drops drying on the heated films as follows. Drops drying on films heated in contact with sodium chloride grew crystal patterns that were more subdivided, more branched than those on a control slide. Those drying on films heated on plain glass exhibited patterns that were less subdivided than controls dried on non-heated polymer film or on glass. Drops from slides dried in a Faraday Cage showed no difference from those dried in a non-screened environment. 

6. Effects of lead mass. The branching pattern of growth is altered in two ways when the drop is dried in the presence of a mass of lead placed 20 cm away. In the first, the subdivision parameter is altered so that more cubes form (Fig 4a). In the second, segments of the drop area show concentric fringes reminiscent of interference fringes (Fig 4b). Smaller areas show deformities where growth is amorphous (Fig 4c). In most experiments a 12 Kg mass was used although there was no significant alteration in the presence of a 250 Kg mass. 

Figure 4. Photomicrographs of the result of dryfng the drop 10-20 cm from a 12 kg mass of lead.
4a.   The fine branching pattern occupying the entire dried drop as shown in Figure le is altered by the appearance of cubes showing early coarse branching. This appearance can occupy 10-25% of the area of the dried drop, Bar = 0.1 mm.	4b.   The drop, most of which occupies the area of the photomicrograph, has had its aspect considerably altered by peripheral zones of crystal rows distributed in a fashion reminiscent of a diffraction pattern.   Bar = 1 mm.
	4c.   A whole drop is depicted in this photomicrograph to show pattern disturbance in which crystal growth is replaced by a bizarre waveform pattern at the right hand edge of the drop. This is probably derivative of the pattern illustrated in 4b. Bar = 1 mm.

III. Further experiments to examine extremely long range effects 

1. The altered growth pattern described in the previous section in the presence of lead is further disturbed when certain chemical reactions are carried out in space in the vicinity. The reactions consisted in the addition of conc. hydrochloric acid first to metallic zinc or, in other experiments to calcium oxide or calcium sulphate. The results were unchanged whether the reaction vessel was open or lidded. Two sites were used to display the effect respectively at a distance of 3 m and 15 m from the drying drop. In the latter case the site was an adjacent laboratory separated from the drying site by a conventional 10 cm brick wall and closed glass-wooden doors. The disturbances took the form of an increase in the numbers and area of amorphous patches which developed among the crystals, as had occurred in the presence of a lead mass illustrated in Figure 4c. 

2. Mutual effects of concurrent drying of two or more drops. During the currency of the observations already described, there was a clear impression of material interference to the growth pattern of drops that were drying on adjacent slides concurrently, yet which were separated on the bench by varying distances of several cm. Experiments were devised to explore this effect using three standard distances respectively 5 cm, 1 m and 15 m, the latter in an adjacent laboratory. The effects were identical at these sites which, for convenience in the following description will be treated as the one site referred to as the distant drop. In each experiment, pairs of slides were allowed to dry concurrently as follows. The distant slide contained a drop of standard albumen-salt solution at the usual pH 6 the experimental slide was a drop from a solution at pH 1. 

On drying, which occurred at each site concurrently within a range of 1-3 m of each other, the experimental slide showed the expected absence of the fern pattern and its replacement by numbers of single cubic crystals. Its counterpart, instead of the usual fern pattern, showed cubes in conformity with the pattern in the experimental slide. This highly reproducible effect was subject to further analysis.

(i) Effect of screens.
Each slide was enclosed in a box with a lid 20 x 12 x 3 cm made of the following material: cardboard 1 mm thick, lead foil 1.5 mm thick, aluminium foil 0.3 mm thick. No differences were noted compared with the absence of the box in that both slides showed the presence of cubes and the absence of fern pattern.

(ii) Effect of gaseous environment. 
Each member of the slide pair was enclosed in gas-tight boxes of 'lucite' 50 x 50 x 20 cm arranged side by side. With each slide at the centre of the box, members of the pair were thus 20 cm distant from each other. The boxes were then filled with gas. Gases were used in separate experiments on each pair at the pH already described. These were oxygen, nitorgen and argon. Only in the case of oxygen was the altered growth pattern on the distant slide observed. With the other two gases, the distant slide showed the normal subdivision pattern.

(iii) Effect of an earthed screen or Faraday Cage.
There was no alteration to the patterns described when one or both members of the pair were effectively screened. 

DISCUSSION 

It has been known to gynaecologists observing the crystal arborisation occurring in dried mucus secretion that the striking patterns were formed by sodium chloride (1)(2). The latter authors showed that other substances including proteins and polysaccharides could cause sodium chloride to form the same complex patterns when dried from appropriate solution mixtures. While we have not measured the increase in surface area of the crystal related to its subdivision, it is clear that there must be an increase of orders of magnitude. If we then use an equation for the free energy of surfaces in a multicomponent system(3). 

dG = - SdT + VaP + YdA + å₁ µ_i dn_i 
where 

• G = free energy 
  S = entropy 
  T - temp abs 
  V - volume 
  P   pressure 
  Y  - Surface Constant ('tension') 
  A - surface area 
  µ_i- chemical potential of the ith component 
  n_i- mole number of the ith component 

which, under conditions of constant temperature and pressure and where changes in the surface area do not alter the composition of the surface, reduces to 

• dGTp - YdA,

then the free energy of any surface is proportional to its area. The question arises as of prime concern in this study, whence the source of this energy which the observations show is somehow connected with additives to the solvent of biological nature and with space?

The underlying belief was that such an approach could illuminate present concepts of the energy supply for other situations where subdivision of material is a prominent feature, namely biological systems in their growth phase. Use of the microscope to observe the subdivision process showed the existence of crystal patterns intermediate between cube and dendrite, reinforcing the idea of a time-related growth gradient process which could be arrested or modified perhaps experimentally. The possibility of modifying the sequence, cube to dendritic form, may then throw some light on the source and nature of the energy required. In this light the dried drop thus becomes a micro-area where inhomogeneities in free energy available for subdivision can be observed.

We can discuss the experimentally imposed modifications under three headings related to their proximity in space to the drying solute, respectively, vicinal that is, the effect of matter in the solvent, effects at a distance equivalent to the thickness of the plastic films, often termed long range effects, and lastly effects at still greater distances up to many metres which one could term extremely long range effects.

Vicinal Effects

In formal biochemical descriptions, proteins are divided into 13 or so groups. Proteins representative of several of these groups permit dendrite formation: the effect is not characteristic of a particular group. Even homopolymers of lysine are effective. Equivalent concentrations of several amino acids were ineffective suggesting that association of the solute with matter as complex as a polymer is necessary. This is attested by the failure of salt to arborise when dried from glucose solutions despite arborisation when dried from dextran solutions(2). That the polymer must have certain special properties is attested by the inability of polyarginine to sustain an arborisation pattern. It may be relevant that of all twenty amino acids, the amino acid arginine shows a special property to structure water not shared by any other amino acid (4). Its polymers may thus lower the free energy to a minimum inconsistent with any form of subdivided crystal patterning.

It is to be noted that concurrent drying of polyarginine in the solvent with a preparation of polylysine in the solvent separated by a few cm in space was able to 'supplement' polyarqinine in the solvent to the extent of producing subdivision. If the protein is denatured by heating, no arborisation occurs, indicating that not only is a more complex molecule necessary as discussed, but that part of this complexity may reside in the folding pattern (quarternary structure) which is considerably altered by heating. Nor does arborisation appear at extremes of pH, which would have the same gross effects on quaternary structure. Within the range 1-10 mg ml ^-1, the higher the protein concentration, the more elaborate the arborisation. When, in the living organism, the protein is under the steroid oestrogen dominance, arborisation occurs. On the other hand, the salt crystals are reduced to an amorphous form when the protein is under dominance of a related steroid, progesterone. In short, alterations of the disposition of these complex molecules in space (their conformation) contingent on binding of the steroid molecule, alters their capacity to elaborate the crystal pattern of sodium chloride.

Long Range Effects

In considering effects on crystal pattern frorn more distant sources than the solvent itself we made use of the ability to vary the substrate on which the drying was carried out. These experiment were prompted by observations reported in a series of papers published some years ago by Distler (5)(6)(7). Crystal patterns forming on the outer surface of thin layers whose inner surface was applied in turn to a substrate of freshly cleaved crystal, provided evidence in the outer crystal pattern of events occurring at the cleavage layer whose distance from the substrate was thus equivalent to the film thickness. Distler used this elegant method to study long range effects imparted by the cleavage face. In the present studies, plastic films of polyvinylchloride and polystyrene were subject to conditions devised to produce an electret in the films. These conditions were set up in films in contact with ~3 g of randomly arranged sodium chloride crystals.

The observation was consistently made, that arborisation subsequently produced on such films was more elaborate than that occurring on films made on glass without the interposition of iodium chloride crystals. Distler's observations clearly show that the effect of alterations to energy flow within the crystal (he used electromagnetic radiations) can be made manifest at a distance from the crystal equivalent to the film thickness. The crystal appears to be conducting the energy pattern which had been imposed on it through space to a distance at least equal to the film thickness. His observations, when an electret was installed in the film, showed that, in its turn, the film was able further to modify the energy pattern which it had received from the substrate and which it, in turn had made available to the growing crystal. The more ornate the pattern experimentally generated in the heated film (now an electret) the more ornate the crystal pattern. In the present studies, the random orientation of the sodium chloride crystals over which the polymer film was laid would be expected to produce an irregularity in polymer dipole array in the molten state compared to the array derived from a polymer brought to the same glass temperature in the absence of an irregular crystal layer, that is on glass (Fig.5). The observation that there is a difference in degree of arborisation on two films with different histories of preparation indicates first that the observed change could be related to irregularity in the orientation of dipoles in the substrate: the greater the misorientation of dipoles the greater the degree of subdivision of this pattern and secondly that this irregularity can be imposed (?by induction) over long range.

Figure 5. Diagrammatic representation of crystal patterns on plastic films brought to glass temperature on two substrates prepared as described in text. The polymer configuration in the films is imaginary.

Crystal pattern (real) Plastic film layer (proposed) Substrate layer (real)

We can now discuss experfments designed to reveal effects from even more remote sources, effects which may be called extremely long range.

Extremely Long Range Effects

The merit of Distler's approach using interposed films, lies in the ability to record events at what physicists and physical chemists would regard as considerable distance or long range from the underlying crystal face. Using the electret draws attention to a flux to which crystals, possibly all matter, is subject, that is, a flux of long range occurring in the absence of imposed energy. The possible existence of such a flux for which there is an increasing theoretical basis (8)(9)(10)(11), caused us to study the effects of materials placed even more distantly than the film thickness in the environment of the growing crystal. From evidence derived from its effect on mammalian cells growing in culture (unpublished) we used a 12 Kg mass of lead placed 10-50 cm distant as described. The effect was an alteration of the branching toward a simpler pattern, the appearance of diffraction rings together with the occurrence of gross patterning defects at foci within the field.

The ubiquity of proton gradients and their property of determining striking modifications to growth patterns in biological growth (12) caused us to test the effect of acids reacting with matter at distances of metres from the drying drop. The mechanism by which the observed effects of acids reacting with zinc and other substances over such long distances can be transmitted through space are, of course, presently unknown. The repeatability of the observation albeit under specffic protocols marks it as a suitable target for further studies to be reported later. Meanwhile, observations on the effect of ambient gases on the transmission of crystalline states across 20 cm of space indicate that properties of oxygen should be further examined. That matter, gaseous or otherwise, cannot be the only aspect in space transmission is attested by the curious very long range effects studied for many years by Rothen in New York (13).

As in Distler's studies, he used the interposition of films of various composition to show a solar effect on the rate of crystallisation of protein. The rate, as measured by thickness of protein films in the process of deposition, varied on a diurnal basis. The existence of a varying intensity flux of importance to crystal growth of the type he has proposed as the result of his observations may have contributed to results of our own observations on extremly long range effects and of the occasional diurnal variation in dendrite pattern development. Some of the long range effects reported by Distler and those from Rothen's observations as well as the present results involving oxygen, indicate that the fluxes may be of classical electromagnetic origin. The observations reported here were reproducible in a shielded cage suggeseing that there are other than electromagnetic forces involved. There is no reason why the two sources electromagnetic and non-electromagnetic cannot be closely correlated in space as the theoretical considerations of Hagan (11) indicate.

Some properties of space conceived as a multidimensional array admit of the emergence of photons (11). It is possible that other important analogies with biological growth are provided by even more physically orientated phenomena than is crystal growth. One of these is dielectric breakdown (14) and another is diffusion limited aggregation, a concept with which the former can be linked (15). In modelling dielectric breakdown, the authors (14) adduce evidence for fractal properties of such a discharge structure. Its branches do not grow with the size of the object and in their stepwise increase they do not overlap since no crossing is possible. The picture is thus much as we have illustrated with crystals. At each branch point, the process starts anew and the pattern is generated by a compromise between the 'tip effect' on the one hand and those branches which shield themselves by growth inside a cage on the other. Although the growth source is at once apparent in such a system (the discharge itself) the definitive patterning in any discharge is apparently conveniently modelled as its interaction with certain properties of space here (14) described as screening. In connection with a thesis on the nature of biological growth which we have developed, the present observations reveal the following about the elaboration of pattern and of growth of sodium chloride crystals, under conditions used in these experiments.

(i) The pattern is much more complicated when the crystals are grown in the esence of many but not all types of protein than when they form from simple aqueous solution. In very general terms the more heterogeneous the amino acid composition, the more elaborate the pattern. The importance of heterogenity inducing elaboration reappeared in experiments with polymer film substrates with differing preparation histories.

(ii) In addition to the presence of protein in the solvent, contributions to pattern formation originate more distantly. They can originate in the substrate (long range) and even more distantly, measurable at least in metres (extremely long range), this can be manifest experimentally on distant but concurrently growing pairs of drying solvent by effects of ion release and of certain materials in the environment such as a lead mass.

(iii) Although some of these effects are demonstrably connected with matter in space such as gas molecules, others could be concerned with the properties of space itself. To this extent they are timely in their support of a present leaning toward the plenary concept of space. From reports in the literature some of which have been cited (7)(8)(9), the pendulum, stationary at the void concept for the last eighty years following the lead given by Einstein and interpretations of the results of the renowned Michelson-Morley experiments, is slowly swinging back to the plenary idea. A study of crystal growth patterns may provide an unexpected method of studying the structure and other properties of space.

(iv) A hint as to the mechanism whereby periodic time changes in the quality of space could be translated into structural change in matter in dynamic equilibrium (such as in the formation of a crystal pattern) is given by our most recent studies. Capacitance changes in the nanofarad range in aqueous solutions of sodium chloride occur when these solutions are exposed to experimental conditions described under extremely long range effects in this paper. The results will be reported later.

REFERENCES

 (1) R. R. Macdonald, J. Obstet. Gynecol. Brit. Commonwlth. 76, 1090, (1969).

 (2) K. J. Beck, U. Budde, A. Neuhaus, K. F. Seifert. Arch. Gynak. 210, 76, (1971).

 (3) G. A. Somorjai. Principles of Surface Chemistry (Prentice-Hall Inc. Englewood, N. J. Cliffs) (1972).

 (4) R. Wolfenden. Science 222, 1087 (1983).

 (5) G. I. Distler. Jour. Crst. Growth, 3,4. 175, (1968).

 (6) G. I. Distler, V. P. Vlasov. Thin Solid Films 3, 333 (1969).

 (7) G. I. Distler, Jour. Crst. Grow-th. 9, 76 (1971).

 (8) B. DeWitt, Phys. Rep. 19C, 297 (1975)

 (9) W. Greiner, J. Hamilton, Amer. Sci. 68, 154, (1980)

(10) P. Davies, T. Birrell, Quantum Fields in Curved Space, (Cambridge Univ. Press, Cambridge) (1982).

(11) B. Hagan, in press

(12) P. Mitchell, Nature, 191, 144 (1961)

(13) A. Rothen, Proc. Nat. Acad. Sci. (US), 72, 2462, (1975)

(14) L. Niemeyer, L. Betroniero, H.J. Weismann, Phys. Rev. Lett. 52, 1033 (1984)

(15) T.A. Witten, L.M. Sander, Phys. Rev. B 27, 5686 (1983).