Advances in Imaging and Electron Physics, Volume 101 (Advances in Imaging and Electron Physics)

ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 101

EDITOR-IN-CHIEF

PETER W. HAWKES CEMESILaboratoire d' Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

ASSOCIATE EDITORS

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California

TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom

Advances in

Imaging and Electron Physics EDITEDBY PETER W. HAWKES CEMES/L..aboratoired Optique Electronique du Centre National de la Recherche Scientifrque Toulouse,France

VOLUME 101

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1

CONTENTS CONTRIBUTORS PREFACE . .

...................... . . . . . . . . . . . . . . . . . . . . . .

vii

ix

Applications of Transmission Electron Microscopy in Mineralogy P. E. CHAMPNESS I. Introduction .................... 11. Analytical Electron Microscopy of Minerals . . . . . . . . 111. Phase Separation (Exsolution) . . . . . . . . . . . . . . IV. HFtTEM and Defect Structures . . . . . . . . . . . . . V. Concluding Remark . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

1 2

6 27

33 33

High-Resolution Electron Microscopy of Quasicrystals KENJIHIRAGA I. Introduction .................... 11. Quasiperiodic Lattices . . . . . . . . . . . . . . . . . 111. Experimental Procedures . . . . . . . . . . . . . . . . IV. Electron Diffraction of Quasicrystals . . . . . . . . . . . V. High-ResolutionElectron Microscopy Images of Quasicrystals . VI. Structure of Icosahedral Quasicrystals . . . . . . . . . . . VII. Structure of Decagonal Quasicrystals and Their Related Crystalline Phases . . . . . . . . . . . . . . . . . . . . . . VIII. Concluding Remarks . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

66 96 96 97

Formal Polynomials for Image Processing ATSUSHI IMIYA I . Introduction .................... I1. Image Polynomials . . . . . . . . . . . . . . . . . III. Quotient Fields of Digital Images . . . . . . . . . . .

101 113

V

. .

37 38 41 42 50 53

99

vi

CONTENTS

IV. Image Polynomial and Pyramid . . . V. Shape Analysis Using Image Polynomials VI. Concluding Remarks . . . . . . . Acknowledgments . . . . . . . . References . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 134 139 140 140

I. Introduction . . . . . . . . . . . . . . . . . . . I1. Wave-Particle Models of Massive Particles . . . . . . . . 111. Wave-Particle Models of Photons . . . . . . . . . . . IV. Electromagnetic Model of Extended Particles . . . . . . . V. Extended Special Relativity and Quantum Mechanics in a Local L-Space . . . . . . . . . . . . . . . . . . . . VI . mo-Wave Model of Charged Particles in Kaluza-Klein Space VII . Extended de Broglie-Bohm Theory . . . . . . . . . . . VIII. Infons? . . . . . . . . . . . . . . . . . . . . . IX Concluding Remarks . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

.

144 148 161 174

The Dual de Broglie Wave MARCINMOLSKI

INDEX

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.

. .

. . . . .

.

198 207 213 231 232 234 234 240

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin.

P. E. CHAMPNEss (l), Department of Earth Sciences, University of Manchester, Manchester M13 9 PL, United Kingdom KENJIHIRAGA (37), Institute for Materials Research, Tohoku University, Katahira, Aoba-ku, Sendai 980-77, Japan ATSUSHI IMIYA(99),Department of Information and Computer Sciences, Faculty of Engineering, Chiba University, Yayoi-cho, Chiba 260, Japan MARCINMOLSKI(144). Department of Theoretical Chemistry, Faculty of Chemistry, Adam Mickiewicz University, ul. Grundwaldzka 6, Poznan PL 60-670, Poland

vii

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PREFACE The four surveys that occupy this volume illustrate widely diverse aspects of imaging and electron physics. We begin with two chapters on applications of electron microscopy. The first, by Pamela Champness, describes the role of this technique in mineralogy, an area of applications that is not as well known as it deserves. The structure and physical properties of minerals are complex and not easy to unravel. The familiar limitationsof x-ray diffraction are largely absent from electron microscopy and the breadth and richness of the information provided by this instrument have been immense. I am sure that this survey, by a microscopist whose “first ion-thinned mineral specimen was a pyroxene ...from Apollo 11” will excite wide interest. The second contribution, a study of the high-resolution electron microscopy of quasicrystals by Kenji Hiraga, is really a short monograph rather than a review article. These curious structures, which were for many years dismissed by critics as eminent as L. Pauling, are now widely studied. The work of Hiraga has been among the most important in the comprehension of these materials; here, he presents not only the underlying crystallography but also explains in detail how the corresponding images and diffraction patterns should be interpreted. Next comes an account by Atsushi Imiya of a highly original new approach to deconvolution in image processing. By associating a polynomial with the graylevel values of (discrete) images, Imiya shows that it is possible to invert the convolutionalrelation that describes many kinds of image formation. The role of these image polynomials is illustrated in several other contexts: morphology, the distance transform and skeletonization. I am very pleased to provide a connected account of these ideas here. The volume ends with a very detailed discussion of the dual de Broglie wave by Marcin Molski. The existence and role of such a wave have been the subject of debate for many years and the idea was vigorously defended by David Bohm. Many other approaches to the idea have appeared and Molski gives a critical account of these developments as well as setting out his own ideas on the subject. I have no doubt that the debate will continue and I am sure that this full presentation will clarify many of the issues. It only remains for me to thank very sincerely all the contributorsfor the care and scholarship that they have brought to their chapters and to list the review articles that are planned for forthcoming volumes. Note that volume 100, a cumulative index, will appear shortly after volume 102. Peter W. Hawkes ix

PREFACE

X

FORTHCOMING CONTRIBUTIONS Nanofabrication Finite-element methods for eddy-current problems Mathematical models for natural images Use of the hypermatrix Image processing with signal-dependent noise The Wigner distribution Hexagon-based image processing Microscopic imaging with mass-selected secondary ions Modern map methods for particle optics ODE methods Microwave tubes in space Fuzzy morphology The study of dynamic phenomena in solids using field emission Gabor filters and texture analysis Miniaturization in electron optics Liquid metal ion sources X-ray optics The critical-voltage effect Stack filtering Median filters The development of electron microscopy in Spain Space-time representation of ultra-wideband signals Contrast transfer and crystal images Optical interconnects

Numerical methods in particle optics

H. Ahmed and W. Chen (vol. 102) R. Albanese and G. Rubinacci (vol. 102) L. Alvarez Leon and J. M. Morel D. Antzoulatos H. H. Arsenault M. J. Bastiaans S.B. M. Bell M. T. Bernius M. Berz and colleagues J. C. Butcher J. A. Dayton E. R. Dougherty and D. Sinha M. Drechsler J. M. H. Du Buf A. Feineman and D. A. Crewe (vol. 102) R. G. Forbes E. Forster and F. N. Chukhovsky A. Fox M. Gabbouj N. C. Gallagher and E. Coyle M. I. Herrera and L. Bni E. Heyman and T. Melamed K. Ishizuka M. A. Karim and K. M. Iftekharuddin (vol. 102) E. Kasper

xi

PREFACE

Surface relief Spin-polarized SEM Sideband imaging Vector transformation SEM image processing Electronic tools in parapsychology Z-contrast in the STEM and its applications Phase-space treatment of photon beams Aspects of mirror electron microscopy Image processing and the scanning electron microscope Representationof image operators Fractional Fourier transforms HDTV Scattering and recoil imaging and spectrometry The wave-particle dualism Digital analysis of lattice images (DALI) Electron holography X-ray microscopy Accelerator mass spectroscopy Applications of mathematical morphology Set-theoretic methods in image processing Focus-deflection systems and their applications Electron gun system for color cathode-ray tubes Study of complex fluids by transmission electron microscopy New developments in ferroelectrics Electron gun optics Very high resolution electron microscopy Morphology on graphs Analytical perturbation methods in charged-particleoptics

J. J. Koenderink and A. J. van Doom K. Koike w. Krakow W. Li N. C.MacDonald R. L. Moms P. D. Nellist and S. J. Pennycook G. Nemes S. Nepijko (vol. 102) E. Oh0 B. Olstad H. M. Ozaktas E. Petajan J. W. Rabalais H. Rauch A. Rosenauer D. Saldin G. Schmahl J. P. F. Sellschop J. Serra M. I. Sezan T. Soma H. Suzuki I. Talmon

J. Toulouse Y. Uchikawa D. van Dyck L. Vincent M. I. Yavor (vol. 103)

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ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 101

Applications of Transmission Electron Microscopy in Mineralogy P. E. CHAMPNESS Depurrinerif q/’Eurth Sciences. Univer.sity of Munchester: Munchesrer M I 3 YPL, U.K.

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

A. Alkali Feldspars . . . . . . . B. Ainphi boles . . . . . . . . IV. HRTEM and Defect Structures . . . A. Biopyriholes and Polysomatic Defects V. Concluding Remark . . . . . . . References . . . . . . . . . .

7 14 21

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28 33 33

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1

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11. Analytical Electron Microscopy of Minerals 111. Phase Separation (Exsolution) . . . .

2 6

I. INTRODUCTION Although transmission electron microscopy (TEM) became a routine tool for the physical metallurgist in the 1960s and the theory of image formation from crystalline materials was well established by then, it was not until the 1970s that the TEM was adopted to any great extent by workers in the earth sciences. The main reason for the long delay was that there was no reliable method for preparing thin foils of nonmetallic materials; studies were restricted to cleavage fragments of layered structures or to powdered fragments sedimented onto carbon films. The latter technique only allowed examination of microstructural features smaller than about 1 p m and spatially related information on a larger scale than this was largely lost. The advent of reliable, commercial, beam-thinning devices in the early 1970s solved the problem of specimen preparation. Foils in which hundreds of square microns are transparent to the electron beam can now be prepared almost routinely. Three-millimeter-diameterdisks can be drilled from petrographic thin sections that are approximately 25 p m thick and thinned with a beam of energetic ions or atoms (usually argon) until perforation. The thin sections can be studied beforehand in the petrographic optical microscope, the scanning electron microscope (SEM), or the electron-microprobe analyzer (EMPA), and regions of interest for TEM study can be chosen. I

Copynght 1998 Acadrmii Prc% I ~ L All right\ ot rcprrnlutlion in m y form rewned i m - 5 m 1 ~ $25 7 IH)

2

P. E. CHAMPNESS

Q

Q

Q

Pyroxene

Amphibole

Mica

0 S i . A l i 0 . 0 ; $OH

Rldrpar

FIGUREI . Partial projections of the linkages of the Si-0 tetrahedra in pyroxenes (single-chain silicates). amphiboles (double-chain silicates), micas (sheet silicates), and feldspars (framework silicates).

It so happens that at almost the same time that beam-thinningmachines came on the market, the first moon-rock samples started arriving on Earth as a result of the Apollo space missions. For a time in the early 1970s, more moon-rock samples had been studied in the TEM than terrestrial ones! My first ion-thinned mineral specimen was a pyroxene (a single-chain silicate, Fig. 1) from Apollo 11! Since those days, the TEM has become an integralpart of much of mineralogicalresearch. In this review I highlight just a few examples that illustrate the impact that TEM has had in mineralogy in the last 25 years. I have chosen to concentrate on two of the commonest silicate groups: the alkali (Na-K) feldspars (framework aluminosilicates, Fig. 1) and the amphiboles (double-chain silicates, Fig. l), although I also describe the important contributionthat high-resolutiontransmission electron microscopy (HRTEiM) has made to our understanding of mixed-chain structures.

11. ANALYTICAL ELECTRONMICROSCOPY OF MINERALS The advent of x-ray analysis in the TEM has allowed us to identify fine-scale mineral phases that would have been impossible or extremely tedious to identify

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

3

by electron diffraction, given the large unit cells, complex chemistry, and low symmetries that are involved in most cases. As will be seen in Section III. B, investigations of phase separation (or exsolution) in the amphibole group have relied very heavily on analytical electron microscopy (AEM), so it is worth outlining here some of the procedures that need to be adopted in the AEM of minerals and some of the precautions that need to be taken. The basis of quantificationof mineral analyses is the thin-film criterion of Cliff and Lorimer (1975) in which x-ray absorptionand secondary x-ray fluorescence are assumed to be negligibleto a first approximationand the ratio of the concentrations of two elements CA/CBis related to the ratio of their measured x-ray intensities I, / Is by the equation

where ~ A isB a sensitivity factor that accounts for the relative efficiency of production and detection of the x-rays. For silicates, the reference element, B, is silicon. Because silicates are composed predominantly of oxygen and specific gravities are normally between 2.5 and 3.5, the thickness, tmmrat which Eq. (1) breaks down and corrections for absorption and fluorescence must be made is larger than for metallic systems. Nord (1982) calculated the value of r,, for Mg/Si, CdSi, and Fe/Si for members of the pyroxene quadrilateral. Figure 2 shows a compilation of the minimum value of tmmfor all three elemental ratios and indicates that analysis must be carried out in areas that are less than 130 to 300 nm thick, depending on the bulk composition, if absorption effects are to be insignificant. As it happens, the maximum thickness for which microstructures in silicates can be observed

FIGURE2. Maximum thickness (in nanometers) of Ca-Mg-Fe pyroxenes for which absorption corrections can be ignored. (Source: af'ter Nord, 1982.)

4

P. E. CHAMPNESS

TABLE I AEM ANALYSESOF N O SILICATES" Pyroxene

2

1 Si02 A203

Ti02 Feo MnO MgO Na2O CaO K20 Total

48.01 4.88

0.00 29.9 1 1.89 15.31 0.00 0.00 0.00 Ioo.00

Mica (biotite)

Si AI" AIV1 Ti Fe2+ Mn

0.13

3

1

2'00

38.13 23.20 1.58

Mg Na Ca K

0.89 0.00 0.00 0.00

0

6

2.01

13.75 0.00 13.58 0.62 0.00 9.14 100.00

4 Si AIN AI"' Ti Fe2+ Mn Mg Na Ca K 0

5'31 2.69

!i}

0.00

1

8.00

5.71

2.83

}:

1.78

1.62 22

Source: Champness (1995);reproduced by permission of Chapman & Hall. The oxide weight percents in columns 1 and 3 were derived assuming a total of 100%. The atomic formulas in columns 2 and 4 were calculated assuming a total of 6 and 22 oxygens and a total of 2 and 8 (Si Al) for the pyroxene and the biotite, respectively. All iron has been assumed to be Fez+.

+

using 100-kV electrons is about 200 nm,so if microscopy can be carried out in an area of the foil at -100 kV, it can be assumed that the foil fulfills the thin-film criterion for elements Z 2 11. For higher voltages or lighter elements, this rule of thumb cannot be used and care must be taken to work in suitably thin areas, or, alternatively, to correct for absorption. Silicates are composed predominantly of oxygen, which cannot be reliably quantified by AEM, even with detectors with ultrathin windows. The method adopted for quantification of anhydroussilicate (or other oxide) phases is to assume that all cations are present as oxides and that the sum of the oxides is 100%. The chemical formula is then recalculated to a suitable number of oxygens: for example, six in the case of the pyroxene (a single-chain silicate, Fig. 1) in Table I because the general formula for pyroxene is M2Si206, where M stands for the cations other than silicon. Problems obviously arise in the case of cations such as Fe that can take a number of valances and where there are elements other than oxygen that cannot be detected. The most common of these is hydrogen, as many silicates are hydrated (Fig. 1). For hydrated samples, if all other cations can be detected, a total can be assumed for the oxide analysis that is appropriate to the mineral type (e.g. 95 wt% for the sheet silicate mica, Fig. 1) or the formula can be normalized to an appropriate number of oxygens (22 for micas, Table I, as the general formula for mica is X ~ Y ~ - ~ Z & O ( O Hwhere ) ~ , X and Y are nontetrahedral cations and Z is Si or Al).

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

5

In the general case it is recommended that where possible, normalizationbe carried out on the basis of the known number of cations in a particular crystallographic site (Peacor, 1992). For instance, in Table I the tetrahedral sites in pyroxene and mica have been assigned 2 and 8 (Si Al), respectively. For the mica, the cations known to occupy the X and Y crystallographicsites have been grouped together to give totals of 1.78 and 5.71, respectively. A more complex assignment of cations to particular sites will be encountered in Section III.B, where the amphibole group is considered in some detail. Perhaps the severest problem encountered in the AEM of silicates is that of specimen damage during analysis. Silicates are known to suffer from radiolysis (i.e., electronic excitation leading to atomic displacement) during electron irradiation. The high current densities used for high-resolution AEM can lead to significant structural and chemical changes which ultimately limit the accuracy of analyses. The degree of sensitivity to damage depends on a number of factors, among which are the type of linkage of the Si-0 tetrahedra, the nature of the cations (Na and K being the most vulnerable to loss), and the presence or absence of hydroxyl ions (Veblen and Buseck, 1983; Hobbs, 1984; Champness and Devenish, 1992). Champness and Devenish (1992) and Devenish and Champness (1993) have shown that all silicates suffer some mass loss at the highest current densities used in AEM, but that there is a threshold of the current density for each element in a particular structure for which no loss occurs. For instance, the threshold values of the current density for which no loss occurs for any element is %lo5A/m2 for calcic pyroxene (diopside) and about 3 x 104 A/m2 for calcic mica (margarite). Notice that both these values are lower than the current density in a focused beam from a LaB6 gun. At the highest current densities available (i.e., those obtainable with a FEG),plagioclase (Na-Ca) feldspar is reduced to the composition of Si02 after 200 s (Fig. 3)!

+

A"

FIGURE3. Energy-dispersive X-ray spectra from plagioclase feldspar:(a) defocused beam rastered over specimen for 200 s; (b) beam focused at an approximate current density of 1.8 x lo8 A/m2 in a dedicated STEM.

6

P. E. CHAMPNESS M k V

“’1

1

OS

.

100 kV E -10

Liquid niaogen temperaNre

-1

Ambienl tcmpnsture 50 kV 0.0

0.1

I0

1.1

1.0

1.1

0.0

1.0

1.n

30

40

Dose, C m-2 (x 106)

FIGURE4. Semilog plot for the loss of Na from plagioclase feldspar at a current density of 1.8 x lo3 N m Z : (a) dependence on voltage; (b) dependence on temperature. (Sourre: after Devenish and Champness, 1993; reproduced by permission of the institute of physics.)

It is clearly important that where possible, the analyst operate below the current density at which damage occurs if quantitativeresults are required. Because of the dependence of the rate of damage on the current density, rather than on the total dose, defocusing the electron beam is more effective in minimizing mass loss than rastering a focused beam over the same area. The effect of mass loss may also be minimized by using the highest voltage available (Fig. 4a) and by using a cooling stage (Fig. 4b).

Iu.

PHASE SEPARATION (EXSOLUTION)

It is in the field of phase transformations that TEM has probably had the widest influence in mineralogy. It had long been known from the study of petrographic thin sections in the polarizing microscope that phase separation (or exsolution) is common in the pyroxenes, amphiboles, and feldspars from slowly cooled rocks such as large igneous intrusions. In the 1950s and 1960%studies by single-crystal x-ray diffraction (XRD) (e.g., Smith and MacKenzie, 1955; Bown and Gay, 1959) were able to indicate the lattice orientations of these intergrowths and also to show that exsolution was present in many minerals from more quickly cooled rocks, although the intergrowth was below the resolution of the light-opticalmicroscope. XRD could not, however, give any indication of the mechanisms of exsolution, nor, in general, of the size of the precipitates or the orientation of their interfaces. This is where TEM has come into its own. During the early days of the investigation of exsolution in silicates by TEM,it became apparent that two mechanisms that are extremely rare in metallic systems are very common in silicates: spinodal decomposition (the gradual evolution of

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

7

sinusoidal compositional waves, without a nucleation stage) and homogeneous nucleation and growth of the equilibrium phase (nucleation without the aid of structural defects). The reasons for this difference in behavior between metals and silicates lies in the fact that whereas the crystal structures of the matrix and equilibrium product phases are different in metallic systems, in most cases the structuresof the two silicate phases involved in exsolution are identical (Aaronson et al., 1974). Added to this, in silicate systems the equilibrium solubility at high temperatures is relatively small and the volume change involved in the transformation is small. These factors result in the depression of the coherent spinodal below the equilibrium solvus being small enough that relatively rapid diffusion can take place when the temperature drops below that of the coherent spinodal. The factors that favor spinodal decomposition also favor homogeneous nucleation, although homogeneous nucleation is the more difficult process. However, because the equilibrium phases in silicates usually have a common structure of Si-0 tetrahedra and only second, third, or even higher, nearest neighbors need be in the “wrong” positions across the interphase interface, the chemical interfacialenergy term is small. In addition, the appreciable decrease in solubility with temperature that occurs in silicates provides a high driving force for nucleation and growth. Nevertheless, the cooling rate needs to be extremely slow, as it is in many plutonic and metamorphic rocks, for homogeneous nucleation to occur before the coherent solvus is reached. My examples of exsolution come from the alkali (Na-K) feldspars and the amphiboles and nicely illustrate the diversity of microstructures in the mineral kingdom. They also provide some very spectacular textures. A. Alkali Feldspars

The feldspars are the commonestsilicatesin the Earth’scrust, making up some 54%. They largely belong to the ternary system NaA&Og (dbite)-K,4lsi308 (orthoclase)-CaAl2Si208 (anorthite), the NaAlSi308-KAlSi308 series being known as the alkali feldspars and the NaAlSi308-CaAl2Si208 being known as the plagioclase feldspars. The alkali feldspars show (almost)complete solid solution at temperaturesabove 660”C,but there is a solvus at lower temperatures which extends to almost pure albite and orthoclase at low temperatures(Fig. 5). For most of the compositionrange, the alkali feldspars are monoclinic C 2 / m above the solvus, but both end members undergo a transition to triclinic C i l symmetry at lower temperatures. For the sodic phase the transition is the result of distortion of the S i / A l - O framework and is rapid

’

The nonstandard space p u p is used so that the monoclinic and triclinic phases have the same unit cells.

8

P. E. CHAMPNESS

0 10 NOA1Si, 0,

20

30

40

50

mole Yo

60

70

80

90 100 KAlSi,08

FIGURE5. Simplified subsolidus phase diagram forthe alkali feldspar binary NaAISi3Os (albite)KAISi308 (orthoclase)at 1 kbar as calculated by Robin (1974).The dashed line is the coherent solvus and the dotted line is the coherent spinodal. (Source: Champness and Lorimer, 1976; reproduced by permission of Springer-Verlag.)

(it is classed as displucive by mineralogists and may well be martensitic), whereas the transition in the potassic phase is slow because it involves Si/Al ordering. The alkali feldspars show coarser precipitation structures (calledperthites) than any other silicates; lamellae can be several millimeters wide in plutonic (slowly cooled) rocks. This fact can be attributed to the relatively high difisivities of K and Na ions within the Si/Al-O framework and the fact that unlike the plagioclase feldspars, precipitation does not require diffusion of the Si and Al. McConnell (1969) was the first to examine the microstructure of a volcanic alkali feldspar (composition 36% K-feldspar) in the TEM. He showed that it consisted of coherent compositional modulations with a wavelength of about 10 nm approximately parallel to (100). The diffraction pattern showed a single reciprocal lattice with strong streaks approximately parallel to a*. This was the first direct evidence that spinodal decomposition is an important mechanism of phase transformation in the alkali feldspars, as had first been suggested by Christie (1968). Since then, natural samples have been homogenized and heat treated to reproduce the modulated structures (Fig. 6) (Owen and McConnell, 1971; Yund etal., 1974). Owen and McConnell were able to show that the wavelengthof the modulation was

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

9

FIGURE6. Natural alkali feldspar(36 mol % K-feldspar) that has been homogenized and annealed at 540°C for 48 h at 1 kbar to produce a modulated structure approximately parallel to (601). Inset is an enlargement of a diffraction spot that shows satellites in a direction perpendicular to the modulations. (Source: Owen and McConnell, 1971; reproduced by permission of Narure.)

characteristicof the annealing temperatureand was larger for higher temperatures, as predicted by spinodal theory. Yund et al. (1974) annealed an initially homogeneous alkali feldspar for several days at 600°C and found that the modulations eventually developed into two separate, lamellar phases approximately parallel to (601). Calculations by Willaime and Brown (1974) of the elastic energy at the boundary between two alkali feldspars where both are monoclinic, or where the Nafeldspar is average monoclinic due to periodic twinning, showed that a minimum occurs at approximately (601). Hence the orientation of the interphase boundary is determined predominantly by minimization of elastic strain. The chemical component of the interphase boundary energy is much less important because the Si/Al-O framework is unchanged across the interface. Although exsolution textures that can be attributed to nucleation and growth (including homogeneous nucleation) have been identified in natural alkali feldspars (e.g., Brown and Parsons, 1988; Snow and Yund, 1988), the interdifision of Na and K is too slow to allow nucleation of exsolution lamellae to occur in alkali feldspars in the laboratory. To circumvent this problem, Kusatz et al. (1987)

10

P. E. CHAMPNESS

carried out exsolution experiments on alkali feldspars in which some of the Si had been substituted with Ge2 to give compositions along the binary NaAlGe2.lSio.9 08-KAlGe2.1 Sb.908.This substitution causes the incoherent and coherent solvi to rise (to almost 900°C for the critical composition of the incoherent solvus), the solidus to be depressed and the displacive transformation to move toward the K-rich side of the phase diagram. Kusatz et al. (1987) found two types of textures in their experiments. Short, widely spaced, lens-shaped lamellae were produced between the incoherent solvus and the coherent spinodal and were ascribed to nucleation and growth, whereas thin,closely spaced, and branching lamellae formed only in the central part of the solvus and were ascribed to spinodal decomposition. In a detailed study of the coarsening of spinodal textures in alkali feldspars, Yund and Davidson (1978), found that the lamellar spacing could be described as being proportional to the cube root of the annealing time at constant temperature by the relation

A = A0 + kt"3,

(2)

where A0 is the spacing at zero time and k is a rate constant for each temperature. An Arrhenius plot of the natural logarithm of k against 1/T, where T is the temperature, showed a linear relationship within experimental error. However, as Yund and Davidson (1978) acknowledged, the t law applies to the coarsening of spherical particles and is not appropriate to the coarsening of lamellae. Brady (1987) proposed that the principal mechanism for coarsening in this case is diffusional exchange between the wedge-shaped terminations of exsolution lamellae as seen in the "EM (Fig. 6) and the large, flat sides of adjacent lamellae. Having derived a formula for the chemical potential gradient due to interfacial energy effects, Brady extended the work of Cline (1971) on the coarsening and stability of lamellar eutectics, to show that the appropriate rate law for lamellar coarsening in silicates is given by A2 = A:

+kt.

(3)

Brady replotted Yund and Davidson's (1978) data on a graph of A2 versus f (Fig. 7) and found an excellent fit which gave an activationenergy for coarseningof 33 kcdmol. Further evidence for the correctness of Brady's model was provided by the fact that the values of Ao, the lamellar wavelength at the beginning of coarsening, as derived from the graphs, increased systematically with temperature, This is a trick that mineralogists often employ. For instance, Ge has been substituted for Si in olivine, Mg2Si04, so that the olivine + spinel transition that occurs at a depth of about 400 km in the earth can be studied in the laboratory (e.g., Rubie and Champness, 1987). The transition occurs at a lower pressure in the germanate because Ge has a smaller ionic radius than Si.

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY 1 1

FIGURE7. Plot of k2 versus time, t , for the coarsening experiments of Yund and Davidson (1978) on alkali feldspars. k is the lamellar wavelength. (Source: Brady, 1987; reproduced by permission of the mineralogical society of America.)

as predicted by the theory of spinodal decomposition. The ho values obtained by Yund and Davidson from the f rate law did not increase in this way. Equations (2) and (3) give different values of predicted lamellar wavelengths for long coarsening times (a difference of more than an order of magnitude for coarsening for lo6 years at 500°C) but give comparable results for rapidly cooled rocks (Brady, 1987). However, attempts to determine the cooling history of relatively quickly cooled rocks from the spacing of the lamellae has met with mixed success. There was good agreement between the lamellar spacings observed in a 5.2-m-wide dike and those predicted from heat-flow calculations and Eq. (2) (Christoffersen and Schedl, 1980), but less good agreement for lamellar spacings in a lava flow (Yund and Chapple, 1980) and in a large rhyolitic ash flow (Snow and Yund, 1988). It is also apparent that SilAl ordering and twinning inhibit coarsening in more slowly cooled rocks (Brown et al., 1983). In some more slowly cooled alkali feldspars, the two-phase lamellar intergrowths have coarsened to the scale of visible light, with the consequence that the scattering of light from their regular interfaces produces iridescence. It was a TEM study by Lorimer and Champness (1973) of two gem-quality varieties of these feldspars, known as moonstones, that led to an understanding of the later stages of coarsening. Fleet and Ribbe (1963)were the first to examine a moonstone in the TEM,using crushed grains. They showed that it contained coherent, lamellar precipitates of triclinic Na-feldspar and monoclinic K-feldspar approximately parallel to (601), the plane of iridescence. The Na-feldspar contained regularly

12

P.E. CHAMPNESS

FIGURE8. Microstructure. of two moonstones: (a) feldspar with bulk composition 57.3 wt% K-feldspar contains wavy lamellae of regularly albite-twinned Na-feldspar approximately parallel to (601); (b) feldspar with bulk composition 53.7 wt% K-feldspar has a coarser microstructure with lozenge-shapedparticles of Na-feldspar with boundaries approximately parallel to (661) and smaller, zigzag lamellae parallel to approximately (601). (Source: Lorimer and Champness. 1973; reproduced by permission of Philosophical Magazine.)

spaced Albite twins,3as had been predicted by Laves (1952) from the presence of superlattice reflections parallel to b* in x-ray diffraction patterns. (The regularity of the twins, Laves suggested, reduces the strain energy of the interface between the two phases, a suggestion that was subsequently verified from calculations of the strain energy by Willaime and Gandais, 1972.) Lorimer and Champness’ sampleshad similar compositions(57.3 and 53.7 wt% K-feldspar) but showed markedly different phase distributions. The first sample, which exhibits a blue iridescence, was shown to contain wavy lamellae of regularly Albite-twinned Na-feldspar approximately parallel to (6Ol), together with apparently monoclinic K-feldspar (Fig. 8a). The other moonstone, which shows a white iridescence, had a coarser microstructurecontaining discrete lozenge-shaped particles of regularly twinned Na-feldspar with boundaries approximately parallel to (661) (Fig. 8b). Significantly, this sample also contained zigzag lamellae of Na-feldsparthat were smaller in size than the lozenge-shapedparticles Albite twins arise. during the triclinic + monoclinic transition in Na-feldspar. They are normal twins with (010)as the twin and composition plane.

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

13

SDinodal decomoosition Coarsening Homogeneous monoclinic solid solution

-

Displacive transition in albite

Al,Si ordering in K-feldspar

Further adjustment of interfaces

Rafting

Further ordering and interface movement

FIGURE9. Sequence of evolution of the microstructure in the moonstones in Fig. 8 (Source: Putnis, 1992; reproduced by permission of Cambridge University Press.)

and therefore must have predated them. Detailed investigation of the K-feldspar showed that it was triclinic and mostly twinned on the diagonal association (basically Albite-twinned, but slightly deformed). The observations above suggest the sequence shown in Fig. 9 for the evolution of the microstructure in the coarser moonstone. After coarsening of the spinodal modulations has produced distinct lamellae parallel to (601), the Na-feldspar becomes triclinic and twins on the Albite law. The periodic twinning relieves the strain at the interphase interface and the Na-feldspar remains monoclinic, on average. As the K-feldspar becomes triclinic, however, the lowest-energy interface becomes approximately (661) (as shown in calculations by Willaime and Brown, 1974) and the interface gradually changes during the coarsening process, producing, first, wavy lamellae and, later, discrete, lozenge-shaped particles. Examination of the phase distribution in the coarser of the two samples examined by Lorimer and Champness (1972) shows that rafting of the Na-rich particles has taken place (Fig. 8b) due to interaction of their strain fields during coarsening. This phenomenon has been reported in metallic systems (Ardell er al., 1966).

14

P. E. CHAMPNESS

Although the presence of a fluid phase is known not to have an affect on lattice diffusion (Yund, 1983) or on the coarsening of coherent lamellae in alkali feldspars (Yund and Davidson, 1978), it has a dramatic effect on the coarsening of alkali feldspar intergrowths as coherency is lost. Almost all plutonic, igneous rocks are affected to a greater or lesser extent by water derived from the magma (deuteric alteration) at temperatures <45OoC.The familiar pink, white or creamy, translucent feldspar crystals in granites owe their distinctive appearance to the presence of numerous, small, tubular micropores that were originally, or in some cases still are, filled with fluid (Worden et al., 1990). Parsons (1978) and Parsons and Brown (1984) showed, using light-optical microscopy, that there was a connection between the turbidity of these feldspars and the development of coarse, irregular intergrowths of two alkali feldspar phases. Worden et al. (1990), in a E M and SEM investigation of the microstructure of alkali feldspars from the Klokken syenitic intrusion, Greenland, showed that micropores are abundant in areas where the microstructure is coarsened but are almost absent from uncoarsened areas. The coarsening is patchy and involves an increase in scale of up to lo3 without a change in the composition of the phases or in the bulk composition of the crystal. It occurs abruptly along an irregular front; the regular intergrowth that contains coherent, lozenge-shaped particles gives way, over a few microns, to a highly coarsened, irregular, semicoherentor incoherent intergrowth (Fig. 10). The pores occur along subgrain boundaries within the phases or along the boundaries between them. It is clear that the coarsening has been facilitated by pervasive dissolution-redepositionin an aqueous fluid. The driving force for the coarsening is the reduction in total surface energy for the feldspar intergrowth, including the release of elastic strain energy. What is not so clear is why the fluid, which would be expected to flow along grain boundaries, gives rise to micropores that migrate into the crystal. B. Amphiboles

Amphiboles have an extremely varied chemistry (the name is derived from the Greekamphibolos, “ambiguous,”in allusion to the great variety of compositionand appearance within this mineral group). Their chemical complexity explains why amphibolesoccur in such a wide variety of igneous, metamorphic,and sedimentary rocks. The standard amphibole formula is taken to contain eight tetrahedral sites and can be expressed as

where the Roman numeral superscripts refer to coordination numbers. The F and C1content in the OH site is normally minor. The structureconsists of double chains of S2Al-O tetrahedra, which run parallel to the z-axis, with cations between them

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

15

FIGURE10. Micrograph of an alteration front showing fully coherent, unaltered exsolution structure on the right and a deuterically coarsened, irregular,semicoherent microstructurecontaining micropores on the left. Ab, albite; Ksp, K-feldspar. (Source: Worden er al., 1990 reproduced by permission of Springer-Verlag.)

that are coordinated to oxygens from the chains and to the OH at the centers of the hexagonal rings of the chains (Fig. 11). The large A site may be vacant or contain varying amounts of NdCa, while the B site in the formula corresponds to the M4 site in the structural diagram and may contain Ca, Na, Al, Fe2+, Mg, or Mn. The M4 site is either six- or eightfold coordinated by oxygen, depending on the chemistry; in the former case the symmetry is orthorhombic, Pnma, and in the latter case the symmetry is monoclinic C2/m (or occasionally P21/rn). C in the formula represents the M1,M2, and M3 sites in the structure, all of which are sixfold coordinated by oxygen (and also by OH in the case of M1 and M3). The cations Fe2+, Mg, Fe3+, Al, Cr, and Ti can occupy these sites. The tetrahedral sites, T, are occupied by Si and Al; the limit of A1 substitutionfor Si appears to be Al2SLj. There is an elaborate scheme for naming the amphiboles (hake, 1978), but a simplified scheme is shown in Table II.

P.E. CHAMPNESS

16

TABLE Il SIMPLIFIED CLASSIFICATION FOR END-MEMBER AMPHIBOLES~ A

Ob 0 0 0

0 0 0 Na Na

0 0 0 Na Na

0

M4 Mgz A12 Mgz

(MI+M2+M3) Mg5

T

Mg5 Mg5

Sin SkA12 Si8

MagnesioanthophyUiteC Magnesiogedrite Magnesiocummingtonite'

Fei+

Fez+

Sis

Gmnerite

~ a 2 ca2 Ca2 Caz Caz Ca2

Mgs Fe;+ M&AI Mg4AI MI35 MgMz

Sin Sin Si7AI Si& Si7AI SkA12

Tremolite Femactinolite Magnesiohomblende Pargasite Edenite Tschermakite

Femmagnesian amphiboles

Calcic amphiboles

Naz

Mg3Ah

Sin

Glaucophane

Na2 Na2

Fe?Fe:+ Fei+. Fe3+

Sin Sin

Riebeckite Arfvedsonite

Alkali amphiboles

CaNa CaNa

Mgs Mg4Fe3+

Sis Si8

Richterite Ferriwinchite

Sodic-calcic amphiboles

a There is complete solid solution between Mg and Fe in the Ml-M4 sites. Mg-rich members have the prefix magnesio- and Fe-rich members have the prefix f e r n (or fern'-). Intermediate members have no prefix. Denotes a vacant cation site. The femmagnesian amphibolesmay be monoclinic(the magnesiocummingtonitsgruneriteseries) or orthorhombic (the magnesioanthophyllite-gedrite series). All other amphiboles are monoclinic.

FIGURE 11. Diagrammaticrepresentation of the structure of the doublethain silicate,amphibole: (a) double chain of Si-0 tetrahedra extending along the c-axisand, below, a representation of the chain viewed end on; (b) arrangement of the double chains viewed along the c-axis. The M1, M2, and M3 cations form chains of edge-sharing octahedra between the apices of the tetrahedra, and the M4 polyhedra form similarchains between the bases of the tetrahedra. The large 10- to 12-foldcoordinated polyhedral positions (the A sites) and the OH sites lie in the rings formed along the double chains. One I-beam has been shaded. (Source: Putnis, 1992; reproduced by permission of Cambridge University Ress.)

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY 17

z

J

C 2/m

Pnmo

FIGURE 12. Schematic representation of the stacking of the double chains in monoclinic (left) and orthorhombic (right) amphiboles projected along the b-axis. Notice that the (+ - - +) sequence in the orthorhombic structure, compared with (+ +) [or (- - -)] for the monoclinic structure results in a doubling of the a-axis for the former compared with the latter structure (-1.9 and -1.0 nm,respectively). (Source: Hawthorne, 1981; reproduced by permission of the Mineralogical Society of America.)

+

+

+

Amphibolesmay be consideredas ordered stackingsequencesof alternate layers of M-O polyhedra and tetrahedra along the x-axis (Fig. 12) and there is a stagger of approximately f c / 3 between adjacent tetrahedral layers. For the monoclinic structures, this stagger is always in the same direction, but in the orthorhombic amphibolesthere is aregularreversal of the stagger. The sequenceis: +c/3, +c/3, -c/3, -c/3, (or simply - -). It is this difference between the monoclinic and orthorhombic structures that results in the coordination of the M4 site being eightfold in monoclinic amphibolesbut sixfold in orthorhombic ones. During the last two decades, considerable effort has been expended toward an understanding of the extent of solid solution and phase separation within and between the different amphibole series. There are miscibility gaps between all pairs of the major amphibole groups in Table 11, but there is also incomplete solid solution between some members of the individual groups, the solvus in the orthorhombicanthophyllitegedrite series below about 600°C (Spear, 1980) being the best documented. Some evidence for incompletesolid solution comes from the coexistence of two amphiboles that grew under equilibrium conditions. However, it can be difficult to establish that equilibriumhas been attained (see the discussion in Smelik et al., 1991). As Robinson et al. (1982) have pointed out, “the presence of one set of amphibole lamellae in another is one of the surest and soundest pieces of evidencefor a . ..miscibility gap.” It is E M that has often provided that evidence; although some of the coarser exsolution textures have been investigated by light-opticalmicroscopy and EMF’A, TEM and AEM have paid a very important role in unraveling phase relations and exsolution mechanisms in the amphiboles because of the small scale of some of the intergrowths.

++

18

P.E.CHAMPNESS

1. Exsolution in Monoclinic Amphiboles It has long been known that exsolution occurs between calcic and the monoclinic ferromagnesian amphiboles and between the members of the orthoamphibole series (see Ross er al., 1969,for a review) because the textures that are produced are large enough to be visible in the polarizing microscope. However, it was not possible to determinethe exact chemical compositionof the precipitates by EMPA, or even to determine their chemical nature at all in some cases, because they are beyond the resolution of the instrument. X-ray single-crystalphotographs indicate that the two sets of exsolutionlamellae that are visible optically in many slowly cooled calcic and monoclinic ferromagnesian amphibolesusually share a common (101)or (100)lattice plane. However, careful light-optical studies by Robinson er al. (1971)showed that the orientations of the lamellar boundaries (habit planes) were not exactly parallel to these planes but could differ from them by lo" or more. Robinson er al. (1971)used the symbols 101 and 100to indicate the irrational orientations. I will use the same convention in this review. The relative cell parameters of Ca-rich and ferromagnesian, monoclinic amphiboles are such that for coherent precipitationof one phase from the other, one principal strain is of opposite sign to the other two (the strain quadric is a hyperboloid). Thus there are two directions perpendicular to the intermediate principal axis of strain (the y-axis) for which the strain is zero. One is 101 and the other is 100. As long as the b-axial lengths are nearly identical, as they are for the phases in question, these two orientationswill provide lamellar interfaces of minimum strain, the exact orientations being determined by the relative values of the a and c repeats! This treatment neglects elastic anisotropy and the chemical component of the interfacial energy. However, calculations of the three-dimensional variation of elastic strain energy of monoclinic pyroxenes (Fletcher and McCallister, 1974), whose structures and phase relations mirror those of the amphiboles, shows that the energy minima are within a few degrees of those calculated from Robinson et al.'s (1971,1977)two-dimensional, geometric model. Thus, as in the case of the alkali feldspars (Section 1II.A). the chemical component of the interphase boundary energy is unimportant because the two structures are identical, except for the cation distribution between the WA1-O double chains. - TEM of calcic and ferromagnesian amphiboles that show optically visible 101 and 100 exsolution lamellae has revealed that the microstructure is more Because the relative values of thea- and c-repeats of monoclinic pyroxenes vary considerably with temperature in the range in which exsolution occurs, the orientation of the lamellae also varies. This variation can be used to estimate the temperature at which exsolution began (Robinson er aL, 1977). However, the cell parameters of calcic and ferromagnesian amphiboles do not vary so drastically with temperature, and the range of exsolution temperatures is lower than for pyroxenes, so thermal histories cannot be estimated for amphiboles from lamellar orientations in the same way as for pyroxenes.

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

19

FIGURE13. Exsolution in monoclinic amphiboles. (a) Exsolution of grunerite (Ca-poor, monoclinic amphibole) from hornblende (Ca-rich, monoclinic amphibole) by nucleation of 101 lamellae on a (100) twin boundary, T-T. Notice the growth ledges along some of the interfaces. (b) Exsolution of hornblende from grunerite in the same rock as (a). Notice the homogeneously distributed I 0 0 platelets of hornblende between the large 101,X-X and 100, Y-Y lamellae and the platelet-free zone adjacent to the large lamellae. (Source: Gittos er al., 1976; reproduced by permission of SpringerVerlag.)

complex than it appears in the light microscope. Gittos et al. (1974, 1976) studied the amphiboles in three metamorphic rocks that contained coexisting grunerite/cummingtonite and hornblende. The large (up to 0.5 pm thick) 101 and 100 exsolution lamellae were found to be coherent with the matrix and had nucleated heterogeneously on twin boundaries or dislocations and thickened by the movement of ledges across the interfaces (Fig. 13a). In addition, the Ca-poor amphibole also contained a much finer, homogeneously distributed set of 100 platelets of hornblende between the lameUae (Fig. 13b). A zone free of the platelets occurred adjacent to each lamella. Gittos et al. (1974) concludedthat the platelets nucleated homogeneously in areas where the calcium supersaturation was high enough, a diffusion profile having been left from the growth of the lamellae. The formation of the platelets or G.P.zones in the Ca-poor, but not in the Ca-rich, amphiboles can be explained by the difference in the shape of the solvus on the two sides of the phase diagram, the Ca-poor side being much steeper than the Ca-rich side (Champness and Lorimer, in preparation).

20

P.E.CHAMPNESS

FIGURE 14. no-stage exsolution involving three different amphiboles: (a) primary exsolution of cummingtonite lamellae from glaucophane parallel to 281 and 281; (b) secondary exsolution of actinolite parallel to 100 in the cummingtonite lamellae, different area of the same specimen as (a). (Source: Smelik and Veblen. 1989; reproduced by permission of the Mineralogical Association of Canada.)

An example of a two-stage exsolution process involving three different monoclinic amphiboles was described by Smelik and Veblen (1994). The matrix phase is the alkali amphibole glaucophaneand the first stage of exsolution consists of coherent cummingtonitelamellae, parallel to the irrational planes 281 and 281, that reach a maximum thickness of 60 to 80 nm (Fig. 14a). The most common mechanism of exsolution appears to have been homogeneous nucleation and growth, although there was some nucleation on dislocations and chain-width errors (see Some of the cummingtonitelamellae contained periodic lamellae up Section N). to 7.5 nm in width of a second amphibole parallel to 100 (Fig. 14b). The periodicity of these lamellae and their thickness were dependent on the thickness of the host cummingtonite lamellae. The cummingtonitelamellae were significantly enriched in Ca compared with normal cummingtonite, and when their compositions were plotted on the ternary Ca-Mg-Fe amphibole composition diagram, they fell well within the actinolitecummingtonitemiscibility gap and are thus metastable (Fig. 15). The secondary exsolution lamellae inside the cummingtonite lamellae were too narrow for

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

tremo

21

lite

M

2

grunerite series FIGURE15. Ca-Fe-Mg amphibole quadrilateral showing the compositions of the primary cummingtonite lamellae in Fig. 14 as determined by AEM. The shaded areas show the normal compositions of natural amphiboles. The compositions of the lamellae fall inside the miscibility gap between Carich and Ca-poor amphiboles. (Source: Smelik and Veblen, 1994; reproduced by permission of the Mineralogical Association of Canada.)

quantitative analysis, but AEM showed that the Ca has segregated almost entirely to one of the phases, while the other phase is richer in Mg and Fe. Thus stable compositions of actinolite and cummingtonite have been produced by the second exsolution process. Smelik and Veblen (1991) showed by calculation that 28i and 28i are the planes of minimum misfit [or optimal phase boundaries (OPBs)] for a coherent intergrowth of glaucophane and cummingtonite (elastic strain was ignored in the calculations). The glaucophanecell parameters used for the calculation were measured by powder x-ray diffraction of grains of the glaucophane in the rock, while the cell parameters for the cummingtonite were derived using the regression equations of Viswanathan and Ghose (1965) and the average composition determined by AEM (the cell parameters vary nearly linearly with composition). The largest difference in the cell parameters was for the b-axis, confirming that the plane(s) of minimum misfit are expected to be close to (010). Smelik and Veblen (1994) showed from calculations of misfit and elastic strain that although the 100boundary between the actinolite and cummingtonite is optimal, the 28T interface between the actinolite and glaucophane has relatively high strain. The periodic nature of the secondary exsolution is a result of the minimization of the total elastic strain associated with the intergrowth of the three amphiboles. The existence,or otherwise, of a miscibility gap between members of the calcicamphibole group has been the subject of considerabledebate over the last 20 years (see Smelik et al., 1991, for a review). Although some authors have argued for the existence of a gap from the presence of primary actinolite and hornblende grains in the same rock, others have argued that these occurrences represent metastable assemblages. Experimental studies have also yielded contradictory

22

P. E. CHAMPNESS

FIGURE16. Microstructure of acalcic amphibole from Wyoming. It showsa two-stage exsolution process involving three different amphiboles. (a) Pervasive tweed exsolution parallel to 132 and I22 between two larger 100 cummingtonite lamellae; (b) high-resolution image of an area showing a coarse tweed. The microstructure is coherent, with no change in the orientation or spacing ot the 0 2 0 lattice fringes. (Souwe: Smelik et d..1991: reproduced by permission of the Mincralogical Society of America.)

results. Recently, unequivocal evidence for the existance of such a miscibility gap has been provided by TEM of calcic amphiboles from metagabbros in Wyoming which contain another example of a two-stage exsolution process that involves three different amphiboles (Smelik et al., 1991). The calcic amphiboles, which range in composition from actinolite to hornblende, contain sparse 101 and 100 lamellae of cummingtonite that are just visible in the light microscope. Between them is a fine, tweedlike structure parallel to two irrational, symmetrically equivalent planes 132 and 132 (Fig. 16a). Diffraction patterns from the tweed structure showed a single reciprocal lattice with four satellites about each spot that are approximately perpendicular to the modulations. HRTEM showed that the interfaces between the elements of the tweed were coherent with no change in orientation or spacing of the lattice fringes between them (Fig. 16b). Smelik el al. (199 1) used EMPA and AEM to investigate the compositions of the phases in Fig. 16 (Table 111). As expected, the only significantexchangesduring the first stage of the exsolution (that producing the cummingtonite lamellae) were Ca ++(Fe, Mg) and Al"',Allv c) Mg", Si", called the tschermakite substitution; the Fe/(Fe Mg) ratio did not change. The tweed structure was coarse enough in places to allow semiquantitative analysis of the individual components by AEM. The tweed was found to consist of two chemically different regions that approached actinolite and hornblende in composition (Table 111). The actinolite regions have higher Si, lower Al, a

+

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY TABLE 111 EMPA AND AEM ANALYSIS OF PHASES IN

Tetrahedral sites Si Al CTsite M(1.2.3) Al Ti Mg Fe?' CM( I , 2.3)

A

23

CALClC AMPHIBOLE".'

7.02 0.99 8.00

7.88 0.12 8.00

I . 10 0.90 8.00

7.62 0.38 8.00

6.52 I .4n 8.00

0.50 0.05 2.63 1.82 5.00

0.03 0.01 3.45 1.51 5.00

0.52 0.03 2.62 1.83 5.00

0.38 0.02 3.05 I .55 5.00

0.76 0.05 I .no 2.40 5.00

I .n7 0.18 0.03 2.08

0.10 0.08

1 .no

1.64

0.1 1

0.18 2.00

0.02 2.01

1.71 0.18 0.08 0.03 2.00

I .nn

0.08

0. I6 0.02 2.00

0.18

0.15 0.01 0.16 0.48

0.34 0.05 0.39 0.47

0.12 0.03 0.15 0.35

0.36 0.14 0.50 0.59

M4 Ca

Na Fe" Mn CM(4) A site Na K CAsite Fe''/(Fe?+

0.06 0.23 0.44

+ Mg)

-

Soime: Smelik e r a/ . (1991). " I . Bulk analysis of composite grains by EMPA (average of 8); 2 AEM analysis ofcumniingtonite lainellae (average of 6): 3 bulk AEM analysis of tweed structure (average of 8); 4 AEM analysis ot actinolite lainellae in tweed structure (average of 20): 5 AEM analysis of hornblende lamellae in tweed stnicture (average of 21). "Amphibole formulas are based on normalization to 23 oxygens and assuming that all Fe is Fez+. For method of allocation of cations to the various crystallographic sites, see Robinson PI (11. (1982). The amphiboles were compositionally zoned (cored);columns I and 2 represent averages of analyses with a wide range. The total Al contents ranged from 0.559 to 2.581 per formula unit.

+

lower Fe2+/(Fe2+ Mg) ratio and an apparently lower A-site occupancy than the hornblende regions. Na also appears to be slightly redistributed, with slightly more NaM4in the actinolite and more NaA in the hornblende. However, the apparent redistribution is probably the result of the difficulty of analyzing Na in the AEM (see Section 11) and the difference in the Fe contents of the two phases, all of it having been assumed to be Fe2+ (although the amphiboles are likely to contain some Fe3+), which may have led to overestimation of the A site. The miscibility gap defined by these compositional differences is shown graphically in Fig. 17. In Fig. 17a total A1 has been plotted against Fe2+/(Fe2+ Mg) for the individual

+

24

P. E. CHAMPNESS

0

0.0

Fe-tr 4

0.2

0.6

0.4

Fe 2+/(Fe 2+ + Mg)

0.8

1.0

I

Pa[]

;

/!***

r

1

2

A,('V'

FIGURE17. Plots of AEM analyses of actinolite (filled diamonds) and hornblende (open squares) regions of the tweed structure, showing the miscibility gap. The open circles connected by a tie line are the average compositions. (a) Plot of total Al versus Fe2+/(Fe2+ Mg); (b) plot of calculated A-site occupancy versus All"; (c) plot of Alto[versus All". End-member abbreviations are as Follows: tr, tremolite; ed, edenite; pa, pargasite; ts. tschermakite. (Source: Smelik et al., 1991; reproduced by permission of the Mineralogical Society of America.)

+

APPLICATIONS OF TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

25

analyses that contributed to the average in Table In, Fig. 17b shows the gap in terms of the calculated A-site occupancy versus AllV,and in Fig. 17c total A1 is plotted against AI". The gap is well defined in each case. The substitutionsthat occurred were A'', AllV * Mgvl, Si"; (Na, K)*, All" t, CIA,Si", called the edenitic substitution; and Fez+ * Mg, with the tschermakite exchange being dominant. Smelik et al. (1991) interpreted the tweed texture as having been produced by spinodaldecompositionbetween two calcic phases at a lower temperaturethan that at which exsolution of the cummingtonite lamellae took place. They attempted to calculate the orientationof planes of minimum misfit for the actinolite-hornblende pair. However, the cell parameters for the two phases are very similar and their variation with temperature and pressure is not accurately known for the compositions involved. The calculations failed to show conclusively that 132 and 152 are the planes of minimum misfit, but it is clear that a range of lamellar orientations may be possible, depending on the exact compositions and unit-cell parameters of the phases involved.

2. Exsolution in Orthorhombic Amphiboles a. Exsolution Between Two Orthoamphiboles. Evidence from exsolution textures for a miscibility gap between the orthorhombic amphiboles anthophyllite and gedrite was first reported by Boggild (1905, 1924), who used light-optical microscopy, but the complexity of the phase distributions has only become apparent in the last 20 years from "EM observations (Gittos et al., 1976; Smelik and Veblen, 1993). As in the case of the monoclinic amphiboles, the habit plane is determined by the differences in the cell parameters of the two phases. The usual plane is (010) because Ab is considerably larger than Aa or Ac [although in absolute terms it is still small, being around 1.5% or less at room temperature; (Smelik and Veblen, 1993)l. However, the relative cell parameters are extremely sensitive to composition and temperature (Smelik and Veblen, 1993); if the amphibole contains relatively large amounts of Ca andor Fe, the habit plane changes from (1 10) to an (hkO) orientation up to 26" from (010) (approximately (120)). This variation in orientation can be seen in different areas of the same zoned crystal and sometimes a fall in temperature has caused the orientation to change within the same area (Gittos et al., 1976; Smelik and Veblen, 1993). There is evidence from TEM studies that heterogeneous nucleation, homogeneous nucleation, and spinodal decomposition can all occur during exsolution of the orthoamphiboles. In samples showing the coarsest textures, nucleation appears to have taken place on (010) chain-width defects (Smelik and Veblen, 1993). In Fig. 18a almost all the (010) lamellae contain a chain-width defect that probably acted as a nucleation site. Later nucleation of (010) platelets has taken place on stacking faults parallel to (100) in regions between the large lamellae that had a higher concentration of solute. These faults have been shown to be narrow strips

26

P. E. CHAMPNESS

FIGUREIS. Microstructures in exsolved orthorhombic amphiboles. (a) sample with bulk composition 60% gedrite showing large (0 10) lamellae of anthophyllite with a coniplex morphology. The lamellae have probably nucleated on (010) chain-width errors. Terminations of the chain-width errors have impeded lamellar growth (arrowed). Between the lamellae are small (010)platelets. sonic of which have nucleated on (100) stacking faults. Others appear to have nucleated homogeneously. Notice the precipitate-free zone adjacent to the large lamellae. (b) HRTEM image of a (100)stacking fault in a homogeneous region of exsolved anthophyllite. The image is taken along [OI 11. The regular alternation of the stacking (+ - - -) along the waxis can be seen (compare with Fig. 12) in the orthorhombic phase. In the stacking fault the stacking is (+ +) (or - - - -), indicating that it is a narrow strip of monoclinic material. The faults are thought to fonn by deformation. ( c ) TEM image of an orthoamphibole that contains curved lamellae straddling (010). The dark lamcllae are getlrite and the light ones are anthophyllite. Note the branching of the lamellae. The electron beam is near [OOI I. ( S o u n ~ s :(a) Gittos PI d..1976: reproduced by permission of Springer Verlag; (b) and (c) Smelik and Veblen. 1993; reproduced by permission of the Mineralogical Society o f America.)

+

++

++

of monoclinic material (Fig. 18b) that predate all the exsolution (because they pass through it undisturbed) and were probably produced by deformation (Smelik and Veblen, 1993). The final stage of exsolution in the sample illustrated in Fig. 18a appears to have been homogeneous nucleation of the (010) platelets in regions devoid of defects. The large larnellae in Fig. 18a show an unusual morphology, in the development of which the (010)lamellar defects appear to have played an important role. Lamellar growth is impeded in the vicinity of the dislocation that forms the termination of the chain-width defect (arrow in Fig. 18a), and an embayment is formed in the lamella; thus the defect is effectively pinning the boundary. Similar embayments form at the terminations of (100)stacking faults (Smelik and Veblen, 1993). It is noticeable that in regions where there are no chain-width terminations, the lamellae are straight (bottom right, Fig. 18a), but that where the orientation deviates from (010) there are terminations. This suggests that the strain produced by the terminations can influence the orientation of the lamellae. This phenomenon

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27

will be aided by the fact that the anisotropy of the misfit is very small (Smelik and Veblen, 1993). Orthoamphiboles with a somewhat finer exsolution texture than that shown in Fig. 18a and compositions near the centroid of the solvus show characteristics that are consistent with spinodal decomposition (Fig. 18c and Gittos etal., 1976, Fig. 6). Although the interfaces of the lamellae are now sharp, the lamellae are long and thin, their distribution is very regular, and they show evidence of branching similar to lamellae produced experimentally by spinodal decomposition in Ge-substituted alkali-feldspars (Kusatz et al., 1987) and in pyroxenes (Buseck et al., 1980). h. Exsolution of a Monoclinic Amphibole from an Orthoarnphibole. Exsolution between orthorhombic and monoclinic amphiboles has been postulated for many years by analogy with the single-chain pyroxenes. In the latter system it is common for ferro-magnesian orthopyroxenes to contain exsolution lamellae of Ca-rich clinopyroxene (and vice versa) parallel to (100). Because of the close chemical and structural similarities between the pyroxenes and amphiboles, one would expect similar exsolution between calcic clinoamphiboles and orthoamphiboles. However, despite the abundance of orthoamphibole-bearing rocks, many of which contain coexisting calcic amphiboles, no such microstructures had been reported, until recently. Smelik and Veblen (1992) found that (100) lamellae of hornblende up to 80 nm wide had exsolved from an orthoamphibole that also contained earlier-formed lamellae of a second orthoamphibole with a habit plane that varied from (010) to ( 120). The hornblende lamellae were semicoherent and had nucleated on (100) stacking faults. Because the faults are narrow strips of monoclinic material, they act as ideal templates for the hornblende structure. Semiquantitative analysis of the matrix and hornblende lamellae showed that the main chemical change is CaM4++ (Mg, Fe, Mn)'", as would be expected from the chemistry of the calcic and ferromagnesian amphiboles (Table 11). However, other coupled substitutions involving the M4, M2, and T sites are also important in the exsolution (Smelik and Veblen, 1992). The analysis also showed that during the first stage of exsolution, the Ca segregated largely to the gedrite rather than to the anthophyllite.

IV. HRTEM AND DEFECTSTRUCTURES

The study of defects by conventional amplitude-contrast imaging, such as the darkfield technique, has revealed a vast amount of information about phase transitions (see Nord, 1992, for a recent review) and deformation structures and mechanisms in minerals (see Green, 1992, for a recent review). However, in this section 1 concentrate on describing how HRTEM has furthered our understanding of the nature of planar defects, in particular polysomatic defects and their role in replacement reactions in the pyroxenes and amphiboles.

P.E. CHAMPNESS

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A. Biopyriboles and Polysomatic Defects

Thompson (1978) defined apolysome as “a crystal . . . that can be regarded as made up of chemically distinct layer modules.” Thus it is distinct from a polytype, in which there is no chemical variation between the layers. A polysomatic series is a group of crystalline compounds (e.g., minerals) that possess the same types of modules in different ratios or sequences, the general term for this structural mixing being polysomatism. As polysomatic defects and small regions of ordered polysomatic structures have finite width, HRTEM can be used to resolve details within them and hence to identify them. In a polysomatic series in which the two types of modules have the same width, there are commonly certain defects that produce virtually no net displacement in the surrounding structure and thus would produce little contrast by conventional bright or dark-field imaging (Veblen, 1992). The pyroxenes, amphiboles, and sheet silicates (e.g., mica and talc) can be regarded as belonging to a polysomatic series known as the biopyriboles, a term derived from biotite (a variety of mica), pyroxene, and amphibole (Johannsen, 1911). Pyribole is the name given to biopyriboles, excluding the sheet silicates. When projected along the c-axis, the amphibole and pyroxene structures can be described in terms of the stacking of I-beams, a pair of Si-0 chains, and the cations between them (Figs. I I and 19), along the 6-axis. The pyroxene I-beam is one chain wide, the amphibole I-beam is two chains wide, and the mica structure has infinitely wide I-beams (Fig. 1). I. New Biopyriboles Theoretically, there is a complete, homologous series from pyroxene to mica, and in the last 20 years there have been numerous reports of natural and synthetic examples of other biopyriboles than the three described above. From light-optical and x-ray diffraction studies, Veblen and Burnham (1978a,b) described four new minerals that were intergrown with anthophyllite and cummingtonite in a metamor-

FIGURE 19. Schematic diagram showing I-heams projected onto the (001) plane in orthopyroxene. orthoamphihole, jimthompsonite, and chesterite. The digits refer to the number of chains i n each I-beam. (Sorrrce: Klein and Hurlbut, 1993; reproduced by permission of John Wiley & Sons, Inc.)

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FIGURE20. HRTEM images, viewed down the c-axis, of anthophyllite (An), jimthompsonite (Jt), and chesterite (Ch). The white spots are the projected positions of the A sites, which are located between the I-beams. The structural interpretation is shown in terms of the I-beams in Fig. 19. Unit cells are indicated. (Source: Veblen and Buseck, 1979; reproduced by permission of the Mineralogical Society of America.)

phosed rock near Chester, Vermont. The new minerals have either triple chains [jimthompsonite, (Mg, Fe)1&0054(OH)6] or both double and triple chains in regular alternation [chesterite,’ (Mg, Fe) loSi12032(OH),J (Fig. 19) and, like the pyroxenes and amphiboles, can occur in both monoclinic and orthorhombic forms. Specimens of the new, ordered pyriboles have been studied extensively by HRTEM (Fig. 20). but of more interest are the reports of ordered pyriboles in materials not previously known to contain them: for instance, in nephrite (actinolite), jade (Jefferson et al., 1978), and altered pyroxene (Nakijima and Ribbe, 1980). Several new, ordered pyribole structures were also discovered by HRTEM in specimens from Chester (Veblen and Buseck, 1979). Structures with the following New minerals are commonly named after the locality at which they were first found (e.g.,chesterite) or after a distinguished scientist [e.g., jimthompsonite (there was already a mineral named thompsonite, hence the use of J. B. Thompson’s forename)].

30

P. E. CHAMPNESS

FIGURE 21. HRTEM images viewed down the c-axis of pyriboles from Chester, Vermont. (a) The ordered sequence (2333). The double-chain slabs are unlabeled. The diffraction pattern is on the right. (b) An area containing triple, quadruple, and quintuple chains and exhibiting extreme chainwidth disorder. (Source: Veblen and Buseck, 1979; reproduced by permission of the Mineralogical Society of America.)

statistically significant,ordered mixed-chain sequences were found: (2233),(233). (232233), (222333), (2332323), (2333) (Fig. 21a), and (433323), where the numbers 2, 3, and 4 indicate the number of chains in each I-beam. The number and complexity of these structures suggest that they are unlikely to be stable, and the reason for their formation remains obscure. Although they are far more abundant than the phases noted above, it is still unclear whether jimthompsonite and chesterite have true stability fields under geological conditions or whether they are always metastable (Droop, 1994).

2. Chain-Width Disorder in Pyriboles Chisholm (1973) was the first to report the existence of chain-width defects (otherwise known as crystallographic shear planes or Wadsley defects) in chain silicates. Chisholm examined a number of different amphibole asbestos samples by electron diffraction and conventional TEM and sunnized that the (010) defects were intercalated slabs of pyroxene or slabs with more than two chains. Since then, HRTEM

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FIGURE22. HRTEM image of riebeckite asbestos (crocidolite) showing fibrils, which contain chain-width errors, separated by low-angle boundaries. (Source: Ahn and Buseck, 1991 : reproduced by permission of the Mineralogical Society of America.)

has shown that the defects are mostly of the triple-chain variety and that they are generally far more common in asbestos that in nonasbestos amphiboles (Fig. 22; see Veblen, 1981, 1992, for reviews). As shown in Section III.B, Chain-width errors can act as nucleation sites for exsolution in orthoamphiboles and can have a strong influence on the growth of the coarser lamellae. Isolated chain-width errors in amphiboles are usually thought to be primary growth features, whereas those that been reported in pyroxenes, jimthompsonite, and chesterite are associated primarily with alteration reactions; see the next section). Some of the pyriboles from Chester, Vermont, are extremely disordered (Fig. 21b).

3 . Polysomatic Reactions in Pyriboles Polysomatic reactions can be defined as reactions that turn one polysome into another. In biopyriboles, any reaction that changes the widths or sequences of the silicate chains is thus a polysomatic reaction. Recent work has shown that polysomatic reaction of pyriboles, on a scale observable with the TEM, is common whenever such minerals are in contact with hydrous fluids at moderate temperatures during retrograde metamorphism. Although bulk processes, in which transformation occurs along a broad reaction front may operate in many cases of polysomatic reaction in pyriboles, most TEM observations have involved materials that have been replaced wholly or in part by a lamellar mechanism. Here a lamella

32

P.E. CHAMPNESS

FIGURE23. (a) HRTEM image of an amphibole lamella in pyroxene. A ledge two chains wide is mowed. (b) An I-beam diagram of the image in (a). (Source: Veblen, 1981; reproduced by permission of the Mineralogical Society of America.)

or zipper of material having a different chain sequence for the matrix nucleates and grows. In most cases the lamellae terminate coherently, but the termination may also be associated with a dislocation. Thickening of the lamellae takes place by the propagation of ledges along the interface. Figure 23a shows a I-RTEM image of an amphibole lamella in pyroxene that is thickening from four to five unit cells wide by the migration of a ledge that has a width of two amphibole chains (one unit cell); Fig. 23b shows an I-beam model of the ledge, showing that it terminates coherently. Random nucleation and later growth of zippers in pyriboles will inevitably result in chain-width errors in the resultant phase. Figure 24 shows a possible mechanism by which such errors can be eliminated (Veblen and Buseck, 1980). The material in the top of the micrograph is perfectly ordered chesterite, while that at the bottom contains chain-width errors. The two regions are separated by an en echelon series of planar faults; migration of these faults toward the bottom of the figure would result in the replacement of disorderedpyribole by ordered chesterite. Veblen and Buseck (1980) have also suggested that the tunnels that exist at the terminations of the zippers in these reaction (e.g., Fig. 23b) provide a route for ultrafast (pipe) diffusion of the chemical species (hydrogenand octahedral cations) that are needed for the change in stoichiometry of the polysomatic reaction. It is thought that chesterite and jimthompsonite usually form as intermediate phases in the retrograde reaction of amphibole to the sheet silicate talc by the mechanisms outlined above, but this does not always appear to be the case (Droop, 1994).

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FIGURE24. Possible mechanism by which chain-width errors can be eliminated. The material in the top of the micrograph is perfectly ordered chesterite, while that at the bottom contains chain-width errors. The two regions are separated by an en echelon series of planar faults; the migration of these faults toward the bottom of the figure would result in the replacement of disordered pyribole by ordered chesterite. (Source: Veblen and Buseck, 1980; reproduced by permission of the Mineralogical Society of America.)

V. CONCLUDING REMARK

Although I have only been able to cover a few of the many applications of TEM to mineralogy in the past 25 years and did not have the space to cover convergentbeam electron diffraction and electron energy-loss spectroscopy (both of which are beginning to be applied to mineralogical problems), I hope I have made clear that TEM has had an impact second only to x-ray diffraction in this century in unraveling the complexities of mineral behavior.

REFEREN cEs Aaronson, H. 1.. Lorimer, G . W., Champness, P. E., and Spooner, E. T. C. (1974). On differences between phase transformations (exsolution) in metals and minerals. Chem. Geol. 1 4 , 7 5 4 0 . Ahn, J. H., and Buseck, P. R. (1991). Microstructures and fiber formation mechanisms of crocidolite asbestos. Am. Mineral. 76, 1467-1478.

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Ardell, A. J., Nicholson, R. B., and Eshelby,J. D. (1966). On the modulated structure of aged Ni-AI alloys. Acra Merall. 14, 1295-1309. B~iggild.0. B. (1905). Mineralogia Groenlandica. MeaX Groenl. 32.400. Beggild, 0. B. (1924). On the labradorizationof the feldspars. K.Dan. Vidensk. Selsk. Mar. Fys. Medd. 6, 1-79. Bown, M. G., and Gay, P. (1959). Identificationof oriented inclusions in pyroxene crystals. Am. Mineral. 44.592-602. Brady, J. B. (1987). Coarsening of fine-scaleexsolution lamellae. Am. Mineral. 72,697-706. Brown, W. L., and Parsons, 1. (1988). Zoned ternary feldspars in the Klokken intrusion; exsolution microtextures and mechanisms. Conrrib. Mineral. Petrol. 98,444454. Brown, W. L., Becker, S. M., and Parsons, I. (1983). Cryptoperthites and cooling rate in a layered syenite pluton: a chemical and TEM study. Conrrib, Mineral. Petrol. 82, 13-25. Buseck, P. R., Nod, G. L., Jr., and Veblen, D. R. (1980). Subsolidus phenomena in pyroxenes. Rev. Mineral. 7, 117-21 I , (C. T. Prewitt, Ed.). Champness, P. E.(1995). Analytical electron microscopy.In: P. J. Potts, J. F.W. Bowles, S . J. B. Reed, and M. R. Cave, Eds. Microprobe Techniques in rhe Earth Sciences. Chapman & Hall, New York, pp. 91-139. Champness. P. E., and Devenish, R. W. (1992). Radiation damage in silicate minerals: implications for AEM. Proc. EUREM '92,Granada. 2,541-545. Champness. P. E., and Lorimer, G. W. (1976). Exsolution in silicates.In: H. R. Wenk eral.. Eds. Electron Microscopy in Mineralogy. Springer-Verlag,New York, pp. 174-204. Chisholm, J. E. (1973). Planar defects in fibrous amphiboles. J. Marer. Sci. 8,475-483. Christie, 0. H. J. (1968). Spinodal precipitation in silicates. 1. Introductory application to exsolution in feldspars. Lirhos 1, 187-192. Christoffersen, P., and Schedl, A. (1980). Microstructure and thermal history of cryptoperthites in a dike from Big Bend, Texas. Am. Mineral. 65,444-448. Cliff, G., and Lorimer, G. W. (1975). The quantitative analysis of thin specimens. J. Microsc. 103. 203-207. Cline, H. E. (1971). Shape instabilities of eutectic composites at elevated temperatures. Acra Metall. 19,481490. Devenish, R. W., and Champness, P. E.(1993). The rate of mass loss in silicate minerals during x-ray analysis. Proc 13th Inr. Cong. on X-ray Optics and Microanalysis, Manchester, 1992. Institute of Physics, London and Bristol, pp. 233-236. Droop, G. T. R. (1994). Triple-chain pyriboles in Lewisian ultramafic rocks.Mineral. Mag. 58, 1-20. Fleet, S. G., and Ribbe, P. H. (1963). Anelectron microscopeinvestigationofarnoonstone.Phil. Mag. 8, 1179-1 187. Fletcher. R. C., and McCallister, R. H. (1974). Spinodal decomposition as a possible mechanism in the exsolution of clinopyroxene. Carnegie Inst. Washington Yearb., 396-399. Gittos. M. E, Lorimer, G. W., and Champness, P. E. (1974). an electron-microscopic study of precipitation (exsolution) in an amphibole (the hornblende-grunerite system). J. Mate6 Sci. 9, 184-192. Gittos, M. F., Lorimer, G. W., and Champness. P. E. (1976). The phase distributionsin some exsolved amphiboles. In: H. R. Wenk er al., Eds. Electron Microscopy in Mineralogy. Springer-Verlag,Berlin, pp. 238-247. Green, H. W., 11. ( 1992). Petrology-high-temperature and deformation-inducedreactions. Rev. Mineral. 27.425-454 (P. R. Buseck, Ed.). Hobbs, L. W. (1984). Radiation effects in analysis by TEM. In: J. N. Chapman and A. J. Craven, Eds. Quanrirarive Electron Microscopy (Scottish Universities Summer School in Physics). SUSSP Publications, Edinburgh, pp. 399-445. Jefferson, D. A., Mallinson, L. G., Hutchison, J. L., and Thomas, J. M. (1978). Multiple-chain and other unusual faults in amphiboles. Contrib. Mineral. Petrol. 66, 1 4 .

APPLICATIONS O F TRANSMISSION ELECTRON MICROSCOPY IN MINERALOGY

35

Johannsen. A. (191 I ) . Petrographic terms for field use. J. Crol. 19. 317-322. Kusatr. B.. Kroll. H.. and Kaiping, A. (1987). Mechanisnius und Kinctik von Entmischungsvorgiiiigen am Beispiel Ge-suhstituicrter Alkalifeldspiite. f i m c h . Minerrrl. 65. 203-248. Laves, F. ( 1952). The phase relations o f t h e alkali feldspars. II. ,I.G d . 60. 549-574. Leake. B. E. ( 1978). Noinenclature of amphiboles. Arm Miric,rd. 63. 1023-1052. Lorimer. G. W.. and Champness, P. E. ( 1973). The origin of the phase distribution in two perthitic alkali fcldspars. Phil. Mlr,?.28, 1391-1403. McConnell. J. D. C. ( 1969). Electron optical study of incipient exsolution and inversion phenomena i n the system NaAISi30x-KAISi30x. Phil. Mtrg. 19. 221-229. Naki,jima. Y.. and Rihhe. P. H. ( I 980). Alteration of pyroxenes from Hokkaido. Japan. to amphihole. clays and other hiopyriholes. Nrus Jdirh. Mirierlil. Morrutsli. 6, 258-268. Nord. G. L.. Jr. ( 1982). Analytical electron microscopy in mineralogy: exsolved phases in pyroxenes. Ulrrcimic,rcj.scf~/~y 8. 109- 120. Nord. G. L,,Jr. ( 1992). Imaging transformation-induced microstructures. Rnj. Miriercrl. 27. 455-508 (P. R. Buscck. Ed.). Owen, D. C., and McConnell. J. D. C. (197 I ). Spinodal decomposition in an alkali feldspar. Ntrtrrrr Phys. S1.i. 230, I I 8- I 19. Parsons. I. (1978). Feldspars and Huids in cooling plutons. Minerd. Mog. 42. 1-17. Parsons. 1. and Brown, W. L. (1984). Feldspars and the thernial history of igneous rocks. / , I : W. L. Brown. Ed. Frlds/)eirs r i m / Fe/d~pctthoid.s.D. Reidel. Dordrecht. The Netherlands. pp. 3 17-37 I . Peacor. D. R. (1992). Analytical electron microscopy: x-ray analysis. 27. I 13-140 Rev. Miriereil (P. R. Buseck. Ed.). Putnis. A. ( 1992). Irirrocltcc~tiori/o Mbiercrl Scie~ic~c~s. Cambridge University Press. Canihridge. Robin, Y.-P. F. (1974). Stress and strain in cryptoperthitic lamellae and the coherent solvus of alkali feldspars. Am. Mirwrcrl. 59, 1299- I3 18. Rohinson. P.. Jaffe. H. W.. Ross, M.. and Klein, C.. Jr. (1971 ). Orientation of exsolution laniellae i n clinopyroxenes and clinoamphiholes: consideration of optimal phase boundaries. Am. Mirimtl. 56. 909-939.

Rohinson. P., Ross. M.. Nord. G. L. Jr.. Smyth. J. R.. and Jaffe. H. W. (1977). Exsolution lamellae in augite and pigeonitc: fossil indicators of lattice parameters at high temperature and pressure. Am. M ~ / I ~ N62. I / .857-873. Rohinson. P.. Spear. F. S..Schumacher. J. C.. Laird, J.. Klein. C.. Evans. €3. W., and Doolan. B. L. ( 1982). Phase relations of metamorphic amphiboles: natural occurrence and theory. Re): Mirirrd. 9R. 1-227 (D. R. Vehlen and P. H. Ribhe. Eds.). Ross. M., Papike. J. J., and Wier Shaw, K. (1969). Exsolution textures in amphiholes as indicators of (/rid suhsolidus thernial histories. In: J. J. Papike. Ed. P y r o w w s trrie/Arrip/iik~lr.s:Crystrtl Cliemistr!~ Plicisr Petri~logV.Mineralogical Society America. Rubie, D. C., and Champness, P. E. (1987). The evolution of microstructure during the transformation of Mg2GeOl olivine to spinel. Bull. Mirirrrd. 110.471480. Snielik. E. A,. and Veblen. D. R. (1989). A five-amphibole assemblage from hlueschists in northern Vermont. AJNI:Mirierd. 74, 960-964. Smelik. E. A., and Veblen, D. R. (1991 ). Exsolution ofcuminingtonite from glaucophane: a new orientation for exsolution lamellae in clinoamphiholes. Am. Mirrerul. 76. 97 1-984. Smelik. E. A,. and Vehlen. D. R. (1992). Exsolution of hornblende and the solubility limitsofcalcium i n orthoamphihole. Scir~ice257. 1669-1672. Smelik. E. A,. and Veblen, D. R. (1993). A transmission and analytical electron microscope study of exsolution inicrostructtires and mechanisms i n the orthoamphiboles anthophyllite and gedrite. AUI. Minered. 78. 5 I 1-532. Smelik. E. A,. and Vehlen, D. R. (1994). Complex exsolution in glaucophane from Tillotson Park. north-central Verinont. C m . Mirierd. 32, 233-255.

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Snielik, E. A,, Nyman, M. W., and Veblen, D. R. (1991). Pervasive exsolution within the calcic amphibole series: TEM evidence for a miscibility gap between actinolite and hornblende in natural samples. Am. Mineral. 76, I 184-1 204. Smith, J. V.. and MacKenzie, W. S. (1955). The alkali feldspars. 11. A simple x-ray technique for the study of alkali feldspars. Am. Mineral. 40,733-747. Snow, E., and Yund, R. A. (1988). Origin of cryptoperthites in the Bishop Tuff and their hearing in its thermal history. J. Geophys. Res. 93, 8975-8984. Spear, F. S. (1980). The gedrite-anthophyllite solvus and the composition limits of orthoamphihole from the Post Ponds Volcanics, Vermont. Am. Mineral. 65, 1103-1 118. Thompson, J. B., Jr. (1978). Biopyriholes and polysomatic series. Am. Mineral. 63, 239-249. Vehlen, D. R. ( 198I ). Non-classical pyriholes and polysomatic reactions in hiopyriholes. Rev. Mineral. 9A. 189-236 (D. R. Vehlen. Ed.). Veblen, D. R. ( 1992). Electron microscopy applied to nonstoichiometry. polysomatism and replacement reactions in minerals. Rev. Mineral. 27, 181-229 (P. R. Buseck, Ed.). Veblen, D. R., and Burnham, C. W. (1978a). New hiopyriboles from Chester, Vermont. 1. Descriptive mineralogy. Am. Minerd. 63. 100&1009. Vehlen, D. R., and Burnham, C. W. (1978b). New hiopyriboles from Chester, Vermont. 11. Crystal chemistry of jimthompsonite, clinojimthompsonite and chesterite, and the amphihole-mica reaction. Am. Mineral. 63, 1053- 1073. Vehlen, D. R., and Buseck, P. R. (1979). Chain-width order and disorder in hiopyriholes. Am. Mineral. 64,687-700. Veblen. D. R., and Buseck, P. R. (1980). Microstructures and reaction mechanisms in hiopyriboles. Am. Mineral. 65,599-623. Veblen, D. R., and Buseck, P. R. (1983). Radiation effects on minerals in the electron microscope. Pmc. Annir. EMSA Meet. 41. 350-353. Viswanathan, K., and Ghose, S. (1965). The effect of Mg2+ substitution on the cell parameters of cummingtonite. Am. Mineral. 50, 1106-1 112. Willaime, C., and Brown, W. L. (1974). A coherent elastic model for the determination of the orientation of exsolution boundaries: application to feldspars. Acra Crystallog,: A 30, 3 16-33 1. Willaime, C.. and Gandais, M. (1972). Study of exsolution in alkali feldspars: calculation of elastic stresses inducing periodic twins. Phys. Status Solidi 9, 529-539. Worden, R. H., Walker, F. D. L., Parsons, I., and Brown, W. L. (1990). Development of microporosity, diffusion channels and deuteric coarsening in perthitic alkali feldspars. Contrih. Mineral. Perrol. 104,507-515. Yund, R. A. (1983). Diffusion in feldspars. Feldspar Mineralogy. Rev. Mineral. 2, 203-222 (P. H. Rihhe. Ed.). Yund, R. A,, and Chapple, W. M. (1980). Thermal histories of two lava flows estimated from cryptoperthite lamellar spacings. Am. Mineral. 6 5 , 4 3 8 4 3 . Yund, R. A., and Davidson, P. (1978). Kinetics of lamellarcoarsening in cryptoperthites. Am. Minerd. 63,470477. Yund, R. A,, McLaren, A. C., and Hohhs. B. E. (1974). Coarsening kinetics of the exsolution microstructure in alkali feldspar. Contrib. Mineral. Petrol. 48.45-55.

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRYSTALS.VOL . in1

High-Resolution Electron Microscopy of Quasicrystals KENJI HIRAGA tnstirure for Muterials Research. Tohoku Universitv Kutuhiru. Aobu.ku. Sendui 980-77. Jupm

1. Introduction . . . 11. Quasiperiodic Lattices

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . One-Dimensional Quasiperiodic Lattices . . . . . . . . . . . . B . Two-Dimensional Quasiperiodic Lattices . . . . . . . . . . . 111. Experimental Procedures . . . . . . . . . . . . . . . . . . IV. Electron Diffraction of Quasicrystals . . . . . . . . . . . . .. A . Good- and Poor-Quality Quasicrystals . . . . . . . . . . . . B . Icosahedral Quasicrystals . . . . . . . . . . . . . . . . . C . Decagonal Quasicrystals . . . . . . . . . . . . . . . . . V. High-Resolution Electron Microscopy Images of Quasicrystals . . . . . A . Icosahedral Quasicrystals . . . . . . . . . . . . . . . . . B . Decagonal Quasicrystals . . . . . . . . . . . . . . . . . VI . Structure of Icosahedral Quasicrystals . . . . . . . . . . . . . . A . Topological Features of Icosahedral Quasicrystalline Lattices . . . . B . Atomic Arrangements of Icosahedral Quasicrystals . . . . . . . . C . Defects in Icosahedral Quasicrystals . . . . . . . . . . . . . VII . Structure of Decagonal Quasicrystals and Their Related Crystalline Phases . A . Framework of Columnar Atom Clusters . . . . . . . . . . . . B . Decagonal Quasicrystals with 0.4-nm Periodicity . . . . . . . . . C . Decagonal Quasicrystals and Crystalline Phases with I .2-nm Periodicity D. Decagonal Quasicrystal and Crystalline Phases with I .6-nrn Periodicity . VIII . Concluding Remarks . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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I . INTRODUCTION The discovery of an icosahedral phase having noncrystallographic symmetry by Shechtman el al . (1984). and the following theoretical explanation as a quasicrystal by Levine and Steinhardt (1984). have had a strong impact on solid state physicists. We had thought for a long time that solids were divided into two structural classes: crystalline with periodic atomic arrangements and amorphous with random atomic arrangements. Also. we had recognized that only crystals with periodic structures produce sharp diffraction peaks. The discovery by Shechtman er a1. brought about a drastic change in attitudes concerning the structure of solids. The quasicrystals show diffraction patterns with noncrystallographic symmetries but nonetheless. 31

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consisting of sharp peaks. That is, the quasicrystals have aperiodic structures producing sharp diffraction peaks. High-resolution electron microscopy has been developed to study aperiodic structures, such as the structures of defects, in solids, so it is the most powerful or only tool for investigating the “real” structure of quasicrystals. Consequently, many high-resolution electron microscopy studies of the quasicrystals have been carried out, and those results have given us valuable information about the structurc of the quasicrystals. Our group has studied high-resolution electron microscopy of quasicrystals from the early stage for about a decade. In this review I discuss the real structure of the quasicrystalline alloys, based primarily on the results of our group. 11. QUASIPERIODIC LAITICES

A. One-Dimensional Quasiperiodic Lattices To make it easy to understand this paper, I will briefly make mention of the projection method, which is one of theoretical ideas that help to explain aperiodic structures showing sharp diffraction peaks. One-dimensional quasiperiodic lattices can be formed by the projection of a two-dimensional square lattice on a straight line with an irrational slope (Fig. I), as can be seen in many papers (e.g., Elser, 1986, and Katz and Duneau, 1986). An orthogonal coordinate system with axes labeled XII and X I is superimposed on the coordinate system of a two-dimensional square lattice, which is rotated by an angle 8 = tan-l(l/r) (r is the golden ratio) with respect to the former coordinate system. X I Iand XI are called a physical subspace and an internal subspace. respectively. Basis vectors of the two-dimensional square lattice, e l and ez, transform to e l l I= cos8 and ell? = sin8 on the physical space, and to ell = - sin8 and el? = cos8 on the internal space. The square lattice points, which are described as the set of n lel +nzez with integers n I and nz. are projected at the points of n l e lIl nzellz on the axis X I Iand at the points of n l e l l n2e12 on the axis X I . A one-dimensional quasiperiodic lattice can be obtained by projecting square lattice points, which are inside a strip parallel to the XiI axis, on the line Xll, as shown in Fig. 1. The region obtained by projecting the strip on the axis XI, called a window, is labeled W in Fig. 1. That is, one can obtain lattice points on the physical subspace as follows. First, a square lattice point is projected on the internal space (i.e., on the axis X l ) . If the projected point is inside the window, the square lattice point is projected on the physical subspace (i.e., on the axis X l ) . If the window is now taken as the size of projection of the square lattice unit on the X I I a, quasiperiodic lattice, called the Fibonacci sequence, with two intervals L and S ( L = TS),is obtained, as shown in Fig. 1.

+

+

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.

4FIGUREI . Construction of a one-dimensional quasiperiodic (Fibonacci) lattice by the projection of a two-dimensional square lattice. By projecting lattice points inside a strip (dashed lines) on the axis X I , a Fibonacci lattice of intervals L and S is obtained.

B. Two-Dimensional Quasiperiodic Lattices Two-dimensionalquasilattices with tenfold symmetry are described by the projection of a five-dimensional hypercubic lattice. In this case the physical subspace is two-dimensional space and the internal subspace is three-dimensional space. Lattice points X in the five-dimensional hypercubic lattice can be described with the basis vectors of e j ( j = 0, 1, 2, 3,4) as follows: X = C4=o njej. Projected points of X on the physical subspace, XI,,are described as XIl = C:=,njellj, where e l l j= (cos(2nj/5), sin(2nj/5)) ( j =0, 1,2,3,4).Three-dimensionalspaceofthe internal subspace is divided into components of the two-dimensional space, X I , and a one-dimensional component perpendicular to the former two-dimensional space, XI:thus XI = C4j=, n j e l j and Xl = C:=,, n,e$, , where elj is (cos(4n j/5),sin(4nj/5)) ( j = O , 1 , 2 , 3 , 4 ) a n d e l ji s e l j = ( l / & ) j ( j = 0 , l , 2 , 3 , 4 ) .

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Here, only lattice points inside the four windows shown in Fig. 2a are projected on the physical subspace, and the Penrose lattice (Fig. 2b) can then be obtained. On the other hand, one can use the same decagonal windows on five e l j = ( 1 / 4 ) j ( j = 0, 1,2,3,4) planes, and then obtain pentagonal Penrose lattices, as shown in Fig. 3. By reducing a size of the windows, some different types of pentagonal lattices are obtained.

*O C

B

'0 cO D

0

FIGURE 2. Windows in three-dimensional internal subspace (a) that enable the Penrose lattice (b) to be constructed. (Source: Yamamoto and Ishihara, 1988.)

(4 FIGURE 3. Three types of pentagonal Penrose lattices (a, b, and c) constructed by windows in (d); (a). (b). and ( c ) are made by windows A, B, and C, respectively.

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As shown in this paper, one can determine experihental lattice points, occupied by some atom clusters, on the physical subspace (i.e., XI,= C4=o n j e l l jby , highresolution electron microscopy. From the lattice points on the physical subspace, one can determine lattice points in the five-dimensional space, X = C4=o n,ej, and then also lattice points on the internal subspace, XI = C4=o n j e l j . A threedimensional quasiperiodic lattice can be constructed from the projection of a sixdimensional hypercubic lattice. In this case the physical subspace and internal subspace are a three-dimensional space. The three-dimensional quasiperiodic lattice is called a three-dimensional Penrose lattice, and it is formed with a threedimensional aperiodic arrangement of two types of rhombohedra with planes of a golden diamond, in which a ratio of a long and short diagonals is the golden ratio. The construction manner and topological characteristics of the three-dimensional Penrose lattice were first discussed by Ogawa (1985). The various quasilattices obtained by the projection method are known to produce diffraction patterns consisting of sharp spots. 111. EXPERIMENTAL PROCEDURES

As an aid to those who would like information about sample preparations and compositions,and details of the experimentalprocedures, in this section we briefly summarize the principal samples mentioned in this paper and sample preparations for high-resolution electron microscopy. Metastable quasicrystals were formed in rapidly solidified (R.S.) alloys, which were prepared using a melt-spinning apparatus with a single copper roller 20 cm in diameter at 2000 to 4000 rev/min. On the other hand, stable quasicrystals were formed in conventionally solidified (C.S.) alloys in an arc furnace under an argon atmosphere and annealed at proper temperatures and then quenched mainly in water. Compositions and sample preparations for the quasicrystals mentioned in this paper are as follows: Icosahedral phases Al-Mn R.S. Al&hl4 alloy Al-Mn-Si R.S. A174Mn20Si6 alloy Al-Fe-Cu C.S. A165Fe&120alloy Al-Ru-Cu C.S. A165RU15CU20alloy Al-Pd-Mn C.S. A17oPd2oMnlo alloy Al-Li-Cu Zone-melted Al-Li-Cu alloy Decagonal phases R.S. A186Mn14 alloy A1-Mn Al-Ni-Fe C.S. A172Ni24Fe4 alloy

42 Al-Cu-Co Al-Ni-Co Al-Pd-Mn A1-Pd

KENJI HIRAGA

C.S. Alh5Cu15Co;?~ alloy C.S. A170Nil5C015 alloy

C.S. A170Pdl~Mn2(~ or A170Pd13Mn17 alloy R.S. Al3Pd alloy

High-resolutionelectron micrographs and electron diffraction patterns presented in this paper were taken with a 200-kV (JEM-200CX) electron microscope having a resolution of 0.23 nm and a 400-kV electron microscope (JEM-4000EX) with resolution 0.17 nm. All images and diffraction patterns, except for those of Al-Mn, Al-Mn-Si, and Al-Fe-Cu icosahedral phases and an Al-Mn-Si decagonal phase, were taken with the 400-kV electron microscope. Samples for electron microscopy were prepared by electrolytic polishing using an ice-cold solution of perchloric acid and methanol, in a 1:9 volume ratio, for A1-Mn and Al-Mn-Si quasicrystals. For the other quasicrystals and crystalline phases, crushed materials were dispersed on holey carbon films. Image contrast of high-resolution electron micrographs is very sensitive to experimental conditions such as sample thickness and defocus value. Particularly, the high-resolution structure images, having information about atomic arrangements, should be taken under strict experimental conditions. Defocus values in observed images can be estimated from Fresnel fringes at the edges of samples or amorphous films stuck on samples. Most high-resolution images presented in this paper were confirmed as taken close to Schemer defocus.

DIFFRACTION OF QUASICRYSTALS Iv. ELECTRON The electron diffraction technique has been used widely to study quasicrystals because of its convenience, high sensitivity for weak reflections, and ability to take diffraction patterns from selected small areas compared with x-ray and neutron diffraction. In this section, we look at structural information obtained from observation of electron diffraction patterns. A. Good- and Poor-Quality Quasicrystals

Figure 4 shows diffraction patterns for some icosahedral and decagonal quasicrystals, taken with the incident beam parallel to the five- and tenfold symmetry axes, respectively. Figure 4a is a pattern taken from an A1-Mn icosahedral phase, first found by Shechtman ef al. (1984) as a metastable phase, whereas Fig. 4b is that from a stable Al-Fe-Cu icosahedral phase, found later by Tsai et al. (1987, 1988) in an A165Fe15Cu20 alloy conventionally solidified and then annealed at 850°C

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FIGURE 4. Electron diffraction patterns showing the structural quality ofqasicrystalline structures. taken with the incident beam parallel to the fivefold symmetry (a, b) and the tenfold symmetry (c. d) axcs. Parts (a) and (b) are of AI-Mn metastable and Al-Fe-Cu stable icosahedral phases, respectively; (c) and (d) are of AI-Mn metastable and Al-Ni-Co stable decagonal phases. respectively.

for 48 h. From the comparison between the two patterns, one can clearly see the difference in the structural quality of the quasicrystals. That is, in the pattern of the Al-Fe-Cu icosahedral phase in Fig. 4b, one can see a number of weak spots, showing the existence of highly ordered correlation in the atomic arrangement, and can see that their positions are located at perfect icosahedral symmetry positions. On the other hand, the pattern of the Al-Mn icosahedral phase (Fig. 4a) shows the disappearance of weak spots due to poor correlation, and systematical shifts of diffraction spots, which can be clearly seen as the deformation of small pentagons formed with weak diffraction spots. The shifts of diffraction spots result from linear phason strain, which will be mentioned later. Also, it can be said from

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the projection method that the appearance of weak spots in diffraction patterns results from the sharp discontinuities of the distribution p~ in the internal subspace (Elser, 1985). That is, if the distribution of atom positions in the quasicrystals in the internal subpace shows the sharp discontinuity, the gl-dependent decay of diffraction intensities become slow and weak spots appear. If the distribution has a broad discontinuity, the gl-dependent decay becomes high and weak spots disappear. The difference in structural quality of decagonal quasicrystals can be seen from the comparison between Fig. 4c and d, which were obtained from an Al-Mn metastable phase and a stable phases in a conventionally solidified Al-Ni-Co alloy. The Al-Mn decagonal phase was found by Bendersky (1985) and by Chattopadhyay et al. (1985). whereas the Al-Cu-Co decagonal phase was found in a later study by Tsai et al. (1989b.c). The appearance of many sharp weak spots at exact decagonal symmetry positions in Fig. 4d shows that compared with the Al-Mn decagonal phase, the Al-Cu-Co decagonal phase is a good-quality or highly ordered quasicrystal without any linear phason strain. As mentioned above, quasicrystals exist in wide structural regions from poorquality to good-quality, so one should use good-quality quasicrystals to investigate the real structural characteristics of the quasicrystalline structures. Like the Al-Fe-Cu phase, good-quality icosahedral quasicrystals, have been found in AlRu(or Os)-Cu (Tsai et al., 1988), Al-Pd-Mn (Tsai et al., 1990). and good-quality decagonal quasicrystals in Al-Ni-Co (Tsai et al., 1989a),Al-Pd-Mn (Beeli et al., 1991), and Al-Ni-Fe (Lemmerz et al., 1994). It may be of interest to note that all the icosahedral and decagonal quasicrystals, except for the Al-Ru(or Os)-Cu and Al-Pd-Mn icosahedral quasicrystals, are found as high-temperature phases, which are stable only at high temperatures and transform to crystals or other quasicrystalline phases at low temperatures.

B. Icosahedral Quasicrystals Figure 5 shows electron diffraction patterns of the stable Al-Cu-Fe icosahedral phase, taken with the incident beams parallel to the two-, three-, and fivefold symmetry axes. They were taken with two different camera lengths to observe diffraction spots on the zeroth Laue zone and those on the higher zones, and Kikuchi patterns. In Fig. 5e and f, the patterns taken with different exposure times are inserted to get a simultaneous view of diffraction spots in higher Laue zones and the Kikuchi pattern. In the patterns one can see a number of diffraction spots, which are very sharp and located at strict decagonal symmetry positions without showing any systematic shifts due to linear phason strain. In Fig. 5 , diffraction patterns in a zeroth Laue zone and Kikuchi bands indicate two-, six-, and tenfold rotational symmetriescorrespondingto the projection symmetry of the icosahedral

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FIGURE 5 . Electron diffraction patterns of an icosahedral quasicrystal in an Al7oFel5Culs alloy conventionally solidified and then annealed for 48 h at I 1 18 K, taken with the incident beam parallel to the twofold [(a) and (d)], threefold [(b) and ( e ) ] .and fivefold [(c) and (01symmetry axes. Patterns of (d). (el, and (0 were taken with a shorter camera length than that in (a), (b), and (c), to obtain diffraction spots in higher Laue zones and Kikuchi patterns. In the central parts of ( e )and (0,pictures taken with different exposure times are inserted. (Source: Hiraga era/., 1988.)

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phase, whereas diffraction spots in the higher Laue zones and Kikuchi lines show two-, thrce-, and fivefold symmetries, which correspond to three-dimensional symmetry. For example, in Fig. 5e one can clearly see the threefold rotational symmetry in the intensity distribution of diffraction spots on higher Laue zones and in the Kikuchi pattern formed with bright and dark lines. On the other hand, pentagons drawn with bright and dark Kikuchi lines are observed in Fig. 5f. The Al-Cu-Fe icosahedral phase is known to be an F-type icosahedral quasicrystal, which is interpreted in terms of face-centered six-dimensional hypercubic lattices (Ishimasa ef al., 1988; Ebalard and Spaepen, 1989). in contrast to P-type quasicrystals derived from a primitive hypercubic lattice, which were observed primarily in the early stage of quasicrystal studies. The difference between the F- and P-type icosahedral quasicrystals can be seen in diffraction patterns taken with the incident beam parallel to the twofold axis, as shown in Fig. 6. Figure 6a and b are electron diffraction patterns of the P-type Al-Li-Cu and F-type AI-FeCu icosahedral quasicrystals, respectively. In the pattern of Al-Fe-Cu (Fig. 6b), there are a number of sharp diffraction spots in addition to the spots appearing in the pattern of Al-Li-Cu (Fig. 6a). Some examples of the extra spots appearing in only the F-type structure are indicated by small white arrows in Fig. 6b. It should be noted that most of the extra reflections appear on the five- and threefold axes. so no extra reflections appear in diffraction patterns taken with the incident beam parallel to the five- and threefold axes. Structural characteristics of the F- and P-type icosahedral quasicrystals were studied by high-resolution electron microscopy images taken with the incident beam parallel to the twofold axis (Hiraga and Shindo, 1989). The transformation

FlciOI
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from the P-type to F-type icosahedral phase was observed by annealing a rapidly solidified Al-Ru-Cu icosahedral quasicrystal (Hiraga et al., 1989b). C. Decagonal Quasicrystals 1. Characteristics of Diffraction Patterns of Decagonal Quasicrystals

Decagonal quasicrystals are two-dimensional quasicrystals with two-dimensional quasiperiodic planes and a one-dimensional periodic axis along the tenfold axis, so their diffraction patterns show reflection planes showing quasicrystalline structures and a periodic array of the reflection planes, as shown in Fig. 7. Figure 7 shows electron diffraction patterns of a stable decagonal quasicrystal with 0.4-nm

FIGURE 7. Electron diffraction patterns of a decagonal phase in an A172Ni24Fe4 alloy conventionally solidified and then annealed at 850°C for SO h, taken with the incident beam parallel to the tenfold axis (a), parallel to the p-axis (b) and y-axis (c) in (a): (d) enlarged pattern of a part of (a). (Source: Hiraga ct d . ,1996.)

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periodicity in an A172Ni24Fe4 alloy conventionally solidified and then annealed at 850°C for 50 h. The diffraction pattern of Fig. 7b, taken with the incident beam perpendicular to the tenfold axis, along the p-axis indicated in Fig. 7a, shows the existence of a period of about 0.4 nm along the vertical direction. On the other hand, in Fig. 7c, taken with the incident beam parallel to the q-axis. one can see an extinction rule that causes diffraction spots showing the period of 0.4 nm to disappear. The extinction rule suggests the existence of a twofold screw axis parallel to the tenfold axis and the space group P lO~/mmc(Yamamoto and Ishihara, 1988). The extinction rule has been widely found in the decagonal quasicrystals with other periods. It should be noted here that diffuse scattering is apparently visible on the background around strong Bragg spots in diffraction patterns of the decagonal quasicrystals, taken with the incident beam parallel to the periodic axis, as can be seen clearly in Figs. 4d and 7d. It results from local disordering from the ideal Penrose tiling in an atomic arrangement. We know that the Penrose lattice, obtained mathematically from the projection of hypercubic lattices in the higher-dimensional space to three- or two-dimensional space, produces diffraction patterns consisting of sharp spots, described as a delta function, and any local modification from the Penrose lattices (i.e., random phason strain) results in the rather high g l dependent decay of diffraction intensities (Section IV.A). Consequently, the local modification in the real structure of the quasicrystals produces diffuse scattering. The diffuse scattering in decagonal quasicrystals spreads out on two-dimensional reciprocal planes perpendicular to the tenfold symmetry axis, whereas that in the icosahedral quasicrystals spreads out over three-dimensional reciprocal space. Therefore, one can clearly see the diffuse scattering in diffraction patterns of decagonal quasicrystals (Fig. 4d) compared with those of icosahedral quasicrystals (Fig. 4b). 2. Polytypes of Decagonal Quasicrystals The decagonal quasicrystals have some polytypes with different periods, such as 0.4 nm, 0.8 nm, 1.2 nm, 1.6 nm, and so on, along the tenfold symmetry axis. Figure 8 shows diffraction patterns of the decagonal quasicrystals with about 0.8-, 1.2-, and 1.6-nm periods. The patterns were taken with the incident beam parallel to the two directions perpendicular to the periodic axis (the p and q directions in Fig. 7). The diffraction patterns, Fig. 8a, c, and e, show 0.8-, 1.2-, and 1.6-nm periodicities along the vertical direction (viz., along the tenfold symmetry axis). On the other hand, in Fig. 8b, d, and f, one can see only diffraction spots showing 0.4-, 0.6-, and 0.8-nm periodicities if the weak diffuse spots indicated by small arrows are ignored. The extinction rule shows that the space group is P lO~/mmc. The diffuse spots, which are often observed in decagonal quasicrystals between Bragg reflection planes, tend to weaken and disappear in good-quality decagonal quasicrystals with sharp stoichiometric compositions (Hiraga et al., 1991a).

FIGURE 8. Electron diffraction patterns of decagonal quasicrystals with three different periodicities, taken with the incident beam perpendicular to the periodic axis. parts (a) and (b) are metastable decagonal phase with 0.8-nm periodicity in a rapidly solidified A113C04 alloy, (c) and (d) are stable in phase with 1.2-nm periodicity in a conventionally solidified A17oPdloMn20 alloy annealed at 800°C for 16 h and then quenched in water, and ( e )and (0 are metastable phase with 1.6-nm periodicity in a rapidly solidified A13Pd alloy.

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Besides the classification of decagonal quasicrystals by different periodicity, polytypes with different symmetry are known. In addition to the main decagonal quasicrystals with central symmetry, noncentral symmetry has been found in some decagonal quasicrystals with 0.4-nmperiodicity, particularly in metastable phases formed in rapidly solidifiedalloys (Tsuda etal., 1993). Decagonal quasicrystals are discussed in Section VII.B.3 in connection with a crystalline approximant phase. Also, decagonal quasicrystals with different sizes of columnar atom clusters or different tilings of the atom clusters have been found. Those structures are covered in Section VII.

v. HIGH-RESOLUTION ELECTRON MICROSCOPY IMAGES OF QUASICRYSTALS Quasicrystals have aperiodic structures despite the presence of sharp spots in diffraction patterns, so high-resolution electron microscopy is the most powerful tool for investigating their real structures. Consequently, many high-resolution rnicroscopy studies of quasicrystals have been carried out and have given us valuable information about the structures and defects of quasicrystalline alloys, although there are limitations due to the limited resolution of an electron microscope, cornpared with diffraction techniques, and due to the use of projected images along the beam axis. In this section, I describe characteristic features of high-resolution images of icosahedral and decagonal quasicrystals. In the early stage of high-resolution electron microscopy of quasicrystals, almost all of the images of quasicrystals observed were taken with 200-kV electron microscopes having resolutions of about 0.23 nrn, so they gave us information about the topological features of lattices of the quasicrystals, but little information about atomic arrangements, because of poor resolution and thick samples. Afterward, high-resolution images, which directly reflect the projected potential of the quasicrystals, have been observed with a 400-kV electron microscope having a resolution of 0.17 nm. They give us much valuable information that enables us to understand atomic arrangements of the quasicrystals. To distinguish between the two types of images, we call a lattice image and a structure image (Hiraga, 1991a,b). In this section I describe characteristic features of the high-resolution lattice images and structure images. A. Icosahedral Quasicrystals

Figure 9 shows a lattice image and the structure image of icosahedral phases, taken with the incident beam parallel to the fivefold axis. They were taken with a 200- and a 400-kV electron microscope, having resolutions of 0.23 and 0.17 nm, respectively. Figure 9a is a typical high-resolution lattice image of an Al-Mn-Si

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F l C i i l K t 9. High-resolution lattice image (a) of a nictastahle Al-Mn-Si icosahedral phase, and siruciure iniage d i i stable Al-Pd-Mn icosahedral phase. Parts ( a ) and (c) were taken with the incident beam parallel to the fivefold syiiimetry axis.

FI(iIIKE 10. (a) Electron diftracticin pattern of the Al-Mn-Si icosahedral phase: (b) Fouriciditfractogram 0 1 the lattice image of Fig. 9a: ( c ) Fotirier diffractograin of the structure image o f Fig. 9h.

icosahedral phase, and Fig. 9b is a structure image of an Al-Pd-Mn icosahedral phase. The lattice image (Fig. 9a) was obtained from a relatively thick region of a few tens of nanometers, whereas the structure image was taken from a thin region, less than about 5 nm. By comparing the two images, one can see image contrast with a higher resolution in Fig. 9b than in Fig. 9a. Figure 10 shows Fourier diffractograms taken from the images, together with an electron diffraction pattern. The diffraction spots indicated by the arrows in the electron diffraction pattern (Fig. IOa) have strong intensity in a kinematical

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approximation just as in x-ray diffraction compared with the other spots. Those strong spots have lattice spacings of about 0.2 nm and result from the nearestneighbor atom pairs. The Fourier diffractogram of Fig. 10b is formed by reflections inside the strong reflections with lattice spacings of about 0.2 nm, which are enhanced by multiple scattering in the relatively thick sample. That is the reason why the lattice image of Fig. 9a has information about quasilattices but little information about atomic arrangements. On the other hand, the diffractogram of Fig. 1Oc reproduces well an intensity distribution of the electron diffraction pattern, so the structure image can be considered to have information about the atomic arrangement. It can be said that the structure images taken under strict conditions and from thin samples reflect faithfully atomic arrangements projected along the fivefold symmetry axis, and the dark and bright regions in the observed images correspond to the high- and low-potential regions, respectively. Detailed interpretation of the lattice and structure images is mentioned again later.

B. Decagonal Quasicrystals Figure 11 shows a high-resolution electron micrograph of an Al-Pd-Mn stable decagonal phase taken with the incident beam parallel to the tenfold axis. It is well

FIGUREI I . High-resolution electron micrograph of an Al-Pd-Mn decagonal phase, taken with the incident beam parallel to the tenfold byinmetry axis. A structure image in a thin region on the left side and lattice image in a thick region on the right side are observed.

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53

known that image contrast of high-resolution electron micrographs is extremely sensitive to instrumental conditions such as defocus value and sample thickness. From the Fresnel fringes at the edge of an amorphous film, indicated by an arrowhead in Fig. 11, one can see that the image was taken with an optimum defocus near the Scherzer defocus of -45 nm. In Fig. 11, also, one can see a contrast change with increasing sample thickness from the left side of the micrograph to the right side. In the thin region on the left side in Fig. 11, a structureimage reflecting projected potential is observed, whereas a lattice image showing adistribution of some atomic clusters, which are represented as ring contrasts, is observed in a thick region of the right region. The structure image can give us valuable information about the atomic arrangement in a local region, and from the lattice image one can obtain the arrangement of some atom clusters in a wide region.

VI.

STRUCTURE OF ICOSAHEDRAL

QUASICRYSTALS

A. Topological Features of Icosahedral Quasicrystalline Lattices In the initial studies of quasicrystals, nearly all the high-resolutionimages reported were lattice images, taken with a 200-kV microscope having a resolution of about 0.23 nm. Figure 12a is a typical high-resolution lattice image, taken from AI-Mn icosahedral quasicrystal, together with an electron diffraction pattern and a Fourier diffractogram of the image. The micrograph was taken in the very early days just after the report by Shechtman et al. (1984) with a 200-kV electron microscope with a resolution of 0.23 nm. In the image of Fig. 12a,we can clearly see a homogeneous distribution of sharp bright dots and their straight array on lines parallel to the fivefold directions. The existence of sharp bright dots in the image also shows that the topological features forming the bright dots are arrayed along lines parallel to the incident beam. The homogeneous distribution of bright dots ruled out models that explain the diffraction patterns with icosahedral symmetry in terms of large unit cells and/or multiply twinned crystals (Field and Fraser, 1984-1985; Pauling, 1985) and showed clearly the existence of long-range quasiperiodicity, producing a diffraction pattern with fivefold rotational symmetry. Characteristics of the bright-dot distribution in the lattice image can be seen clearly on the enlarged image in Fig. 13a. In this image one can see the characters of the distribution of the bright dots: They form pentagons of various sizes as well as being arrayed on straight lines parallel to the fivefold directions. These characteristics can also be seen in the projection of the three-dimensional Penrose lattice along the fivefold symmetry axis, as shown in Fig. 14a. Open

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KENJI HIRAGA

FI(iIIKE 12. High-resolution lattice image ( a ) and electron diffraction pattern (h) of the AI-Mn icosahedral phase. laken with the incident heam parallcl to the fivefold symmetry axis. Part ( c ) is an optical diffractogram of the ohserved image (a). (SOIWW:Hiraga PI d.. 1985.)

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FIGUKE13. (a) High-resolution lattice image of the AI-Mn-Si icosahedral phase; (b) pentagonal tiling constructed by the connection of bright dots in part (a).

circles correspond to vertices of fundamental rhombohedra1 units forming the three-dimensional Penrose lattices. In Fig. 14a one can see that the open circles are arrayed on straight lines along the fivefold directions indicated with arrows and form pentagons of various sizes associated with a scaling of the golden ratio, indicated by lines. In the image observed, a pentagonal tiling can also be constructed by connecting the bright dots by lines, as indicated in Fig. 13b(Hiraga et al., 1987). Assuming that a lattice spacing of the icosahedral quasicrystal (corresponding to the edge length of the fundamentalrhombohedra)is 0.46 nm, the pentagonal tiling in the observed image correspondsto a tiling of the pentagon, indicated by a small arrow in Fig. 14a.

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KENJI HIRAGA

(a) (b) 14. (a) Projection of three-dimensional Penrose tiling along the fivefold symmetry axis. Open circles are vertices of the fundamental rhombohedra; note that the circles lie along the fivefold directions and also form pentagons of various sizes. Compare the arrangement of circles with that of the bright dots in Fig. 13. (b) One-dimensional sequence of lattice planes (arrays of bright dots) in Fig. 13 is described by replacing d, and d1 with the unit vectors in the y and x directions, respectively. Note that the slope of lines in part (b) is I/s. (Soume: Hiraga, 1991a.) FIGURE

One-dimensional sequences of lattice planes (bright-dot rows) in Fig. 13a can be described as an arrangement of two intervals, indicated by dl and d, (d/ = td,$). If the distancesd, and d, are replaced with the unit vectors in the x and y directions, respectively,one obtains the diagram of Fig. 14b, the slope of which is 1/t (Hiraga, 199la). This means that the one-dimensional sequence of lattice planes correspond to a quasiperiodic lattice which can be obtained by the projection of square lattices onto a line with a slope of l / r , as shown in Fig. 1. In quasicrystals with linear phason strain, a change in the average slope from 1/r was observed (Hiraga et al., 1989a). Here it should be noted that lattice images such as Fig. 13a show features of lattices averaged over a thickness of a few tens of nanometers. Topological features of the averaged quasilattices in the icosahedral quasicrystals are in good agreement with those of the three-dimensional Penrose lattices. The lattice images also give us information about linear phason strains and dislocations, but they provide little information about random phason strain corresponding to local modification, because the images observed are projections along the incident beam.

B. Atomic Arrangements of Icosahedral Quasicrystals Figure 15 shows high-resolution structure images of Al-Pd-Mn and Al-Li-Cu icosahedral phases, taken with the incident beam parallel to the fivefold axis. In the images we notice characteristic image contrast distributions consisting of

FIGURE15. High-resolution structure images of Al-Pd-Mn (a) and Al-Li-Cu (b) icosahedral phases, taken with the incident beam parallel to the fivefold axis. Note the decagonal contrasts consisting of 10 bright dots surrounding a bright ring. indicated by black circles.

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KENJI HIRAGA

FIGURE16. (a) Fundamental icosahedral cluster; (b) tetrahedral arrangement on the icosahedral cluster; (c) octahedral arrangement on the icosahedral atom cluster; (d) triacontahedral atom cluster constructed by tetrahedral arrangements (b); (e) Mackay icosahedral atom cluster formed by octahedral arrangement (c).

10 bright dots surrounding a bright ring and a central dark dot, as enclosed by circles. This image contrast distribution is called decagonal contrast in this section. The decagonal contrast can be interpreted as an atom cluster with icosahedral symmetry. It is well known that atom clusters with icosahedral symmetry are formed with nearly close-packed atomic arrangements. Figure 16d is a triacontahedral atom cluster, considered to be a structure unit of the Frank-Kasper type of icosahedral phase, exemplified by an Al-Li-Cu icosahedral phase (Henley and Elser, 1986; Audier et al., 1986). Figure 16e is called a Mackay icosahedral atom cluster, which is suggested to be an important structure unit in the structure of Al-Mn (Elser and Henley, 1985; Guyot and Audier, 1985), probably also in the structure of AlFe-Cu and Al-Pd-Mn icosahedral phases. They can be formed from the basic icosahedral atom cluster of Fig. 16a. There are two ways that atoms can be put on the icosahedral cluster. One is a tetrahedral arrangement in which one atom is put on three atoms, and the other is an octahedral arrangement in which three atoms

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59

Ow0

0430 o b . ( a 0

O%O

O@O

0% (a)

OO@@B 0 00 0 0

0

(h)

(c)

17. Atomic arrangenienfs ofthe triacontahedral alomic cluster ( a )and Mackay icosahetlral clusfcr (b). projected along the fivefold symmetry axis. ( c ) High-resolution structure image calcularcd from an arrangement of the Mackay atom cluster (h). FlGLlKE

are rotated and put on three atoms of the icosahedral cluster. By putting atoms with the tetrahedral arrangement, the triacontahedral atom cluster of Fig. 16d is constructed, and the octahedral arrangement leads to the icosahedral atom cluster of Fig. 16e. Atomic arrangements of the triacontahedral and icosahedral atom clusters, projected along the fivefold axis, are shown in Fig. 17a and b, respectively. In these atomic arrangements, one can see a characteristic atom arrangement, similar for both clusters, in the central part: that is, double decagonal atom rings surrounding a central atom and the large decagonal ring. From those atom arrangements, one can expect high-resolution image contrast, as shown in Fig. 17c (Hiraga, 1990, 1991 a, b; Hiraga and Shindo, 1990). As mentioned before, in the structure images taken from thin zones, high-potential regions (i.e., atom positions) are seen as dark regions, and low-potential regions without atoms are seen as bright regions. The central atom becomes a central dark dot, the double decagonal atom ring becomes a dark ring, the ring channel between the central atom and the double decagonal rings becomes a bright ring, and the 10 channels outside become 10 bright dots. The calculated image contrast is in good correspondence with the decagonal contrast in Fig. 15. It is suggested that the triacontahedral and icosahedral atom clusters occupy special positions in the three-dimensional Penrose lattice (i.e., the 12-fold positions proposed by Henley, 1986) (Yamamoto and Hiraga, 1988). The 12-fold positions can be considered to be the special sites on which the atom clusters are located to give a dense packing in space by the three types of linkages of Fig. 18. Figure 19 shows a projection of the 12-foldpositions on the three-dimensional Penrose lattice along the fivefold axis. An atomic arrangement, which is formed by placing the triacontahedral atom clusters at the 12-fold positions indicated with arrows in Fig. 19a, is shown in Fig. 19b, and the expected contrast distribution of bright regions is shown in Fig. 19c. The image contrast distributions are observed in the observed structure image of Fig. 15b.

VDVXIIH IfNm

09

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61

C. Defects in Icosahedral Quasicrystals

1. Linear Phason Strain By close examination of the image of Fig. 12 with oblique viewing along the vertical direction, one can see frequent displacements of the bright-dot arrays. The displacements are often observed in the rapidly solidified quasicrystals and are interpreted in terms of quenched linear phason strain (Socolar et al., 1986). The linear phason strain is considered to be related to the growth process of quasicrystalline domains (Hiraga and Hirabayashi, 1987a). Figure 20a is an ordinary bright-field electron micrograph showing dendritic growth of the Al-Mn-Si decagonal phase. From the morphology of the dendrite, the quasicrystalline grain is supposed to grow from the left side to the right side, as indicated by arrows. Figure 20b shows a high-resolution lattice image of the rectangular area indicated in Fig. 20a. In the lattice image, bright dots aligned along the fivefold directions p , q , r, s, and t are distributed homogeneously over the whole region. With close examination, by viewing Fig. 20b obliquely along the fivefold directions, however, one notices that the bright-dot rows are frequently displaced. The density of the displacements of bright-dot rows depends on the directions, as illustrated schematically in Fig. 21. That is, no shift appears along the t direction, which is nearly parallel to the growth direction of the quasicrystalline grain, and a few shifts appear along the s and p directions, whereas a fairly high density of the shifts exists along the q and r directions. We also notice that the shifts along the q and r directions are concentrated at the upper region C of Fig. 20c, but almost disappear at the lower area D near the edge of the observed quasicryatalline grain. In Fig. 20c and d, two optical diffractograms taken from the high-density area C and the low-density area D are inserted. Diffraction spots in the pattern of Fig. 20d are located at nearly the exact positions with fivefold symmetry, but those of Fig. 20c are clearly shifted from the positions of fivefold symmetry along the vertical directions, as indicated by arrows. The shifts of spots in reciprocal space as well as displacementsof bright-dot rows in real space can be interpreted in terms of anisotropic phason strain (i.e., linear phason strain) (Socolaretal., 1986). If the linear phason strain persists along one of the fivefold directions, the diffraction spots shift along the correspondingdirection in the reciprocal space. In the image of Fig. 20b, the linear phason strain lies almost perpendicular to the t direction, that is, approximately to the quasicrystal growth direction. It is natural that the frozen linear phason strain is relaxed in the vicinity of the edges of grown quasicrystals. This is clear in the optical diffractogram of Fig. 20d, taken from the D area near the edge of the quasicrystalline domain. In the pattern, all the spots are located very close to exact fivefold symmetry positions, showing no linear phason strain. The frozen linear phason strain can be relaxed

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FIGURE 20. (a) Conventional transmission electron micrograph showing the growth morphology of an icosahedral quasicrystal in the melt-quenched A174Mn2[&,alloy. Black arrows show growth directions of the quasicrystallinc domains. (h) High-resolution lattice image of the rectangle region in (a). A large black arrow shows the growth direction. Parts (c) and (d) are optical diffractograms taken from regions C and D in part (h).

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63

FIGURE21. Schematic illustration of displacements of lattice planes (lines of bright dots) in region C of Fig. 20b.

and virtually disappears by annealing at high temperatures for stable quasicrystal phases (Hiraga, 1989; Guryan et al., 1989). Figure 22 shows electron diffraction patterns of an as-cast Al-Ru-Cu and annealed Al-Ru-Cu icosahedral phases at about 850°C. In Fig. 22a, the diffraction spots are seen to be systematically displaced from the icosahedral symmetry positions. That is, we can apparently see zigzag arrays of spots, particularly weak spots, by viewing along the J and K directions and along the I and L directions with smaller amounts of displacements. In contrast, the diffraction spots located on the lines parallel to the H direction show a straight-line array. The displacements can also be seen as the deformation of pentagons formed by the diffraction spots, particularly for small pentagons. The displacements of diffraction spots in Fig. 22a disappear completely in the diffraction pattern of Fig. 22b. That is, the frozen linear phason strain in the ascasted alloy is perfectly relaxed and disappears by annealing at a high temperature of about 850°C. The diffraction spots displaced from the icosahedral symmetry positions in Fig. 22a are extremely sharp. The result shows that the linear phason strain exists as homogeneous strain with long-range correlation.

2. Dislocations The existenceof dislocations in icosahedralquasicrystals was first found by observations of high-resolution lattice images of the Al-Mn-Si icosahedral phase (Hiraga and Hirabayashi, 1987b). Figure 23a and b are high-resolution lattice images showing dislocations. We may determine the Burgers vector on the plane perpendicular to the fivefold symmetry axis by counting the number of lattice fringes around the dislocations with two types of fringe distances, a’, and d,, in Fig. 23. In Fig. 23a, lattice fringes along the A and B directions are shown by black and white lines, respectively. These fringes make closed circuits STUS and PQRP surrounding a dislocation core, where TU and QR coincide with the directions A and B, respectively. The difference in the number of lattice fringes along the A

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FIGURE 22. Electron diffraction patterns of the Al-Ru-Cu icosahedral quasicrystals in an as-cast alloy (a) and annealed alloy at 8SO"C (b), taken with the incident beam parallel to the fivefold symmetry axis. Note systematic displacements of diffraction spots in part (a) by viewing obliquely along thc H. L, J, K, and L directions, and deformed pentagons of weak diffraction spots. The displacements disappear completely by annealing at about 8SO"C, in part (b). (Source: Hiraga rt d . , 1989a.)

direction between the paths from S to T and from S to U is measured as lOd, - 6 4 , whereas that along the B direction between the paths from P to Q and from P to R is 2 4 - 3ds. The number of lattice fringes in the upper side is always larger than that in the lower side. Taking account of the relation d, = rd,$,the differences in fringe numbers, A , along the A , B , C, D,and E directions may be determined respectively as

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65

FIGURE23. High-resolution lattice images of the Al-Mn-Si icosahedral quasicrystal showing the existence of dislocations. Lattice fringes with distances d/ and d, are marked by dark and white lines to count the number of fringes along the burgers circuits around dislocation cores. (Source: Hiraga and Hirabayashi, 1987b.)

66

KENJI HIRAGA

In Fig. 23b there are two dislocations, X and Y ,whose Burgers vectors have the same magnitude with opposite signs. This is evident from the fact that the difference in fringe numbers between H-1 and J - I around dislocation X is (5 - 3 ~ ) d , ~ , whereas that between I-K and I-L around dislocation Y is (5 - 3r)ds. The differences in fringe numbers along the five directions around Y may be measured as follows: A A = A B = (5 - 3t)d,$= t P 4 d s , A c = A E = (5r - 8)d, = r - ’ A A . A D = 0.

This type of dislocation is similar to that predicted by Levine er al. (1986) in the context of their density-wave description of icosahedral quasicrystals. Equations (1) and (2) describe the Burgers vectors on the plane perpendicular to the fivefold symmetry axis. AE = AA = (5 - 3 ~ ) d , ~ , A s = A D = (5T - 8)d,y, A c = 0.

(3)

are derived from Eq. (2) by rotating the axis by an angle of r/5. The sum of Eqs. (2) and (3) actually corresponds to the relations of Eq. (1). This fact implies that the dislocation in Fig. 23a is composed of two elemental dislocations with the equivalent Burgers vectors; the cores of the two dislocations are barely distinguishable, with bright and dark diffraction contrast.

VII. STRUCTURE OF DECAGONAL QUASICRYSTALS A N D THEIRRELATED CRYSTALLINE PHASES Since the discovery of decagonal quasicrystals by Bendersky (1985) and Chattopadhyay er al. (1985) in rapidly solidified Al-rich manganese alloys, many decagonal phases have been found as metastable or stable phases in Al-based binary and ternary alloys. High-resolution electron micrographs of decagonal quasicrystals, taken with the incident beam parallel to the periodic axis, are easily interpreted, compared with those of icosahedral quasicrystals, because of periodic structures along the incident beam, and they make it possible to determine two-dimensional quasiperiodic structures directly. Current high-resolution electron microscopy studies of the decagonal quasicrystals have shown that their structures are interpreted as two-dimensional quasiperiodic arrangements formed with definite linkages of large columnar atom clusters having decagonal symmetry. Also, it was found that there are not only various sizes of the atom clusters, but also various tilings of the atom clusters. In this

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67

(a) (h) FKilIRE 24. Framework of a decagonal atom cluster (a) and its linkages (b).

Section I present detailed results for some decagonal quasicrystals and to discuss structural characteristics of all the decagonal quasicrystals.

A . Framework of Columnar Atom Clusters Figure 24 shows the framework of the projection of the columnar atom cluster and its linkages. The basic framework is formed with two types of rhombic tiles, called fat and skinny rhombuses, with an edge length of 0.25 nm. The atom cluster is divided into a decagon and 10pentagons (labeled D) with an edge length of 0.47 nm. In the framework, decagons of three sizes A, B, and C, inflated with r scaling, are shown. The three decagons and the D pentagon are fundamental atomic clusters forming the structures of the decagonal quasicrystals and crystalline approximants, and they are joined by the edge-sharing linkage L and interpenetrating linkage S , as shown in Fig. 24b. The bond distance L is about 2 nm and S is 1.2 nm (= L / r ) for the A-sized decagon. The interpenetrating linkage S for the A decagon corresponds to the edge-sharing linkage L of the smaller decagon B, as shown in Fig. 24b. Structures of decagonal quasicrystals and their related crystalline phases can be described as tilings of the A-, B-, and C-sized decagons, and the D pentagons, with the edge-sharing and interpenetrating linkages. For example, the structure of an Al-Pd-Mn decagonal quasicrystal with 1.2-nm periodicity is formed by the L linkage of the A decagonal clusters (Hiraga and Sun, 1993a), whereas that of an Al-Ni-Co decagonal quasicrystal with 0.4-nm periodicity is described by the L and S linkages of the A decagonal clusters (Hiraga et al., 1991a; Hiraga et al., 1994a). A structure formed by the L and S linkages of the B decagonal clusters was found in an Al-Ni-Fe decagonal quasicrystal with 0.4-nm periodicity (Hiraga er al., 1996), and that by the L linkage of the smallest C decagonal clusters is in an A1-Pd decagonal quasicrystal with 1.6-nm periodicity (Hiraga et al., 1994b). Details of those structures is given below.

68

KENJI HIRAGA

B . Decagonal Quasicrystals with 0.4-nmPeriodicity 1. Structure of Al-Ni-Co Decagonal Phases A cluster model of the decagonal quasicrystals was first established by highresolution observations of stable Al-Ni-Co decagonal phases with 0.4-nm periodicity, and an atomic arrangement in the clusters was discussed (Hiraga et al., 199 1a). Figure 25 shows the structure model for the columnar atom cluster in the Al-Ni-Co decagonal quasicrystal and a high-resolution image calculated from the model. The structure model consists of two layers, z = 0 and z = stacked along the periodic axis, and atomic arrangements in the two layers are rotated by n radian relative to each other. One notices in the model that all atoms are located at comers of two types of rhombic tiles in the A-sized decagonal cluster. The decagonal atom cluster can be divided into a decagon and 10 pentagons, with an edge length of a = 0.47 nm. Atomic arrangements in the 10 pentagons and at the center of the decagon are described as an atomic column of close-packed pentagonal atomic arrangements, as shown in Fig. 26.

i,

FIGURE 25. Structure model and calculated high-resolution image of a decagonal atom cluster in the Al-Ni-Co decagonal quasicrystal. (a) Atomic arrangement in a z = 0 layer. Open and filled circlcs are Al and transition metals. Meshed circles are mixtures of Al and transition metals. (b) Projection of two layers of z = O and z = f . (c) Image calculated from the structure model under a condition o f 4 nm thickness and 45 nm under focus.

26. Pentagonal atom column located at the center and in pentagons in Fig. 2.5 FIGURE

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69

The calculated high-resolutionimage of the atom cluster is easily understood by consideringthat high-resolution structure images taken from thin regions represent atom positions as dark regions and channels without atoms as bright regions. A projected atomic arrangement inside the decagon with an edge length of 0.47 nm is represented as a wheel-like contrast distribution, and atomic arrangements in the 10 pentagons produce ring contrast distributions consisting of a dark ring surrounding a bright ring and a central dark dot in the high-resolution image of Fig. 2%. The wheel-like contrast and ring contrast distributions are henceforward called wheel-like contrast and ring contrast. For the Al-Ni-Co decagonal quasicrystals, there exist two types of decagonal phases, which appear in different temperature ranges. A high-temperature phase appears in an A170Ni15Co15 alloy annealed at 800°C and then quenched in water, and a low-temperature phase in the alloy annealed at about 550°C (Hiraga et al., 199 1a). Figure 27a shows a high-resolution structure image of the high-temperature phase of the Al-Ni-Co decagonal quasicrystal, taken with the incident beam parallel to the periodic axis. In the image one can see image contrasts corresponding to the atom clusters, which are characterized by a central wheel-like contrast and 10-ring contrast, indicated with black circles. From the observed image, one can determine the arrangement of the ring contrasts and wheel-like contrasts directly and then form an arrangement of the framework of decagons and pentagons with an edge length of 0.47 nm, shown in Fig. 27b. Figure 27b is an illustration of the framework of the decagons and pentagons, obtained from Fig. 27a. In the illustration one can see that most of the atom clusters, which consist of 10 pentagons surrounding a decagon, are joined by sharing two pentagons, namely, the L linkage with a distance of 2 nm in Fig. 24b, and some are joined by interpenetrating with a distance of 2 / nm ~ = 1.2 nm (i.e., the S linkage). Figure 28a shows a high-resolution structure image of the low-temperature phase. In the image one can see contrast distributionsof the atom clusters similar to those seen in Fig. 27a, but most of the atom clusters are deformed from a decagonal arrangement, as indicated with black circles in Fig. 28a. One can clearly see that the deformation results from a rhombic tiling of an atom cluster arrangement in Fig. 28b, which is a schematic illustration of a framework of the decagons and pentagons, obtained from Fig. 28a. In the illustration one notices that the atom clusters are joined by sharing two pentagons, but most of the decagonalframeworks are broken by the appearance of a rhombic tiling, indicated with dashed lines. That is, the rhombic tiling produces a different linkage with a short diagonal distance [4sin(rr/5) nm] of the fat rhombus. This linkage, which does not exist in the pentagonal tiling, results in partially deformed atom clusters. It should be noted here that the clear image contrasts corresponding to large atom clusters in Figs. 27a and 28a show the solid ordered atomic arrangements in the atom clusters, without any disordering along the periodic axis. The existence

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KENJI HIRAGA

FIGURE27. (a) High-resolution structure image of a high-temperaturephase in an Al7oNi 1sCols alloy annealed at 8W'C for I h and then quenched in water; (h) framework of decagons and pentagons, obtained from an arrangement of the ring contrasts (indicated with hlack circles) in Part (a). Tiling of the atom clusters is shown by dotted lines. (Source: Hiraga et d., 1994h.)

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71

FIGURE 28. (a) High-resolution structure image of a low-temperature phase in an A170Ni15Col5 alloy annealed at 550°C for 72 h; (b) framework of decagons and pentagons, obtained from an arrangement of the ring contrasts (indicated with black circles) in part (a). Tiling of the atom clusters is shown by dotted lines. Note the deformed decagons, resulting in rhombic tiling of the atom clusters. (Source: Hiraga ei d., 3994b.)

72

KENJI HIRAGA

of the strong correlation along the periodic axis, which can generally be observed in all decagonal quasicrystals, rules out the random tiling model, in which entropy, resulting from the randomness, stabilizes quasicrystals. Because Burkov (1991) showed that the entropy term dominates the energy term in the decagonal quasicrystal if the layers perpendicular to the periodic axis are uncorrelated. The solid ordered atomic arrangements in the atom clusters are common to all decagonal quasicrystals, and some structural randomness from the ideal pentagonal lattice appears in two-dimensional arrangements of the atom clusters in a decagonal quasicrystal. The local structures in Al-Ni-Co quasicrystals were determined from the structure images, whereas arrangements of the atom clusters in wide regions can be obtained from observed lattice images. Figure 29 shows high-resolution lattice images of the high-temperature (a) and low-temperature (b) Al-Ni-Co decagonal phases taken from thicker regions (a few tens of nanometers) than those of Figs. 27a and 28a. In the images, one notices the characteristic contrast of a bright ring with a central bright dot. The bright ring represents a decagon placed at the center of the atom cluster in Fig. 25. From the arrangements of the ring contrasts in Fig. 29, one can determine the arrangements of atom clusters in wide regions. Figure 30a shows a schematic arrangement of the atom clusters obtained from a wider region of Figs. 29a and 30b is a tiling obtained by connecting the centers of atom clusters in Fig. 30a by bonds of length 2 nm. Gaps in some parts of the tiling are due to indistinguishable rings, which result from the lack of correlation along the direction perpendicular to the image plane. The tiling of Fig. 30b appears to be considerably disordered from an ideal quasicrystalline lattice, but the atom clusters arrayed with strong long-range correlation can be seen by obliquely viewing Fig. 30a along the fivefold directions. Some arrays of atom clusters are on lines longer than 100 nm. This highly ordered arrangement of the atom clusters can be seen more clearly in Fig. 30c, where the positions of the atom clusters are projected on the internal space in the manner noted in Section 1I.B. Most of the atom cluster positions are inside the decagonal window that forms the pentagonal Penrose lattice, corresponding to the A decagon shown in Fig. 3d. On the other hand, the low-temperature Al-Ni-Co decagonal phase cannot be said to be a highly ordered quasicrystal, as can be seen in an arrangement of atom clusters and tiling in Fig. 3 1, which were obtained from the lattice image of Fig. 29b. Compared with Fig. 30a, the arrays of atom clusters in Fig. 31a show frequent displacements, and the atom cluster positions projected on the internal subspace (Fig. 3 lc) are remarkably scattered outside the decagonal window. Also, the directional distribution in the internal subspace shows the existence of linear phason strain. Although the two tilings obtained, Figs. 30b and 31b, in the high- and lowtemperature Al-Ni-Co decagonal phases are remarkably different from the pentagonal and rhombic tilings of Figs. 2b and 3a, they have the characteristics of

FIGURE 29. High-resolution lattice images of the high-temperature phase (a)and low-temperature phase (b) of the Al-Ni-Co icosahedral quasicrystals. The images were observed from relatively thick regions to enhance bright rings showing positions of the decagonal atom clusters.

74

KENJl HIRAGA

.,

,'................................... .',.::.:.*:.*..' ..: ........ . . . . :.. . . . . :.. ..:...... :.:,. .............. .. ..

. . . .... . . . .. *

FIGURE30. (a) Positions of the decagonal atom clusters, obtained from Fig. 29a. (b) Tiling constructed by the connection of the cluster positions in part (a) with bonds of 2.0-nm length. (c) Distribution of the cluster positions on the internal subspace and the decagonal window to construct a pentagonal Penrose lattice.

pentagonal and rhombic tilings. Consequently, the different tilings of pentagonal and rhombic atom clusters, suggest that different types of diffraction patterns will be seen for the high- and low-temperature phases. Figures 32a and b show electron diffraction patterns of the high- and lowtemperature Al-Ni-Co decagonal phases taken with the incident beam parallel to the periodic axis. The electron diffraction pattern of the high-temperature phase in Fig. 32a shows a number of weak spots, indicating a highly ordered quasicrystal. It should be noted that the distribution of diffraction spots is similar to diffraction patterns of icosahedral quasicrystals, except for the relative intensity of diffraction spots, as can be seen in Fig. 4b. On the other hand, for the pattern

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRYSTALS

c

-: .. -. .

75

..

FIGURE 31. (a) Positions of the decagonal atom clusters, obtained from Fig. 2Yb; (b) tilings constructed by the connection of the cluster positions in pan (a) with bonds of 2.0-nm length: (c) distribution of the cluster positions on the internal subspace and the decagonal window to construct a pentagonal Penrose lattice.

of the low-temperature phase in Fig. 32b, the intensity distribution is strikingly different from that in Fig. 32a. In the pattern one can see extra spots, which are located at different positions from the diffraction spots in Fig. 32a, as well as a great change of the intensity distribution. The most characteristic extra spots can be seen at the centers of pentagons, as indicated by arrows in Fig. 32b. Features of the two types of diffraction patterns result from the difference in the tilings of atom clusters in the high- and low-temperature phases, as shown in Figs. 30b and 3 1b. The different features in the diffraction patterns can be reproduced in terms of the rhombic and pentagonal Penrose tilings constructed using the projection method (Hiraga et al., 1994a). In Fig. 32c and d, diffraction patterns obtained by placing atoms at each comer of tiles in Figs. 3a and 2b, respectively, are shown. Apart from intensities of diffraction spots, one can see that the calculated diffraction patterns of Figs. 32c and d represent well the features of the observed patterns of Fig. 32a and b, respectively. For example, the extra reflections located at the centers of pentagons in Fig. 32b can be seen in Fig. 32d, as indicated by

16

KENJl HIRAGA

c. .

a

0

.

. a 0

0 - 0 0 .

.

.. 0

0

. O

0

FIGURE32. Electron diffraction patterns of the high-temperature phase (a) and low-temperature phase (b) of Al-Ni-Co icosahedral quasicrystals. Diffraction patterns in parts (c) and (d) are made by placing atoms at vertices of the pentagonal Penrose lattice (Fig. 3a) and of the rhombic Penrose lattice (Fig. 2b). respectively. (Sorrrcr: Hiraga P I a/., 1994a.)

arrows. The extra reflections appearing for the rhombic tiling result from neighboring pairs, which do not exist in the pentagonal tiling, corresponding to a short diagonal distance of the rhombic tile. The neighboring bond vectors are characterized by vectors described as the sum u; + u j , where ui and u j are unit vectors forming nearest-neighbor pairs in the pentgonal and rhombic tilings, and the extra reflections are found to be indexed by the sum u; u, (Edagawa et al., 1992). It should be emphasized here that the extra reflections in the low-temperature phase are caused by the appearance of the new bond relationship in the rhombic tiling and are not due to the ordering of an atomic arrangement. Also, structures of Al-MnSi (Hiraga et al., 1987) and Al-Fe (Fung et al., 1986) metastable decagonal quasicrystals may be characterized by the rhombic tiling (Hiraga, 199lc; Yamamoto and Ishihara, 1988). It should also be noted that the pentagonal tiling of Fig. 3a is similar to that of the projection of the three-dimensional Penrose tiling along the fivefold symmetry axis in Fig. 14a. The fact shows that the diffraction pattern of Fig. 32a resembles those of the icosahedral quasicrystals (Fig. 5c).

+

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRYSTALS

77

Phase transformationof decagonal quasicrystalswas also found in an Al-Cu-Co decagonal phase with 0.4-nm periodicity. In this case, a high-temperaturephase is a high-quality decagonal quasicrystal characterized by a pentagonal tiling, and it is transformed to a microcrystalline structure based on rhombic tiling by annealing at 550°C for 46 h (Hiraga et al., 1991b; Hiraga, 1991~).It should be mentioned here that a later study of the Al-Cu-Co decagonal quasicrystal by He et al. (1991) could not find a transformation in the same composition alloy. However, their quasicrystal of a high-temperature phase cannot be said to be a high-quality decagonal quasicrystal and already shows features of a rhombic tiling partially. The result shows that the high-quality Al-Cu-Co quasicrystal is formed under strict conditions, particularly under a sharp stoichiometric composition, and only that undergoes a clear structure change. 2. Al-Ni-Fe Decagonal Phase Figure 33 shows a high-resolution structure image of the Al-Ni-Fe decagonal quasicrystal, taken with the incident beam parallel to the periodic axis. In the image one can see an arrangement of the wheel-like contrasts with a diameter of about 1.2 nm, as indicated by dark circles. The atom clusters are connected by edge-sharing and interpenetrating linkages, which have 1.2- and 0.76-nm bond lengths, respectively. Also, in the gaps between the wheel-like contrast features, ring contrast and bright spots with a triangular shape, indicated by small arrows, can be recognized. A preliminary atomic model of the Al-Ni-Fe decagonal phase was proposed on the basis of the structure of the Al-Ni-Co decagonal quasicrystal for the decagonal atom cluster and for the occasional gaps (Hiraga et al., 1996). This model is shown in Fig. 34a. in which the atomic arrangement of the decagonal atom clusters is similar to that of the inner part of the large decagonal atom cluster in the Al-Ni-Co decagonal quasicrystal (Fig. 25). The atomic arrangements in the gaps were chosen to be consistent with the observed image contrast of Fig. 33. Figure 34b shows a computer simulation image calculated from the atomic model, in which the rectangular region of Fig. 34a is shown in a periodic arrangement. The calculation was made assuming a defocus value of -45 nm and a sample thickness of 3 nm at an accelerating voltage of 400 kV. The calculated image can be seen to reproduce well the characteristic features of the wheel-like contrast of the decagonal atom clusters and the triangular bright spots and a ring contrast inside the gaps in the observed image of Fig. 33. In the structure model, all atoms are located at ideal positions (i.e., at the vertices of a rhombic tiling) and at special positions associated with the golden ratio. Although the calculated image of the structure model corresponds well to the observed image, it should be noted that it is impossible to determine precise atomic positions and to distinguish clearly between A1 and transition metal atoms from the observed high-resolution

78

KENJl HIRAGA

FIGURE 33. High-resolution structure image of an AI-Ni-Fe decagonal quasicrystal, taken with the incident beam parallel to the tenfold symmetry axis. The atom clusters, forming structures of the decagonal quasicrystals, are indicated by black circles. (Source: Hiraga er a / . , 1996.)

image. This implies that detailed structure analysis must be ascertained in the future by single-quasicrystalx-ray diffraction. I will now discuss a tiling showing an arrangement of decagonal atom clusters. Figure 35a is a tiling formed by connecting the centers of the decagonal atom clusters in a wider region, about 35 x 35 nm, than that of Fig. 33. In the tiling, the bond length is 1.2 nm, at which the B-sized decagonal atom clusters in Fig. 24a touch each other directly; and the short distance of 0.76 nm, which corresponds to the interpenetrating linkage, also appears in several places. This experimental tiling certainly shows characteristic features of the real arrangement of the atom clusters in the Al-Ni-Fe decagonal quasicrystal. The tiling obtained can be said to resemble the pentagon Penrose tiling formed by regular pentagons and thin rhombuses, although there are also a few fat rhombuses and irregular hexagons. The distribution of atom clusters on the internal subspace (Fig. 35b) shows that the Al-Ni-Fe decagonal phase is a highly ordered quasicrystal, similar to the Al-Ni-Co high-temperature decagonal phase. The highly ordered Al-Ni-Fe decagonal phase has a sharp stoichiometric composition of about Al72Ni24Fe4. Besides the decagonal phase, metastable Al-Ni-Fe decagonal quasicrystals are

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRYSTALS

79

FIGURE34. (a) Atomic arrangement in decagonal atom clusters and in some gaps. Large and small circles shows atom positions on 7 = 0 and z = planes, respectively, being perpendicular to the tenfold axis. Open and filled circles correspond to Al and transition metal atoms, respectively. One decagonal atom cluster is utlined by solid lines. (b) High-resolution image calculated from periodic arrangement of the atomic model (a). A rectangle with black lines in (b) corresponds to a rectangle unit of (a). (Source: Hiraga el d.,1996.) b

..

. . , a

..*..

.*

. .. .. .. ... .... ... FIGURE 35. (a) Tiling of decagonal atom clusters, obtained by connecting centers of the wheellike contrast features in Fig. 33; (b) distribution of atom cluster positions, projected onto the internal subspace, and a decagonal window to construct the pentagon Penrose tiling. (Source: Hiraga et al., 1996.)

found in rapidly solidfied alloys with a wide composition range, and most of them were found to have noncentral symmetry, as noted in the next section (Tanaka et al., 1993).

3. Crystalline Approximant Phase It is well known that there exist some crystalline phases around quasicrystalline alloys. They have local atomic arrangements similar to those in quasicrystals, so

80

KENJl HIRAGA

they are called crystalline approximants. Thus structuresof the crystalline approximants are considered to be an important key for understanding the local structures of quasicrystals. Here I mention a crystalline phase, which seems to be important for an interpretation of some unsolved points in the decagonal quasicrystals with 0.4-nm periodicity. The crystalline phase having about A173C023Pb composition is called a W-phase (Yubuta et al., 1996). Figures 36a-c show electron diffraction patterns of the W-phase in the as-melted A173Co22Pd5 alloy, taken with the incident beam parallel to the three principal axes, together with corresponding patterns of the decagonal quasicrystal formed in the melt-quenched A173Co22Pd5 alloy. In the diffraction patterns, one can see that the characteristicsof intensity distributions in those of the W-phase are similar to those for the decagonal quasicrystal, although positions of individual diffraction spots are, of course, different in the two diffraction patterns. Also, one should note the pseudo-fivefold and fivefold symmetric distributions of diffraction spots in Figs. 36a and d, in which relatively strong fivefold spots are indicated by small arrows. The fivefold and pseudo-fivefold symmetric distributionsshow a departure from Friedel’s law, for which the intensity of F(hkl) is equal to that of F(-h k - 1 ) in a kinematical approximation. The departure from Friedel’s law results from the dynamic effect of the electron beam in thick regions and indicates the existence of noncentral symmetry in these structures. Also, in general, diffraction patterns of noncentral symmetric decagonal quasicrystals are characterized by the appearance of diffraction spots on the lines indicated by large arrowheads in Fig. 36f (Saito et al., 1992). In particular, the appearanceof the spot indicated by a small white arrow in Fig. 36f is an important characteristic in an understanding of the structure of noncentral symmemc decagonal quasicrystals, as discussed later. Those spots in Fig. 36f disappear by the extinction rule, which corresponds to the space group P lOs/mmc in the central symmetric decagonal quasicrystals with 0.4-nm periodicity, as can be seen in Fig. 7. This feature of the diffraction patterns of noncentral symmetric decagonal quasicrystals in Fig. 36f can also be seen in the pattern of Fig. 3612. In many decagonal quasicrystalswith 0.4-nm periodicity, particularlyin noncentral symmetric phases and/or metastable phases formed in rapidly solidified alloys, diffuse scattering on the lines between Bragg reflection planes showing 0.4-nm periodicity has been observed, as indicated by small arrowheads in Figs. 35e and f, although it tends to become weak or disappear in good-quality decagonal phases (Hiraga et al., 1991a). A general feature of diffuse scattering is that lines in a diffraction pattern taken in the direction shown in Fig. 36f are stronger than those in a pattern taken in the direction shown in Fig. 36e. This feature of the diffuse scattering can also be seen in the distribution of diffraction spots in the W-phase patterns in Fig. 36b and c. From the analogy between the diffraction patterns of the W-phase and decagonal quasicrystalsmentioned above, it can be concluded that the W-phase is an important

FIGURE36. Electron diffractionpatterns of the w-phase in the as-melted A173C022Pd~alloy, taken with the incident beam parallel to the a-axis (a), b-axis (b), and c-axis (c), and those of the decagonal quasicrystal in the rapidly solidified A173C~2Pdsalloy, taken with the incident beam parallel to the periodic axis (d), the p direction (e), and the q direction (0,indicated in (d). (Source: Yubuta er 01.. 1996.)

82

KENJI HIRAGA

crystalline approximant to an understanding of the structure of noncentral symmetric decagonal quasicrystals with 0.4-nm periodicity and also to a discussion of the origin of the diffuse scattering often observed in the decagonal quasicrystals with 0.4-nm periodicity. The W-phase has an orthorhombic structure with lattice parameters a = 0.82 nm, b = 2.06 nm, and c = 2.35 nm and the space group Pmn 21. It should be mentioned here that in diffraction patterns of the W-phase, diffuse scattering along the b*-axis has frequently been observed, as can be seen in Fig. 36a and c. Figure 37 shows a high-resolution structure image of the W-phase, taken with the incident beam along the a-axis (i.e., the pseudo-fivefold axis). In the image one can notice periodic zigzag arrays of large wheel-like contrasts and small ring contrasts, as indicated by arrowheads and small arrows, respectively. Each of the ring contrasts is surrounded by five triangular bright dots, so a star-shaped contrast is formed. The features of the image contrasts can be seen clearly by comparison with the inserted calculated image, as noted later. A unit cell of the structure of the W-phase is indicated by a rectangle in the middle of Fig. 37. In the upper part of Fig. 37, one can see a defect region consisting of straight arrays of three wheel-like contrasts instead of the zigzag arrays, as indicated by arrowheads. In the defect, one can draw an orthorhombic unit cell with lattice parameters of 1.2 and 3.9 nm, as outlined by lines in the upper part of Fig. 37. The partial appearance of the straight arrays in the zigzag arrays of the wheel-like contrasts as defects causes diffuse scattering along the b*-axis in the diffraction patterns of Fig. 36a and c. and shows the possibility of formation of a crystalline phase with the orthorhombic structure. From the topological features in the observed high-resolution image of Fig. 37, we can propose structural frameworks of the W-phase and the defect region, as shown in Fig. 38. The frameworks are formed with decagons and pentagons with an edge length of 0.48 nm, which correspond to the wheel-like contrasts and ring contrasts in the observed image of Fig. 37, respectively. The frameworks of the decagons and pentagons resemble those of Al-Ni-Co and Al-Ni-Fe decagonal quasicrystals. Consequently, the structure determination of the W-phase is expected to give us valuable information to interpret local atomic arrangements in the decagonal quasicrystals with 0.4- and 0.8-nmperiodicities, and to understand the reason for the diffuse scattering observed in the decagonal quasicrystals with 0.4-nm periodicity. A provisional structure model of the W-phase was proposed on the basis of an ordered arrangement of transition metal atoms, and the structure of the decagonal quasicrystals with 0.4-nm periodicity was discussed, based on the model, in the paper by Yubuta et al. (1996). However, the structure model is not discussed in this paper, because detailed structure analysis of the W-phase by x-ray diffraction with a single crystal is now in progress.

FIGURE37. High-resolution structure image of the w-phase, taken with the incident beam parallel to the a-axis. An image calculated from a structure model is inserted. In the main part, one can see zigzag arrays of wheel-like contrasts, indicated by arrowheads, and small ring contrasts, indicated by small arrows, along the h-axis. A unit cell of the w-phase is drawn by a rectangle in the middle part. In the upper part one can see a defect consisting of straight arrays of three wheel-like contrasts, indicated by arrowheads, instead of their zigzag arrays in the w-phase. In the defect, a unit cell outlined by a rectangle can be taken. (Source: Yubuta et a/., 1996.)

84

KENJI HIRAGA

a

FIGURE 38. Frameworks of decagons and pentagons in the w-phase and defcct region. obtained from Fig. 37. It should be noted that the origin of the unit cell in part (a) is shifted along the h- and c-axes from that of the unit cell in Fig. 37.

C . Decagonal Quasicrystals and Crystalline Phases with 1.2-nm Periodicity 1. Fundamental Structure Units

Structures of the Al-Pd-Mn decagonal quasicrystal and its related crystalline phases with 1.2-nm periodicity can be interpreted in terms of two-dimensional arrangements formed with a definite linkage of two types of fundamental atom columns, which have a twofold screw relationship, as shown in Fig. 39a and b (Hiraga, 1996). The atom columns are composed of the pentagonal atom column of Fig. 39c, which is formed with stacking of pentagonal arrangements of atoms and central atoms along the columnar axis, and decagonal atom rings surrounding the pentagonal atom column. The atomic arrangements in the atom columns were determined from structure analysis of the A13Mn phase by single-crystal x-ray diffraction (Li and Kuo, 1992a; Hiraga et al., 1993). The atom columns are connected by edge sharing of pentagons with an edge length of 0.47 nm, as shown in the projected atomic arrangement at the bottom of Fig. 39a and b, and make up two-dimensional arrangements. Here it should be noted that the pentagon with a 0.47-nm edge length has the same size as that of the D pentagon in Fig. 24a. Figure 40 shows some structure units, which are important to an understanding of the structures of the decagonal and crystalline phases, formed by the edge sharing of the pentagons of the atom columns. The hexagonal (H), star-shaped pentagonal (P), decagonal (D), and ship-shaped octagonal (0)units with an edge length of 0.65 nm are the ones that form aperiodic or periodic tilings in the decagonal and

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRYSTALS

85

T

19nm

7 nm FIGURE39. (a) and (b) Fundamental atom columns in decagon: .psicryst; and crystalline phases with 1.2-nm periodicity and projected atomic arrangements along the columnar axis in the bottom side; (c) pentagonal atom column inside the (a) column. Open and filled circles are Al and transition metals, respectively. (Source: Hiraga, 1995.)

0.65 nm

1 H-unit

P-unit

D-unit

e 0-unit

FIGURE 40. Four units formed with edge sharing of pentagons. The edge lengths of the pentagons and units are 0.47 and 0.65 nni. respectively.

crystalline phases (Hiraga, 1996). Exact atomic arrangements in the H, P, and D units can be seen in Fig. 43 (Hiraga and Sun, 1993a).

2. Structure of Al-Pd-Mn Decagonal Quasicrystal Figure 41 is a high-resolution structure image taken with the incident beam parallel to the periodic axis. In the image one can see small ring contrasts consisting of a dark ring surrounding a bright ring and a central dark dot. The ring contrasts correspond to the projection of the atom columns of Fig. 39 along the columnar axis. An arrangement of the ring contrasts in the image is drawn schematically in Fig. 42. The ring contrasts form decagons, star-shaped pentagons, and squashed hexagons, which correspond to the D, P, and H units, respectively, in Fig. 40. All the decagonal atom clusters are joined with a definite linkage (i.e., by sharing two ring contrasts), and gaps in an arrangement of the decagonal clusters are perfectly filled

86

KENJI HIRAGA

FIGURE 41. High-resolution structure image of the Al-Pd-Mn decagonal phase, taken with the incident beam parallel to the periodic axis. An image calculated from an atomic arrangement in a decagonal atom cluster (Fig. 43) is inserted.

FIGURE 42. Schematic illustration showing an arrangement of the ring contrasts (small circles) and decagonal atom clusters (large circles), obtained from Fig. 41.

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRY STALS

0.:2=3/4

0:2 4 . 8 2

87

.o:Z10,56

FIGURE43. Atomic arrangement in decagonal, pentagonal, and hexagonal units in Fig. 40. Open and filled circles are A1 and transition metals, respectively. (Source: Hiraga and Sun, 1993a.)

up with the star-shaped pentagons and hexagons, without any overlaps and without gaps, as can be seen in Fig. 42. Thus the determination of atomic arrangements in the three polygons leads to a solution for the structure of the Al-Pd-Mn decagonal quasicrystal. Figure 43 is a structure model in the D, P, and H units, which was proposed from the high-resolution structure image of Fig. 41 with the aid of computer simulation as well as from the structure of the AI3Mn crystalline phase. The atomic arrangements in the P and H units are unequivocally determined by placing the atom columns of Fig. 39 in the pentagonal framework, and the atomic arrangement near the center of the D-unit, which remained ambiguous, was proposed from the observed high-resolution structure image. In Fig. 43, atomic arrangements only on the layers from z = f to z = perpendicular to the periodic axis are drawn, because layers of both z = f and z = are mirror planes. An image contrast of the decagonal atom cluster, calculated from the model, was inserted in the observed image of Fig. 4 1. One can see a good correspondence between the calculated and observed contrasts. The Al-Pd-Mn decagonal quasicrystal formed by annealing alloys around Al7oPdloMn2" composition has strong linear phason strain, because it grew up from crystalline phases (Hiraga et al., 1991c; Hiraga, 1993). Thus, to examine characteristics of a tiling of the decagonal atom clusters in a wide region for the Al-Pd-Mn decagonal quasicrystal with little linear phason strain, a high-resolution lattice image of the Al-Pd-Mn decagonal phase, which was partially grown from the icosahedral phase, was observed. Figure 44 is the lattice image of the Al-Pd-Mn decagonal phase grown from the icosahedral phase by annealing an Al7oPd2oMnlo alloy. The image was taken

i

i

88

KENJl HIRAGA

FIGURE 44. High-resolution lattice image of the Al-Pd-Mn decagonal quasicrystal, grown from an icosahedral phase, in an Al7oPdzoMnlo alloy annealed at 800°C for 56 h, taken with the incident beam parallel to the periodic axis. (Source: Hiraga and Sun, 1993b.)

with the incident beam parallel to the periodic axis. Ring contrasts in Fig. 44, which are slightly different in a thin region on the bottom side and in a thick region on the upper side, show the positions of the decagonal atom clusters. From the arrangement of the ring contrasts in Fig. 44,we can directly determine the arrangement of the atom clusters in a wide region, then form a tiling of the atom clusters, as shown in Fig. 45. A tiling constructed by connecting the L linkages of a bond length of 2 nm with lines is formed of many kinds of polygons in addition to pentagons, as shown in Fig. 45a. Here a few thin rhombuses are observed, but they may be

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRYSTALS

b

89

.. . . .. .. . . .. a

.

. . . FIGURE 45. (a) Tiling constructed by connecting ring contrasts (decagonal atom clusters) in Fig. 44; (b) distribution of the atom cluster positions on the internal subspace and a decagonal window to construct the pentagonal Penrose tiling.

considered as defects in this tiling. It should be noted that the tiling in Fig. 45a may also be represented as a space-filling tiling using three types of tiles: decagons (D),star-shaped pentagons (P), and squashed hexagons (H) (Hiraga and Sun, 1993b). From Fig. 45b, which shows the distribution of the atom cluster positions projected on the internal subspace, the Al-Pd-Mn decagonal phase is said to be a comparatively high-ordered decagonal quasicrystal with little phason strain. On the other hand, the Al-Pd-Mn decagonal quasicrystal grown up from crystalline phases was found to have strong linear phason strain because of the influence of directional structure in the crystalline phases. 3. Crystalline Approximant Phases Many periodic tilings can be formed with the structure units of Fig. 40. Figure 46 shows seven examples of periodic tilings, observed in crystalline approximants with 1.2-nm periodicity. Unit cells indicated by dotted lines in Fig. 46 can be estimated from the edge length of the D, P, and H units, 0.65 nm, as 0.76 nm and 2.35 nm (a), 1.48 nm and 1.25 nm (b), 2.38 nm and 2.00 nm (c), 2.38 nm and 3.28 nm (d, e). 2.0 nm and 6.1 nm (f), and 3.8 nm and 5.23 nm (g). The tiling of Fig. 46a, called a n-phase, is observed to coexist with the Al3Mn phase (Li and Kuo, 1994; Hiraga, 1995), and the tiling of Fig. 46b is in the Al3Mn phase. The tilings of Fig. 46c, d, and e. formed with the H and P units, are in the crystalline phase of AlCuFeCr (Li et al., 1995), AlCuFeCr (Li et al., 1995), and AlCrPd. called an 0 phase (Sun et al., 1995), respectively. Also, the tilings of

90

KENJI HIRAGA

a

b

C

d

e

1

0.65 nm FIGURE46. Periodic tilings of H. P, D, and 0 units in structures of some crystalline phases. (Source: Hiraga. 1995; Li and Hiraga. 1996.)

Fig. 46f and g, including the D units, are observed in Al-Pd-Mn alloys, as phases coexisting with the Al-Pd-Mn decagonal phases (Hiraga, 1995). Figure 47 shows high-resolution structure images of the ALMn phase and 0 phase. In the images one can clearly see tilings with an H unit (a), and H and P units (b), formed with ring contrasts corresponding to the projections of the pentagonal atom columns (Fig. 39). The image contrasts in Fig. 47 are in a good correspondence to the calculated images inserted. From this type of images, one can directly determine two-dimensional arrangements of atom columns. D . Decagonal Quasicrystal and Crystalline Phases with 1 h-nm Periodicity 1. Structure of Al-Pd Decagonal Phase

A fundamental atom column in the AI-Pd decagonal quasicrystal and its related crystalline phases with 1.6-nm periodicity was derived from the structure of the ALPd crystalline phase (Matsuo and Hiraga 1994), shown in Fig. 48a. The strata of the atom column of Fig. 48a are connected with edge sharing of the decagon with an edge length of 0.24nm, as shown in Fig. 48c, and make up two-dimensional tilings of decagons. Thus the tilings are formed with the L-type linkage of C-sized atom clusters, shown in Fig. 24.

FIGURE47. High-resolution structure images of the A13Mn phase (a) and 0-phase (b). Tilings of the H and P units are indicated by white lines. Calculated images from structure models are inserted in each image.

92

KENJI HIRAGA

b

a

C

Unm

I

f t

0.76nm FIGURE48. (a) Decagonal atom column forming structures of decagonal quasicrystals and crystalline phases with 1.6nm periodicity; (b) atomic arrangementson successive layers along the columnar axis. Open and closed circles correspond to A1 and transition metal atoms. Some polygons constructed by sharing an edge of the decagonal atom columns. Only atoms on the top layer are shown. (Source: Hiraga era/., 1994b.)

Figure 49 shows a high-resolution structure image of the A1-Pd decagonal quasicrystalin a melt-quenched Al3Pd alloy, taken with the incident beam parallel to the periodic axis. Bright-ring contrasts in the image correspond to the projection of the decagonal atom columns of Fig. 48a along the columnar axis. In the image the ring contrasts are arranged with an interval of 0.76nm. Figure 50a shows a schematic illustration of an arrangement of the decagonal atom columns, obtained from Fig. 49, and Fig. 50b is a tiling drawn by connecting the ring contrasts in Fig. 49 by bonds of 0.76-nm length, by ignoring the differences in contrast of the bright rings. In the illustrations one can see clearly an arrangement of the decagonal atom clusters and the manner of tiling. In Fig. 50 there are some areas with no distinguishablering contrasts. The absence of ring contrasts or weak ring contrasts may be caused by the lack of correlation of atomic arrangements along the incident beam.The sample thickness in the image of Fig. 49 is possibly about 10 nm, so disordering of atomic arrangements in the thickness results in weak contrasts or the absence of ring contrasts, because the quasicrystal was formed in a rapidly solidified alloy. However, most ring contrasts show apparent contrast, showing long-range correlation of atomic arrangements along the tenfold axis. Although there are some gaps due to the lack of ring contrasts in Fig. 50, one can see features of the arrangement of the decagonal atom clusters. Most of the atom clusters are on straight lines parallel to the tenfold directions, as can clearly be seen by viewing Fig. 50a obliquely along the fivefold directions. Almost all the

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRYSTALS

93

FIGURE 49. High-resolution structure image of the AI-Pd decagonal phase. Bright-ring contrasts correspond to the projection of the decagonal atom column of Fig. 48a along the columnar axis. (Source: Hiraga er a/., 1994b.)

atom clusters except for central ones of decagons are connected with a bond length of 0.76 nm. In the tiling of the atom clusters, there are many types of polygons (i.e., a pentagon, squashed hexagon, decagon, etc.). Atomic arrangements in the polygons can be interpreted by edge-sharing linkage of the decagonal atom columns, as in Fig. 48c. The Al-Pd decagonal quasicrystal is a metastable phase formed in a rapidly solidified alloy. Thus a distribution of decagonal atom cluster positions on the internal space is rather scattered compared with those of the Al-Ni-Co (Fig. 30c) and Al-Ni-Fe (Fig. 35b) decagonal phases, but it shows that the Al-Pd decagonal phase is a highly ordered quasicrystal with little linear phason strain. Although the tiling of Fig. 50b has some gaps with no atom clusters, it shows features of the pentagonal tiling of Fig. 3c.

94

KENJl HIRAGA

........................... ... a .......................... ........................... ................................ .... .. ............... ,.;.. ......... ......... ;. .. ;. ................... ;:':*. ................................ ........ ;:..................:. ;:. :. ........................................... ............................. ... ...................................... ..'......*'.*..'.::. ............... ... ................................ ..................................... .................................... .*.:. ;:........................ .................. :.....: ................. ....................................... .................................. ,; ................................ ................... ;:.:. ::.: ......;. .................................... ;,..,......,.,..... .................. ................................. ........................ :<*

.

C

.

:*

_

.. . . .............. *

.

FIGURE 50. (a) Schematic illustration of an arrangement of the decagonal atom clusters, obtained from Fig. 49; (b) tiling of the decagonal atom columns constructed by connecting the bright rings in Fig. 49 with lines; (c) distribution of the atom cluster positions on the internal subspace and a decagonal window to construct the pentagonal Penrose tiling.

2. Crystalline Approximant Phases Crystalline phases with simple structures formed with the decagonal atom column of Fig. 48a were found in conventionally solidified alloys around an A13Pd composition. Figure 5 1 shows high-resolution structure images of crystalline phases observed in A13Pd and A175Pd20Mn.5 alloys. Figure 5 la is the A13Pd phase and Fig. 51b is another crystalline phase. In the images the decagonal atom columns are represented as bright rings arranged with a nearest-neighbor distance of 0.76 nm. From the images one can obtain two simple tilings of decagons, as shown in Fig. 52. The atomic arrangement in Fig. 52a was determined by single-crystal x-ray diffraction. Besides the two simple structures, various types of arrangements with the edge sharing of pentagons can be formed. Actually, many types of structures were

HIGH-RESOLUTION ELECTRON MICROSCOPY OF QUASICRYSTALS

95

FIGURE5 I. High-resolution structure images of two crystalline phases with I .6-nm periodicity. The bright circles in the image correspond to the projection of decagonal atom columns of Fig. 48a along the columnar axis. (Source: Hiraga. 1995.)

M

W

t

t

0.76 nm

(a) (b) 52. Tilings of decagonal atom clusters obtained from Fig. 51. (Source: Hiraga. 1995.) FIGURE

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KENJl HIRAGA

(a) (b) (C) (4 FIGURE53. Periodic tilings with edge sharing of decagons obtained in some crystalline phases. (Source: Hiraga, 1995.)

observed as modulated structures of the AbPd phase in an Al-Pd-transition metal alloy system. Figure 53 shows five tilings of the modulated structures.These tilings show the possibility of appearance of various tilings with more complex links. VIII. CONCLUDING REMARKS

In this review, current results of our group about high-resolution electron microscopy studies on quasicrystals have been summarized. From the results it can be concluded that structures of icosahedral and decagonal quasicrystals can be described as aperiodic arrangements of some atom clusters, which have the same symmetry as those of the quasicrystals, with definite linkages. There is variety in aperiodic arrangement of the atom clusters and atomic arrangements in the atom clusters, depending on quasicrystalline alloys. High-resolution electron microscopy is the most powerful tool for studying the varied structures. However, it should be noted that high-resolution electron microscopy has many limitations for carrying out precise structure analysis about atomic arrangements, because of its limited resolution compared with diffraction and the use of images projected along the incident beam. To proceed to the more precise structure analysis of the quasicrystals,combining high-resolution electron microscopy with the diffraction method becomes more important. It should be mentioned that the atomic arrangements of Al-Ni-Co, Al-Pd-Mn, and Al-Pd decagonal quasicrystals presented in this paper were proposed using the results of x-ray diffraction. ACKNOWLEDGMENTS I sincerely thank Professors M. Hirabayashi, D. Shindo, M. Matsuo, and F, J. Lincoln, Drs. W. Sun and A. Yamamoto, and Mrs. E. Abe, K. Yubuta, and K.-T. Park for their cooperation in carrying out this work. This work has been supported

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by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan.

REFERENCES Audier, M., Sainfort, P., and Dubost, B. (1986). Phil. Mug. B 54, L105-LI 1 I. Beeli, C., Nissen, H.-U., and Robadey, J. (1991). Phil. Mug. Lett. 63, 87-95. Bendersky, L. (1985). Phys. Rev. Lett. 55, 1461-1463. Burkov, S. E. (1991). J . Stat. Phys. 65,395401. Chattopadhyay, K., Raganathan, S . , Subbanna. G.N., and Thangaraj, N. (1985). Scr. Metull. 19, 767-771. Ebalard, S., and Spaepen, F.(1989). J. Muter. Res. 4 , 3 9 4 3 . Edagawa, K., Ichihara, M., Suzuki, K., and Takeuchi, S. (1992). Phil. Mug. Leti. 66, 19-25. Elser. V. (1985). Phys. Rev. B 32.48924898. Elser, V. (1986). Acta Crysiullogr A. 42, 3 6 4 3 . Elser, V., and Henley, C. L. (1985). Phys. Rev. Lett. 55,2883-2886. Field, R. D., and Fraser, H. L., (1984-1985). Mater. Sci. Eng. 68, L17-L2 I . Fung, K. K., Yang, C. Y., Zhou, Y. Q., Zhao, J. G.,Zhan, W. S., and Shen, B. G.(1986). Phys. Rev. Lett. 56,2060-2063. Guryan. C. A., Goldman, A. I., Stephens, P. W., Hiraga, K., Tsai, A. P., Inoue, A., and Masumoto, T. ( 1989). P hys. Rev. Lett. 62,2409-24 12. Guyot, P., and Audier, M. (1985). Phil. Mug. B 52, L15-Ll9. He, Y., Chen, H., Poon, S. J., and Shiflet, G.(1991). Phil. Mug. Lett. 64,307-315. Henley, C. L. (1986). Phys. Rev. B 34,797-816. Henley, C. L., and Elser, V. (1986). Phil. Mag. Lett. B 53, L59-L66. Hiraga, K. (1989). Muter. Res. Soc. Symp. Proc. 1989 139, 125-134. Hiraga. K. (1990). In: T. Fujiwara and T. Ogawa, Eds. Quusicrysrals. Springer Series in Solid-State Sciences, Vol. 93. Springer-Verlag, Berlin, pp. 68-77 (Proc. 12th Tuniguchi Symp., Shimu, Mie Prefecture. Japan, 1989). Hiraga, K. ( 1991a). J. Electron Microsc. 40, 8 1-9 I . Hiraga, K. (1991b). In: D. P. DiVincenzo and P. J. Steinhardt, Eds. Quusicrystuls: The State of the Art. Directions in Condensed Matter Physics, Vol. 1 I. World Scientific, Singapore, pp. 95-1 10. Hiraga, K. (1991~).Sci. Rep. Res. Inst. Tohoku Univ. A 36, 115-127. Hiraga, K. (1993). J . Non-Crystalline Solids. 153/154,28-32. Hiraga, K . (1995). Proc. Int. Conf. on Aperiodic Crystallography (Aperiodic '941, Les Diublereis, Switzerland 1994 (G. Chapuis and W. Paciorek, Eds.). World Scientific, Singapore, pp. 341-350. Hiraga, K., and Hirabayashi, M. (1987a). J. Electron Microsc. 36.353-360. Hiraga, K., and Hirabayashi, M. (1987b). Jpn. J. Appl. Phys. 26, L155-Ll58. Hiraga, K., and Shindo, D. (1989). Jpn. J . Appl. Phys. 28,2556-2560. Hiraga, K., and Shindo, D. (1990). Muter. Trans. JIM 31,567-572. Hiraga, K., and Sun. W. (1993a). Phil. Mug. Leii. 67, 117-123. Hiraga, K., and Sun, W. (1993b). J . Phys. Soc. Jpn. 62, 1833-1836. Hiraga, K., Hirabayashi, M., Inoue, A., and Masumoto, T. (1985). Sci. Rep. Res. Inst. Tohoku Univ. A 32,309-3 14. Hiraga, K., Hirabayashi, M., Inoue, A., and Masumoto, T. (1987). J. Microsc. 146,245-260. Hiraga, K., Zhang, B . 2 , Hirabayashi, M., Inoue, A., and Masumoto, T. (1988). Jpn. J. Appl. Phys. 27, L951-L953. Hiraga, K., Lee, K. H., Hirabayashi, M., Tsai, A. P., Inoue, A., and Masumoto, T. (1989a). Jpn. J . Appl. Phys. 28,L162kL1627.

98

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Hiraga, K.. Hirabayashi, M.. Tsai, A. P.. Inoue. A., and Masumoto, T. (1989b). Phil. Mug. Lett. 60, 201-205. Hiraga, K., Lincoln, E J., and Sun. W. (1991a). Mu/er. Truns. JIM 32.308-314. Hiraga, K., Sun, W.. and Lincoln, F. J. (1991b).Jpn. J . Appl. Phys. 30. L302-L305. Hiraga, K., Sun, W., Lincoln, F. J., Kaneko, M., and Matsuo. Y. (1991~).Jpn. J . Appl. Phys. 30, 2028-2034. Hiraga, K., Kaneko. M., Matsuo. Y., and Hashimoto, S. (1993). Phil. Mug. B 67. 193-205. Hiraga, K.. Sun, W.. and Yamamoto, A. (1994a). Muter. Trans. JIM 35.657462. Hiraga, K., Abe, E.. and Matsuo. Y. (1994b). Phil. Mug. Let/. 70, 163-168. Hiraga, K.. Yubuta, K.. and Park, K.-T. (1996). J. Muter. Res. 11, 1702-1705. Ishimasa, T., Fukano, Y., and Tsuchimori. M. (1988). Phil. Mug. Lett. 58, 157-167. Katz, A., and Duneau, M. (1986). J . Phys. 47, 181-196. Lemmerz, U.. Grushko. B.. Freiburg, C., and Jansen, M. (1994). Phil. Mug. Lett. 69, 141-146. Levine, D.. and Steinhardt, P. J. (1984). Phys. Rev. Lett. 53.2477. Li. X.2.. and Hiraga, K. (1996). Sci. Rep. Res. Inst. Tohoku Univ. A 42,213-218. Li, H. L.. and Kuo, K. H. (1994). Phil. Mug. Lett. 70,5542. Li, X.Z.. Dong, C., and Dubois. J. M. (1995). J . Appl. Crystullogr. 28.96-104. Matsuo, Y., andHiraga, K. (1994). Phil. Mug. Lett. 70, 155-161. Ogawa, T. (1985). J. Phys. SOC.Jpn 54,3205-3208. Pauling. L. (1985). Nature 317,512. Saito, M., Tanaka. M., Tsai. A. P., Inoue. A., and Masumoto, T. (1992). Jpn. J. Appl. Phys. 31, L109-LI 12. Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W. (1984). Phys. Rev. Lett. 53, 1951-1953. Socolar, J. E. S., Lubensky, T. C., and Steinhardt, P. J. (1986). Phys. Rev. B 34,3345-3360. Sun, W., Yubuta. K.. and Hiraga, K. (1995). Phil. Mug. B 71,7140. Tanaka, M., Tsuda. K., Terauchi, M., Fujiwara, A., Tsai, A. P., Inoue, A., and Masumoto, T. (1993). J . Non-Crysrulline Solids 153/154,98-102. Tsai, A. P., Inoue. A.. and Masumoto, T. (1987). Jpn. J . Appl. Phys. 26, L1505-L1507. Tsai. A. P., Inoue, A.. and Masumoto, T. (1988). Jpn. J . Appl. Phys. 27, L1587-LI590. Tsai. A. P., Inoue, A., and Masumoto, T. (1989a). Muter. Trans. JIM 30, 150-154. Tsai, A. P., Inoue. A., and Masumoto, T. (1989b). Muter. Trans. JIM 30,300-304. Muter. Truns. JIM 30,463-473. Tsai, A. P., Inoue. A.. and Masumoto, T. (1989~). Tsai, A. P., Inoue, A., Yokayama, Y., andMasumot0.T. (1990). Phil. Mug. Lett. 61,9-14. Tsuda, K.. Saito, M., Terauchi. M., Tanaka, M.. Tsai, A. P., Inoue, A., and Masumoto, T. (1993). Jpn. J . Appl. Phys. 32, 129-134. Yamamoto, A.. and Ishihara. K. N. (1988). Acru Crystullogr. A 44.707-714. Yamamoto, A., and Hiraga, K. (1988). Phys. Rev. B 37,62074214. Yubuta. K.. Sun. W., and Hiraga. K. (1997). Phil. Mug. 75,273-284.

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ADVANCES IN IMAGING AND ELECIRON PHYSICS VOL. 101

Formal Polynomials for Image Processing ATSUSHI IMIYA Department of Information and Computer Sciences. Faculty of Engineering. Chiba University* Yayoi.cho. Chiba 260. Japan

I . Introduction . . . . . . . . . . . . . I1. Image Polynomials . . . . . . . . . . A . Mathematical Preliminaries . . . . . . B. Image Polynomials of Real Values . . . C . Image Polynomials over a Binary Boolean Set . . D . Image Polynomials for Color Images 111. Quotient Fields of Digital Images . . . . . A. Quotient FieldsofPolynomials . . . . . B. RegularityofImagePolynomials . . . . C. InversionoftheToeplitzEquation . . . . D . Numerical Examples . . . . . . . . E. Generalized Inverse of the Toeplitz Operator IV. Image Polynomial and Pyramid . . . . . . A . Subpixel Image . . . . . . . . . . B . Subpixel Superresolution . . . . . . . C . Pyramid Transform . . . . . . . . . D . InversionofFyramidTransform . . . . E . Numerical Examples of Superresolution . V. Shape Analysis Using Image Polynomials . . A . Morphological Operation . . . . . . . B . Distance Transform . . . . . . . . . C . Skeletonization . . . . . . . . . . VI . Concluding Remarks . . . . . . . . . Acknowledgments . . . . . . . . . . References . . . . . . . . . . . . .

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99 101 102 103 107 Ill 113 113 114 115 119 122 125 125

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I . INTRODUCTION Linear shift-invariant operations are an important class of operations in image processing since they linearly approximate many classes of operations (Craig and Brown. 1986). For instance. if the region of interest (ROI) is restricted to around the center of the optical axis of an optical imaging system. the system can be described as a shift-invariant linear operator. This class of operations is expressed mathematically as the convolution between an input image and the point spread function (PSF) of a system (Bracewell. 1995; Stark. 1987). Shift-invariantblurring 99

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Copyright @ 1998 Academic Press lnc All rights of reproduction in any forni reserved. 1076-5670/97125.W

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ATSUSHI IMIYA

is also expressed as the convolution between an original image and the PSF of a blurring system. For the restoration of clear images from blurred ones when the PSF of the blurring system is known, a deconvolution procedure should be devised (Bracewell, 1995). Deconvolution is a special case of solving an integral equation of the second kind. The convolution equation is sometimes called the Toeplitz equation (Jdjd, 1992; Kailath et al., 1978; Nagy, 1995a,b)because discretizationof the equation derives the Toeplitz matrix as the system matrix, which expresses the transformation of an imaging system (Bracewell, 1995). The other mathematical treatment of deconvolution is based on Fourier analysis (Prost and Gouttie, 1977; Sanz and Huang, 1984) since the Fourier transform of the convolution equation yields the product of spectra of the PSF and an input image. This mathematical property implies that division of the spectrum of a blurred image by that of the PSF yields a clear image (Bracewell, 1995; Prost and Gouttie, 1977; Sanz and Huang, 1984). For the analysis of deconvolution, there are two main methods, numerical analysis and digital processing. In the former, equations are solved using samples and a continuous solution is approximated by interpolation of the discrete data. In the latter, only discrete functions and discrete transformationsare considered (Graham et al., 1989; Hayahara and Haruki, 1981). In this paper we deal with digital data processing. This method is called digital signal processing for time sequences and digital image processing for arrays, since samples of time-varying signals and images form sequences and two-dimensional arrays, respectively (Bracewell, 1995). The Z-transform for digital signals is the dual of the Fourier-Laplace transform for continuous signals. Another discrete version of the Fourier transform is the discrete Fourier transform (DFl') (Bracewell, 1995; Johnson and Bums, 1985). The Z-transform is used for the analysis of mathematical properties of a PSF and for solving convolution equations of the discrete type. On the other hand, the DFT is a method for the numerical approximation of Fourier transform of analog signals. The two-dimensional Z-transform is an extension of the usual Z-transform to arrays of samples on planes. Z-transforms of discrete functions are formally dealt with as polynomials of several arguments. Also, for the inversion of the Toeplitz equation, properties of polynomials are essential (JhjA, 1992). There is, however, no inverse for the multiplication operation of polynomials since the set of polynomials forms a ring with respect to addition and multiplication. The theory of formal power series also provides a method for dealing with the shift-invariant linear operations on discrete sequences and arrays or discrete-time series and discrete images (Agui et al., 1982a,b; Imiya el al., 1987). Formal power series can be used efficiently to solve difference equations with constant coefficients. The discretized convolution equations can be expressed as difference equations with constant coefficients. These mathematical relations between Ztransforms and polynomials mean that we can express the system equations and solve the convolution equations of signal and image processing using the theory

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of formal power series instead of the theory of Z-transforms. In this paper we use formal power series for discrete signal and image processing. Formal power series enable us to define a generalized inverse (Campbelland Meyer, 1991)for the multiplication of polynomials because formal power series permit us to define a quotient field over an appropriate set of coefficients. Moreover, we define the formal power series over the binary Boolean set. Binary digital images are expressed as arrays of elements of a binary Boolean set. Since the Minkowski addition of a pair of binary digital sets in Euclidean spaces is equivalent to convolution of Boolean-valued functions defined in Euclidean space, formal power series over the binary Boolean set enable us to define an operation that has properties similar to those of Z-transforms for binary digital signals and images. Thus the formal power series over the binary Boolean set acts as the Z-transform of binary signals and images. The Z-transform is derived as a discrete approximation of the Fourier-Laplace transform. Thus the operand of transformation should be a real-valued function or a complex function. This property does not yield a Z-transform of signals and images of which values are elements of the binary Boolean set. The theory of power series, however, yields a Z-transform of Boolean functions. This is one of the most significant differences between the Z-transform and the theory of formal power series. Furthermore, the Z-transform of a discrete function is an analytic function on the complex domain. Thus in calculation of the Z-transform and analysis based on the 2-transform, the results of complex analysis play an important role. However, in the context of the formal power series, the algebraic properties of polynomials play the main role. Thus the theory of formal power series also clarifies the algebraic structure of digital signal and image processing. As an application of the expression of digital images in the manner of formal power series, we prove that we can solve the Toeplitz equation within a finite number of iterations if the transformation kernel of the system satisfies some natural assumptions concerning image processing. Furthermore, we show that the inversion operation of a pyramid (Kropatsch, 1985; Tanimoto, 1976) is again a pyramid. This property is used to derive an algorithm that yields images with subpixel resolution from a set of images. Moreover, as an application of the formal power series over the binary Boolean set, we express some fundamental shape analysis algorithms (Heijmans, 1994; Serra, 1982) in terms of polynomials.

11. IMAGE POLYNOMIALS

First, we consider images of which gray scales are real numbers and which are functions with finite supports. Second, we also consider images of which gray scales are elements of the binary Boolean set. For the expression of a gray-scale

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image, a plane is sufficient, but for the expression of a color image, more than two planes are required; for example, the red plane, green plane, and blue plane if a color image is decomposed into RGB channels. Thus we also consider gray-scale images of which gray scales are vectors, that is, we deal with images defined on a collection of planes. A. Mathematical Preliminaries

In this section we first give brief mathematical notations of two-dimensionalvector space and define the images that we deal with. Setting x-y to be an orthogonal coordinate system on the Euclidean plane R2, we write a vector on R2 as a = (a,/3)T, where xTis the transpose of the vector x. Points for which both coordinates are integers are called lattice points on R2 and the set of all lattice points is denoted by Z2. An image defined on R2 is denoted by f ( x , y ) , where f ( x , y ) is the gray scale of an image at a point x = ( x , Y ) ~ We . use U and for the union of sets and intersection of sets, respectively. Furthermore, expressing the complement of a set A as A\B is equivalent to A B. Let F be a set of points in a vector space and a b be the sum of two vectors a and b. Images that take constant values in small regions called pixels are called digital images (Gialdia, 1986). Our pixels are unit square domains Unln, U r n n = { ( X , y ) T I m - Z1 ( x L r n + ; , n - ~ l y 5 n + 2 }1 . (1)

n

+

x,

n

of which centroids are lattice points. Setting

1

if x E int(U,,),

where aU,, is the boundary of Umnr (3)

and

v,,,

= { ( ~ + E , ~ + E ) ~ I E = ~ ; } ,

the average of f ( x , y ) in Urn, is obtained as

(4)

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

103

Denoting f ( m , n ) such that ( m , n)T E Z2 by f m n , we call fmn the gray scale of pixel ( m , nlT. Furthermore, binary digital images are black or white inside each small square domain (Gialdia, 1986; Voss, 1993). We allocate 0 to white pixels and 1 to black pixels. A binary digital image p is the sum of um,(x, y),

where

p(m,n) =

1

0

if U,,, is black, if U,,,, is white.

(7)

A binary digital image expressed as a sum of urn,,( x , y ) defines a finite set F,

F = ( x l x E Z2,x = ( m ,n)T,p ( m , n ) = l ) ,

(8)

because a function with a finite support is black at a finite number of pixels. Conversely, when we substitute coefficients

into Eq. (6), a finite subset F of Z2defines a binary digital image. Therefore, there is one-to-one mapping between binary digital images and subsets of Z2. Thus there is one-to-one correspondencebetween the set of pixels and 2 ' . This means that we can deal with digital images as functions on Z2as shown in Fig. 1.

B. Image Polynomials of Real Values Setting

u = (I,O)T, u-1 = (-l,O)T,

v = (0, 1)T, v-' - (0,

we define products and powers of vectors as

u * u = (2, o)T, u-' .u = (O,O)T, v .v = (0,2)T, v-' . v = (O,O)T, u . v = v * u = ( l , l T) ,

104

ATSUSHI IMlYA

FIGURE1. Relation between a function that is constant in pixels and a discrete function.

and

= urn* u, u-" = (u-I)",

Um+l

for m 2 1 and n 2 1. These definitions imply the property

urn V" = (m,n)T

(17)

This means that there is one-to-one correspondencebetween u"' v" and elements for integers m and n of the set of lattice points Z2. In the following, replacing urn v" by u'"u", we express the gray scales of pixel ( m ,n)T as fmnumu" since u and u act as the generators of vectors in Z2. This expression can be interpreted that the translation of the value fmn at the origin of the coordinate system by u'" u" yields the gray scale at a pixel ( m ,n)T. Next, set F ( u , U) = fmnUmd'

+ g,,,~,,~~~'d"'

(18)

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

105

to be the superpositionof images f m n U m U ” and gm~,,/um’un’. which have gray scales and gmlnf at pixels (m,n)T and (m’, r ~ ’ ) respectively. ~, If m = m’ and n = n’, Eq. (18) becomes fmn

F(u, u ) = ( f m n

+

gmn)UmUn.

(19)

Furthermore, the product of gray scales is defined by fmnUmun

*

g m , n d G n ‘= fmn

‘

g, ,n ,Um+m’un+n‘.

(20)

These relations enable us to express digital images as formal power series, that is, a series

uniquely corresponds to a digital gray-scale image. We call this expression of a digital image an image polynomial (Agui ef al., 1982a,b). Note that we do not consider the convergence of the power series expressed by Eq. (21) and that the series is used as a formal description of digital images. Image polynomials have the following properties. Setting

cc ..

F“(u,u) =

..

f-nmUmUn,

and

the relations F’(u, U ) = u P u P F ( u u), , F”(u, U) = F ( u , u p ’ ) ,

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ATSUSHl IMIYA

and F”’(U, u)

= F(u-1, u-I)

(28)

hold. Equations (26). (27), and (28) show that the multiplication of a monomial, the exchange of arguments, and the inversion of arguments of a polynomial correspond to a translation, a 90” rotation with respect to the origin of the coordinate system, and the reflection of an image with respect to the origin, respectively. These relations imply that fundamental geometric operations for digital images are achieved by operations on the arguments of image polynomials. Furthermore, these operations are easily accomplished by the pointer operations in the memory of digital computers. Setting

TPq(u,V ) = u P v 4 ,

(29)

we have m

m

Thus Tpq(u,u ) acts as the translation operation. The convolution of two digital images is defined by

without considering the convergenceof the summation.Concerning the multiplication of image polynomials and convolution of two functions, we note the following property.

Property 1. Setting

rn=-m n = - m

m=-m

m

n=-m m

and o o m i=-m j=-m

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

107

the relation

H ( u , U) = F ( u , u ) G ( u , U) holds. This is a well-known relation between convolution and the product of polynomials (Jdjd, 1992; Oppenheim and Schafer, 1975).

This property, called the convolution theorem, also plays an important role in the theory of the Z-transform. However, it is possible to derive the convolution theorem of digital images without using an integral transform for a formal analysis of the image transform. A considerationof the one-to-onecorrespondencebetween vector ( m , n)Tand monomial urnu" is sufficientin the theory of image polynomials, althoughfor a definition of the Z-transform theory of distributions,which describes the properties of Dirac's delta function, the Heaviside function and similar factors are needed. A family of linear shift-invariant operations are expressed using convolution between a PSF and an input image. This fundamental property provides a method of studying linear shift-invariant systems using image polynomials. In Section I11 we consider image restoration using image polynomials. The set of image polynomials with real coefficients forms a commutative ring (Allenby, 1991; Birkoff and Bartte, 1970; Hayabara and Haruki, 1981; van Lint and van der Geer, 1988) for addition and multiplication of polynomials. This mathematical property can be generalized to a set of polynomials of which coefficients are elements of a finite field such as a Galois field (Allenby, 1991). Thus it is possible to define a set of image polynomials of which coefficients are elements of an abstract field. These polynomials act as Z-transforms for images of which coefficients are neither real nor complex numbers. This is the most significant difference between the theory of image polynomials and that of Z-transforms.

C. Image Polynomials over a Binary Boolean Set

In this section we define image polynomials for the analysis of binary digital images. The binary images play an important role in shape analysis. Some mask operations for binary shape analysis are shift invariant. Thus the mask operations may be expressed as convolution-like operations. The relations between mask operations and convolutions have been considered by several authors (Bracewell, 1985; Serra and Soille, 1994). The analysis of mask operations for binary images as shift-invariant operation is employed by embedding the binary values in a set of gray scales. However, image polynomials over the binary Boolean set enable us to describe the mathematical structure of mask operations for binary digital images without using this embedding of singal values.

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A set F @ G defined by

is called the Minkowski addition of F and G (Heijmans, 1994; Serra, 1982; Stoyan et al., 1995). The following property combines convolution and Minkowski addition of supports of a pair of functions.

Property 2. For a pair of bounded closed sets F and G on R2, let

and

Setting

h ( x ,y ) has the property

The digitized version of this property is used for computation of the Minkowski addition of two finite sets in Z2because binary images satisfy the conditions of this property if the binary values of images are embedded in a set of gray scales. For the execution of this method, it is necessary to define an operation that determines a threshold. Thus if fmn and g m n are gray scales of binary digital images, then for fmn

=

gmti

=

{

1

0

if (m,n)T E A, otherwise,

and 1

0

if ( m , n)T E B, otherwise,

(43 )

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

109

the two-dimensional digital function h m n , which is defined by c

o

w

i=-w j = - x

has nonnegative gray values at each pixel. This property implies that for the generation of a binary image, we require a function T such that 1 ifh,, > 0, (45 ) T(hmn) = otherwise. These properties imply that for the analysis of binary shape using the usual convolution, nonlinear operations such as the threshold operation should be defined if we embed binary images in the collection of gray-scale images. However, in the remainder of this section we show that nonlinear operations are not necessary if we define image polynomials of which coefficients are elements of the binary Boolean set. A mathematically clear definition of binary images is obtained by applying binary logic; that is, the black and white pixels are expressed as 1 and 0 of the binary Boolean set 8, respectively (Birkoff and Bartte, 1970). The sum v, the product A, and the negation 7 for elements of 23 are defined as

{

a A b = min(a, b),

(46) (47)

ii= 1 - a .

(48)

a v b = max(a, b),

and

Then convolution of Boolean-valued binary digital images is defined by

v v m

=

hmn

w

{fm-in-j

,=-w ,=-w

Furthermore, setting

F = ( ( m , n)T G = { ( m , n)T

and

H = ( ( m , n)T the following theorem is obtained.

Theorem 1. The relation

H=F@G holds.

A gij~.

(49)

110

ATSUSHI IMlYA

Proof.

H = { ( m ,n>TI h,,,, = 11 = { ( m ,nlT I 3 ( i , jlT,j t i - i t i - j

A gij

= 1)

= [ ( m ,n)T I (m - i , n - j)TE F, ( i , j)TE G } = [ ( m ,n)T I ( m , nIT = ( p

+ i , q + AT,( p ,q)T E F, ( i , JIT E GI

=F$G. Q.E.D

Since operations v and A for elements of the binary Boolean set correspond to

+ and . of the real field, respectively, in the following we use +, ., and zr-w

for V, A, and v:-~, a binary image as

respectively. Hence we can define the image polynomial of 0

0

0

0

m=-00 11=-00

as in the case of the usual gray-scale images. We call image polynomials of binary coefficients binary image polynomials. Theorem 1 implies that for a pair of binary image polynomials F ( u , u ) and G ( u , u ) , H(u, u ) = F ( u , u ) G ( u ,u ) (55) corresponds to the Minkowski addition of finite sets F and G in Z2. Thus, setting

m=-m 11=--00

we have

F @ G = { ( m ,n)T I h,t,t, = 1 ) .

(57)

From the definition of binary image polynomials, we obtain the following theorem.

Theorem 2. The set of binary image polynomials of which coeflcients are elements of 8 has the properties of a commutative ring.

Proof. Setting 0 ( u , u ) and I ( u , u ) to be the binary image polynomials of which all coefficients are zero and 1, respectively, the following relations are satisfied:

+ O(u,u ) = O(u,u) + F(u, u), F ( u , U) + G ( u , u ) = G ( u , u ) + F ( u , u ) , F ( u , U ) + ( G ( u ,U ) + H(u, u ) ) = ( F ( u , u ) + G ( u , u ) ) + H(u, u ) , F(u, u)

F ( u , u)l(u, u ) = I ( u , u ) F ( u , u ) = F ( u , u ) ,

F ( u , v ) G ( u , u ) = G ( u , u ) F ( u ,u ) , F ( u , u ) * ( G ( u ,u ) . H ( u , u ) ) = ( F ( u ,u)G(u, u ) ) . H ( u , u ) .

(58) (59)

(60) (61)

(62) (63)

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

111

Furthermore, the relation

implies that ( F ( u ,U)

+ G ( u , u ) ) H ( u , u ) = F ( u , u ) H ( u , u ) + G(u, u ) H ( u , u ) .

(65) Q.E.D.

If we set

in=-cc

n=-oo

binary polynomial F ( u , u ) defines the image polynomial corresponding to the complement of set F. Thus we define subtraction of binary image polynomials as

F(u, U) - G(u,U ) = F ( u , u)G(u,u), which corresponds to the set minus operation of binary point sets; that is, Eq.(67) defines the binary image polynomial that corresponds to F\G. Moreover, we set M

M

for a binary image polynominal since fi(u, u ) corresponds to F, which is the reflection of F.

D. Image Polynomials for Color Images In this section we define image polynomials for color images (Agui el al., 1982b), but we do not deal with the expression of color space. We only consider algebraic treatments of multicolor images as a collection of gray-scale images and a collection of binary images. One method for mathematical expression of color images is decomposition of color into basic colors. A well-known method is RGB decomposition, which expresses a color image as a collection of three gray-scale images: the red-, green-, and blue-band gray-scale images. Thus properties of each pixel are described as a function of gray scales of these three images. There are many possibilities for the decomposition of colors. Setting

to be a vector that is obtained as the result of the decomposition of a color into the basic colors, where each fAn is the gray scale of a color, we define a color of a

112

ATSUSHI IMIYA

pixel (m,n)T as Cnin

2

(70)

=c(f,p!n( f m n ? . . . ) f i n ) .

Function c depends on the decomposition of colors; for example, Cmn is given as a linear combination off;,, such as Cmn

= W I fAn

W2fin

+

k

*

(71)

Wk f m n

*

for an appropriate set of coefficients (w ~ ) ; = We ~ , define a color image polynomial as a formal power series with vector coefficients,

Furthermore, we define

where

i=-m j=-m

and

.

m n T

hmnumd'= ( h ~ , ~ " ' u ~ , h .. ~ ~,h;,u ~ ~ u u" ,)

.

(76)

Then the set of color image polynomials has the properties of a ring, as does the set of image polynomials of monochromatic gray-scale images. Next, as an example we consider binary color images that consist of eight colors. We show that binary color images that consist of eight colors define the image polynomials of which coefficients are elements of a finite field. Setting red, green, and blue to be

R = (O,O, G = (0,1, O)T, B = (l,O,O)T,

(77)

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

113

vectors of the other five colors are

Y = R + G = (0, 1, l)T, C = G + B = (0,1, l)T, M = B + R = (0, 1, l)T, W = R + G + B = (1, 1, l)T, K = (0, 0, O)T, where Y,C , M ,W, and K express vectors for yellow, cyan, magenta, white, and black, respectively. For a binary vector (a,, a2, a mapping

k=

+ a22 + a3

2’

(79)

defines one-to-one mapping between B3 and the Galois field GF(23). Thus we can express eight color images as image polynomials of which coefficients are elements of GF(23). However, we cannot define multiplication of colors image polynomials because addition and multiplicationdo not satisfy the arithmetic rules among colors.

In.

QUOTIENT RELDS OF

DIGITALIMAGES

This section covers image polynomials of which coefficients are real numbers. Using a relation between the multiplication of polynomials and the Toeplitz equation, we develop an image restoration algorithm. The generalized inverse of the Toeplitz operator is also defined using the properties of the quotient field. A. Quotient Fields of Polynomials For a pair of polynomials F ( u , u ) and G ( u , u ) , G ( u , u ) / F ( u ,u ) is generally a rational function. However, if there exists a polynomial H ( u , u ) such that F ( u , v ) H ( u , U ) = G ( u , u),

(80)

we can define G ( u , u ) / F ( u , u ) as a polynomial. Denoting the solution of EQ. (80) by G ( u , u ) / F ( u ,u ) , the set of image polynomials satisfies the properties of a quotient field. Next, we define an operation D as M

N

p=o q=o

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ATSUSHI IMIYA

Thus the coefficients of DF(u, u ) are zero outside ROI S, where

Since the coefficients of an image polynomial correspond to the gray scales of an image that is defined on a plane, we also use the term ROI for a set of the coefficients of an image polynomial. Furthermore, we define

using the solution of Eq.(80). B. Regularity of Image Polynomials

In this section we show that a class of Toeplitz equations can be solved after a finite number of iterations. For the polynomial H ( u , u ) such that

m=-m n=-m

we assume that hp4 is not zero for a pair of integers p and 4 . Setting 1 rpq(u,U ) = - u - ~ u - ~ hP4

and

A @ ,u ) = rpq(u, u)H(u, v ) ,

(86)

D&u, u ) = A ( u , u )

(87)

if the relation

holds, we say that H ( u , u ) is ( p ,4)-regular. In the rest of this section we assume that for the equation F(u, v ) H ( u , u ) = G(u, v ) , the relations DF(u, U) = F ( u , u), DH(u, U) = H ( u , u), DG(u, U) = G(u, U)

(88)

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

115

hold and H ( u , u ) is ( p , q)-regular for a pair of positive integers p and q . If ROI is sufficiently large, almost all digital systems and images satisfy Eqs. (89), (90), and (91). Furthermore, a class of systems is ( p , q)-regular for an appropriate pair of p and q . For example, the Laplacian operator in the eight-neighborhood

+ U - ' + UU-' + U-' - 8 + u + U - ' U + u + u u

L(u,U) = u - ~ u - '

(92)

is (-1, -1)-regular.

C. Inversion of the Toeplitz Equation 1, Geometric Properties of the Product of Image Polynomials The division of a polynomial by a polynomial, in general, does not yield a polynomial, since in the commutative ring of polynomials there is no inverse operation for multiplication of polynomials. This means that F ( u , u ) , which solves an equation

F ( u , u ) H ( u , U ) = G ( u ,U ) (93) for a given pair of polynomials G ( u , u ) and H ( u , u ) , is not a polynomial. However, if we can find a polynomial F* such that D ( F * ( u , u ) - F ( u , u ) ) = 0,

(94) we can adopt D F * ( u ,u ) as the solution of Eq. (93). In this section we show that using a product of finite number of polynomials, an F*(u, u ) that satisfies Eq. (94) can be constructed. Multiplying both sides of Eq. (93) by t,,(u, u ) , we obtain G(u, u ) = A ( u , u ) . F ( u , u),

where Gcu, u ) = t,,(u,

u)C(u,u),

A(u,u ) = t,,(u,

u)H(u,u)

Setting

A ( u , u ) + K ( u , u ) = 1, Eq.(95) becomes

+

F ( u , u ) =&, u ) K ( u , u ) F ( u ,u). Equation (98) derives the following iteration form fork p 0:

+ K ( u , u ) . Fk(U, u ) ,

F d u , u ) = 0. For the iteration form of Eq. (99). we obtain the following lemma. Fk+l ( u , u ) =G(u, u )

Lemma 1. Fork 2 0, the relation Fk(u, u ) = F ( u , U ) 4-K ( u , u)k(Fo(u,V ) - F ( u , V ) )

holds.

116

ATSUSHI IMIYA

Proof. Equation (99) yields Fk(U, u ) =&u,

=&u,

+

v)/(l - K ( u , u ) ) K ( u , u)k[Fo(U,u ) -&u, u)/(l - K ( u , u ) ) ] u ) / f i ( u , u ) K ( u , u)k[Fo(u,u ) - 6 ( u , u ) / f i ( u , u ) ]

= F(u, V )

+

+ K ( u , ~ ) ~ ( F o U( )u -, F ( u , u)).

(101) Q.E.D

Lemma 1 shows that Fk(u,u ) depends primarily on K ( u , v ) ~ Therefore, . we . the polynomial Q ( u , u ) , of which examine geometric properties of K ( u , u ) ~ For coefficients are zero in ROI,

for a pair of appropriate positive integers (a,fl). Q ( u , u ) satisfies ~ the following lemma.

Lemma 2. The coeflcients of Q ( u , u ) are ~ zem on

Proof. We prove the lemma using mathematical induction. If the coefficients of Q(u, u ) are zero for ( m , n ) E S(l), p+p'=m

q+q'=n

-

is zero for ( p , q ) and (p', 4'). which are elements of S( 1). Furthermore, since p+p'=m q+q'=n

where qkn is the coefficient of Q ( u , u ) ~and , qki' is zero if ( m , n) E S ( k

+ 1). Q.E.D.

For Fq. (102), if all a , a,and B are 1, we set

Then the following lemma is obvious.

Lemma 3. All coeflcients o f K ( u , u ) are ~ zem on U ( k ) . If all coefficientsof Fo(u, u ) are zero outside ROI,we have the followinglemma.

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

117

Lemma 4. The relation D(FM+N+I(U, v ) - F ( u , u)l = 0

holds.

Proof. From Lemma 3, if Fo(u,u ) = 0, we have

This relation leads to the equation

+

+

Since the coefficients of F ( u , u ) are zero for m n < 0 and M N < rn and the coefficients of K ( u , u ) ~ + ~ + I are zero in U(M N l), we have

+ +

D(FM+N+I(u, U ) - F(u, u)) = D K ( u , u ) ~ + ~ + I F (U)u = , 0.

+ n,

(111) Q.E.D.

Lemma 4 implies that the minimum number of iterations required to solve a Toeplitz equation depends on the degrees of polynomials that correspond to ROI in which images are defined. Figures 2 and 3 illustrate the product of two images and the power of an image, respectively. Figure 3 shows the properties of the multiproduct of an image which are proven in Lemmas 1, 2, 3, and 4. Figure 4 shows the relation between ROI and the support of K ( u , u ) " ' + ~ + ' ,that is ROI and the support of K ( u , u ) ~ + ~ + ' have no common point.

2. Iterative Method for Inversion of the Toeplitz Equation Next, investigating the geometric properties of Eq.(99), we derive a more efficient algorithm that reduces the memory capacity required for computation in each step

yM.yK =yp#J 1 3 1

1 2

0

x o

1 1

x o

FIGURE 2. Product of two images.

1 1 2

X

118

ATSUSHI IMIYA

0

X

k

=k*k

FIGURE 3. Powei. of an image.

N 0

P

M

FIGURE 4. Relation between ROI and the support of K ( u , u ) ~ + ~ + '

of iterations. From Eq. (97), fork ? 0, we have F ( u , u ) - F k ( U , u ) = K ( u , U ) k ( F ( U , u ) - FdU, u ) ) .

(1 12)

This relation leads to

F~+I(u, u ) - F . ( u , U ) = K ( u , V Y ( K ( UV, ) - I)(Fo(u, U ) - F ( u , u)). (113) -

Since the coefficients of the right-hand side of Q. ( 1 12) exist in a region U(k), the coefficients of F ( u , u ) and DFk(u, u ) differ at the lattice pointsin S n U ( k ) . Furthermore, the coefficients of Fk(u, u ) and Fk+l(u,u ) differ in U ( k ) . These properties of the coefficients imply the following properties.

Property 3.

P-3.1. The coeflcients

+

fmn that satisfy the relation m n = k are computed in the kth iteration step. P-3.2. Once the coeflcients are obtained, they do not change in the later iteration step.

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

119

P-3.3. The coeflcients obtained in each step are equivalent to the desired coeflcients. P-3.4. The coeflcients generated in in each step do not affect the coeficients in S in the successive iteration step.

s

These four properties lead to the following algorithm, which solves E!q. (93): 1. Set

kmn =

("

-hmn

2.0. Set F to be a polynomial such that:f 2.1. For (mk, nk)T E U(k) f l S,

2.2.

f,M,+N+1.0

= fmn

m=p,n=q otherwise. = 0, (m, n)T $ S .

-

As shown in Section III.B, the Laplacian operator is a (- 1, - 1)-regular polynomial. Thus we can solve the digital Poisson equation, which is expressed

as gmn

+fm-ln +fm-~n+l

= fm-In-1

+ fm-1 + fm-1

n

- gfmn

n+l

+

fmn+l

+ fmn+l + f m + l n + l ,

defined on ROI on the digital plane.

D. Numerical Examples The algorithm proposed in Section, using only multiplication of polynomials, sometimes generates apolynomial of which absolute values of coefficients are very large in each iteration step. Furthermore, these coefficients with large absolute values cannot always be expressed in computer memories. A method for avoiding this overflow phenomenon is embedding of coefficients in a finite field, because our gray scales are quantized. Thus, using an appropriate finite field for quantized levels of gray scales of images, we can treat a set of gray scales as elements of a finite set. Figure 5 illustratesresults of image restoration by the iteration method proposed. Parts (a), (b), (c), (d), and (e) are the outputs of iteration steps 1, 10,45,55, and

120

ATSUSHI IMIYA

FIGURE5. Outputs of iterations (a) I , (b) 10, (c) 45. (d) 55, and (e) 65, from top to bottom.

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

121

(e) FIGURE 5.

(Continued.)

65,respectively. In Fig. 5 a blurring system is defined by

+ u-2 + 2 u - l + u + u31v-3 + ( ~ - u3 - ~ +2ulV-2 + (u-2 + 3 + u3)u-l + (u-3 + u-' - 2 + 2u + 3u3) + (2u-2 - 324-' + 1 + 2u2)u + (u-3 + 3u-' + 2u + u 3 ) 2 + (2u-3 + u-l - u + u2 + 4 X ' ) U ' . (1 15)

~ ( uv ,) = (u-3

The theoretical number of iterations for solving this example is 79 since the size of ROI is 39 x 39. However, the support of the image is smaller than 39 x 39. Thus the algorithm reconstructs the original image after 67 iterations. In this example we embed the quantized gray scales into GF(97) since the gray scales are quantized into 32 levels from 0 to 3 1. In Tables I and I1 we show values of pixels on the line rn = n of iteration 51, results obtained both with embedding and without embedding. If we do not embed gray scales in a finite field, as shown in the tables, it is necessary to compute products of polynomials using long-precision arithmetic. This implies that a large amount of memory is required for computation. However, using the properties of finite fields, we use only the arithmetic of positive integers for image restoration.

122

ATSUSHI IMIYA

TABLE I RESTORATIONRESULTOBTAINED WITHOUT EMBEDDWG GRAYSCALES IN A FINITE FIELD

21st step 0.0 6.6 12,12 18,18 24.24 30.30

0 2 3 2327541844944 563678476360680177 1730903404414932073767

41st step

090 6.6 12,12 18.18

24.24 30,30

0 2 3 2 0 38462198570023657718415360

67th step 0

0.0 6.6 12.12 18.18 24.24 30.30

n

n

1

68th step 0.0 6.6 12.12 18.18 24.24 30,30

0

E. Generalized Inverse of the Toeplitz Operator For a ( p , q)-regularsystem H ( u , u ) , a system H - ( u , u ) such that D(H--(u,u ) H ( u , u ) F ( u , u ) - F ( u , u ) ) = 0

(116)

FORMAL POLYNOMIALS FOR IMAGE PROCESSING TABLE I1 RESTORATION RESULT GRAY OBTAINED BY EMBEDDING SCALES IN FINITEFIELD

Position m of (m. m)T f(m, m) 21st step

08 6.6 12.12

0 2 3

18,18

50

24.24 30.30

37 49

4 1 st step 0 2 3 2 0 44

0.0 6.6 12.12 18.18

24.24 30.30 67th step 0.0 0

2 3 0 0 2

6.6 12.12 18.18 24,24 30.30 68th step

0.00 6.6 12,12 18.18 24.24 30,30

2 3 0 0 2

123

124

ATSUSHI IMIYA

is a generalized inverse of H ( u , u ) (Campbell and Meyer, 1991). The iterative method proposed in Section is a representation of a computation procedure of this generalized inverse. In this section we derive a closed form for the expression of H - ( u , u). Fork 2 0, setting k k & ( U , v ) = t - p q ( U , V) (hpqt-p-q(U, V ) - H ( u , u ) ) (117) we obtain the following theorem.

Theorem 3. A generalized inverse ofa ( p , q)-regular system H ( u , u ) is

= DK(u, u ) ~ + ~ + ' F (U)u , = 0.

(119)

Furthermore, since we can apply operator D term by term, we obtain Eq.(1 18). Q.E.D.

Theorem 3 implies that a generalized inverse is constructed from { D K ( u ,~ ) ~ ) k M f g N Since . the set of image polynomials has the properties of a commutative ring, Theorem 3 yields the following theorem.

Theorem 4. Fork 2 0, setting Gk(U, v ) = D [ t p q ( u , u ) R k ( U , u)G(u,v)I

(120)

we can restore F ( u , u ) using

c

MfN

F ( u , U) =

k=O

Gk(U,

v).

(121)

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

125

Theorem 4 implies that we can restore the clear original image from a blurred image using only convolution if the blurring system is ( p , q)-regular. Furthermore, once ( & ( U , IJ)),M=+,~are determined, {Gk(u,IJ)),M=+,~are computed in parallel. Thus Eq.(120) represents a parallel method for the inversion of the Toeplitz equation. For example, a generalized inverse of Laplacian is

1

L - ( u , I J )= D u u

M+N

.

].

U - " U - ~ ( U I J - L(u, IJ))"

n=O

(122)

Thus the solution of the Poisson equation, G ( u , I J ) = L ( u , v ) F ( u ,U )

(123)

F ( u , I J ) = D ( L - ( u , u)G(u, u ) )

(124)

in S, is obtained by

as an image polynomial.

n! IMAGE POLYNOMIAL AND PYRAMID Image restoration clarifies degraded images, superresolution schemes, however, recover high-frequency portions of images which are eliminated by observation systems. Images taken using digital camera systems, which observe data through a CCD array, are digital images. This section introduces a superresolutionscheme for digital gray-scale images. The method recovers a gray-scale digital image with subpixel accuracy from four gray-scale digital images. We also clarify a mathematical relation between superresolution processes of digital images and a pyramid transform (Jolion and Rosenfeld, 1994; Kropatsch, 1985; ter Haar Romeny, 1994). Using this relation, we construct a spatial network which recovers digitalimages with subpixel accuracy (Imiya, 1994). The pyramid transformwhich compresses image data has been well studied as an application of the wavelet theory. Superresolutionof a digital image is obtained by data extrapolation which is the inverse procedure of data compression (Imiya, 1994). A . Subpixel Image

Setting m k(m) = 2k-1 -

fork 2 1 and ( m , n) E Z2, we define lattice points of the kth order by

z:~,= {(E(m),E(n>)TI ( m , n)T E 2').

(126)

126

ATSUSHI IMIYA

m

FIGURE6. Relation between Z;,) and Z&

Therefore, Z:l) is equivalent to Z2. Thus we call Z:z) the set of sublattice points. Figure 6 illustrates the relation between Zt,) and Z$, . Let a pixel of the kth order, of which the length of each edge is 1/2k-1,be

We call U")(a, b) and U@)(a,b) a pixel and a subpixel respectively. Vertexes of U(k)(a,b) are located at (a, b)T, (a 1/2k-1,b)T, (a 1/2k-1,b 1/2k-')T, and (a, b 1/2k-1)T.Then we define the vertex set of a pixel of kth order by

+

U'k'(&),

+

+

+

P(n)) and U ' k + l ) ( k ( m ) k, + l ( n ) ) satisfy the relations

127

FORMAL POLYNOMIALS FOR Ih4AGE PROCESSING

m

m+;

m+1

m+f

FIGURE 7. Relations of eqs. (129) and (130).

and

+

~ ' ~ " ' ( k ( 2 r n 11, k+1(2n

+ 1)) = W)(E(rn), E(n)) nu ( ~ ) ( E+( ~I ) , EW nU(k'(P(rn+ I), E(n + 1)) nU'k)(E(rn),E(n + 1)). (130)

Figure 7 illustrates the relation expressed by Eqs. (129) and (130). Thus a pixel of the kth order can be expressed as a union of four pixels of the (k 1)th order. Conversely, a pixel of the (k 1)th order can be expressed as an intersection of four pixels of the kth order. Next, for k 2 1 and (a, b)T E R2, we define the function uLy(x, y) by

+

+

$

if x E int (Uk'(a, b ) ) i f x E a W ) ( a ,b ) \ ~ ' ~ ' ( ab ,) if x E # ) ( a , b )

o

i f x E Uck'(a,b ) ,

1

where

(131)

128

ATSUSHI IMIYA

functions of the kth order. A digital image of the kth order is defined by

fi:&(n)

is a real value. Furthermore, we call f (I)(x,y) and where each gray-level f ( 2 ) ( ~ ,y) a digital image and a subdigital image, respectively. In the following we are concerned with digital images that are always zero outside the regions

U

D ( ~=)

Vk)(h(rn), E(n)).

( 134)

(k(m).k(n))TeW

Here, set S(k)is an appropriate finite subset of ROI S for each k 2 1. X , and Equation (129) leads to a relation between functions U ~ : , ~ ( ~ , ( y) @+I)

U-k + l ( m ’ ) m ( n ’ ) ( X 1 ’)’

Therefore, this equation shows that a base function of the kth order is a linear sum of four base functions of the (k 1)th order. Therefore, if the condition

+

+

is held, Eq. (135) enables us to express a digital image of the (k 1)th order as a digital image of the kth order. Conversely, Eq.(135) also enables us to express a digital image of the kth order as a digital image of the (k 1)th order.

+

B. Subpixel Superresolution Let

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

129

and

For {a,/I) = (0, I}, (1, l), and { 1,0),L$ is the translation of Lhk,' by vector (a,/I)T. These sets of points define four types of pixel arrays. A pixel array that corresponds to L$ is obtained by translating the pixel array of which centroids are points in z : k ) using vector (a,/I>T. n u s pixels of which centroids are points in L$ and L$L, overlap if the centroids are neighboring. Using these properties of pixels, we can derive a digital gray-scale image of which centroids of pixels are lattice points of (k 1)th order from four digital gray-scalq images of which centroids of pixels are lattice points of kth order. For proving these properties of digital images, it is sufficient to derive an algorithm that yields a subpixel digital image from four digital images. Let f ( x , y ) be a real-valued function defined in R2. The average of f ( x , y ) in U ( k ) ( ab) , is obtained by

+

[I [I

1 =2-2(k-1)

f ( x , y ) u f i ( x - H(rn), y - H(n))dx dy. (141)

These systems of linear recursive formulae enable us to recover the averages of f ( x , y ) in U'k)(u,b) from those in U(k+l)(u, b). From the viewpoint of practical

130

ATSUSHI IMIYA

fz2

application, we are only interested in the solution of systems of linear recursive n / 2 by computing systems of linear formulae for k = 1. Thus we can obtain recursive formulae iteratively for ( m , n)T E Z2. The second method is to solve the systems of linear recursive formulas at each subpixel and to define the closed form of the solution. Supposing that

we obtain the following theorem.

Theorem 5. Let 01 and /?be 0 or 4. Then, defining (1)

- fm+un+l/2+pv

(1)

(148)

(1)

(1)

- fm+crn+py

(149)

g2' = fm+l/2+crn+1/2+p

hG,B = fm+l/2+crn+B

the solution of the systems of linear recursive formulas (1431, (144), (1451, and (146) is m

n

i=O j = O

Equation (147) is valid i f we prepare workspaces around the ROI.

If an image is observed using a CCD array of which each aperture function is u$)(x, y ) . we can obtain the digital image of the first order; that is, we obtain !:f

for each (m, n)T E Z2 in the ROI. Therefore, Theorem 5 indicates that four digital images recover a digital image with subpixel accuracy if we observe four mutually y and f (I)(x, y shifted images f (')(x, y ) , f ( ' ) ( x $, y ) , f ( I ) ( x (1) (1) Furthermore, for each (m,nIT E Z2, f:]:,12 n , fm+1/2 ,,+1/2, and f m n + l / 2 are obtained by arrays shifted by vectors O)T, and (0, respectively. Thus, from the viewpoint of implementation of an observation system, these four images canbeobservedusingfourarrayswhichobtain f " ) ( x , y ) , f ( " ( x - i , y ) , f ( ' ) ( x yand f (')(x, y respectively.

+

+ i, + i),

(i, (4, i)T

i,

i),

+ i).

i)T,

i),

C. Pyramid Transform Let O2be the set of all points z = ( m , n)T such that both m and n are odd integers, and E2the set of all points z = (m, n)T such that both m and n are even integers. Furthermore, let U"'(a, b ) be a pixel of zeroth order centered at (a, b)T,of which the length of each edge is 2, and f,':' denote the gray level of U(O)(a,b) which is obtained as the average of f (x , y ) in U'O)( a , b ). In the following, we are concerned with digital images of which the minimum unit is U'O)(a,b).

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

131

FIGURE8. Relation of eq. (151).

A pyramid transform,

determines the gray level of U(0)(2rn,2n) from those of U(O'(2m - 1,2n - l), U'O) (2rn+1,2n-1),U(0)(2m+1, 2n+1),andU(o)(2m-l, 2n+1). Inthecaseofthe pyramid transforms, the parameter k in f::) expressesthe number of the application of a transform. Figure 8 illustrates the relation represented by Eq. (151). If we shift the ROI, we also obtain the equations

and

If we leave out the factor f use from them, Eqs. (151), (152). (153), and (154) correspond to Eqs. (143), (144), (143, and (146), respectively. Therefore, we can invert the pyramid transform of Eq. (15 1) by computing systems of linear recursive formulas, similarly to image recovery with subpixel accuracy. It should be noted, however, that the image pyramid transform does not change the sizes of pixels in either the forward or backward process, whereas subpixel image recovery does.

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ATSUSHI IMIYA

D. Inversion of Pyramid Transform Discrete functions xi!,) and f::,) are zero in 0' and E2, respectively. Therefore, if we set

+ u)(u-' + v ) ,

P ( u , u ) = $(u-I

(155)

the pyramid transform in Section 1V.C is given by F'"(u, u ) = P ( u , v)F'O'(u, u ) ,

( 156)

where c

a

m

and

because f " ) ( x , y) and f ( O ) ( x ,y ) are discrete functions defined on E2 and 02, respectively. The formal inverse of P ( u , u ) is

since P ( u , u ) Q ( u , U ) = Q ( u , u ) P ( u , U) = 1 .

Furthermore, since m

M

m =O

m=O m

m =O m

m=l

we obtain

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

133

where A(u, V ) = (u-' - u)(v-' - v ) ,

(163)

c o w

m=l n=l

If the discrete function f;:) is zero outside region S, the output function f i k ) is also zero outside region S. Therefore, a finite series M

N

m = l n=l

which is a finite truncation of ll (u, v), is adequate to recover f;: we obtain the following theorem.

from f;','. Thus

Theorem 6. Zfweset DQ(u, V ) = 4Dn(u, ~ ) A ( uv, ) ,

(166)

the solution of Eq. (156) is

F"'(U,

V)

= DQ(u, v ) F ( ' ) ( uu). ,

(167)

E. Numerical Examples of Superresolution Operator n integratestwo-variable discretefunctions.Therefore, operatorDQ(u, v) leads to the equation

where (1) 2rn+22n+2

=

(1)

g 2 m 2n

(169)

h2m2n

= f ~ r n + 2 2 n - f2rn2n.

(170)

f

(1)

f

- 2m 2n+27 (1)

Hence Eqs. (168), (169), and (170) correspond to Eqs. (148), (149), and (150), respectively,if we compare image recovery with subpixel accuracy and the inversion of the pyramid transform defined by Eq. (151). Next, operator A corresponds to a weighted pyramid transform, of which the weight function wmnis defined by Wmn

=

-1

ifm-n=2 ifm-n=-2

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ATSUSHI IMlYA

c c c

c

FIGURE 9. Signal flow graph of a two-step process.

I

optics

CCD FIGURE 10. Camera system that takes four digital images through a one-aperturelens system and four imaging planes which are arranged on four walls of an imaging box.

for (m,n)T E E2. Therefore, the transform of EQ.(151) is inverted by a two-step algorithm. This algorithm first computes a weighted pyramid transform. Second, results of the previous transform are integrated on the digital array. This twostep process define the signal flow graph of Fig. 9, where C indicates discrete integration along both orthogonal axes on image arrays. Figure 10 illustrates a camera system that takes four digital images through a one-aperture-lens system and four imagingplanes that are arranged on four walls of (1) f c l ) ( x + animaging box. Thefourimagingplanesofthecameraobserve(f1 )( I ) ( x ,y), Y), y+i),andf“’(x, Y+;) andproduce f,!,;, fm+1pn+1/2* and Therefore, the spatial network of Fig. 9 and the camera system of i

y

f(”(x+i, f,i!,)+1,2.

f m + 1 p n 9

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

135

Fig. 10 enable us to invert the image pyramid transform. Moreover, a mathematical correspondance between the inversion of the image pyramid transform and subpixel superresolutionleads to the conclusion that the network and camera system enable us to recover digital gray-scale images with subpixel accuracy if we change the numbering of array elements on input and output image planes of the network. Figure 11 shows an example of superresolution for a RGB color image. The size of the original image is 640 x 480 pixels. The colors are decomposed to RGB expression, and each gray-scale image is expressed using 24 bits. This image is taken from the University of Tsukuba image database. In Fig. 11 we show (a) an original image, (b) the image measured using a low-resolution camera, and (c) the restored image form.

V. SHAPEANALYSIS USINGIMAGE POLYNOMIALS As an application of binary image polynomials, we express distance transform and skeletonizationof binary digital images in the manner of image polynomials. These two transforms are fundamentalfor shape analysis and digital image coding. Because these two transforms are achieved by local shift-invariant operations for binary digital images, it is possible to describe algorithms in the manner of image polynomials. A. Morphological Operation

For a fixed vector a, a set F, which is defined by

F, =

Ub+ a), xeF

is called the translation of F by a. Furthermore, a set # that is defined by

P = {-x

I x E F)

is called the symmetry of F. Moreover, a set F 8 G defined by

is called the Minkowski subtraction of F and G. There is a relation

(173)

136

FIGURE1 1 . Example of superresolution for a RGB color image: (a) an original image, (b) the image measured using a low-resolution camera, and (c) the restored image. The size of the original image is 640 x 480 pixels. The colors are decomposed to RGB expression using 24 bits. This image is taken from the University of Tsukuba Image Database.

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

137

between the Minkowski addition and the Minkowski subtraction. The Minkowski addition and subtraction are fundamental concepts of mathematical morphology. Furthermore, set-theoretical operations and binary operations for a pair of point sets can be used to define more complex morphological operations for the shape analysis of binary images. B. Distance Transfonn Setting d(x, y) to be a distance measure on R2, we define

F(r) = (x I mind(x, y) = r

x E F,y E F),

(176)

for a bounded closed set F.Thus, setting

D(r) = {x I 1x1 = r), F(r) is computed as F(r) = lim(F 8 D(r - S))\(F 8 D(r 6-+0

+ 6)).

(178)

Figure 12 illustrates the distance transform, which is expressed using morphological operations. n')T on Z2, For a pair of points x = (m,n)T and y = (m', d4(x, y) = Im - m'l

+ ( n- n'(

(179)

and d8(x, y) = max(lm - m'l, In - n'l)

( 180)

are L1 and L , distances, respectively. Furthermore, N4

= I(m, nIT I ( f l ,

WT, (0,0lT,(0,

(181)

and

Ns = { ( m ,n)T I ( f l , o)T,(0,olT,(0,f1IT, ( f l , f l ) T }

( 182)

are unit disks if we adopt L1 and L , metrics, respectively. Since we have only discrete values for the diameters of circles, Eq. (178) becomes

F(n) = E"-'(F, N)\E"(F, N),

( 183)

where E(A, B) is the Minkowski subtraction of B from A and N expresses one of Nq and N8. According to the relations between the Minkowski addition of a pair of finite point set on R2 and multiplicationof image polynomials, we have the relation ~

F"(u,U) = F(u, U) ( N ( u , u))"-' -,-

( 184)

138

ATSUSHI IMIYA

FIGURE12. Distance transform expressed using morphological operations.

where N ( u , u ) is N4(u,u ) = u

+ u-I + 1 + u + u - ' ,

or N8(U, u )

= (u-'

+ 1 + u)(v-' + 1 + u ) ,

if we adopt L I or L, metrics, respectively. C. Skeletonization

First we give the definition of skeletonization on R2.For x E F and a positive real number r > 0,the set D(r), E F satisfies the relation

D(r)x = ty I IY - XI = f ) . Setting S ( r ) to be the union of all x such that

(187)

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

I39

FIGURE13. Relation between a skeleton and circles that cover the original shape.

S ( r ) is computed as

where D ( A , B) is the Minkowski addition of A and B. Equation (188) implies that S ( r ) is the closure of centroids of circles that cover F with the radius r . Furthermore, Eq. (188) implies that we can reconstruct F from S ( r ) for a given r . Figure 13 illustrates the relation between the skeleton and circles that cover the original shape. In Z2, we can obtain S ( n ) as

S ( n ) = E"-'(F, N)\D(E"(F, N),N).

(190)

Equation (190) is expressed as

S"(U,IJ)= F(u,IJ)* ( N ( u , IJ))"-'- P(u,U) * N ( u , IJ)". N ( u , IJ). (191) S"(u, IJ) is called the nth-order skeleton of F ( u , IJ).

VI. CONCLUDING REMARKS

In this work we summarized the role of formal power series for image processing and image analysis. The formal power series which is defined from the generator functions plays an important role in combinatorial analysis (Sachakov, 1996). In combinatorial analysis, formal power series are defined through the Taylor expansion of generator functions. The theory of generator functions that was proposed in the field of probability is equivalent to the theory of the Z-transform. However, in image processing, formal power series are defined using gray scales of pixels. Furthermore, the theory permits us to define Z-transforms of digital signals and

140

ATSUSHI IMIYA

images algebraicallywithout using any propertiesof complex analysis (Agui et al., 1992; Imiya et al., 1987). This is an advantage of formal power series against 2transforms since most operationsand calculationsbased on Z-transforms for signal and image processing are formal algebraic calculations based on the properties of the commutativering of polynomials which are derived as Z-transforms of discrete functions. Thus, the results obtained in Section III showed that we can solve an image restoration problem using only algebraic methods. The problem that we dealt with in Section 111covers most image restoration problems. Furthermore, the equivalence between the division of polynomials and the inversion of the Toeplitz matrix is used for the fast numerical inversion of the Toeplitz matrix (Jhjh, 1992). This inversion method is based on the division of one-valued polynomials. We showed in this paper the two-dimensional version of this equivalence. However, our method for solving equations does not require any matrix inversion or quotient of polynomials, even in the initial stage of computation. Using the theory of formal power series, we defined a Z-transformlike operation for binary digital signals and images that permits us to describe some fundamental morphological operations in the context of commutative ring. The formal analysis of linear differential equations goes back to Heaviside for circuit analysis (Mikusidski, 1963; Yosida, 1981). The Z-transform was proposed as the discrete version of Fourier-Laplace transform for the analysis of difference equation that describes digital properties of systems. Mikusidski also proposed an operational theory for the analysis of differential equations (Mikusiiiski, 1963). Yosida proposed a new treatment of Mikusidski’s theory (Yosida, 1981). In this new treatment, a quotient field in the operator space plays important roles. Hayabara and Haruki published a monograph on the operational analysis of linear difference equations and discussed complex analysis (Hayabara and Haruki, 1981) in Japanese. Their theory is based on formal power series. Although in their book they do not consider any applications to engineering problems, our theory owns much to their treatment of difference equations. However, the convergence properties of inversion of the Toeplitz equation, definition of generalized inverses of polynomials, and 2-transform-like operation for binary digital images are reported for the first time in the present work. ACKNOWLEDGMENTS A part of this research was carried out while the author was at the Department of Electrical and Computer Engineering, Kanazawa University, and parts of Sections 111 and V are based on joint work with T. Nakamura at Kanazawa University. Numerical examples in Section III were prepared by T. Nakamura while he was at Kanazawa University. The author expresses his thanks to Professor Emeritus T. Takeba, who supported the early stage of this research as chairman of the

FORMAL POLYNOMIALS FOR IMAGE PROCESSING

141

Software EngineeringDivision of the department. A numerical example in Section

IV was prepared by M. Oguri using the University of Tsukuba image database; the author thanks Professor Y. Ohta of the University of Tsukuba, who designed the database. The final manuscript was prepared while the author was visiting the Departmentof Applied Mathematics, University of Hamburg. He expresses many thanks to Professor Dr. Ulrich Eckhardt for his hospitality. While the author was at Kanazawa University, this research was supported by the Inoue Science Foundation and Okawa Foundation. While staying in Germany, the author was supported by the Program for Overseas Research of the Ministry of Education, Culture, and Science of Japan. REFERENCES Agui. T., Yamanouchi, T., and Nakajima, M. (1982a). An algebraic description of painted digital pictures. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4.627434. Agui, T., Nakajima, M.. and Arai, Y. (1982b). An algebraic approach to the generation and description of binary pictures. lEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 635-641. Allenby, R. B. J. T. (1991). Rings, Fields and Groups: An Introduction to Abstract Algebra, 2nd ed. Edward Arnold, London. Birkoff, G., and Bame, T. C. (1970). Modern Applied Algebra. McGraw-Hill. New York. Bracewell, R. N. (1995). 7bo-Dimensional Imaging. Prentice Hall, Upper Saddle River, N.J. Campbell, S. L., and Meyer, C. D., Jr., (1991). Generalized Inverses of Linear Transformations. Dover, New York. Craig, I. J. D., and Brown, J. C. (1986). Inverse Problems in Astronomy. Adam Hilger, Bristol, England. Crimmins. T. R., and Brown, W. E. (1985). Image algebra and automatic shape recognition. IEEE Trans. Aerosp. Electron. Sysr. AES-21,60-69. Gialdia, C. R. (1986). The universal image algebra. Adv. Electron. Electron Phys. 67, 121-182. Graham, R. L., Knuth, D. E., and Patashnik, 0. (1989). Concrete Mathematics. Addison-Wesley, Reading, Mass. Hayabara, S., and Haruki, S . (1981). Atarashii Ensanshiho To Risan Kansu Kuiseki (A New Operational Calculus and Discrete Analytic Functions). Maki Shoten, Tokyo. Heijmans, H. J. A. M. (1994). Morphological Image Operators. Advances in Electronicsand Electron Physics, Supplement 25. Academic Press, London. Imiya, A. (1994). Subpixel superresolutionand inversion of image pyramid. In: I., Plander. Ed. Artij c i a l Intelligence and Information-Control Systems of Robots '94, Proc. 6th Inr. Conf on Art$cial Intelligence and Information-Control Systems of Robots. World Scientific,Singapore. Imiya, A., Kodera, S., and Takebe, T. (1987). Image polynomial and its applicationsto image restoration. Proc. SPIE 804.45-52. Jhjh, J. (1992). An Introduction to Parallel Algorithms. Addision-Wesley, Reading, Mass, Johnson, H. W., and Bums, C. S. (1985). On the structure of efficient DFT algorithms. IEEE Trans. Acoust. Speech Signal Process. ASSP-33,248-254. Jolion. J.-M., and Rosenfeld, A. (1994). A Pyramid Framework for Early Ksion. Kluwer, Dordrecht, The Netherlands. Kailath, T., Vieria, A., and Morf, M. (1978). Inversion of Toeplitz operators, innovations, and orthogonal polynomials. SIAM Rev. 20, 106-1 19. Kropatsch, W. G. (1985). A pyramid that grows by the power of 2. Pattern Recognition k t t . 3, 315-322.

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Mikusidski, J. (1963). Rachunek Operatordw, (Japanese Edition). Syokabo, Tokyo. Nagy, J. G. (1995a). terative techniques for the solution of Toeplitz systems. SIAM News 28, Augustkptember. Nagy, J. G . (1995b). Applications of Toeplitz systems. SIAM News 28, October. Oppenheim, A. V., and Schafer, R. W. (1975). Digital Signal Processing. Prentice Hall, Upper Saddle River, N.J. Prost. R., and Gouttie, R. (1977). Deconvolution when the convolution kernel has no inverse. IEEE Trans. Acousr. Speech Signal Process. ASSP-25,542-549. Sachakov, V. N. ( 1996). Combinatorial Methods in Discrete Mathematics. Cambridge University Press, Cambridge. Sanz,J. L. C., and Huang, T. S. (1984). A unified approach to noniterative linear signal restoration, IEEE Trans. Acoust. Speech, Signal Process. ASSP-32,403-409. Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London. Serra, I., and Soille, P., Eds. (1994). Mathematical Morphology and Its Applications to Image Processing. Kluwer. Dordrecht, The Netherlands. Stark, H.. Ed. (1987). Image Recovery-Theory and Application. Academic Press, San Diego, Calif. Stoyan, D.. Kendall, W. S.. and Mecke, J. (1995). Stochastic Geometry and Applications, 2nd ed. Wiley, Chichester, West Jussey, England. Tanimoto, S. L. (1976). Pictorial feature distortion in a pyramid. Comput. Graph. Image Process. 5, 333-352. ter Haar Romeny. B. M. (1994). Geometry-Driven Difusion in Computer Vision. Kluwer, Dordrecht, The Netherlands. van Lint, J. H., and van der Geer, G. (1988). Inrroduction to Coding Theory and Algebraic Geometry. BirkhBuser, Basel, Switzerland. Voss, K. (1993). Discrete Images, Objects, and Functions in Z". Springer-Verlag, Berlin. Yosida, K. ( I98 1 ). Ensansiho-Hitotsu no Chokansumn (Operational Calculus-A Theory of Distributions). Tokyodaigaku Syupannkai (The University of Tokyo Press), Tokyo.

.

ADVANCES IN LMAGING AND ELECTRON PHYSICS VOL . 101

The Dual de Broglie Wave MARCIN MOLSKI Deparfment of Theoretical Chemistry. Faculfy of Chemistry. Adam Mickiewicz Universify Grunwaldzka 6. PL 60-780 Poutah. Poland

. . . . . . .. A. De Broglie-Bohm Wave Theory . . . . . . . . . . B. Mackinnon Soliton . . . . . . . . . . . . . . C. Jennison-Drinkwater Electromagnetic Theory . . . . .

I . Introduction

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11. Wave-ParticleModelsofMassiveParticles

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. . . . . . . . . . . . . . . .

D. Corben TachyonicTheory . . . . . . . . . . . . . . . E. Horodecki-Kostro Model and the -0-Wave Hypothesis . . . . F. Das Model and Pseudovelocity . . . . . . . . . . . . . G. ElbazModelandLorentzTransformations . . . . . . . . . H. Generalized Barut Approach . . . . . . . . . . . . . . I . Conclusions . . . . . . . . . . . . . . . . . . . . 111. Wave-Particle Model of Photons . . . . . . . . . . . . . . A. Photon as a Bradyon-Tachyon Compound . . . . . . . . . B . Conversion of Light into B- and D-Waves . . . . . . . . . C. lko-WaveModelofLongitudinalPhotons . . . . . . . . . D. Massless Photons and Ponderable Matter . . . . . . . . . E. Extended Proca Theory . . . . . . . . . . . . . . . . F. Conclusions . . . . . . . . . . . . . . . . . . . . IV. Electromagnetic Model of Extended Particles . . . . . . . . . A. Three-Dimensional Rectangular Space Cavity . . . . . . . . B. Three-DimensionalSpherical Space Cavity . . . . . . . . . C . One-DimensionalLinear Time Cavity . . . . . . . . . . . D. Three-Dimensional Spherical Time Cavity . . . . . . . . . E. Two-Dimensional Square Space-Time Cavity . . . . . . . . F. Conclusions . . . . . . . . . . . . . . . . . . . . V. Extended Special Relativity and Quantum Mechanics in a Local L-Space A. Special Relativity in L-Space . . . . . . . . . . . . . . B. Quantum Mechanics in L-Space . . . . . . . . . . . . . C. Conclusions . . . . . . . . . . . . . . . . . . . . VI . Two-Wave Model of Charged Particles in Kaluza-Klein Space . . . A. Kaluza-Klein Field Theory . . . . . . . . . . . . . . B. Charged Particle as a Five-DimensionalTachyonic Bootstrap . . C. Conclusions . . . . . . . . . . . . . . . . . . . . VII. Extended de Broglie-Bohm Theory . . . . . . . . . . . . . A. Tachyo-kinematicEffect . . . . . . . . . . . . . . . B. Particle on a Line . . . . . . . . . . . . . . . . . . C. Particle in a Scalar Field . . . . . . . . . . . . . . . D. Uncertainty Principle . . . . . . . . . . . . . . . . E. Many-Body Problem . . . . . . . . . . . . . . . . . 143

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Copyright @ 1998 Academic Press Inc

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All rights of repduction in any form reserved.

1076-567W$25.00

144 F. Conclusions . . VIII. Infons? . . . . . IX. Concluding Remarks Acknowledgments . . . . References

MARCIN MOLSKl

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230 231 232 234 234

swiatlu zalamanemu w krysztale czasu To the light refracting in the crystal of time W.Sedlak (1985)

I.

INTRODUCTION

The idea that not only an ordinary de Broglie wave (B-wave) but also a matter wave of the second kind is associated with a moving particle has a long and fascinating history. The first suggestion of this matter emanated from Louis de Broglie, who between 1924 (the date of publication of his doctoral thesis) and 1927 developed a theory of double solution known also as a causal theory, or, in its simplified form, as a pilot-wave theory (de Broglie, 1960, p. 97). According to this concept, we associate with a uniformly moving particle waves of two types (de Broglie, 1960, p. 99): a probabilistic B-wave of continuous amplitude and subjective character, and a singularity wave of finite amplitude and objective character, centered about a point (or a pointlike region) at which the associated particle is localized. The B-wave, commonly employed in conventional wave mechanics, represents no physical reality and has only statistical meaning, whereas the singularity wave is a true physical representation of a physical entity, a purricle. This particle is localized in space as in the classical picture, but is also incorporated in an extended wave phenomenon. According to the phase connection principle (de Broglie, 1960, p. 99) the B-wave has the same phase as the singularity wave; one may then state that a particle during its motion is guided in space-time by the B-wave playing a role of a pilot wuve. The theory of a double solution was extended by Bohm (1952a,b, 1953), Bohm and Vigier (1954), Bohm and Bub (1966a,b), Bohm and Hiley (1973, and Bohm et al. (1987) to the form known as the theory of hidden variubles, which with de Broglie’s causal theory laid the basis for the Paris school of interpretation of quantum mechanics, competitive with the Copenhagen school (Bohr, Born, Heinsenberg, and others). According to the former authors, the fundamental constituents of matter (e.g., electrons, protons, neutrons, etc.) are both waves and particles following continuous and causally determined trajectories. Thus they are composed of a particle satisfying certain equations of motion and a wave satisfying a SchrWnger equation. Both particle and wave (represented by the wave function) are taken to be objectively real whether or not they are observed. In particular, Bohm and Vigier (1954) suggested that a particle can execute a random movement (resemblingBrownian motion) as a result of interactions with a subquantum level characterized by unknown (hidden) parameters. In

THE DUAL. DE BROGLIE WAVE

145

this connection the probability density for the particle to be at a certain position is a steady-stare distribution ultimately resulting from random movements of the particle. This has raised the possibility that quantum mechanics can be understood essentially in terms of ontological assumptions concerning the nature of individual systems so that the epistemological and statistical content would then take on a secondary role as in Newtonian mechanics. ( B o b et al., 1987)

Contrary to the Paris school, proponents of the Copenhagen interpretation assume that there is no way to describe or to understand any physical microprocess or microentity in intuitive and imaginative terms as we must use classical language and classical concepts (as a result of our macroexperience) to do so. Hence quantum theory yields only statisticalpredictions of results of measurements of quantum systems (Bohr, 1934, 1958). In particular the square of the modulus of the wave function associated with a particle is interpreted as the density of the probability of finding the particle at a point in space-time. Despite critical opinions by de Broglie, quoted below, this approach has been intensively developed since 1926, and is dominant in the scientific community. As the work of other scientists led to further progress in wave mechanics, it became daily more evident that the @ wave (B-wave) with its continuous amplitude could be used only in statistical predictions. And so, little by little, there was an increasing trend towards the purely probabilistic interpretation, of which Born, Bohr and Heisenberg were the chief advocates. I was surprised at this development, which did not seem to me to fulfill the explanatory aim of the theoreticalphysics.... (de Broglie, 1960, p. V) The @ wave (B-wave) usually employed in wave mechanics cannot be a physical reality; its normalization is arbitrary; its propagation, in the general case, is supposed to take place in an obviously fictitious configuration space, and the success of its probabilistic interpretation shows clearly that it is merely a representation of probabilities dependent upon the state of our knowledge and suddenly modified by every new piece of information. (de Broglie, 1960, p. 91)

Further investigation in the field, leading directly to the concept of a dual de Broglie wave, was undertaken by Kostro (1978, 1985a,b, 1988), who proposed a three-wuveparticle model in the framework of the Paris interpretationof quantum mechanics and Einstein’s theory of relativity, in which the existence of a covariant erher is assumed. According to Kostro’s model, one assumes that three waves are associated with a moving particle: one internal nondispersive standing Compton wave (C-wave) and two external ones, one ordinary B-wave having a superluminal phase velocity, and a wave of the second kind endowed with a subluminal phase velocity. The C-wave is responsible for inertial properties of ponderable objects, whereas a matter wave of the second kind is assumed to result from interaction between Einstein’s hidden subquantum ether (excited by the B-wave) and a particle associated with C-wave. The extra wave is supposed to be a carrier of the

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MARCIN MOLSKI

particle inertia (connected with the C-wave) through space-time in which the way is prepared for it by the pilot B-wave (Kostro, 1985a.b). Kostro’s model was modified and extended significantly by Horodecki ( 1981, l982,1983a,b, 1984, 1988a,b,c, 1991). who introduced to the wave theory of matter the notion of a dual de Broglie D-Wave (Horodecki, 1981). According to this concept, a matter wave of the second kind (D-wave) appearingin Kostro’s approach is assumed to have a subluminalphase velocity equal to the B-wave group velocity identified with the classical velocity of an associated particle; the group velocity of the D-wave is assumed to be superluminal and equal to the phase velocity of the B-wave, a cross correspondence. Consequently, according to the wave picture a moving massive particle is represented by a nonlinear nondispersive wave packet (called a C-wave) that involves an internal characteristic spectrum of matter waves, including B- and D-waves. Because the C-wave may be considered a nonlinear superposition of B- and D-waves, according to Horodecki’s model the number of independent waves associated with a moving particle decreases to two, providing a conceptual background for a two-wave particle model, a nonlinear wave hypothesis, and an extended space-time description of matter (Horodecki, 1988a, 1991). According to the latter concept, a massive particle in motion is described approximately in terms of a timelike component, characterized by a timelike fourmomentum associated with a B-wave and a spacelike component characterized by a spacelike four-momentum associated with a D-wave (Horodecki, 1988a, 1991). Because a group velocity of the D-wave according to Horodecki’s model is assumed to be superluminal, the spacelike particle constituent becomes endowed with typical tachyonic properties (Horodecki, 1988a). A two-wave particle model including B- and D-waves was investigated by Das (1984.1986, 1988, 1992) and Elbaz (1985, 1986,1987, 1988); they started from two points representing the corpuscular and wave aspects of the problem. Exploiting the fundamental concept that “the particle picture and the wave picture are merely two aspects of one and the same physical reality” (Jammer, 1974), Das (1986,1988) pointed out that the correspondences energy momentum velocity

frequency wavelength t ) group velocity ? t) phasevelocity, t ) t )

familiar for particle-wave duality,are incomplete,as they fail to include parameters of like number in the particle and wave pictures. To restore parametric symmetry, Das (1986, 1988) assumed that a moving massive particle might be endowed not only with an ordinary velocity but also with a nonobservable superluminal pseudovelocity equal to the phase velocity of the associated B-wave. Because a particlehaving a pseudovelocity must also have a nonobservablepseudomomentum and pseudoenergy, and because these must be associated with a pseudowave via

THE DUAL DE BROGLIE WAVE

147

extended wave-particle corresponences,Das deduced a matter wave of a new type called the transformed Compton wave. Detailed analysis indicated that it was endowed with the same wave characteristic as the D-wave of Horodecki’s model. A similar incompleteness,but in conventional wave theory, was found by Elbaz (1983, who considered transformational properties of a B-wave phase under a Lorentz boost. He realized that accordingto de Broglie’s theory, a B-wave function of a moving particle is derived by means of a time Lorentz transformation to a timelike periodic element associated with a particle at rest (de Broglie, 1960, p. 4). Proceeding analogously, one may derive a matter wave of the second kind, applying a space Lorentz transformation to the spacelike periodic element associated with the same particle. The amplitude wave (Elbaz, 1985) obtained in this way has wave characteristics identical to those of the transformed Compton wave of Das’s theory and the D-wave of Horodecki’s model. Because attributionad hoc of a spacelike periodic element to a particle at rest is artificial and groundlessform a physical point of view, Elbaz (1985) assumed that a photonlike standing wave, being a superposition of purely timelike and spacelike waves, is associated with a particle at rest . This system of waves under time and space Lorentz transformations produces B- and D-waves associated with a uniformly moving particle. All these concepts gained strong support in independentinvestigations by Mackinnon (1978, 198la,b, 1988), Jennison and Drinkwater (1977), Jennison (1978, 1983, 1988), Jennison er al. (1986), Corben (1977, 1978a,b) and Recami (1986, p. 111). On the basis of de Broglie’s phase connectionprinciple and a particular superposition of de Broglie waves, Mackinnon (1978,198 la,b) constructedanondispersive wave packet that fails to spread with time and constitutes a particlelike solitary wave endowed with characteristics identical to those of a C-wave according to Horodecki-Kostro theory. Jennison (1983) showed that Mackinnon’s soliton represented a massive particle having a sharp and finite boundary and was entirely consistent with a model of an electromagnetic phase-locked cavity (Jennison and Drinkwater, 1977; Jennison, 1978, 1983, 1988; Jennison er al., 1986). According to this concept, the inertia of all finitely bounded material particles originates in an echo effect of a feedback process occurring for c-velocity waves, which is intrinsic to phase-locked particles (Jennison, 1983). The electromagnetic model of a massive particle proposed by Jennison is compatible with a fundamental assumption (Elbaz, 1985) that there is a photonlike standing wave associated with a particle at rest. Controversial ideas of a superluminal pseudovelocity proposed by Das, an extended space-time description of matter developed by Horodecki, and a dual de Broglie wave appearing in all models considered are consistent with a tachyonic theory of elementary particle structure developed by Corben (1977, 1978a,b). According to this concept, a moving particle is considered to be a composite object comprising both a bradyonic component (timelike, slower-than-light particle) and

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MARCIN MOLSKI

a tachyonic component (spacelike, faster-than-light particle’) associated with a system of time- and spacelike waves having similar, and under some conditions (Molski, 1991, 1993a) the same, wave characteristics as those of B- and D-waves according to the Horodecki-Kostro, Das, and Elbaz models. A complete synthesis of all these approaches is presented here. The two-wave particle model, an extended space-time description of matter, a nonlinear wave hypothesis, a tachyonic theory of elementary particle structure, and an electromagnetic concept of matter become unified to yield a consistent wave-corpuscular model of both massless photons and massive particles. We are concerned with the formulation of extended special relativity and quantum mechanics in a local L-space, which seems the simplest and most natural way to introduce tachyon mechanics and a D-wave concept. Moreover, we consider an internal structure of charged particles and reinvestigate standard problems of quantum theory in the framework of extended de Broglie-Bohm theory, including a D-wave concept and a tachyonic theory of matter.

11. WAVE-PARTICLE MODELSOF MASSIVEPARTICLES

To increase our insight into wave-particle models of matter presented in the preceding section, we consider their brief mathematical formulation, confined to fundamental information in the field. A. De Broglie-Bohm Wave Theory

The starting point of all models considered is a wave-particle dualism of matter represented by the correspondence

in which pw = ( E / c ,p) = mod’, up

= ( y c , yv),

y = (1 - p ) - ” 2 , kp

= ( w / c , k),

w = 2nv,

k = 2n/A.

(2)

p = v/c,

(3) (4) (5)

In the abundant literatureon tachyons, as an introduction we recommend papers by Bilaniuk er al. (1962), Bilaniuk and Sudarshan( I 969). and Kreisler ( I 973); experimental aspects of the problem were considered by Feinberg (1970), Meszaros (1986). and Clay (1988). Resources for this subject are presented by Feldman (1974). Perepelitsa (1986), and Recami (1986).

THEDUAL DE BROGLIE WAVE

149

In these equations p, denotes the timelike four-momentum of a particle endowed with rest mass mo,moving in four-dimensional space-time x p = (ct, x , y, z) of - -- at velocity v ; k@ is the wave four-vector of an associated signature B-wave of frequency u and wavelength A. These four-vectors fulfill the invariant relations

+

2 2 P,P@ = moC , V,V@

=c2,

(6)

k,k, = (%/c)’,

(7)

in which W O / C = moc/h = ~ I T / A o A0 ; is the Compton wavelength, and the B-wave is additionally characterized by a group (phase) velocity given by dw 0 vgvp= c2, (8) dk =k c. To derive a wave equation governing propagation of a B-wave, one applies a quantization procedure, vg = - < c,

pp

’

+pP

= ihar,

(9)

to Eq. (6)to produce a timelike Klein-Gordon wave equation [a,W a,ap

+ ( m ~ c / h +) ~=] 0,

= a; - a,‘ = a; - A =

(10)

m,

(1 1)

with a timelike B-wave solution +(xc”) = A exp [ i k , x p ] .

(12)

A wave equation for the field of a charged particle is obtained on substituting into Eq. (10)

a,

--+ D, = a,

+ ie/(hc)A,,

(13)

+

(14)

yielding

+

[D,D p ( m ~ c / h ) ~=] 0,

in which Ah = ( V / c ,A) is an electromagnetic four-potential. In a nonrelativistic regime ( v << c ) Eq. (14) reduces to a time-dependent SchrWnger equation [ih:

: ,’I +

- eV - (2mo)-’ fi - -A

+

(

= 0,

which, after generalization to a 3 ( N 1)-dimensionalequation [see Eq. (512)], forms a fundamental equation of conventional quantum mechanics employing statistical interpretation of a B-wave

+.

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MARCIN MOLSKI

According to de Broglie-Bohm theory, alternative to that above, one assumes that a wave function of a massive particle endowed with rest mass MOis given in a factorized form (de Broglie, 1960; Bohm, 1952a,b), q ( x @ )= R ( x @ )exp[(i/h)S(x’)],

(16)

and satisfies a timelike Klein-Gordon equation,

[o+

(MOC/W2] @

= 0,

(17)

in which R(x’) and S(x’) are real functions to describe the amplitude and phase of an associated matter wave. Introducing Eq. (16) into Eq. (17) and then separating the latter into real and imaginary parts, one obtains two equations (de Broglie, 1960, p. 112): [80S(x@)I2- [VS(x’)I2 = M;c2 &R(x’)8oS(x’)

+h 2 n R ( ~ ’ ) / R ( ~ ’ ) ,

- VR(x’)VS(x’) = - 1 / 2 R ( x ” ) u S ( ~ @ ) ,

(18) (19)

of which the first is Jacobi’s relativistic equation for a particle endowed with a variable rest mass (de Broglie, 1960, p. 116),

This variable rest mass has interesting properties (de Broglie, 1960, p. 135); for example, the energy of a particle endowed with a mass mo, moc2

=

JW’

remains finite for u = c and real for u > c; consequently, the kinematics of such an object are subject to no limitation involving the velocity of light as an upper bound. In the nonrelativisticregime, the following approximations hold (de Broglie, 1960, p. 121): mo 2: MO

+ Q(x’)/c2,

E

2: Moc2

+ Q ( P +) 1/2M0v2,

(22)

h2 U R ( x ” ) 2Mo R(x’)

Q(x’) = --,

in which the quantity Q(x’) is generally interpreted as a quantumpotential related to a quantumJield existing even in the absence of a field of classical type (e.g., gravitational or electromagnetic).

THE DUAL DE BROGLIE WAVE

15 1

In the case of a charged particle moving in an electromagneticfield A’, Eqs. (1 8) and (19) take the form 2

[%S(x”) - -eV ] - b S ( x ’ ) C

+ ‘A] C

2

= Mic2

+ h 2 ~ R ( x ’ ) / R ( x c ” ) ,(24)

According to de Broglie-Bohm theory, the guidance formula (de Broglie, 1960, pp. 107, 113)

plays an important role in determining the relativistic velocity of a particle interacting with an external electromagnetic field. For a nonrelativistic many-body problem, Eqs. (16), (24), and (25) become generalized to the form ( B o b and Hiley, 1975) * ( x , t ) = R ( x ) exp[(i/h)S(x,

r)l,

(27)

in which x = (a,), a = x , y, z, denotes the set of 3N position coordinates. Equation (28) is a Hamilton-Jacobi equation for a system of N particles, on which not only a classical potential V ( x )but also a quantum potential Q ( x ) act. The latter introduces an additional action that is a major discriminant between classical and quantal theories (Bohm er al., 1978). Equation (29) describes the conservation of probability density p with currents

determined in configuration space M(1,3N). Hence velocities of particles in the system are determined by a nonrelativistic counterpart of a guidance formula

MARCIN MOLSKI

From the context of these considerations, the physical meaning of amplitudal and exponential parts of a wave function (27) emerges: R ( x ) determines both the probability density and the quantum potential, whereas S(x, r) determines the mean momenta of the particles and their velocities.

B. Mackinnon Soliton The explicit form of R ( x ” ) and S ( x ” ) functions for a freely moving particle was obtained by Mackinnon (1978, 1981a,b). The point of departure for this concept was relativistic invariance of a phase S ( x ” ) of matter wave under a Lorentz transformation. According to Mackinnon (1978): The motion of a particle in space-time does not depend on the motion relative to it of any observer or of any frame of reference. Thus, if the particle has an internal vibration of the type hypothesized by de Broglie, the phase of that vibration at any point in space-time must appear to be the same to all observers, i.e. the same in all frames of reference. Each observer or reference frame will have its own de Broglie wave for the particle. The phase of the particle’s vibration must, by definition,be the same as that of all possible de Broglie waves at the point where the particle is. By superimposing all these possible de Broglie waves, a wave packet is formed centred in space on the particle.

The superposition of B-waves detected by a set of observers moving relative to the laboratory frame at velocity v E (-c, +c) in the x’-direction produces a wave packet

+ + x:,

r , = d(x1- ~ x o ~ y xz 2

(34)

in which m = moc/h, which has no spread with time and propagates at a group velocity equal to the velocity of an associated particle. Mackinnon’s construction constitutes a particlelike solitary wave and fulfills all requirements of de BroglieBohm theory provided that they make the identifications

THE DUAL DE BROGLLE WAVE

153

One readily verifies that in Minkowski M (1,3) space Mackinnon’s soliton satisfies a photonlike d’Alembert equation (Mackinnon, 1981b).

O@(X’”) = 0,

(36)

whereas in M(1, 1) space it is a solution of a nonlinear field equation,

considered first by Gueret and Vigier (1982). Mackinnon’s model was reinvestigated by Czachor (1989), who showed that one can construct a wave packet competitive to Eq. (33), which for a particle at rest takes the form sin(mx’) sin(mx2) sin(mx3) @(xq = exp[imxO]. mx1 mx2 mx3 Equation (38) reveals that Mackinnon’s and Czachor’s constructions become identical only in M(1,l) space, whereas in quadridimensional M(1,3) space they differ significantly. C. Jennison-Drinkwater Electromagnetic Theory An interesting wave-particle model of matter is Jennison-Drinkwater electromagnetic theory (Jennison andDrinkwater, 1977; Jennison 1978,1983,1988; Jennison et al., 1986). according to which a massive particle is considered a relativistic phase-locked cavity with an internal standing electromagnetic wave. They showed that such trapped radiation has inertial properties of ponderable matter, and that all particles having inertial mass may be considered to consist of trapped radiation. According to Jennison and Drinkwater (1977): Movement of the perfectly reflecting wall of the cavity into the radiation falling upon it from the internal waves will create a small excess force from the radiation, for it will appear Doppler shifted to the blue and the rate of energy flow is increased relative to the equilibrium value when the wall was at rest. Thus one of Newton’s laws appears naturally at this stage.. . .

In Jennison’s (1988) mathematical formulation the excess force SF and kinetic energy T of a node moving at velocity u = Bc are given by SF = C

[

(

+

g --

T = -CnA ( 1B)”2 2

1-/?

+

”‘”1

( 11+B

(39)

,

I54

MARCIN MOLSKI

in which C is a proportionality factor, and nA, n = 1,2, . .. , denote a length of the trapped wave. According to Eqs. (39) and (40). one obtains

SF =

2T

2v

c2(1 - p)'/2z'

in which 6r = nA/c is a feedback echo period. As T is half the total internal energy of the system of trapped waves, whereas the second term is a Galilean acceleration A, Eq. (41) may be rewritten in the form

Equation (42) is merely a relativisticNewton's law for a particle endowed with rest mass mo moving at velocity u; a system of trapped electromagnetic waves is thus endowed with a rest mass. This result is fully consistent with de Broglie's fundamental hypothesis: that wave properties of an elementary particle are manifested in both moving and rest frames.

D. Corben Tachyonic Theory A trend is developed to assume that tachyons play a role in the structure of elementary particles (Hamamoto, 1974; Corben, l977,1978a,b; Rosen and Szamosi, 1980; Horodecki, 1988a; Recami, 1986, p. 111 and references cited therein). Corben (1977,1978a.b) showed that a free bradyon with rest mass mo and a free tachyon with rest mass2 mb can trap each other in a relativistically invariant way. If mo > mb, the compound particle is invariably a bradyon with rest mass

M~ = drn; - m f ,

(43)

* = JIJI'.

(44)

described by a wave function

satisfying a wave equation

[awa" + ( M ~ C / A 9 ) ~= ] 0,

(45)

with respect to an invariant interaction condition

a,+ar*'

= 0.

Such a bradyon-tachyon compound is considered two coupled particles associated with a timelike JI wave and a spacelike JI' wave being solutions of Klein-Gordon Bilaniuk and Sudarshan (1969) stated that the notion resr mass is somewhat a misnomer for tachyons, as a superluminal particle has no rest frame; they suggested the term proper mass.

THE DUAL DE BROGLIE WAVE

155

and Feinberg (1967) wave equations

+

The and +’ fields interact and lock to form a plane wave Q, which is timelike whenmo > mb; internal motion of associated bradyonic and tachyonic constituents can be either of oscillatory type or of Kepler type (Recami, 1986, p. 116). A single timelike state can become locked with at most three spacelike states, leading to a timelike state of mass (Corben, 1978a)

The number 3 reflects the condition expressed in Eiq. (46) and the number of spatial dimensionsof the M(1,3)-spacein which a bradyon-tachyon compound is formed. One cannot combine two time(space)like states in this way, because application of condition (46) to such states leads to imaginary momenta and exponentially increasing (not normalizable) wave functions. ’ b o bradyons or two tachyons cannot thus trap each other to yield a bound system of particles. This tachyonic theory holds under substitution (13) to produce a wave equation for a bradyon-tachyon compound interacting with an electromagnetic field A,, [D,D ,

+ ( M ~ c / ~Q )=~0,]

D,+D,+’

= 0.

(50)

(51)

The appearance of tachyonic components and an associated superluminal velocity is no inconsistency of Corben’s theory: Some wave equations based on perfectly valid representations of the Lorentz group lead to space-like solutions, thereby forcing us to deal with tachyons, if only to get rid of them. If relativity theory allows the existence of tachyons, and if they exist one can study them; if they do not exist we learn that something is wrong with our understanding of relativity theory. (Corben, 1978a)

According to well-known models of Salam-Weinberg type, gauge symmetry is spontaneously broken, filling the vacuum with tachyons. In this case such objects are Eggs-field particles that can be considered formally tachyons converted into bradyons (Recami, 1986, p. 117). A created tachyonic field may be interpreted as a component of a virtual massive photon (or Higgs-like particle), of which the four-momentum is orthogonal to the four-momentum of the bradyonic component (Horodecki, 1988a). Such authors as Barut and Nagel (1977) showed that spacelike particle states may play an important role in second-order processes such as the Compton effect, electromagnetic polarizabilities, and Delbriick scattering.

156

MARCIN MOLSKI

E. Horodecki-Kostro Model and the lho-Wave Hypothesis The Horodecki-Kostro model was developed in the framework of de Broglie’s postulate of wave-particle duality, the many-wave hypothesis (Kostro, 1978, 1985a,b; Horodecki, 1981, 1982, 1983), and Einstein’s theory of relativity, in which the existence of a unitary information ether (Horodecki, 1991) is assumed. This approach provides a conceptual background for an extended space-time description of matter, a two-wave particle model, and a nonlinear wave hypothesis. According to an extended space-time description of matter, a massive particle is described in terms of a timelike component characterized by an ordinary timelike four-momentum pp = ( E / c , p) associated with a B-wave, and a spacelike component described by a spacelike four-momentum p’p = ( E ’ / c ,p’) related to a D-wave. It implies a double, two-wave structure of a massive particle, characterized by the generalized equation (1) (Horodecki, 1988a, 1991) p @= hkp,

p‘@= hk’p,

(52)

k’@= (w’/c, k’)

(53)

in which k p = (w/c, k),

are wave four-vectors of B- and D-waves, respectively,and k’ = 2n/I’, w’ = 2n u’, in which I’and u’ denote the wavelength and frequency of a D-wave. The time and spacelike four-momenta satisfy the following conditions: p @= r n o u p , up

= ( Y C , yv),

p‘@= rnou’p, U’P

(54)

= (yu, yc),

(55)

PpP‘fi = 0,

(57)

in which c/c is a unit vector along the direction of the particle three-velocity. On applying quantization law (9) to Eqs. (56)-(57), one obtains the following time and spacelike wave equations:

[spa" + ( r n o ~ / h ) ~ = ] 0,

- ( r n o ~ / h ) ~ ]= o

(58)

+ ( x p ) = A exp [ i k , x @ ] , + ’ ( x p ) = A‘exp [ik&xp]

(60)

[apap

e’

and a field interaction condition

in which

are B- and D-wave functions, respectively.

THE DUAL DE BROGLIE WAVE

Substitution of wave solutions y? and dispersion relations for B- and D-waves,

k,k” = (o/c)~ - k2 = (oo/c)~,

157

+‘ into wave equations (58) produces

klk” = ( w ’ / c ) ~- k’2 = -(w/c)~, (61)

and a wave velocity interrelation

Cross correspondences (63) and (64)imply particle relations (Horodecki, 1991)

p = hk = ~ U ’ C / C= E’c/c,

p’ = hk’ = ~ U C / C= E c / c ,

(65)

and the frequency relations 2 wo = w-

*

w+,

w- = w - w’,

w+ = w

+ w‘.

(66)

These equations form a conceptual background for a nonlinear wave hypothesis (Horodecki, 1988a, 1991) according to which a massive particle in motion represents a nonlinear nondispersive wave packet (C-wave) that involves an internal (characteristic)spectrum of matter waves. A de Broglie oscillation of frequency w excites in the vacuum medium an oscillation of frequency w’, and Doppler-shifted frequencies w- and w+ arise (Horodecki, 1988a, 1991). The quantities w and w’ are interpreted as harmonics of the internal spectrum of a de Broglie extended particle, being a nonlinear C-wave excitation of the vacuum field identified by Horodecki with a unitary information ether (Horodecki, 1991). E Das Model and Pseudovelocity According to Das’s model (Das, 1986, 1988, 1992), one assumes that a massive particle in motion is simultaneously associated with velocity of two types: the actual velocity v = c drldxo, and an unobservable velocity v‘ = c dxo/dr (called a pseudovelocity),interrelated by d = c 2.

(67)

Accordingto Das, the wave-particle dualism of matter representedin E = h u, p = h/A, v, = v , and up = c 2 / u > c clearly demonstratesthat the idea of pseudovelocity v’ is not trivial and is indeed necessary to complete the set of correspondences:

MARCIN MOLSKI

158

u + E , )c + p , ug e u, and up * u’ among parameters involved in wave ( u , A , ug, u p ) and particle (E, p , u , u’) characteristics.

Employing Eq. (67), we find that the spacelikefour-velocity and four-momentum appearing in the Horodecki-Kostro model can be identified with the four-pseudovelocity and four-pseudomomentum given by U’P = ( y ’ c , y’v’) = ( y u , y c )

y’ = (8’2

- l)-I/z,

p’

= V’/C,

p’@= (mocy’,mov’y’) = (moyu, rnoyc).

(70)

Taking advantage of the extended wave-particle dual correspondences given in Eq.(52), one introduces a matter wave of the second kind (the transformed Compton wave) characterized by a wave four-vector

identical to that obtained for a D-wave in the Horodecki-Kostro model. Hence one may identify the transformed Compton wave introduced by Das with a D-wave in Horodecki-Kostro theory. Consequently, Eqs. (63) and (64)become written in the forms B = v D --

u
ugD=upB=v’>c.

G. Elbaz Model and Lorentz Transformations

The starting point for an extended wave model of a particle developed by Elbaz (1985, 1986, 1987, 1988) was an observation that the fundamental de Broglie hypothesis of wave-particle duality is incomplete by itself. On introducing the concept of a B-wave, de Broglie (1960, p. 4) assumed that in a rest frame one attributes to a particle endowed with energy E = mgc2 a periodic phenomenon of frequency 00 = E / h and phase S = o 0 x o / c . When a particle moves uniformly in the XI-direction at velocity u = c d x l / d x orelative to a laboratory frame, the phase of an associated wave undergoes a relativistic time Lorentz transformation X0

S(X’) = W O X ’ / C

4

+ (XO - p x ’ ) y ,

(73)

S(X’, x’) = WO(X’ - / ~ X ’ ) Y / C = koxo - k l x ’ , (74)

yielding an ordinary B-wave characterized by

THE DUAL DE BROGLIE WAVE

-

159

Although Lorentz transformations act on space and time coordinates, in these calculations only a time transformation is used; the space transformation x1

(XI - /3xO)y

(76)

is ignored (Elbaz, 1985). Following this conclusion, Elbaz assumed that a standing wave of photonlike type is associated with a particle at rest (Elbaz, 1985): q ( x 0 , X I ) = + ( x o ) + ' ( x ' ) = exp [iooxo/c] exp [ i w o x ' / c ] , OW(X0,

x ' ) = 0,

(77) (78)

in which the spatial part of Eq.(77) is called by Elbaz an amplitude wave. When a particle is moving, the phase of an associated amplitude wave transforms according to Eq. (76): S'(X') = OOX'/C

-

S'(x0, X I ) = % ( X I - /3x0)/c = k ' , ~ ' k&xo, (79)

kl, = W ' / C = w y / 3 = OOY',

k' = W O Y / C = ~ ~ / I ' Y ' / C ,

(80)

providing the wave function +/'(xo, X I ) = exp [i ( k i x ' - k h x o ) ] ,

(81)

whereas the time-dependent component + ( x o ) of the standing W ( x o ,X I ) wave transforms into an ordinary B-wave component, +(xo, X I ) = exp [i(koxo - k l x ' ) ] .

(82)

+

The type of superposition of +' and waves associated with a moving particle is photonlike; the d' Alembert wave equation (78) remains satisfied. Equations (65), (71), and (80) reveal that a D-wave, a transformed Compton wave, and an amplitude wave introduced by Elbaz are endowed with the same wave characteristics. H. Generalized Barut Approach Mackinnon's soliton theory was recently generalized by Barut (1990). who constructed exact localized oscillating energy solutions,

of the massless d'Alembert equation

160

MARCIN MOLSKI

that fail to spread and that behave like a single massive particle. The factor f ( w 0 ) appearing in Eq. (83) ensures that the solution has a finite energy (Barut, 1990). The time-independent function F ( r ) appearing in Eq. (83) satisfies the purely spacelike equation [A

+ (00/c)2] F0-1 = 0

(85)

and hence has the form

C~mr-'J2JI+~/2(oor/c)P;f(cosO) exp[im4].

F(r) =

(86)

Im

In the moving frame the solution (83) for 1 = 0 becomes

in which rv = [r2

+ 8' y2x,2 - 2y2x0p r + y 2 ( p

w=wy,

+

k=myp,

r12] 112,

~=v/c.

(88) (89)

For 1 = 0 and one-dimensionalmotion along the x'-axis, solution (87) reduces to the form

which differs from Mackinnon's soliton by the factor f ( o / y ) . 1. Conclusions

This summary of the most important concepts in the field clearly shows that all are consistent in many points and lead to the following final conclusions. 1. A massive particle in motion may be considered a system composed of a bradyonic component associated with a timelike B-wave and a tachyonic component connected with a spacelike D-wave. These particle constituents can trap each other in a relativistically invariant way to produce a bradyon-tachyon compound described with a superposition of the associated matter waves. 2. Models of a massive particle by Horodecki and Kostro, by Das, and by Elbaz are viewed as special cases of a more general tachyonic Corben theory that reduces to the former when masses of bradyonic and tachyonic constituents are the same. 3. Superposition of B- and D-waves forms a nonlinear nondispersive wave packet (C-wave), identified by Horodecki and Kostro and by Das with Mackinnon's soliton, of which propagation is governed by nonlinear propagation law (37).

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161

4. A particle at rest is associated with a system of photonlike standing waves. This assumption is compatible with both Elbaz’s model and Jennison-Drinkwater theory, in which a particle is considered a system of standing electromagnetic waves trapped in a phase-locked cavity. Radiation thus trapped has inertial properties of ponderable matter; and, vice versa, all massive objects are considered to consist of trapped radiation. 5 . The B- and D-waves are identified with exponentialand amplitudal waves appearing in de Broglie-Bohm theory; Eq.(17) for M O= 0 reduces to d’Alembert’s equation describing propagation of a C-wave identified with Mackinnon’s soliton. De Broglie-Bohm theory and a two-wave concept might thus be unified into one generalized theory. This problem is considered in detail in Section VII.

III. WAVE-PARTICLE MODELOF PHOTONS Wave-particle models of a massive particle considered in the preceding section may be extended to include massless objects of luminal type (photons, neutrinos, and gravitons)called luons. Photons (spins = 1) and gravitons (s = 2) belong to particles of integer spin in the class bosons, in contradiction to neutrinos (s = which are fermions. Consequently, their formulation is based on second-order (Klein-Gordon) and first-order (Dirac) equations, respectively. Considerations presented in this section are limited to photons. Characteristicsof a photon and its conjugated wave are given approximately by a four-momentum and a wave four-vector.

i),

related through waveparticle correspondences p; = A f f i ,

vg = v, = c,

(92)

in which w = 2nv, and f = 2n/A,; v, and A, are the frequency and wavelength of the electromagneticwave, and E, and p , denote the energy and momentum of an associated photon. Electric- and magnetic-field components of an electromagnetic wave obey Maxwell’s equations,

V E = 4np,

V x H - 4nc-’j - aoE = 0

V.H=O,

VxE+aoH=O,

(93) (94)

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MARCIN MOLSKI

in which j and p denote current and density of electric charge. A direct consequence of Eqs. (93) and (94) are the wave equations

a,a% + 4n (.-'ad + v p ) = 0,

+ 4nc-'V x j = 0,

i3,a.H

When p = j = 0, valid for a space free of charge and current densities, Eqs. (95) and (96) become t l p a p W ( x f l ) = 0,

Q ( x p ) = Aoexp [if#],

(98)

in which A0 = EOor HOdenotes the amplitude of an electromagnetic wave. In a description of the electromagnetic field one may use the four-potential AP = ( V / c ,A), whose components satisfy (Jackson, 1975)

H = V x A, a,WA"(xfi) = 0,

E = - V * V -&A,

A"@,) = AI; exp [if,x"],

(99) a = 0, 1,2,3. (100)

For an electromagnetic wave propagating in the x3-direction, Eq. (100) and its solutions reduce to a two-component equation (a= 1,2), A"(xo, x3) = A: exp [i (foxo - f3x3)>1,

a,aWA"(xo, x3) = 0,

(101)

A. Photon as a Bradyon-Tachyon Compound

Detailed analysis of Corben's tachyonic theory reveals that photons may be treated as a special case of bradyon-tachyon compounds, of which the particle constituents are endowed with the same rest mass (Molski, 1991,1993a. l994,1995a,b). When the bradyonic constituent is at rest and the tachyonic component attains a limiting infinite velocity, the associated field is periodic in time and independent of position, in contrast to a field that is static in time and periodic in space. Hence Eqs. (45)-(48) reduce to the simple forms

+'

+

[a: + (moc/h)*]+(xo) = 0, [A

+ (moc/h)2]$(r)' = 0,

+(xo) = exp[irnocxo/h],

(102)

+'(r) = exp[fimocr/h],

(103)

a,+(xo)#'$'(r) = 0, a,apW(xp)

= 0,

Q(x") = +(xo)+'(r) = exp [ i f P x " ] ,

p~ = (moc, moc),

fp

= (rnoch-', rnoch-I),

f, f, = ( w / c ) 2 - f 2 = 0,

in which c/c denotes the unit vector along a direction of wave propagation.

(104) (105) (106) (107)

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163

From Eq.(107) one obtains

The results indicate that a superposition Q of time- and spacelike waves and @' propagates at a phase (group) velocity equal to the velocity of light, whereas the associated compound of a bradyon at rest and a tachyon at infinite speed, termed transcendent (Recami, 1986, p. 23), is endowed with zero rest mass. Such a bradyon-tachyon compound may thus be treated formally as an object of luminal type; and, vice versa, photons may be consideredobjects composed of both bradyonic and tachyonic components endowed with the same rest mass. This result, 1993), c o n k e d in independent investigations by Dutheil(1984,1989,1990a,b, plays an important role in our considerations,presented below. B. Conversion of Light into B- and D-Waves Taking into account Eqs. (54)and ( 5 3 , one derives the relativistic mass formulas (Molski, 1991) mu = moy,

m: = moy' = mo/?y,

m: = (mu - rn:)(rn,

+ m:),

m: = /?mu,

(109)

(1 10)

that relate masses of time and spacelike components of a particle. Equation (1 10) shows clearly that mu and m: may be termed an internal mass spectrum of a particle, so that a massive particle in motion can be regarded as a composite object comprised of both bradyonic and tachyonic components. Equation (1 10) leads to a conclusion that at a luminal velocity, B=1,

m:=m,,

mo=O;

(1 11)

hence lurons appear to be objects without mass. This case features a great resemblance to conversion of particle-antiparticle pairs (of identical masses) into photons, accompaniedby disappearanceof the rest mass. According to a two-wave model, in the reverse phenomenon (e.g., Delbriick scattering), in addition to an ordinary bradyonic component associated with a B-wave, a tachyonic component connected with a D-wave is created. To prove this hypothesis, we consider a one-dimensional phase-locked cavity filled with standing electromagnetic waves, such as according to a JennisonDrinkwater model (Jennison and Drinkwater, 1977; Jennison 1978). When the motive reflector element on the wall of the cavity is moved forward at a velocity V I relative to the laboratory, the internal frequency wg undergoes a Doppler shift

164

MARCIN MOLSKI

according to the formula w1=w0(-)1

+

1 /2

UI/C

.

1 - U'/C The internal frequency 00 is received at the motive reflecting element at a frequency w1, whereas in the laboratory system it is reflected at the frequency

+

+

1 UI/C 1 - UI/C

~=wl(-1 ) " ~U=l / C% ( - )

1 - UI/C

=m(-) 1 + u / c

1/ 2

,

(113)

1- u/c

in which u = 2Ul/(l+ u:/c2>

is the velocity of the following mirror element (placed on the opposite wall of the cavity) moving at velocity v = c/? = d x 3 / d x o ; the frequency w is restored to the original value 00 (Jennison and Drinkwater, 1977; Jennison, 1978). If we restrict our considerations to one-dimensional motion along the x3-axis, the electromagnetic wave characterized with a four-potential A, = (0, A X ,A)', 0) propagates inside the cavity with a Doppler-shifted frequency w, in accordance with a luminal Maxwell equation a,aw(xo, x 3 ) = 0,

w ( x 0 , x 3 ) = A; exp [ ~ o , c - ' ( x ~- x O ) ] ,

(115)

Exponential part of Eq. (1 15) is written in the alternative form = exp [iooc-'y(l+ p ) ( x 3 - x o ) ] = exp [iw0c-'y(/?x3 - x o ) ] exp[iwc-'y(x3 - / ? x O ) ] ,

(117)

which with use of the Jennison-Drinkwater (1978) result rno = wo h/c2 becomes = exp [i(rnoc/h)y(/?x3- x o ) ] exp [i(rnoc/h)y(x3 - /?xo)]

= exp [ih-' (p3x3 - pox')] exp [ih-' ( p i x 3 - pbxo)] = + $ I .

(1 18)

Substitution of Eq. (1 18) into wave equation (1 15) yields

a,a,++'

= +lapap+

which on differentiating @ and provides the set of equations

+ +a,ap+l + 2a,+a~+'

= 0,

(1 19)

+' with respect to time and space coordinates

THE DUAL DE BROGLIE WAVE

165

These are merely field equations for a timelike B-wave and a spacelike D-wave appearing in a two-wave particle model (Horodecki, 1988a. 1991; Das, 1984, 1986, 1988, 1992; Elbaz, 1985, 1986, 1987, 1988) and a tachyonic theory of matter (Corben, 1977,1978a.b). Hence conclude that radiation trapped in a phaselocked cavity under motion of the motive reflector wall is Doppler-shifted and transforms into a system composed of B- and D-waves that lock to the form of the luminal wave. According to a corpuscular interpretation, photons in a phaselocked cavity undergo conversion into bradyon-tachyon components coupled to each other in a relativistically invariant way. The compound particle has photonlike characteristics: It moves at the velocity of light and has zero rest mass. The same holds for a cavity at rest. In such a case trapped radiation may be considered a system of purely timelike and purely spacelike waves having the same wave characteristics as those of the amplitudal wave considered by Elbaz (1985). A massive particle at rest may consequently be associated with an intrinsic system of standing luminal-type waves characterized by the wave four-vector

identical to that given in Eq. (77). This conclusion is consistent with models of Elbaz, of Jennison and Drinkwater, and of Corben, and with a Horodecki's twowave hypothesis of the structure of matter. C. Two-Wave Model of Longitudinal Photons

The preceding considerations are extensible to include radiation trapped in waveguides that are two-dimensional phase-locked cavities with the third dimension open. Thus one can investigatenot only the staircase dynamics of trapped radiation (Jennison and Drinkwater, 1977) but also one-dimensional kinematic phenomena of energy-momentum transfer mediated by longitudinal photons. Free photons are relativistic particles par excellence moving in vacuum at the velocity of light, characterized by zero rest mass and spin J = 1. The electromagnetic vector fields associated with such photons (known as transverse photons) are perpendicular to the wave three-vector determining the direction of wave propagation. The transverse photons associated with a transverse electromagnetic (TEM) wave are exactly massless; the extraterrestrial limit on the photon mass obtained g; an experiment by Williams by Goldhaber and Nieto (1971) is mo I4 x et al. (1971) yielded a similar result, mo 5 1.6 x g. In contrast, photons that have longitudinal components of electromagnetic fields cannot be massless (de Broglie, 1951; Perkins, 1982, p. 96). The latter case may be realized, for example, inside waveguides during transmission of transverse electric (TE)and transverse magnetic (TM) waves.

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MARCIN MOLSKl

We consider a rectangular vacuum waveguide of transverse dimensions a' and a2 with perfectly conducting walls and an interior free of charge. If a TE wave

is excited in the waveguide and propagates in the x'-direction. Maxwell's and Helmholtz's equations for the longitudinal magnetic H, component take the form (Jackson, 1975) 8pi3fl@,!,(xo) exp [i(knx3 - k0xo)] = 0,

= 1,2,

no = 0,1,2,. . . ,

( A + K ~ ~ ) + ; J X ~ )= 0, K ; ~ = ki

(Y

- k:,

(122) (123) (124)

in which ko = o / c = 2nv/c and k' = 2n/A; v and A denote the frequency and wavelength of the TE wave. An explicit form of solution of Helmholtz's equation, satisfying boundary con-

With a criticalfrequency (cutoff frequency) Wn,

= CKna

9

the group (phase) velocity of TE wave is defined as (Jackson, 1975) ~g

= cdko/dk = ~ [ -l ( W n , / o )

2 112

]

2 -112

up = cko/k = c[l - ( w n a / o ) ]

5 C, 2 c,

whereas Maxwell's equation (122) takes the form (apV

+ K ~ ~ ) @ ( x')X ' , = 0,

@(xo,x3) = exp [i(k3x3 - koxO)]

These equations yield thus pertinent conclusions. Phase and group velocities of transferred radiation satisfy a relation upug = c2 identical to that for a B-wave associated with a particle moving at velocity v g . The wave equation (131) is identical to the timelike Klein-Gordon equation for a particle endowed with rest , moving in the x'-direction. As the x3-directioncoincides with mass rn;, = h ~ ,/c

167

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the direction of propagation of the longitudinal H,component of the TE wave, we identify m:a with a mass of the associated longitudinal photon. Such a photon has bradyon-type characteristics (Molski, 1991).

E = poc = tto = mjlacz[l - (ug/c)2]-’/2,

( 132)

and the same relativistic properties as those of ordinary ponderable matter; for instance, it cannot move at the velocity of light because this case holds for w = 00; for w = unW, the longitudinal photon is at rest ug = 0, up = 00, and in the case of the luminal TEM wave, A+’(x”) = 0, K = 0, ug = up = c; the associated (transverse) photon is exactly massless. These astonishing conclusions become clarified in the context of the JennisonDrinkwater theory considered in Section: Solutions of Helmholtz’s equation (K # 0) describe an electromagnetic wave standing between the x z and yz planes of a waveguide. An application of the boundary conditions given in Eq.(125) leads to the appearance of standing-wave modes, directly related to the quantization of an associated mass,

Such trapped radiation has inertial properties of ordinary matter, with the possibility of excitation of various inertial states, in full accordance with JennisonDrinkwater theory. This concept is consistent with de Broglie’s (1960, p. 102) theory of constrained particle states. According to this concept, the constrained states present an illuminating analogy to the circumstances one encounters for photons enclosed in a waveguide, whose motions correspond to rest masses which vary according to the form of the waveguide and the type of waves propagated-rest masses much greater than the normal rest mass of the photon, which is zero or undetectably small. (de Broglie, 1960 p. 102)

Equations (122), (123), and (131) reveal that they are compatible with Corben’s of Helmholtz’s tachyonic theory and a two-wave particle model. The solution equation, being static in time and periodic in space, may be interpreted (Molski, 1993a.b) in terms of a D-wave associated with two transcendent tachyonsof infinite speed, endowed with rest masses n t t n l / a ~and nAnz/az, moving back and forth in oscillatory motion between the xz and yz planes of a waveguide. The exponential timelike solution is interpreted as a B-wave connected with a bradyon of rest

+Au

+

MARCIN MOLSKI

168

mass m& moving at velocity ug along the x3-dimension of the waveguide. Such time- and spacelike fields interact to form a Q,,= wave (interpreted according to the nonlinear wave hypothesis as a C-wave),

n (5) 2

q n m ( x f i )= HO

exp [i (k3x3 - k o x 0 ) ] ,

cos

(136)

ff=l

which satisfies the luminal Maxwell equation (122) and is associated with a bradyon-tachyon compound of zero rest mass:

Mo =

\I

" c($) =o. 2

(m;a)z -

ff=l

This conclusion becomes more comprehensible if we consider the simplest case of a TE1,o wave ( 9 , , , = l . n 2 = 0 wave) propagating at group velocity ug = 0. Then according to Eq. (133), k3 = 0 and ko = K ~,,whereas longitudinal and transverse components of the TE1,o wave become (Jackson, 1975)

(T')

H, = zf0cos - exp [ i K l s o X o ] ,

Hx -

ik3a3 lr

HOsin

($)

exp[i~l,~x']= 0,

If a waveguide with a T E l . 0 wave excited inside moves relative to the laboratory frame in the x'-direction at velocity u = c/l = dx'/dxO, the associated field components undergo relativistic transformations: XO

-x ' p

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169

producing

Amplitudal and exponential functions appearing there are solutions of space- and timelike wave equations

+La

to be identified with a D-wave associated with a tachyonic photon component of rest mass nA/(a'c),and a B-wave connected with a bradyonic photon constituent of rest mass ~ 1 . h0 / c . The interaction condition

+

ap+;vafi+

=o

(147)

is fulfilled in both rest frames. The case of a TM wave endowed with a longitudinal E, component of the electromagnetic field (Jackson, 1975)

satisfying boundary conditions \u(x" = a") = 0, provides the same conclusions. This analysis is extensible to include TE and TM waves transferred inside a cylindrical waveguide of internal radius ro. The longitudinal E, component of a TM wave is a solution of the wave equation (Jackson, 1975)

(a; - a,2 - , - - l a , , - r -2 a+2 - a ~ ) \ u m n ( x o , r , # , x 3=) o ,

( 149)

satisfying the boundary condition @ (ro) = 0 and given in terms of Bessel functions Jm ( r ) , Qmn(Xo,

r, $J,x 3 ) = EoJ,,, ( x m , , r ; l r )

.

exp[firn#] exp [i (k3x3 - koxo)], (150)

in which Xm,, rn = 0, 1,2,3,. . , n = 1,2,3,. .. denote the roots of Bessel function J m ( X m n ) = 0.

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MARCIN MOLSKI

Proceeding analogously to the case of a rectangular waveguide, one decomposes the luminal superposition given in Eq. (150) into space- and timelike solutions (8:

+ r-'% + r-2ai + xi,ri2)+k,(r,

4) = 0,

(151)

+k,n(r*4, x 3 ) = Jm ( x m n r i ' r ) exp [ztimbl,

(a:

- a;

+

K~,)+(XO,

(152)

x 3 ) = 0,

(153)

+(xo, x 3 ) = exp [i (koxo - k 3 x 3 ) ] ,

(154)

in which components of the B-wave four-vector k@ = (kO,0, 0, k 3 ) obey the dispersion formula 2 -2 ko2 - k32 -- K,,2 - x,,ro .

(155)

The solution +k,, of Helmholtz's equation (151) is interpreted as a transcendent D-wave associated with a tachyon of infinite speed and rest mass hx,,/cro moving in combined radial and circular motions about the bradyonic constituent of rest C with a B-wave propagating in the x3-direction. These mass ~ K , , , ~ / associated time- and spacelike fields interact to form a Q,, wave (C-wave) propagating as an excitation of luminal type associated with a bradyon-tachyon compound of zero rest mass,

+

The results indicate that this two-wave model of longitudinalphotons is consistent not only with a two-wave hypothesis of matter, a nonlinear wave hypothesis, and a tachyonic theory of matter but also with de Broglie-Bohm theory. Comparison of Eqs. (136) and (146) with Eq. (16) allows us to identify the amplitudinal part of TE and TM waves with R ( x @ )function, whereas

+'

S(x'L) = pox0 - p 3 2 .

(157)

When the guidance formula given in Eq.(26) is applied to Eq. (157) one finds that

+'

+

This result shows that the amplitude wave and the exponential wave are also interpretablein terms of singular and probabilisticwaves appearing in de Broglie's theory of double solution. Consequently, a two-dimensional region of space ( x , y ) or (r, 4) confined by walls of a waveguide represents an extended two-dimensional massive photon moving in the open x3-directionof a waveguide at velocity u = ug. The most important conclusion is that in the process of conversion of massless transverse photons into ponderable longitudinal photons, not only is an ordinary

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171

timelike component created but also a spacelike component associated with a tachyonic field called a matter wave of the second kind @-wave). D. Massless Photons and Ponderable Matter

A process of conversion of massless photons into ponderable matter occurring in a phase-locked cavity may be explained in a purely relativistic framework (Molski, 1991). Free photons and an associated coherent beam of electromagnetic rays are exactly massless, but for photons moving in various directions,the situation differs. We consider N identical photons in a system characterized by the total fourmomentum N

ni hwlc

P," = Nhwlc,

(159)

i=l

in which ni is a unit vector along the photon motion. The photon system as a whole has a nonzero rest mass

4

N

p o = ~ w c - NZ ~ -

where

C cos(eij),

cos(eij) = ni nj,

(160)

i. j = 1

eij = cos-'(ni, nj) # 0,

i

+j.

(161)

Such a cloud of electromagnetic radiation composed of massless photons is endowed with a nonzero rest mass, generates its own gravitational field, and is affected by external gravitational fields. For a two-photon system, Eq. (160) reduces to po = h ~ c - ~ [ 2 (-1 C O S ( ~ ~ , ~ ) ) I ~ / ~ ,

(162)

which indicates that a nonzero rest mass 2 hw/c2 may be attributed to an associated standing electromagnetic wave (01.2 = n). Hence we understand the result of Jennison and Drinkwater, who proved that standing electromagneticwaves trapped in a phase-locked cavity have inertial properties of ponderable matter. E. Extended Pmca Theory The preceding results indicate that the conventional Proca theory3(Jackson, 1975) of massive timelike longitudinal photons is extensible to include their spacelike m a ' s works are listed by Goldhaber and Nieto (1971).

172

MARCIN MOLSKl

constituents. Maxwell’s equations for timelike massive photons, which play a fundamental role in Proca’s theory, read V .E = 4np - K’V,

V*H=O,

V x H = 41rc-’j - K’A,

(163)

VxE=-aoH,

(164)

in which K = rnoc/h and mo denotes the mass of a timelike photon constituent, whereas A,, is a four-potential, defined in Eq. (99). Proca’s equations for a massive vector fields have the form (Jackson, 1975) (a,,ag

+ K’)A,

apF,,,

a,, jj’

4n . = --J,,, c

4n . = -J , ,

+ K’A,

c

Fpv= apAv - &A,.

= 0,

(167)

For a free massive electromagneticfield there exists a conservedenergy-momentum density

E=

81r

+ +

[E2 H2 K’(A’ 1 4RC

P = -[EX

+ A:)],

H+K’AoA],

so that

aoE + v . P = 0.

(170)

To extend this Proca theory to include spacelikeobjects is difficult. A fundamental requirement of such a generalized theory is invariance of the electromagnetic tensor FF’’ and the four-potential A” under superluminal Lorentz transformations. Such operations change a timelike tangent vector into a spacelike vector, and vice versa (they invert the quadratic-form sign), and form a new extended group G including subluminal (ortho- and antichronous) Lorentz transformations (Recami,1986, p. 54). However, extended relativistic theories including superlumind inertial frames present interpretive difficulties unless they are formulated in pseudo-Euclidean M ( n , n) space having space and time dimensions of the same number (Recami, 1986, pp. 39, 118). Hence only a two-dimensional representation of the G-group, acting in M(1, 1) space, has a clear physical interpretation (Maccarone and Recami, 1982). If one assumes Ffi”and A,, to be G-invariant.

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173

the ordinary Maxwell equations retain their form for spacelike objects (Recami and Mignani, 1974, p. 277). According to a two-wave model of matter, one can write basic equations for the extended Maxwell and Proca theory, including spacelike components of massive photons (Molski, 1991), V . E = ~ ~ ~ + K ~VVx H , =4nc-’j+~~A V * H= 0, (apap

V x E = -&H,

-K

(172) (173)

4n .

~ ) A= ~- j p , C

being the spacelike counterpart of timelike equations (163)-(167). Proceeding as in the timelike case, one writes the energy-momentumdensity of free electromagnetic fields associated with the spacelike component of a massive photon - 1[ E 2 + H 2 - ~ ’ ( A 2 + & ) ] , 877 1 ’ P I = -[E x H - K~AoA], 4nc

(177)

which imply the continuity equation

a,,€’

+ v . PI= 0.

(179)

For spacelike objects, Eq.(177) and (178) have a physical meaning only for E2

+ €I22 K ’ ( A ~+ A:).

E x H # K~AoA;

(180)

the case

E’

+ H~= K ’ ( A ~+ A:)

(181)

corresponds to an infinite group velocity allowed for tachyons (Molski, 1991). A simple extension of Proca’s original theory developed here is confirmed by Horodecki and Horodecki (1995), who showed that the timelike Proca equation and its spacelike counterpart are derivable from Dirac electrodynamics with spontaneous gauge symmetry breaking, and that Proca’s extended theory is fully consistent with Corben’s tachyonic theory and a two-wave particle model.

174

MARCIN MOLSKI

E Conclusions A photon is considered to be a composite object having both bradyonic and tachyonic components endowed with equal rest masses, which trap each other in a relativistically invariant way to yield a particle endowed with zero rest mass. According to the wave picture, a photon is associated with a superposition of timeand spacelike waves that lock to form a luminal-type wave propagating at a phase (group) velocity equal to the velocity of light in vacuo. A longitudinal Doppler effect for an electromagneticwave emitted from a moving source or reflected from a moving mirror is explained as a result of transformational properties of time and spacelike fields incorporated in a light wave. A massless transverse photon imprisoned in a phase-locked cavity or waveguide undergoes conversion into a bradyon-tachyon pair of which the timelike constituent is interpreted as a massive longitudinal photon. According to a wave picture, trapped electromagnetic fields are considered a nonlinear system of Band D-waves that lock to form a C-wave propagating as an excitation of luminal type. The timelike component of a longitudinal photon is endowed with the same properties as those of ordinary ponderable matter-it has rest mass and moves at a subluminal velocity that cannot exceed the velocity of light. Consequently, such objects are considered relativistic models of massive particles (Molski, 1993b). Relativistic equations ( 132)-(139) are derived using neither a Lorentz transformation nor reference to geometrical properties of space-time, whereas the KleinGordon equation (131) is obtained without applying a standard quantization formalism. One can hence derive fundamental equations of special relativity and quantum mechanics in a purely electromagnetic framework (Molski, 1993b). Interpretation of wave-particle phenomena appearing in a phase-locked cavity and a waveguide interior is consistent with a two-wave particle model, a nonlinear wave hypothesis, Corben’s tachyonic theory, and models Jennison and Drinkwater and of Elbaz for a massive particle. The appearance of a spacelike constituent in the spectrum of mass associated with a longitudinal photon indicates that Proca’s conventional theory is extensible to include spacelikephoton states. Such a generalizationis made in the framework of a two-wave hypothesis of matter or Dirac electrodynamics with spontaneous breaking of gauge symmetry.

Iv.ELECTROMAGNETIC MODELOF EXTENDED PARTICLES Although according to standard concepts of physics one attributes a pointlike character to material objects, this approach generates unphysical effects of infinite

THE DUAL DE BROGLIE WAVE

175

self-energy and self-field; to overcome these difficulties, Dirac distributions or a renormalization procedure is used. To avoid these complicated and abstract operations, several models are proposed (Jehle, 1971, 1972, 1975; Post, 1982, 1986; Jennison, 1983; Jennison et al., 1986; Elbaz, 1987, 1988, 1995; Molski, 1991, 1993a, 1994, 1995a,b) in which an elementary particle is treated as an extended structure, not as a pointlike object. Because radiation trapped in a waveguide is endowed with kinematical properties of ordinary matter and associated longitudinal photons with two-dimensional spatial extension, such a system accurately reproduces fundamental properties of extended particles, although neglecting their extension in a third space dimension. Therefore, the approach described above should be generalized to include electromagnetic fields trapped in three-dimensional resonator cavities, which seems the best geometrical representation of these objects (Molski, 1993a, 1995a.b).

A. Three-Dimensional Rectangular Space Cavity We considerarectangularelectromagneticspace cavity @-cavity) of size a, (a= 1,2,3) with perfectly conducting walls and an interior free of charge. Under these assumptions the imprisoned electromagnetic fields satisfy the luminal Maxwell equation (Jackson, 1975)

in which mn,c is a resonator frequency and equation (A

+ m ~ a ) $ A o ( ~=u )0,

are solutions of Helmholtz’s

IZ, = 0,

1,2,. . ..

(1W

If the TE mode, endowed with the longitudinal H, component of a magnetic field, is excited in a S3-cavity,solutions of Eq. (184) satisfying suitable boundary conditions take the form (Jackson, 1975)

Equation (185) in this form ensures that boundary conditions are satisfied not only for the longitudinal component of the TE wave, but also for the transversal ones, E l and HL.to be obtained from Eq.(185) and Maxwell’s equations (93) and (94).

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MARCIN MOLSKI

With the results obtained in Section 111, a nonzero rest mass

becomes attributed to electromagnetic fields imprisoned in an SN-cavity. This associated mass has inertial properties of ordinary ponderable matter, with the possibility of quantization of various inertial states (Jennison and Drinkwater, 1977; Jennison, 1978). Consequently, all intrinsically stable massive objects are considered modes of imprisoned radiation. If we trap an electromagnetic field within a rectangular S3-cavity,spatial field transmission in the external cavity domain vanishes; Eq. (182) reduces to a timedependent formula

(a: + miu)+(xo) = 0,

+(xo) = exp [im,,xo]

(187)

describing propagation of outer or radiating fields (Elbaz, 1988) associated with a massive electromagneticS3-cavityalong the xo-axis. Because in four-dimensional space-time the direction of field propagation coincides with the worldline along which the associated object moves, the xO-axisappears incidentally as a worldline of the S”-cavity. Consequently, the timelike Klein-Gordon equation for outer fields associated with a freely moving S3-cavity in the x3-direction at velocity u = CB = c dx3/dx0 is obtained on application of Lorentz’s time transformation to Eq.(187), yielding

(%ap +mi,,)exp [ik,xp]

= 0,

Simultaneously under a uniform motion, the trapped internal S3-cavity fields, called inner or selfjelds (Elbaz, 1988), transform into time and spacelike fields (Molski, 1991-1 995b).

$Aa+

of which superposition yields a TE wave (C-wave) q n , , = propagating in accordance with wave equation (182) as an excitation of luminal type. Functions (190) and (191) satisfy space and timelike Klein-Gordon equations

177

THE DUAL DE BROGLIE WAVE

consequently, from Eq. (182) one obtains the invariant interaction condition a p + ; J x q a q b ( x p ) = 0.

(193)

Functions (190) and (191) are also given in covariant forms:

in which p p is the bradyonic four-momentum and p'p, p i , and p i denote the tachyonic four-momentum and momenta of transcendent tachyons, respectively. Equations (191) and (192) are fully compatible with a two-wave description of matter and a tachyonic theory of the structure of an elementary particle. In is interpreted as a superposition of three independent spacelike particular, states (D-waves),

+Aa

associated with tachyonic particle constituents related to three mass states m:. = nttc-'n,a;'. When a cavity is at rest, these spacelike objects move back and forth between cavity walls at infinite speed along the x,-axes, whereas associated fields are described with functions periodic in space and independent of time. In contrast, @ ( x o ) is interpreted as a B-wave periodic in time and independent of position and orthorhombic is associated with a bradyonic constituent at rest of mass m:--an lattice with spacings h / p l ,h / p , , and h / p , (Corben, 1978a). Interaction between such fields is analogous to scattering of a wave at a diffraction grating (Corben, 1978a). Three values of luffice spacings in three directions of space correspond to masses of three spacelike states that combine with one timelike state (Corben, 1978b; Recami, 1986, p. 113). In particle terms we interpret it as conversion into a bradyon-tachyon bound system of particles that trap each other in a relativistically invariant way, yielding compound particles endowed with zero rest mass:

178

MARCIN MOLSKI

A TE mode excited inside an $-cavity is then treated as a system of B- and D-wave that lock to form a photonlike C-wave *,,a propagating as excitation of luminal type. This tachyonicinterpretationof an SN-cavityinterior agrees with that of Corben (1977, 1978a.b) and with other theoretical results in the field (Recami, 1986. p. 111 and references cited therein). Thereby a free bradyon can trap at most three free tachyons (Corben 1978b). or in wave terms, that at most three spacelike states are superimposable on one timelike state to produce another particle [see Eq.(48), cf. also Pagels. 1976; Hoh, 19761. This approach is applied to a TM mode for which the Lorentz transformed E, component of electric field, satisfying suitable boundary conditions, takes the form

Thus these considerations hold true for a TM mode excited inside a rectangular S'-cavity.

B. Three-DimensionalSpherical Space Cavity Extended particles in a rectangular geometry, treated as electromagnetic S'-cavities, contradict intuitive expectations and a standard concept attributing to all particles a spherical geometry rather than a rectangular one. Therefore, we treat a construction, in an electromagnetic framework, of an improved model of extended particles, which takes into account not only their wave-corpuscular character but also a spherical geometry. To proceed we consider a spherical cavity of internal radius a , which has perfectly conducting walls and an interior free of charge. If a TE wave is excited inside the cavity, its field component Eg propagates according to Maxwell's equation (Jackson, 1975).

{a: - [a,? + r-2( sinV'a6 sinBa6 - ~ i n B - ~ ) ] } r E= + 0,

(203)

Eg = Eo(m,rr)-'Jfl/(mfl/r)0(8): exp [im,/xo],

(204)

.

in which m,,lc, n, 1 = 0 , 1 , 2 , . . is the resonator frequency of the TE,/ wave and oY(f3)denotes associated Legendre polynomials with m = f l ; J,,/(m,/r) are spherical Bessel functions of order f satisfying

[a:

+ m:/ - Z(1+

l)r-2]J,l(m,/r) = 0,

J,/(O) = 0.

(205)

THE DUAL DE BROGLIE WAVE

179

The remaining field components H, and He are calculated on making use of Maxwell's equations (93) and (94), providing

H, = i Eol(l+ l)(mnlr)-2Jnl(m,,lr)@(0)l exp [imnrxo],

(206)

He = -iEo(m,r)-2r-'a,J,l(m,,r)0(0): exp [im,lxo].

(207)

Imposing on a electric Eg component of TE wave the boundary condition E4 ( a ) = 0, one obtains the resonator frequency

in which x , ~denotes root n of a spherical Bessel function Jnr(xnl) = 0. In the simplest case of the TE,o wave, the amplitudal part of Eq. (204) becomes

J,o(mnor) = sin(m,or),

(209)

and the resonator frequency and field components of the TE wave take the forms nn m,o = -, a

H, = 0. Equation (212) reveals that He, having a normal orientation relative to the cavity surface, satisfies the boundary condition a,r He ( a ) = 0, providing the same values of the resonator frequency as those of a field component E4. Proceeding as in the case of a rectangular cavity, we attribute a nonzero rest mass hnlr m:o = ca

to radiation trapped inside a spherical cavity, which depends on the mode characteristic n and the cavity dimension a. The mass mlo associated with imprisoned fields is quantized, fully in accordance with the results of Jennison and Drinkwater (1977) and Jennison (1978). Hence the electromagnetic spherical cavity may be employed as a wave-corpuscular model of extended massive particles endowed with spherical geometry. Exploiting this idea, we identify the fields trapped inside the cavity (r E < 0, a ) ) with inner (self) fields propagating as excitation of a luminal type, whereas outside the cavity (r E (a, 00)) spatial propagation of fields vanishes and Maxwell's equation (203) reduces to a purely timelike equation,

(a: + mio) exp [imnoxo]= 0,

(214)

180

MARCIN MOLSKl

describing the propagation of outer fields in the external cavity domain along the xo-axis being, incidentally, the cavity worldline. A wave equation governing propagation of outer fields, associated with a cavity moving in the x’-direction at velocity u = c d x l / d x o ,is obtained on application of Lorentz’s time transformation,producing (a,W

+ rn;,)

exp [ik,xfl] = 0,

(215)

kp = {rn,o(l - /?2)-’/2, rn,op(l - p2)-1/2,0, O } .

(216) Equation (2 15) is merely a Klein-Gordon wave equation describing propagation of a timelike B-wave associated with a freely moving particle of rest mass rno = rn,o h / c and spin zero. To obtain the Lorentz transformed inner fields associated with a moving cavity interior, we note that not only is @‘noa solution of the wave equation (203) but it also satisfies

Jm,

r= with a d’Alembertian in terms of Cartesian coordinates

0= a;

(218)

- a; - a; - a.;

(219) Simultaneous application of Lorentz time and space transformations to Eq. (21 1) provides inner fields in the form

+ +

rv = d ( x 1 - bx0)2/(1- 8 2 ) x; x z . (221) Equation (220) shows that inner fields associated with a moving cavity satisfy

181

THE DUAL DE BROGLIE WAVE

identical with those in Corben’s (l977,1978a,b) tachyonic theory of matter, a twowave particle model and Mackinnon’s soliton theory. +Ao and fields are then identified with D- and B-waves, respectively; their superposition yields a C-wave Qn0 propagating as an excitationof luminal type. According to a particle picture, it is interpretable as conversion of a photon into a bradyon-tachyon system of which the particle constituents trap each other in a relativistically invariant way, yielding a photon-type particle endowed with rest mass zero, as

+

For a sphericalcavity a tachyonic componentcreated has colaritudal(O),azimuthal (#), and radial ( r ) degrees of freedom; it can move on the surface of a sphere centered at the bradyonic constituent, but the radial degree of freedom also enables the tachyon to travel between infinitesimally close spherical surfaces. Hence such a bradyon-tachyon system created in a spherical cavity structurally resembles a hydrogen-type (H-type) compound. According to Mackinnon’s soliton model, a Qn0 field is interpretable as a nonlinear nondispersive wave packet that has no spread with time and that constitutes a particlelike solitary wave. Gueret and Vigier (1982) showed that such a soliton wave follows a geodetic in an external gravitational field; so it behaves as a singularity of the gravitational field (test particle). Because Mackinnon’s soliton is endowed with a three-dimensional spatial extension and an inertia property, it is proposed as a wave-corpuscular model of extended massive particles (Mackinnon, 1978, 1981a,b, 1988). Because in M(1, 1) space propagation of a Qn0 wave is governed by a nonlinear propagation law (Gueret and Vigier, 1982), it may be interpreted in the framework of a nonlinear wave hypothesis as a C-wave involving B- and D-waves as internal spectrum waves associated with the cavity interior. Consistent with all models mentioned in Section II,a wave picture of the interior of a spherical cavity emerges. Radiation trapped inside the cavity undergoes conversion into a timelike B-wave and a spacelike D-wave +,,o that lock to form a solitary photonlike wave Qn0. Such imprisoned inner fields form a nondispersive wave packet that has no spread with time and that travels at a group velocity equal to that of the moving cavity. We focus our attention on correlation between geometrical characteristics of spherical cavities and of extended particles. For this purpose, on the basis of the rest-mass formula given by Eq.(213). we derive the relationship

+

in which d is a cavity diameter and A0 = h/rnE0cdenotes a Compton wavelength characterizing the massive cavity at rest. Equation (229) indicates that for the fundamental T E l o mode the diameter of a spherical cavity at rest is equal to the Compton wavelength of an associated mass

182

MARCIN MOLSKI

and correspondsto the width of a wave packet measured at the first zero points of a function W 1 0 ( x o ,a ) . Hence standard parameters characterizingextended particles are strongly correlated with the geometry of spherical cavities. The geometry of the cavity and its associated mass vary in the moving frame. To find suitable relations, we consider for interpretative simplicity propagation of inner fields Q,," projected onto M(1, 1) space, which is governed by the wave equation

(230)

-

Equation (230) reveals that the radius of a moving cavity undergoes a relativistic deformation and compresses in the direction of motion, a

(23 1)

a J v ,

in strict conformity with a transformation of the Compton wavelength,

Simultaneously,the mass associated with a cavity varies according to the equation

which indicates that radiation trapped in a spherical cavity behaves as ordinary ponderable matter, and the relativistic mass problem may be considered in a purely geometrical framework (Molski, 1991, 1993a). For a de Broglie wavelength associated with a moving cavity 1 = h / p l , PI being a component of the cavity three-momentum,one obtains

for which p' = p - I ; the de Broglie wavelength measured in diffraction experiments provides important information about the geometry of particles considered as spherical cavities. Because the width between the first zeros of the amplitude of the bradyonic soliton (230) is equal to

A X ' = 2aJ-

=

h m:&?'y'

- -h

the uncertainty principle for a spacelike momentum is

Ap; '

(235)

THE DUAL DE BROGLIE WAVE

183

We investigate geometrical characteristics of a bradyon-tachyon H-type compound created inside a cavity. Because the spacelike field associated with a tachyonic component is delocalized in space (Recami, 1986, p. 56), one obtains useful information about the internal structure of a particle system by calculating the radial distribution of a tachyonic field in space surrounding a bradyonic constituent. Taking advantage of the formula

dp(r) =4 ~ r ~ J , o ( r ) ~ d r ,

(237)

and introducing into Eq. (237) the radial function given by Eq. (209) for u = 0, we obtain

which provides the probability of finding a tachyonic constituent within a spherical shell of radius r and thickness d r , or within volume 4 n r 2 d r at radius r . The function given by Eq. (238)becomes zero both at the bradyonic constituent and at the cavity wall and has n maxima at points

+

2k 1 r k = ( y ) a ,

k = 0 , 1 , 2,..., n - 1 .

(239)

For the fundamental mode (n = 1) the radial distribution attains a maximum for ro = a / 2 ;the greatest probability to finding a tachyon moving in space surrounding a bradyon occurs at half the cavity radius. These considerations are extensible to a TM wave endowed with field components ( H # ,E o ,E r ) . In this case the boundary conditions (Jackson, 1975)

&(a) = 0,

a,rH@(a)= 0,

(240)

imposed onto a TM,o wave yield a resonator frequency

mnlc =

c(n

+ 1/2)n.

a then the geometrical correlation given in Eq. (229)is absent. C. One-DimensionalLinear Time Cavity Derivation of a timelike Klein-Gordon equation for outer fields associated with a massive electromagnetic $-cavity is divided into two stages. On the basis of Maxwell’s and Helmholtz’s equations, one obtains a time-dependent wave equation (214)describing a massive cavity at rest; application of Lorentz’s time transformation yields a timelike Klein-Gordon equation (2 15) for outer fields associated with a moving cavity. Because bradyon-tachyon symmetry is strictly related to time-space symmetry (Recami, 1986, pp. 34,35), one derives (Molski, 1993a) within an electromagnetic

MARCIN MOLSKI

184

framework a spacelike Klein-Gordon equation employing mirror operations of time imprisonment of electromagnetic fields and Lorentz’s space transformation. Although a time-trapped field seems speculative, this possibility deserves attention, as it plays an important role in quantum theory of spacelike states. For example, Horodecki (1988b) considered in a nonrelativistic regime time quantization of spacelike fields trapped in an impulselike rectangular well, and Vyiin (1977a) investigated quantization of spacelike states on a closed timeline. As the time- and spacelike representations of a Poincark group are SO3 and S02,1,respectively, tachyons are not localizable in our ordinary space (Duffey, 1975,1980;Vygin, 1977a.b) and appear more to resemble fields than do particles of fmite spatial extension (Recami, 1986,p. 59). Tachyons invariably admit reference frames (called criticulframes) in which they appear (at a speed u = 00) as points in time extended simultaneously in space along a line (Recami, 1986, p. 56). Hence the concept of a tachyon as a time cavity with trapped electromagneticfields seems reasonable from a physical point of view. However, we refrain from comment on how such time imprisonment may be implemented practically. To realize this concept, we consider an electromagneticwave characterizedwith the four-potential Ap = (0,AX,AY, 0), which propagates in the fx3-direction. Nonzero components of A” satisfy a luminal Maxwell equation A ~ ( x =~ A; ) exp [i(f3x3 r foxo)],

a,apAu = 0,

(242)

in which f p = { w / c , 0, 0, f 3 } denotes the wave four-vector of an electromagnetic field with f3 = fo = w / c . After such a wave becomes trapped in a onedimensional T’ cavity of dimension ao, placed on the xo-axis, the imprisoned fields obey the equation apavQ(xp) = 0,

Q(xp) = A;$,,.,,(x0) exp [if3x3],

(243)

in which $,,.,,(x’), no = 0, 1,2, ... are solutions of the time-dependent Helmholtz equation

(a:

+m~o)$no(xo)= 0.

(244)

Taking into account a boundary condition for time (Horodecki, 1988b), $,,JXO

= aO) = 0,

(245)

similar to the spatial one, we find solutions of Helmholtz’s equation (244) in the form

If a nonzero rest mass

THE DUAL DE BROGLIE WAVE

185

is associated with radiation trapped in time, Maxwell’s equation (243) for outer fields associated with a T ‘-cavity exterior reduces to (8;

+ mio) exp [ikax3]= 0,

k3 = m 0n o c K1 .

In the next step, application of Lorentz’s space transformation to Eq. (243) yields a spacelike Klein-Gordon equation (Feinberg, 1967),

(0- m,2,,)+’<x@>,

+’(x”) = exp [ikLx”] = 0,

(249)

describing propagation of spacelike outer fields associated with a freely moving TI -cavity at superluminalvelocity u’. Simultaneously,inner fields associated with a T -cavity transform to yield

’

in which an amplitude function

+,, satisfies a timelike Klein-Gordon equation

Equation (25 1) reveals that an electromagneticfield trapped inside a T ‘-cavity can be considered to be a system of D- and B-waves, but by comparison with spacetrapped radiation the amplitude wave has timelike characteristics, whereas the exponential one is a wave of spacelike type. Such trapped electromagnetic fields reproduce fundamental properties of tachyons (ie., ponderability, localization in time, and superluminal kinematic). This approach fails to reproduce the threedimensional extension of tachyons, which seems to be a genuine property of all particles, independent of their time- or spacelike characteristics. D. Three-Dimensional Spherical Eme Cavity

To construct an electromagnetic model of three-dimensional tachyons, we generalize the approach proposed by Horodecki (1983a), who constructed a onedimensional tachyonic soliton dual to the soliton constructed by Mackinnon. In the superluminalrest frame the tachyonic soliton has the form (Horodecki, 1983a) W(x ,x ) =

sin(mocxO/h) exp[imocx’ / h ] , mocxO/h

(253)

186

MARCIN MOLSKI

whereas in a Lorentz frame moving at a velocity u = cj? = c dx ' / d x o ,it becomes

in which y = (1 - j?')-'/'. Such a nondispersive solitary wave, viewed as a dual C-wave (Horodecki, 1983a), does not spread during its superluminal motion; it may be viewed as an extended model of one-dimensionaltachyons. To generalize Horodecki's approach to three dimensions, one finds ordinary M(1,3) space-time inadequate, as it assumes the existence of a single time dimension. Several authors (Cole, 1977, 1979, 1980a.b; Demers, 1975; Recami, 1986, p. 130 and references cited therein; VySin, 1995) support the idea that a theory of relativity involves three time dimensions. Cole (1980a) showed that sixdimensional Lorentz transformations are derivable in which two extra variables are interpreted as time coordinates. According to Cole (1980b), the metric of such pseudo-Euclidean M(3,3) space has the form 6

ds2 =

dyj d y J = c2(dr:

+ d r i + dr:)

- dx: - dx;

- dx;,

(255)

j=1

whereas a link with a standarddescriptionin M(1,3) space is achieved by assuming the particle three-velocity to be dr dr

v = C--,

dr = ldtl =

d

m

.

(256)

in which d t / d t determines the direction of a time displacement in six-dimensional space. These relations indicate that a transition from vector time of Cole's theory to scalar time appearing in our physical M(1,3) space can be realized in the same manner as passing from Cartesian coordinates to spherical ones: ( x ' , x2, x 3 )

-, (r, o,4),

r = dx: + x i

+x f .

(257)

Hence a subspace M(3,O) has spherical symmetry, resulting in the space-time metric form ds2 = dp2

+ p2(dp2+ sin219) - dx: - dx; - dx:, p = ct = c d t ; + t; + r f ,

crl = p sin(I9) cos(p),

cr2 = p sin(I9) sin(p),

cr3 = p cos(I9).

(258) (259) (260)

We consider a spherical electromagnetic wave trapped in a three-dimensional spherical time T'-cavity of dimension po, which satisfies a six-dimensional Maxwell equation

+

[p-2app2ap P - ~ sin ( 8-'a,~sin I9ao

+ sin I9-'8:

-

a:)]

Q ( y i ) = 0.

(261)

THE DUAL DE BROGLIE WAVE

187

For the simplest case of purely radial solutions and one-dimensionalmotion along the XI-axis,one obtains

q,o, XI)

=

sin(m,!p) mn'P

exp [ i m , ~ ~ ' ]n', = 1 , 2 , 3 , . . . ,

in which the amplitudal function is a solution of a time-dependent Helmholtz equation (p-'aPp2ap

+ m;f)+,,r(p)= 0,

(263)

n'n m,,l = m:, A/c = -, PO

and we attribute rest mass m!, to radiation trapped in time. Bearing in mind Eq. (259), we write the function (262) alternatively as

which is a three-dimensional version of a tachyonic soliton (253) constructed by Horodecki (1983a). In M(1, 1) space, Eq. (265) reduces to Eq. (253). According to Eqs. (253) and (265), an extended tachyon may be viewed as a lump of energy localized in a closed region of time. Total imprisonment of energy carried by an electromagnetic wave produces a transcendent tachyon endowed with zero energy and nonzero momentum m:,c. As the width between the first zeros of the time-dependent amplitude of the tachyonic soliton (254) is equal to Axo = 2poJ1-gz

h

hc AE'

= -- -

mjj,cy

it implies the uncertainty principle AEAt = h

of conventional quantum mechanics. E. Two-Dimensional Square Space-Time Cavity

Because wave-particle duality appears as a genuine property of matter, including objects both massive and of zero rest mass, we construct in an electromagnetic framework an extended wave-corpuscular model of photons. The starting point is the fact that photons may be viewed as a special case of bradyon-tachyon compounds of which the constituents endowed with the same rest mass couple with each other in a relativistically invariant way (see Section 1II.A). Such a compound particle has photonlikecharacteristics;it moves at the velocity of light, has zero rest

MARCIN MOLSKI

188

mass, and is associated with a system of waves propagating as an excitation of luminal type. With time- and spacelikeobjects as electromagneticSN-and T1-cavities, respectively, one combines these two approaches to construct a wave-corpuscular model of a photon treated as an extended object and not as a pointlike structure. We consider a simple model in which bradyons and tachyons are considered to be S’ and T’phase-locked cavities of dimensions a3 and ao, respectively, with internally trapped electromagneticfields. In the case of photons, the notion cavity wall has only an auxiliary value, as massive objects (walls) cannot travel at the velocity of light. Proceeding as in Section IV.C, we generate the following equations: S’ -cavity: apa”Az#n,(X3)

exp [ifoxo] = 0,

(A + m i , ) # n , , ( ~ 3 = > 0,

n3

..

= 0, 1 , 2 , . ,

According to results of preceding sections, nonzero rest masses m:, and mXo are attributed to space- and time-trapped electromagnetic fields, whereas associated inner fields are given in the general form w p ’ )=4(X3)*(XO),

W(XP)

=#(XO)@(X3).

(276)

With this solution one may build a nonlinear superposition of time- and spacelike fields 52(xfi)

= Q(X@)Q’(Xfl) = # ( x ’ ” ) x ( x p ) ,

# W )= 4 ( X 0 M ( X 3 ) ,

X ( X 9

= *(x0)*’(x3),

(0) (7) m 3 x 3 exp [i (foxo q= f 3 x 3 ) ]

52 ( x p ) = A: sin m o x o sin

(277) (278)

(279)

THE DUAL DE BROGLIE WAVE

189

associated with a bradyon-tachyon system of particles treated as one-dimensional phase-locked cavities. This function and its field components 4 and x satisfy the lurninal wave equations =o

a,aw(xq a,afix(x”) = 0,

a,ah$(x,)

(280) = 0,

(281)

provided that associated bradyonic and tachyonic constituents are endowed with the same rest mass,

and that an interaction condition a , ~ ( x ~ ) a ~ x ( x= c ”0)

(283)

is fulfilled. A requirement for equality of rest masses of the interacting bradyon and tachyon given by Q. (282) implies correspondences 03

= ao,

n3

= 110,

(284)

so that dimensions of the space cavity must equal those of the time cavity and so that trapped radiations must have the same mode characteristics. Introducing in explicit form the wave functions x and 4 into Eq. (283), and employing Eqs. (271) and (275), we obtain

or in an equivalent form,

Equation (286) is satisfied for x 3 f x0 = x 3 f c t = a 3 = a o .

(287)

This condition determines the initial position of a wave x = exp[i(koxo 7 k3x3)] propagating in the 7x3-direction at a constant phase velocity equal to the velocity of light (Coulson and Jeffrey, 1977). Hence the x wave propagates in an external space-time cavity domain: It can be identified with outer or radiating fields, in contrast to a 4 wave representing inner fields associated with a space-time cavity interior. Consequently, extended photons appear in our formalism as twodimensionalspace-time (S’T ’) square cavitiesof equal space and time dimensions. This condition is a consequence of identical mass of bradyonic and tachyonic photon constituents. As such imprisoned inner fields are endowed with an effective

190

MARCIN MOLSKI

zero rest mass, outer fields obey the luminal Maxwell equation, in agreement with our knowledge of the field. Hence, an electromagnetic square S' TI -cavity is considered a wave-corpuscular model of a photon as it reproduces its fundamental properties: zero rest mass, nonpoint extended structure, and luminal kinematics. If we take into consideration the de Broglie relation pp = hf p that is valid for massive and massless particles, one obtains

E = p 0c = hnna- I c,

p'3 = hnna-I,

(288)

or in an equivalent form, 0 2 E = hw = m,c = nhwo,

w0 = n a - l c ,

(289)

These formulas for a E (0; 00) reproduce the full spectrum of waves associated with photons, indicating a rigorous connection between wave and geometric characteristics of associated luminal particles viewed as an S 1TI-cavity: values of parameters such as frequency or wavelength are related to SIT1-cavitydimensions. Moreover, these formulas not only provide a value of the elementary energy-momentum quantum (for n = 1) but also predict the possibility of the existence of its integral multiple (multiphoton) for n = 2,3, .. . . Multiphotons appear, for example, in Planck's emission, in photoelectric effect, and in Compton scattering, and can spontaneously decay to the ground state (n = I), producing n spatially separated but coherent single photons. Chin and Lambropoulos (1984) showed that when an isolated atom in an excited state decays spontaneously to its ground state, the emitted light must be a single photon having a well-defined energy and frequency and that the process must occur within a characteristic duration comparable with the period of oscillations of the emitted photon. Hence the spatial dimensions of the photon must be on the order of its wavelength. Equation (275) reveals that a tachyonic component of a photon, endowed with infinite momentum p 3 = h f = m!,]c and zero energy, is responsible for momentum carried by a luminal wave. In contrast, Eq. (27 1) indicates that energy of a c-velocity wave is rigorously connected to a bradyonic component endowed with zero momentum and energy E = m!,c2. The equation p p = hf p was first considered for photons by Einstein and extended by de Broglie to include massive particles, providing a conceptual background for a fundamental hypothesis of wave-particle duality and a formulation of quantum mechanics. These results indicate that wave-particle duality for photons may be accounted for by de Broglie relations (1) plus an energy-mass relation giving w = h-'m:lc2, f = h-'rn:,,c. Hence a tachyonic theory can explain that zero rest mass M0 = = 0 is attributed to photons, but we associate with photons a nonzero mass mo = hu/c2 via de Broglie's and Einstein's relations.

d-

191

THE DUAL DE BROGLIE WAVE

We focus attention on transformational properties of fields associated with an extended photon under a Lorentz transformation. To this effect, we assume that a photon treated as an S' TI-cavity is observed in a reference frame moving in the fx'-direction at velocity u = cp = c dx'ldx'. The Lorentz-transformed fields and the corresponding wave equation take the form

fv

=f

m o p

(293)

z/l'rs'

in which fu denotes the Doppler-shifted frequency of a luminal wave associated with a photon. Equation (291) reveals that in the moving frame both time and space dimensions of an S1T 1-cavity undergo relativistic compression

the masses of associated bradyonic and tachyonic components both increase but remain equal to each other. For this reason a photon and its associated luminal waves observed in a moving frame invariably travel at the velocity of light. In the limiting case ( p + 1) photons become pointlike objects of infinite energy and momentum. Thus the concept of a photon as an object of invariant properties in all inertial reference frames is invalid; the zero rest mass and the velocity of light attributed to a photon remain invariant in moving and rest frames, but their geometric characteristics become modified according to Eq. (294). Summarizing these considerations, we indicate the possibility of verifying this model of a photon. To accomplish this objective, we consider photons (treated as S1TI-cavities) in a beam incident on a periodic crystal-like structure and reflected from its planes. In such circumstances, interaction between a nonpoint luminal particle and a periodic obstacle may be considered an elastic collision; the angle of reflection equals the angle of incidence. Cavity dimensions are invariant under reflection, as an elastic collision excludes dissipative processes. Values of the angle 00 leading to a large reflected intensity of a photon beam are determined on making use of Bragg's law, 21 sin 00 = nk, (Kittel, 1966),in which 1 is a distance between crystal planes and 00 is the angle measured from crystal planes to incident and reflected beams. When we use Eq. (271) in the form

'f = f' = 2n/k, = nn3a3-1

,

(295)

192

MARCIN MOLSKI

in which A, is the wavelength of a luminal wave, after simple mathematical operations we find that 2423

= n31,,

(296)

and Bragg’s formula (n = n3) takes the form a3 = 1 sin&,

(297)

relating only geometrical parameters of a periodic structure and interacting photons, Equation (297) indicates that Bragg’s reflection may appear for only photons of internal space dimensionsa3 5 1. This geometric version corresponds to a wave condition (Kittel, 1966) &I2 5 1

(298)

obtained for n3 = 1, which clearly shows that Bragg’s reflection may be interpreted in a purely geometrical framework in which a photon is considered to be extended a nonpoint object of internal space size a3 corresponding to the S ’ T ’ cavity dimension. Equation (297) is valid for only photons emitted from a source at rest relative to the laboratory frame in which a crystal is placed. Because the geometry of photons emitted from a moving source varies according to Eq.(294). for this case Eq. (297) becomes replaced with a3

Jg = I sine,,

(299)

in which 8, is the angle measured from crystal planes to incident and reflected beams of Doppler-shifted photons. Dividing EQ.(297) by Eq. (299), one obtains sin 00 --

sine,,

1

-d

m =y;

hence the angle B,,. measured in a dynamical Bragg experiment, varies according to

e,, = sin-] [(I - u2/c2)s i n ~ o ] .

(301)

The formula conjugate to Eq. (300) provides an excellent way to determine the velocity of a moving source given by sin2eo sin2e,, These formulas are obtained employing only geometric properties of photons treated as extended relativistic objects. So far our considerations are confined to photons interacting with ordinary timelike periodic obstacles; Eqs. (275) and (295) reveal that similar interactions

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193

occur for a hypothetical periodic structure composed of spacelike objects. From Eq. (275) in the form f3

= 2n/& = f O = nnoa,' ,

(303)

one obtains 2ao = no& and the spacelike counterpart of Eq. (297),

(304)

a. = 1'sin8;,

(305)

in which 1' is the spacing within a spacelike periodic obstacle. The results indicate that a photon is a universal object that may interact with particles of both bradyonic and tachyonic types. In particular, photons interact with a hypothetical periodiclike structure composed of spacelike objects. If such an interaction is viewed as an elastic collision between an extensivephoton treated as an S' T -cavity and a periodic obstacle, one may evaluate the reflection angle leading to a large reflected intensity governed by the law 21' sin 0; = nAp,

(306)

which is a spacelike counterpart of Bragg's law in ordinary timelike form. This electromagneticmodel of photons is also applied to a geometric interpretation of the cosmologicalred shift observed for spectral lines of distant galaxies and nebulae. For this purpose we consider an isotropic and homogeneous expanding space-time characterized by the Robertson-Walker metric

( + :)i[dr2+ r2(d02+ sin2

ds2 = d x i - R ( x ' ) ~1

8 d ~ $ ~ ) ] , (307)

E-

in which R(xo)is an expansion parameter; E = +1,0, -1 for spherical, flat, and hyperbolic geometries of the universe. Maxwell's equation for electromagneticwaves propagating in space-time conformally coupled to (307) reads (Parker, 1972)

(spa" - 2r)A, = 0,

(308)

in which x o is replaced by a conformal time variable xoR(xo)-'.Solutions of wave equation (308) take the forms (Parker, 1972) A, = exp [fifoxoR(xo)-'] F(r, 8, qj),

(309)

fo = (f2 - 2€)'/2,

(3 10)

=n2-1,

~ = + 1 n = l , 2 , 3 ,...,

f 2 = - A F ( r , 8, C$)/F(r,8 , # )

(311) E

= -1

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MARCIN MOLSKI

Introducing the frequency formula (271) into Eq. (309), we find that

[

A, = exp fi-x a3;:9

O]

F(r, 8,q5),

(312)

indicating that frequencies of electromagnetic modes in an expanding universe decrease as the universe expands. This red shift is familiar for spectral lines of distant galaxies and nebulae in the expansion process. Equation (312) reveals that decreasing frequency modes result from stretching of the space dimension of an S' TI-cavity (representing a photon) due to an expansion a3 + unR(x"),

R(0)= 1.

(313)

This interpretation may be applied not only to an electromagnetic field but also to all massless fields of arbitrary nonzero spin s; the reason is that wave equations governing propagation of such fields are all conformally invariant (Parker, 1972). For example, one may consider for s = the two-component neutrino equation, and for s = 2 the vacuum solutions of Einstein's field equations. This geometric interpretation of the cosmological red shift hence applies to photons, neutrinos, and gravitons. These considerations are readily generalized to include an extended model of a photon considered as a superposition of bradyonic and tachyonic solutions given by Eqs. (217) and (265)projected onto a M(1, 1) space. Proceeding as in the case of a square S' TI-cavity, one may construct a nonlinear superposition of waves of luminal type, which in the moving frame takes the form

(3 14) 0

m,, = m,c/h = nlr/ro,

0

m,,! = m,,c/h = n'Tr/po.

(315)

in which m! and m!, denote masses of associated time- and spacelike S1T '-cavity constituents. Amplitudal and exponential functions appearing in Eq. (3 14), and their superposition satisfy the luminal wave equation provided that Q,,,,l,

r0

= po,

n = n',

x -3 f x 0 = r0 = po.

(3 16)

These restrictions are identical to those obtained for the square S' TI-cavity.

E Conclusions The electromagnetic approach enables one to construct wave-corpuscular models of extensive massive and massless particles, and improves our understanding

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of their internal structure. In this context, timelike objects appear as threedimensional electromagnetic cavities with space-trapped electromagnetic fields. Such imprisoned radiation undergoes conversion into a system of B- and D-waves that lock to form photonlike inner fields occupying the cavity (particle) interior. According to the particle picture, the photon becomes converted into a bradyontachyon system, of which constituents are self-trappedin a relativisticallyinvariant way, yielding a compound particle of photon type. The three-dimensional space cavity has a bradyon-type characteristic, i.e., it behaves like a massive subluminal extensive object associated with time-like outer fields interpreted as matter waves of the first kind (B-waves). Exploiting a mirror method in a similar manner, one may construct an electromagnetic model of a spacelike object considered to be a cavity with time-trapped electromagnetic fields. Such imprisoned radiation is endowed with a rest mass depending on cavity dimensions, whereas the outer spacelike fields associated with the cavity exterior can be interpreted as matter waves of the second kind (D-waves). In both cases space and time cavities with trapped radiation reproduce well the fundamental properties of ponderable matter, i.e., inertia, mass quantization, three-dimensional spatial extension, and time(space)-like kinematic characteristics. Hence they may be consideredwave-corpuscular models of massive particles. This approach is consistent in many points with well-known theoretical results, and predicts, for example, the following: 1. Conversion of massless photons into ponderable matter in accordance with Jennison-Drinkwater (1977) theory. 2. A factorized form of the particle wave function identical with that in the de Broglie-Bohm model. 3. The presence of B- and D-waves in the spectrum of states associated with the cavity (particle) interior, which is compatible with a two-wave particle model. 4. Formation of a nonlinear nondispersive wave packet (C-wave) by fields trapped inside a moving cavity, in full accordance with a nonlinear wave hypothesis and Mackinnon’s (1978, 1981a,b) theory. 5 . The appearance of bradyonic and tachyonic constituents in the mass spectrum associated with imprisoned radiation, which is consistent with an extended space-time description of matter developed by Horodecki (1988a, 1991) and Corben’s (1977, 1978a.b) tachyonic theory. In particular, the electromagnetic model provides an explanation of why a particlelike solitary wave obtained by Mackinnon satisfies the luminal d’Alembert’s equation but fails to satisfy the timelike Klein-Gordon equation (Gueret and Vigier, 1982). Mackinnon’s equation governs the propagation of inner fields associated with the cavity (particle) interior. Because the spectrum of inner fields includes

196

MARCIN MOLSKl

both time- and spacelike waves in the form of a nonlinear superposition, its propagation is governed by the luminald’Alembert’s equation. As outer fields associated with the cavity exterior are endowed with a timelike characteristic, they propagate according to the Klein-Gordon equation for massive subluminal particles. Both equations are correct, but in distinct cavity (particle) domains. This approach also permits constructionof a relativistic wave-corpuscular model of a photon that may be treated as square two-dimensional electromagnetic spacetime cavities. Such imprisoned inner fields have characteristics of luminal type: zero rest mass is attributed to them, and the associated outer fields propagate as c-velocity excitations. According to a particle interpretation, a photon is considered an object composed of both bradyonic and tachyonic components, of the same rest mass, trapping each other in a relativistically invariant way. Such a compound particle is endowed with a characteristic of luminal type; it behaves as a massless object moving at the velocity of light. This model is consistent in many points with the tachyonic model of photons proposed by Dutheil (1984, 1989, 1990a,b, 1993). Therefore, photons are treated as objects of time and spacelike faces, which live on the border between two kinematically inpenetrable subluminal and superluminal worlds. This view is consistent with the well-known fact that photon worldlines separate the Minkowski cone into domains of time and spacelike kinematic characteristics. Hence photons seem to be universal objects interacting with both bradyons and tachyons; for example, photons should interact with a hypothetical periodic structure constructed from spacelike objects. If such an interaction is viewed as an elastic collision between a nonpoint photon and a periodic obstacle, one can evaluate the reflection angle leading to a large reflected intensity using a spacelike Bragg’s law so far unknown in the domain of theoretical physics. The proposed model of extended photons also predicts the following properties: 1. Quantization of photon energy and momentum experimentally confirmed in the blackbody radiation law and Compton scattering. 2. The possibility of existence of multiphotons appearing in Planck’s emission, the photoelectric effect, Compton scattering, and multiphoton ionization of atoms. 3. Photons with finite dimensions, which may be viewed as localized packets of electromagneticenergy of a size comparable to the photon wavelength, this property being consistent with the photoelectric effect, Compton scattering, and the Hunter-Wadlinger (1988) model of photons. 4. A strict correlation among wave, corpuscular, and geometric characteristics of extended photons. 5 . The possibility of a purely geometrical interpretation of a time and spacelike Bragg’s law and a cosmological red shift predicted for neutrinos, gravitational waves and light.

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197

The first three points are consistent with best established characteristic of a quantum of light formulated by Einstein and quoted by Diner et al. (1984): The energy of a light ray spreading out from a point source is not continuously distributed in space, but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units.

The results obtained in this section have importantmethodologicalconsequences worth discussing. The timelike Klein-Gordon equation and its spacelike counterpart for a particle considered to be an electromagnetic cavity are derived in a framework of Maxwellian electromagnetism and the special theory of relativity. Hence equations fundamental in quantum mechanics have an electromagnetic nature; consequently, ‘‘a de Broglie wave may prove to be as real as a classical electromagnetic wave, and [d’Alembert’s] equation may prove to be of more importance to quantum mechanics than has hitherto been supposed” (Mackinnon, 1981a). Conversely, this electromagneticapproach permits one to consider the relativistic mass problem in a purely geometric framework and explains why the mass of an extensive timelike particle increases with velocity. An internal photonlike structure of an extended, massive particle is consistent with Weinberg’s (1975), Winterberg’s (1978), and Sedlak’s (1986) concepts treating a photon as a fundamental particle of nature. The following onto logical gradient of fundamental aspects of the existence of matter: wave + geometry + particle becomes a consequence of the fact that corpuscular properties of matter can be derived from a wave theory of light with suitable boundary conditions (geometry). The electromagneticconcept of matter permits also one to construct a quantum theory without Planck’s constant h. According to this idea (Barut, 1992), it is possible to formulate quantum mechanics without fundamental constants such as h, mo,ore, as a purely wave theory, provided that a quantum system is characterized . (1992) showed that all wellby a fundamental characteristic frequency ~ 0 Barut known equations in conventional quantum theory, such as Schrodinger’sor Dirac’s equation and their eigenvalues can be expressed in terms of wo instead of a mass h q / c 2 . A nonrelativistic Schriidinger equation for a freely moving particle and for a particle moving in a Coulombic field take the form

in which w is the Rydberg frequency w=001112/2,

(Y

e2 ch

=-

(3 19)

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MARCIN MOLSKI

and a! is the fine-structure constant. The eigenvalues of the wave equation (318) are given as a frequency spectrum

The solutions for space-trapped electromagnetic fields become viewed as a special case of solutions (83) obtained by Barut (1990). whereas associated masses become expressed in terms of the resonator frequency WO. Consequently, the electromagnetic approach enables one to formulate a relativistic quantum mechanics of extended spin-0 particles without the Planck constant h. An important general conclusion is that a pointlike characteristic attributed to material objects seems to be only a rough mathematical approximation of real properties of particles appearing in nature. The wave-particle duality of matter, including both massive particles and radiation, becomes enriched with the geometrical aspect of their internal structure, because the wave, corpuscular, and geometric pictures are merely three aspects of the same physical reality.

v. EXTENDEDSPECIAL RELATIVITY AND QUANTUM MECHANICS M A LOCALL-SPACE

Our research on the D-wave so far is confined to classical electromagnetism, which seems to be the best framework in which to construct an extended relativity embracing superluminal motion and an extended quantum theory, including spacelike particle states. The fundamental problem arises as to whether there exists a physically well-established theory other than Maxwellian electromagnetism that can verify an extended wave-particle model of matter and answer the main question: Can a D-wave exist in nature? A solution of this problem was obtained in the framework of special relativity plus the de Broglie postulate on wave-particle duality (Horodecki, 1988a) or by extension of de Broglie’s theory (Elbaz, 1985; Das, 1992). Detailed analysis indicates that they do not reclassify the basic problem from the category hypothesis to a physically well-established theory. Hence we seek to reformulate a conventional theory of special relativity and quantum mechanics in the framework of local space (L-space) (Molski, 1995c), which appears to be the best framework to introduce concepts of the D-wave and spacelike particle states. A. Special Relativity in L-Space

Our point of departure is a notion of absolute space introduced into physics by Newton to explain inertia and then generalized to a four-dimensional absolute

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199

space-timeby Einstein within special and general theories of relativity. The notion absolute means that "the space-time is physically real but also independent in its physical properties, having a physical effect, but itself not influenced by physical conditions" (Einstein, 1955). The absolutenessof space-timeseems well established within general relativity; de Sitter's and Godel's solutions of Einstein's field equations indicate that a spacetime continuum can exist without mass and that all masses in the universe rotate within an absolute space-time (Einstein, 1955). For special relativity the situation differs; its formulation does not refer explicitly to the notion of absoluteness that appears in this theory as a hidden property of space-time. Because a transition from general to special relativity can be made easily (Einstein, 1955), it becomes obvious that in such an operation the basic equations of the latter are completely deprived of information about absoluteness. To consider this problem mathematically, we assume that a test particle of rest mass rn0 follows a geodesic in four-dimensionalspace-time endowed with a metric G,,, and signature (+ - --). The metric G,,, evaluated by solving Einstein's equation (Einstein, 1955),

rigorously determines the particle kinematics in curved space; its four-velocity v, is calculated on integrating the equations of motion,

in which s is a four-dimensional affine parameter along a classical trajectory and vi", denotes an initial value of the four-velocity. When we consider a particle moving with a four-velocity u: in the local flat space-time called by Einstein a Galilean region, relation (322) reduces to G,,0

+ v:

= const.

(323)

Because Gi,, may in general contain both diagonal and off-diagonal nonzero elements, a transition to the fundamental metric of Einstein's relativity (E-metric), 3

S,,dx'dx" = dxi - dx: - dxi - d x f ,

ds2 =

(324)

,v=0

is realized with the operations (1) diagonalization procedure GE, + ,g: SiS,,, and (2) scaling ,g: --+ S,, in which

B,2

= ,g ,,

0

800 = 1,

611 = 822 = 833 = -1,

S, = Oforp

+ v.

=

(325)

These relations indicate that a correlation between the kinematic characteristics of a particle in uniform translational motion and a geometry of a local flat space

200

MARCIN MOLSKI

(L-space) may be given by elements of the metric GE,, or after diagonalization, by the diagonal elements gi,, and scaling leading to the E-metric S, completely removes the possibility of finding geometric and kinematic relationships. To find such a geometry-kinematicconnection,we note that the two-dimensional L-interval obtained after diagonalization of the metric G i v ds2 = (Bo)2dxi- (/11)2d~:

(326)

is invariant4under generalized Lorentz transformations (Molski, 199%) 0

x = 1

x =

BBIZ.'

BOZO-

podfi1.F'

(327)

'

- BBoZo

g , d m' in which ' = Bc is the velocity of a reference frame. Dividing Eq. (3 Eq. (327), one obtains XI p- = b(Z'/Zo)- B

p = - -BI,

p- = - Sl .

xo 1 - BB(K1/Ko)" PO BO This equation indicatesthe conclusionthat if we assume Bxl appearing in Eq. (329) for p < 1 and /I> 1 to be treated as a contracted or expanded light distance XI equal to the light time xo = ct, then XI/XO = Zl/Zo = 1, and Eq. (329) reduces to

These equations are only transformationalrelations for /?measured in two inertial frames, Their form is identical with that of the velocity addition theorem, provided that we identify (Molski, 1995~)

B =BI/BO

-

s =B1/Bo

VIIC,

-

ijI/C.

(331)

In the case of four-dimensional space-time the L-metric (326) takes the form 3

ds2 = Bidxi -

C pidxi,

(332)

k= I

which is given equivalently as

(333) k=l

In the case of &,-space,

PI) H

(xh, r', jJh, the quadratic form of Eq. (326). (xo, r, jJo,

the term invariance implies that we seek a transformation relating suitable quantities in two inertial frames and preserving

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20 1

in which we use the relation dxk = dxo = ds # 0 valid in L-space. Employing Eqs. (333) and (331) generalized to the form

B = ( v I / c , v2/c7 % I C )

vk/c = Bk/BO,

(334)

we express POand B k in terms of V k :

allowing us to correlate geometric characteristics of L-space g:, with kinematic characteristics of a particle in uniform translational motion. By comparison of Eq. (335) with Eq. (3), we see that parameters Bp become expressed in terms of components of the particle four-velocity Bp

= v,/c.

(336)

In the framework of L-space we derive not only conventional relativity for timelike particles but also its form extended to include spacelike objects. Thus we introduce the time(space)likeLagrangian

+

in which q = 1 for timelike and q = - 1 for spacelike particles, respectively (Recami, 1986, p. 50). With the four-momentum formula

for q = f l we obtain a time(space)like four-momentum appearing in the Horodecki-Kostro and Das models [see Eqs. (52). ( 5 3 , and (70)]. Hence the L-space concept is compatible not only with a postulated extended space-time description of matter (Horodecki, 1989a, 1991) but also with subluminal and superluminal theories of relativity (Recami, 1986, p. 21). The L-space theory also works for limiting cases Vk = 0 and v k = 00. Then the interval (326) reduces to

lim ds2 = d x i - ( 0 . dx1)2,

Uk’O

lim ds2 = ( 0 . d x d 2 - dx:,

Uk+bo

(339)

whereas generalized Lorentz transformations (327) and (328) take the forms

202

MARCIN MOLSKI

Under transformations (340) and (34I), the purely timelike and purely spacelike intervals (339) transform to

ds2 = dxi - (0 . dX1)2 -+ dS2 = SidXi - BfdX:,

-

ds2 = (0 dxo)2- dx: + dd2 =&dii - $ d i : .

(342) (343)

Bearing in mind Eq. (336), we write the operation (342) alternatively in the form ds’ = dx,2 + dS2 = (1 - B2)-1’2(d i 2 - B 2 d i : ) , (344)

,

discussed in detail in Section VII. The concept of L-space allows one to explain the phenomenon of inertia. As the geometry of L-space rigorously determines kinematic characteristics of a particle in uniform translational motion, such a space is explicitly absolute, and inertness of massive objects appears as a consequence of the relation

which may be viewed as a relativistic formulation of the first principle of newtonian mechanics.

B. Quantum Mechanics in L-Space We consider matter waves associated with a particle uniformly moving at velocity v in an isotropic (PI = 8 2 = 8 3 ) L-space characterized by the line element

ds2 = Bidxi - /3:dr2,

(346)

in which

BIlBO

= B =VIC.

(347)

Taking into account de Broglie’s (1924) idea that “nature treats all particles in the same way with regard to particle-wave duality, whether their rest mass is zero or not,” one assumes that d’Alembert’s equation constructed in L-space,

g;v8p8,,exp [ik,xc’] = 0,

(348)

not only describes the propagation of electromagnetic waves (B = 1) but also governs the propagation of a B-wave (B < 1) and a D-wave (j3 > 1). According to this hypothesis, by differentiating the wave function appearing in Eq. (348), we obtain the dispersion formula

g;”k,k,, = 0,

(349)

p2k2 0 0 - B?k2 = 0.

(350)

equivalent to

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Taking advantage of Eq. (347) and the relativistic equation v/c2 = p/E,

(35 1 )

valid for subluminal and superluminal particles, from Eq. (350) we obtain

This result expresses the wave-particle correspondences for timelike ( u < c), photonlike ( u = c), and spacelike ( u > c) objects, derived here with an accuracy proportionality to the factor A. In the next step we consider the wave equation (348) in the space dS2 = ds2&*, conformally coupled to (346)

introducing into it the useful relation

derived from Eq. (338), which is valid for bradyons ( q = + l ) , photons ( q = O), and tachyons ( q = - 1). We thus produce a wave equation that after differentiation of the wave function twice with respect to space coordinates yields

+

Equation (355) for q = 1, 0, - 1 gives the Klein-Gordon, Maxwell, and Feinberg ( 1967) wave equations, respectively, describing propagation of matter waves associated with the subluminal, luminal, and superluminal particles moving at a velocity cp. This general wave equation includes wave equations governing propagation of B- and D-waves. With this approach they are obtained with no standard quantization formalism and without referring to quantum-mechanical operators. The derived wave equation (355) is generalized to include a charged particle interacting with an electromagnetic field A,. To proceed we consider the guidance formula (26) and a relation (de Broglie, 1965, p. 112) 2

+

2

[aoS(xU) - e V ] - [ V S ( x @ ) !A] C

that is rewritten in the equivalent form

C

= qrn;c2

(357)

204

MARCIN MOLSKI

Substituting Eqs. (358) and (13) into Eq. (348), we obtain g["D,D, exp[ih-'S(xP)] = 0.

(359)

From this compact wave equation one may derive, by differentiation of the wave function twice, its expanded form

+

[ D ~ D P q ( m o ~ / h exp[ih-'~(xP)l= )~] 0.

(360)

To consider the Dirac equation for a charged particles of spin we assume that Eq. (359) in the equivalent form (S2Di - D: - 0 2 2 - D f ) @ P= 0 is satisfied by

@P

(36 1 )

components of Dirac's bispinor function

JI = (@o,

@ I , $2, $3)

= ( U O , U I , u2, ~3)exp[ih-'S(x~)l,

(362)

in which the uP denote components of bispinor amplitude. We find the first-order differential operator a!$

Do

+ a!' D' + a2 D2 + a 3 D 3 ,

(363)

which when multiplied on the left (right) side by an operator D' - CY~D'- u3D3 uODO - IY~

(364)

produces the operator part of Eq. (361). Then we arrive at the matrix relations U,'CY,

+ CX,U$

= 28B2,

+

u ~ ( Y ,CX,,(Y,,

a0+(-)a,- a!P a!+(-) O

= 28,

p, v

# 0,

= 0, p # 0,

(365) (366)

in which aPr8, and 0 are 4 x 4 matrices; the second is a unit matrix. Because

B2 = B-B+

= B+B-,

(367)

in which e

DoS(X') = &S(X') - -V, C

we achieve the matrix representation

(370)

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205

or equivalently,

in which a = fa*.The matrices appearing in the equations above are defined as __ follows~ 0

0

0 -1O 0

0

0

;),

-

a-=.[

0 - 1

1

0

0

; -; ; 0

0

!),

0

(373)

0

(. [. ; ;-i). 0 0 0 1

a1 = 0 1 O 0 00) ' 1 0 0 0 0

0

0 - i

0

a3=

0 -1

0

0 -1

(374)

1

(375)

0

Introducing the explicit form (371) of the matrix a : into Eq. (363),

and then differentiating $ once with respect to the time xO-coordinate,we obtain Dirac's equation,

+

( D o a1D'

+ a2D2 + a3D3 + i f i m o c a / h ) + = 0,

(377)

valid for timelike ( q = +1) and spacelike ( q = - 1) particles (Molski, 1992). Similarly, from Eq. (364) we derive

( D 0 - a l D ' - a 2 D 2 - a j D 3 -i,/ilmoca/h)$

=O;

(378)

hence the product of operator parts of Eqs. (378) and (377) yields the second-order operator appearing in Eq. (360). Matrices a,,are conventionally denoted (p, a,,a Y a z ) ,

206

MARCIN MOLSKI

Finally, we write compactly Dirac’s equation (363) in the highly symmetric form

d‘D,$(x’*) = 0,

(379)

in which no mass term i,/ilmoccr/h appears explicitly. C. Conclusions These results indicate that the L-space concept is the best way to formulate extended special relativity and wave theory, including spacelike particle states. The tachyon kinematic and D-wave mechanics emerge in a simple and natural manner without introductory assumptions. The L-space concept provides a theoretical and conceptual basis for an extended space-time description of massive particles and a two-wave hypothesis (Horodecki, 1989a, 1991; Das, 1989, 1992). If we require space-time to be explicitly absolute, we obtain via generalized Lorentz transformations the velocity-dependent L-metric, and then by generalization of Maxwell’s equation, wave equation (353). which includes the Maxwell, Klein-Gordon, and Feinberg equations as special cases. Additionally, de Broglie wave-particle correspondences are obtained with an accuracy proportionality to the factor A . In a conventional formulation of a wave theory of matter, these equations (for mo # 0) are obtained on postulating the following: 1. De Broglie wave-particle correspondences p , = hkp 2. The form of quantum-mechanical operators i), = i ha, and f, = x, 3. Jordan’s‘ rules p, + i),, x, + f, 4.A general form ofthe wave equation P(i),, f,)$ = o

On comparison of these approacheswe concludethat formulationof wave theory in L-space lessens the number of its basic postulates and allows a consistent waveparticle description of micro-objects, whether or not they possess rest mass. For instance, putting u = c into Eq. (353). we find Maxwell’s equation, whereas for u # c, Eq. (353) gives the Klein-Gordon or Feinberg equation. Instead of three wave equations for time-, photon-, and spacelike objects, we need only one, d’Alembert’s equation (353). to describe matter waves of all types that may appear in nature. This achievment confirms de Broglie’s fundamental insight in assuming a similarity between massless photons and particles with nonzero rest mass, and indicates that propagation of de Broglie waves in L-space seems to have much in common with propagation of ordinary electromagnetic waves. For a product of two noncomutating operators, Jordan’s rule must be supplemented by a symmetrization rule to produce a Hermitian operator; it fails for products of three operators, leading to the Temple paradox (Julg, 1993).

207

THE DUAL DE BROGLIE WAVE

The proposed approach generates methodological consequences. It is well known that from the chronologicaland methodological points of view, wave theory and classical mechanics, including both general and special relativity, developed independent of each other. Whereas classical mechanics is based on geometric space-time properties, the basis of wave theory has no connection with space-time features. Consequently, its fundamental equations, such as the wave equation, the correspondence rule, or the form of quantum-mechanical operators, are not derived or deduced from geometric space-time properties. Then L-space theory allows one to construct a wave theory in a purely geometric framework with an initial supposition that physical space is an explicitly absolute four-dimensional space-time continuum. The Dirac equation derived for spacelike fermions raises a question about tachyon spin. Although this problem seems speculative, it is widely discussed (Recami, 1986, p. 115 and references cited therein) and deserves attention for the reason that spacelike objects play a role in internal structure of particles endowed with spin (Costa de Beauregard, 1972; Corben, 1995). For example, in quantum electrodynamics based on Joos-Weiberg higher-spin wave equations, some solutions for integer-spin particles are of the tachyonic type, whereas the Bargmann-Wigner equation holds not only for time- but also for spacelike particles (Recami, 1986, p. 115). Tachyons are generally considered to be spinless (scalar) particles, or objects with infinite number of polarization states (Recami, 1986, p. 115); contrary opinions are also published (Corben, 1977).

VI. NO-WAVE MODELOF CHARGED PARTICLES IN

KALUZA-KLEIN

SPACE

The classical, semiclassical, and quanta1 models of a charged particle are well investigated but they still animate discussions. The fundamental problem to be solved is to explain how charge and mass are structurally related inside a particle, and the nature of the force binding charge and mass into charged matter to give a stable system of finite self-energy and self-field. To answer this question, several models are proposed that, according to the particular method, are developed in the framework of Maxwellian electromagnetism and Newtonian gravity (Visser, 1989; Robinson, 1995), theories exploiting the Casimir effect (Boyer, 1968), electromagnetic approach (Jehle, 1971, 1972, 1975; Post, 1982, 1986), and unified field theories of the Kaluza-Klein type (Kaluza, 1921; Klein, 1926). Kaluza-Klein theory treats gauge fields and gravitation on a five-dimensional M4 x S' manifold (five-space),of which M4 denotes an ordinary four-dimensional space (four-space) and S' is a compact space with the topology of an extra dimension as a circle of radius R.

208

MARCIN MOLSKl

In the five-space the position of a particle is described with coordinates in a set x i = ( x ~x ,4

+ 2scnR),

i = 0, 1 , 2 , 3 , 4

(380)

that for any integer n denotes the same point; the geometry of five-space is characterized with a metric (Toms, 1984) d t 2 = gijdx'dxj = g,,,dXfldX" - (KA,,dxp K

+d

~ ~ ) ~ (381) ,

=C - 2 d m ,

(382)

in which A,, denotes the electromagnetic four-potential and G the gravitational constant; g,,, describes Einstein's gravitation. A. Kaluza-Klein Field Theory

According to this theory, an electric charge acquires a purely kinematic interpretation (Rayski, 1965); namely, it can be related to a momentum p4 canonically conjugate to a compact extra x4 variable. To investigate this concept in detail, we consider a complex scalar field * ( x i ) associated with a charged particle of rest mass Mo,moving in five-space (381). The transition to a field description is made via a Lagrangian formalism starting with the action (Toms, 1984)

in which g5 = det gi, . Restricting consideration to a q ( x i ) field in the absence of gravitation, from Eq.(383) under a variational principle ASS = 0, we find a wave equation in five-space, [(a,, - ~

-

~ , , a ~ ) ( aKp m q )

- a;

+ ( M ~ C / A ) ~ ] Q (=X0, ~)

(384)

which for A" = 0 reduces to a simple form

[05 + (Moc/h)2]* ( x i )

= 0,

05 = aiai = a; - a; - a; - a; - a:.

(385) (386)

As a consequence of assuming of a closed extra space, according to Kaluza-Klein

theory every field must be a periodic function of the fifth coordinate. To derive the field equation in four-space we make a Fourier analysis of the \Ir field,

.

+m

THE DUAL DE BROGLIE WAVE

209

hence W is expanded into a complete set of harmonics with four-dimensional fields taken as expansion coefficients. Introducing Q. (387) into Eq. (383) and then integrating the extra variable x4 with orthonormality properties of the harmonics, we obtain the action in four-space (Toms, 1984),

nz-m

,=O

J

in which g4

= detg,,,

D , = a,

+ in KR-’A,.

(390)

This procedure is called dimensional reduction. Employing a variational principle, AS4 = 0, and assuming absence of gravitation, from Eq. (388) we derive the field equation in four-space,

[(a, + inKR-lA,)(alL +inKR-’A’*) + ( m ~ c / h ) ~ ] @ ( x=’ 0, )~

(391)

which in the absence of an electromagnetic field reduces to

[a$,

+ (moc/h)*]+(x%

= 0.

(392)

The derived equation (391) has a form identical to that of a wave equation well known in electrodynamics,

[(ac, + i ( q / h c ) A , ) ( P + i(q/hc)A’) + ( m ~ c / h ) ~ ] @ ( x ’=) ,0, ,

(393)

to describe a charged particle of 0-spin interacting with an A , field, provided that charge q associated with the particle is quantized in discrete units of elementary charge e, ChK R

q = ne = n-.

(394)

This result enables us to express the mass term mb appearing in Eq. (389) and the radius of the compact manifold in terms of the fine-structure constant a,the Planck mass m p [ ,and the Planck length 1,1, as follows: R = h(l6~rG)’/~/ec = a-’/21pl, mb = e(16~rG)-’/~ = a’/2m,l, (395) a = e2/hc= 1/137.03604,

1,1 = J 1 6 ~ r G h c -x~ 1.6 x 10-33cm,

mpl = d

(396)

m

2.2 x lOW5g. (397)

210

MARCIN MOLSKI

Detailed analysis of the transition from Eq. (384) to Eq. (390) indicates that one may formally associate with charge an operator of infinitesimal displacement of the closed extra x4-coordinate q + a 4 = -ih&,

(398)

having eigenfunctions \I,( x i ) and eigenvalues q ~ ( 1 6 n G ) - '= / ~- i h & Q / q ,

(399)

given in an alternative form as nmbc = -iha4Q/Q.

(400)

These equationsallows akinematic interpretationof electric charge as amomentum p4 o( q conjugate to the compact extra x4 variable. As from Eq. (389) one can derive the relation

l4 = mbcR = nh,

(40 1 )

electric charge is also interpretable as an angular momentum I, conjugate to an extra angular coordinate 8 of period 2n (Rayski, 1965). B. Charged Particle as a Five-Dimensional Tachyonic Bootstrap As all charged particles appearing in nature (excluding quarks) are endowed with charge q = ne, n = 1 , 0, - 1, expansion (387) becomes reduced to

+

@ ( x i ) = (2nR)-'/*{ $ ( x p ) + exp[+ix4/R]

+ +(x@)- exp[-ix4/R]},

(402)

which contains the functions

including the mass term mb = crl'zmp,. Under detailed analysis, Eqs. (385) and (392) and the field component given by Eqs. (403) and (404) indicate that these equations can be viewed as a fivechmensional version of Corben's tachyonic theory (Molski, 1997). In particular, @ ( x p ) may be interpreted as a B-wave associated with a bradyon of mass m0 moving in a four-space M 4 and described by the wave equation

[O + ( m o c ~ @) ~( x]p ) = 0,

(405)

whereas +'(x4) is a purely spacelike D-wave associated with a transcendent tachyon of mass mb and infinite speed moving in a compact space S' about a

THE DUAL DE BROGLIE WAVE

21 1

bradyonic constituent and satisfying the wave equation

[a$ - ( m b c / ~ +(x4) )~]

= 0.

(406)

These two free objects can trap each other in a relativistically invariant way, yielding in the five-space M4 x S’ a bradyon-tachyon compound of mass [see Eq. (389)] MO

= dm: -mt,

mb = crI/2mpl,

(407)

described by the wave function (408)

* ( x i ) = +(x9$7x4),

satisfying the five-dimensional wave equation [ 0 5

+ (Moc/h)2]* ( x i )

= 0.

(409)

with respect to the invariant interaction condition a;$(x9a;$’(X4)

= 0.

(410)

According to a particle picture, the quantity called charge (in conventional theory) is a purely spacelike object endowed with mass mb = a ’ / 2 m p land momentum p4 = &m&. which moves in an S1 space of internal radius R = c r - ’ / 2 ! p j . Such ultraheavy tachyons might be created (Molski, 1993c) from hypothetical particles called maximons (Markov, 1982)in the process of expansion of the universe (Minn, 1990). The quantity crA’2m,,1 with a negative sign appears in a classical model of an electron, coupling ordinary electromagnetism with a self-interacting version of newtonian gravity, and is called the bare mass of an electron (Visser, 1989). Although the fine-structure constant and Planck mass are both quanta1 concepts, their combination crA/2m,j is purely classical (independent of h). If bradyonic and tachyonic constituents of a charged particle interact through a field similar to a gravitational field (Recami, 1986, p. 112 and references cited therein), the motion of a charge at infinite speed corresponds to a fundamental state of a system of particles in which the trapping force holding charge on a circular orbit tends to zero. Hence a charged particle in Kaluza-Klein space may be considered to be a relativistically invariant tachyonic bootstrap (Corben, 1978a; Chew, 1968,1970) of two free particles in the foim of a Kepler-type (Recami, 1986 p. 116) system. According to a field interpretation, a free complex scalar field \i, is viewed as a superposition of and $‘ fields that interact and lock to form a $$’ field whose characteristics depend on masses mo and mb of interacting particles timelike, photonlike, spacelike,

mo =- cr’f2mpl mo = a’j2mpl . mo < a ‘ f 2 m p ,

212

MARCIN MOLSKI

For timelike particles appearing in our physical four-space, the condition mo

<< a’/2mpl

(412)

invariably holds; consequently, in M 4 x S’-space a bradyon-tachyon system is endowed with an imaginary rest mass

and @ ( x i ) is spacelike. This peculiar property of a @ field according to Kaluza-Klein theory is merely a tachyonic interpretation of a large mass problem: It is impossible to describe a test particle with a charge e and effective mass less than the Planck mass, unless the test particle follows a space-like geodesic in five dimensions (Gegenberg and Kunstater, 1984). Despite much effort, this problem cannot be solved satisfactorily (Gegenberg and Kunstater, 1984).

C. Conclusions

According to the standard paradigm of physics, charge and mass are attributed to the same material object but without explaining how charge and mass are structurally related inside the particle. The tachyonicmodel permits penetration into the particle interior to answer this question. Moreover, we explain the nature of the force binding charge and mass into a stable system. From the context of our analysis the following conclusions emerge. Kinematics of charge and mass are treated separately in our approach, and charge kinematics are subject to no limitation involving the velocity of light as an upper bound. Elementary charge in a particle moves at a superluminal velocity. Motion of a charge inside a charged particle generates a ring current circulating azimuthally at a superluminal angular velocity. An elementary manifestation of electric charge is simultaneously of a stationary and dynamic nature, as a transcendent tachyon representing it transports not energy E = 0 but momentum p 4 = mbc. Those conclusion agree with results obtained by Post (1982.1986), who considered a charged particle to be a ring current involving elementary charge circulating azimuthally as a tabular sheath of distributed charge. Application of the Post model enabled retrieval of the Dirac moment of an electron up to the first-order QED anomaly a/2n. and produced conclusions identical to those above. The results are also compatible with those reported by Davidson and Owen (1986), who attempted to clarify a geometrical relationship between mass and charge

THE DUAL DE BROGLIE WAVE

213

considering charged particles in the Kaluza-Klein space to be tachyonic modes of higher dimensions. With a proposed tachyonic model of charged objects we assume that a test particle follows a spacelike five-dimensional geodesic in Kaluza-Klein space, but without conflict with known laws or experimental facts; superluminal motion along a five-dimensional spacelike geodesic appears to be a genuine property of the system under consideration (Recami, 1986, p. 91 and references cited therein).

VII. EXTENDED DE BROGLIE-BOHM THEORY Detailed analysis of the de Broglie-Bohm, Horodecki-Kostro, and Corben concepts presented in Section I1 indicates a possibility of their unification into one generalized theory. The point of departure to realize this objective is an important property stipulated by Corben (1978a)-that two time (space)like waves cannot be locked one to the other, or, according to a particle picture, that two bradyons (tachyons) cannot trap each other in a relativistically invariant way to yield particles in a bound system. If they did, the momenta of such interacting particles would become imaginary, leading to nonnormalizable solutions (Corben, 1978a). Consideringthe de Broglie-Bohm factorized function (16) to be a superpositionof amplitudal and exponential waves, we recognize that if one of them is spacelike, the other must be timelike. We assume that for a freely moving particle, these functions satisfy time- and spacelike wave equations

[IJ

+ ( r n ~ c / h exp[(i/h)S(xP)l )~] = 0,

[IJ- ( r n b ~ / h R) ~( x] p ) = 0, (414)

whereas their superposition is a solution of Eq. (17). In such circumstances one obtains from Eq. (20)

in full agreement with Corben’s theory. For M O = 0, Eqs. (414) and (415) reproduce the fundamental equations of Mackinnon’s soliton theory. Exploiting this analogy we link the quantum potential (23) with a spacelike solution (414) to generate an equation

valid in a nonrelativistic approximation.

214

MARCIN MOLSKI

In the case of a tirne-independent spacelike field R ( x a ) , the wave equation (414) reduce to [A

+ ( r n b ~ / h R(x") ) ~ ] = 0,

(418)

whereas the quantum potential (23) takes the form

This result indicates that carriers of a field R(xP) and a source of quantum potential Q(xP) can be spacelike particles. This result plays a vital role in an extended de Broglie-Bohm theory developed in succeeding parts of this article. A. Tachyo-kinemaricEffect

We reinvestigate Corben's theory of particles composed of both bradyonic and tachyonic components (see Section 1I.D). If a bradyon at rest endowed with rest mass mo and described by

[a; + ( r n o ~ / h )+(xo) ~ ] = 0,

+(xo) = exp [irnocxo/h]

(420)

absorbs a tachyon (antitachyon)of a rest mass mb and infinite speed, characterized by [A

+ ( r n ; , ~ / h+(r)' ) ~ ] = 0,

+'(r) = exp[fimbc. r/h],

(421 )

the bound system of particles has a mass

MO = d m i -

rnt

and is described by the wave function

~ ( x p= ) +(xo)+'(r) = exp [i/h(rnocxO fmbc r)],

(423)

satisfying the wave equation

[apap + (Mo~/h)~] W(X')

= 0,

(424)

under an invariant interaction condition

aP+(xo)aP+'(r)= 0. According to the generalized Stiickelberg-Feynman switching principle (Recami, 1986, pp. 32,33), the signs f in the wave function (421) correspond to the tachyon and antitachyon, respectively, moving in opposite space directions.

THE DUAL DE BROGLIE WAVE

215

Wave equation (424) and its solution (423) have the equivalent form

which when compared to a standard form of the Klein-Gordon equation

provides the identification

first mentioned by Corben (1978b). According to a corpuscular picture, relation (428) is a result of a four-momentum conservation principle

implying that a free bradyon absorbing a free tachyon (antitachyon) not only alters its mass according to Eq. (422) but also attains a velocity the value of which depends on masses of interacting particles. This phenomenon, henceforth called the tachyo-kinematic efect, allows us to reinterpret the kinematic and wavemechanical properties of ordinary timelike particles in terms of Corben's tachyonic theory.

Timelike Theorem. Each slower-than-light particle of rest mass MO is considered a bound system of a bradyon at rest endowed with mass mo and a transcendent tachyon (antitachyon) of mass m; < mo. These two free objects trap each other moving at a velocity to yield a timelike particle of rest mass MO = J v = cmb/mo determined by masses of interacting particles. As the phase of a B-wave associated with a moving microobject is expressed in

Eq. (426) only with masses of interacting time- and spacelike particles we derive Lorentz's time transformation making use of mass-velocity relation (428). To prove this thesis, we consider a definition of a mass center of a bound system of N particles endowed with masses mi and described by coordinates in the set ff;, a! = x , y , z :

216

MARCIN MOLSKI

in which Mo denotes the total mass of the system. This equation for a bradyontachyon compound formed in two-dimensional pseudo-Euclidean M ( l,l)-space reduces to the form

given equivalently as XCM

=

x o f (mb/mo)x'

M .

Taking advantage of Eq. (428). we arrive at Lorentz's time transformation,

which describes the position of the mass center of a bradyon-tachyon compound moving in M ( 1 , 1)-space. To obtain Lorentz's space transformation, we consider similarly the process necessary to form a superluminal bradyon-tachyon compound. For this purpose we assume that a tachyon at infinite speed and of rest mass mb absorbs a bradyon (antibradyon) of rest mass mo. If mo < mb, the compound particle moves at a superluminal velocity, whereas Eqs. (420)-(424)take the form

+ ( m ~ c / h $) (~x o] ) = 0, [A + ( m b ~ / h $09' ) ~ ] = 0,

[a;

$ ( x o ) = exp [ f i m o c x o / h ] ,

$'(r) = exp[im;c . r / h ] ,

q(xP) = $(xo)$'(r) = exp [ i / h ( m b c .r fmocx')],

[aPaL(- ( M h ~ / h ) *W'(xP) ]

= 0.

(434) (435)

(437) (438)

According to a switchingprincipk (Recami, 1986, pp. 32,33), the signs f that appear in the timelike wave function (434)correspond to a bradyon and antibradyon, respectively, moving in opposite directions of time. Equation (438), given equivalently as

THE DUAL DE BROGLIE WAVE

2 17

takes a form identical to that of a spacelike Feinberg (1967) equation [see Eq.(58)]

for a D-wave associated with a superluminal particle moving at a velocity cp'. Then we identify

These results enable us to formulate the spacelike theorem.

Spacelike Theorem. Each faster-than-light particle of rest mass Mt, is considered a bound system of a bradyon (antibradyon) at rest endowed with mass mo and a transcendent tachyon of mass mb > mo. These twofree objects trap each other to yield a spacelike particle of rest mass Mh = d-2 moving at a velocity v' = cmb/mo determined by masses of interacting particles. Following the same procedure as for Lorentz's time transformation, in the first step we determine the mass center of a bradyon-tachyon system moving in M(1, 1)-space, ECM

=

mbx' fmoxo

JR'

which is given alternatively as .fCM

=

x'

f (rno/mb)x"

d

w

(443) *

Taking Eq. (441) into account, we arrive at Lorentz's space transformation, x' WCM

=

fpxo

J1-s"

in which /J = j?-',to describe the position of the mass center of a bradyon-tachyon compound moving at superluminal velocity cp' = c/J-' in M(1, 1)-space-time. Hence

pp'

=1

VU!

= c2,

(445)

so d becomes identified with the Das pseudovelocity [see Eq. (67)]. According to our approach, it is related to a tachyonic component of a particle, not with the particle itself.

218

MARCIN MOLSKI

As a bradyon can trap at most three tachyons [see Eq. (49)], with the foregoing approach we derive Lorentz's time transformation acting in M(1,3)-space. To accomplish this we consider a system of bradyon at rest and three transcendent tachyons (antitachyons), the mass center of which is described with the equation XCM

=

mOxo f (mix'

+ m;x2 + mix')

Jrn; - mi2 - m$ - mf

(446)

in which m i , m i , mi < mo.

(447)

The four-dimensional version of Lorentz's time transformation is obtained from Eq. (446),

in which we introduce

B = W , / m o ,m;/mo, rn;/mo) = ( v l / c ,v2/c, vnlc).

(449)

Detailed analysis of a transition from wave equation (420) to (424),

+

+

[a: ( m o c / h ~+~( x] o ) -+ [ a , ~ ( ~ o c / h q) (~x ]f i ) = 0, (450) reveals that such an operation changes a mass term mo + Mo, and this condition contradicts the fact that the rest mass of a micro-object is relativistically invariant. To avoid this inconsistency, we reconsider the basic wave equation (424) in the form

[n+ (m; - m$)c2/h2]exp [i/h(mocxof mbc . r ) ] = 0.

(451)

If we make the conformal transformation

+ P = x p d 1 - m t / m i , mb < mo, of space-time coordinates x" and wave equation (451), we obtain xp

ds2 = dx,dxfi

-+ dS2 = d i , d . V ,

[n+ (mt - mL2)c2/h2]exp [i/h(m,cZo f m:.c. F ) ] = 0,

(452) (453)

(454) in which m, and m: are conformal masses of bradyonic and tachyonic constituents of a particle

Hence m,2 -mF = m i

(456)

THE DUAL DE BROGLIE WAVE

2 19

and Eq. (454) takes the well-known form

Comparing Eq. (456) with the four-momentum relation (6) in the form

( E / c 2 ) 2-

= mi,

(458)

we make the important identifications m, = E / c 2 , mi = p / c ,

(459)

to correlate suitablequantitiesappearingin ordinary and extended specialrelativity. Equation (457) shows that formulation of the problem in conformally coupled space-time (453) preserves invariance of the bradyonic rest mass and kinematic mass relation (428). The latter is a consequence of invariance of the three-velocity under the conformal transformation (452) dr dxo

-

di: - m:. - mb dIo m, mo'

With Eq. (428), conformally coupled coordinates 2' in the form

attain a purely relativistic interpretation: They are Lorentz-contractedcoordinates xp of a laboratory frame.

This procedure also holds for a spacelikebradyon-tachyon compound described by wave equation (439), but in this case transformation (452) takes the form xfi --+

ifi = x p J 1 -

mi/rn$,

mb > mo.

(462)

Consequently, time (space) Lorentz transformation of the phase of a B(D)-wave

is equivalent to two operations: tachyon (bradyon) absorption s(x0) = mOcxo + S ( X P ) = mOcxo fmbcxl, ~ ' ( x ' )= m&xl + ~ ' ( x p )= mbcx' f mOcxo

(464)

MARCIN MOLSKI

220

and transformation (452) (or (462) for a spacelike bradyon-tachyon compound)

yielding the same final result as application of the time (space) Lorentz transformation in Eq. (463). A transition to conformal masses and space-time coordinates does not affect the tachyo-kinematic interpretation of Lorentz transformations. Then Eqs. (43 1) and (442) take the form XCM

=

.fcM =

m,io

J-

fmLZ1 - i0 f (m:/rnc)Z-'

-

m : . i l fm , ~ '

J -

J1-

Z'

=

(m:/m,)2'

f( r n , / r n : ) ~ O

J I - (m,/mk)2'

(466) (467)

so by virtue of Eq. (460) from Eqs. (466) and (467). we obtain time and space Lorentz transformations. A transition to conformally coupled space-time becomes more comprehensible if we consider it in a framework of the L-space concept. If a bradyon at rest absorbs a tachyon of infinite speed, the compound system attains a velocity according to Eq. (428). Consequently, associated with a bradyon the L-space undergoes transformation (344) to the form

ds2 = d x i + dS2 = [1 - ( r n b / m ~ ) ~ ] - " ' ( d Z-i (mb/mo)2dZf). (468) The right side here reveals that an L-metric associated with a moving bradyontachyon compound is expressed in terms of conformal coordinates (452). From Eqs. (468) and (452) we recover the original form of the L-interval before conformal transformation:

The necessity to formulate the basic problem in conformally coupled spacetime (453) has important interpretive consequences. During formation of a bradyon-tachyon compound, three effects appear: alteration of the rest mass of a bradyon on absorbing a tachyon, alteration of the motionless state of a bradyon that attains a velocity on absorption of a tachyon, and Lorentz contraction of spatial and temporal dimensions. These effects together produce an entity that in conventional relativistic mechanics is called a moving particle in the space time.

221

THE DUAL DE BROGLIE WAVE

So far we treated relativistic wave mechanics of freely moving time- and spacelike objects. To derive a nonrelativistic counterpart of wave equation (457), we note that for u << c, or equivalently, mb << mo. the following approximations hold:

+ m:/(2mo) + 3m:/(8mi) +. . . mo + m” /(2mo), m: = mb + m!/(2mi) + 3 m $ / ( 8 m i ) + m , = mo

0

%

(470)

. - a

x mb.

(47 1)

Consequently, in the nonrelativistic regime, Eq. (454) takes the form7 exp { -i/h [ ( T

+ moc2)1 F p - r] } = 0,

(472)

in which (473) are interpreted, according to a conventional approach, as kinetic energy T = (rn011)’/2rnoand momentum p = mov of a particle. Equations (459),(472),and (473)form a basis to reinterpret conventional quantum mechanics in terms of a tachyonic theory of matter, presented as follows. B. Particle on a Line We consider the simplestquantum-mechanicalproblem, a particle trapped in a onedimensional infinitely deep potential well of length a. Then the time-dependent Schrodinger equation (472) has solutions

wk(r,x)

= (2/a)’/’sin(knx/a) exp [-ih-’

(mbc)’ (khn)’ Tk=--2mo 2moa2’

(mot' + T k ) f ] ,

k = l , 2 , 3 , ...

(474) (475)

that satisfy the boundary condition $(x = 0, a) = 0. The amplitude function appearing in Eq.(474)is a solution of the time-independent Schrodinger equation (A

+ ~ ~ ) ( 2 / a sin(knx/a) >’/~ = 0,

in which

’ To keep the notation simple we r e t m to the original variables

xP

instead of .V

(476)

MARCIN MOLSKI

222

Taking Eq. (475) into account, we rewrite Eq. (476) as [A

+ (m;c/h)'] (2/a)'/' sin(knx/a) = 0,

(478)

which represents a purely spacelike equation for a tachyon at infinite speed and endowed with a quantized nonrelativistic mass

khrr

mi = -, k = l , 2 , 3,... ca

(479)

and momentum

kAn

pk = fm;c = &-. a

Accordingly, we show that a particle on a line can be considered to be a compound of a bradyon at rest endowed with mass m, = mo Tk/c2 and a tachyon of mass m: = mi moving at infinite speed back and forth between the potential walls. This controversial conclusion becomes more comprehensible when we consider the transformational properties of the field associated with a particle trapped on a line. Thus we assume that a line with a particle trapped thereon moves in the x -directionat a velocity V relative to the laboratory frame. Space (time)likefields associated with a particle undergo a space (time) Lorentz transformation, yielding

+

'

in which B = V / c ; and $ transform as typical space- and timelike fields that satisfy wave equations

and

for subluminal and superluminal particles endowed with masses rn: and m,, respectively. In conventional quantum mechanics, wave function (474) is identified with an ordinary B-wave of only statistical meaning. According to the Copenhagen interpretation it determines the probability of finding a particle between the points x and x d x , which is equal to

+

d P = **9dx = (2/a) sin'(nkx/a) d x .

(485)

THE DUAL DE BROGLIE WAVE

223

Equation (485) shows that fork 2 2 the wave function has nodes. In particular, for k = 2 the node appears at the middle of the line; hence the probability of finding a particle between x = a / 2 and x = a/2 dx is zero. If a particle really moves between the ends of the line, a question arises as to how it moves from one side of the node to the other. Several solutions to this problem are proposed (Nelson, 1990), of which proposals of Bohm (1952a,b) and White (1934) merit detailed analysis. In the de Broglie-Bohm theory, the wave function (474) is given in a factorized form (16) in which

+

If we consider nonrelativistic guidance formula (32). by virtue of Eq. (486) one finds that u = 0; hence a particle trapped on the line is at rest,' so it cannot traverse the nodes. According to White's (1934) interpretation, the infinitesimal interval in which dx in a stationary state is equal to Q Q * dx = the particle is between x and x p d x . Hence its average velocity during crossing left to right, and vice versa, is given by u = d x / d t o< p - ' ; at the nodal point at which p = 0, u = 00 (i.e., the particle traverses the node infinitely quickly). This result remains contradictory to a classical picture assuming a subluminal velocity for a particle moving between the ends of a line. Detailed analysis of these concepts indicates that they describe a basic problem from two points of view, related to the time- and spacelike constituents of a particle. If a moving particle is a system of a bradyon at rest and a transcendent tachyon, the following consequences apply. An amplitudal function R ( x ) = (2/a)'I2 sin(knx/a) is a purely spacelike wave associated with the tachyonic Component of a particle, endowed with mass mi = knh/ac and momentum P k = f m ; c = f k n h / a . An exponential function @ ( t ) = exp[-ih-'(moc2 Tk)t]describes a bradyonic component of a particle, remaining at rest and having rest mass m, = mo Tk/c2. A superposition of these fields Q ( r , x ) = R(x)*(r) satisfies the nonrelativistic time-dependent Schrodinger equation for a particle of mass rno moving at velocity u << c on a line. These conclusions explain the results of de Broglie and Bohm and of White. The guidance formula imposes zero velocity for a bradyonic constituent of the particle; the node is traversed by its tachyonic component but not by the particle itself. The problem of transit of a particle from one side of the node to the other consequently disappears.

+

+

+

*

This result was criticized by Einstein (de Broglie, 1960, p. 136), but Bohm (1952a.b) asserted that it holds only for a particle moving in the box with perfectly rigid walls. It does not apply to a particle in a real box, since the thermal motion of the walls prevents the particle from being at rest.

MARCIN MOLSKI

224

The constant velocity attributed to a particle trapped on a line is explained on comparison of Eq. (475) with Eq. (417) to yield the important relation

Hence the source of quantum potential on a line is a tachyonic constituent of the particle, and a quantum force F generated on the line by the quantum potential Q equals zero: Because neither external nor internal forces act on the particle, it moves at constant velocity. C. Particle in a Scalar Field

The proposed interpretation of a particle on a line also holds for a particle interacting with an external scalar field endowed with a potential U(r). Then the relativistic four-momentum formula

(E-

- p2/c2 = m i

(488)

takes the equivalent form (m, - u/c 2 ) 2 - rn: = m i ,

(489)

in which m, and m i are defined in Eq. (459). In the nonrelativisticregime Eq. (489) becomes

m: a m ; =

AC JG.

(491)

Hence the associated tachyonic component is endowed with variable rest mass rnb. depending on the particle’s position, in contrast with the bradyonic component that for stationary systems (E = const) is endowed with constant rest mass m,. Consequently, the time dependence of particle velocity in conventional mechanics corresponds to space dependence of mass of a tachyonic particle constituent, rnov(r) = mb(r)c.

(492)

Because by differentiating Eq. (492) with respect to time, we obtain (493) and taking into account the relation valid for conserved fields, dv = -VU = F, dt

mo-

(494)

225

THE DUAL DE BROGLIE WAVE

we find an important identity

as a direct consequence of Q. (490). These results indicate that two representations of the particle dynamic are possible: a conventional one for interactions with a field of potential U and another assuming the existence of a tachyonic field T interacting with a particle. According to a classical picture, both descriptions are equivalent and yield an equation of motion (493) of the same form as Eq. (494). In a quantal description, the Schrodinger equation for a particle interacting with an external scalar field takes the form ( i h k - moc 2 - U ( r )

+-

R ( r ) exp [ - i l k ( &

+ moc2)t] = 0,

(496)

h2m0 2A>

in which the amplitudal term is a solution of the spacelike wave equation [A

+ (m;c/h)*]R ( r ) = 0

(497)

for a tachyonic component endowed with variable rest mass

Hence E k denotes a quantized energy of the particle system, resulting in mass quantization of a tachyonic component. Taking Eq. (498) into account, one may introduce a quantal counterpart of a classical potential T = - ( m b ~ ) ~ / 2 m o ,

(499) which differs from quantum potential (29) in the de Broglie-Bohm theory by sign. Consequently, forces generated by Tk(r) and Q ( r ) are equal but act in opposite directions. To summarize these considerations we propose a dynamical theorem.

Dynamical Theorem. Quanta1 interaction of a particle with a field endowed with potential U ( r ) is interpreted as a result ofparticle interaction with a quantum potential Tk ( r )associated with a transcendent tachyon of quantized variable mass rn; = ( h / c ) , / - A R ( r ) / R ( r ) . To show how the dynamical theorem works, we consider a particle moving in oscillatory motion along the x-axis in a field endowed with harmonic potential U ( x ) = 1/2moo2x2,in which o denotes the frequency of vibrations. The solution

MARCIN MOLSKI

226

of wave equation (496) takes the form mow

q(r,x)k = Akexp [--i/ft(Ek +moc2)rl~k(x)exp[ - x x

2

]

= 0,

(500)

in which H k ( x ) , k = 0, 1,2, 3, . . . are Hermite polynomials, whereas

are the normalization factor and vibrational energy, respectively. The amplitudal part of Eq.(500) is a solution of a spacelike wave equation (497),

including quantized variable rest mass 2rno[hw(k

+ 1/21 - 1/2mow2x2]

(503)

C2

Wave equation (502) and its solutions transform as typical spacelike expressions, turning in a frame moving at velocity V = c B along the fx-axis, to

1

[O - (mi~/h)~]AkHk[(x f Bxo)G]exp

mi =

2mo(hw(n

+ 1/2) - 1/2mow2[(xf Bx0)Gl2} C2

in which G = (1 - B 2 ) - ' / 2 . Comparison of Eqs. (502) and (505) shows that tachyonic variable rest mass is not relativisticallyinvariant but varies in the moving frame according to formula (505). Superposition (500) of a spacelike field (502) and a timelike field described by the exponential part of wave function (500) produces a field of timelike type, associated with a bradyon-tachyon system moving at a velocity

Equation (506) shows that at the turning points

+

h(2k 1) (507) mow the tachyonic mass and the velocity of a system of bradyon and tachyon reduce to zero, whereas in the region beyond the turning points, which is unattainable in

x = f

I

THE DUAL DE BROGLIE WAVE

227

classical mechanics, the tachyon acquires an imaginary mass so that the compound particle moves in the forbidden region at an imaginary velocity v = ic {mow2x2 -my(2k

+ I)

(508)

As a particle endowed with such a velocity travels areal distanced x in an imaginary interval i d t , we conclude that a tachyonic interpretation of the tunnel effect is equivalent to that employing extrapolation to imaginary time (Bjorken and Drell, 1964; Freed, 1972; Jackiw and Reebi, 1976). The proposed tachyonic interpretation of particle interaction holds true for all scalar fields endowed with a potential U ( r ) .Thus the proposed model is applicable to describe electromagnetic, strong, and gravitational interactions, provided that we can solve a suitable wave equation to determine a spacelike field R ( r ) .

D. Uncertainty Principle We examine Heisenberg's uncertainty relations AEAtZh,

Ap,Aa?h,

CY=X,Y,Z

(509)

h -. C

(5 10)

which with Eq. (459) become o h Am,Ax 1 -,

Am'zAa ?

C

Relations (509) can be interpreted to be a result of delocalization in space-time of conformal masses of bradyonic and tachyonic constituents of a particle. This conclusion becomes clear when we recall that free bradyons invariably admit subluminal reference frames (rest frames) of a particular class from which they appear as points in space extended in time along a line. Free tachyons invariably admit subluminal reference frames from which they appear as points in time extended in space along a line (Recami, 1986, p. 56). The reason is that localization groups (little groups) of time- and spacelike representationsof the Poincark group are SO3 and S02.1;hence tachyons (bradyons) are not localizable in space (time) (Recami, 1986, p. 59). If a region of possible motion of a particle is restricted to size a, the spatial uncertainty relation enables us to estimate qualitatively its tachyonic mass Am;

h -,

ac

in accordance with mass formula (479) obtained for a particle trapped on a line.

228

MARCIN MOLSKI

E. Many-Body Problem

A fundamental question emerging from this discussion is whether our approach can be generalized to include the most general case of the many-body problem. To answer this question, we consider a stationary system of N interacting particles of mass ms described with a generalized Schriidinger equation (15)

in which

is the total mass of the system, whereas the wave function tL.es the form Q ( x , t ) = R ( x ) exp [-i/h(Ek

+moc2)t].

(5 14)

In Eq.(514) E k denotes the overall quantized energy of a system depending on quantum numbers k = (kl,k2, . . .), whereas the amplitudal function is a solution of a spacelike wave equation

Because Eq.(512)is formulatedin configuration space M ( 1,3N),we can interpret it within a generalized Corben’s (1977) approach, assuming that several spacelike states with mass m&,can combine with a timelike state mo to produce a subluminal particle of mass

I

N s=l cr=x.y.z

Accordingly, proceeding along lines of the one-body problem, we define the mass of tachyonic component of a system, (517) that allows calculation of local velocity ua, of particle s:

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229

directly from amplitudefunction R(x), and not from the phase S ( x , t ) via guidance formula (32). We also derive a quantum potential Tk(x),

and with Eq. (517), the classical potential T(x)=

-xN

s=l

(m:d2 - U(x)-E. 2mS

According to de Broglie-Bohm theory, the amplitudal function R(x) determines the quantum potential Q ( x ) and probability density for particles to be at certain positions in space-time (Bohm etal., 1987). Equation (518) clearly shows that in our approach R ( x ) additionally determines the velocities of particles, which depend on the state of the entire quantum system described by R(x). Although particles follow continuous and causally determined trajectories, obtained by integration of Eq.(5 18). they execute random movements, resulting in a statistical interpretation of R(x). For points at which R(x) = 0, the quantum potential Tk(x) = 00 and the particle velocity attains an infinite value according to Eq. (5 18). The probability of finding a particle there equals zero; conversely, if R(x) at some point takes a large value, the particle velocity tends to zero and the probability of finding it there increases. This interpretationis fully compatiblewith our intuitive (classical) understanding of the notion probubilify. If wave function (514) is factorizable, N

N

s=l

s= I

and the basic wave equation (512) decomposes into N independent equations [ i h k - msc2 - Us(rs)

+ -2mS As h2

1

Rs(rs) exp [-i/h(Ek,,

+ mSc2)t] = 0,

interpretable within a framework of the one-body problem considered in Section VI1.C. In such circumstances the quantum potential Tk(x) reduces to the sum of terms

Consequently, we consider an N-particle system as N bradyons of mass E k , / c 2

m , , interacting independently with N tachyons endowed with mass ml .

+

230

MARCIN MOLSKI

For a nonfactorizable wave function (5 14) this interpretation is inapplicable. In such circumstances the system of interacting particles forms a holistic structure [unbroken wholeness (Bohm and Hiley, 1975)], which cannot be considered in terms of separate and independent parts. However, having solved Schrodinger's equation (512) in an approximation scheme, one may determine a spacelike field R ( x ) and an associated quantum potential Tk(x),and then calculate velocities and trajectories of particles in the system according to formula (5 18).

E Conclusions

Our results indicate that we may construct in the framework of a two-wave model of a particle and tachyonic theory of matter a generalized de Broglie-Bohm theory in which an amplitudal wave R ( x ) is identified with a spacelike D-wave, and an exponential wave + ( x , t ) = exp[ih-'S(x, r ) ] is identified with a timelike B-wave. Thesetwo wavesinteractandlocktoformatimelikewaveq ( x , r ) = R ( x ) + ( x , r ) , conventionally employed in the causal interpretation of the quantum mechanics. Carriers of the spacelike field R ( x ) are tachyons of infinite speed and generally of variable rest mass, which are also a source of a quantum potential Tk(x) and its classical counterpart T ( x ) . The former has properties identical to those of the quantum potential Q ( x ) in de Broglie-Bohm theory; however, Tk ( x ) generates quantum forces acting in directions opposite that of forces generated by Q ( x ) . This fact has important ontological and methodological consequences. First, if Tk(x) # 0, a quantum system cannot appear in a motionless state, as de Broglie-Bohm theory predicts for a particle trapped in the box (Bohm, 1952a.b) or an electron in state ns of a hydrogen atom (de Broglie, 1960, p. 126). Consequently, even though a guidance formula attributes zero velocity to a quantum system, its components can move at a velocity determinedwith an amplitudal wave R ( x ) and not with phase S ( x , t) of an exponential wave associated with the system. Second, in the classical limit (h + 0) the de Broglie-Bohm quantum potential Q ( x ) may be neglected; then a particle is influenced only by the classical potential V ( x )(Bohm and Hiley, 1975). In an extended approach one may distinguish a quantum potential Tk(x) and its classical counterpart T ( x ) ,both well defined but in separate ontological domains. As the potentials Tk(x) and Q ( x ) differ only in sign, the former includes all well-known properties of the latter. For example, the quantum potential produces no vanishing interaction between components of quantal systems; even distant objects may still have strong and direct interconnections. As the quantum potential depends on R ( x ) , being a generally nonfactorizable function of position coordinates of all components of a system, it also depends on the quantum state of the entire system. In such circumstances a quantum many-body system cannot properly be analyzed into independentlyexisting parts, with fixed and determinate dynamical relationships between them (Bohm and Hiley, 1975).

THE DUAL DE BROGLIE WAVE

23 1

As with the spacelike field R ( x ) are associated transcendent tachyons playing the role of carriers of particle interactions, the quantum potential Tk(x) introduces instantaneous superluminal interactions at a distance interconnecting all particles in the quantum system. For this reason the behavior of each constituent may depend nonlocally on all others, no matter how distant they may be. According to Bohm et al. (1987): If one reflects deeply and seriously on this subject one can see nothing basically irrational about such an idea. Rather it seems to be most reasonable to keep an open mind on the subject and therefore to allow oneself to explore this possibility. If the price of avoiding non-locality is to make an intuitive explanation impossible, one has to ask whether the cost is not too great.

The proposed tachyonic interpretation of particle interactions can be applied not only to microparticles but also to large-scale systems, including the universe as a whole. Then all macro- and micro-objects are seen to be in direct connection, depending, in an irreducible way, on the state of the entire system. According to this picture, transcendent tachyons form a universal background for phenomena of energy-momentum transfer between parts of a system, and the universe appears as a multidimensional self-trappingtachyonic bootstrap (Chew, 1968, 1970; Corben, 1978a). This conclusion is fully consistent with the results of experiments (Freedman and Clauser, 1972; Aspect et al., 1982) that test Bell's theorem (Bell, 1964). Bell-Kochen-Specker theorem (Bell, 1966; Kochen and Specker, 1967) and other theoretical results in the field (Bell, 1982,1987; Stapp, 1971,1972,1975,1977). It is also compatible with the concept of synchronicity first proposed by Jung (1953, which indicates a possibility of extension of the interpretativearea even to living systems.

VIII. INFONS? The results obtained in Sections 1I.A and VI1.A show that one can propose that all particles appearing in nature may be considered to be systems composed of a bradyon at rest and a tachyon at infinite speed. Then if we express time- and spacelike four-momenta of particle constituents in the form pP = hot, I),

the associated B- and D-waves are given as

I = (O,O, O),

(524)

232

MARCIN MOLSKI

Consequently, the bradyon-tachyon system, representing a particle, becomes described with a superposition of time- and spacelike fields Q(XP)

= exp [ih-’(mocx’ fmbc. r)] I ( X P ) ,

I * - I* = 0.

(528)

(530)

This result indicates that the Q(xP) wave is invariably coupled with an additional photonlike I ( x P ) wave associated with an object having zero energy and zero momentum. This additional wave is characterized with an infinite wavelength and zero frequency. Characteristicsidentical to these were attributed to hypothetical particles called infons (Stonier, 1990). According to this concept, infons possess neither mass nor energy and consist only of information, and may be envisaged to be photons that appear to have stopped oscillating, so that their wavelengths become infinitely stretched. Equation (528) reveals that such a hypothetical particle may be attached to objects of all types, whether or not their mass is zero and whether they move at subluminal,luminal, or superluminalvelocity. Consequently, our tachyonic theory of matter provides a conceptual background for the hypothesis of infons assumed to be a particular form of information manifesting itself, for example, in variation of organization (Stonier, 1990).

IX.CONCLUDING REMARKS The concepts of a D-wave and an associated spacelike particle constituent, even if speculative, deserve attention, as they may play an important role in the internal structureof microparticles; their incorporationinto quantum mechanics and special relativity enables us to understand many problems of contemporaryphysics. These include extended wave-corpuscular models of massive and massless particles, the internal structure of micro-objects, the nature of mass and charge and the force binding them in a particle, a formulation of extended special relativity, and quantum mechanics, including spacelike particle states and superluminal particle interactions. The former two problems were considered in the framework of Maxwellian ,electromagnetismand Einstein’s special relativity. They are sufficient to construct simple extended wave-corpuscular models of massive and massless particles from light.

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233

The idea that mass has an electromagnetic origin was developed long ago (Jackson, 1975) but still animates discussions (Weinberg, 1975; Sedlak, 1986; Winterberg, 1987). In cosmological models of the universe, for example in KleinAlfven and Charon’s models, and in the standard model, one assumes that all particles originated from photons of high energy at the beginning of the universe. Although the proposed approach seems controversial, it is fully consistent with classical electromagnetism and special relativity, whereas interpretation of the obtained results employs well-known concepts in the field: a two-wave particle model, an extended space-timedescription of matter, a nonlinear wave hypothesis, a tachyonic theory of elementary particle structure, and an electromagnetictheory of matter. The third problem, concerning the internal structure of charged particles, is considered in a framework of five-dimensional Kaluza-Klein theory, which seems to be the best framework to unify gauge fields and gravitation. The proposed model of a charged particle as a relativistic self-trapping tachyonic bootstrap is consistent with Post’s model formulated in ordinary four-dimensional space, but it advances in trying to explain the nature of electric charge and the force binding charge and mass to produce a stable charged particle. The fourth problem is realized in a framework of the L-space concept and extended de Broglie-Bohm theory. The L-space concept enables one to formulate extended special relativity and quantum mechanics in the simplest and most natural manner, whereas the extended de Broglie-Bohm theory generates a bootstrap model of physical reality, including nonlocal superluminal interactions. The proposed tachyonic interpretation of wave mechanics and the results presented should be treated as an outline reporting only the most important ideas in the field worth of further investigation. However, the preliminary results, which seem only the tip of an iceberg, are sufficient to suppose that interpretation of wave mechanics in this manner might be useful to explain, for instance, the EinsteinPodolslq-Rosen (1935) paradox or interactions at a distance (Kocis, 1994), appearing, for example, in Pfleegor-Mandel(1967), Freedman-Clauser (1972), and Aspect-Dalibard-Roger (1982) photon experiments. According to the proposed connection, the tachyonic components of quantum systems are responsible for the existence of non-message-bearing quantum correlations over a spacelike interval involving holistic features of nature (Horodecki, 1994). This part of the article is a realization of Recami’s (1986, p. 160) program: For future research, it looks, however, even more interesting to exploit the possibility of reproducing quantum mechanics at the classical level by means of tachyons. In this respect even the appearance of imaginary quantities in the theories of tachyons can be a relevant fact, to be further studied.

The results presented here and our proposed interpretations indicate that a Dwave concept may be key to open a door to an unknown physical reality that seems to be faster than hitherto supposed.

234

MARCIN MOLSKI ACKNOWLEDGMENTS

This work is dedicated to the memory of Professors W. Sedlak and R. Dutheil. Thanks are due to Professors E. Recami, H. C. Corben, and V. VySin and Dr. L. Kostro for making it possible to become acquainted with research work indispensable for this article. I am indebted also to Dr. M. Wnuk who brought the infon concept to my attention, and to Professor J. Konarski and Dr. R. Horodecki for constructive discussions. Special thanks are also due to Professor Ch. Marcinkowski for stimulating correspondence, and to my family Z. Z. Skupniewicz and Skupniewicz-Molska for support.

REFERENCES Aspect, A., Dalibard. J., and Roger,’G.(1982). Experimental test of Bell’s inequalities using timevarying analyzers. Phys. Rev. Lerr. 49, 1804. Barut, A. 0. (1990). E = hw. Phys. Lett. A 143,349. Barut, A. 0. (1992). Formulation of wave mechanics without the Planck constant h. Phys. Lett. A 171, 1. Barut, A. O., and Nagel, J. (1977). Interpretation of space-like solutions of infinite-componentwave equations and Grodsky-Streater “no-go” theorem. J. Phys. A 10, 1233. Bell, J. S. (1964). On the Einstein-Podolsky-Rosen paradox. Physics 1, 195. Bell. J. S. (1966). On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38,447. Bell, J. S. (1982). On the impossible pilot wave. Found. Phys. 12,989. Bell, J. S. (1987). Speakable and Unspeakable in Quantum Mechanics. cambridge University Press, Cambridge. Bilaniuk, 0. M. P., Deshpande, V. K., and Sudarshan, E. C. G. (1962). “Meta”re1ativity. Am. J. Phys. 30.7 18. Bilaniuk, 0.M. P., and Sudarshan, E. C. G. (1969). Particles beyond the light barrier. Phys. Today. May, pp. 43-5 I. Bjorken. J. D.. and Drell. S. D. (1964). Relativistic Quantum Mechanics, Vol. 1. McGraw-Hill, New York, p. 86. Bohm, D. (1952a). A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85, 166. Bohm, D. (1952b). A suggested interpretation of the quantum theory in terms of “hidden” variables. 11. Phys. Rev. 85, 180. Bohm, D. (1953). Proof that probability density approaches 1+,1’ in causal interpretation of the quantum theory. Phys. Rev. 89.458. Bohm, D., and Bub. J. (1966a). A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory. Rev. Mod. Phys. 38,453. Bohm, D., and Bub, J. (1966b). A refutation of the proof by Jauch and Piron that hidden variables can be excluded in quantum mechanics. Rev. Mod. Phys. 38.470. Bohm, D., and Hiley, B.J. (1975). On the inuitive understanding of nonlocality as implied by quantum theory. Found. Phys. 5.93. Bohm. D., Hiley, B. J., and Kaloyerou, P. N. (1987). An ontological basis for the quantum theory. Phys. Rep. 144.321-375.

THE DUAL DE BROGLIE WAVE

235

Bohm, D., and Vigier, J. P. (1954). Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations. Phys. Rev. 96,208. Bohr. N. (1934). Atomic Theory and the Description of Nature. Cambridge University Press, Cambridge. Bohr, N. (1958). Atomic Physics and Human Knowledge. Wiley, New York. Boyer, T.H. (1968). Quantum electromagnetic zero-point energy of a conducting spherical shell and the Casimir model for a charged particle. Phys. Rev. 174, 1764. Chew, G. F. (1968). Bootstrap: a scientific idea? Science 161, May 23, p. 762. Chew, G. F. (1970). Hadron bootstrap: triumph or frustration? Phys. Today 23, October. p. 23. Chin, S. L. and Lambropoulos, P. (1984). Multiphoton Ionisation of Atoms. Academic Press, New York. Clay, R. W. (1988). A search for tachyons in cosmic ray showers. Aust. J. Phys. 41.93. Cole, E. A. B. (1977). Superluminal transformations using either complex space-time or real spacetime symmetry. Nuovo Cimento A 40, 171. Cole, E. A. B. (1979). Emission and absorption of tachyons in six-dimensional relativity. Phys. Lett. A 75.29. Cole, E. A. B. (1980a). Comments on the use of three time dimensions in relativity. Phys. Lett. A 76, 37 1. Cole, E. A. B. (1980b). Particle decay in six-dimensional relativity. J. Phys. A 13, 109. Corben, H. C. (1977). Relativistic selftrapping for hadrons. Lett. Nuovo Cimento 20,645. Corben, H. C. (1978a). Electromagnetic and hadronic properties of tachyons. In: E. Recami, Ed. Tachyons, Monopoles and Related Topics. North-Holland, Amsterdam, pp. 3 14I. Corben, H. C. (1978b). The 4, f’,FI K spectrum. Lett. Nuovo Cimento 22, 116. Corben H. C. (1995). The quantization of relativistic classical mechanics. Phys. Essays 8,321. Costa de Beauregard, 0. (1972). Noncollinearity of velocity and momentum of spinning particles. Found. Phys. 2, I 11. Coulson, C. A,, and Jeffrey, A. (1977). Waves: A Mathematical Approach to the Common Types of Wave Motion. Longman, London, p. 15. (Polish edition). Czachor, M. (1989). Mackinnon’s soliton reexamined. Phys. Lett. A 139, 193. Das, S. N. (1984). De Broglie wave and Compton wave. Phys. Lett. A 102,338. Das, S. N. (1986). A two-wave hypothesis of massive particles. Phys. Lett. A 117,436. Das, S . N. (1988). The pseudovelocity and its consequences. Phys. Lett. A 129,281. Das, S. N. (1992). De Broglie wave theory and the two-wave description of matter. Nuovo Cimento B 107. 1185. Davidson, A,, and Owen, D. A. (1986). Elementary particles as higher-dimensional tachyons. Phys. Lett. B 177, 77. de Broglie, L. (1924). T h b e 1924. Masson, Pans. de Broglie, L. (1951). Pmblimes de la propagation quidbe des ondes electromagndiques. 2nd ed. Gauthier-Villars, Paris, pp. 34-36. de Broglie, L. ( 1960). Non-linear Wave Mechanics: A Causal Interpretation. Elsevier, Amsterdam. Demers, P. (1975). SymCtrisationde la longueur et du temps dans un espace de Lorentz C3 en algdbre lineaire, pouvant servir en thtorie trichromatique des couleurs. Can. J. Phys. 53, 1687. Diner, S., Fargue, D., Lochak. G., and Selleri, F. (1984). The Wave-Particle Dualism. D. Reidel, Dordrecht, The Netherlands. Duffey, G. H. (1975). Tachyons and superluminal wave groups. Found. Phys. 5,349. Duffey, G. H. (1980). Reconciling causality with superluminal travel groups. Found. Phys. 10,959. Dutheil, R. (1984). Sur un model de particule dont le rtfkrentiel propre est du type I.M.F. Bull. SOC. Sci. L i g e 53, 129. Dutheil, R. (1989). Theorie de la relativiti et mkanique quantique dans la region du genre espace. Derouaux Ordina, Lidge, Belgium. Dutheil, R. (1990a). Relativitk et mkcanique dans la kgion du genre espace. Ann. Fond. L. de Broglie.

236

MARCIN MOLSKI

15,449.

Dutheil, R. (1990b). Pr6ons. bradyons at tachyons. Ann. Fond. L. de Bmglie. 15.47 I . Dutheil R. (1993). Sur une interpretationdes propridtes de la polarisation des ondes Cvanscentes.Ann. Fond. L de Broglie 13,239. Einstein, A. (1955). The Meaning of Relativity. Princeton University Press, Princeton. N.J. (Polish edition). Einstein, A., Podolsky, B., and Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47,777. Elbaz, C. (1985). On de Broglie waves and compton waves of massive particles. Phys. Lett. A 109.7. Elbaz, C. (1986). Some physical properties of the amplitude function of material particles. Phys. Lett. A 114.445.

Elbaz, C. (1987). Some inner physical properties of material particles. Phys. Lett. A 123,205. Elbaz, C. (1988). On self-field electromagnetic properties for extended material particles. Phvs. Lett. A 127,308. Elbaz. C. (1995). Classical mechanics of an extended material particle. Phys. Lett. A 204,229. Feinberg, G. (1967). Possibility of faster-than-lightparticles. Phys. Rev. 159, 1089. Feinberg, G. (1970). Particles that go faster than light. Sci. Am. 222.68. Feldman. L. M. (1974). Short bibliography on faster-than-light particles (tachyons) Am. J. Phys. 42, 179. Freed, K. F. (1972). Path integrals and semiclassical tunneling, wavefunctions,and energies.J. Chem. Phys. 56,692. Freedman, S . J., and Clauser, J. F. (1972). Experimentaltest of local hidden variable theory. Phys. Rev. Lett. 28.938. Gegenberg. J., and Kunstater, G. (1984). The motion of chargedparticles in Kaluza-Klein space-time. Phys. Lett. A 106.410. Goldhaber, A. S., and Nieto, M. M. (1971). Terrestrial and extraterrestrial Limits on the photon mass. Rev. Mod. Phys. 43, 277. Gueret, Ph., and Vigier, J. P. (1982). Nonlinear Klein-Gordon equation carrying a nondispersivesolitonlike singularity. Found. Phys. 35,256. Hamamoto, S. (1974). Subluminal particle as a composite system of superluminal particles. Prog. Theor. Phys. 51, 1977. Hoh, F. C. (1976). Quark theory with internal coordinates. Phys. Rev. D 14,2790. Horodecki, R. (1981). De Broglie wave and its dual wave. Phys. Lett. A 87.95. Horodecki. R. (1982). Dual wave equation. Phys. Lett. A 91,269. Horodecki. R. (1983a). Superluminal singular dual wave. Lett. Nuovo Cimento 36,509. Horodecki, R. (1983b). The extended wave-particle duality. Phys. Lett. A 96, 175. Horodecki, R. (1984). Wave-particle duality and extended special relativity. Nuovo Cimento E 80, 217. Horodecki, R. (1988a). Is a massive particle a compound bradyon-pseudotachyon system?Phys. Lett. A 133, 179. Horodecki, R. (1988b). Extended wave description of a massive spin-0 particle. Nuovo Cimento B 102.27. Horodecki, R. (1988~).Information concept of the aether and its application in the relativistic wave mechanics and quantum cybernetics. In: Problems in Quantum Physics-Recent and Future Experiments and Interpretations. World Scientific, Singapore, pp. 582-595. Horodecki, R.(1991). Unitary information ether and its possible applications. Ann. Phys. 7,479. Horodecki. R. (1994). Informationallycoherent quantum systems. Phys. Lert. A 187, 145. Horodecki, R., and Horodecki, P. (1995). Dirac electrodynamics with gauge symmetry breaking and the tachyonic theory of elementary particle structure. Hadronic J. 18, 161. Hunter, G.. and Wadlinger, R. L. P. (1988). Finitephotons:thequantaofaction. In: Problemsin Quun-

THE DUAL DE BROGLIE WAVE

237

rum Physics-Recent and Future Experiments and Interpretations. World Scientific, Singapore, pp. 149-162. Jackiw, R., and Rebbi, C. (1976). Vacuum periodicity in a Yang-Mills quantum theory. Phys. Rev. Lett. 37, 172.

Jackson, 1. D. (1975). Classical Electrodynamics, 2nd ed. Wiley, New York (Polish edition). Jammer, M. (1974). The Philosophy of Quantum Mechanics. Wiley-Interscience, New York, p. 68. Jehle, H. (1971). Relationship of flux quantization to charge quantization and the electromagnetic coupling constant. Phys. Rev. D 3,306. Jehle, H. (1972). Flux quantization and particle physics. Phys. Rev. D 6,441. Jehle, H. (1975). Flux quantization and fractional charges of quarks. Phys. Rev. D 11,2147. Jennison, R. C. (1978). Relativistic phase-locked cavities as particle models. J. Phys. A 11, 1525. Jennison, R. C. (1983). Wave-mechanical inertia and the containment of fundamental particles of matter. J. Phys. A 16,3635. Jennison, R. C. (1988). The non-particulate nature of matter and the universe. In: Problems in Quantum Physics-Recent and Future Experiments and Interpretations. World Scientific, Singapore, pp. 163-186. Jennison, R. C., and Drinkwater, A. J. (1977). An approach to the understanding of inertia from the physics of the experimental method. J. Phys. A 10, 167. Jennison, R. C., Jennison. M. A. C., and Jennison, T. M. C. (1986). A class of relativistically rigid proper clocks. J. Phys. A 19,2249. Julg, A. ( I 993). Courants, amers, 6cueils en microphysique. p. 2 11. Fondation Louis de Broglie, Paris. Jung, C. G. (1955). Synchronicity: acausal principle. In: The Inrepretation of Nature and the Psyche. Ballinger, New York, p. 90. Kaluza, Th. (1921). On the unification problem in physics. Sitzungsber: Preuss. Wiss., p. 966. English translation in An Introduction to Kaluza-Klein Theories. World Scientific, Singapore, pp. 1-9. Kittel, C. (1966). Introduction to Solid State Physics. Wiley, New York (Polish edition), p. 53. Klein, 0. (1926). Quantum theory and five-dimensional relativity. Z. Phys. 37,895. English translation in An introducrion to Kaluza-Klein Theories. World Scientific, Singapore, pp. 10-22. Kochen. S., and Specker, E. P.(1967). The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59. Kocis. L. (1994). Interference at very low intensities-a review of experiments confirming quantum mechanics. Phys. Lett. A 187,40. Kostro, L. (1978). A wave model of the elementary particle: a three waves hypothesis. Unpublished paper quoted by Kostro (1985a.b). Kostro. L. (1985a). A three wave model of the elementary particle. Phys. Lett. A 107,429. Kostro. L. (1985b). Planck's constant and the three waves (TWs) of Einstein's covariant ether. Phys. Lett. A 112,283. Kostro, L. (1988). Mackinnon soliton on top of Einstein's relativistic ether. In: Problems in Quantum Physics-Recent and Future Experiments and Interpretations. World Scientific, Singapore, pp. 608619.

Kreisler, M. N. (1973). Are there faster-than-light particles? Am. Sci. 61. 201. Logan, B. A,, and Ljubicic, A. (1976). Background counting rates and possible tachyonic decays. Am. J. Phys. 44,789. Maccarone, G. D., and Recami, E. (1982). Lett. Nuovo Cimento 34,251. Mackinnon, L. (1978). A nondispersive de Broglie wave packet. Found. Phys. 8, 157. Mackinnon, L. (1981a). Particle rest mass and the de Broglie wave packet. Lett. Nuovo Cimento31,37. Mackinnon, L. ( 198 1b). A fundamental equation in quantum mechanics'?Lett. Nuovo Cimento 32,3 1 1. Mackinnon, L. (1988). The non-dispersive wave packeet and its significance for quantum mechanics. In: Problems in Quantum Physics-Recent and Future Experiments and In terpretations. World Scientific, Singapore, pp. 225-246.

238

MARCIN MOLSKI

Markov, M. A. (1982). Proceedings of the Workshop “The Very Early Universe.” University Press. Cambridge, p. 363. Mathews, P.M., and Seetharaman, M. (1973). Method of characteristics and causality of field propagation. Phys. Rev. D 8, 1815. Meszaros. A. (1986). Indirect experimental evidence against the tachyons. As/rophys. Space Sci. 123,490. Minn, H. 1990. Creation from “nothing or anything”. Nuovo Cimenfo B 105,901. Molski, M. (1991). Extended wave-particle description of longitudinal photons. J. Phys. A 24,5063. Molski, M. (1992). Classical and field description of space-like objects in a 5-dimensional space with an extra dimension coupled to the object rest mass. Hadronic. J. 15, 135. Molski, M. (l993a). Tachyonic properties of space- and time-trapped electromagnetic fields. J. Phys. A 26, 1765. Molski, M. (l993b). An electromagneticapproach to special relativity and quantum mechanics. Phys. Essays 6, 143. Molski, M. (1993~).Ancient cosmological tachyons in the present-day world. Hudronic J. 16,207. Molski, M. (1 994). Electromagnetic model of extended spin-0 particles. Ann. Fond. L. de. Broglie. 19, 361. Molski, M. (1995a). Extended wave-particle description of luminal-type objects. Ann. Fond. L. d e Bmglie. 20,45. Molski, M. (1995b). Does a dual de Broglie wave exist? Ann. Fond. L. de Broglie. 20, 181. Molski. M. (1995~).Special relativity and space-time geometry. Phys. Essays 8.601. Molski, M. (1997). Is electric charge a superluminal particle? Phys. Essays 10, No. I , in press. Nelson, P. G.(1990). How do electrons get across nodes? J. Chem. Ed. 67,643. Pagels, H. (1976). Collective model of the hadrons. Phys. Rev. D 14,2747. Parker, L. (1972). Backscatteringcaused by the expansion of the universe. Phys. Rev. D 5,2905. Perepelitsa, V. F. (1986). Yu. B. Molchanov, Ed. In: Philosophical Problems of /he Hypothesis of Superlurninal Speeds. Nauka, Moscow. Perkins, D. H. (1982). Intmducrion / o High Energy Physics, 2nd ed. Addison-Wesley, London (Polish version). Pfleegor, R. L., and Mandel, L. (1967). Interference of independent photon beams. Phys. Rev. 159, 1084. Post, E. J. (1982). Can microphysical structure be probed by period integrals’?Phys. Rev. D 25, 3223. Post. E. J. (1986). Linking and enclosing magnetic flux. Phys. Lett. 119A,47. Rayski, J. (1965). Unified field theory and modem physics. Ac/a Phys. Pol. 27,89. Recami, E. (1986). Classical tachyons and possible applications. Riv. Nuovo Cimenio 9, 1-1 78. Recami, E.. and Mignani, R. (1974). Classical theory of tachyons (special relativity extended to superluminal frames and objects). Riv. Nuovo Cimenro 4,209. Robinson, T. R. (1995). Mass and charge distributions of the classical electron. Phvs. Lett. 200A, 335. Rosen, N.,and Szamosi. G. (1980). Classical particles with unusual properties. Nuovo Cimento 568.3 13. Sedlak, W. (1985). zycie Jest dwiurtem. PAX, Warsaw, p. I . Sedlak, W.( I 986). Nu poczqiku Byto Jednak dwiatto. PWN, Warsaw. Stapp. H.(1971). S-matrix interpretation of quantum theory. Phys. Rev. 0 3 , 1303. Stapp, H. (1972). The Copenhagen interpretation and the nature of space-time. Am. J. Phys. 40, 1098. Stapp, H. (1975). Bell’s theorem and world process. Nuovo Cimenro B 29,270. Stapp, H. ( 1977). Are superluminal connections necessary? Nuovo Cimento B 40, 191. Stonier, T. ( 1990). Information and /he Internal Structure of /he Universe. Springer-Verlag,London, p. 126.

THE DUAL DE BROGLIE WAVE

239

Toms, D. J. (1984). Kaluza-Klein theories. In: An Introduction to Kaluza-Klein Theories. World Scientific, Singapore, pp. 185-232. Visser, M. (1989). A classical model for the electron. Phys. Lett. A 139,99. VySin. V. ( I 977a). Nonrelativistic reduction and interpretation of the MeinGordon equation of tachyons. Nuovo Cirnento A 40,113. VySin. V. (1977b). Propagator of spinless tachyons. Nuovo CimentoA 40, 125. VySin, V. ( 1995). Geometrical approach to superluminal transformations in six-dimensional special relativity. In: Collection of Papers of International Scientific Symposium of Technical University. Ostrawa. Czech Republic, pp. 16&166. Weinberg, S. (1975). Light as a fundamental particle. Phys. Today, June, pp. 32-37. White, H. E. (1934). Introduction to Atomic Spectra. Wiley, New York, Sec. 4.10. Williams, E. R., Faller, J. E., and Hill, H. A. (1971). New experimental test of Coulomb’s law: a laboratory upper limit on the photon rest mass. Phys. Rev. Lett. 26, 720. Winterberg, F. (1987). Lorentz invariance as a dynamic symmetry. Z. Naturforsch. A 42, 1428.

Index

Absolute space, 198-199 Albite twins, 12 Alkali feldspars coarsening and iridescence in moonstone, 11-13 coarsening in, 10-1 1 exsolution and, 7-14 fluid phase in. 14 homogeneous nucleation in, 9-10 precipitation structures (perthites) in, 8 spinodal decomposition in, 8-9. 10 Al-Ni-Co phases, structure of, 68-77 Al-Ni-Fe phases, structure of, 77-79 AI-Pd-Mn, structure of, 85-89 ACPd phase, structure of, 90-94 Amphiboles classification of, 15-16 exsolution in, 14-27 standard formula for, 14-15 Amphiboles, monoclinic AEM and EMPA studies of, 22-25 exsolution in, 18-25 exsolution of, from an orthoamphibole, 27 gaps between members of calcic, 21-25 tweed structure, 22-23,25 two-stage exsolution, 20-21 Amphiboles, orthorhombic exsolution between two, 25-27 exsolution of monoclinic amphibole from, 27 gaps in, 25 homogeneous nucleation in, 25-26 spinodal decomposition in, 27 Amplitude wave, 147. 159 Analytical electron microscopy (AEM) analysis of calcic amphiboles using, 22-25

analysis problems using, 2-6 specimen damage using, 5-6

Bare mass of an electron, 2 I 1 Barut generalized approach, 159-160 Binary Boolean set. image polynomials versus, 107-1 1 1 Binary image polynomials, 110 Biopyriboles new, 28-30 origin of term, 28 polysomatic defects and, 27-32 Bradyon-tachyon compounds, photons as, 162-163 Bragg’s law, 191-192 B waves. See de Broglie wave (B-wave) Causal theory, 144 Chain-width disorder in pyriboles, 30-3 1 Compton wave (C-wave). 145 transformed, 147 Crystalline approximant phase, structure of, 79-83.89-90.94-96 Crystallographic shear planes, 30 Convolution equation, 100 Convolution theorem, 106-107 Corben tachyonic theory, 154- I55 d’Alembert wave equation, 159 Dark-field technique, 27,28 Das model. pseudovelocity and, 157-158 de Broglie, Louis, 144 de Broglie wave (B-wave) background of, 144-148 conversion of light into, 163-165 introduction of dual, 146 three-wave particle model, 145-146 de Broglie-Bohm wave theory, 148-152

240

24 1

INDEX de Broglie-Bohm wave theory, extended background of, 213-214 dynamical theorem, 225-227 many-body problem, 228-230 particle in a scalar field, 224-227 particle on a line, 221-224 tachyokinematic effect, 216221 uncertainty principle, 227 Decagonal contrast, 58 Decagonal quasicrystals. See Quasicrystals, decagonal Deconvolution. 100 Digital data processing of deconvolution, 100 Digital image processing, 100 Digital images, formula for, 102 Digital signal processing, 100 Dimensional reduction, 209 Dirac bispinor function, 204 delta function, 107 distributions, 175 equation, 205-206.207 Discrete Fourier transform (DFT), 100 Dislocations, icosahedral quasicrystals, 63-66 Distance transform, 136-138 Dual de Broglie wave (D-wave) conversion of light into, 163-165 introduction of, 146

Edenitic substitution, 25 Elbaz model, Lorentz transformations and, 158-1 59 Electromagnetic model of extended particles, 174 conclusions, 194-198 degrees of freedom, I8 I inner or self fields, 176 lattice spacings, 177 one-dimensional linear time cavity, 183-1 85 outer or radiating fields, 176 three-dimensional rectangular space cavity, 175-178 three-dimensional spherical space cavity, 1 78- 183 three-dimensional spherical time cavity, 185-1 87 two-dimensional square space-time cavity, 187-194 Electron diffraction of quasicrystals.

See Quasicrystals, electron

diffraction of Electron-microprobe analyzer (EMPA), 1 analysis of calcic amphiboles using, 22 Exsolution, 3 in alkali feldspars, 7-14 in amphiboles, 1 6 2 7 homogeneous nucleation and growth of equilibrium phase, 7 spinodal decomposition, 6-7 x-ray diffraction study of, 6 Extended space-time description of matter, 146

Feinberg wave equation, 155 Fibonacci sequence, 38 Formal power series, theory of, 100-101, 105 Fourier analysis, 100 Fourier-Laplace transform, 100, 101 Frank-Kasper type of icosahedral phase, 58 Friedel’s law, 80 Galilean region, 199 Gray scales, product of, 104 Guidance formula, IS1 Hamilton-Jacobi equation, 15 1 Heaviside function, 107, 140 Helmholtz’s equation, 166, 175, 184, 187 Higgs-field particles, 155 High-resolution transmission electron microscopy (HRTEM), 2 See also Quasicrystals, high-resolution electron microscopy of biopyriboles and polysomatic defects studied using, 27-32 Homogeneous nucleation and growth of equilibrium phase, 7 in alkali feldspars, 9-10 orthorhombic amphiboles, 25-26 Horodecki-Kostro theory, 147 two-wave hypothesis and, 156-157 lcosahedral quasicrystals. See Quasicrystals, icosahedral Image polynomials for color images, 1 11-1 13 convolution, 99-100 convolution theorem, 106-107 digital processing of deconvolution, 100 distance transform. 1 3 6 138 formal power series, 100-101. 105

242

INDEX

Image polynomials (cum.) generalized inverse of Toeplitz operator, 121-125 geometric properties of the product of, 115-1 17 inversion of Toeplitz equation, 115-1 19 iterative method for inversion of Toeplitz equation, 118-1 19 mask operations, 107 mathematical preliminaries, 102-103 morphological operation, 136 numerical analysis of deconvolution, 100 numerical examples, 119-121 properties of, 105-107 pyramid transform and, 125-134 quotient fields of, 1 I3 of real values, 103-107 regularity of, 114 shape analysis using, 134-138 skeletonization, 138 versus binary Boolean set. 107-1 1 I Image polynomials. pyramid transform and description of pyramid transform, 130-1 3 1 inversion of pyramid transform, 131-132 numerical examples of superresolution. 133- I34 subpixel image, 125-128 subpixel superresolution. 128-130 Infons, 23 1-232

Jacobi’s relativistic equation, 150 Jennison-Drinkwater electromagnetic theory, 153-154 Kaluza-Klein space, two-wave model of, 207 conclusions, 2 12-2 13 field theory, 208-2 10 five-dimensional tachyonic bootstrap, 210-212 Klein-Gordon wave equation, 149, 150, 154-155. 166 one-dimensional linear time cavity, 185 three-dimensional rectangular space cavity, 176 three-dimensional spherical space cavity, 180 Linear phason strain, 43,44 in icosahedral quasicrystals. 61-63 Local space conclusions, 206-207 quantum mechanics in, 202-206 special relativity in, 198-202

Lorentz transformations EIbaz model and, 158-159 local space and, 201-202 one-dimensional linear time cavity, 185 tachyokinematic effect, 216-220 three-dimensional rectangular space cavity, 176 three-dimensional spherical space cavity, 180 two-dimensional square space-time cavity, 191 Luxons, 161, 163

Mackey icosahedral atom cluster, 58 Mackinnon soliton, 152-153, 181 Barut generalized approach. 159-160 Many-body problem, 228-230 Maximons, 21 1 Maxwell’s equations, 161-162, 166, 172 one-dimensional linear time cavity, 184. 185 three-dimensional rectangular space cavity, 175 three-dimensional spherical space cavity, 178, I79 three-dimensional spherical time cavity, 186 two-dimensional square space-time cavity, 193 Minkowski addition, 107-108, 137, 138 Minkowski subtraction, 136, 137 Moving particle, 220 Nonlinear wave hypothesis, 146 Particles See o h Electromagnetic model of extended

particles; Wave-particle models. massive particles and in a scalar field, 224-227 on a line, 221-224 Phase connection principle, 144 Phase separation. See Exsolution Photons. See Wave-particle models, photons and Pilot-wave theory, 144 Plagioclase feldspars, 7 , 8 Polysomatic defects, biopyriholes and, 27-32 Polysomatic reactions, in pyriboles. 3 1-32 Polysomatic series, defined, 28 Polysomatism, use of term. 28 Polysome, defined, 28 Polytype, defined, 28 Proca theory extended, 171-173 Pseudovelocity. 146 Das model and, 157-158

INDEX Pyramid transform. See Image polynomials, pyramid transform and Pyriboles chain-width disorder in. 30-31 polysomatic reactions in, 3 1-32 use of term, 28

Quantum mechanics, in local space, 202-206 Quasicrystals, decagonal Al-Ni-Co phases, structure of, 68-77 Al-Ni-Fe phases, structure of, 77-79 Al-Pd-Mn, structure of, 85-89 AI-Pd phase, structure of, 90-94 characteristics of diffraction patterns in, 4743 columnar atom cluster framework, 67 crystalline approximant phase, structure of, 79-83.89-90.94-96 defocus values, 42.53 high-resolution electron microscopy of, 52-53 polytypes of, 48-50 structure of, 6 6 9 6 structure of, and crystalline phases with I .2-nm periodicity, 84-90 structure of, and crystalline phases with 1.6-nm periodicity, 90-96 structure of, with 0.4-nm periodicity, 68-83 W-phase, 80,82 Quasicrystals, electron diffraction of characteristics of diffraction patterns in decagonal, 4 7 4 8 decagonal, 47-50 good versus poor quality, 42-44 icosahedral, 4 4 4 7 linear phason strain, 43.44 polytypes of decagonal, 48-50 Quasicrystals, high-resolution electron microscopy of decagonal, 52-53.6696 defocus values, 42.53 experimental procedures, 4 1 4 2 icosahedral, 50-66 one-dimensional lattices, 38 two-dimensional lattices, 3 9 4 1 Quasicrystals. icosahedral atomic arrangements of, 56-60 decagonal contrast, 58 defects in, 61-66 dislocations, 6346

243

electron diffraction of, 44-47 high-resolution electron microscopy of, 50-52 linear phason strain, 43, 4 4 , 6 1 4 3 structure of, 53-66 topological features of lattices, 53-56 Quotient fields, of polynomials, 113

Radiolysis, 5 RGB decomposition, 1 1 I Robertson-Walker metric, 193 Scanning electron microscope (SEM), 1 Schrijdingerequation, 144. 149, 197,221,223, 225,228 Shape analysis, using image polynomials, 134-138 Skeletonization, 138 Space cavity See also Kaluza-Klein space, two-wave model of; Local space three-dimensional rectangular, 175-178 three-dimensional spherical, 178-1 83 two-dimensional square space-time cavity, 187-194 Special relativity, in local space, 198-202 Specimen damage using AEM, 5-6 Spinodal decomposition, 6-7 in alkali feldspars, 8-9, 10 in orthorhombic amphiboles, 25-26 Steady-state distribution, 145 Stuckelberg-Feynman switching principle, 214,216 Switching principle, 214, 216 Tachyokinematic effect, 2 14 spacelike theorem, 2 17-22 I timelike theorem, 215-217 Tachyons critical frames, 184 photons as bradyon-, 162- I63 spin, 207 transcendent, 163, 177, 187,210,218,223, 23 1 Tachyonic theory, Corben, 154-155 Theory of hidden variables, 144 Thin-film criterion for mineral analysis, 3 4 Three-wave particle model, 145-146 Time cavity one-dimensional linear, 183-185 three-dimensional spherical, 185-1 87 two-dimensional square space-, 187-194

244

INDEX

Toeplitz equation, 100, 113, I14 generalized inverse of operator, 121-1 25 inversion of, 115-1 19 iterative method for inversion of, 118-1 19 Transformed Compton wave, 147 Transmission electron microscopy (TEM) See also High-resolution transmission electron microscopy (HRTEM) exsolution in alkali feldspars, 7-14 exsolution in amphiboles, 14-27 thin-film criterion for mineral analysis, 3-4 used in the earth sciences, 1-2 Transverse electric waves (TE) three-dimensional rectangular space cavity and, 175-178 three-dimensional spherical space cavity, 178-183 two-wave model of longitudinal photons and, 165- I69 Transverse magnetic waves (TM), 165 three-dimensional rectangular space cavity and, 178 three-dimensional spherical space cavity, 183 two-wave model of longitudinal photons and, 165. 169-171 Transverse photons, 165 Tschermakite substitution, 22,25 h o - w a v e particle model. 146 See also Kaluza-Klein space, two-wave model of of longitudinal photons, 165-171

Uncertainty principle, 227

I S B N 0-12-034743-2

Variable rest mass,150 Velocity addition theorem, 200 Wadsley defects, 30 Wave-particle models, massive particles and Barut generalized approach, 159-160 comparisons of, 160-161 Corben tachyonic theory, 154-1 55 Das model and pseudovelocity, 157-158 de Broglie-Bohm wave theory, 148-152 Elbaz model and Lorentz transformations, 158-159 Horodecki-Kostro theory, 147, 156-1 57 Jennison-Drinkwater electromagnetic theory, 153-154 Mackinnon soliton, 15 2- I53 two-wave hypothesis. 156-157 Wave-particle models, photons and conclusions, 174 conversion of light into B and D waves, 163-165 massless photons and ponderable matter, 17 1 photons as brad yon-tachyon compounds, 162-163 Proca theory extended, 171-173 two-wave model of longitudinal photons, 165-1 71 Wave theory, description of different waves, 144-148 W-phase, 80.82 X-ray diffraction study of exsolution, 6 2-transforms, 100, 101, 107, 140