Phononics: Interface Transmission Tutorial Book Series

By Léonard Dobrzyński, El Houssaine El Boudouti, Abdellatif Akjouj and 4 more

Phononics: Interface Transmission Tutorial Book Series provides an investigation of modern systems that includes a discrete matrix description. Classical continuous systems relying on the use of differential equations are recalled, showing that they generally have a specific limit on their corresponding modern matrix formulation.

A detailed description of the mathematical languages that enables readers to find the composite system linear transmission properties is provided in the appendix. The physical model is described with exacting detail, and the bibliography is built to cite—in chronological order—all the scientists that have contributed over many years.

Each volume is written with the aim of providing an up-to-date and concise summary of the present knowledge of interface transmission science, thus fostering the exchange of ideas among scientists interested in different aspects of interface transmission.

The book serves as an introduction to advanced graduate students, researchers, and scientists with little study on the subject, and is also useful to help keep specialists informed on general progress in the field.

• Offers a unique approach on phononics from the interfacial transmission point-of-view
• Teaches the modern physics of interface transmission, in particular, phononics through composite systems
• Authored and edited by world-leading experts on interface transmission

• Language: English
• Publisher: Elsevier Science
• Release date: Sep 14, 2017
• ISBN: 9780128099315

Léonard Dobrzyński

Léonard Dobrzynski is Emeritus Research Professor at CNRS, Lille University, France. His research interests focus on interface science, phononics, magnonics, and resonance.

Book preview

Phononics

Interface Transmission Tutorial Book Series

First Edition

Leonard Dobrzynski

El Houssaine El Boudouti

Abdellatif Akjouj

Yan Pennec

Housni Al-Wahsh

Gaëtan Lévêque

Bahram Djafari-Rouhani

Series Editor

Leonard Dobrzynski

Table of Contents

Cover image

Title page

Copyright

Contributors

Preface

Acknowledgments

Chapter 1: Interface Response Theory

Abstract

1 Introduction

2 Composite Matrix Structure

3 Discrete Systems

4 Continuous Systems

5 Discrete and Continuous Systems

Chapter 2: Phonon Monomode Circuits

Abstract

1 Introduction

2 Diatomic Chain

3 Triatomic Chain

4 A Simple Atomic Multiplexer

5 Classical Analog of Fano and EIT Resonances in a Phononic Waveguide

6 Resonant Tunneling of Acoustic Waves Between Two Slender Tubes

7 Phononic Spectral Gaps in Serial Stub Tubes

8 Stopping and Filtering Phonons in Serial Loop Tubes

9 Quasiperiodic Phononic Circuits

10 Summary and Conclusion

Chapter 3: Phonons in Supported Layers

Abstract

1 Introduction

2 General Equations for a Phononic Material

3 The Case of Fluids

4 Resonant Guided Phonons in Supported Slab

5 Resonant Guided Phonons in Supported Bilayer

6 Localized and Resonant Guided Phonons in an Adsorbed Layer on a 1D Phononic Crystal

7 Relation to Experiments

8 Summary and Conclusions

Chapter 4: One-Dimensional Phononic Crystals

Abstract

1 Introduction

2 Shear-Horizontal Acoustic Waves in Semiinfinite PnCs

3 Shear-Horizontal Acoustic Waves in Finite PnCs

4 Sagittal Acoustic Waves in Finite Solid-Fluid PnCs

5 Omnidirectional Reflection and Selective Transmission in Layered Media

6 Conclusions

Chapter 5: Transmission in 2D Phononic Crystals and Acoustic Metamaterials

Abstract

1 General Introduction

2 Methods of Calculations

3 2D Bulk Phononic Crystal

4 2D AMMs

5 2D Phononic Crystal Plate

6 AMMs Interface

Index

Copyright

Elsevier

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Notices

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ISBN: 978-0-12-809948-3

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Contributors

Numbers in Parentheses indicate the pages on which the author’s contributions begin.

Abdellatif Akjouj     (1, 19, 79, 139, 271), The National Center for Scientific Research-Lille University, Lille; UMR CNRS 8520, Faculty of Science, Villeneuve d’Ascq, France

Housni Al-Wahsh     (1, 19, 79, 139, 271), Faculty of Engineering, Benha University, 11241 Cairo, Egypt

Bahram Djafari-Rouhani     (1, 19, 79, 139, 271), The National Center for Scientific Research-Lille University, Lille; UMR CNRS 8520, Faculty of Science, Villeneuve d’Ascq, France

Leonard Dobrzynski     (1, 19, 79, 139, 271), The National Center for Scientific Research-Lille University, Lille; UMR CNRS 8520, Faculty of Science, Villeneuve d’Ascq, France

El Houssaine El Boudouti     (1, 19, 79, 139, 271), Laboratory of Physics of Matter and Radiation, Faculty of Science, Mohammed I University, 60000 Oujda, Morocco

Gaëtan Lévêque     (1, 19, 79, 139, 271), The National Center for Scientific Research-Lille University, Lille; UMR CNRS 8520, Faculty of Science, Villeneuve d’Ascq, France

Yan Pennec     (1, 19, 79, 139, 271), The National Center for Scientific Research-Lille University, Lille; UMR CNRS 8520, Faculty of Science, Villeneuve d’Ascq, France

Preface

Leonard Dobrzynski, Villeneuve d’Ascq, France

The Interface Transmission Tutorial Book Series reviews a few simple and pedagogical model examples of interface transmission in phononics, magnonics, plasmonics, photonics, electronics, and polaritonics. It brings together rewritten, updated, and amended previous fundamental publications of its authors on all these topics, within a unified description. Each volume topic in the series is organized in a similar manner, enabling readers to move easily from one field to another. The series also offers a tutorial introduction to rapidly expanding novel applications of physics domains. Nevertheless this book series cannot be an exhaustive review of interface transmission in any of the physics fields introduced here. This would be an endless task. The series leaves space for other books by colleagues whose achievements do not appear here. The first volume of this tutorial book series deals with examples of interface transmission of phonons and of elastic waves. The transmission of a phonon or of an elastic wave, from one medium into a second medium, is a simple example of interface transmission. The propagation of surface and interface phonon in slabs and multilayer systems in directions parallel to their interfaces is another example of interface transmission. A system interface is the space of all its external and internal interfaces. Phononics is a science devoted to the investigation of the vibrations of composite materials and systems built out of different homogenous parts bounded by interfaces. The physical models are described with enough details so the reader need not urgently consult the references. The unified mathematical languages used to understand simple physical phonon models may be generalized in order to be used for the investigations of more complex systems in any other science.

Acknowledgments

We wish to express our appreciation and gratitude to many colleagues— professors, researchers, and doctors—for fruitful discussions during the last two decades. We cite in particular Victor Velasco, George Fytas, P. Zielinski, Jerome Vasseur, Abdelkrim Khelif, Pierre Deymier, Vincent Laude, Abdelkrim Nougaoui, Driss Bria, El Hassane Lahlaouti, Tarik Mrabti, Rayisa Moiseyenko, Ossama El Abouti, Youssef El Hassouani, Hassan Aynaou, Dirk Schneider, Hocine Larabi, Elena Alonso-Redondo, Abdellatif Gueddida, Clivia Sotamayor Torres, Anne-Christine Hladky, Ilyasse Quotane, Bernard Bonello, Mohamed Alami, Daniel Torrent, Said El-Jallal, Samira Amoudache, Yabin Jin, and Mourad Oudich.

It is a pleasure to thank our publishers and in particular Anita Koch, Acquisition Editor; Amy Clark, Editorial Project Manager; and Anitha Sivaraj, Production Project Manager.

Finally, we dedicate this book to our families in partial compensation for taking so much of our time away from them.

Chapter 1

Interface Response Theory

Leonard Dobrzynski*; Abdellatif Akjouj*; El Houssaine El Boudouti†; Housni Al-Wahsh‡; Bahram Djafari-Rouhani*; Gaëtan Lévêque*; Yan Pennec*    * The National Center for Scientific Research-Lille University, Lille; UMR CNRS 8520, Faculty of Science, Villeneuve d’Ascq, France

† Laboratory of Physics of Matter and Radiation, Faculty of Science, Mohammed I University, 60000 Oujda, Morocco

‡ Faculty of Engineering, Benha University, 11241 Cairo, Egypt

Abstract

A general theory of interface response in linear composite systems is presented in the first chapter of this book on phononics. Specific examples, fully developed in the next chapters of this book, are shortly presented in this chapter. Other examples using this general interface response theory appear also in the following books of this interface transmission tutorial book series. This theory is first presented for discrete systems with a matrix mathematical formulation. Then the case of continuous systems described by differential equations is given. In this case the response functions are the classical Green’s functions. Finally the case of partly discrete and partly continuous composite systems is also discussed.

Keywords

Surface; Interface; Composite; Scattering; General theory; Phase shift; Density of states; Green’s functions

Contents

1 Introduction

2 Composite Matrix Structure

3 Discrete Systems

3.1 Infinite Matrix Inverse

3.2 System Matrix Inverse

3.3 System Eigenvalues

3.4 Response to an Action

3.5 Discrete System Eigenvectors

3.6 Total Density of States

3.7 Local Density of States

3.8 Variation of the Total Density of States

3.9 Conservation of the Number of States

3.10 Variations of Additive Functions

4 Continuous Systems

4.1 Composite System Diffusion Matrix

4.2 Continuous System Eigenvalues

4.3 Response to an Action

4.4 Continuous System Eigenvectors

4.5 Densities of States

5 Discrete and Continuous Systems

References

1 Introduction

This tutorial book on Interface Transmission addresses Phononics in composite materials. Phononics starts with the prediction of surface elastic waves by Rayleigh [1]. The experimental excitation of Rayleigh waves at high frequencies and at low temperatures begins with Arzt and Dransfeld [2].

Any system is fully defined only once its surface and interface boundaries are clearly defined. Almost all bodies and systems may be considered to be build out from different homogenous parts with internal interfaces bounding them, see for example, Friedel [3]. So the concept of a composite system, build out of homogenous parts related by internal interfaces and bounded by external surfaces, is a very general one, see for example, Dobrzyński [4].

The knowledge of the algebra of matrix and differential equations is the only basic requirement for those willing to study this tutorial. Different mathematical and numerical simulation methods are used in this book. One of the most popular method is the surface Green’s function one, introduced by Green [5] for bulk continuous systems described by differential equations. For those interested by the historical background of this method, let us cite some early papers, introducing surface Green’s functions [4–18].

One approach used in this book considers discrete materials, for example, within atomic studies of surface and interface phonons. Analyzes are in this case done with the help of matrices. This discrete approach has in general a continuous limit where the classical Green’s function method is recovered. This approach called interface response theory is described in the first chapter as well for discrete and continuous composite systems. Some simple examples within simple one-dimensional phononics models are given in this tutorial. Other mathematical approaches are also used in particular in the studies of two-dimensional phononics. They are presented here with some examples of fiber composite materials.

This book is written with the aim to be understandable by graduate university students without any urgent need to consult the original references.

This chapter called interface response theory [4, 15–18] gives a detailed demonstration of a universal theory. This mathematical method is used in the first chapters of this book. The readers interested to learn and use this general method are advised to read this general and abstract theory and at each step redo the simple examples given in the first chapters of this book.

Here time is supposed to be a continuous variable as well as its Fourier transformed entity which when multiplied by the Planck constant is energy. Energy is a discrete quantized variable in some systems and a continuous one in others. Space variables are also discrete or continuous depending on the investigated system. Mathematicians build respectively discrete and continuous languages, for example, matrix and differential ones enabling to describe the physical world. As an illustration of this very general duality, let us cite for example, the lattice dynamics and the elasticity theory. Lattice dynamics uses matrix equations and elasticity theory differential approaches. In the limit when the phonon wavelengths are large as compared to the interatomic distances, the elasticity results are the limits of those of the lattice dynamics.

2 Composite Matrix Structure

Consider a general composite matrix defined in the space D and build out of N subsystems defined in their corresponding spaces Di, with 1 ≤ i ≤ N. The N parts are bounded together by interface domains Mi ∈ Di. Each part i is in general interacting with J other parts. The interface space Mi is in general formed by J subinterfaces Mij, with 1 ≤ j ≤ J.

3 Discrete Systems

3.1 Infinite Matrix Inverse

The calculation of the inverse matrix Gi corresponding to the matrix operator Hi of each part i starts from the following equation

   (1)

As a simple example consider the case of the tridiagonal dynamical matrix Hi associated with an infinite chain of atoms with first nearest neighbor atom interactions βi. Its inverse Gi can be obtained in closed form, see chapter 2 of this volume. An illustration of this structure is given in Fig. 1.

Fig. 1 An infinite chain of atoms with first nearest neighbor atom interactions β i . The interatomic distance is a 0 .

3.2 System Matrix Inverse

Let us introduce another matrix operator Vi such that

   (2)

As a simple example take Vi to be the two by two perturbation matrix which removes the interactions between two nearest neighbor atoms of the above considered infinite chain and transform it into two semiinfinite chains bounded by atoms 0 and 1, see chapter 2. An illustration of this structure is given in Fig. 2.

Fig. 2 Two semiinfinite chains of atoms with first nearest neighbor atom interactions β i . The interatomic distance is a 0 .

The inverse matrix gi corresponding to hi by

   (3)

From Eqs. (1)–(3), one obtains easily the following relation

   (4)

where

   (5)

Define then for each submatrix i the cleavage operator Vi defined in its interface space Mi enabling to cut out from the infinite system the needed finite submatrix hs defined in the Di space, such that the corresponding operator hi the following block diagonal form

   (6)

The semiinfinite matrix blocks ha and hb play no role in what follows, as they have no interaction with the block hsi of interest. The inverse of hi has then also a block diagonal form

   (7)

The Ai operator has then the following form

   (8)

where Asi is defined in the space Di and is a truncated part of Ai.

In the same manner define in the space Di the truncated part Gsi of Gi

   (9)

With these notations Eq. (4) reads

   (10)

This presentation shows that gsi can be obtained in its space Di from

   (11)

Note that all the above considerations done for a finite system can be redone easily for a semiinfinite one.

Note also that this procedure using truncated infinite matrices reduces the size of the matrices. This simplifies greatly the calculations, mostly when the matrix sizes are big.

Proceeding with the above simple example of the semiinfinite atomic chain, it is possible to obtain explicitly all the matrix elements of gsi for the semiinfinite and for the finite linear atomic chain, see chapter 2 of this volume. An illustration of this structure is given in Fig. 3.

Fig. 3 One semiinfinite chain of atoms with first nearest neighbor atom interactions β i . The interatomic distance is a 0 .

Let us now put together all the N independent blocks hsi (1 ≤ i ≤ N) inside a matrix hs defined in the space D of the system of interest. The system operator h is defined by the addition of the matrix hs build out of independent blocks hsi and the interaction operator VI coupling together all the matrix blocks hsi

   (12)

The inverse matrix g is defined inside the space D by

   (13)

As hs is build out of N independent matrix blocks, its inverse gs is also build out of N disconnected blocks in the space D

   (14)

From the above three equations, it is easy to obtain the following relation

   (15)

Multiply now this equation by (I +As) where As is built by putting together the N noninteracting matrix blocks As

   (16)

where G is build out from the N independent blocks Gsi. Finally one obtains in the D space

   (17)

where A is the interface operator

   (18)

Eq. (17) is the general equation of the matrix method presented in this review.

Define now the interface space M for the system under investigation as the space in which exists the before defined nonzero elements of the interaction operator VI (Eq. 12) and the N cleavage operators Vi (Eq. 2). The operator A has only nonzero matrix elements between the points of the interface space M and the points of the D space of the system (see Eqs. 5, 8, 18). Let us introduce a rectangular matrix A(MD) and use similar notations for all the other operators. Eq. (17) can now be written in the following form

   (19)

A particular form of this equation is

   (20)

or

   (21)

where

   (22)

Eqs. (19), (22) enable us to obtain another form of the general equation presented above

   (23)

This relation is very helpful in order to obtain the matrix elements of the system inverse matrix g.

Proceeding with the above simple example of the semiinfinite atomic chain, consider now the interface between two different semiinfinite chains and find the interface response function, see chapter 2 of this volume. An illustration of this structure is given in Fig. 4.

Fig. 4 A simple interface between two semiinfinite chains of atoms with first nearest neighbor atom interactions respectively β 1 and β 2 within the chains and β I at the interface. The interatomic distance is a 0 .

3.2.1 Another Useful General Relation for the Interface Elements of the Green’s Function

The general equation (17) can be written in particular within the interface space as

   (24)

where from Eq. (18)

   (25)

When the interface coupling VI(MM) vanishes, Eq. (24) becomes

   (26)

Using the above three equations, one obtains

   (27)

This is a very simple and useful relation between the matrix elements of the composite Green’s function and the surface elements of its free surface components.

3.3 System Eigenvalues

such that the eigenvalue E of a system with wave function u , i.e.,

   (28)

where I is the identity matrix. This matrix operator including two-body particle interactions is the starting entity enabling the studies of all linear properties of a given system. It enables to calculate all the eigenvalues and eigenfunctions associated with this operator and then all the system properties. The elements of this matrix may or may not have differential forms. When these elements are differential, they address some continuous properties of the system under investigation. When these elements are not differential, they deal with some discrete system properties. By continuous and discrete properties, we mean properties continuous or discrete in space. We address first the case of purely discrete systems. In order to use the above results obtained for a system matrix, let us take for the matrices H and h defined above the following expressions

   (29)

and

   (30)

and ϵ is an infinitesimally small number. For these matrices their corresponding inverses G and g are defined by

   (31)

and

   (32)

These definitions enables to see at once that the eigen energies of the composite material can be obtained from the poles of g.

Now when looking at Eqs. (23), (31), one sees also that some eigen-energies of the composite material are given by

   (33)

The poles of G give the eigen values of the infinite systems out of which the composite system was build. This is straightforward when the bulk Gi(k) are written in the Brillouin reciprocal space k, taking due account of the translational periodicity of a periodical infinite system i. In real space the Gi can be obtained by a Fourier transformation of the Gi(k) and then the eigen values of the infinite system i are not appearing directly as its poles. Let us note also that infinite systems have in general an infinite number of eigen values falling inside finite or semiinfinite bands of eigen values. So the eigen values provided by Eq. (33) are, for infinite or semiinfinite composite systems, localized states falling outside the bulk bands of the infinite systems out of which the system was build. When the system is finite then the number of its eigen values is finite. All these eigen values are given by Eq. (33). Most of them fall inside the bulk bands as defined above and some of them outside and are localized near the system interfaces.

In order to understand fully these general considerations, it is recommended to use the simple example of a finite atomic chain and to calculate its eigenvalues, see chapter 2 of this book and Ref. [17]. An illustration of this structure is given in Fig. 5.

Fig. 5 One finite chain of N atoms with first nearest neighbor atom interactions β i . The interatomic distance is a 0 .

3.4 Response to an Action

Consider now an action F, acting on the system. The response u, to this action is defined by

   (34)

where h is given by Eq. (30). The evaluation of the response u may be done with the help of the inverse matrix g as defined by Eq. (13). With the notations defined above, one obtains

   (35)

Using then the general equation (23), one obtains

   (36)

where

   (37)

and

   (38)

are the responses of the reference system made out of the noninteracting bulk pieces of the system. Note that the action F may be localized on only one part of the system and even on only one point.

Let us stress that the knowledge of U(D), [Δ(MM)], and A(MD) enables to obtain through the above equations the response of the system to any action applied on it. This is true for any value of E. However let us precise the particular case of E equal to an eigen value of the system. In that case the response is the excitation of an eigen vector corresponding to this eigen value. The following subsection shows how the above equations should be used in order to obtain the eigenvectors.

3.5 Discrete System Eigenvectors

3.5.1 Finite system

For a finite system, as stressed above, all the eigen values are given by Eq. (33). Then in order to avoid divergences for E equal to an eigen value, one has to use rather than Eq. (36) the following one

   (39)

which provides the unnormalized corresponding eigen vector. An appropriate normalization factor can be calculated afterwards.

In order to understand fully these general considerations, it is recommended to redo the simple example of a finite atomic chain, see Ref. [17].

3.5.2 Infinite System

For an infinite system, when the action is applied inside one homogeneous part i of the system and for E equal to an eigen value of this subsystem i, U(D) has to be constructed with the help of the corresponding bulk eigenvalue inside this subsystem i and zeros inside all other subsystems. Then Eq. (36) may be used in order to obtain all transmitted and reflected waves induced by this action.

The procedure defined above for a finite composite system has to be used in order to get the eigen vector corresponding to a localized state falling outside the bulk bands of the reference system. The reference system is defined here as that constituted by all corresponding infinite subsystems.

In order to understand fully these general considerations, it is recommended to redo the simple example of a semiinfinite linear atomic chain, see Ref. [17].

3.6 Total Density of States

Let us define the density of states of an infinite system i per unit eigen value E and unit space volume by

   (40)

where Tr is the usual notation for the trace of a matrix, δ is the system matrix.

Define

   (41)

and use the symbolic relationship

   (42)

to show that

   (43)

stand for imaginary part of what follows.

In the same manner as in Eq. (40), the total density of states of any system per unit eigen value and unit volume is defined as

   (44)

is the system matrix. As above, after having defined

   (45)

one can write

   (46)

3.7 Local Density of States

Considering Eq. (46), one sees that the total density of states n(E) is the sum of local densities of states n(E, x), where x is the position in the real space D of a given system point

   (47)

with

   (48)

In particular Eqs. (48), (21) enable to calculate the local density of states at an interface point x

   (49)

where x, x′∈ M, M being the interface space of the system.

3.8 Variation of the Total Density of States

Another useful entity is the variation of the total density of states between the reference system containing all finite or semiinfinite pieces with free surfaces, represented by gs+(E) and the final system represented by g+(E). This reference system is defined in connection with Eqs. (14), (15). Then the variation of the total density of states between the final system and the reference one as defined here is with the help of Eq. (46)

   (50)

Let us define

   (51)

where

   (52)

Then Eq. (15) provides

   (53)

and in the same manner as for Eqs. (21), (23) one may write

   (54)

and

   (55)

With the help of these last two equations, one obtains also

   (56)

The variation of the total density of states can now be written as

   (57)

or with the help of the theorem of the cyclic invariance of the trace

   (58)

Take the derivative versus E of Eq. (32) to obtain

   (59)

use the general property of any operator or matrix B = g(MM)

   (60)

and do the same transformations for gs(MM). Now Eq. (58) can be rewritten as

   (61)

Use of Eq. (54) provides finally the simple result

   (62)

where

   (63)

Let us stress that the above result gives the variation of the total density of states between the final composite system and a reference system formed by its pieces with free surfaces and interfaces in function of a phase shift η(E).

As a simple example, take again the infinite atomic linear chain, remove one interaction between two nearest neighbor atoms and obtain the phase shift between the final (two semiinfinite chains) and the reference system (one infinite linear chain) to be π/2 within the bulk phonon band and 0 outside.

3.9 Conservation of the Number of States

When the number of independent degrees of freedom is not changed when going from the reference system to the real composite one, then the total number of states is conserved. This implies that

   (64)

This is the most usual case. One may however wish to compare two systems which differ in their degrees of freedom (addition or removal of particles, for example). This can also be studied within the frame of this theory, but one must count in the second member of the above equation the number Ng of degrees of freedom gained or lost.

In particular when the variation of the density of states takes the simple form of Eq. (62), then the above considerations imply

   (65)

or

   (66)

In particular when the number of states is conserved, the above equation shows that the phase shift η(E. Eq. (63) shows that the phase shift η(Ethe system under investigation has no band, nor eigen-energy. So in this case

   (67)

3.10 Variations of Additive Functions

Define an additive function F of the composite system by

   (68)

and FR of the reference system by

   (69)

In these expressions f(E) may be any function of E and of other parameters like the temperature T and the pressure P. F and FR can be, for example, usual thermodynamical functions, like the free energy, the specific heat, the entropy, etc.

Define the variation of the additive functions between the final and the reference systems by

   (70)

So the general result given by Eq. (62) for the variation of the density of states between a composite system and a reference system formed out of pieces with free surfaces enables to calculate easily the variation of any additive function.

   (71)

Integrating by part, one obtains

   (72)

When there is no change of the number of degrees of freedom between the reference and final systems (Ng = 0), then with the help of Eq. (67)

   (73)

These variations may be important physical entities like, for example, the surface and interface energies, specific heats, entropies, etc.

As a simple example, calculate the surface specific heat within the simple cubic lattice model, see for example Ref. [7]. An illustration of this structure is given in Fig. 6.

Fig. 6 A simple cubic lattice of atoms with first nearest neighbor atom interactions β i . The interatomic distance is a 0 .

4 Continuous Systems

Consider now a continuous infinite subsystem such that its properties may be described as function of continuous space positions x. Consider its system matrix Hi(x) whose elements are differential operators. The inverse of such a matrix is

   (74)

where δ(x −x′) is the Kronecker delta function, such that,

   (75)

The Green’s function Gi(x −x′) is function of the difference in the two space positions because the system is infinite and homogenous and therefore invariant by translation. The corresponding Green’s function Gi(x −x′) is defined by

   (76)

, ϵ is the Hamiltonian operator of a continuous space variable x.

Consider now the same continuous system but with free surfaces, its Green’s function is defined by

   (77)

where

   (78)

The cleavage operator Vi(x) which cuts out the free surface system out of the infinite one has now a differential form and acts only in the surface space. The surface response operator used in the theory for discrete media (see Eq. 5) becomes for continuous media

   (79)

The matrix equation (19) of the theory for discrete media becomes for the continuous media the following integral equation

   (80)

where the integration is done over the surface space. This equation enables to calculate the Green’s function elements of the free surface elements of the composite system. The solution of this integral equation can in general be obtained by numerical discretization methods which transform it in a matricial equation. Then with the same notations as those defined above for discrete systems, it is possible, with the help of Eq. (11) to write within the surface space Mi

   (81)

where

   (82)

Let us underline that in continuous systems there are no particles and therefore no particle interactions. The continuous systems are described by macroscopic coefficients relating other macroscopic entities. The free surfaces of such continuous systems are described by the vanishing of some macroscopic entities at the surfaces.

Let us now consider the interface between different continuous systems. Such an interface is classically described by the continuity at the interface of some macroscopic entities, without any other interface interactions, so the interface coupling operator VI of the above theory for discrete systems is obviously zero for continuous systems. So the limit of Eq. (27) for continuous systems is

   (83)

It is then possible to obtain g(MM)]−1 by addition of the [gs(MM)]−1 at the different subinterfaces. The notation for the interfaces and subinterfaces are here as defined at the beginning of Section 2. So

   (84)

   (85)

   (86)

These are the results for the interface elements of the composite system Green’s function. In what follows is outlined how to obtain the other elements of this Green’s function.

4.1 Composite System Diffusion Matrix

For any discrete and continuous system, it is possible to define a diffusion matrix T(MM) by the following relation

   (87)

can then by written in a matrix form g(DD).

A special form of the above equation is

   (88)

Multiplication of this equation from left and right by (g(MM))−1 provides

   (89)

which provides when substituted in Eq. (87)

   (90)

The diffusion matrix T(MM) is useful for solving many problems. The Green’s function g(DD) can be calculated with the help of Eq. (90) as well for continuous as discrete composite systems. However for discrete systems, it is in general easier to use Eq. (23).

4.2 Continuous System Eigenvalues

Recalling the considerations developed in Section 3.3 for discrete systems, one sees from Eq. (90) that the new poles of the Green’s function g(DD) are those of g(MM). So the new eigenvalues of the composite continuous system, called also interface states can be obtained from the following equation

   (91)

Let us recall also that for finite composite systems, the above equation provides all system eigenvalues.

4.3 Response to an Action

Following the considerations developed in Section 3.4 for discrete systems, one sees that the deformation u(D) of the continuous composite system can be obtained from the deformation U(D) of the reference system from the following relation

   (92)

4.4 Continuous System Eigenvectors

4.4.1 Finite System

For a finite continuous system, as stressed above for discrete systems, here also all eigenvalues are given by Eq. (91) and then in order to obtain the system eigenvectors, one needs only the third term of Eq. (92), namely

   (93)

4.4.2 Infinite System

For an infinite continuous system, when the action is applied inside one homogeneous part i of the system and for E equal to an eigen value of this subsystem i, U(D) has to be constructed with the help of the corresponding bulk eigenvalue inside this subsystem i and zeros inside all other subsystems. Then Eq. (92) may be used in order to obtain all transmitted and reflected waves induced by this action.

The procedure defined above for a finite composite system has to be used in order to get the eigen vector corresponding to a localized state falling outside the bulk bands of the reference system. The reference system is defined here as that constituted by all corresponding infinite subsystems.

4.5 Densities of States

For continuous systems the densities of states can be obtained in the same manner as explained above for the discrete systems.

The total density of states is given by Eq. (46)

   (94)

The local density of states at a given space point x is given by Eq. (48)

   (95)

As for the variation of the total density of states between the composite system and its reference one formed by its pieces with free surfaces and interfaces. Let us notice that Eq. (56) is also valid for continuous systems and can be obtained easily from Eq. (90). Then one obtains in the same manner as above (Eq. 61) the variation of the total density of states, namely

   (96)

where

   (97)

As about the conservation of the number of states and variations of additive functions, the results obtained above for discrete systems remain valid for continuous systems.

5 Discrete and Continuous Systems

Composite systems formed out of partly discrete and continuous systems exist also. Let us cite as a simple example electrons in a crystal (described by the tight-binding approach) with an interface with vacuum (described by a free electron model).

The above Green’s function analyze of such discrete and continuous systems may be done with the help of the above formalism. There is just one important point to keep in mind when dealing with such systems, namely the Green’s function definitions for discrete systems (Eq. 1) differs from those for the continuous systems (Eq. 74) by the δ(x −x′) factor. The obvious consequence of this difference is that the Green’s functions for discrete and continuous systems do not have the same dimensions when the corresponding Hamiltonians have both the dimensions of energy. Indeed the integral over space of a delta Kronecker